Research Article Free Vibration Analysis of Moderately ... · Research Article Free Vibration...
26
Research Article Free Vibration Analysis of Moderately Thick Rectangular Plates with Variable Thickness and Arbitrary Boundary Conditions Dongyan Shi, 1 Qingshan Wang, 1 Xianjie Shi, 2 and Fuzhen Pang 3 1 College of Mechanical and Electrical Engineering, Harbin Engineering University, Harbin 150001, China 2 Institute of Systems Engineering, China Academy of Engineering Physics, Mianyang 621900, China 3 College of Shipbuilding Engineering, Harbin Engineering University, Harbin 150001, China Correspondence should be addressed to Dongyan Shi; [email protected] Received 27 February 2014; Revised 30 May 2014; Accepted 16 June 2014; Published 14 July 2014 Academic Editor: Lei Zuo Copyright © 2014 Dongyan Shi et al. is is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited. A generalized Fourier series solution based on the first-order shear deformation theory is presented for the free vibrations of moderately thick rectangular plates with variable thickness and arbitrary boundary conditions, a class of problem which is of practical interest and fundamental importance but rarely attempted in the literatures. Unlike in most existing studies where solutions are oſten developed for a particular type of boundary conditions, the current method can be generally applied to a wide range of boundary conditions with no need of modifying solution algorithms and procedures. Under the current framework, the one displacement and two rotation functions are generally sought, regardless of boundary conditions, as an improved trigonometric series in which several supplementary functions are introduced to remove the potential discontinuities with the displacement components and its derivatives at the edges and to accelerate the convergence of series representations. All the series expansion coefficients are treated as the generalized coordinates and solved using the Rayleigh-Ritz technique. e effectiveness and reliability of the presented solution are demonstrated by comparing the present results with those results published in the literatures and finite element method (FEM) data, and numerous new results for moderately thick rectangular plates with nonuniform thickness and elastic restraints are presented, which may serve as benchmark solution for future researches. 1. Introduction In comparison with vibration analysis of plates with uniform thickness, far less attention has been paid to the vibration problems of plates with variable thickness, which are com- monly widely used in engineering applications as a stand- alone structure or a constituent structural component. By carefully designing the thickness distribution, a substantial increase in stiffness, buckling, and vibration capacities of the plate may be obtained over its uniform thickness counter- part. In addition, the moderately thick plates with variable thickness in these practical applications oſten work in com- plex environments and suffer from arbitrary edge restraints. erefore a thorough dynamic study of moderately thick rec- tangular plates with variable thickness and arbitrary edge restraints is essential to assess and use the full potentials of plates. Over past decades, extensive investigations have been carried out to determine the vibration characteristics (natural frequencies, mode shapes, and so on) of moderately thick rectangular plates. A comprehensive review on the rele- vant studies done before 1995 was presented by Liew et al. [1]. e Rayleigh-Ritz method based on polynomials with prop- erties corresponding to those of Timoshenko beam functions was used by Chung et al. [2] to study the free vibrations of orthotropic rectangular Mindlin plates with edges elastically restrained against rotation. Later, Cheung and Zhou [3] extended this solution for the free vibration problems of moderately rectangular plates. Wang [4] presented an exact formula for the vibration frequencies of simply supported Mindlin plates with the corresponding simply supported thin plate frequencies. Saha et al. [5] utilized the variational method to investigate the vibration characteristics of iso- tropic Mindlin plates with edges elastically restrained against rotation and translation. Gorman [6] presented accurate eigenvalues for shear-deformable plates resting on uniform elastic foundations with modified superposition-Galerkin Hindawi Publishing Corporation Shock and Vibration Volume 2014, Article ID 572395, 25 pages http://dx.doi.org/10.1155/2014/572395
Transcript of Research Article Free Vibration Analysis of Moderately ... · Research Article Free Vibration...
Research ArticleFree Vibration Analysis of Moderately Thick Rectangular Plateswith Variable Thickness and Arbitrary Boundary Conditions
Dongyan Shi1 Qingshan Wang1 Xianjie Shi2 and Fuzhen Pang3
1 College of Mechanical and Electrical Engineering Harbin Engineering University Harbin 150001 China2 Institute of Systems Engineering China Academy of Engineering Physics Mianyang 621900 China3 College of Shipbuilding Engineering Harbin Engineering University Harbin 150001 China
Correspondence should be addressed to Dongyan Shi shidongyanhrbeueducn
Received 27 February 2014 Revised 30 May 2014 Accepted 16 June 2014 Published 14 July 2014
Academic Editor Lei Zuo
Copyright copy 2014 Dongyan Shi et al This is an open access article distributed under the Creative Commons Attribution Licensewhich permits unrestricted use distribution and reproduction in any medium provided the original work is properly cited
A generalized Fourier series solution based on the first-order shear deformation theory is presented for the free vibrations ofmoderately thick rectangular plates with variable thickness and arbitrary boundary conditions a class of problem which is ofpractical interest and fundamental importance but rarely attempted in the literatures Unlike in most existing studies wheresolutions are often developed for a particular type of boundary conditions the current method can be generally applied to a widerange of boundary conditionswith noneed ofmodifying solution algorithms andproceduresUnder the current framework the onedisplacement and two rotation functions are generally sought regardless of boundary conditions as an improved trigonometricseries in which several supplementary functions are introduced to remove the potential discontinuities with the displacementcomponents and its derivatives at the edges and to accelerate the convergence of series representations All the series expansioncoefficients are treated as the generalized coordinates and solved using the Rayleigh-Ritz techniqueThe effectiveness and reliabilityof the presented solution are demonstrated by comparing the present results with those results published in the literatures and finiteelement method (FEM) data and numerous new results for moderately thick rectangular plates with nonuniform thickness andelastic restraints are presented which may serve as benchmark solution for future researches
1 Introduction
In comparison with vibration analysis of plates with uniformthickness far less attention has been paid to the vibrationproblems of plates with variable thickness which are com-monly widely used in engineering applications as a stand-alone structure or a constituent structural component Bycarefully designing the thickness distribution a substantialincrease in stiffness buckling and vibration capacities of theplate may be obtained over its uniform thickness counter-part In addition the moderately thick plates with variablethickness in these practical applications often work in com-plex environments and suffer from arbitrary edge restraintsTherefore a thorough dynamic study of moderately thick rec-tangular plates with variable thickness and arbitrary edgerestraints is essential to assess and use the full potentials ofplates
Over past decades extensive investigations have beencarried out to determine the vibration characteristics
(natural frequencies mode shapes and so on) of moderatelythick rectangular plates A comprehensive review on the rele-vant studies done before 1995 was presented by Liew et al [1]The Rayleigh-Ritz method based on polynomials with prop-erties corresponding to those of Timoshenko beam functionswas used by Chung et al [2] to study the free vibrations oforthotropic rectangular Mindlin plates with edges elasticallyrestrained against rotation Later Cheung and Zhou [3]extended this solution for the free vibration problems ofmoderately rectangular plates Wang [4] presented an exactformula for the vibration frequencies of simply supportedMindlin plates with the corresponding simply supported thinplate frequencies Saha et al [5] utilized the variationalmethod to investigate the vibration characteristics of iso-tropic Mindlin plates with edges elastically restrained againstrotation and translation Gorman [6] presented accurateeigenvalues for shear-deformable plates resting on uniformelastic foundations with modified superposition-Galerkin
Hindawi Publishing CorporationShock and VibrationVolume 2014 Article ID 572395 25 pageshttpdxdoiorg1011552014572395
2 Shock and Vibration
method Subsequently Gorman [7 8] utilized the superposi-tion method to investigate the vibration problems of elasticrestrained and point supported Mindlin plates In his super-position method the solution satisfied the differential equa-tions exactly but approximated the boundary conditionsXiang [9] andXiang andWei [10] employed the Levy solutionapproach in conjunction with the domain decompositiontechnique to analytically solve the free vibrations of rectangu-lar Mindlin plates with two opposite edges simply supportedIn [10] the influence of the step thickness ratios on thevibration behavior of rectangular Mindlin plates is studiedYeh et al [11] proposed a hybrid method which combines thefinite difference method and the differential transformationmethod to analyse the free vibration of clamped and simplysupported rectangular thin plates Xiang et al [12] extendedthe DSC-Ritz element method to solve the free vibrationanalysis of moderately thick rectangular plates with mixedsupporting edges Thai et al [13ndash15] and Nguyen-Thoi et al[16ndash19] have recently proposed the isogeometric analysis(IGA) and smoothed finite element method (S-FEM) toanalyse the cracked Mindlin plate laminated composite andsandwich plates In their research the structural boundarywas taken as uniform classic boundary conditions In addi-tion Nguyen-Thanh et al [20] utilized the alpha finite ele-ment method (120572-FEM) in static free vibration and bucklinganalyses ofMindlin-Reissner plates using triangular elementsonly
A Rayleigh-Ritz formulation based on pb-2 functions wasutilized by Liew et al [21] to study the vibration behaviors ofMindlin plates with elastic restrained edges Zhou et al [2223] investigated the similar vibration problems in terms of aset of static Timoshenko beam functions In their study thefree vibrations of rectangularMindlin plateswith internal linesupports were also solved Shen et al [24] presented a new setof admissible functions which satisfied both geometrical andnatural boundary conditions for the free and forced vibrationproblems of moderately thick plates with four free edgesXing and Liu [25] presented the closed-form solutions forvibration problems of Mindlin plates with any combinationsof simply supported and clamped edge conditions Apartfrom aforementioned solutions some other methods suchas finite strip element method [26] spline strip method[27] finite element method [28] meshless method [29] Ritzmethod [30] differential quadraturemethod (DQM) [31ndash33]and Green function method [34 35] had been developed todetermine the vibration behavior of moderately thick plates
It is noteworthy that most of the previous researches onmoderately thick rectangular plates are confined to the uni-form thickness and classical boundary conditions such asfree simply supported clamped edges and their combina-tions Only few studies have been found in the published lit-erature about the vibrations of moderately thick rectangularplates with variable thickness and elastic restraints A linearfinite strip element method based on Mindlinrsquos plate theorywas proposed by Gagnon et al [26] to solve the vibrationproblems of rectangular thick plates in which the thicknesscan vary in any direction Huang et al [36] developeda discrete method for solving the vibration problems oforthotropic rectangular plates with variable thickness and
general classical boundary supports Extended Kantorovichmethod was utilized to investigate the free vibrations ofrectangular thick plates with variable thickness and differentclassical boundary conditions by Shufrin and Eisenberger[37] Eftekhari and Jafari [38] proposed an efficient and accu-rate variational formulation for the vibration problems ofvariable thin and thick plates with elastic edge restraints
Although a large number of studies have been carried outbased on Mindlinrsquos plate theory and methods it appears thatthe information available about the vibration characteristicsof nonuniform thickness moderately thick rectangular platesis very limited Most of the contributions to moderately thickrectangular plates with classic boundary supports and elasticedges are confined to uniform thickness or bilinearly vary-ing thickness However the engineering practices containa variety of possible boundary conditions such as elasticrestraints and nonlinearly varying thickness The existingresults are simply too scarce for engineering applications andcomparative studies Moreover most of the available solutionprocedures in the open literature are often only customizedfor a specific set of restraint conditions which may not beappropriate for practical application because there are hun-dreds of different combinations of boundary conditions for aplate It is desirable to develop a unified and efficient methodwhich is capable of dealing with more complicated problemsinvolving arbitrary elastic edge restraints and nonuniformthickness
In view of those technical limitations and practical needsthis investigation sets out to present a modified Fourier solu-tion technique for the free vibration analysis of moderatelythick rectangular plates with variable thickness and arbitraryboundary conditions and to provide a unified and reason-able accurate alternative to other analytical and numericaltechniques This paper can be considered as an extension ofthe modified Fourier series method previously developed formodeling plates [39ndash41] and shells [42 43]The one displace-ment and two rotation functions are invariably expressedas the superposition of a 2D Fourier cosine series and foursupplementary functions in the form of the product of apolynomial function and a single cosine series expansionwith all these unknown expansion coefficients treated as thegeneralized coordinates and determined using the Rayleigh-Ritz procedure The change of the boundary conditions canbe easily achieved by only varying the stiffness of the threesets of boundary springs along all edges of the rectangularplates without involving any change to the solution proce-dureThe current results are checked againstwith FEMresultsor existing results published in the literature for both uniformand nonuniform (linear and nonlinear variation) thicknessplate cases with good agreement achieved
2 Theoretical Formulations
To perform the free vibration analysis of moderately thickrectangular plates with thickness varying in two directionssubjected to the general elastic boundary conditions thecombination of the artificial spring technique together withRayleigh-Ritzmethod is feasible Consider amoderately thickrectangular plate with the dimension of 119886 times 119887 times ℎ(119909 119910)
Shock and Vibration 3
x
z
y
a
b
h(x y)
Figure 1 The general elastic boundary conditions of moderatelythick rectangular plates with varying thickness in two directions
and the coordinate of the moderately thick rectangularplate with elastically retrained edges is depicted in Figure 1Three groups of boundary restraining springs (translationrotational and torsional springs) are arranged at all sidesof the plate to separately simulate the boundary force Byassigning the stiffness of the boundary springs various valuesit is equivalent to impose different boundary force on themidsurface of the plate For example the clamped boundarycondition can be readily obtained by setting the spring coef-ficients into infinity (a very large number in practical calcu-lation) for the translation rotations and torsional restrainingsprings along each edge
Based on the Mindlin plate theory the displacementsvectors with three directions are
119906 (119909 119910 119911 119905) = 119911120595119909(119909 119910 119905)
V (119909 119910 119911 119905) = 119911120595119910(119909 119910 119905)
119908 (119909 119910 119911 119905) = 119908 (119909 119910 119905)
(1)
where 119906 V and 119908 represent the 119909 119910 and 119911 directiondisplacement functions and the 120595
119909and 120595
119910are the slop due
to bending along in the respective planes The relationship 119908with the slops 120595
119909and 120595
119910is 120595119909= minus119889119908119889119909 and 120595
119910= minus119889119908119889119910
For the moderately thick rectangular plates making useof the strain-stress relationship defined in elasticity theorythe normal shear strains and transverse shear strains can beexpressed as follows
120576119909119909
120576119910119910
120574119909119910
120574119909119911
120574119910119911
=
119911120597120595119909
120597119909
119911120597120595119910
120597119910
119911(
120597120595119909
120597119910
+
120597120595119910
120597119909
)
120595119909+
120597119908
120597119909
120595119910+
120597119908
120597119910
[
[
[
[
[
[
[
[
120590119909119909
120590119910119910
120591119909119910
120591119909119911
120591119910119911
]
]
]
]
]
]
]
]
=
119864
2 (1 minus 1205832)
[
[
[
[
[
[
[
[
2 2120583 0 0 0
2120583 2 0 0 0
0 0 1 minus 120583 0 0
0 0 0 120581 (1 minus 120583) 0
0 0 0 0 120581 (1 minus 120583)
]
]
]
]
]
]
]
]
times
[
[
[
[
[
[
[
[
120576119909119909
120576119910119910
120574119909119910
120574119909119911
120574119910119911
]
]
]
]
]
]
]
]
(2)
where 120576119909119909 120576119910119910 and 120574
119909119910are the normal and shear strains in the
119909119910 and 119911 coordinate systemThe transverse shear strains 120574119909119911
and 120574119910119911
are constant through the thickness The 120590119909119909
and 120590119910119910
are the normal stresses in the 119909 119910 directions 120591119909119911 120591119910119911 and 120591
119909119910
are shear stresses in the 119909 119910 and 119911 coordinate system The 119864is Youngrsquos modulus 120583 is Poissonrsquos ration and 120581 is the shearcorrection factor to account for the fact
In terms of transverse displacements and slope thebending and twisting moments and the transverse shearingforces in plates can be expressed as
119872119909119909= int
ℎ2
minusℎ2
120590119909119909119911 119889119911 = 119863(
120597120595119909
120597119909
+ 120583
120597120595119910
120597119910
)
119872119910119910= int
ℎ2
minusℎ2
120590119910119910119911 119889119911 = 119863(
120597120595119910
120597119910
+ 120583
120597120595119909
120597119909
)
119872119909119910= int
ℎ2
minusℎ2
120591119909119910119911 119889119911 =
119863 (1 minus 120583)
2
(
120597120595119909
120597119910
+
120597120595119910
120597119909
)
119876119909= 120581int
ℎ2
minusℎ2
120591119909119911119889119911 = 120581119866ℎ (119909 119910) (120595
119909+
120597119908
120597119909
)
119876119910= 120581int
ℎ2
minusℎ2
120591119910119911119889119911 = 120581119866ℎ (119909 119910) (120595
119910+
120597119908
120597119910
)
(3)
where ℎ(119909 119910) is the thickness function 119863 = 119864ℎ(119909 119910)3
(12(1 minus 1205832
)) is the flexural rigidity and 119866 = 119864(2(1 + 120583))
is the shear modulusThe boundary conditions for a general supported moder-
ately thick rectangular plate can be expressed as the followingforms based on the force equilibrium relationship on the foursides
on 119909 = 0
1198961199090119908 = minus119876
119909 1198701199090120595119909= minus119872
119909119909 1198701199101199090
120595119910= minus119872
119909119910
on 119909 = 119886
119896119909119886119908 = 119876
119909 119870119909119886120595119909= 119872119909119909 119870119910119909119886
120595119910= 119872119909119910
4 Shock and Vibration
on 119910 = 0
1198961199100119908 = minus119876
119910 1198701199100120595119910= minus119872
119910119910 1198701199091199100
120595119909= minus119872
119909119910
on 119910 = 119887
119896119910119887119908 = 119876
119910 119870119910119887120595119910= 119872119910119910 119870119909119910119887
120595119909= 119872119909119910
(4)
where 1198961199090
and 119896119909119886
(1198961199100
and 119896119910119887) are linear spring constants
1198701199090
and 119870119909119886
(1198701199100
and 119870119910119887) are the rotational spring con-
stants and 1198701199101199090
and 119870119910119909119886
(1198701199091199100
and 119870119909119910119887
) are the torsionalspring constants at 119909 = 0 and 119886 (119910 = 0 and 119887) respectivelyTherefore arbitrary boundary conditions of the plate can begenerated by assigning the linear springs rotational springsand torsional springs at proper stiffness For instance aclamped boundary (C) is achieved by simply setting thestiffness of the entire springs equal to infinity (which is repre-sented by a very large number 1014) Inversely a free bound-ary (F) is gained by setting the stiffness of the entire springsequal to zero
Thus the total potential energy of the spring restrainedplate which is composed of two parts namely the strainenergy of the moderately thick rectangular plates and thepotential energy stored in the boundary springs can beexpressed as
119880 =
1
2
int
119886
0
int
119887
0
119863[(
120597120595119909
120597119909
+
120597120595119910
120597119910
)
2
minus 2 (1 minus 120583)
times (
120597120595119909
120597119909
120597120595119910
120597119910
minus
1
4
(
120597120595119909
120597119910
+
120597120595119910
120597119909
)
2
)]
+ 120581119866ℎ (119909 119910)
times [(120595119909+
120597119908
120597119909
)
2
+ (120595119910+
120597119908
120597119910
)
2
] 119889119909119889119910
+
1
2
int
119886
0
[(11989611991001199082
+ 11987011991001205952
119910+ 1198701199091199100
1205952
119909)
10038161003816100381610038161003816119910=0
+ (1198961199101198871199082
+ 1198701199101198871205952
119910+ 119870119909119910119887
1205952
119909)
10038161003816100381610038161003816119910=119887
] 119889119909
+
1
2
int
119887
0
[(11989611990901199082
+ 11987011990901205952
119909+ 1198701199101199090
1205952
119910)
10038161003816100381610038161003816119909=0
+ (1198961199091198861199082
+ 1198701199091198861205952
119909+ 119870119910119909119886
1205952
119910)
10038161003816100381610038161003816119909=119886
] 119889119910
(5)
As the springs are considered with no mass while retain-ing certain stiffness the total kinetic energy of themoderatelythick rectangular plates is
119879 =
1205881205962
2
int
119887
0
int
119886
0
[
[
ℎ (119909 119910)1199082
+ ℎ3
(119909 119910) (1205952
119909+ 1205952
119910)
12
]
]
119889119909119889119910
(6)
where 120588 is the mass density and 120596 denotes the natural fre-quency of the plate
In view of satisfying arbitrarily supported boundary con-ditions of the moderately thick rectangular plate the admis-sible functions expressed in the form of the improved Fourierseries are introduced to remove the potential discontinuitieswith the functions and their derivatives Thus the moder-ately thick rectangular plate displacements and rotation areexpressed as
120595119909(119909 119910) =
infin
sum
119898=0
infin
sum
119899=0
119860119898119899
cos 120582119886119898119909 cos 120582
119887119899119910
+
2
sum
119897=1
120577119897
119887(119910)
infin
sum
119898=0
119886119897
119898cos 120582119886119898119909
+
2
sum
119897=1
120577119897
119886(119909)
infin
sum
119899=0
119887119897
119899cos 120582119887119899119910
(7)
120595119910(119909 119910) =
infin
sum
119898=0
infin
sum
119899=0
119861119898119899
cos 120582119886119898119909 cos 120582
119887119899119910
+
2
sum
119897=1
120577119897
119887(119910)
infin
sum
119898=0
119888119897
119898cos 120582119886119898119909
+
2
sum
119897=1
120577119897
119886(119909)
infin
sum
119899=0
119889119897
119899cos 120582119887119899119910
(8)
119908 (119909 119910) =
infin
sum
119898=0
infin
sum
119899=0
119862119898119899
cos 120582119886119898119909 cos 120582
119887119899119910
+
2
sum
119897=1
120577119897
119887(119910)
infin
sum
119898=0
119890119897
119898cos 120582119886119898119909
+
2
sum
119897=1
120577119897
119886(119909)
infin
sum
119899=0
119891119897
119899cos 120582119887119899119910
(9)
where 120582119886119898
= 119898120587119886 120582119887119899= 119899120587119887 and 119860
119898119899 119861119898119899 and 119862
119898119899are
the Fourier coefficients of two-dimensional Fourier seriesexpansions for the displacements and rotation functionsrespectively 119886119897
119898 119887119897119898 119888119897119898 119889119897119898 119890119897119898 and 119891119897
119898are the supplemented
coefficients of the auxiliary functions where 119897 = 1 2 Thespecific expressions of the auxiliary functions 120577119897
119886and 120577
119897
119887are
defined as
1205771
119886(119909) =
119886
2120587
sin(1205871199092119886
) +
119886
2120587
sin(31205871199092119886
)
1205772
119886(119909) = minus
119886
2120587
cos(1205871199092119886
) +
119886
2120587
cos(31205871199092119886
)
1205771
119887(119910) =
119887
2120587
sin(120587119910
2119887
) +
119887
2120587
sin(3120587119910
2119887
)
1205772
119887(119910) = minus
119887
2120587
cos(120587119910
2119887
) +
119887
2120587
cos(3120587119910
2119887
)
(10)
As shown in (7)ndash(9) the supplementary functions 1205771119886(119909)
1205772
119886(119909) 1205771
119887(119910) and 120577
2
119887(119910) are used for the displacement and
rotation expressions The theoretical meaning of introduc-ing these terms into the Fourier series is to remove the
Shock and Vibration 5
potential discontinuities and their derivatives throughout thewhole plate structure including the boundaries and then toeffectively enhance the convergence of the results To ensurethis continuity of selection expressions and correspondingderivatives at any point on the plate the first-order derivativesof the 119909 and119910 directions should exist as indicated by (4)Thisrequirement is guaranteed by the selected supplementaryfunctions because it is easy to verify that
1205771
119886(0) = 120577
1
119886(119886) = 120577
11015840
119886(119886) = 0 120577
11015840
119886(0) = 1
1205772
119886(0) = 120577
2
119886(119886) = 120577
21015840
119886(0) = 0 120577
21015840
119886(119886) = 1
(11)
Similar conditions exist for the 119910-related polynomials1205771
119887(119910) and 120577
2
119887(119910) It has to be mentioned that although the
solution is theoretically exact for the superposition of infinitenumbers of Fourier terms in actual calculation we truncatethe infinite series to 119872 and 119873 to obtain the results withacceptable accuracy
Since the energy expressions and admissible function ofthe plate have been established the remaining task is to deter-mine the Fourier expanded coefficients and supplemented
coefficients in (7)ndash(9) The Lagrangian energy functional (119871)of the plate is written as
119871 = 119879 minus 119880 (12)
Then the Lagrangian expression is minimized by takingits derivatives with respect to these coefficients
120597119871
120597120599
= 0 120599 =
119860119898119899
119886119897
119898119887119897
119899
119861119898119899
119888119897
119898119889119897
119899
119862119898119899
119890119897
119898119891119897
119899
(13)
Since the displacements and rotation components of theplate are chosen as 119872 and 119873 to obtain the results withacceptable accuracy a total of 3 lowast (119872 + 1) lowast (119873 + 1) + 6 lowast
(119872 +119873 + 2) equations are obtainedThey can be summed up in a matrix form
(K minus 1205962M)E = 0 (14)
The unknown coefficients in the displacement expres-sions can be expressed in the vector form as E where
E =
11986000 11986001 119860
11989810158400 11986011989810158401 119860
11989810158401198991015840 119860
119872119873 1198861
0 119886
1
119872 1198862
0 119886
2
119872 1198871
0 119887
1
119873 1198872
0 119887
2
119873
11986100 11986101 119861
11989810158400 11986111989810158401 119861
11989810158401198991015840 119861
119872119873 1198881
0 119888
1
119872 1198882
0 119888
2
119872 1198891
0 119889
1
119873 1198892
0 119889
2
119873
11986200 11986201 119862
11989810158400 11986211989810158401 119862
11989810158401198991015840 119862
119872119873 1198901
0 119890
1
119872 1198902
0 119890
2
119872 1198911
0 119891
1
119873 1198912
0 119891
2
119873
119879
(15)
In (14)K is the stiffness matrix for the plate andM is themass matrix They can be expressed separately as
K =
[
[
[
[
[
[
[
[
[
K1-1 K
1-2 K1-3 sdot sdot sdot K
1-9
K1198791-2 K
2-1 K2-3 sdot sdot sdot K
2-9
K1198791-3 K119879
2-3 K3-3 sdot sdot sdot K
3-9
d
K1198791-9 K119879
2-9 K1198793-9 sdot sdot sdot K
9-9
]
]
]
]
]
]
]
]
]
M =
[
[
[
[
[
[
[
[
[
M1-1 M
1-2 M1-3 sdot sdot sdot M
1-9
M1198791-2 M
2-1 M2-3 sdot sdot sdot M
2-9
M1198791-3 M119879
2-3 M3-3 sdot sdot sdot M
3-9
d
M1198791-9 M119879
2-9 M1198793-9 sdot sdot sdot M
9-9
]
]
]
]
]
]
]
]
]
(16)
The specific expressions for the elements in (16) aregiven in Appendix AMoreover all the necessary expressionswhich will be used in the calculations of the eigenvalues andeigenvectors are given in Appendix B
Obviously the natural frequencies and eigenvectors cannow be readily obtained by solving a standard matrix eigen-problem Since the components of each eigenvector are actu-ally the expansion coefficients of the modified Fourier seriesthe corresponding mode shape can be directly determinedfrom (14) In other words once the coefficient eigenvectorE is
determined for a given frequency the displacement functionsof the plate can be determined by substituting the coefficientsinto (9) When the forced vibration is involved by addingthe work done by external force in the Lagrangian energyfunction and summing the loading vector F on the right sideof (14) the characteristic equation for the forced vibration ofthe moderately thick rectangular plates is readily obtained
3 Numerical Examples and Discussion
In this section a systematic comparison between the cur-rent solutions and theoretical results published by otherresearchers or finite element method (FEM) results is car-ried out to validate the excellent accuracy reliability andfeasibility of the modified Fourier method A comprehensivestudy on the effects of elastic restraint parameters andvarying thickness in two directions is also reported Newresults are obtained for plates subjected to general elasticboundary restraints with nonlinear variable thickness in bothdirections The discussion is arranged as follows Firstly theconvergence of the modified Fourier solution is checked Inaddition the influence of the stiffness of boundary springcomponents is studied Secondly the uniform thicknessplates with various combinations of classical boundary con-ditions elastic boundary conditions and different structureparameters are examined Thirdly the nonuniform thicknessplate with linear variation in both directions various com-binations of classical boundary conditions conditions and
6 Shock and Vibration
Table 1 Frequency parameters Ω = 1205961198872
(120588ℎ119863)12
1205872 for square moderately thick plates with uniform thickness and different boundary
conditions
Boundary conditions M N Model sequence1 2 3 4 5 6 7 8
C-C-C-C
119872 = 119873 = 2 3375 7191 7191 9856 1605 1628 1747 1747119872 = 119873 = 4 3299 6308 6308 8853 1046 1056 1263 1263119872 = 119873 = 6 3294 6282 6282 8801 1037 1047 1254 1254119872 = 119873 = 8 3292 6278 6278 8797 1036 1046 1253 1253119872 = 119873 = 10 3292 6276 6276 8794 1036 1046 1253 1253119872 = 119873 = 12 3292 6276 6276 8793 1036 1046 1252 1252119872 = 119873 = 14 3292 6276 6276 8793 1036 1046 1252 1252119872 = 119873 = 16 3292 6275 6275 8792 1036 1045 1252 1252Reference [4] 3292 6276 6276 8792 1036 1046 1252 1252
E-E-E-E
119872 = 119873 = 2 06153 10861 10861 17898 27709 28417 35010 35010119872 = 119873 = 4 06150 10799 10799 17785 27695 28405 34914 34914119872 = 119873 = 6 06149 10796 10796 17773 27694 28403 34898 34898119872 = 119873 = 8 06149 10796 10796 17771 27693 28403 34893 34893119872 = 119873 = 10 06149 10796 10796 17770 27693 28403 34892 34892119872 = 119873 = 12 06149 10796 10796 17770 27693 28403 34892 34892119872 = 119873 = 14 06149 10796 10796 17770 27693 28403 34892 34892119872 = 119873 = 16 06149 10796 10796 17770 27693 28403 34891 34891Reference [33] 06153 10861 10861 17898 27709 28417 35010 35010
FEM 06150 17097 17097 17774 27694 28406 34898 34898
different structure parameters are examined Then numer-ical solutions for moderately thick rectangular plates withelastic boundary conditions are presented The effects of theelastic restraint parameters are also investigated Finally thenonuniform thickness plate with nonlinear variation in bothdirections and arbitrary boundary conditions is also studied
31 Convergence and Stiffness Value Study Since the Fourierseries is numerically truncated and only finite terms arecounted in actual calculations the proposed solution shouldbe understood as a solution with arbitrary precision Inthis subsection a uniform thickness square moderately thickrectangular plate with completely clamped boundary condi-tion (C-C-C-C) and four edges equally elastically restrainedagainst linear spring constants and rotation spring constantssupports (E-E-E-E 119870
119879= 119896119894(1198873
119863) 119870119877
= 119870119894(119887119863)
119870119879
= 10 and 119870119877
= 5) has been selected to demon-strate the convergence and accuracy of the modified Fouriermethod In Table 1 the first eight frequency parametersΩ = 120596119887
2
(120588ℎ119863)12
1205872 for the considered uniform thickness
square moderately thick rectangular plate with C-C-C-C andE-E-E-E boundary conditions are examinedThe table showsthat the proposedmethod has fast convergence behaviorThemaximum discrepancy in the worst case between the 6 times 6
truncated configuration and the 8times8 one is less than 0064In order to fully illustrate the convergence of the presentmethod Figures 3 4 and 5 present the 1st and 8th frequencyparameters Ω = 120596119887
2
(120588ℎ119863)12
1205872 with various truncated
numbers 119872 = 119873 subjected to different boundary conditionand aspect ratios A highly desired convergence characteristicis observed such that (a) sufficiently accurate results can beobtained with only a small number of terms in the seriesexpansions and (b) the solution is consistently refined asmore
terms are included in the expansions However this shouldnot constitute a problem in practice because one can alwaysverify the accuracy of the solution by increasing the trunca-tion number until a desired numerical precision is achievedAs a matter of fact this ldquoquality controlrdquo scheme can be easilyimplemented automatically In modal analysis the naturalfrequencies for higher-order modes tend to converge slower(see Table 1) Thus an adequate truncation number shouldbe dictated by the desired accuracy of the largest naturalfrequencies of interest In view of the excellent numericalbehavior of the current solution the truncation numbers willbe simply set as119872 = 119873 = 12 in the following calculations
As far as the accuracy of the present method is con-cerned the converged solutions of the present method arein excellent agreement with both the results reported byreference data and the finite element results For C-C-C-C boundary conditions the max discrepancy between thepresent results and the reference data does not exceed 0011for the worst case and in most cases is 0 Comparing theresults with exact solutions [4] it is observed that eight termsare sufficient to obtain accurate resultsMoreover with regardto the E-E-E-E boundary condition the max discrepancybetween the present results and the reference data does notexceed 031 for the worst case and in most cases is 01Regarding the results with DQM solutions [33] it can be seenthat the six terms are sufficient to obtain enough accurateresults In addition it is clear that the results of the presentapproach with just 663 DOFs (119872 times 119873 = 12 times 12) canpredict the vibration characteristics accurately Most of themare identical to those obtained from finite element method(FEM) with 10201 DOFs (S4R 001m times 001m) That isto say it needs only 662 DOFs compared with FEM toobtain the same precision solutions for the considered case
Shock and Vibration 7
1
2
3
4
5
6
The 1st orderThe 3rd order
101 103 105 107 109 1011 1013
Ω
Ki (Nm)
(a)
2
3
4
5
6
101 103 105 107 109 1011 1013
The 1st orderThe 3rd order
Ω
Ki (Nmrad)
(b)
3
4
5
6
101 103 105 107 109 1011 1013
Kij (Nmrad)
The 1st orderThe 3rd order
Ω
(c)
Figure 2 The effect of boundary spring stiffness on the natural frequencies Ω (a) translation spring (b) rotation spring and (c) torsionalspring
On the same hardware (Intel i7-39GHz) the computing timeof the present formulation for the solution (119872times119873 = 12times12)implemented in optimized MATLAB scripts is about 2125 swhereas the finite element solution consumes 34578 s that isat least 16 times more CPU time than the present method forthe same problem
As mentioned earlier in the current modeling frame-work all the classical boundary conditions and their com-binations can be conveniently viewed as special cases whenthe stiffness for the normal and tangential boundary springsbecomes zero andor infinitely large Thus the effects of thestiffness of the translation (119896
119894) rotation (119870
119894) and torsional
springs (119870119894119895) on the modal characteristics should be inves-
tigated As shown in the Figures 2(a)ndash2(c) the first and thethird frequency parameters are separately obtained by vary-ing the stiffness of one group of the boundary springs from
extremely large (1014) to extremely small (100) while assigningthe other group of the springs infinite stiffness (1014) It canbe found in Figure 2(a) that the frequency parameter almostkeeps at a level when the stiffness of the translation springs islarger than 1012 or smaller than 107 In Figure 2(b) the influ-ences of the rotation springs on frequency parameters aregiven It is shown that the frequency curves change greatlywithin the stiffness range (106 to 1010) while out of this rangethe frequency curves separately keep level In Figure 2(c) theinfluences of the torsional springs on frequency parametersare given It is shown that the frequency curves almost changewhen the stiffness changes in the whole range
Based on the analysis it can be found that the torsionalsprings almost have no effect on the structure Also the rela-tionship between the rotation springs and twisting momentcan be seen from the boundary condition expression Then
8 Shock and Vibration
32
36
40
44
48
52
2
3
4
5
6
7
8
Ω Ω
0 2 4 6 8 10 12 14 16 18M = N
ab = 1
ab = 32
(a) The 1st order
0 2 4 6 8 10 12 14 16 180
20
40
60
80
0
50
100
150
200
250
300
M = N
Ω Ω
ab = 1
ab = 32
(b) The 8th order
Figure 3 The effect of numerically truncated numbers119872 = 119873 on the natural frequenciesΩ for C-C-C-C boundary condition
0 2 4 6 8 10 12 14 16 18125
130
135
140
145
150
088
090
092
094
096
098
Ω Ω
M = N
ab = 1
ab = 32
(a) The 1st order
0 2 4 6 8 10 12 14 16 18
55
60
65
70
75
80
40
45
50
55
60
65
70
75
80Ω Ω
M = N
ab = 1
ab = 32
(b) The 8th order
Figure 4 The effect of numerically truncated numbers119872 = 119873 on the natural frequenciesΩ for F-F-F-F boundary condition
the twisting moments have small effect on the vibration char-acteristics of the structure However in this paper in order toget a more accurate prediction of the vibration characteristicsof the structure the twisting moment on boundary edges istaken into account In the latter study in addition to the factthat free boundary is not unexpected considering the torsionspring the other boundary conditions are introduced into atorsion spring and the spring stiffness takes infinity FormFigure 2 analysis it also concluded that the translation springhas a wider influence range than the rotation spring on thefrequency parameters that is for the translation springs thestable frequency parameters appearwhen the stiffness ismorethan 1012 or less 107 while for the rotation springs when thestiffness value is assigned more than 1010 the frequency para-meters become smoothThus it is suitable to use 1014 to sim-ulate the infinite stiffness value in the model validation partsand in the following examples
32 Uniform Thickness Moderately Thick Plates with Classicaland Elastic Boundary Conditions In this subsection themodified Fourier solution is applied to deal with vibrationproblems of uniform thickness moderately thick rectangu-lar plates subject to the classical boundary conditions andarbitrary elastic boundary conditions In present work threegroups of continuously distributed boundary springs areintroduced to simulate the given or typical boundary condi-tions As mentioned earlier the stiffness of these boundarysprings can take any value from zero to infinity to bettermodel many real-world restraint conditions Taking edge 119909 =0 for example the corresponding spring stiffness for the threetypes of classical boundaries and elastic boundaries is
completely free 119876119909= 0119872
119909119909= 0 and119872
119909119910= 0
1198961199090= 0 119870
1199090= 0 119870
1199101199090= 0 (17a)
Shock and Vibration 9
0 2 4 6 8 10 12 14 16 18
166
168
170
172
152
156
160
164
168
Ω Ω
ab = 1
ab = 32
M = N
(a) The 1st order
0 2 4 6 8 10 12 14 16 1850
55
60
65
70
4
5
6
7
8
9
10
Ω Ω
M = N
ab = 1
ab = 32
(b) The 8th order
Figure 5 The effect of numerically truncated numbers 119872 = 119873 on the natural frequencies Ω with four edges elastically restrained againsttranslation and rotation (119870
119879= 119896119894(1198873
119863) 119870119877= 119870119894(119887119863) 119870
119879= 100 and 119870
119877= 100)
simply supported 119908 = 0119872119909119909= 0 and 120595
119910= 0
1198961199090= 1014
1198701199090= 0 119870
1199101199090= 1014
(17b)
completely clamped 119908 = 0 120595119909= 0 and 120595
119910= 0
1198961199090= 1014
1198701199090= 1014
1198701199101199090
= 1014
(17c)
arbitrarily elastic boundary 119908 = 0 120595119909
= 0 120595119910
= 0119876119909
= 0119872119909119909
= 0 and119872119909119910
= 0
1198961199090= Γ1199081199090
1198701199090= Γ1199091199090
1198701199101199090
= 1014
(17d)
The appropriateness of the three classical boundariesdefined in (17a)ndash(17c) will be proved by several examplesgiven in following the arbitrary elastic boundaries are alsodefined in (17d) and the Γ
119908119894(Γ119908119894= 1198961198941198960 1198960= 1 times 10
9Nm2119894 = 1199090 1199091198861199100119910119887) and Γ
119909119894(Γ119909119894= 11987011989411987001198700= 1times10
9Nmrad119894 = 1199090 119909119886 1199100 119910119887) elastic restraint parameters representcorresponding spring stiffness For the sake of simplicitya four-letter string is employed to represent the restraintcondition of a plate such that F-C-S-E identifies the platewithedges 119909 = 0 119910 = 0 119909 = 119886 and 119910 = 119887 having free clampedshear-supported restrained and elastic boundary conditionsrespectively
As for the first case a uniform thickness moderately thickplate with different classical boundaries and structure param-eters is investigated here In Table 2 the comparison of thefirst eight frequency parametersΩ = 120596119887
2
(120588ℎ119863)12
1205872 of the
considered plate is presented The S-S-S-S C-F-F-F S-S-F-FF-F-F-F and S-C-S-C boundary conditions are performed inthe comparison Excellent agreements are observed betweenthe solutions obtained by the modified Fourier method thereferential data and finite element method (FEM) results forthe uniform thickness moderately thick rectangular platesIt is also verified that the definition of the three types of
classical boundaries in (17a)ndash(17c) is appropriate In additionthe elastic boundary conditions (17d) are also verified Inthe next two examples we will account for the vibrationof moderately thick plate with elastic edge supports Thefirst model considered is an S-S-S-S square moderately thickplate with all edges elastically rotationally restrained Thatis 1198701199090
= 119870119909119886
= 1198701199100
= 119870119910119887
= Γ119909 and all the
other restraining springs are set to have an infinite stiffness(namely represented by 1014 in numerical calculation) Thesix frequency parameters Ω = 120596119887
2
(120588ℎ119863)12
1205872 are given
in Table 3 for several different restraining coefficient valuesthe finite element method (FEM) results calculated usingABAQUSare also listed in this table as a referenceThe secondmodel concerns a complete square moderately thick platewith all edges elastically restrained That is 119896
1199090= 119896119909119886
=
1198961199100
= 119896119910119887
= Γ119908and 119870
1199090= 119870119909119886
= 1198701199100
= 119870119910119887
= Γ119909 The
six frequency parameters Ω = 1205961198872
(120588ℎ119863)12
1205872 are given in
Table 4 for several different restraining coefficient values thefinite element method (FEM) results are also listed in Table 4as a reference It can be clearly seen that the comparison isextremely good which implies that the current method isable tomake correct predictions for themodal characteristicsof moderately thick plate with not only classical boundaryconditions but also elastic edge restraints
The excellent agreements between the solution obtainedby the modified Fourier method and the referential datafor the moderately thick plate subjected to the combina-tions of classical boundary conditions and elastic boundaryconditions given in Tables 2ndash4 indicate that the proposedmethod is sufficiently accurate to deal with uniform thicknessmoderately thick plate with arbitrary boundary conditions
33 Linearly VariationThickness Moderately Thick Plates withClassical and Elastic Boundary Conditions In the theoreticalformulations this paper concerns the varying thicknessmoderately thick plates with classical and elastic boundary
10 Shock and Vibration
Table 2 Frequency parameters Ω = 1205961198872
(120588ℎ119863)12
1205872 for uniform thickness moderately thick plates with different classical boundaries and
structure parameters
Boundary condition ab Model sequence1 2 3 4 5 6 7 8
S-S-S-S
11931 4605 4605 7064 8605 8605 10792 10792
(1933a) (4611) (4611) (7067) (8607) (8607) (10799) (10780)(1931b) (4605) (4605) (7064) (8605) (8605) (10792) (10792)
321408 2648 4127 4605 5261 7064 7153 8184(1410a) (2651) (4128) (4608) (5267) (7074) (7155) (8188)(1408b) (2648) (4127) (4605) (5261) (7064) (7151) (8184)
C-F-F-F
10348 0817 2035 2583 2862 4816 5478 5774
(0348a) (0817) (2038) (2586) (2865) (4821) (5504) (5793)(0348b) (0816) (2034) (2582) (2860) (4811) (5477) (5772)
320155 0502 0941 1665 2292 2613 3303 3498
(0154a) (0502) (0941) (1663) (2295) (2618) (3305) (3500)(0154b) (0501) (0940) (1662) (2292) (2612) (3298) (3494)
S-S-F-F
10333 1678 1875 3559 4720 4946 6474 6632
(0335a) (1680) (1878) (3600) (4721) (4949) (6478) (6642)(0333b) (1677) (1874) (3557) (4718) (4945) (6472) (6631)
320222 0938 1631 2363 2573 4082 4482 4753
(0223a) (0940) (1632) (2364) (2575) (4092) (4483) (4759)(0222b) (0938) (1631) (2361) (2571) (4077) (4479) (4752)
F-F-F-F1 1290 1919 2363 3235 3235 5605 5605 5646
(1291a) (1920) (2365) (3239) (3239) (5607) (5607) (6648)
32 0865 0948 1961 2166 2461 2843 3551 4026(0865a) (0949) (1963) (2168) (2465) (2844) (3556) (4034)
S-C-S-C1 2700 4971 5990 7973 8787 10250 11334 12024
(2701a) (4971) (5993) (7975) (8788) (10253) (11335) (12028)
32 2348 3253 4972 5646 6489 7381 7974 9930(2348a) (3255) (4974) (5647) (6590) (7384) (7801) (9932)
aResults in parentheses are taken from FEMbResults in parentheses are taken from [11]
conditions The varying thickness function ℎ(119909 119910) can beexpressed as ℎ
0(1 + 120572119909
119904
)(1 + 120573119910119905
) in which the ℎ0 120572 and
120573 represent the initial thickness gradient in 119909 direction andgradient in 119910 direction When the indexes 119904 and 119905 take thevalue 119904 = 119905 = 1 the analyticalmodel imitates the linearly vari-ation thickness moderately thick plates structure In order tounify the description and facilitate the analytical calculationsof the involved integrals all the thickness variation functionscan be expanded into either 1D or 2D Fourier cosine seriesresulting in
ℎ (119909 119910) =
infin
sum
119898=0
infin
sum
119899=0
ℎ0119902119898119899
cos 119898120587119909119886
cos119898120587119910
119887
(18)
where119902119898119899
=
4
119886119887
int
119887
0
int
119886
0
(1 + 120572119909119904
) (1 + 120573119910119905
) cos 119898120587119909119886
cos119898120587119910
119887
119889119909 119889119910
(19)
In order to prove the validity of the present methodfor the vibration of linearly variation thickness moderately
thick plates with arbitrary boundary conditions the typicalclassical boundary conditions as the first case will be con-sidered In Table 5 the comparison of the first six frequencyparameters Ω = 120596119886
2
(120588ℎ01198630)12 of the moderately thick
plates with linearly varying thickness is presentedThe S-S-S-S C-F-F-F S-S-F-F C-C-C-C and S-C-S-C boundary condi-tions are performed in the comparison Excellent agreementsare observed between the solutions obtained by the modifiedFourier method and finite element method (FEM) results forthemoderately thick plates with linear variation thickness Toinvestigate the influence of the aspect ratio on the uniformthickness and nonuniform thickness moderately thick platesthe effect on the frequency parameters for plates with S-S-S-Sboundary conditions is presented in Figure 6 The thicknessfunctions are ℎ
0and ℎ0(1+05times119909)(1+05times119910) respectively It
is seen from Figure 6 that the influence of aspect ratios on thefrequency parameters for nonuniform thickness moderatelythick plates is more complicated
In the next two examples we also account for the vibra-tions of moderately thick plate with linear variation thicknessand elastic edge supports The first model considered is an S-S-S-S square moderately thick plate with all edges elastically
Shock and Vibration 11
Table 3 Frequency parameters Ω = 1205961198872
(120588ℎ119863)12
1205872 for S-S-S-S moderately thick plates (119886119887 = 1) with uniform thickness and elastic
rotation support (1198701199090= 119870119909119886= 1198701199100= 119870119910119887= Γ119909)
Γ119909
aℎ0119886
Model sequence1 2 3 4 5 6
0
005 1945 4834 4834 7591 9515 9519(1942b) (4836) (4836) (7584) (9547) (9552)
01 1856 4500 4500 6829 8499 8513(1858b) (4510) (4510) (6843) (8539) (8554)
015 1758 4114 4114 6040 7421 7445(1761b) (4126) (4126) (6060) (7462) (7486)
02 1656 3733 3733 5335 6468 6499(1660b) (3747) (3747) (5357) (6508) (6538)
05
005 3541 7071 7071 10227 12318 12399(3546b) (7091) (7091) (10253) (12386) (12468)
01 3274 6242 6242 8719 10302 10421(3282b) (6267) (6267) (8754) (10370) (10488)
015 2947 5361 5361 7290 8465 8606(2956b) (5387) (5387) (7327) (8524) (8666)
02 2624 4599 4599 6148 7057 7199(2634b) (4624) (4624) (6182) (7107) (7250)
1
005 3541 7072 7072 10228 12320 12401(3547b) (7093) (7093) (10254) (12388) (12470)
01 3278 6248 6248 8727 10311 10429(3286b) (6274) (6274) (8762) (10378) (10497)
015 2957 5374 5374 7305 8478 8620(2966b) (5400) (5400) (7341) (8537) (8680)
02 2641 4617 4617 6166 7070 7215(2651b) (4643) (4643) (6200) (7121) (7268)
15
005 3542 7073 7073 10229 12320 12401(3547b) (7093) (7093) (10254) (12388) (12471)
01 3280 6251 6251 8729 10313 10432(3287b) (6276) (6276) (8765) (10381) (10500)
015 2960 5379 5379 7309 8482 8625(2969b) (5405) (5405) (7346) (8541) (8685)
02 2647 4623 4623 6172 7074 7221(2657b) (4649) (4649) (6206) (7125) (7274)
2
005 3542 7073 7073 10229 12321 12402(3547b) (7093) (7093) (10255) (12389) (12471)
01 3280 6252 6252 8730 10315 10434(3288b) (6277) (6277) (8766) (10382) (10501)
015 2962 5381 5381 7312 8484 8628(2971b) (5407) (5407) (7348) (8543) (8688)
02 2650 4627 4627 6175 7076 7224(2660b) (4652) (4652) (6210) (7128 (7277)
aΓ119909= 1198701198941198700 (1198700= 1 times 10
9 Nmrad 119894 = 1199090 119909119886 1199100 119910119887)bResults in parentheses are taken from FEM
12 Shock and Vibration
Table 4 Frequency parameters Ω = 1205961198872
(120588ℎ119863)12
1205872 for moderately thick plates (119886119887 = 1) with uniform thickness and all edge elastic
restraints (1198701199090= 119870119909119886= 1198701199100= 119870119910119887= Γ119909and 119896
1199090= 119896119909119886= 1198961199100= 119896119910119887= Γ119908)
Γ119909
aΓ119908
aℎ0119886
Model sequence1 2 3 4 5 6
05 0
005 0986 1967 3893 3893 4850 4850(0988b) (1971) (3897) (3897) (4852) (4852)
01 0916 1835 3515 3522 4375 4375(0917b) (1837) (3519) (3528) (4341) (4341)
015 0789 1635 2983 3027 3771 3771(0792b) (1638) (2988) (3029) (3775) (3775)
02 0646 1444 2497 2601 3249 3249(0652b) (1448) (2508) (2621) (3254) (3254)
1 05
005 3417 3417 4431 5294 5461 6448(3419b) (3419) (4435) (5303) (5478) (6456)
01 1593 1593 2401 3849 3861 4729(1611b) (1611) (2408) (3852) (3866) (4767)
015 1122 1122 1913 3272 3290 4038(1127b) (1227) (1924) (3284) (3314) (4069)
02 0890 0890 1640 2781 2829 3476(0899b) (0899) (1654) (2787) (2834) (3497)
15 1
005 4325 4325 5619 6273 6571 7619(4334b) (4334) (5681) (6299) (6642) (7642)
01 2013 2013 2812 4088 4116 4993(2024b) (2024) (2827) (4122) (4187) (4994)
015 1338 1338 2108 3430 3443 4196(1342b) (1342) (2112) (3467) (3482) (4200)
02 1038 1038 1772 2936 2965 3613(1042b) (1042) (1786) (2937) (2977) (3624)
2 2
005 5231 5231 6919 7630 8002 9220(5238b) (5238) (6931) (7637) (8011) (9241)
01 2594 2594 3439 4470 4544 5428(2597b) (2597) (3442) (4495) (4557) (5438)
015 1653 1653 2403 3613 3632 4392(1662b) (1662) (2414) (3627) (3646) (4404)
02 1238 1238 1949 3077 3099 3751(1244b) (1244) (1964) (3094) (3116) (3788)
aΓ119909= 1198701198941198700 (1198700= 1 times 10
9 Nmrad 119894 = 1199090 119909119886 1199100 119910119887) Γ119908= 1198961198941198960 (1198960= 1 times 10
9 Nm 119894 = 1199090 119909119886 1199100 119910119887)bResults in parentheses are taken from FEM
05 10 15 20 25 300
2
4
6
8
10
12
14
16
ab
The 1st orderThe 2nd orderThe 3rd order
The 4th orderThe 5th orderThe 6th order
Ω
(a)
05 10 15 20 25 30
30
60
90
120
150
180
ab
The 1st orderThe 2nd orderThe 3rd order
The 4th orderThe 5th orderThe 6th order
Ω
(b)
Figure 6 The effect of aspect ratio 119886119887 on the natural frequenciesΩ for S-S-S-S boundary condition (a) uniform thickness and (b) nonuni-form thickness
Shock and Vibration 13
Table 5 Frequency parameter Ω = 1205961198862
(120588ℎ01198630)12 for moderately thick plates with linearly varying thickness in different boundary
conditions (ℎ(119909 119910) = ℎ0(1 + 05119909)(1 + 05119910))
Boundary condition 119886119887 ℎ0119886
Model sequence1 2 3 4 5 6
S-S-S-S
101 27261 62397 62926 92423 111537 111732
(27263a) (62410) (63842) (92445) (111867) (111921)
02 23103 47010 47191 65018 75831 75865(23489a) (47209) (47349) (65057) (76605) (76473)
3201 59338 99101 138798 151102 166710 204855
(59874a) (99249) (138177) (151362) (166644) (205381)
02 41175 63765 63765 73598 83956 89084(41586a) (63313) (63669) (73661) (83918) (89547)
C-F-F-F
101 4173 10331 27407 37826 38809 64095
(4856a) (10680) (27513) (37203) (39135) (64858)
02 3983 8702 21403 29443 30286 46705(4118a) (9306) (21990) (29670) (30602) (47156)
3201 4868 15488 35620 63523 89969 100089
(4887a) (15320) (35175) (63353) (89159) (100783)
02 4469 9763 22338 42188) 50104 56331(4944a) (9636) (22087) (42177) (50580) (56559)
S-S-F-F
101 4117 24795 26216 48567) 68195 69667
(4789a) (24149) (26751) (49324) (68842) (70575)
02 3820 20516 21924 37523 48980 50327(4103a) (20041) (22125) (37221) (48894) (50622)
3201 7935 39454 65063 92877 92877 136176
(7856a) (40145) (65162) (92462) (93022) (136336)
02 6389 26149 35627 49539 56844 65170(6406a) (26536) (35626) (49831) (57144) (65981)
C-C-C-C
101 44471 80299 80681 109422 126871 128207
(45102a) (80271) (80656) (110015) (126328) (128888)
02 32162 52781 52933 69410 78474 79482(32240a) (52629) (53166) (70063) (79059) (80266)
3201 82975 115587 151992 161579 176601 212330
(83139a) (115838) (152200) (162193) (177058) (212849)
02 48577 66022 83468 90830 97474 98157(49187a) (66259) (84065) (91749) (97827) (98777)
S-C-S-C
101 36847 66664 76965 101045 113594 125551
(37901a) (68568) (78356) (101200) (113978) (125651)
02 27673 48469 51220 67027 76358 78241(27897a) (48685) (51491) (67447) (77530) (79573)
3201 77378 106848 150079 154569 173120 208391
(78003a) (107058) (150991) (155171) (173099) (208402)
02 45895 64732 73393 83282 90325 97089(45803a) (64131) 73656 83801 90518 97015
aResults in parentheses are taken from FEM
restrained That is 1198701199090
= 119870119909119886
= 1198701199100
= 119870119910119887
= Γ119909 The six
frequency parameters Ω = 1205961198862
(120588ℎ01198630)12 are given in
Table 6 for several different restraining coefficient valuesthe finite element method (FEM) results calculated usingABAQUS are also presented as a referenceThe secondmodelis a complete square moderately thick plate with all edgeselastically restrainedThat is 119896
1199090= 119896119909119886= 1198961199100= 119896119910119887= Γ119908and
1198701199090= 119870119909119886
= 1198701199100= 119870119910119887= Γ119909 The six frequency parameters
Ω = 1205961198862
(120588ℎ01198630)12 are given in Table 7 for several different
restraining coefficient values the finite element method(FEM) results are also listed as a reference It can be clearlyseen that the comparison is extremely good which impliesthat the current method is able to make correct predictionsfor the modal characteristics of linearly varying thickness
14 Shock and Vibration
Table 6 Frequency parameter Ω = 1205961198862
(120588ℎ01198630)12 for S-S-S-S moderately thick plates (119886119887 = 1) with linearly varying thickness and elastic
rotation support (ℎ(119909 119910) = ℎ0(1 + 05119909)(1 + 05119910) 119870
1199090= 119870119909119886= 1198701199100= 119870119910119887= Γ119909)
Γ119909
aℎ0119886
Model sequence1 2 3 4 5 6
0
005 28809 70080 70891 108967 134884 135295(28897b) (70334) (71129) (109112) (135254) (135772)
01 27261 62398 62926 92423 111540 111735(26606b) (61784) (62303) (91193) (111323) (111648)
015 25231 54192 54505 77118 91228 91312(25204b) (55297) (55595) (77503) (91871) (91941)
02 23103 47010 47191 65018 75832 75866(23947b) (47071) (47208) (65404) (76531) (76589)
05
005 23103 47010 47191 65018 75832 75866(23210b) (47245) (47358) (65241) (76122) (76345)
01 48377 94650 95597 135484 162635 163501(48524b) (94891) (95992) (135845) (167012) (164024)
015 36239 70939 71298 99997 118584 119009(36647b) (71214) (71687) (100024) (119114) (119457)
02 28749 56937 57037 79214 93009 93162(29142b) (57034) (57544) (79654) (93668) (93754)
1
005 49651 96759 97619 138089 165368 166304(49755b) (97245) (97964) (138544) (165864) (166878)
01 38951 73735 74251 102748 121079 121709(39021b) (74145) (74664) (102874) (121613) (121817)
015 30630 58406 58533 80432 93984 94255(30638b) (58684) (58644) (80874) (94121) (94478)
02 25548 48636 48674 66165 76708 76824(25687b) (48788) (48992) (66544) (76824) (76944)
15
005 50118 97561 98374 139096 166421 167382(50247b) (97877) (98661) (139247) (166875) (167662)
01 40310 75216 75783 104229 122402 123164(40366b) (75542) (75921) (104557) (122874) (123385)
015 31837 59352 59530 81245 94611 94983(31902b) (59451) (59871) (81375) (94784) (95105)
02 26302 49119 49156 66528 76954 77130(26418b) (49354) (49334) (66921) (77010) (77470)
2
005 50360 97983 98769 139631 166979 167954(50472b) (98005) (99114) (140154) (167188) (167986)
01 41130 76146 76724 105165 123229 124082(41421b) (76642) (78114) (105661) (123278) (124974)
015 32687 60025 60245 81832 95052 95508(32778b) (60244) (60549) (82146) (95114) (95842)
02 26893 49492 49539 66815 77138 77370(26922b) (49874) (49924) (66987) (77367) (77588)
aΓ119909= 1198701198941198700 (1198700= 1 times 10
9 Nmrad 119894 = 1199090 119909119886 1199100 119910119887)bResults in parentheses are taken from FEM
Shock and Vibration 15
Table 7 Frequency parameterΩ = 1205961198862
(120588ℎ01198630)12 for moderately thick plates (119886119887 = 1) with linearly varying thickness and elastic rotation
and translation support (ℎ(119909 119910) = ℎ0(1 + 05119909)(1 + 05119910) 119870
1199090= 119870119909119886= 1198701199100= 119870119910119887= Γ119909 and 119896
1199090= 119896119909119886= 1198961199100= 119896119910119887= Γ119908)
Γ119909
aΓ119908
aℎ0119886
Model sequence1 2 3 4 5 6
05 0
005 14751 14917 28232 55251 55397 69512(14780b) (14822) (29252) (55476) (55505) (70904)
01 11811 11910 23857 44462 45037 55453(11818b) (11897) (24711) (44376) (45257) (56511)
015 8487 8514 20011 34783 36335 44970(8500b) (8544) (20738) (34404) (36663) (45625)
02 6073 6073 17431 28679 30805 38010(6096b) (6106) (18039) (28129) (31165) (38311)
1 05
005 21225 32173 32173 44381 64061 65222(20708b) (32097) (32285) (45110) (64182) (65355)
01 8140 16878 16941 28100 48663 49015(7943b) (16838) (16970) (28888) (48700) (49171)
015 4493 11859 11878 22429 38352 39247(4383b) (11867) (11891) (23125) (38177) (39488)
02 2932 8690 8691 18919 31108 32600(3540b) (8791) (8791) (16229) (27520) (27889)
15 1
005 27819 40919 41430 54595 70792 72896(27146b) (41008) (41434) (55256) (70895) (73051)
01 11285 20068 20230 31101 51046 51429(11012b) (20025) (20270) (31849) (51126) (51569)
015 6292 13912 13922 24132 40500 41120(6139b) (13912) (13934) (24808) (40426) (41316)
02 4121 10321 10331 20051 32796 33921(4021b) (10329) (10357) (20635) (32540) (34178)
2 2
005 34615 51972 53088 68255 81344 84510(33806b) (52100) (53157) (68891) (81433) (84761)
01 15416 24659 24829 35615 53879 54494(15045b) (24625) (24888) (36303) (53970) (54626)
015 8740 16420 16481 26353 42417 42946(8529b) (16414) (16499) (26996) (42398) (43114)
02 5761 12127 12127 21434 34286 35192(5622b) (12134) (12148) (21998) (34110) (35415)
aΓ119909= 1198701198941198700 (1198700= 1 times 10
9 Nmrad 119894 = 1199090 119909119886 1199100 119910119887) Γ119908= 1198961198941198960 (1198960= 1 times 10
9 Nm 119894 = 1199090 119909119886 1199100 119910119887)bResults in parentheses are taken from FEM
moderately thick plate with not only classical boundaryconditions but also elastic edge restraints
As the last case of this subsection the influence of thegradient 120572 and 120573 on the fundamental frequency parametersfor a linearly varying thickness moderately thick plate isinvestigated The model is a square moderately thick platewith all edges elastically restrainedThat is 119896
1199090= 119896119909119886= 1198961199100=
119896119910119887= Γ119908= 2 and 119870
1199090= 119870119909119886
= 1198701199100= 119870119910119887= Γ119909= 2 The six
frequency parameters Ω = 1205961198862
(120588ℎ01198630)12 are given in
Table 8 for several different 120572 and 120573 values the finite elementmethod (FEM) results are also listed as a reference Againgood agreement can be observed Through Table 8 it is alsofound that the frequency parameter increases with increasinggradient parameters
The above studies are given as linearly varying thicknessmoderately thick plates with several of boundary condition
and different structure parameters In the next section vibra-tion results for the plates subjected to nonlinear variationthickness will be presented
34 Nonlinearly Varying Thickness Moderately Thick Plateswith Classical and Elastic Boundary Conditions In Sec-tion 33 the linearly varying thickness moderately thickplates were studied However in the practical engineeringapplications the varying thickness of amoderately thick platemay not always be linear variation in nature A variety ofpossible thickness varying cases may be encountered in prac-tice Therefore the moderately thick plates with nonlinearvariation thickness subjected to general elastic edge restraintsare examined in this subsection For the sake of brevity theindexes 119904 and 119905 will be chosen as 2 to imitate the nonlinearlyvarying thickness moderately thick plates structure in this
16 Shock and Vibration
Table 8 Frequency parameterΩ = 1205961198862
(120588ℎ01198630)12 for elastic support moderately thick plates (119886119887 = 1) with different gradient (119896
1199090= 119896119909119886=
1198961199100= 119896119910119887= Γ119908= 2 119870
1199090= 119870119909119886= 1198701199100= 119870119910119887= Γ119909= 2)
120572 120573 ℎ0119886
Model sequence1 2 3 4 5 6
120572 = 120573 = 01
005 31275 52615 52657 68954 76146 79850(31303a) (52664) (52675) (69033) (76174) (79948)
01 17270 25244 25250 33949 46050 46625(17271a) (25255) (25258) (34089) (46122) (46708)
015 10190 16266 16274 24164 37327 37526(10189a) (16269) (16280) (24323) (37401) (37622)
02 6780 12258 12260 19685 31452 31754(6780a) (12265) (12270) (19851) (31514) (31864)
120572 = 120573 = 02
005 32530 52947 53141 68954 77034 80589(32577a) (53014) (53203) (69172) (77088) (80773)
01 16882 24979 24979 34154 48057 48575(16883a) (24983) (25001) (34433) (48134) (48667)
015 9804 16284 16309 24683 38844 39085(9803a) (16283) (16322) (24980) (38909) (39191)
02 6499 12290 12295 20133 32354 32770(6498a) (12296) (12307) (20420) (32387) (32895)
120572 = 120573 = 03
005 33516 52845 53308 68704 78203 81561(33563a) (52939) (53389) (69055) (78271) (81789)
01 16419 24799 24830 34527 50061 50583(16419a) (24790) (24869) (34947) (50143) (50688)
015 9430 16334 16378 25236 40193 40506(9429a) (16329) (16395) (25660) (40243) (40625)
02 6235 12288 12293 20572 33102 33663(6235a) (12296) (12306) (20965) (33090) (33811)
120572 = 120573 = 04
005 34215 52463 53264 68430 79654 82865(34252a) (52575) (53346) (68921) (79732) (83113)
01 15921 24695 24793 35027 52011 52571(15921a) (24674) (24842) (35583) (52098) (52688)
015 9074 16388 16444 25797 41379 41789(9074a) (16381) (16464) (26336) (41404) (41928)
02 5990 12236 12238 21002 33735 34461(5989a) (12249) (12250) (21487) (33653) (34641)
120572 = 120573 = 05
005 34615 51972 53088 68255 81344 84510(33806a) (52100) (53157) (68891) (81433) (84761)
01 15416 24659 24829 35615 53879 54494(15045a) (24625) (24888) (36303) (53970) (54626)
015 8740 16420 16481 26353 42417 42946(8529a) (16414) (16499) (26996) (42398) (43114)
02 5761 12127 12127 21434 34286 35192(5622a) (12134) (12148) (21998) (34110) (35415)
aResults in parentheses are taken from FEM
section that is 119904 = 119905 = 2 Also the thickness varying functionsare also be expressed as Fourier cosine series according to (18)and (19)
In order to validate the accuracy and reliability of theproposed method for predicting the vibration behavior ofnonlinearly varying thickness moderately thick plates with
arbitrary boundary conditions the typical classical boundaryconditions viewed as the special cases of elastically restrainededges will be considered The comparison of the first six fre-quency parameters Ω = 120596119886
2
(120588ℎ01198630)12 for the moderately
thick plates with nonlinearly varying thickness is presentedin Table 9 The S-S-S-S C-F-F-F S-S-F-F C-C-C-C and
Shock and Vibration 17
Table 9 Frequency parameter Ω = 1205961198862
(120588ℎ01198630)12 for moderately thick plates with linearly varying thickness in different boundary
conditions (ℎ(119909 119910) = ℎ0(1 + 05119909
2
)(1 + 051199102
))
Boundary condition 119886119887 ℎ0119886
Model sequence1 2 3 4 5 6
S-S-S-S
101 23462 54999 55573 82950 100900 101187
(23741a) (55114) (55983) (83146) (101004) (101347)
02 18606 42725 42971 60040 71167 71761(18936a) (43014) (43148) (60247) (71642) (71993)
3201 53055 65251 85404 106700 114880 143531
(53467a) (65472) (85984) (107211) (115687) (144007)
02 17439 21531 32170 44987 59438 64179(17874a) (21748) (32651) (45264) (60188) (64879)
C-F-F-F
101 3658 9317 24135 32816 35433 59179
(3749a) (9547) (24034) (33627) (36308) (60142)
02 3520 7986 19591 26959 27491 43882(3611a) (8176) (20075) (27125) (27625) (43984)
3201 3901 9963 28984 69187 73686 89360
(3930a) (10095) (29381) (69578) (73994) (90242)
02 3292 4439 13385 23947 40850 44758(3386a) (4498) (14012) (24068) (41221) (45142)
S-S-F-F
101 3557 21268 22762 43524 60253 62429
(3649a) (21441) (22796) (43916) (60622) (62630)
02 3171 17822 19512 33616 45042 46833(3489a) (18006) (20430) (33815) (45247) (49004)
3201 6738 31658 52071 75625 86604 94903
(6957a) (31882) (52518) (75823) (86939) (95610)
02 3653 12930 20955 26517 44097 48058(6807a) (13160) (21312) (26864) (44422) (48873)
C-C-C-C
101 41020 75196 75614 104142 119683 120949
(41405a) (75452) (75882) (104711) (119637) (121137)
02 31237 51686 51868 68411 76868 77864(32010a) (52960) (52145) (69335) (77481) (78099)
3201 95823 128727 163657 173751 190890 223595
(95364a) (128932) (163588) (172350) (190350) (222948)
02 53306 71372 89384 102404 104660 120908(53587a) (71759) (90103) (102503) (105174) (121275)
S-C-S-C
101 33715 60298 71446 94187 103802 117928
(34148a) (60786) (71828) (94515) (104263) (119326)
02 26143 45499 49458 64253 72941 76191(26789a) (46622) (50680) (64329) (73474) (76843)
3201 73228 115845 137445 148166 168342 182542
(73761a) (116121) (137788) (148530) (168698) (182921)
02 41302 53648 65630 72202 81632 88105(41304a) (53757) (65842) (72794) (82033) (88474)
aResults in parentheses are taken from FEM
S-C-S-C boundary conditions are performed in the compar-ison The results adequately demonstrated the great accuracyof the modified Fourier method
We now turn to elastically restrained moderately thickplates The first one involves an S-S-S-S moderately thicksquare plate with a uniform rotational restraint along eachedge that is 119870
1199090= 119870119909119886
= 1198701199100
= 119870119910119887
= Γ119909 The calculated
frequency parameters Ω = 1205961198862
(120588ℎ01198630)12 are given in
Table 10 together with the FEM results Since the elasticallyrestrained plates with nonlinear variation thickness are rarelyinvestigated the FEMresults are used as the referenceA goodagreement is observed between the current and FEM resultsThe second example concerns amoderately thick square plateelastically supported along all edgesThe stiffness of the linear
18 Shock and Vibration
Table 10 Frequency parameterΩ = 1205961198862
(120588ℎ01198630)12 for S-S-S-S moderately thick plates (119886119887 = 1) with linearly varying thickness and elastic
rotation support (ℎ(119909 119910) = ℎ0(1 + 05119909
2
)(1 + 051199102
) 1198701199090= 119870119909119886= 1198701199100= 119870119910119887= Γ119909)
Γ119909
aℎ0119886
Model sequence1 2 3 4 5 6
0
005 23462 54999 55573 82950 100900 101187(23741b) (55114) (55983) (83146) (101004) (101347)
01 18606 42725 42971 60040 71167 71761(18936b) (43014) (43148) (60247) (71642) (71993)
015 53055 65251 85404 106700 114880 143531(53467b) (65472) (85984) (107211) (115687) (144007)
02 17439 21531 32170 44987 59438 64179(17874b) (21748) (32651) (45264) (60188) (64879)
05
005 6904 13374 13514 19400 22904 23056(9909b) (13548) (13651) (19564) (23088) (23379)
01 5316 10237 10314 14499 17155 17264(5374b) (10448) (10651) (14764) (17188) (17879)
015 4136 8201 8220 11474 13578 13671(4174b) (8348) (8651) (11764) (13688) (13879)
02 3449 6915 6924 9549 11225 11327(3474b) (6948) (6951) (9764) (11388) (11579)
1
005 7040 13617 13745 19726 23233 23402(7074b) (13748) (13851) (19964) (23488) (23879)
01 5712 10696 10793 14991 17594 17744(5774b) (10748) (10851) (15054) (17788) (17879)
015 4489 8507 8540 11759 13803 13922(4514b) (8548) (8611) (11864) (13988) (14119)
02 3677 7084 7090 9692 11330 11445(3684b) (7148) (7191) (9764) (11588) (11879)
15
005 7089 13708 13828 19847 23355 23530(7124b) (13748) (13951) (19967) (23588) (23879)
01 5895 10923 11024 15239 17813 17988(5874b) (10748) (11651) (15264) (17818) (17979)
015 4697 8694 8737 11938 13790 14083(4774b) (8748) (8951) (11864) (13688) (14879)
02 3836 7201 7210 9792 11403 11530(3874b) (7748) (7651) (9864) (11588) (11879)
2
005 7114 13754 13871 19911 23418 23597(7174b) (12848) (13851) (19864) (23588) (23879)
01 6001 11060 11159 15391 17944 18136(6004b) (11148) (11151) (15464) (17988) (18379)
015 4836 8821 8872 12061 14037 14189(4874b) (8848) (8951) (12164) (14188) (14879)
02 3954 7289 7301 9871 11457 11594(3974b) (7348) (7651) (9964) (11488) (11679)
aΓ119909= 1198701198941198700 (1198700= 1 times 10
9 Nmrad 119894 = 1199090 119909119886 1199100 119910119887)bResults in parentheses are taken from FEM
and rotational restraints is set equal to 1198961199090
= 119896119909119886
= 1198961199100
=
119896119910119887
= Γ119908and 119870
1199090= 119870119909119886
= 1198701199100
= 119870119910119887
= Γ119909 respectively
The six frequency parametersΩ = 1205961198862
(120588ℎ01198630)12 are shown
in Table 11 for several different restraining coefficient valuesIt can also be noticed that the Fourier series method is able
to predict the modal characteristics of nonlinearly varyingthickness moderately thick plate with not only classicalboundary conditions but also elastic edge restraints correctly
Finally the influence of the gradient on the fundamentalfrequency parameters of a nonlinearly varying thickness
Shock and Vibration 19
Table 11 Frequency parameterΩ = 1205961198862
(120588ℎ01198630)12 for moderately thick plates (119886119887 = 1) with linearly varying thickness and elastic rotation
and translation support (ℎ(119909 119910) = ℎ0(1 + 05119909
2
)(1 + 051199102
) 1198701199090= 119870119909119886= 1198701199100= 119870119910119887= Γ119909 and 119896
1199090= 119896119909119886= 1198961199100= 119896119910119887= Γ119908)
Γ119909
aΓ119908
aℎ0119886
Model sequence1 2 3 4 5 6
05 0
005 12926 13041 25090 47935 47943 61761(12931b) (13053) 25099 (47947) (47952) (61788)
01 10900 10988 21545 39980 40147 50322(10904b) (10992) (21557) (39984) (40166) (50334)
015 8245 8298 18081 31503 32617 40950(8247b) (8230) (18092) (31508) (32622) (40955)
02 6106 6124 15639 25631 27601 34458(6110b) (6125) (15642) (25637) (27617) (34462)
1 05
005 21575 31615 31644 42573 57889 58607(21580b) (31617) (31644) (42581) (57893) (58610)
01 8451 16024 16210 25969 43839 43867(8451b) (16025) (16217) (25972) (43844) (43870)
015 4682 11479 11480 20642 35172 35598(4684b) (11480) (11481) (20644) (35178) (35604)
02 3059 8621 8648 17320 28487 29638(3062b) (8619) (8644) (17327) (28492) (29641)
15 1
005 27713 40203 40978 52951 65352 66815(27714b) (40207) (40988) (52972) (65368) (66822)
01 11667 19322 19615 29122 46115 46183(11674b) (19327) (19619) (29137) (46120) (46188)
015 6544 13416 13481 22405 37238 37430(6547b) (13420) (13488) (22410) (37241) (37434)
02 4295 10175 10190 18543 30330 31056(4296b) (10177) (10194) (18544) (30336) (31060)
2 2
005 33453 50854 52088 66311 76610 78947(33455b) (50852) (50097) (66321) (76627) (78971)
01 15832 24130 24372 33924 49118 49362(15830b) (24127) (24374) (33931) (49147) (49382)
015 9060 15956 16099 24754 39134 39249(9064b) (15961) (16102) (24755) (39138) (39452)
02 5992 11955 11971 20034 31908 32392(5991b) (11957) (11978) (20042) (31910) (32398)
aΓ119909= 1198701198941198700 (1198700= 1 times 10
9 Nmrad 119894 = 1199090 119909119886 1199100 119910119887) Γ119908= 1198961198941198960 (1198960= 1 times 10
9 Nm 119894 = 1199090 119909119886 1199100 119910119887)bResults in parentheses are taken from FEM
moderately thick plate is investigated The plate is elasticallyrestrained in which the stiffness of the boundary springsis taken as 119896
1199090= 119896119909119886
= 1198961199100
= 119896119910119887
= Γ119908
= 2 and1198701199090
= 119870119909119886
= 1198701199100
= 119870119910119887
= Γ119909= 2 respectively The six
frequency parameters Ω = 1205961198862
(120588ℎ01198630)12 are presented in
Table 12 for several different slop values It can be seen thatthe fundamental frequency parameters will decrease with theincrease of the parameter ℎ
0119886
In the above examples it has been demonstrated thatthe presented method can be universally applied to nonlin-early varying thickness moderately thick plates with severalboundary conditions and different structure parametersNew results are obtained for plates with nonlinear variationthickness in both directions subjected to general elastic
boundary restraints which may be used for benchmarkingof researchers in the field In addition it is interesting to seethat the nature frequency decreases with the increase of theindex for thickness function
4 Conclusions
In this paper a modified Fourier method has been pre-sented to study the free vibration behaviors of moderatelythick rectangular plates with variable thickness and arbi-trary boundary conditionsThe first-order shear deformationplate theory is adopted to formulate the theoretical modelThe displacements and rotation components of the plateregardless of boundary conditions are invariantly expressedas the superposition of a 2D Fourier cosine series and four
20 Shock and Vibration
Table 12 Frequency parameterΩ = 1205961198862
(120588ℎ01198630)12 for elastic support moderately thick plates (119886119887 = 1) with different gradient (119896
1199090= 119896119909119886=
1198961199100= 119896119910119887= Γ119908= 2 119870
1199090= 119870119909119886= 1198701199100= 119870119910119887= Γ119909= 2)
120572 120573 ℎ0119886
Model sequence1 2 3 4 5 6
120572 = 120573 = 01
005 30187 51102 51141 67066 73954 77412(30188a) (51105) (51144) (67068) (73961) (77477)
01 16995 24719 24724 33071 44343 44819(16996a) (24725) (24729) (33084) (44347) (44822)
015 10077 15872 15883 23443 35937 36065(10081a) (15877) (15892) (23447) (35941) (36084)
02 6713 11953 11958 19086 30372 30604(6713a) (11957) (11965) (19092) (30378) (30612)
120572 = 120573 = 02
005 31206 51436 51614 67058 74398 77595(31208a) (51438) (51619) (67064) (74402) (77596)
01 16786 24503 24508 33144 45583 45923(16790a) (24510) (24513) (33147) (45587) (45932)
015 9821 15865 15906 23766 36897 36990(9824a) (15870) (15907) (23768) (36894) (36997)
02 6523 11978 11992 19356 30912 31162(6524a) (11981) (11997) (19362) (30916) (31174)
120572 = 120573 = 03
005 32097 51462 51900 66832 74991 77863(32098a) (51466) (51907) (66849) (74996) (77872)
01 16507 24332 24383 33330 46806 47074(16511a) (24327) (24380) (33339) (46810) (47086)
015 9562 15885 15964 24103 37748 37828(9562a) (15892) (15977) (21105) (37751) (37832)
02 6338 11992 12013 19603 31338 31634(6340a) (11994) (12011) (19612) (31344) (31639)
120572 = 120573 = 04
005 32851 51239 52046 66547 75735 78303(33455a) (50852) (50097) (66321) (76627) (78971)
01 16182 24209 24343 33598 47990 48230(33455a) (50852) (50097) (66321) (76627) (78971)
015 9307 15919 16034 24436 38491 38579(33455a) (50852) (50097) (66321) (76627) (78971)
02 6161 11986 12008 19828 31666 32040(33455a) (50852) (50097) (66321) (76627) (78971)
120572 = 120573 = 05
005 33453 50854 52088 66311 76610 78947(33455a) (50852) (50097) (66321) (76627) (78971)
01 15832 24130 24372 33924 49118 49362(15830a) (24127) (24374) (33931) (49147) (49382)
015 9060 15956 16099 24754 39134 39249(9064a) (15961) (16102) (24755) (39138) (39452)
02 5992 11955 11971 20034 31908 32392(5991a) (11957) (11978) (20042) (31910) (32398)
aResults in parentheses are taken from FEM
supplementary functions in the form of the product of apolynomial function and a single cosine series expansionto ensure and accelerate the convergence of the solutionAt each edge of the plate the general restraint conditionsare implemented by introducing one group of linear springsand two groups of rotational springs which are continuouslydistributed and determined by the stiffness of these springs
Instead of seeking a solution in strong forms in the previousstudies all the Fourier coefficients will be treated equallyand independently as the generalized coordinates and solveddirectly from the Rayleigh-Ritz technique The change of theboundary conditions can be easily achieved by only varyingthe stiffness of the three sets of boundary springs along alledges of the rectangular plates without involving any change
Shock and Vibration 21
to the solution procedure The convergence of the presentsolution is examined and the excellent accuracy is validatedby comparison with existing results published in the lit-erature and FEM data Excellent agreements are obtainedfrom these comparisons The proposed method providesa unified means for extracting the modal parameters andpredicting the vibration behaviors of moderately thick plateswith variable thickness variation functions and arbitraryelastic edge restraints A variety of free vibration results formoderately thick rectangular plates with different thicknessvariation functions and boundary conditions are presentedNew results for free vibration of moderately thick rectangularplates with various thickness variation functions and edgeconditions are presented which may be used for benchmark-ing of researchers in the field
Appendices
A Representative Calculation forStiffness and Mass Matrices
To illuminate the particular expression of the mass andstiffness matrixes clearly and tersely four new variables aredefined as 119904 = 119898 lowast (119873 + 1) + 119899 + 1 119905 = 119898
1015840
lowast (119873 + 1) + 1198991015840
+ 1119901 = (119897minus1)lowast (119872+1)+119898
1015840
+1 and 119902 = (119897minus1)lowast (119873+1)+1198991015840
+1The first row elements of K andM are given below
K1-1119904119905
=
1
2
int
119886
0
int
119887
0
[119863 (119909 119910)
times 21205821198861198981205821198861198981015840 sin 120582
119886119898119909 sin 120582
1198861198981015840119909 cos 120582
119887119899119910
times cos 1205821198871198991015840119910 + (1 minus 120583) 120582
1198871198991205821198871198991015840 cos 120582
119886119898119909
times cos 1205821198861198981015840 sin 120582
119887119899119910 sin 120582
1198871198991015840119910
+ 120581119866ℎ (119909 119910) cos 120582119886119898119909 cos 120582
1198861198981015840119909 cos 120582
119887119899
times119910 cos 1205821198871198991015840119910119889119909119889119910] 119889119909 119889119910
+ int
119886
0
1198701199091199100
cos 120582119886119898119909 cos 120582
1198861198981015840119909
+(minus1)119898+1198981015840
119870119909119910119887
cos 120582119886119898119909 cos 120582
1198861198981015840119909 119889119909
+ int
119887
0
1198701199090cos 120582119887119899119910 cos 120582
1198871198991015840119910
+(minus1)119899+1198991015840
119870119909119886cos 120582119887119899119910 cos 120582
1198871198991015840119910 119889119910
K1-2119904119901
=
1
2
int
119886
0
int
119887
0
[119863 (119909 119910)21205821198861198981205821198861198981015840 sin 120582
119886119898119909 sin 120582
1198861198981015840119909
times
2
sum
119897=1
120577119897
119887(119910) cos 120582
1198871198991015840119910
minus (1 minus 120583) 1205821198871198991015840 cos 120582
119886119898119909 cos 120582
1198861198981015840119909
times
2
sum
119897=1
1205771198971015840
119887(119910) sin 120582
1198871198991015840119910 + 120581119866ℎ (119909 119910)
times ( cos 1205821198861198981015840119909 cos 120582
1198871198991015840119910
times
2
sum
119897=1
120577119897
119887(119910) cos 120582
119886119898119909)]119889119909119889119910
+ int
119886
0
1198701199091199100
cos 120582119886119898119909 cos 120582
1198861198981015840119909
2
sum
119897=1
120577119897
119887(0)
+(minus1)1198991015840
119870119909119910119887
cos 120582119886119898119909 cos 120582
1198861198981015840119909
2
sum
119897=1
120577119897
119887(119887) 119889119909
+ int
119887
0
1198701199090
2
sum
119897=1
120577119897
119887(119910) cos 120582
1198871198991015840119910
+(minus1)119898+1198981015840
119870119909119886
2
sum
119897=1
120577119897
119887(119910) cos 120582
1198871198991015840119910119889119910
K1-3119904119902
=
1
2
int
119886
0
int
119887
0
[119863 (119909 119910) minus 21205821198861198981015840 sin 120582
1198861198981015840119909 cos 120582
1198871198991015840119910
times
2
sum
119897=1
1205771198971015840
119886(119909) cos 120582
119887119899119910
+ (1 minus 120583) 1205821198871198991205821198871198991015840 cos 120582
1198861198981015840119909
times sin 1205821198871198991015840119910
2
sum
119897=1
120577119897
119886(119909) sin 120582
119887119899119910
+ 120581119866ℎ (119909 119910) cos 1205821198861198981015840119909 cos 120582
1198871198991015840119910
times
2
sum
119897=1
120577119897
119886(119909) cos 120582
119887119899119910]119889119909119889119910
+ int
119886
0
1198701199091199100
cos 1205821198861198981015840119909
2
sum
119897=1
120577119897
119886(119909)
+(minus1)119899+1198991015840
119870119909119910119887
cos 1205821198861198981015840119909
2
sum
119897=1
120577119897
119886(119909) 119889119909
+ int
119887
0
1198701199090
2
sum
119897=1
120577119897
119886(0) cos 120582
1198871198991015840119910
+(minus1)119898+1198981015840
119870119909119886
2
sum
119897=1
120577119897
119886(0) cos 120582
1198871198991015840119910119889119910
K1-4119904119905
=
1
2
int
119886
0
int
119887
0
119863(119909 119910) 21205821198871198991205821198861198981015840 sin 120582
1198861198981015840119909 cos 120582
1198871198991015840119910
times cos 120582119886119898119909 sin 120582
119887119899119910 minus 2(1 minus 120583)
119887119899
times 1205821198861198981015840 sin 120582
1198861198981015840119909 cos 120582
1198871198991015840119910 cos 120582
119886119898119909
times sin 120582119887119899119910 + (1 minus 120583) 120582
1198861198981205821198871198991015840 cos 120582
1198861198981015840119909
times sin 1205821198871198991015840119910 sin 120582
119886119898119909 cos 120582
119887119899119910 119889119909 119889119910
22 Shock and Vibration
K1-5119904119901
=
1
2
int
119886
0
int
119887
0
119863(119909 119910) minus 21205821198861198981015840 sin 120582
1198861198981015840119909 cos 120582
1198871198991015840119910
times
2
sum
119897=1
1205771198971015840
119887(119910) cos 120582
119886119898119909
+ 2 (1 minus 120583) 1205821198861198981015840 sin 120582
1198861198981015840119909 cos 120582
1198871198991015840119910
times
2
sum
119897=1
1205771198971015840
119887(119910) cos 120582
119886119898119909
+ (1 minus 120583) 1205821198861198981205821198871198991015840 cos 120582
1198861198981015840119909 sin 120582
1198871198991015840119910
times
2
sum
119897=1
120577119897
119887(119910) sin 120582
119886119898119909119889119909119889119910
K1-6119904119902
=
1
2
int
119886
0
int
119887
0
119863(119909 119910)1205821198871198991205821198861198981015840 sin 120582
1198861198981015840119909 cos 120582
1198861198991015840119910
times
2
sum
119897=1
120577119897
119886(119909) sin 120582
1198871198991015840119910
minus 2 (1 minus 120583) 1205821198871198991205821198861198981015840 sin 120582
1198861198981015840119909 cos 120582
1198861198991015840119910
times
2
sum
119897=1
120577119897
119886(119909) sin 120582
1198871198991015840119910
+ 1205821198871198991015840 cos 120582
1198861198981015840119909 sin 120582
1198871198991015840119910
times
2
sum
119897=1
1205771198971015840
119886(119909) cos 120582
119887119899119910119889119909119889119910
K1-7119904119905
= minus
1
2
int
119886
0
int
119887
0
120581119866ℎ (119909 119910) 120582119886119898
cos 1205821198861198981015840119909 cos 120582
1198871198991015840119910
times sin 120582119886119898119909 cos 120582
119887119899119910119889119909119889119910
K1-8119904119901
= minus
1
2
int
119886
0
int
119887
0
120581119866ℎ (119909 119910) 1205821198861198981015840 cos 120582
1198861198981015840119909 cos 120582
1198871198991015840119910
times
2
sum
119897=1
120577119897
119887(119910) sin 120582
1198861198981015840119909 119889119909 119889119910
K1-9119904119902
=
1
2
int
119886
0
int
119887
0
120581119866ℎ (119909 119910) cos 1205821198861198981015840119909 cos 120582
1198871198991015840119910
times
2
sum
119897=1
1205771198971015840
119886(119909) cos 120582
119887119899119910119889119909119889119910
M1-1119904119905
=
1
24
120588int
119886
0
int
119887
0
ℎ3
(119909 119910) cos 120582119886119898119909 cos 120582
119887119899119910
times cos 1205821198861198981015840119909 cos 120582
1198871198991015840119910119889119909119889119910
M1-2119904119901
=
1
24
120588int
119886
0
int
119887
0
ℎ3
(119909 119910)
2
sum
119897=1
120577119897
119887(119910) cos 120582
119886119898119909
times cos 1205821198861198981015840119909 cos 120582
1198871198991015840119910119889119909119889119910
M1-3119904119902
=
1
24
120588int
119886
0
int
119887
0
ℎ3
(119909 119910)
2
sum
119897=1
120577119897
119886(119909) cos 120582
119887119899119910
times cos 1205821198861198981015840119909 cos 120582
1198871198991015840119910119889119909119889119910
M1-4119904119905 = M
1-5119904119901 = M1-6119904119902
= M1-7119904119905 = M
1-8119904119901 = M1-9119904119902 = 0
(A1)
B Useful Integrals for the Moderately ThickRectangular Plates with Variable Thicknessand Arbitrary Boundary Conditions
Consider
int
119871
0
cos 120582119898119909 cos 120582
119899119909 119889119909 =
0 119898 = 119899
119871
2
119898 = 119899
119871 119898 = 119899 = 0
int
119871
0
sin 120582119898119909 cos 120582
119899119909119889119909
=
minus
(minus1 + (minus1)119898+119899
) 119871119898
(1198982minus 1198992) 120587
119898 = 119899
0 119898 = 119899
int
119871
0
sin 120582119898119909 sin 120582
119899119909 119889119909 =
0 119898 = 119899
119871
2
119898 = 119899
0 119898 = 119899 = 0
int
119871
0
cos 120582119898119909120577119897
119886(119909) 119889119909
=
41198712
31205872 119897 = 1 119898 = 0
minus
41198712
(41198982
minus 3)
(9 minus 401198982+ 16119898
4) 1205872
119897 = 1 119898 = 0
minus
41198712
31205872 119897 = 2 119898 = 0
41198712
(41198982
minus 3) (minus1)119898
(9 minus 401198982+ 16119898
4) 1205872
119897 = 2 119898 = 0
Shock and Vibration 23
int
119871
0
sin 120582119898119909120577119897
119886(119909) 119889119909
=
0 119897 = 1 119898 = 0
161198712
119898(minus1)119898
(9 minus 401198982+ 16119898
4) 1205872
119897 = 1 119898 = 0
0 119897 = 2 119898 = 0
161198981198712
(9 minus 401198982+ 16119898
4) 1205872
119897 = 2 119898 = 0
int
119871
0
cos 1205821198981199091205771198971015840
119886(119909) 119889119909
=
0 119897 = 1 119898 = 0
161198711198982
(minus1)119898
(9 minus 401198982+ 16119898
4) 120587
119897 = 1 119898 = 0
0 119897 = 2 119898 = 0
161198711198982
(9 minus 401198982+ 16119898
4) 120587
119897 = 2 119898 = 0
int
119871
0
sin 1205821198981199091205771198971015840
119886(119909) 119889119909
=
0 119897 = 1 119898 = 0
4119871119898 (minus3 + 41198982
)
(9 minus 401198982+ 16119898
4) 120587
119897 = 1 119898 = 0
0 119897 = 2 119898 = 0
119871119898(
1
120587 minus 41198982120587
+
3
9120587 minus 41198982120587
) (minus1)119898
119897 = 2 119898 = 0
int
119871
0
1205771198971
119886(119909) 1205771198972
119886(119909) 119889119909 =
1198713
41205872 1198971= 1 1198972= 1
minus
21198713
31205873 1198971= 1 1198972= 2
minus
21198713
31205873 1198971= 2 1198972= 1
1198713
41205872 1198971= 2 1198972= 2
int
119871
0
1205771198971
119886(119909) 1205771198971015840
2
119886(119909) 119889119909 =
0 1198971= 1 1198972= 1
minus
1198712
8120587
1198971= 1 1198972= 2
1198712
8120587
1198971= 2 1198972= 1
0 1198971= 2 1198972= 2
int
119871
0
1205771198971015840
1
119886(119909) 1205771198971015840
2
119886(119909) 119889119909 =
5119871
16
1198971= 1 1198972= 1
minus
119871
2120587
1198971= 1 1198972= 2
minus
119871
2120587
1198971= 2 1198972= 1
5119871
16
1198971= 2 1198972= 2
int
119871
0
119909119899 cos 120582
119898119909 119889119909
=
1198711+119899
1 + 119899
119898 = 0
(1198861+119899HypergeometricPFQtimes [ 12 + 1198992 12 32 + 1198992
minus (14)1198982
1205872
]) times (1 + 119899)minus1
119898 gt 0
(B1)
Conflict of Interests
The authors declare that there is no conflict of interestsregarding the publication of this paper
Acknowledgments
The authors would like to thank the anonymous reviewersfor their very valuable comments The authors also gratefullyacknowledge the financial support from the National NaturalScience Foundation of China (no 51209052) HeilongjiangProvince Youth Science Fund Project (no QC2011C013) andHarbin Science and Technology Development InnovationFoundation of Youth (no 2011RFQXG021)
References
[1] K Liew Y Xiang and S Kitipornchai ldquoResearch on thick platevibration a literature surveyrdquo Journal of Sound and Vibrationvol 180 no 1 pp 163ndash176 1995
[2] J H Chung T Y Chung and K C Kim ldquoVibration analysisof orthotropic mindlin plates with edges elastically restrainedagainst rotationrdquo Journal of Sound and Vibration vol 163 no 1pp 151ndash163 1993
[3] Y K Cheung and D Zhou ldquoVibrations of moderately thickrectangular plates in terms of a set of static Timoshenko beamfunctionsrdquoComputers and Structures vol 78 no 6 pp 757ndash7682000
[4] CMWang ldquoNatural frequencies formula for simply supportedMindlin platesrdquo Journal of Vibration and Acoustics vol 116 no4 pp 536ndash540 1994
[5] K N Saha R C Kar and P K Datta ldquoFree vibration analysisof rectangular Mindlin plates with elastic restraints uniformlydistributed along the edgesrdquo Journal of Sound andVibration vol192 no 4 pp 885ndash902 1996
24 Shock and Vibration
[6] D J Gorman ldquoAccurate free vibration analysis of shear-deformable plates with torsional elastic edge supportrdquo Journalof Sound and Vibration vol 203 no 2 pp 209ndash218 1997
[7] D J Gorman ldquoFree vibration analysis of Mindlin plates withuniform elastic edge support by the superposition methodrdquoJournal of Sound andVibration vol 207 no 3 pp 335ndash350 1997
[8] D J Gorman ldquoAccurate free vibration analysis of point sup-ported mindlin plates by the superposition methodrdquo Journal ofSound and Vibration vol 219 no 2 pp 265ndash277 1999
[9] Y Xiang ldquoVibration of rectangular Mindlin plates resting onnon-homogenous elastic foundationsrdquo International Journal ofMechanical Sciences vol 45 no 6-7 pp 1229ndash1244 2003
[10] Y Xiang and G W Wei ldquoExact solutions for buckling andvibration of stepped rectangular Mindlin platesrdquo InternationalJournal of Solids and Structures vol 41 no 1 pp 279ndash294 2004
[11] Y L Yeh M J Jang and C C Wang ldquoAnalyzing the freevibrations of a plate using finite difference and differential trans-formationmethodrdquoAppliedMathematics andComputation vol178 no 2 pp 493ndash501 2006
[12] Y Xiang S K Lai L Zhou and C W Lim ldquoDSC-Ritz elementmethod for vibration analysis of rectangular Mindlin plateswith mixed edge supportsrdquo European Journal of MechanicsmdashASolids vol 29 no 4 pp 619ndash628 2010
[13] H Nguyen-Xuan C H Thai and T Nguyen-Thoi ldquoIsogeo-metric finite element analysis of composite sandwich platesusing a higher order shear deformation theoryrdquo Composites BEngineering vol 55 pp 558ndash574 2013
[14] C H Thai A J M Ferreira E Carrera and H Nguyen-XuanldquoIsogeometric analysis of laminated composite and sandwichplates using a layerwise deformation theoryrdquo Composite Struc-tures vol 104 pp 196ndash214 2013
[15] C H Thai A Ferreira S Bordas T Rabczuk and HNguyen-Xuan ldquoIsogeometric analysis of laminated compositeand sandwich plates using a new inverse trigonometric sheardeformation theoryrdquo European Journal of Mechanics A Solidsvol 43 pp 89ndash108 2014
[16] H Luong-Van T Nguyen-Thoi G R Liu and P Phung-VanldquoA cell-based smoothed finite elementmethod using three-nodeshear-locking free Mindlin plate element (CS-FEM-MIN3) fordynamic response of laminated composite plates on viscoelasticfoundationrdquo Engineering Analysis with Boundary Elements vol42 pp 8ndash19 2014
[17] T Nguyen-Thoi T Rabczuk T Lam-Phat V Ho-Huu andP Phung-Van ldquoFree vibration analysis of cracked Mindlinplate using an extended cell-based smoothed discrete sheargap method (XCS-DSG3)rdquo Theoretical and Applied FractureMechanics 2014
[18] H Nguyen-Xuan G R Liu and C a Thai-Hoang ldquoAnedge-based smoothed finite element method (ES-FEM) withstabilized discrete shear gap technique for analysis of Reissner-Mindlin platesrdquo Computer Methods in Applied Mechanics andEngineering vol 199 no 9ndash12 pp 471ndash489 2010
[19] P Phung-Van T Nguyen-Thoi T Bui-Xuan and Q Lieu-XuanldquoA cell-based smoothed three-node Mindlin plate element(CS-FEM-MIN3) based on the 119862
0 -type higher-order sheardeformation for geometrically nonlinear analysis of laminatedcomposite platesrdquo Computational Materials Science 2014
[20] NNguyen-Thanh T RabczukHNguyen-Xuan and S BordasldquoAn alternative alpha finite element method with discrete sheargap technique for analysis of isotropicMindlin-Reissner platesrdquoFinite Elements inAnalysis andDesign vol 47 no 5 pp 519ndash5352011
[21] KM Liew Y Xiang and S Kitipornchai ldquoTransverse vibrationof thick rectangular plates-I Comprehensive sets of boundaryconditionsrdquo Computers and Structures vol 49 no 1 pp 1ndash291993
[22] D Zhou ldquoVibrations of Mindlin rectangular plates with elasti-cally restrained edges using static Timoshenko beam functionswith the Rayleigh-Ritz methodrdquo International Journal of Solidsand Structures vol 38 no 32-33 pp 5565ndash5580 2001
[23] D Zhou SH Lo F T KAu andYK Cheung ldquoVibration anal-ysis of rectangular Mindlin plates with internal line supportsusing static Timoshenko beam functionsrdquo International Journalof Mechanical Sciences vol 44 no 12 pp 2503ndash2522 2002
[24] H S Shen J Yang and L Zhang ldquoFree and forced vibrationof Reissner-Mindlin plates with free edges resting on elasticfoundationsrdquo Journal of Sound and Vibration vol 244 no 2 pp299ndash320 2001
[25] Y Xing and B Liu ldquoClosed form solutions for free vibrations ofrectangular Mindlin platesrdquo Acta Mechanica Sinica vol 25 no5 pp 689ndash698 2009
[26] P Gagnon C Gosselin and L Cloutier ldquoA finite strip elementfor the analysis of variable thickness rectangular thick platesrdquoComputers amp Structures vol 63 no 2 pp 349ndash362 1997
[27] T Mlzusawa ldquoVibration of rectangular mindlin plates withtapered thickness by the spline strip methodrdquo Computers andStructures vol 46 no 3 pp 451ndash463 1993
[28] H Nguyen-Xuan and T Nguyen-Thoi ldquoA stabilized smoothedfinite element method for free vibration analysis of Mindlin-Reissner platesrdquo Communications in Numerical Methods inEngineering with Biomedical Applications vol 25 no 8 pp 882ndash906 2009
[29] K M Liew L X Peng and S Kitipornchai ldquoVibration anal-ysis of corrugated Reissner-Mindlin plates using a mesh-freeGalerkin methodrdquo International Journal of Mechanical Sciencesvol 51 no 9-10 pp 642ndash652 2009
[30] Y Hou GWWei and Y Xiang ldquoDSC-Ritz method for the freevibration analysis of Mindlin platesrdquo International Journal forNumerical Methods in Engineering vol 62 no 2 pp 262ndash2882005
[31] F- Liu and K M Liew ldquoVibration analysis of discontinuousMindlin plates by differential quadrature element methodrdquoJournal of Vibration and Acoustics Transactions of the ASMEvol 121 no 2 pp 204ndash208 1999
[32] P Malekzadeh G Karami and M Farid ldquoA semi-analyticalDQEM for free vibration analysis of thick plates with twoopposite edges simply supportedrdquoComputerMethods inAppliedMechanics and Engineering vol 193 no 45ndash47 pp 4781ndash47962004
[33] P Malekzadeh and S A Shahpari ldquoFree vibration analysisof variable thickness thin and moderately thick plates withelastically restrained edges by DQMrdquo Thin-Walled Structuresvol 43 no 7 pp 1037ndash1050 2005
[34] R E Diaz-Contreras and S Nomura ldquoGreenrsquos function appliedto solution ofMindlin platesrdquoComputers and Structures vol 60no 1 pp 41ndash48 1996
[35] T Sakiyama and M Huang ldquoFree vibration analysis of rect-angular plates with variable thicknessrdquo Journal of Sound andVibration vol 216 no 3 pp 379ndash397 1998
[36] M Huang X Q Ma T Sakiyama H Matuda and C MoritaldquoFree vibration analysis of orthotropic rectangular plates withvariable thickness and general boundary conditionsrdquo Journal ofSound and Vibration vol 288 no 4-5 pp 931ndash955 2005
Shock and Vibration 25
[37] I Shufrin and M Eisenberger ldquoVibration of shear deformableplates with variable thicknessmdashfirst-order and higher-orderanalysesrdquo Journal of Sound and Vibration vol 290 no 1-2 pp465ndash489 2006
[38] S A Eftekhari and A A Jafari ldquoAccurate variational approachfor free vibration of variable thickness thin and thick plates withedges elastically restrained against translation and rotationrdquoInternational Journal of Mechanical Sciences vol 68 pp 35ndash462013
[39] W L Li X Zhang J Du and Z Liu ldquoAn exact series solutionfor the transverse vibration of rectangular plates with generalelastic boundary supportsrdquo Journal of Sound amp Vibration vol321 no 1-2 pp 254ndash269 2009
[40] X Shi D Shi W L Li and Q Wang ldquoA unified method forfree vibration analysis of circular annular and sector plateswith arbitrary boundary conditionsrdquo Journal of Vibration andControl 2014
[41] D Y Shi X J Shi W L Li and Q S Wang ldquoFree transversevibrations of orthotropic thin rectangular plates with arbitraryelastic edge supportsrdquo Journal of Vibroengineering vol 16 no 1pp 389ndash398 2014
[42] Y Chen G Jin and Z Liu ldquoFree vibration analysis of circu-lar cylindrical shell with non-uniform elastic boundary con-straintsrdquo International Journal of Mechanical Sciences vol 74pp 120ndash132 2013
[43] L Dai T Yang J Du W L Li and M J Brennan ldquoAn exactseries solution for the vibration analysis of cylindrical shellswith arbitrary boundary conditionsrdquo Applied Acoustics vol 74no 3 pp 440ndash449 2013
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Shock and Vibration
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2 Shock and Vibration
method Subsequently Gorman [7 8] utilized the superposi-tion method to investigate the vibration problems of elasticrestrained and point supported Mindlin plates In his super-position method the solution satisfied the differential equa-tions exactly but approximated the boundary conditionsXiang [9] andXiang andWei [10] employed the Levy solutionapproach in conjunction with the domain decompositiontechnique to analytically solve the free vibrations of rectangu-lar Mindlin plates with two opposite edges simply supportedIn [10] the influence of the step thickness ratios on thevibration behavior of rectangular Mindlin plates is studiedYeh et al [11] proposed a hybrid method which combines thefinite difference method and the differential transformationmethod to analyse the free vibration of clamped and simplysupported rectangular thin plates Xiang et al [12] extendedthe DSC-Ritz element method to solve the free vibrationanalysis of moderately thick rectangular plates with mixedsupporting edges Thai et al [13ndash15] and Nguyen-Thoi et al[16ndash19] have recently proposed the isogeometric analysis(IGA) and smoothed finite element method (S-FEM) toanalyse the cracked Mindlin plate laminated composite andsandwich plates In their research the structural boundarywas taken as uniform classic boundary conditions In addi-tion Nguyen-Thanh et al [20] utilized the alpha finite ele-ment method (120572-FEM) in static free vibration and bucklinganalyses ofMindlin-Reissner plates using triangular elementsonly
A Rayleigh-Ritz formulation based on pb-2 functions wasutilized by Liew et al [21] to study the vibration behaviors ofMindlin plates with elastic restrained edges Zhou et al [2223] investigated the similar vibration problems in terms of aset of static Timoshenko beam functions In their study thefree vibrations of rectangularMindlin plateswith internal linesupports were also solved Shen et al [24] presented a new setof admissible functions which satisfied both geometrical andnatural boundary conditions for the free and forced vibrationproblems of moderately thick plates with four free edgesXing and Liu [25] presented the closed-form solutions forvibration problems of Mindlin plates with any combinationsof simply supported and clamped edge conditions Apartfrom aforementioned solutions some other methods suchas finite strip element method [26] spline strip method[27] finite element method [28] meshless method [29] Ritzmethod [30] differential quadraturemethod (DQM) [31ndash33]and Green function method [34 35] had been developed todetermine the vibration behavior of moderately thick plates
It is noteworthy that most of the previous researches onmoderately thick rectangular plates are confined to the uni-form thickness and classical boundary conditions such asfree simply supported clamped edges and their combina-tions Only few studies have been found in the published lit-erature about the vibrations of moderately thick rectangularplates with variable thickness and elastic restraints A linearfinite strip element method based on Mindlinrsquos plate theorywas proposed by Gagnon et al [26] to solve the vibrationproblems of rectangular thick plates in which the thicknesscan vary in any direction Huang et al [36] developeda discrete method for solving the vibration problems oforthotropic rectangular plates with variable thickness and
general classical boundary supports Extended Kantorovichmethod was utilized to investigate the free vibrations ofrectangular thick plates with variable thickness and differentclassical boundary conditions by Shufrin and Eisenberger[37] Eftekhari and Jafari [38] proposed an efficient and accu-rate variational formulation for the vibration problems ofvariable thin and thick plates with elastic edge restraints
Although a large number of studies have been carried outbased on Mindlinrsquos plate theory and methods it appears thatthe information available about the vibration characteristicsof nonuniform thickness moderately thick rectangular platesis very limited Most of the contributions to moderately thickrectangular plates with classic boundary supports and elasticedges are confined to uniform thickness or bilinearly vary-ing thickness However the engineering practices containa variety of possible boundary conditions such as elasticrestraints and nonlinearly varying thickness The existingresults are simply too scarce for engineering applications andcomparative studies Moreover most of the available solutionprocedures in the open literature are often only customizedfor a specific set of restraint conditions which may not beappropriate for practical application because there are hun-dreds of different combinations of boundary conditions for aplate It is desirable to develop a unified and efficient methodwhich is capable of dealing with more complicated problemsinvolving arbitrary elastic edge restraints and nonuniformthickness
In view of those technical limitations and practical needsthis investigation sets out to present a modified Fourier solu-tion technique for the free vibration analysis of moderatelythick rectangular plates with variable thickness and arbitraryboundary conditions and to provide a unified and reason-able accurate alternative to other analytical and numericaltechniques This paper can be considered as an extension ofthe modified Fourier series method previously developed formodeling plates [39ndash41] and shells [42 43]The one displace-ment and two rotation functions are invariably expressedas the superposition of a 2D Fourier cosine series and foursupplementary functions in the form of the product of apolynomial function and a single cosine series expansionwith all these unknown expansion coefficients treated as thegeneralized coordinates and determined using the Rayleigh-Ritz procedure The change of the boundary conditions canbe easily achieved by only varying the stiffness of the threesets of boundary springs along all edges of the rectangularplates without involving any change to the solution proce-dureThe current results are checked againstwith FEMresultsor existing results published in the literature for both uniformand nonuniform (linear and nonlinear variation) thicknessplate cases with good agreement achieved
2 Theoretical Formulations
To perform the free vibration analysis of moderately thickrectangular plates with thickness varying in two directionssubjected to the general elastic boundary conditions thecombination of the artificial spring technique together withRayleigh-Ritzmethod is feasible Consider amoderately thickrectangular plate with the dimension of 119886 times 119887 times ℎ(119909 119910)
Shock and Vibration 3
x
z
y
a
b
h(x y)
Figure 1 The general elastic boundary conditions of moderatelythick rectangular plates with varying thickness in two directions
and the coordinate of the moderately thick rectangularplate with elastically retrained edges is depicted in Figure 1Three groups of boundary restraining springs (translationrotational and torsional springs) are arranged at all sidesof the plate to separately simulate the boundary force Byassigning the stiffness of the boundary springs various valuesit is equivalent to impose different boundary force on themidsurface of the plate For example the clamped boundarycondition can be readily obtained by setting the spring coef-ficients into infinity (a very large number in practical calcu-lation) for the translation rotations and torsional restrainingsprings along each edge
Based on the Mindlin plate theory the displacementsvectors with three directions are
119906 (119909 119910 119911 119905) = 119911120595119909(119909 119910 119905)
V (119909 119910 119911 119905) = 119911120595119910(119909 119910 119905)
119908 (119909 119910 119911 119905) = 119908 (119909 119910 119905)
(1)
where 119906 V and 119908 represent the 119909 119910 and 119911 directiondisplacement functions and the 120595
119909and 120595
119910are the slop due
to bending along in the respective planes The relationship 119908with the slops 120595
119909and 120595
119910is 120595119909= minus119889119908119889119909 and 120595
119910= minus119889119908119889119910
For the moderately thick rectangular plates making useof the strain-stress relationship defined in elasticity theorythe normal shear strains and transverse shear strains can beexpressed as follows
120576119909119909
120576119910119910
120574119909119910
120574119909119911
120574119910119911
=
119911120597120595119909
120597119909
119911120597120595119910
120597119910
119911(
120597120595119909
120597119910
+
120597120595119910
120597119909
)
120595119909+
120597119908
120597119909
120595119910+
120597119908
120597119910
[
[
[
[
[
[
[
[
120590119909119909
120590119910119910
120591119909119910
120591119909119911
120591119910119911
]
]
]
]
]
]
]
]
=
119864
2 (1 minus 1205832)
[
[
[
[
[
[
[
[
2 2120583 0 0 0
2120583 2 0 0 0
0 0 1 minus 120583 0 0
0 0 0 120581 (1 minus 120583) 0
0 0 0 0 120581 (1 minus 120583)
]
]
]
]
]
]
]
]
times
[
[
[
[
[
[
[
[
120576119909119909
120576119910119910
120574119909119910
120574119909119911
120574119910119911
]
]
]
]
]
]
]
]
(2)
where 120576119909119909 120576119910119910 and 120574
119909119910are the normal and shear strains in the
119909119910 and 119911 coordinate systemThe transverse shear strains 120574119909119911
and 120574119910119911
are constant through the thickness The 120590119909119909
and 120590119910119910
are the normal stresses in the 119909 119910 directions 120591119909119911 120591119910119911 and 120591
119909119910
are shear stresses in the 119909 119910 and 119911 coordinate system The 119864is Youngrsquos modulus 120583 is Poissonrsquos ration and 120581 is the shearcorrection factor to account for the fact
In terms of transverse displacements and slope thebending and twisting moments and the transverse shearingforces in plates can be expressed as
119872119909119909= int
ℎ2
minusℎ2
120590119909119909119911 119889119911 = 119863(
120597120595119909
120597119909
+ 120583
120597120595119910
120597119910
)
119872119910119910= int
ℎ2
minusℎ2
120590119910119910119911 119889119911 = 119863(
120597120595119910
120597119910
+ 120583
120597120595119909
120597119909
)
119872119909119910= int
ℎ2
minusℎ2
120591119909119910119911 119889119911 =
119863 (1 minus 120583)
2
(
120597120595119909
120597119910
+
120597120595119910
120597119909
)
119876119909= 120581int
ℎ2
minusℎ2
120591119909119911119889119911 = 120581119866ℎ (119909 119910) (120595
119909+
120597119908
120597119909
)
119876119910= 120581int
ℎ2
minusℎ2
120591119910119911119889119911 = 120581119866ℎ (119909 119910) (120595
119910+
120597119908
120597119910
)
(3)
where ℎ(119909 119910) is the thickness function 119863 = 119864ℎ(119909 119910)3
(12(1 minus 1205832
)) is the flexural rigidity and 119866 = 119864(2(1 + 120583))
is the shear modulusThe boundary conditions for a general supported moder-
ately thick rectangular plate can be expressed as the followingforms based on the force equilibrium relationship on the foursides
on 119909 = 0
1198961199090119908 = minus119876
119909 1198701199090120595119909= minus119872
119909119909 1198701199101199090
120595119910= minus119872
119909119910
on 119909 = 119886
119896119909119886119908 = 119876
119909 119870119909119886120595119909= 119872119909119909 119870119910119909119886
120595119910= 119872119909119910
4 Shock and Vibration
on 119910 = 0
1198961199100119908 = minus119876
119910 1198701199100120595119910= minus119872
119910119910 1198701199091199100
120595119909= minus119872
119909119910
on 119910 = 119887
119896119910119887119908 = 119876
119910 119870119910119887120595119910= 119872119910119910 119870119909119910119887
120595119909= 119872119909119910
(4)
where 1198961199090
and 119896119909119886
(1198961199100
and 119896119910119887) are linear spring constants
1198701199090
and 119870119909119886
(1198701199100
and 119870119910119887) are the rotational spring con-
stants and 1198701199101199090
and 119870119910119909119886
(1198701199091199100
and 119870119909119910119887
) are the torsionalspring constants at 119909 = 0 and 119886 (119910 = 0 and 119887) respectivelyTherefore arbitrary boundary conditions of the plate can begenerated by assigning the linear springs rotational springsand torsional springs at proper stiffness For instance aclamped boundary (C) is achieved by simply setting thestiffness of the entire springs equal to infinity (which is repre-sented by a very large number 1014) Inversely a free bound-ary (F) is gained by setting the stiffness of the entire springsequal to zero
Thus the total potential energy of the spring restrainedplate which is composed of two parts namely the strainenergy of the moderately thick rectangular plates and thepotential energy stored in the boundary springs can beexpressed as
119880 =
1
2
int
119886
0
int
119887
0
119863[(
120597120595119909
120597119909
+
120597120595119910
120597119910
)
2
minus 2 (1 minus 120583)
times (
120597120595119909
120597119909
120597120595119910
120597119910
minus
1
4
(
120597120595119909
120597119910
+
120597120595119910
120597119909
)
2
)]
+ 120581119866ℎ (119909 119910)
times [(120595119909+
120597119908
120597119909
)
2
+ (120595119910+
120597119908
120597119910
)
2
] 119889119909119889119910
+
1
2
int
119886
0
[(11989611991001199082
+ 11987011991001205952
119910+ 1198701199091199100
1205952
119909)
10038161003816100381610038161003816119910=0
+ (1198961199101198871199082
+ 1198701199101198871205952
119910+ 119870119909119910119887
1205952
119909)
10038161003816100381610038161003816119910=119887
] 119889119909
+
1
2
int
119887
0
[(11989611990901199082
+ 11987011990901205952
119909+ 1198701199101199090
1205952
119910)
10038161003816100381610038161003816119909=0
+ (1198961199091198861199082
+ 1198701199091198861205952
119909+ 119870119910119909119886
1205952
119910)
10038161003816100381610038161003816119909=119886
] 119889119910
(5)
As the springs are considered with no mass while retain-ing certain stiffness the total kinetic energy of themoderatelythick rectangular plates is
119879 =
1205881205962
2
int
119887
0
int
119886
0
[
[
ℎ (119909 119910)1199082
+ ℎ3
(119909 119910) (1205952
119909+ 1205952
119910)
12
]
]
119889119909119889119910
(6)
where 120588 is the mass density and 120596 denotes the natural fre-quency of the plate
In view of satisfying arbitrarily supported boundary con-ditions of the moderately thick rectangular plate the admis-sible functions expressed in the form of the improved Fourierseries are introduced to remove the potential discontinuitieswith the functions and their derivatives Thus the moder-ately thick rectangular plate displacements and rotation areexpressed as
120595119909(119909 119910) =
infin
sum
119898=0
infin
sum
119899=0
119860119898119899
cos 120582119886119898119909 cos 120582
119887119899119910
+
2
sum
119897=1
120577119897
119887(119910)
infin
sum
119898=0
119886119897
119898cos 120582119886119898119909
+
2
sum
119897=1
120577119897
119886(119909)
infin
sum
119899=0
119887119897
119899cos 120582119887119899119910
(7)
120595119910(119909 119910) =
infin
sum
119898=0
infin
sum
119899=0
119861119898119899
cos 120582119886119898119909 cos 120582
119887119899119910
+
2
sum
119897=1
120577119897
119887(119910)
infin
sum
119898=0
119888119897
119898cos 120582119886119898119909
+
2
sum
119897=1
120577119897
119886(119909)
infin
sum
119899=0
119889119897
119899cos 120582119887119899119910
(8)
119908 (119909 119910) =
infin
sum
119898=0
infin
sum
119899=0
119862119898119899
cos 120582119886119898119909 cos 120582
119887119899119910
+
2
sum
119897=1
120577119897
119887(119910)
infin
sum
119898=0
119890119897
119898cos 120582119886119898119909
+
2
sum
119897=1
120577119897
119886(119909)
infin
sum
119899=0
119891119897
119899cos 120582119887119899119910
(9)
where 120582119886119898
= 119898120587119886 120582119887119899= 119899120587119887 and 119860
119898119899 119861119898119899 and 119862
119898119899are
the Fourier coefficients of two-dimensional Fourier seriesexpansions for the displacements and rotation functionsrespectively 119886119897
119898 119887119897119898 119888119897119898 119889119897119898 119890119897119898 and 119891119897
119898are the supplemented
coefficients of the auxiliary functions where 119897 = 1 2 Thespecific expressions of the auxiliary functions 120577119897
119886and 120577
119897
119887are
defined as
1205771
119886(119909) =
119886
2120587
sin(1205871199092119886
) +
119886
2120587
sin(31205871199092119886
)
1205772
119886(119909) = minus
119886
2120587
cos(1205871199092119886
) +
119886
2120587
cos(31205871199092119886
)
1205771
119887(119910) =
119887
2120587
sin(120587119910
2119887
) +
119887
2120587
sin(3120587119910
2119887
)
1205772
119887(119910) = minus
119887
2120587
cos(120587119910
2119887
) +
119887
2120587
cos(3120587119910
2119887
)
(10)
As shown in (7)ndash(9) the supplementary functions 1205771119886(119909)
1205772
119886(119909) 1205771
119887(119910) and 120577
2
119887(119910) are used for the displacement and
rotation expressions The theoretical meaning of introduc-ing these terms into the Fourier series is to remove the
Shock and Vibration 5
potential discontinuities and their derivatives throughout thewhole plate structure including the boundaries and then toeffectively enhance the convergence of the results To ensurethis continuity of selection expressions and correspondingderivatives at any point on the plate the first-order derivativesof the 119909 and119910 directions should exist as indicated by (4)Thisrequirement is guaranteed by the selected supplementaryfunctions because it is easy to verify that
1205771
119886(0) = 120577
1
119886(119886) = 120577
11015840
119886(119886) = 0 120577
11015840
119886(0) = 1
1205772
119886(0) = 120577
2
119886(119886) = 120577
21015840
119886(0) = 0 120577
21015840
119886(119886) = 1
(11)
Similar conditions exist for the 119910-related polynomials1205771
119887(119910) and 120577
2
119887(119910) It has to be mentioned that although the
solution is theoretically exact for the superposition of infinitenumbers of Fourier terms in actual calculation we truncatethe infinite series to 119872 and 119873 to obtain the results withacceptable accuracy
Since the energy expressions and admissible function ofthe plate have been established the remaining task is to deter-mine the Fourier expanded coefficients and supplemented
coefficients in (7)ndash(9) The Lagrangian energy functional (119871)of the plate is written as
119871 = 119879 minus 119880 (12)
Then the Lagrangian expression is minimized by takingits derivatives with respect to these coefficients
120597119871
120597120599
= 0 120599 =
119860119898119899
119886119897
119898119887119897
119899
119861119898119899
119888119897
119898119889119897
119899
119862119898119899
119890119897
119898119891119897
119899
(13)
Since the displacements and rotation components of theplate are chosen as 119872 and 119873 to obtain the results withacceptable accuracy a total of 3 lowast (119872 + 1) lowast (119873 + 1) + 6 lowast
(119872 +119873 + 2) equations are obtainedThey can be summed up in a matrix form
(K minus 1205962M)E = 0 (14)
The unknown coefficients in the displacement expres-sions can be expressed in the vector form as E where
E =
11986000 11986001 119860
11989810158400 11986011989810158401 119860
11989810158401198991015840 119860
119872119873 1198861
0 119886
1
119872 1198862
0 119886
2
119872 1198871
0 119887
1
119873 1198872
0 119887
2
119873
11986100 11986101 119861
11989810158400 11986111989810158401 119861
11989810158401198991015840 119861
119872119873 1198881
0 119888
1
119872 1198882
0 119888
2
119872 1198891
0 119889
1
119873 1198892
0 119889
2
119873
11986200 11986201 119862
11989810158400 11986211989810158401 119862
11989810158401198991015840 119862
119872119873 1198901
0 119890
1
119872 1198902
0 119890
2
119872 1198911
0 119891
1
119873 1198912
0 119891
2
119873
119879
(15)
In (14)K is the stiffness matrix for the plate andM is themass matrix They can be expressed separately as
K =
[
[
[
[
[
[
[
[
[
K1-1 K
1-2 K1-3 sdot sdot sdot K
1-9
K1198791-2 K
2-1 K2-3 sdot sdot sdot K
2-9
K1198791-3 K119879
2-3 K3-3 sdot sdot sdot K
3-9
d
K1198791-9 K119879
2-9 K1198793-9 sdot sdot sdot K
9-9
]
]
]
]
]
]
]
]
]
M =
[
[
[
[
[
[
[
[
[
M1-1 M
1-2 M1-3 sdot sdot sdot M
1-9
M1198791-2 M
2-1 M2-3 sdot sdot sdot M
2-9
M1198791-3 M119879
2-3 M3-3 sdot sdot sdot M
3-9
d
M1198791-9 M119879
2-9 M1198793-9 sdot sdot sdot M
9-9
]
]
]
]
]
]
]
]
]
(16)
The specific expressions for the elements in (16) aregiven in Appendix AMoreover all the necessary expressionswhich will be used in the calculations of the eigenvalues andeigenvectors are given in Appendix B
Obviously the natural frequencies and eigenvectors cannow be readily obtained by solving a standard matrix eigen-problem Since the components of each eigenvector are actu-ally the expansion coefficients of the modified Fourier seriesthe corresponding mode shape can be directly determinedfrom (14) In other words once the coefficient eigenvectorE is
determined for a given frequency the displacement functionsof the plate can be determined by substituting the coefficientsinto (9) When the forced vibration is involved by addingthe work done by external force in the Lagrangian energyfunction and summing the loading vector F on the right sideof (14) the characteristic equation for the forced vibration ofthe moderately thick rectangular plates is readily obtained
3 Numerical Examples and Discussion
In this section a systematic comparison between the cur-rent solutions and theoretical results published by otherresearchers or finite element method (FEM) results is car-ried out to validate the excellent accuracy reliability andfeasibility of the modified Fourier method A comprehensivestudy on the effects of elastic restraint parameters andvarying thickness in two directions is also reported Newresults are obtained for plates subjected to general elasticboundary restraints with nonlinear variable thickness in bothdirections The discussion is arranged as follows Firstly theconvergence of the modified Fourier solution is checked Inaddition the influence of the stiffness of boundary springcomponents is studied Secondly the uniform thicknessplates with various combinations of classical boundary con-ditions elastic boundary conditions and different structureparameters are examined Thirdly the nonuniform thicknessplate with linear variation in both directions various com-binations of classical boundary conditions conditions and
6 Shock and Vibration
Table 1 Frequency parameters Ω = 1205961198872
(120588ℎ119863)12
1205872 for square moderately thick plates with uniform thickness and different boundary
conditions
Boundary conditions M N Model sequence1 2 3 4 5 6 7 8
C-C-C-C
119872 = 119873 = 2 3375 7191 7191 9856 1605 1628 1747 1747119872 = 119873 = 4 3299 6308 6308 8853 1046 1056 1263 1263119872 = 119873 = 6 3294 6282 6282 8801 1037 1047 1254 1254119872 = 119873 = 8 3292 6278 6278 8797 1036 1046 1253 1253119872 = 119873 = 10 3292 6276 6276 8794 1036 1046 1253 1253119872 = 119873 = 12 3292 6276 6276 8793 1036 1046 1252 1252119872 = 119873 = 14 3292 6276 6276 8793 1036 1046 1252 1252119872 = 119873 = 16 3292 6275 6275 8792 1036 1045 1252 1252Reference [4] 3292 6276 6276 8792 1036 1046 1252 1252
E-E-E-E
119872 = 119873 = 2 06153 10861 10861 17898 27709 28417 35010 35010119872 = 119873 = 4 06150 10799 10799 17785 27695 28405 34914 34914119872 = 119873 = 6 06149 10796 10796 17773 27694 28403 34898 34898119872 = 119873 = 8 06149 10796 10796 17771 27693 28403 34893 34893119872 = 119873 = 10 06149 10796 10796 17770 27693 28403 34892 34892119872 = 119873 = 12 06149 10796 10796 17770 27693 28403 34892 34892119872 = 119873 = 14 06149 10796 10796 17770 27693 28403 34892 34892119872 = 119873 = 16 06149 10796 10796 17770 27693 28403 34891 34891Reference [33] 06153 10861 10861 17898 27709 28417 35010 35010
FEM 06150 17097 17097 17774 27694 28406 34898 34898
different structure parameters are examined Then numer-ical solutions for moderately thick rectangular plates withelastic boundary conditions are presented The effects of theelastic restraint parameters are also investigated Finally thenonuniform thickness plate with nonlinear variation in bothdirections and arbitrary boundary conditions is also studied
31 Convergence and Stiffness Value Study Since the Fourierseries is numerically truncated and only finite terms arecounted in actual calculations the proposed solution shouldbe understood as a solution with arbitrary precision Inthis subsection a uniform thickness square moderately thickrectangular plate with completely clamped boundary condi-tion (C-C-C-C) and four edges equally elastically restrainedagainst linear spring constants and rotation spring constantssupports (E-E-E-E 119870
119879= 119896119894(1198873
119863) 119870119877
= 119870119894(119887119863)
119870119879
= 10 and 119870119877
= 5) has been selected to demon-strate the convergence and accuracy of the modified Fouriermethod In Table 1 the first eight frequency parametersΩ = 120596119887
2
(120588ℎ119863)12
1205872 for the considered uniform thickness
square moderately thick rectangular plate with C-C-C-C andE-E-E-E boundary conditions are examinedThe table showsthat the proposedmethod has fast convergence behaviorThemaximum discrepancy in the worst case between the 6 times 6
truncated configuration and the 8times8 one is less than 0064In order to fully illustrate the convergence of the presentmethod Figures 3 4 and 5 present the 1st and 8th frequencyparameters Ω = 120596119887
2
(120588ℎ119863)12
1205872 with various truncated
numbers 119872 = 119873 subjected to different boundary conditionand aspect ratios A highly desired convergence characteristicis observed such that (a) sufficiently accurate results can beobtained with only a small number of terms in the seriesexpansions and (b) the solution is consistently refined asmore
terms are included in the expansions However this shouldnot constitute a problem in practice because one can alwaysverify the accuracy of the solution by increasing the trunca-tion number until a desired numerical precision is achievedAs a matter of fact this ldquoquality controlrdquo scheme can be easilyimplemented automatically In modal analysis the naturalfrequencies for higher-order modes tend to converge slower(see Table 1) Thus an adequate truncation number shouldbe dictated by the desired accuracy of the largest naturalfrequencies of interest In view of the excellent numericalbehavior of the current solution the truncation numbers willbe simply set as119872 = 119873 = 12 in the following calculations
As far as the accuracy of the present method is con-cerned the converged solutions of the present method arein excellent agreement with both the results reported byreference data and the finite element results For C-C-C-C boundary conditions the max discrepancy between thepresent results and the reference data does not exceed 0011for the worst case and in most cases is 0 Comparing theresults with exact solutions [4] it is observed that eight termsare sufficient to obtain accurate resultsMoreover with regardto the E-E-E-E boundary condition the max discrepancybetween the present results and the reference data does notexceed 031 for the worst case and in most cases is 01Regarding the results with DQM solutions [33] it can be seenthat the six terms are sufficient to obtain enough accurateresults In addition it is clear that the results of the presentapproach with just 663 DOFs (119872 times 119873 = 12 times 12) canpredict the vibration characteristics accurately Most of themare identical to those obtained from finite element method(FEM) with 10201 DOFs (S4R 001m times 001m) That isto say it needs only 662 DOFs compared with FEM toobtain the same precision solutions for the considered case
Shock and Vibration 7
1
2
3
4
5
6
The 1st orderThe 3rd order
101 103 105 107 109 1011 1013
Ω
Ki (Nm)
(a)
2
3
4
5
6
101 103 105 107 109 1011 1013
The 1st orderThe 3rd order
Ω
Ki (Nmrad)
(b)
3
4
5
6
101 103 105 107 109 1011 1013
Kij (Nmrad)
The 1st orderThe 3rd order
Ω
(c)
Figure 2 The effect of boundary spring stiffness on the natural frequencies Ω (a) translation spring (b) rotation spring and (c) torsionalspring
On the same hardware (Intel i7-39GHz) the computing timeof the present formulation for the solution (119872times119873 = 12times12)implemented in optimized MATLAB scripts is about 2125 swhereas the finite element solution consumes 34578 s that isat least 16 times more CPU time than the present method forthe same problem
As mentioned earlier in the current modeling frame-work all the classical boundary conditions and their com-binations can be conveniently viewed as special cases whenthe stiffness for the normal and tangential boundary springsbecomes zero andor infinitely large Thus the effects of thestiffness of the translation (119896
119894) rotation (119870
119894) and torsional
springs (119870119894119895) on the modal characteristics should be inves-
tigated As shown in the Figures 2(a)ndash2(c) the first and thethird frequency parameters are separately obtained by vary-ing the stiffness of one group of the boundary springs from
extremely large (1014) to extremely small (100) while assigningthe other group of the springs infinite stiffness (1014) It canbe found in Figure 2(a) that the frequency parameter almostkeeps at a level when the stiffness of the translation springs islarger than 1012 or smaller than 107 In Figure 2(b) the influ-ences of the rotation springs on frequency parameters aregiven It is shown that the frequency curves change greatlywithin the stiffness range (106 to 1010) while out of this rangethe frequency curves separately keep level In Figure 2(c) theinfluences of the torsional springs on frequency parametersare given It is shown that the frequency curves almost changewhen the stiffness changes in the whole range
Based on the analysis it can be found that the torsionalsprings almost have no effect on the structure Also the rela-tionship between the rotation springs and twisting momentcan be seen from the boundary condition expression Then
8 Shock and Vibration
32
36
40
44
48
52
2
3
4
5
6
7
8
Ω Ω
0 2 4 6 8 10 12 14 16 18M = N
ab = 1
ab = 32
(a) The 1st order
0 2 4 6 8 10 12 14 16 180
20
40
60
80
0
50
100
150
200
250
300
M = N
Ω Ω
ab = 1
ab = 32
(b) The 8th order
Figure 3 The effect of numerically truncated numbers119872 = 119873 on the natural frequenciesΩ for C-C-C-C boundary condition
0 2 4 6 8 10 12 14 16 18125
130
135
140
145
150
088
090
092
094
096
098
Ω Ω
M = N
ab = 1
ab = 32
(a) The 1st order
0 2 4 6 8 10 12 14 16 18
55
60
65
70
75
80
40
45
50
55
60
65
70
75
80Ω Ω
M = N
ab = 1
ab = 32
(b) The 8th order
Figure 4 The effect of numerically truncated numbers119872 = 119873 on the natural frequenciesΩ for F-F-F-F boundary condition
the twisting moments have small effect on the vibration char-acteristics of the structure However in this paper in order toget a more accurate prediction of the vibration characteristicsof the structure the twisting moment on boundary edges istaken into account In the latter study in addition to the factthat free boundary is not unexpected considering the torsionspring the other boundary conditions are introduced into atorsion spring and the spring stiffness takes infinity FormFigure 2 analysis it also concluded that the translation springhas a wider influence range than the rotation spring on thefrequency parameters that is for the translation springs thestable frequency parameters appearwhen the stiffness ismorethan 1012 or less 107 while for the rotation springs when thestiffness value is assigned more than 1010 the frequency para-meters become smoothThus it is suitable to use 1014 to sim-ulate the infinite stiffness value in the model validation partsand in the following examples
32 Uniform Thickness Moderately Thick Plates with Classicaland Elastic Boundary Conditions In this subsection themodified Fourier solution is applied to deal with vibrationproblems of uniform thickness moderately thick rectangu-lar plates subject to the classical boundary conditions andarbitrary elastic boundary conditions In present work threegroups of continuously distributed boundary springs areintroduced to simulate the given or typical boundary condi-tions As mentioned earlier the stiffness of these boundarysprings can take any value from zero to infinity to bettermodel many real-world restraint conditions Taking edge 119909 =0 for example the corresponding spring stiffness for the threetypes of classical boundaries and elastic boundaries is
completely free 119876119909= 0119872
119909119909= 0 and119872
119909119910= 0
1198961199090= 0 119870
1199090= 0 119870
1199101199090= 0 (17a)
Shock and Vibration 9
0 2 4 6 8 10 12 14 16 18
166
168
170
172
152
156
160
164
168
Ω Ω
ab = 1
ab = 32
M = N
(a) The 1st order
0 2 4 6 8 10 12 14 16 1850
55
60
65
70
4
5
6
7
8
9
10
Ω Ω
M = N
ab = 1
ab = 32
(b) The 8th order
Figure 5 The effect of numerically truncated numbers 119872 = 119873 on the natural frequencies Ω with four edges elastically restrained againsttranslation and rotation (119870
119879= 119896119894(1198873
119863) 119870119877= 119870119894(119887119863) 119870
119879= 100 and 119870
119877= 100)
simply supported 119908 = 0119872119909119909= 0 and 120595
119910= 0
1198961199090= 1014
1198701199090= 0 119870
1199101199090= 1014
(17b)
completely clamped 119908 = 0 120595119909= 0 and 120595
119910= 0
1198961199090= 1014
1198701199090= 1014
1198701199101199090
= 1014
(17c)
arbitrarily elastic boundary 119908 = 0 120595119909
= 0 120595119910
= 0119876119909
= 0119872119909119909
= 0 and119872119909119910
= 0
1198961199090= Γ1199081199090
1198701199090= Γ1199091199090
1198701199101199090
= 1014
(17d)
The appropriateness of the three classical boundariesdefined in (17a)ndash(17c) will be proved by several examplesgiven in following the arbitrary elastic boundaries are alsodefined in (17d) and the Γ
119908119894(Γ119908119894= 1198961198941198960 1198960= 1 times 10
9Nm2119894 = 1199090 1199091198861199100119910119887) and Γ
119909119894(Γ119909119894= 11987011989411987001198700= 1times10
9Nmrad119894 = 1199090 119909119886 1199100 119910119887) elastic restraint parameters representcorresponding spring stiffness For the sake of simplicitya four-letter string is employed to represent the restraintcondition of a plate such that F-C-S-E identifies the platewithedges 119909 = 0 119910 = 0 119909 = 119886 and 119910 = 119887 having free clampedshear-supported restrained and elastic boundary conditionsrespectively
As for the first case a uniform thickness moderately thickplate with different classical boundaries and structure param-eters is investigated here In Table 2 the comparison of thefirst eight frequency parametersΩ = 120596119887
2
(120588ℎ119863)12
1205872 of the
considered plate is presented The S-S-S-S C-F-F-F S-S-F-FF-F-F-F and S-C-S-C boundary conditions are performed inthe comparison Excellent agreements are observed betweenthe solutions obtained by the modified Fourier method thereferential data and finite element method (FEM) results forthe uniform thickness moderately thick rectangular platesIt is also verified that the definition of the three types of
classical boundaries in (17a)ndash(17c) is appropriate In additionthe elastic boundary conditions (17d) are also verified Inthe next two examples we will account for the vibrationof moderately thick plate with elastic edge supports Thefirst model considered is an S-S-S-S square moderately thickplate with all edges elastically rotationally restrained Thatis 1198701199090
= 119870119909119886
= 1198701199100
= 119870119910119887
= Γ119909 and all the
other restraining springs are set to have an infinite stiffness(namely represented by 1014 in numerical calculation) Thesix frequency parameters Ω = 120596119887
2
(120588ℎ119863)12
1205872 are given
in Table 3 for several different restraining coefficient valuesthe finite element method (FEM) results calculated usingABAQUSare also listed in this table as a referenceThe secondmodel concerns a complete square moderately thick platewith all edges elastically restrained That is 119896
1199090= 119896119909119886
=
1198961199100
= 119896119910119887
= Γ119908and 119870
1199090= 119870119909119886
= 1198701199100
= 119870119910119887
= Γ119909 The
six frequency parameters Ω = 1205961198872
(120588ℎ119863)12
1205872 are given in
Table 4 for several different restraining coefficient values thefinite element method (FEM) results are also listed in Table 4as a reference It can be clearly seen that the comparison isextremely good which implies that the current method isable tomake correct predictions for themodal characteristicsof moderately thick plate with not only classical boundaryconditions but also elastic edge restraints
The excellent agreements between the solution obtainedby the modified Fourier method and the referential datafor the moderately thick plate subjected to the combina-tions of classical boundary conditions and elastic boundaryconditions given in Tables 2ndash4 indicate that the proposedmethod is sufficiently accurate to deal with uniform thicknessmoderately thick plate with arbitrary boundary conditions
33 Linearly VariationThickness Moderately Thick Plates withClassical and Elastic Boundary Conditions In the theoreticalformulations this paper concerns the varying thicknessmoderately thick plates with classical and elastic boundary
10 Shock and Vibration
Table 2 Frequency parameters Ω = 1205961198872
(120588ℎ119863)12
1205872 for uniform thickness moderately thick plates with different classical boundaries and
structure parameters
Boundary condition ab Model sequence1 2 3 4 5 6 7 8
S-S-S-S
11931 4605 4605 7064 8605 8605 10792 10792
(1933a) (4611) (4611) (7067) (8607) (8607) (10799) (10780)(1931b) (4605) (4605) (7064) (8605) (8605) (10792) (10792)
321408 2648 4127 4605 5261 7064 7153 8184(1410a) (2651) (4128) (4608) (5267) (7074) (7155) (8188)(1408b) (2648) (4127) (4605) (5261) (7064) (7151) (8184)
C-F-F-F
10348 0817 2035 2583 2862 4816 5478 5774
(0348a) (0817) (2038) (2586) (2865) (4821) (5504) (5793)(0348b) (0816) (2034) (2582) (2860) (4811) (5477) (5772)
320155 0502 0941 1665 2292 2613 3303 3498
(0154a) (0502) (0941) (1663) (2295) (2618) (3305) (3500)(0154b) (0501) (0940) (1662) (2292) (2612) (3298) (3494)
S-S-F-F
10333 1678 1875 3559 4720 4946 6474 6632
(0335a) (1680) (1878) (3600) (4721) (4949) (6478) (6642)(0333b) (1677) (1874) (3557) (4718) (4945) (6472) (6631)
320222 0938 1631 2363 2573 4082 4482 4753
(0223a) (0940) (1632) (2364) (2575) (4092) (4483) (4759)(0222b) (0938) (1631) (2361) (2571) (4077) (4479) (4752)
F-F-F-F1 1290 1919 2363 3235 3235 5605 5605 5646
(1291a) (1920) (2365) (3239) (3239) (5607) (5607) (6648)
32 0865 0948 1961 2166 2461 2843 3551 4026(0865a) (0949) (1963) (2168) (2465) (2844) (3556) (4034)
S-C-S-C1 2700 4971 5990 7973 8787 10250 11334 12024
(2701a) (4971) (5993) (7975) (8788) (10253) (11335) (12028)
32 2348 3253 4972 5646 6489 7381 7974 9930(2348a) (3255) (4974) (5647) (6590) (7384) (7801) (9932)
aResults in parentheses are taken from FEMbResults in parentheses are taken from [11]
conditions The varying thickness function ℎ(119909 119910) can beexpressed as ℎ
0(1 + 120572119909
119904
)(1 + 120573119910119905
) in which the ℎ0 120572 and
120573 represent the initial thickness gradient in 119909 direction andgradient in 119910 direction When the indexes 119904 and 119905 take thevalue 119904 = 119905 = 1 the analyticalmodel imitates the linearly vari-ation thickness moderately thick plates structure In order tounify the description and facilitate the analytical calculationsof the involved integrals all the thickness variation functionscan be expanded into either 1D or 2D Fourier cosine seriesresulting in
ℎ (119909 119910) =
infin
sum
119898=0
infin
sum
119899=0
ℎ0119902119898119899
cos 119898120587119909119886
cos119898120587119910
119887
(18)
where119902119898119899
=
4
119886119887
int
119887
0
int
119886
0
(1 + 120572119909119904
) (1 + 120573119910119905
) cos 119898120587119909119886
cos119898120587119910
119887
119889119909 119889119910
(19)
In order to prove the validity of the present methodfor the vibration of linearly variation thickness moderately
thick plates with arbitrary boundary conditions the typicalclassical boundary conditions as the first case will be con-sidered In Table 5 the comparison of the first six frequencyparameters Ω = 120596119886
2
(120588ℎ01198630)12 of the moderately thick
plates with linearly varying thickness is presentedThe S-S-S-S C-F-F-F S-S-F-F C-C-C-C and S-C-S-C boundary condi-tions are performed in the comparison Excellent agreementsare observed between the solutions obtained by the modifiedFourier method and finite element method (FEM) results forthemoderately thick plates with linear variation thickness Toinvestigate the influence of the aspect ratio on the uniformthickness and nonuniform thickness moderately thick platesthe effect on the frequency parameters for plates with S-S-S-Sboundary conditions is presented in Figure 6 The thicknessfunctions are ℎ
0and ℎ0(1+05times119909)(1+05times119910) respectively It
is seen from Figure 6 that the influence of aspect ratios on thefrequency parameters for nonuniform thickness moderatelythick plates is more complicated
In the next two examples we also account for the vibra-tions of moderately thick plate with linear variation thicknessand elastic edge supports The first model considered is an S-S-S-S square moderately thick plate with all edges elastically
Shock and Vibration 11
Table 3 Frequency parameters Ω = 1205961198872
(120588ℎ119863)12
1205872 for S-S-S-S moderately thick plates (119886119887 = 1) with uniform thickness and elastic
rotation support (1198701199090= 119870119909119886= 1198701199100= 119870119910119887= Γ119909)
Γ119909
aℎ0119886
Model sequence1 2 3 4 5 6
0
005 1945 4834 4834 7591 9515 9519(1942b) (4836) (4836) (7584) (9547) (9552)
01 1856 4500 4500 6829 8499 8513(1858b) (4510) (4510) (6843) (8539) (8554)
015 1758 4114 4114 6040 7421 7445(1761b) (4126) (4126) (6060) (7462) (7486)
02 1656 3733 3733 5335 6468 6499(1660b) (3747) (3747) (5357) (6508) (6538)
05
005 3541 7071 7071 10227 12318 12399(3546b) (7091) (7091) (10253) (12386) (12468)
01 3274 6242 6242 8719 10302 10421(3282b) (6267) (6267) (8754) (10370) (10488)
015 2947 5361 5361 7290 8465 8606(2956b) (5387) (5387) (7327) (8524) (8666)
02 2624 4599 4599 6148 7057 7199(2634b) (4624) (4624) (6182) (7107) (7250)
1
005 3541 7072 7072 10228 12320 12401(3547b) (7093) (7093) (10254) (12388) (12470)
01 3278 6248 6248 8727 10311 10429(3286b) (6274) (6274) (8762) (10378) (10497)
015 2957 5374 5374 7305 8478 8620(2966b) (5400) (5400) (7341) (8537) (8680)
02 2641 4617 4617 6166 7070 7215(2651b) (4643) (4643) (6200) (7121) (7268)
15
005 3542 7073 7073 10229 12320 12401(3547b) (7093) (7093) (10254) (12388) (12471)
01 3280 6251 6251 8729 10313 10432(3287b) (6276) (6276) (8765) (10381) (10500)
015 2960 5379 5379 7309 8482 8625(2969b) (5405) (5405) (7346) (8541) (8685)
02 2647 4623 4623 6172 7074 7221(2657b) (4649) (4649) (6206) (7125) (7274)
2
005 3542 7073 7073 10229 12321 12402(3547b) (7093) (7093) (10255) (12389) (12471)
01 3280 6252 6252 8730 10315 10434(3288b) (6277) (6277) (8766) (10382) (10501)
015 2962 5381 5381 7312 8484 8628(2971b) (5407) (5407) (7348) (8543) (8688)
02 2650 4627 4627 6175 7076 7224(2660b) (4652) (4652) (6210) (7128 (7277)
aΓ119909= 1198701198941198700 (1198700= 1 times 10
9 Nmrad 119894 = 1199090 119909119886 1199100 119910119887)bResults in parentheses are taken from FEM
12 Shock and Vibration
Table 4 Frequency parameters Ω = 1205961198872
(120588ℎ119863)12
1205872 for moderately thick plates (119886119887 = 1) with uniform thickness and all edge elastic
restraints (1198701199090= 119870119909119886= 1198701199100= 119870119910119887= Γ119909and 119896
1199090= 119896119909119886= 1198961199100= 119896119910119887= Γ119908)
Γ119909
aΓ119908
aℎ0119886
Model sequence1 2 3 4 5 6
05 0
005 0986 1967 3893 3893 4850 4850(0988b) (1971) (3897) (3897) (4852) (4852)
01 0916 1835 3515 3522 4375 4375(0917b) (1837) (3519) (3528) (4341) (4341)
015 0789 1635 2983 3027 3771 3771(0792b) (1638) (2988) (3029) (3775) (3775)
02 0646 1444 2497 2601 3249 3249(0652b) (1448) (2508) (2621) (3254) (3254)
1 05
005 3417 3417 4431 5294 5461 6448(3419b) (3419) (4435) (5303) (5478) (6456)
01 1593 1593 2401 3849 3861 4729(1611b) (1611) (2408) (3852) (3866) (4767)
015 1122 1122 1913 3272 3290 4038(1127b) (1227) (1924) (3284) (3314) (4069)
02 0890 0890 1640 2781 2829 3476(0899b) (0899) (1654) (2787) (2834) (3497)
15 1
005 4325 4325 5619 6273 6571 7619(4334b) (4334) (5681) (6299) (6642) (7642)
01 2013 2013 2812 4088 4116 4993(2024b) (2024) (2827) (4122) (4187) (4994)
015 1338 1338 2108 3430 3443 4196(1342b) (1342) (2112) (3467) (3482) (4200)
02 1038 1038 1772 2936 2965 3613(1042b) (1042) (1786) (2937) (2977) (3624)
2 2
005 5231 5231 6919 7630 8002 9220(5238b) (5238) (6931) (7637) (8011) (9241)
01 2594 2594 3439 4470 4544 5428(2597b) (2597) (3442) (4495) (4557) (5438)
015 1653 1653 2403 3613 3632 4392(1662b) (1662) (2414) (3627) (3646) (4404)
02 1238 1238 1949 3077 3099 3751(1244b) (1244) (1964) (3094) (3116) (3788)
aΓ119909= 1198701198941198700 (1198700= 1 times 10
9 Nmrad 119894 = 1199090 119909119886 1199100 119910119887) Γ119908= 1198961198941198960 (1198960= 1 times 10
9 Nm 119894 = 1199090 119909119886 1199100 119910119887)bResults in parentheses are taken from FEM
05 10 15 20 25 300
2
4
6
8
10
12
14
16
ab
The 1st orderThe 2nd orderThe 3rd order
The 4th orderThe 5th orderThe 6th order
Ω
(a)
05 10 15 20 25 30
30
60
90
120
150
180
ab
The 1st orderThe 2nd orderThe 3rd order
The 4th orderThe 5th orderThe 6th order
Ω
(b)
Figure 6 The effect of aspect ratio 119886119887 on the natural frequenciesΩ for S-S-S-S boundary condition (a) uniform thickness and (b) nonuni-form thickness
Shock and Vibration 13
Table 5 Frequency parameter Ω = 1205961198862
(120588ℎ01198630)12 for moderately thick plates with linearly varying thickness in different boundary
conditions (ℎ(119909 119910) = ℎ0(1 + 05119909)(1 + 05119910))
Boundary condition 119886119887 ℎ0119886
Model sequence1 2 3 4 5 6
S-S-S-S
101 27261 62397 62926 92423 111537 111732
(27263a) (62410) (63842) (92445) (111867) (111921)
02 23103 47010 47191 65018 75831 75865(23489a) (47209) (47349) (65057) (76605) (76473)
3201 59338 99101 138798 151102 166710 204855
(59874a) (99249) (138177) (151362) (166644) (205381)
02 41175 63765 63765 73598 83956 89084(41586a) (63313) (63669) (73661) (83918) (89547)
C-F-F-F
101 4173 10331 27407 37826 38809 64095
(4856a) (10680) (27513) (37203) (39135) (64858)
02 3983 8702 21403 29443 30286 46705(4118a) (9306) (21990) (29670) (30602) (47156)
3201 4868 15488 35620 63523 89969 100089
(4887a) (15320) (35175) (63353) (89159) (100783)
02 4469 9763 22338 42188) 50104 56331(4944a) (9636) (22087) (42177) (50580) (56559)
S-S-F-F
101 4117 24795 26216 48567) 68195 69667
(4789a) (24149) (26751) (49324) (68842) (70575)
02 3820 20516 21924 37523 48980 50327(4103a) (20041) (22125) (37221) (48894) (50622)
3201 7935 39454 65063 92877 92877 136176
(7856a) (40145) (65162) (92462) (93022) (136336)
02 6389 26149 35627 49539 56844 65170(6406a) (26536) (35626) (49831) (57144) (65981)
C-C-C-C
101 44471 80299 80681 109422 126871 128207
(45102a) (80271) (80656) (110015) (126328) (128888)
02 32162 52781 52933 69410 78474 79482(32240a) (52629) (53166) (70063) (79059) (80266)
3201 82975 115587 151992 161579 176601 212330
(83139a) (115838) (152200) (162193) (177058) (212849)
02 48577 66022 83468 90830 97474 98157(49187a) (66259) (84065) (91749) (97827) (98777)
S-C-S-C
101 36847 66664 76965 101045 113594 125551
(37901a) (68568) (78356) (101200) (113978) (125651)
02 27673 48469 51220 67027 76358 78241(27897a) (48685) (51491) (67447) (77530) (79573)
3201 77378 106848 150079 154569 173120 208391
(78003a) (107058) (150991) (155171) (173099) (208402)
02 45895 64732 73393 83282 90325 97089(45803a) (64131) 73656 83801 90518 97015
aResults in parentheses are taken from FEM
restrained That is 1198701199090
= 119870119909119886
= 1198701199100
= 119870119910119887
= Γ119909 The six
frequency parameters Ω = 1205961198862
(120588ℎ01198630)12 are given in
Table 6 for several different restraining coefficient valuesthe finite element method (FEM) results calculated usingABAQUS are also presented as a referenceThe secondmodelis a complete square moderately thick plate with all edgeselastically restrainedThat is 119896
1199090= 119896119909119886= 1198961199100= 119896119910119887= Γ119908and
1198701199090= 119870119909119886
= 1198701199100= 119870119910119887= Γ119909 The six frequency parameters
Ω = 1205961198862
(120588ℎ01198630)12 are given in Table 7 for several different
restraining coefficient values the finite element method(FEM) results are also listed as a reference It can be clearlyseen that the comparison is extremely good which impliesthat the current method is able to make correct predictionsfor the modal characteristics of linearly varying thickness
14 Shock and Vibration
Table 6 Frequency parameter Ω = 1205961198862
(120588ℎ01198630)12 for S-S-S-S moderately thick plates (119886119887 = 1) with linearly varying thickness and elastic
rotation support (ℎ(119909 119910) = ℎ0(1 + 05119909)(1 + 05119910) 119870
1199090= 119870119909119886= 1198701199100= 119870119910119887= Γ119909)
Γ119909
aℎ0119886
Model sequence1 2 3 4 5 6
0
005 28809 70080 70891 108967 134884 135295(28897b) (70334) (71129) (109112) (135254) (135772)
01 27261 62398 62926 92423 111540 111735(26606b) (61784) (62303) (91193) (111323) (111648)
015 25231 54192 54505 77118 91228 91312(25204b) (55297) (55595) (77503) (91871) (91941)
02 23103 47010 47191 65018 75832 75866(23947b) (47071) (47208) (65404) (76531) (76589)
05
005 23103 47010 47191 65018 75832 75866(23210b) (47245) (47358) (65241) (76122) (76345)
01 48377 94650 95597 135484 162635 163501(48524b) (94891) (95992) (135845) (167012) (164024)
015 36239 70939 71298 99997 118584 119009(36647b) (71214) (71687) (100024) (119114) (119457)
02 28749 56937 57037 79214 93009 93162(29142b) (57034) (57544) (79654) (93668) (93754)
1
005 49651 96759 97619 138089 165368 166304(49755b) (97245) (97964) (138544) (165864) (166878)
01 38951 73735 74251 102748 121079 121709(39021b) (74145) (74664) (102874) (121613) (121817)
015 30630 58406 58533 80432 93984 94255(30638b) (58684) (58644) (80874) (94121) (94478)
02 25548 48636 48674 66165 76708 76824(25687b) (48788) (48992) (66544) (76824) (76944)
15
005 50118 97561 98374 139096 166421 167382(50247b) (97877) (98661) (139247) (166875) (167662)
01 40310 75216 75783 104229 122402 123164(40366b) (75542) (75921) (104557) (122874) (123385)
015 31837 59352 59530 81245 94611 94983(31902b) (59451) (59871) (81375) (94784) (95105)
02 26302 49119 49156 66528 76954 77130(26418b) (49354) (49334) (66921) (77010) (77470)
2
005 50360 97983 98769 139631 166979 167954(50472b) (98005) (99114) (140154) (167188) (167986)
01 41130 76146 76724 105165 123229 124082(41421b) (76642) (78114) (105661) (123278) (124974)
015 32687 60025 60245 81832 95052 95508(32778b) (60244) (60549) (82146) (95114) (95842)
02 26893 49492 49539 66815 77138 77370(26922b) (49874) (49924) (66987) (77367) (77588)
aΓ119909= 1198701198941198700 (1198700= 1 times 10
9 Nmrad 119894 = 1199090 119909119886 1199100 119910119887)bResults in parentheses are taken from FEM
Shock and Vibration 15
Table 7 Frequency parameterΩ = 1205961198862
(120588ℎ01198630)12 for moderately thick plates (119886119887 = 1) with linearly varying thickness and elastic rotation
and translation support (ℎ(119909 119910) = ℎ0(1 + 05119909)(1 + 05119910) 119870
1199090= 119870119909119886= 1198701199100= 119870119910119887= Γ119909 and 119896
1199090= 119896119909119886= 1198961199100= 119896119910119887= Γ119908)
Γ119909
aΓ119908
aℎ0119886
Model sequence1 2 3 4 5 6
05 0
005 14751 14917 28232 55251 55397 69512(14780b) (14822) (29252) (55476) (55505) (70904)
01 11811 11910 23857 44462 45037 55453(11818b) (11897) (24711) (44376) (45257) (56511)
015 8487 8514 20011 34783 36335 44970(8500b) (8544) (20738) (34404) (36663) (45625)
02 6073 6073 17431 28679 30805 38010(6096b) (6106) (18039) (28129) (31165) (38311)
1 05
005 21225 32173 32173 44381 64061 65222(20708b) (32097) (32285) (45110) (64182) (65355)
01 8140 16878 16941 28100 48663 49015(7943b) (16838) (16970) (28888) (48700) (49171)
015 4493 11859 11878 22429 38352 39247(4383b) (11867) (11891) (23125) (38177) (39488)
02 2932 8690 8691 18919 31108 32600(3540b) (8791) (8791) (16229) (27520) (27889)
15 1
005 27819 40919 41430 54595 70792 72896(27146b) (41008) (41434) (55256) (70895) (73051)
01 11285 20068 20230 31101 51046 51429(11012b) (20025) (20270) (31849) (51126) (51569)
015 6292 13912 13922 24132 40500 41120(6139b) (13912) (13934) (24808) (40426) (41316)
02 4121 10321 10331 20051 32796 33921(4021b) (10329) (10357) (20635) (32540) (34178)
2 2
005 34615 51972 53088 68255 81344 84510(33806b) (52100) (53157) (68891) (81433) (84761)
01 15416 24659 24829 35615 53879 54494(15045b) (24625) (24888) (36303) (53970) (54626)
015 8740 16420 16481 26353 42417 42946(8529b) (16414) (16499) (26996) (42398) (43114)
02 5761 12127 12127 21434 34286 35192(5622b) (12134) (12148) (21998) (34110) (35415)
aΓ119909= 1198701198941198700 (1198700= 1 times 10
9 Nmrad 119894 = 1199090 119909119886 1199100 119910119887) Γ119908= 1198961198941198960 (1198960= 1 times 10
9 Nm 119894 = 1199090 119909119886 1199100 119910119887)bResults in parentheses are taken from FEM
moderately thick plate with not only classical boundaryconditions but also elastic edge restraints
As the last case of this subsection the influence of thegradient 120572 and 120573 on the fundamental frequency parametersfor a linearly varying thickness moderately thick plate isinvestigated The model is a square moderately thick platewith all edges elastically restrainedThat is 119896
1199090= 119896119909119886= 1198961199100=
119896119910119887= Γ119908= 2 and 119870
1199090= 119870119909119886
= 1198701199100= 119870119910119887= Γ119909= 2 The six
frequency parameters Ω = 1205961198862
(120588ℎ01198630)12 are given in
Table 8 for several different 120572 and 120573 values the finite elementmethod (FEM) results are also listed as a reference Againgood agreement can be observed Through Table 8 it is alsofound that the frequency parameter increases with increasinggradient parameters
The above studies are given as linearly varying thicknessmoderately thick plates with several of boundary condition
and different structure parameters In the next section vibra-tion results for the plates subjected to nonlinear variationthickness will be presented
34 Nonlinearly Varying Thickness Moderately Thick Plateswith Classical and Elastic Boundary Conditions In Sec-tion 33 the linearly varying thickness moderately thickplates were studied However in the practical engineeringapplications the varying thickness of amoderately thick platemay not always be linear variation in nature A variety ofpossible thickness varying cases may be encountered in prac-tice Therefore the moderately thick plates with nonlinearvariation thickness subjected to general elastic edge restraintsare examined in this subsection For the sake of brevity theindexes 119904 and 119905 will be chosen as 2 to imitate the nonlinearlyvarying thickness moderately thick plates structure in this
16 Shock and Vibration
Table 8 Frequency parameterΩ = 1205961198862
(120588ℎ01198630)12 for elastic support moderately thick plates (119886119887 = 1) with different gradient (119896
1199090= 119896119909119886=
1198961199100= 119896119910119887= Γ119908= 2 119870
1199090= 119870119909119886= 1198701199100= 119870119910119887= Γ119909= 2)
120572 120573 ℎ0119886
Model sequence1 2 3 4 5 6
120572 = 120573 = 01
005 31275 52615 52657 68954 76146 79850(31303a) (52664) (52675) (69033) (76174) (79948)
01 17270 25244 25250 33949 46050 46625(17271a) (25255) (25258) (34089) (46122) (46708)
015 10190 16266 16274 24164 37327 37526(10189a) (16269) (16280) (24323) (37401) (37622)
02 6780 12258 12260 19685 31452 31754(6780a) (12265) (12270) (19851) (31514) (31864)
120572 = 120573 = 02
005 32530 52947 53141 68954 77034 80589(32577a) (53014) (53203) (69172) (77088) (80773)
01 16882 24979 24979 34154 48057 48575(16883a) (24983) (25001) (34433) (48134) (48667)
015 9804 16284 16309 24683 38844 39085(9803a) (16283) (16322) (24980) (38909) (39191)
02 6499 12290 12295 20133 32354 32770(6498a) (12296) (12307) (20420) (32387) (32895)
120572 = 120573 = 03
005 33516 52845 53308 68704 78203 81561(33563a) (52939) (53389) (69055) (78271) (81789)
01 16419 24799 24830 34527 50061 50583(16419a) (24790) (24869) (34947) (50143) (50688)
015 9430 16334 16378 25236 40193 40506(9429a) (16329) (16395) (25660) (40243) (40625)
02 6235 12288 12293 20572 33102 33663(6235a) (12296) (12306) (20965) (33090) (33811)
120572 = 120573 = 04
005 34215 52463 53264 68430 79654 82865(34252a) (52575) (53346) (68921) (79732) (83113)
01 15921 24695 24793 35027 52011 52571(15921a) (24674) (24842) (35583) (52098) (52688)
015 9074 16388 16444 25797 41379 41789(9074a) (16381) (16464) (26336) (41404) (41928)
02 5990 12236 12238 21002 33735 34461(5989a) (12249) (12250) (21487) (33653) (34641)
120572 = 120573 = 05
005 34615 51972 53088 68255 81344 84510(33806a) (52100) (53157) (68891) (81433) (84761)
01 15416 24659 24829 35615 53879 54494(15045a) (24625) (24888) (36303) (53970) (54626)
015 8740 16420 16481 26353 42417 42946(8529a) (16414) (16499) (26996) (42398) (43114)
02 5761 12127 12127 21434 34286 35192(5622a) (12134) (12148) (21998) (34110) (35415)
aResults in parentheses are taken from FEM
section that is 119904 = 119905 = 2 Also the thickness varying functionsare also be expressed as Fourier cosine series according to (18)and (19)
In order to validate the accuracy and reliability of theproposed method for predicting the vibration behavior ofnonlinearly varying thickness moderately thick plates with
arbitrary boundary conditions the typical classical boundaryconditions viewed as the special cases of elastically restrainededges will be considered The comparison of the first six fre-quency parameters Ω = 120596119886
2
(120588ℎ01198630)12 for the moderately
thick plates with nonlinearly varying thickness is presentedin Table 9 The S-S-S-S C-F-F-F S-S-F-F C-C-C-C and
Shock and Vibration 17
Table 9 Frequency parameter Ω = 1205961198862
(120588ℎ01198630)12 for moderately thick plates with linearly varying thickness in different boundary
conditions (ℎ(119909 119910) = ℎ0(1 + 05119909
2
)(1 + 051199102
))
Boundary condition 119886119887 ℎ0119886
Model sequence1 2 3 4 5 6
S-S-S-S
101 23462 54999 55573 82950 100900 101187
(23741a) (55114) (55983) (83146) (101004) (101347)
02 18606 42725 42971 60040 71167 71761(18936a) (43014) (43148) (60247) (71642) (71993)
3201 53055 65251 85404 106700 114880 143531
(53467a) (65472) (85984) (107211) (115687) (144007)
02 17439 21531 32170 44987 59438 64179(17874a) (21748) (32651) (45264) (60188) (64879)
C-F-F-F
101 3658 9317 24135 32816 35433 59179
(3749a) (9547) (24034) (33627) (36308) (60142)
02 3520 7986 19591 26959 27491 43882(3611a) (8176) (20075) (27125) (27625) (43984)
3201 3901 9963 28984 69187 73686 89360
(3930a) (10095) (29381) (69578) (73994) (90242)
02 3292 4439 13385 23947 40850 44758(3386a) (4498) (14012) (24068) (41221) (45142)
S-S-F-F
101 3557 21268 22762 43524 60253 62429
(3649a) (21441) (22796) (43916) (60622) (62630)
02 3171 17822 19512 33616 45042 46833(3489a) (18006) (20430) (33815) (45247) (49004)
3201 6738 31658 52071 75625 86604 94903
(6957a) (31882) (52518) (75823) (86939) (95610)
02 3653 12930 20955 26517 44097 48058(6807a) (13160) (21312) (26864) (44422) (48873)
C-C-C-C
101 41020 75196 75614 104142 119683 120949
(41405a) (75452) (75882) (104711) (119637) (121137)
02 31237 51686 51868 68411 76868 77864(32010a) (52960) (52145) (69335) (77481) (78099)
3201 95823 128727 163657 173751 190890 223595
(95364a) (128932) (163588) (172350) (190350) (222948)
02 53306 71372 89384 102404 104660 120908(53587a) (71759) (90103) (102503) (105174) (121275)
S-C-S-C
101 33715 60298 71446 94187 103802 117928
(34148a) (60786) (71828) (94515) (104263) (119326)
02 26143 45499 49458 64253 72941 76191(26789a) (46622) (50680) (64329) (73474) (76843)
3201 73228 115845 137445 148166 168342 182542
(73761a) (116121) (137788) (148530) (168698) (182921)
02 41302 53648 65630 72202 81632 88105(41304a) (53757) (65842) (72794) (82033) (88474)
aResults in parentheses are taken from FEM
S-C-S-C boundary conditions are performed in the compar-ison The results adequately demonstrated the great accuracyof the modified Fourier method
We now turn to elastically restrained moderately thickplates The first one involves an S-S-S-S moderately thicksquare plate with a uniform rotational restraint along eachedge that is 119870
1199090= 119870119909119886
= 1198701199100
= 119870119910119887
= Γ119909 The calculated
frequency parameters Ω = 1205961198862
(120588ℎ01198630)12 are given in
Table 10 together with the FEM results Since the elasticallyrestrained plates with nonlinear variation thickness are rarelyinvestigated the FEMresults are used as the referenceA goodagreement is observed between the current and FEM resultsThe second example concerns amoderately thick square plateelastically supported along all edgesThe stiffness of the linear
18 Shock and Vibration
Table 10 Frequency parameterΩ = 1205961198862
(120588ℎ01198630)12 for S-S-S-S moderately thick plates (119886119887 = 1) with linearly varying thickness and elastic
rotation support (ℎ(119909 119910) = ℎ0(1 + 05119909
2
)(1 + 051199102
) 1198701199090= 119870119909119886= 1198701199100= 119870119910119887= Γ119909)
Γ119909
aℎ0119886
Model sequence1 2 3 4 5 6
0
005 23462 54999 55573 82950 100900 101187(23741b) (55114) (55983) (83146) (101004) (101347)
01 18606 42725 42971 60040 71167 71761(18936b) (43014) (43148) (60247) (71642) (71993)
015 53055 65251 85404 106700 114880 143531(53467b) (65472) (85984) (107211) (115687) (144007)
02 17439 21531 32170 44987 59438 64179(17874b) (21748) (32651) (45264) (60188) (64879)
05
005 6904 13374 13514 19400 22904 23056(9909b) (13548) (13651) (19564) (23088) (23379)
01 5316 10237 10314 14499 17155 17264(5374b) (10448) (10651) (14764) (17188) (17879)
015 4136 8201 8220 11474 13578 13671(4174b) (8348) (8651) (11764) (13688) (13879)
02 3449 6915 6924 9549 11225 11327(3474b) (6948) (6951) (9764) (11388) (11579)
1
005 7040 13617 13745 19726 23233 23402(7074b) (13748) (13851) (19964) (23488) (23879)
01 5712 10696 10793 14991 17594 17744(5774b) (10748) (10851) (15054) (17788) (17879)
015 4489 8507 8540 11759 13803 13922(4514b) (8548) (8611) (11864) (13988) (14119)
02 3677 7084 7090 9692 11330 11445(3684b) (7148) (7191) (9764) (11588) (11879)
15
005 7089 13708 13828 19847 23355 23530(7124b) (13748) (13951) (19967) (23588) (23879)
01 5895 10923 11024 15239 17813 17988(5874b) (10748) (11651) (15264) (17818) (17979)
015 4697 8694 8737 11938 13790 14083(4774b) (8748) (8951) (11864) (13688) (14879)
02 3836 7201 7210 9792 11403 11530(3874b) (7748) (7651) (9864) (11588) (11879)
2
005 7114 13754 13871 19911 23418 23597(7174b) (12848) (13851) (19864) (23588) (23879)
01 6001 11060 11159 15391 17944 18136(6004b) (11148) (11151) (15464) (17988) (18379)
015 4836 8821 8872 12061 14037 14189(4874b) (8848) (8951) (12164) (14188) (14879)
02 3954 7289 7301 9871 11457 11594(3974b) (7348) (7651) (9964) (11488) (11679)
aΓ119909= 1198701198941198700 (1198700= 1 times 10
9 Nmrad 119894 = 1199090 119909119886 1199100 119910119887)bResults in parentheses are taken from FEM
and rotational restraints is set equal to 1198961199090
= 119896119909119886
= 1198961199100
=
119896119910119887
= Γ119908and 119870
1199090= 119870119909119886
= 1198701199100
= 119870119910119887
= Γ119909 respectively
The six frequency parametersΩ = 1205961198862
(120588ℎ01198630)12 are shown
in Table 11 for several different restraining coefficient valuesIt can also be noticed that the Fourier series method is able
to predict the modal characteristics of nonlinearly varyingthickness moderately thick plate with not only classicalboundary conditions but also elastic edge restraints correctly
Finally the influence of the gradient on the fundamentalfrequency parameters of a nonlinearly varying thickness
Shock and Vibration 19
Table 11 Frequency parameterΩ = 1205961198862
(120588ℎ01198630)12 for moderately thick plates (119886119887 = 1) with linearly varying thickness and elastic rotation
and translation support (ℎ(119909 119910) = ℎ0(1 + 05119909
2
)(1 + 051199102
) 1198701199090= 119870119909119886= 1198701199100= 119870119910119887= Γ119909 and 119896
1199090= 119896119909119886= 1198961199100= 119896119910119887= Γ119908)
Γ119909
aΓ119908
aℎ0119886
Model sequence1 2 3 4 5 6
05 0
005 12926 13041 25090 47935 47943 61761(12931b) (13053) 25099 (47947) (47952) (61788)
01 10900 10988 21545 39980 40147 50322(10904b) (10992) (21557) (39984) (40166) (50334)
015 8245 8298 18081 31503 32617 40950(8247b) (8230) (18092) (31508) (32622) (40955)
02 6106 6124 15639 25631 27601 34458(6110b) (6125) (15642) (25637) (27617) (34462)
1 05
005 21575 31615 31644 42573 57889 58607(21580b) (31617) (31644) (42581) (57893) (58610)
01 8451 16024 16210 25969 43839 43867(8451b) (16025) (16217) (25972) (43844) (43870)
015 4682 11479 11480 20642 35172 35598(4684b) (11480) (11481) (20644) (35178) (35604)
02 3059 8621 8648 17320 28487 29638(3062b) (8619) (8644) (17327) (28492) (29641)
15 1
005 27713 40203 40978 52951 65352 66815(27714b) (40207) (40988) (52972) (65368) (66822)
01 11667 19322 19615 29122 46115 46183(11674b) (19327) (19619) (29137) (46120) (46188)
015 6544 13416 13481 22405 37238 37430(6547b) (13420) (13488) (22410) (37241) (37434)
02 4295 10175 10190 18543 30330 31056(4296b) (10177) (10194) (18544) (30336) (31060)
2 2
005 33453 50854 52088 66311 76610 78947(33455b) (50852) (50097) (66321) (76627) (78971)
01 15832 24130 24372 33924 49118 49362(15830b) (24127) (24374) (33931) (49147) (49382)
015 9060 15956 16099 24754 39134 39249(9064b) (15961) (16102) (24755) (39138) (39452)
02 5992 11955 11971 20034 31908 32392(5991b) (11957) (11978) (20042) (31910) (32398)
aΓ119909= 1198701198941198700 (1198700= 1 times 10
9 Nmrad 119894 = 1199090 119909119886 1199100 119910119887) Γ119908= 1198961198941198960 (1198960= 1 times 10
9 Nm 119894 = 1199090 119909119886 1199100 119910119887)bResults in parentheses are taken from FEM
moderately thick plate is investigated The plate is elasticallyrestrained in which the stiffness of the boundary springsis taken as 119896
1199090= 119896119909119886
= 1198961199100
= 119896119910119887
= Γ119908
= 2 and1198701199090
= 119870119909119886
= 1198701199100
= 119870119910119887
= Γ119909= 2 respectively The six
frequency parameters Ω = 1205961198862
(120588ℎ01198630)12 are presented in
Table 12 for several different slop values It can be seen thatthe fundamental frequency parameters will decrease with theincrease of the parameter ℎ
0119886
In the above examples it has been demonstrated thatthe presented method can be universally applied to nonlin-early varying thickness moderately thick plates with severalboundary conditions and different structure parametersNew results are obtained for plates with nonlinear variationthickness in both directions subjected to general elastic
boundary restraints which may be used for benchmarkingof researchers in the field In addition it is interesting to seethat the nature frequency decreases with the increase of theindex for thickness function
4 Conclusions
In this paper a modified Fourier method has been pre-sented to study the free vibration behaviors of moderatelythick rectangular plates with variable thickness and arbi-trary boundary conditionsThe first-order shear deformationplate theory is adopted to formulate the theoretical modelThe displacements and rotation components of the plateregardless of boundary conditions are invariantly expressedas the superposition of a 2D Fourier cosine series and four
20 Shock and Vibration
Table 12 Frequency parameterΩ = 1205961198862
(120588ℎ01198630)12 for elastic support moderately thick plates (119886119887 = 1) with different gradient (119896
1199090= 119896119909119886=
1198961199100= 119896119910119887= Γ119908= 2 119870
1199090= 119870119909119886= 1198701199100= 119870119910119887= Γ119909= 2)
120572 120573 ℎ0119886
Model sequence1 2 3 4 5 6
120572 = 120573 = 01
005 30187 51102 51141 67066 73954 77412(30188a) (51105) (51144) (67068) (73961) (77477)
01 16995 24719 24724 33071 44343 44819(16996a) (24725) (24729) (33084) (44347) (44822)
015 10077 15872 15883 23443 35937 36065(10081a) (15877) (15892) (23447) (35941) (36084)
02 6713 11953 11958 19086 30372 30604(6713a) (11957) (11965) (19092) (30378) (30612)
120572 = 120573 = 02
005 31206 51436 51614 67058 74398 77595(31208a) (51438) (51619) (67064) (74402) (77596)
01 16786 24503 24508 33144 45583 45923(16790a) (24510) (24513) (33147) (45587) (45932)
015 9821 15865 15906 23766 36897 36990(9824a) (15870) (15907) (23768) (36894) (36997)
02 6523 11978 11992 19356 30912 31162(6524a) (11981) (11997) (19362) (30916) (31174)
120572 = 120573 = 03
005 32097 51462 51900 66832 74991 77863(32098a) (51466) (51907) (66849) (74996) (77872)
01 16507 24332 24383 33330 46806 47074(16511a) (24327) (24380) (33339) (46810) (47086)
015 9562 15885 15964 24103 37748 37828(9562a) (15892) (15977) (21105) (37751) (37832)
02 6338 11992 12013 19603 31338 31634(6340a) (11994) (12011) (19612) (31344) (31639)
120572 = 120573 = 04
005 32851 51239 52046 66547 75735 78303(33455a) (50852) (50097) (66321) (76627) (78971)
01 16182 24209 24343 33598 47990 48230(33455a) (50852) (50097) (66321) (76627) (78971)
015 9307 15919 16034 24436 38491 38579(33455a) (50852) (50097) (66321) (76627) (78971)
02 6161 11986 12008 19828 31666 32040(33455a) (50852) (50097) (66321) (76627) (78971)
120572 = 120573 = 05
005 33453 50854 52088 66311 76610 78947(33455a) (50852) (50097) (66321) (76627) (78971)
01 15832 24130 24372 33924 49118 49362(15830a) (24127) (24374) (33931) (49147) (49382)
015 9060 15956 16099 24754 39134 39249(9064a) (15961) (16102) (24755) (39138) (39452)
02 5992 11955 11971 20034 31908 32392(5991a) (11957) (11978) (20042) (31910) (32398)
aResults in parentheses are taken from FEM
supplementary functions in the form of the product of apolynomial function and a single cosine series expansionto ensure and accelerate the convergence of the solutionAt each edge of the plate the general restraint conditionsare implemented by introducing one group of linear springsand two groups of rotational springs which are continuouslydistributed and determined by the stiffness of these springs
Instead of seeking a solution in strong forms in the previousstudies all the Fourier coefficients will be treated equallyand independently as the generalized coordinates and solveddirectly from the Rayleigh-Ritz technique The change of theboundary conditions can be easily achieved by only varyingthe stiffness of the three sets of boundary springs along alledges of the rectangular plates without involving any change
Shock and Vibration 21
to the solution procedure The convergence of the presentsolution is examined and the excellent accuracy is validatedby comparison with existing results published in the lit-erature and FEM data Excellent agreements are obtainedfrom these comparisons The proposed method providesa unified means for extracting the modal parameters andpredicting the vibration behaviors of moderately thick plateswith variable thickness variation functions and arbitraryelastic edge restraints A variety of free vibration results formoderately thick rectangular plates with different thicknessvariation functions and boundary conditions are presentedNew results for free vibration of moderately thick rectangularplates with various thickness variation functions and edgeconditions are presented which may be used for benchmark-ing of researchers in the field
Appendices
A Representative Calculation forStiffness and Mass Matrices
To illuminate the particular expression of the mass andstiffness matrixes clearly and tersely four new variables aredefined as 119904 = 119898 lowast (119873 + 1) + 119899 + 1 119905 = 119898
1015840
lowast (119873 + 1) + 1198991015840
+ 1119901 = (119897minus1)lowast (119872+1)+119898
1015840
+1 and 119902 = (119897minus1)lowast (119873+1)+1198991015840
+1The first row elements of K andM are given below
K1-1119904119905
=
1
2
int
119886
0
int
119887
0
[119863 (119909 119910)
times 21205821198861198981205821198861198981015840 sin 120582
119886119898119909 sin 120582
1198861198981015840119909 cos 120582
119887119899119910
times cos 1205821198871198991015840119910 + (1 minus 120583) 120582
1198871198991205821198871198991015840 cos 120582
119886119898119909
times cos 1205821198861198981015840 sin 120582
119887119899119910 sin 120582
1198871198991015840119910
+ 120581119866ℎ (119909 119910) cos 120582119886119898119909 cos 120582
1198861198981015840119909 cos 120582
119887119899
times119910 cos 1205821198871198991015840119910119889119909119889119910] 119889119909 119889119910
+ int
119886
0
1198701199091199100
cos 120582119886119898119909 cos 120582
1198861198981015840119909
+(minus1)119898+1198981015840
119870119909119910119887
cos 120582119886119898119909 cos 120582
1198861198981015840119909 119889119909
+ int
119887
0
1198701199090cos 120582119887119899119910 cos 120582
1198871198991015840119910
+(minus1)119899+1198991015840
119870119909119886cos 120582119887119899119910 cos 120582
1198871198991015840119910 119889119910
K1-2119904119901
=
1
2
int
119886
0
int
119887
0
[119863 (119909 119910)21205821198861198981205821198861198981015840 sin 120582
119886119898119909 sin 120582
1198861198981015840119909
times
2
sum
119897=1
120577119897
119887(119910) cos 120582
1198871198991015840119910
minus (1 minus 120583) 1205821198871198991015840 cos 120582
119886119898119909 cos 120582
1198861198981015840119909
times
2
sum
119897=1
1205771198971015840
119887(119910) sin 120582
1198871198991015840119910 + 120581119866ℎ (119909 119910)
times ( cos 1205821198861198981015840119909 cos 120582
1198871198991015840119910
times
2
sum
119897=1
120577119897
119887(119910) cos 120582
119886119898119909)]119889119909119889119910
+ int
119886
0
1198701199091199100
cos 120582119886119898119909 cos 120582
1198861198981015840119909
2
sum
119897=1
120577119897
119887(0)
+(minus1)1198991015840
119870119909119910119887
cos 120582119886119898119909 cos 120582
1198861198981015840119909
2
sum
119897=1
120577119897
119887(119887) 119889119909
+ int
119887
0
1198701199090
2
sum
119897=1
120577119897
119887(119910) cos 120582
1198871198991015840119910
+(minus1)119898+1198981015840
119870119909119886
2
sum
119897=1
120577119897
119887(119910) cos 120582
1198871198991015840119910119889119910
K1-3119904119902
=
1
2
int
119886
0
int
119887
0
[119863 (119909 119910) minus 21205821198861198981015840 sin 120582
1198861198981015840119909 cos 120582
1198871198991015840119910
times
2
sum
119897=1
1205771198971015840
119886(119909) cos 120582
119887119899119910
+ (1 minus 120583) 1205821198871198991205821198871198991015840 cos 120582
1198861198981015840119909
times sin 1205821198871198991015840119910
2
sum
119897=1
120577119897
119886(119909) sin 120582
119887119899119910
+ 120581119866ℎ (119909 119910) cos 1205821198861198981015840119909 cos 120582
1198871198991015840119910
times
2
sum
119897=1
120577119897
119886(119909) cos 120582
119887119899119910]119889119909119889119910
+ int
119886
0
1198701199091199100
cos 1205821198861198981015840119909
2
sum
119897=1
120577119897
119886(119909)
+(minus1)119899+1198991015840
119870119909119910119887
cos 1205821198861198981015840119909
2
sum
119897=1
120577119897
119886(119909) 119889119909
+ int
119887
0
1198701199090
2
sum
119897=1
120577119897
119886(0) cos 120582
1198871198991015840119910
+(minus1)119898+1198981015840
119870119909119886
2
sum
119897=1
120577119897
119886(0) cos 120582
1198871198991015840119910119889119910
K1-4119904119905
=
1
2
int
119886
0
int
119887
0
119863(119909 119910) 21205821198871198991205821198861198981015840 sin 120582
1198861198981015840119909 cos 120582
1198871198991015840119910
times cos 120582119886119898119909 sin 120582
119887119899119910 minus 2(1 minus 120583)
119887119899
times 1205821198861198981015840 sin 120582
1198861198981015840119909 cos 120582
1198871198991015840119910 cos 120582
119886119898119909
times sin 120582119887119899119910 + (1 minus 120583) 120582
1198861198981205821198871198991015840 cos 120582
1198861198981015840119909
times sin 1205821198871198991015840119910 sin 120582
119886119898119909 cos 120582
119887119899119910 119889119909 119889119910
22 Shock and Vibration
K1-5119904119901
=
1
2
int
119886
0
int
119887
0
119863(119909 119910) minus 21205821198861198981015840 sin 120582
1198861198981015840119909 cos 120582
1198871198991015840119910
times
2
sum
119897=1
1205771198971015840
119887(119910) cos 120582
119886119898119909
+ 2 (1 minus 120583) 1205821198861198981015840 sin 120582
1198861198981015840119909 cos 120582
1198871198991015840119910
times
2
sum
119897=1
1205771198971015840
119887(119910) cos 120582
119886119898119909
+ (1 minus 120583) 1205821198861198981205821198871198991015840 cos 120582
1198861198981015840119909 sin 120582
1198871198991015840119910
times
2
sum
119897=1
120577119897
119887(119910) sin 120582
119886119898119909119889119909119889119910
K1-6119904119902
=
1
2
int
119886
0
int
119887
0
119863(119909 119910)1205821198871198991205821198861198981015840 sin 120582
1198861198981015840119909 cos 120582
1198861198991015840119910
times
2
sum
119897=1
120577119897
119886(119909) sin 120582
1198871198991015840119910
minus 2 (1 minus 120583) 1205821198871198991205821198861198981015840 sin 120582
1198861198981015840119909 cos 120582
1198861198991015840119910
times
2
sum
119897=1
120577119897
119886(119909) sin 120582
1198871198991015840119910
+ 1205821198871198991015840 cos 120582
1198861198981015840119909 sin 120582
1198871198991015840119910
times
2
sum
119897=1
1205771198971015840
119886(119909) cos 120582
119887119899119910119889119909119889119910
K1-7119904119905
= minus
1
2
int
119886
0
int
119887
0
120581119866ℎ (119909 119910) 120582119886119898
cos 1205821198861198981015840119909 cos 120582
1198871198991015840119910
times sin 120582119886119898119909 cos 120582
119887119899119910119889119909119889119910
K1-8119904119901
= minus
1
2
int
119886
0
int
119887
0
120581119866ℎ (119909 119910) 1205821198861198981015840 cos 120582
1198861198981015840119909 cos 120582
1198871198991015840119910
times
2
sum
119897=1
120577119897
119887(119910) sin 120582
1198861198981015840119909 119889119909 119889119910
K1-9119904119902
=
1
2
int
119886
0
int
119887
0
120581119866ℎ (119909 119910) cos 1205821198861198981015840119909 cos 120582
1198871198991015840119910
times
2
sum
119897=1
1205771198971015840
119886(119909) cos 120582
119887119899119910119889119909119889119910
M1-1119904119905
=
1
24
120588int
119886
0
int
119887
0
ℎ3
(119909 119910) cos 120582119886119898119909 cos 120582
119887119899119910
times cos 1205821198861198981015840119909 cos 120582
1198871198991015840119910119889119909119889119910
M1-2119904119901
=
1
24
120588int
119886
0
int
119887
0
ℎ3
(119909 119910)
2
sum
119897=1
120577119897
119887(119910) cos 120582
119886119898119909
times cos 1205821198861198981015840119909 cos 120582
1198871198991015840119910119889119909119889119910
M1-3119904119902
=
1
24
120588int
119886
0
int
119887
0
ℎ3
(119909 119910)
2
sum
119897=1
120577119897
119886(119909) cos 120582
119887119899119910
times cos 1205821198861198981015840119909 cos 120582
1198871198991015840119910119889119909119889119910
M1-4119904119905 = M
1-5119904119901 = M1-6119904119902
= M1-7119904119905 = M
1-8119904119901 = M1-9119904119902 = 0
(A1)
B Useful Integrals for the Moderately ThickRectangular Plates with Variable Thicknessand Arbitrary Boundary Conditions
Consider
int
119871
0
cos 120582119898119909 cos 120582
119899119909 119889119909 =
0 119898 = 119899
119871
2
119898 = 119899
119871 119898 = 119899 = 0
int
119871
0
sin 120582119898119909 cos 120582
119899119909119889119909
=
minus
(minus1 + (minus1)119898+119899
) 119871119898
(1198982minus 1198992) 120587
119898 = 119899
0 119898 = 119899
int
119871
0
sin 120582119898119909 sin 120582
119899119909 119889119909 =
0 119898 = 119899
119871
2
119898 = 119899
0 119898 = 119899 = 0
int
119871
0
cos 120582119898119909120577119897
119886(119909) 119889119909
=
41198712
31205872 119897 = 1 119898 = 0
minus
41198712
(41198982
minus 3)
(9 minus 401198982+ 16119898
4) 1205872
119897 = 1 119898 = 0
minus
41198712
31205872 119897 = 2 119898 = 0
41198712
(41198982
minus 3) (minus1)119898
(9 minus 401198982+ 16119898
4) 1205872
119897 = 2 119898 = 0
Shock and Vibration 23
int
119871
0
sin 120582119898119909120577119897
119886(119909) 119889119909
=
0 119897 = 1 119898 = 0
161198712
119898(minus1)119898
(9 minus 401198982+ 16119898
4) 1205872
119897 = 1 119898 = 0
0 119897 = 2 119898 = 0
161198981198712
(9 minus 401198982+ 16119898
4) 1205872
119897 = 2 119898 = 0
int
119871
0
cos 1205821198981199091205771198971015840
119886(119909) 119889119909
=
0 119897 = 1 119898 = 0
161198711198982
(minus1)119898
(9 minus 401198982+ 16119898
4) 120587
119897 = 1 119898 = 0
0 119897 = 2 119898 = 0
161198711198982
(9 minus 401198982+ 16119898
4) 120587
119897 = 2 119898 = 0
int
119871
0
sin 1205821198981199091205771198971015840
119886(119909) 119889119909
=
0 119897 = 1 119898 = 0
4119871119898 (minus3 + 41198982
)
(9 minus 401198982+ 16119898
4) 120587
119897 = 1 119898 = 0
0 119897 = 2 119898 = 0
119871119898(
1
120587 minus 41198982120587
+
3
9120587 minus 41198982120587
) (minus1)119898
119897 = 2 119898 = 0
int
119871
0
1205771198971
119886(119909) 1205771198972
119886(119909) 119889119909 =
1198713
41205872 1198971= 1 1198972= 1
minus
21198713
31205873 1198971= 1 1198972= 2
minus
21198713
31205873 1198971= 2 1198972= 1
1198713
41205872 1198971= 2 1198972= 2
int
119871
0
1205771198971
119886(119909) 1205771198971015840
2
119886(119909) 119889119909 =
0 1198971= 1 1198972= 1
minus
1198712
8120587
1198971= 1 1198972= 2
1198712
8120587
1198971= 2 1198972= 1
0 1198971= 2 1198972= 2
int
119871
0
1205771198971015840
1
119886(119909) 1205771198971015840
2
119886(119909) 119889119909 =
5119871
16
1198971= 1 1198972= 1
minus
119871
2120587
1198971= 1 1198972= 2
minus
119871
2120587
1198971= 2 1198972= 1
5119871
16
1198971= 2 1198972= 2
int
119871
0
119909119899 cos 120582
119898119909 119889119909
=
1198711+119899
1 + 119899
119898 = 0
(1198861+119899HypergeometricPFQtimes [ 12 + 1198992 12 32 + 1198992
minus (14)1198982
1205872
]) times (1 + 119899)minus1
119898 gt 0
(B1)
Conflict of Interests
The authors declare that there is no conflict of interestsregarding the publication of this paper
Acknowledgments
The authors would like to thank the anonymous reviewersfor their very valuable comments The authors also gratefullyacknowledge the financial support from the National NaturalScience Foundation of China (no 51209052) HeilongjiangProvince Youth Science Fund Project (no QC2011C013) andHarbin Science and Technology Development InnovationFoundation of Youth (no 2011RFQXG021)
References
[1] K Liew Y Xiang and S Kitipornchai ldquoResearch on thick platevibration a literature surveyrdquo Journal of Sound and Vibrationvol 180 no 1 pp 163ndash176 1995
[2] J H Chung T Y Chung and K C Kim ldquoVibration analysisof orthotropic mindlin plates with edges elastically restrainedagainst rotationrdquo Journal of Sound and Vibration vol 163 no 1pp 151ndash163 1993
[3] Y K Cheung and D Zhou ldquoVibrations of moderately thickrectangular plates in terms of a set of static Timoshenko beamfunctionsrdquoComputers and Structures vol 78 no 6 pp 757ndash7682000
[4] CMWang ldquoNatural frequencies formula for simply supportedMindlin platesrdquo Journal of Vibration and Acoustics vol 116 no4 pp 536ndash540 1994
[5] K N Saha R C Kar and P K Datta ldquoFree vibration analysisof rectangular Mindlin plates with elastic restraints uniformlydistributed along the edgesrdquo Journal of Sound andVibration vol192 no 4 pp 885ndash902 1996
24 Shock and Vibration
[6] D J Gorman ldquoAccurate free vibration analysis of shear-deformable plates with torsional elastic edge supportrdquo Journalof Sound and Vibration vol 203 no 2 pp 209ndash218 1997
[7] D J Gorman ldquoFree vibration analysis of Mindlin plates withuniform elastic edge support by the superposition methodrdquoJournal of Sound andVibration vol 207 no 3 pp 335ndash350 1997
[8] D J Gorman ldquoAccurate free vibration analysis of point sup-ported mindlin plates by the superposition methodrdquo Journal ofSound and Vibration vol 219 no 2 pp 265ndash277 1999
[9] Y Xiang ldquoVibration of rectangular Mindlin plates resting onnon-homogenous elastic foundationsrdquo International Journal ofMechanical Sciences vol 45 no 6-7 pp 1229ndash1244 2003
[10] Y Xiang and G W Wei ldquoExact solutions for buckling andvibration of stepped rectangular Mindlin platesrdquo InternationalJournal of Solids and Structures vol 41 no 1 pp 279ndash294 2004
[11] Y L Yeh M J Jang and C C Wang ldquoAnalyzing the freevibrations of a plate using finite difference and differential trans-formationmethodrdquoAppliedMathematics andComputation vol178 no 2 pp 493ndash501 2006
[12] Y Xiang S K Lai L Zhou and C W Lim ldquoDSC-Ritz elementmethod for vibration analysis of rectangular Mindlin plateswith mixed edge supportsrdquo European Journal of MechanicsmdashASolids vol 29 no 4 pp 619ndash628 2010
[13] H Nguyen-Xuan C H Thai and T Nguyen-Thoi ldquoIsogeo-metric finite element analysis of composite sandwich platesusing a higher order shear deformation theoryrdquo Composites BEngineering vol 55 pp 558ndash574 2013
[14] C H Thai A J M Ferreira E Carrera and H Nguyen-XuanldquoIsogeometric analysis of laminated composite and sandwichplates using a layerwise deformation theoryrdquo Composite Struc-tures vol 104 pp 196ndash214 2013
[15] C H Thai A Ferreira S Bordas T Rabczuk and HNguyen-Xuan ldquoIsogeometric analysis of laminated compositeand sandwich plates using a new inverse trigonometric sheardeformation theoryrdquo European Journal of Mechanics A Solidsvol 43 pp 89ndash108 2014
[16] H Luong-Van T Nguyen-Thoi G R Liu and P Phung-VanldquoA cell-based smoothed finite elementmethod using three-nodeshear-locking free Mindlin plate element (CS-FEM-MIN3) fordynamic response of laminated composite plates on viscoelasticfoundationrdquo Engineering Analysis with Boundary Elements vol42 pp 8ndash19 2014
[17] T Nguyen-Thoi T Rabczuk T Lam-Phat V Ho-Huu andP Phung-Van ldquoFree vibration analysis of cracked Mindlinplate using an extended cell-based smoothed discrete sheargap method (XCS-DSG3)rdquo Theoretical and Applied FractureMechanics 2014
[18] H Nguyen-Xuan G R Liu and C a Thai-Hoang ldquoAnedge-based smoothed finite element method (ES-FEM) withstabilized discrete shear gap technique for analysis of Reissner-Mindlin platesrdquo Computer Methods in Applied Mechanics andEngineering vol 199 no 9ndash12 pp 471ndash489 2010
[19] P Phung-Van T Nguyen-Thoi T Bui-Xuan and Q Lieu-XuanldquoA cell-based smoothed three-node Mindlin plate element(CS-FEM-MIN3) based on the 119862
0 -type higher-order sheardeformation for geometrically nonlinear analysis of laminatedcomposite platesrdquo Computational Materials Science 2014
[20] NNguyen-Thanh T RabczukHNguyen-Xuan and S BordasldquoAn alternative alpha finite element method with discrete sheargap technique for analysis of isotropicMindlin-Reissner platesrdquoFinite Elements inAnalysis andDesign vol 47 no 5 pp 519ndash5352011
[21] KM Liew Y Xiang and S Kitipornchai ldquoTransverse vibrationof thick rectangular plates-I Comprehensive sets of boundaryconditionsrdquo Computers and Structures vol 49 no 1 pp 1ndash291993
[22] D Zhou ldquoVibrations of Mindlin rectangular plates with elasti-cally restrained edges using static Timoshenko beam functionswith the Rayleigh-Ritz methodrdquo International Journal of Solidsand Structures vol 38 no 32-33 pp 5565ndash5580 2001
[23] D Zhou SH Lo F T KAu andYK Cheung ldquoVibration anal-ysis of rectangular Mindlin plates with internal line supportsusing static Timoshenko beam functionsrdquo International Journalof Mechanical Sciences vol 44 no 12 pp 2503ndash2522 2002
[24] H S Shen J Yang and L Zhang ldquoFree and forced vibrationof Reissner-Mindlin plates with free edges resting on elasticfoundationsrdquo Journal of Sound and Vibration vol 244 no 2 pp299ndash320 2001
[25] Y Xing and B Liu ldquoClosed form solutions for free vibrations ofrectangular Mindlin platesrdquo Acta Mechanica Sinica vol 25 no5 pp 689ndash698 2009
[26] P Gagnon C Gosselin and L Cloutier ldquoA finite strip elementfor the analysis of variable thickness rectangular thick platesrdquoComputers amp Structures vol 63 no 2 pp 349ndash362 1997
[27] T Mlzusawa ldquoVibration of rectangular mindlin plates withtapered thickness by the spline strip methodrdquo Computers andStructures vol 46 no 3 pp 451ndash463 1993
[28] H Nguyen-Xuan and T Nguyen-Thoi ldquoA stabilized smoothedfinite element method for free vibration analysis of Mindlin-Reissner platesrdquo Communications in Numerical Methods inEngineering with Biomedical Applications vol 25 no 8 pp 882ndash906 2009
[29] K M Liew L X Peng and S Kitipornchai ldquoVibration anal-ysis of corrugated Reissner-Mindlin plates using a mesh-freeGalerkin methodrdquo International Journal of Mechanical Sciencesvol 51 no 9-10 pp 642ndash652 2009
[30] Y Hou GWWei and Y Xiang ldquoDSC-Ritz method for the freevibration analysis of Mindlin platesrdquo International Journal forNumerical Methods in Engineering vol 62 no 2 pp 262ndash2882005
[31] F- Liu and K M Liew ldquoVibration analysis of discontinuousMindlin plates by differential quadrature element methodrdquoJournal of Vibration and Acoustics Transactions of the ASMEvol 121 no 2 pp 204ndash208 1999
[32] P Malekzadeh G Karami and M Farid ldquoA semi-analyticalDQEM for free vibration analysis of thick plates with twoopposite edges simply supportedrdquoComputerMethods inAppliedMechanics and Engineering vol 193 no 45ndash47 pp 4781ndash47962004
[33] P Malekzadeh and S A Shahpari ldquoFree vibration analysisof variable thickness thin and moderately thick plates withelastically restrained edges by DQMrdquo Thin-Walled Structuresvol 43 no 7 pp 1037ndash1050 2005
[34] R E Diaz-Contreras and S Nomura ldquoGreenrsquos function appliedto solution ofMindlin platesrdquoComputers and Structures vol 60no 1 pp 41ndash48 1996
[35] T Sakiyama and M Huang ldquoFree vibration analysis of rect-angular plates with variable thicknessrdquo Journal of Sound andVibration vol 216 no 3 pp 379ndash397 1998
[36] M Huang X Q Ma T Sakiyama H Matuda and C MoritaldquoFree vibration analysis of orthotropic rectangular plates withvariable thickness and general boundary conditionsrdquo Journal ofSound and Vibration vol 288 no 4-5 pp 931ndash955 2005
Shock and Vibration 25
[37] I Shufrin and M Eisenberger ldquoVibration of shear deformableplates with variable thicknessmdashfirst-order and higher-orderanalysesrdquo Journal of Sound and Vibration vol 290 no 1-2 pp465ndash489 2006
[38] S A Eftekhari and A A Jafari ldquoAccurate variational approachfor free vibration of variable thickness thin and thick plates withedges elastically restrained against translation and rotationrdquoInternational Journal of Mechanical Sciences vol 68 pp 35ndash462013
[39] W L Li X Zhang J Du and Z Liu ldquoAn exact series solutionfor the transverse vibration of rectangular plates with generalelastic boundary supportsrdquo Journal of Sound amp Vibration vol321 no 1-2 pp 254ndash269 2009
[40] X Shi D Shi W L Li and Q Wang ldquoA unified method forfree vibration analysis of circular annular and sector plateswith arbitrary boundary conditionsrdquo Journal of Vibration andControl 2014
[41] D Y Shi X J Shi W L Li and Q S Wang ldquoFree transversevibrations of orthotropic thin rectangular plates with arbitraryelastic edge supportsrdquo Journal of Vibroengineering vol 16 no 1pp 389ndash398 2014
[42] Y Chen G Jin and Z Liu ldquoFree vibration analysis of circu-lar cylindrical shell with non-uniform elastic boundary con-straintsrdquo International Journal of Mechanical Sciences vol 74pp 120ndash132 2013
[43] L Dai T Yang J Du W L Li and M J Brennan ldquoAn exactseries solution for the vibration analysis of cylindrical shellswith arbitrary boundary conditionsrdquo Applied Acoustics vol 74no 3 pp 440ndash449 2013
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Shock and Vibration
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International Journal of
Shock and Vibration 3
x
z
y
a
b
h(x y)
Figure 1 The general elastic boundary conditions of moderatelythick rectangular plates with varying thickness in two directions
and the coordinate of the moderately thick rectangularplate with elastically retrained edges is depicted in Figure 1Three groups of boundary restraining springs (translationrotational and torsional springs) are arranged at all sidesof the plate to separately simulate the boundary force Byassigning the stiffness of the boundary springs various valuesit is equivalent to impose different boundary force on themidsurface of the plate For example the clamped boundarycondition can be readily obtained by setting the spring coef-ficients into infinity (a very large number in practical calcu-lation) for the translation rotations and torsional restrainingsprings along each edge
Based on the Mindlin plate theory the displacementsvectors with three directions are
119906 (119909 119910 119911 119905) = 119911120595119909(119909 119910 119905)
V (119909 119910 119911 119905) = 119911120595119910(119909 119910 119905)
119908 (119909 119910 119911 119905) = 119908 (119909 119910 119905)
(1)
where 119906 V and 119908 represent the 119909 119910 and 119911 directiondisplacement functions and the 120595
119909and 120595
119910are the slop due
to bending along in the respective planes The relationship 119908with the slops 120595
119909and 120595
119910is 120595119909= minus119889119908119889119909 and 120595
119910= minus119889119908119889119910
For the moderately thick rectangular plates making useof the strain-stress relationship defined in elasticity theorythe normal shear strains and transverse shear strains can beexpressed as follows
120576119909119909
120576119910119910
120574119909119910
120574119909119911
120574119910119911
=
119911120597120595119909
120597119909
119911120597120595119910
120597119910
119911(
120597120595119909
120597119910
+
120597120595119910
120597119909
)
120595119909+
120597119908
120597119909
120595119910+
120597119908
120597119910
[
[
[
[
[
[
[
[
120590119909119909
120590119910119910
120591119909119910
120591119909119911
120591119910119911
]
]
]
]
]
]
]
]
=
119864
2 (1 minus 1205832)
[
[
[
[
[
[
[
[
2 2120583 0 0 0
2120583 2 0 0 0
0 0 1 minus 120583 0 0
0 0 0 120581 (1 minus 120583) 0
0 0 0 0 120581 (1 minus 120583)
]
]
]
]
]
]
]
]
times
[
[
[
[
[
[
[
[
120576119909119909
120576119910119910
120574119909119910
120574119909119911
120574119910119911
]
]
]
]
]
]
]
]
(2)
where 120576119909119909 120576119910119910 and 120574
119909119910are the normal and shear strains in the
119909119910 and 119911 coordinate systemThe transverse shear strains 120574119909119911
and 120574119910119911
are constant through the thickness The 120590119909119909
and 120590119910119910
are the normal stresses in the 119909 119910 directions 120591119909119911 120591119910119911 and 120591
119909119910
are shear stresses in the 119909 119910 and 119911 coordinate system The 119864is Youngrsquos modulus 120583 is Poissonrsquos ration and 120581 is the shearcorrection factor to account for the fact
In terms of transverse displacements and slope thebending and twisting moments and the transverse shearingforces in plates can be expressed as
119872119909119909= int
ℎ2
minusℎ2
120590119909119909119911 119889119911 = 119863(
120597120595119909
120597119909
+ 120583
120597120595119910
120597119910
)
119872119910119910= int
ℎ2
minusℎ2
120590119910119910119911 119889119911 = 119863(
120597120595119910
120597119910
+ 120583
120597120595119909
120597119909
)
119872119909119910= int
ℎ2
minusℎ2
120591119909119910119911 119889119911 =
119863 (1 minus 120583)
2
(
120597120595119909
120597119910
+
120597120595119910
120597119909
)
119876119909= 120581int
ℎ2
minusℎ2
120591119909119911119889119911 = 120581119866ℎ (119909 119910) (120595
119909+
120597119908
120597119909
)
119876119910= 120581int
ℎ2
minusℎ2
120591119910119911119889119911 = 120581119866ℎ (119909 119910) (120595
119910+
120597119908
120597119910
)
(3)
where ℎ(119909 119910) is the thickness function 119863 = 119864ℎ(119909 119910)3
(12(1 minus 1205832
)) is the flexural rigidity and 119866 = 119864(2(1 + 120583))
is the shear modulusThe boundary conditions for a general supported moder-
ately thick rectangular plate can be expressed as the followingforms based on the force equilibrium relationship on the foursides
on 119909 = 0
1198961199090119908 = minus119876
119909 1198701199090120595119909= minus119872
119909119909 1198701199101199090
120595119910= minus119872
119909119910
on 119909 = 119886
119896119909119886119908 = 119876
119909 119870119909119886120595119909= 119872119909119909 119870119910119909119886
120595119910= 119872119909119910
4 Shock and Vibration
on 119910 = 0
1198961199100119908 = minus119876
119910 1198701199100120595119910= minus119872
119910119910 1198701199091199100
120595119909= minus119872
119909119910
on 119910 = 119887
119896119910119887119908 = 119876
119910 119870119910119887120595119910= 119872119910119910 119870119909119910119887
120595119909= 119872119909119910
(4)
where 1198961199090
and 119896119909119886
(1198961199100
and 119896119910119887) are linear spring constants
1198701199090
and 119870119909119886
(1198701199100
and 119870119910119887) are the rotational spring con-
stants and 1198701199101199090
and 119870119910119909119886
(1198701199091199100
and 119870119909119910119887
) are the torsionalspring constants at 119909 = 0 and 119886 (119910 = 0 and 119887) respectivelyTherefore arbitrary boundary conditions of the plate can begenerated by assigning the linear springs rotational springsand torsional springs at proper stiffness For instance aclamped boundary (C) is achieved by simply setting thestiffness of the entire springs equal to infinity (which is repre-sented by a very large number 1014) Inversely a free bound-ary (F) is gained by setting the stiffness of the entire springsequal to zero
Thus the total potential energy of the spring restrainedplate which is composed of two parts namely the strainenergy of the moderately thick rectangular plates and thepotential energy stored in the boundary springs can beexpressed as
119880 =
1
2
int
119886
0
int
119887
0
119863[(
120597120595119909
120597119909
+
120597120595119910
120597119910
)
2
minus 2 (1 minus 120583)
times (
120597120595119909
120597119909
120597120595119910
120597119910
minus
1
4
(
120597120595119909
120597119910
+
120597120595119910
120597119909
)
2
)]
+ 120581119866ℎ (119909 119910)
times [(120595119909+
120597119908
120597119909
)
2
+ (120595119910+
120597119908
120597119910
)
2
] 119889119909119889119910
+
1
2
int
119886
0
[(11989611991001199082
+ 11987011991001205952
119910+ 1198701199091199100
1205952
119909)
10038161003816100381610038161003816119910=0
+ (1198961199101198871199082
+ 1198701199101198871205952
119910+ 119870119909119910119887
1205952
119909)
10038161003816100381610038161003816119910=119887
] 119889119909
+
1
2
int
119887
0
[(11989611990901199082
+ 11987011990901205952
119909+ 1198701199101199090
1205952
119910)
10038161003816100381610038161003816119909=0
+ (1198961199091198861199082
+ 1198701199091198861205952
119909+ 119870119910119909119886
1205952
119910)
10038161003816100381610038161003816119909=119886
] 119889119910
(5)
As the springs are considered with no mass while retain-ing certain stiffness the total kinetic energy of themoderatelythick rectangular plates is
119879 =
1205881205962
2
int
119887
0
int
119886
0
[
[
ℎ (119909 119910)1199082
+ ℎ3
(119909 119910) (1205952
119909+ 1205952
119910)
12
]
]
119889119909119889119910
(6)
where 120588 is the mass density and 120596 denotes the natural fre-quency of the plate
In view of satisfying arbitrarily supported boundary con-ditions of the moderately thick rectangular plate the admis-sible functions expressed in the form of the improved Fourierseries are introduced to remove the potential discontinuitieswith the functions and their derivatives Thus the moder-ately thick rectangular plate displacements and rotation areexpressed as
120595119909(119909 119910) =
infin
sum
119898=0
infin
sum
119899=0
119860119898119899
cos 120582119886119898119909 cos 120582
119887119899119910
+
2
sum
119897=1
120577119897
119887(119910)
infin
sum
119898=0
119886119897
119898cos 120582119886119898119909
+
2
sum
119897=1
120577119897
119886(119909)
infin
sum
119899=0
119887119897
119899cos 120582119887119899119910
(7)
120595119910(119909 119910) =
infin
sum
119898=0
infin
sum
119899=0
119861119898119899
cos 120582119886119898119909 cos 120582
119887119899119910
+
2
sum
119897=1
120577119897
119887(119910)
infin
sum
119898=0
119888119897
119898cos 120582119886119898119909
+
2
sum
119897=1
120577119897
119886(119909)
infin
sum
119899=0
119889119897
119899cos 120582119887119899119910
(8)
119908 (119909 119910) =
infin
sum
119898=0
infin
sum
119899=0
119862119898119899
cos 120582119886119898119909 cos 120582
119887119899119910
+
2
sum
119897=1
120577119897
119887(119910)
infin
sum
119898=0
119890119897
119898cos 120582119886119898119909
+
2
sum
119897=1
120577119897
119886(119909)
infin
sum
119899=0
119891119897
119899cos 120582119887119899119910
(9)
where 120582119886119898
= 119898120587119886 120582119887119899= 119899120587119887 and 119860
119898119899 119861119898119899 and 119862
119898119899are
the Fourier coefficients of two-dimensional Fourier seriesexpansions for the displacements and rotation functionsrespectively 119886119897
119898 119887119897119898 119888119897119898 119889119897119898 119890119897119898 and 119891119897
119898are the supplemented
coefficients of the auxiliary functions where 119897 = 1 2 Thespecific expressions of the auxiliary functions 120577119897
119886and 120577
119897
119887are
defined as
1205771
119886(119909) =
119886
2120587
sin(1205871199092119886
) +
119886
2120587
sin(31205871199092119886
)
1205772
119886(119909) = minus
119886
2120587
cos(1205871199092119886
) +
119886
2120587
cos(31205871199092119886
)
1205771
119887(119910) =
119887
2120587
sin(120587119910
2119887
) +
119887
2120587
sin(3120587119910
2119887
)
1205772
119887(119910) = minus
119887
2120587
cos(120587119910
2119887
) +
119887
2120587
cos(3120587119910
2119887
)
(10)
As shown in (7)ndash(9) the supplementary functions 1205771119886(119909)
1205772
119886(119909) 1205771
119887(119910) and 120577
2
119887(119910) are used for the displacement and
rotation expressions The theoretical meaning of introduc-ing these terms into the Fourier series is to remove the
Shock and Vibration 5
potential discontinuities and their derivatives throughout thewhole plate structure including the boundaries and then toeffectively enhance the convergence of the results To ensurethis continuity of selection expressions and correspondingderivatives at any point on the plate the first-order derivativesof the 119909 and119910 directions should exist as indicated by (4)Thisrequirement is guaranteed by the selected supplementaryfunctions because it is easy to verify that
1205771
119886(0) = 120577
1
119886(119886) = 120577
11015840
119886(119886) = 0 120577
11015840
119886(0) = 1
1205772
119886(0) = 120577
2
119886(119886) = 120577
21015840
119886(0) = 0 120577
21015840
119886(119886) = 1
(11)
Similar conditions exist for the 119910-related polynomials1205771
119887(119910) and 120577
2
119887(119910) It has to be mentioned that although the
solution is theoretically exact for the superposition of infinitenumbers of Fourier terms in actual calculation we truncatethe infinite series to 119872 and 119873 to obtain the results withacceptable accuracy
Since the energy expressions and admissible function ofthe plate have been established the remaining task is to deter-mine the Fourier expanded coefficients and supplemented
coefficients in (7)ndash(9) The Lagrangian energy functional (119871)of the plate is written as
119871 = 119879 minus 119880 (12)
Then the Lagrangian expression is minimized by takingits derivatives with respect to these coefficients
120597119871
120597120599
= 0 120599 =
119860119898119899
119886119897
119898119887119897
119899
119861119898119899
119888119897
119898119889119897
119899
119862119898119899
119890119897
119898119891119897
119899
(13)
Since the displacements and rotation components of theplate are chosen as 119872 and 119873 to obtain the results withacceptable accuracy a total of 3 lowast (119872 + 1) lowast (119873 + 1) + 6 lowast
(119872 +119873 + 2) equations are obtainedThey can be summed up in a matrix form
(K minus 1205962M)E = 0 (14)
The unknown coefficients in the displacement expres-sions can be expressed in the vector form as E where
E =
11986000 11986001 119860
11989810158400 11986011989810158401 119860
11989810158401198991015840 119860
119872119873 1198861
0 119886
1
119872 1198862
0 119886
2
119872 1198871
0 119887
1
119873 1198872
0 119887
2
119873
11986100 11986101 119861
11989810158400 11986111989810158401 119861
11989810158401198991015840 119861
119872119873 1198881
0 119888
1
119872 1198882
0 119888
2
119872 1198891
0 119889
1
119873 1198892
0 119889
2
119873
11986200 11986201 119862
11989810158400 11986211989810158401 119862
11989810158401198991015840 119862
119872119873 1198901
0 119890
1
119872 1198902
0 119890
2
119872 1198911
0 119891
1
119873 1198912
0 119891
2
119873
119879
(15)
In (14)K is the stiffness matrix for the plate andM is themass matrix They can be expressed separately as
K =
[
[
[
[
[
[
[
[
[
K1-1 K
1-2 K1-3 sdot sdot sdot K
1-9
K1198791-2 K
2-1 K2-3 sdot sdot sdot K
2-9
K1198791-3 K119879
2-3 K3-3 sdot sdot sdot K
3-9
d
K1198791-9 K119879
2-9 K1198793-9 sdot sdot sdot K
9-9
]
]
]
]
]
]
]
]
]
M =
[
[
[
[
[
[
[
[
[
M1-1 M
1-2 M1-3 sdot sdot sdot M
1-9
M1198791-2 M
2-1 M2-3 sdot sdot sdot M
2-9
M1198791-3 M119879
2-3 M3-3 sdot sdot sdot M
3-9
d
M1198791-9 M119879
2-9 M1198793-9 sdot sdot sdot M
9-9
]
]
]
]
]
]
]
]
]
(16)
The specific expressions for the elements in (16) aregiven in Appendix AMoreover all the necessary expressionswhich will be used in the calculations of the eigenvalues andeigenvectors are given in Appendix B
Obviously the natural frequencies and eigenvectors cannow be readily obtained by solving a standard matrix eigen-problem Since the components of each eigenvector are actu-ally the expansion coefficients of the modified Fourier seriesthe corresponding mode shape can be directly determinedfrom (14) In other words once the coefficient eigenvectorE is
determined for a given frequency the displacement functionsof the plate can be determined by substituting the coefficientsinto (9) When the forced vibration is involved by addingthe work done by external force in the Lagrangian energyfunction and summing the loading vector F on the right sideof (14) the characteristic equation for the forced vibration ofthe moderately thick rectangular plates is readily obtained
3 Numerical Examples and Discussion
In this section a systematic comparison between the cur-rent solutions and theoretical results published by otherresearchers or finite element method (FEM) results is car-ried out to validate the excellent accuracy reliability andfeasibility of the modified Fourier method A comprehensivestudy on the effects of elastic restraint parameters andvarying thickness in two directions is also reported Newresults are obtained for plates subjected to general elasticboundary restraints with nonlinear variable thickness in bothdirections The discussion is arranged as follows Firstly theconvergence of the modified Fourier solution is checked Inaddition the influence of the stiffness of boundary springcomponents is studied Secondly the uniform thicknessplates with various combinations of classical boundary con-ditions elastic boundary conditions and different structureparameters are examined Thirdly the nonuniform thicknessplate with linear variation in both directions various com-binations of classical boundary conditions conditions and
6 Shock and Vibration
Table 1 Frequency parameters Ω = 1205961198872
(120588ℎ119863)12
1205872 for square moderately thick plates with uniform thickness and different boundary
conditions
Boundary conditions M N Model sequence1 2 3 4 5 6 7 8
C-C-C-C
119872 = 119873 = 2 3375 7191 7191 9856 1605 1628 1747 1747119872 = 119873 = 4 3299 6308 6308 8853 1046 1056 1263 1263119872 = 119873 = 6 3294 6282 6282 8801 1037 1047 1254 1254119872 = 119873 = 8 3292 6278 6278 8797 1036 1046 1253 1253119872 = 119873 = 10 3292 6276 6276 8794 1036 1046 1253 1253119872 = 119873 = 12 3292 6276 6276 8793 1036 1046 1252 1252119872 = 119873 = 14 3292 6276 6276 8793 1036 1046 1252 1252119872 = 119873 = 16 3292 6275 6275 8792 1036 1045 1252 1252Reference [4] 3292 6276 6276 8792 1036 1046 1252 1252
E-E-E-E
119872 = 119873 = 2 06153 10861 10861 17898 27709 28417 35010 35010119872 = 119873 = 4 06150 10799 10799 17785 27695 28405 34914 34914119872 = 119873 = 6 06149 10796 10796 17773 27694 28403 34898 34898119872 = 119873 = 8 06149 10796 10796 17771 27693 28403 34893 34893119872 = 119873 = 10 06149 10796 10796 17770 27693 28403 34892 34892119872 = 119873 = 12 06149 10796 10796 17770 27693 28403 34892 34892119872 = 119873 = 14 06149 10796 10796 17770 27693 28403 34892 34892119872 = 119873 = 16 06149 10796 10796 17770 27693 28403 34891 34891Reference [33] 06153 10861 10861 17898 27709 28417 35010 35010
FEM 06150 17097 17097 17774 27694 28406 34898 34898
different structure parameters are examined Then numer-ical solutions for moderately thick rectangular plates withelastic boundary conditions are presented The effects of theelastic restraint parameters are also investigated Finally thenonuniform thickness plate with nonlinear variation in bothdirections and arbitrary boundary conditions is also studied
31 Convergence and Stiffness Value Study Since the Fourierseries is numerically truncated and only finite terms arecounted in actual calculations the proposed solution shouldbe understood as a solution with arbitrary precision Inthis subsection a uniform thickness square moderately thickrectangular plate with completely clamped boundary condi-tion (C-C-C-C) and four edges equally elastically restrainedagainst linear spring constants and rotation spring constantssupports (E-E-E-E 119870
119879= 119896119894(1198873
119863) 119870119877
= 119870119894(119887119863)
119870119879
= 10 and 119870119877
= 5) has been selected to demon-strate the convergence and accuracy of the modified Fouriermethod In Table 1 the first eight frequency parametersΩ = 120596119887
2
(120588ℎ119863)12
1205872 for the considered uniform thickness
square moderately thick rectangular plate with C-C-C-C andE-E-E-E boundary conditions are examinedThe table showsthat the proposedmethod has fast convergence behaviorThemaximum discrepancy in the worst case between the 6 times 6
truncated configuration and the 8times8 one is less than 0064In order to fully illustrate the convergence of the presentmethod Figures 3 4 and 5 present the 1st and 8th frequencyparameters Ω = 120596119887
2
(120588ℎ119863)12
1205872 with various truncated
numbers 119872 = 119873 subjected to different boundary conditionand aspect ratios A highly desired convergence characteristicis observed such that (a) sufficiently accurate results can beobtained with only a small number of terms in the seriesexpansions and (b) the solution is consistently refined asmore
terms are included in the expansions However this shouldnot constitute a problem in practice because one can alwaysverify the accuracy of the solution by increasing the trunca-tion number until a desired numerical precision is achievedAs a matter of fact this ldquoquality controlrdquo scheme can be easilyimplemented automatically In modal analysis the naturalfrequencies for higher-order modes tend to converge slower(see Table 1) Thus an adequate truncation number shouldbe dictated by the desired accuracy of the largest naturalfrequencies of interest In view of the excellent numericalbehavior of the current solution the truncation numbers willbe simply set as119872 = 119873 = 12 in the following calculations
As far as the accuracy of the present method is con-cerned the converged solutions of the present method arein excellent agreement with both the results reported byreference data and the finite element results For C-C-C-C boundary conditions the max discrepancy between thepresent results and the reference data does not exceed 0011for the worst case and in most cases is 0 Comparing theresults with exact solutions [4] it is observed that eight termsare sufficient to obtain accurate resultsMoreover with regardto the E-E-E-E boundary condition the max discrepancybetween the present results and the reference data does notexceed 031 for the worst case and in most cases is 01Regarding the results with DQM solutions [33] it can be seenthat the six terms are sufficient to obtain enough accurateresults In addition it is clear that the results of the presentapproach with just 663 DOFs (119872 times 119873 = 12 times 12) canpredict the vibration characteristics accurately Most of themare identical to those obtained from finite element method(FEM) with 10201 DOFs (S4R 001m times 001m) That isto say it needs only 662 DOFs compared with FEM toobtain the same precision solutions for the considered case
Shock and Vibration 7
1
2
3
4
5
6
The 1st orderThe 3rd order
101 103 105 107 109 1011 1013
Ω
Ki (Nm)
(a)
2
3
4
5
6
101 103 105 107 109 1011 1013
The 1st orderThe 3rd order
Ω
Ki (Nmrad)
(b)
3
4
5
6
101 103 105 107 109 1011 1013
Kij (Nmrad)
The 1st orderThe 3rd order
Ω
(c)
Figure 2 The effect of boundary spring stiffness on the natural frequencies Ω (a) translation spring (b) rotation spring and (c) torsionalspring
On the same hardware (Intel i7-39GHz) the computing timeof the present formulation for the solution (119872times119873 = 12times12)implemented in optimized MATLAB scripts is about 2125 swhereas the finite element solution consumes 34578 s that isat least 16 times more CPU time than the present method forthe same problem
As mentioned earlier in the current modeling frame-work all the classical boundary conditions and their com-binations can be conveniently viewed as special cases whenthe stiffness for the normal and tangential boundary springsbecomes zero andor infinitely large Thus the effects of thestiffness of the translation (119896
119894) rotation (119870
119894) and torsional
springs (119870119894119895) on the modal characteristics should be inves-
tigated As shown in the Figures 2(a)ndash2(c) the first and thethird frequency parameters are separately obtained by vary-ing the stiffness of one group of the boundary springs from
extremely large (1014) to extremely small (100) while assigningthe other group of the springs infinite stiffness (1014) It canbe found in Figure 2(a) that the frequency parameter almostkeeps at a level when the stiffness of the translation springs islarger than 1012 or smaller than 107 In Figure 2(b) the influ-ences of the rotation springs on frequency parameters aregiven It is shown that the frequency curves change greatlywithin the stiffness range (106 to 1010) while out of this rangethe frequency curves separately keep level In Figure 2(c) theinfluences of the torsional springs on frequency parametersare given It is shown that the frequency curves almost changewhen the stiffness changes in the whole range
Based on the analysis it can be found that the torsionalsprings almost have no effect on the structure Also the rela-tionship between the rotation springs and twisting momentcan be seen from the boundary condition expression Then
8 Shock and Vibration
32
36
40
44
48
52
2
3
4
5
6
7
8
Ω Ω
0 2 4 6 8 10 12 14 16 18M = N
ab = 1
ab = 32
(a) The 1st order
0 2 4 6 8 10 12 14 16 180
20
40
60
80
0
50
100
150
200
250
300
M = N
Ω Ω
ab = 1
ab = 32
(b) The 8th order
Figure 3 The effect of numerically truncated numbers119872 = 119873 on the natural frequenciesΩ for C-C-C-C boundary condition
0 2 4 6 8 10 12 14 16 18125
130
135
140
145
150
088
090
092
094
096
098
Ω Ω
M = N
ab = 1
ab = 32
(a) The 1st order
0 2 4 6 8 10 12 14 16 18
55
60
65
70
75
80
40
45
50
55
60
65
70
75
80Ω Ω
M = N
ab = 1
ab = 32
(b) The 8th order
Figure 4 The effect of numerically truncated numbers119872 = 119873 on the natural frequenciesΩ for F-F-F-F boundary condition
the twisting moments have small effect on the vibration char-acteristics of the structure However in this paper in order toget a more accurate prediction of the vibration characteristicsof the structure the twisting moment on boundary edges istaken into account In the latter study in addition to the factthat free boundary is not unexpected considering the torsionspring the other boundary conditions are introduced into atorsion spring and the spring stiffness takes infinity FormFigure 2 analysis it also concluded that the translation springhas a wider influence range than the rotation spring on thefrequency parameters that is for the translation springs thestable frequency parameters appearwhen the stiffness ismorethan 1012 or less 107 while for the rotation springs when thestiffness value is assigned more than 1010 the frequency para-meters become smoothThus it is suitable to use 1014 to sim-ulate the infinite stiffness value in the model validation partsand in the following examples
32 Uniform Thickness Moderately Thick Plates with Classicaland Elastic Boundary Conditions In this subsection themodified Fourier solution is applied to deal with vibrationproblems of uniform thickness moderately thick rectangu-lar plates subject to the classical boundary conditions andarbitrary elastic boundary conditions In present work threegroups of continuously distributed boundary springs areintroduced to simulate the given or typical boundary condi-tions As mentioned earlier the stiffness of these boundarysprings can take any value from zero to infinity to bettermodel many real-world restraint conditions Taking edge 119909 =0 for example the corresponding spring stiffness for the threetypes of classical boundaries and elastic boundaries is
completely free 119876119909= 0119872
119909119909= 0 and119872
119909119910= 0
1198961199090= 0 119870
1199090= 0 119870
1199101199090= 0 (17a)
Shock and Vibration 9
0 2 4 6 8 10 12 14 16 18
166
168
170
172
152
156
160
164
168
Ω Ω
ab = 1
ab = 32
M = N
(a) The 1st order
0 2 4 6 8 10 12 14 16 1850
55
60
65
70
4
5
6
7
8
9
10
Ω Ω
M = N
ab = 1
ab = 32
(b) The 8th order
Figure 5 The effect of numerically truncated numbers 119872 = 119873 on the natural frequencies Ω with four edges elastically restrained againsttranslation and rotation (119870
119879= 119896119894(1198873
119863) 119870119877= 119870119894(119887119863) 119870
119879= 100 and 119870
119877= 100)
simply supported 119908 = 0119872119909119909= 0 and 120595
119910= 0
1198961199090= 1014
1198701199090= 0 119870
1199101199090= 1014
(17b)
completely clamped 119908 = 0 120595119909= 0 and 120595
119910= 0
1198961199090= 1014
1198701199090= 1014
1198701199101199090
= 1014
(17c)
arbitrarily elastic boundary 119908 = 0 120595119909
= 0 120595119910
= 0119876119909
= 0119872119909119909
= 0 and119872119909119910
= 0
1198961199090= Γ1199081199090
1198701199090= Γ1199091199090
1198701199101199090
= 1014
(17d)
The appropriateness of the three classical boundariesdefined in (17a)ndash(17c) will be proved by several examplesgiven in following the arbitrary elastic boundaries are alsodefined in (17d) and the Γ
119908119894(Γ119908119894= 1198961198941198960 1198960= 1 times 10
9Nm2119894 = 1199090 1199091198861199100119910119887) and Γ
119909119894(Γ119909119894= 11987011989411987001198700= 1times10
9Nmrad119894 = 1199090 119909119886 1199100 119910119887) elastic restraint parameters representcorresponding spring stiffness For the sake of simplicitya four-letter string is employed to represent the restraintcondition of a plate such that F-C-S-E identifies the platewithedges 119909 = 0 119910 = 0 119909 = 119886 and 119910 = 119887 having free clampedshear-supported restrained and elastic boundary conditionsrespectively
As for the first case a uniform thickness moderately thickplate with different classical boundaries and structure param-eters is investigated here In Table 2 the comparison of thefirst eight frequency parametersΩ = 120596119887
2
(120588ℎ119863)12
1205872 of the
considered plate is presented The S-S-S-S C-F-F-F S-S-F-FF-F-F-F and S-C-S-C boundary conditions are performed inthe comparison Excellent agreements are observed betweenthe solutions obtained by the modified Fourier method thereferential data and finite element method (FEM) results forthe uniform thickness moderately thick rectangular platesIt is also verified that the definition of the three types of
classical boundaries in (17a)ndash(17c) is appropriate In additionthe elastic boundary conditions (17d) are also verified Inthe next two examples we will account for the vibrationof moderately thick plate with elastic edge supports Thefirst model considered is an S-S-S-S square moderately thickplate with all edges elastically rotationally restrained Thatis 1198701199090
= 119870119909119886
= 1198701199100
= 119870119910119887
= Γ119909 and all the
other restraining springs are set to have an infinite stiffness(namely represented by 1014 in numerical calculation) Thesix frequency parameters Ω = 120596119887
2
(120588ℎ119863)12
1205872 are given
in Table 3 for several different restraining coefficient valuesthe finite element method (FEM) results calculated usingABAQUSare also listed in this table as a referenceThe secondmodel concerns a complete square moderately thick platewith all edges elastically restrained That is 119896
1199090= 119896119909119886
=
1198961199100
= 119896119910119887
= Γ119908and 119870
1199090= 119870119909119886
= 1198701199100
= 119870119910119887
= Γ119909 The
six frequency parameters Ω = 1205961198872
(120588ℎ119863)12
1205872 are given in
Table 4 for several different restraining coefficient values thefinite element method (FEM) results are also listed in Table 4as a reference It can be clearly seen that the comparison isextremely good which implies that the current method isable tomake correct predictions for themodal characteristicsof moderately thick plate with not only classical boundaryconditions but also elastic edge restraints
The excellent agreements between the solution obtainedby the modified Fourier method and the referential datafor the moderately thick plate subjected to the combina-tions of classical boundary conditions and elastic boundaryconditions given in Tables 2ndash4 indicate that the proposedmethod is sufficiently accurate to deal with uniform thicknessmoderately thick plate with arbitrary boundary conditions
33 Linearly VariationThickness Moderately Thick Plates withClassical and Elastic Boundary Conditions In the theoreticalformulations this paper concerns the varying thicknessmoderately thick plates with classical and elastic boundary
10 Shock and Vibration
Table 2 Frequency parameters Ω = 1205961198872
(120588ℎ119863)12
1205872 for uniform thickness moderately thick plates with different classical boundaries and
structure parameters
Boundary condition ab Model sequence1 2 3 4 5 6 7 8
S-S-S-S
11931 4605 4605 7064 8605 8605 10792 10792
(1933a) (4611) (4611) (7067) (8607) (8607) (10799) (10780)(1931b) (4605) (4605) (7064) (8605) (8605) (10792) (10792)
321408 2648 4127 4605 5261 7064 7153 8184(1410a) (2651) (4128) (4608) (5267) (7074) (7155) (8188)(1408b) (2648) (4127) (4605) (5261) (7064) (7151) (8184)
C-F-F-F
10348 0817 2035 2583 2862 4816 5478 5774
(0348a) (0817) (2038) (2586) (2865) (4821) (5504) (5793)(0348b) (0816) (2034) (2582) (2860) (4811) (5477) (5772)
320155 0502 0941 1665 2292 2613 3303 3498
(0154a) (0502) (0941) (1663) (2295) (2618) (3305) (3500)(0154b) (0501) (0940) (1662) (2292) (2612) (3298) (3494)
S-S-F-F
10333 1678 1875 3559 4720 4946 6474 6632
(0335a) (1680) (1878) (3600) (4721) (4949) (6478) (6642)(0333b) (1677) (1874) (3557) (4718) (4945) (6472) (6631)
320222 0938 1631 2363 2573 4082 4482 4753
(0223a) (0940) (1632) (2364) (2575) (4092) (4483) (4759)(0222b) (0938) (1631) (2361) (2571) (4077) (4479) (4752)
F-F-F-F1 1290 1919 2363 3235 3235 5605 5605 5646
(1291a) (1920) (2365) (3239) (3239) (5607) (5607) (6648)
32 0865 0948 1961 2166 2461 2843 3551 4026(0865a) (0949) (1963) (2168) (2465) (2844) (3556) (4034)
S-C-S-C1 2700 4971 5990 7973 8787 10250 11334 12024
(2701a) (4971) (5993) (7975) (8788) (10253) (11335) (12028)
32 2348 3253 4972 5646 6489 7381 7974 9930(2348a) (3255) (4974) (5647) (6590) (7384) (7801) (9932)
aResults in parentheses are taken from FEMbResults in parentheses are taken from [11]
conditions The varying thickness function ℎ(119909 119910) can beexpressed as ℎ
0(1 + 120572119909
119904
)(1 + 120573119910119905
) in which the ℎ0 120572 and
120573 represent the initial thickness gradient in 119909 direction andgradient in 119910 direction When the indexes 119904 and 119905 take thevalue 119904 = 119905 = 1 the analyticalmodel imitates the linearly vari-ation thickness moderately thick plates structure In order tounify the description and facilitate the analytical calculationsof the involved integrals all the thickness variation functionscan be expanded into either 1D or 2D Fourier cosine seriesresulting in
ℎ (119909 119910) =
infin
sum
119898=0
infin
sum
119899=0
ℎ0119902119898119899
cos 119898120587119909119886
cos119898120587119910
119887
(18)
where119902119898119899
=
4
119886119887
int
119887
0
int
119886
0
(1 + 120572119909119904
) (1 + 120573119910119905
) cos 119898120587119909119886
cos119898120587119910
119887
119889119909 119889119910
(19)
In order to prove the validity of the present methodfor the vibration of linearly variation thickness moderately
thick plates with arbitrary boundary conditions the typicalclassical boundary conditions as the first case will be con-sidered In Table 5 the comparison of the first six frequencyparameters Ω = 120596119886
2
(120588ℎ01198630)12 of the moderately thick
plates with linearly varying thickness is presentedThe S-S-S-S C-F-F-F S-S-F-F C-C-C-C and S-C-S-C boundary condi-tions are performed in the comparison Excellent agreementsare observed between the solutions obtained by the modifiedFourier method and finite element method (FEM) results forthemoderately thick plates with linear variation thickness Toinvestigate the influence of the aspect ratio on the uniformthickness and nonuniform thickness moderately thick platesthe effect on the frequency parameters for plates with S-S-S-Sboundary conditions is presented in Figure 6 The thicknessfunctions are ℎ
0and ℎ0(1+05times119909)(1+05times119910) respectively It
is seen from Figure 6 that the influence of aspect ratios on thefrequency parameters for nonuniform thickness moderatelythick plates is more complicated
In the next two examples we also account for the vibra-tions of moderately thick plate with linear variation thicknessand elastic edge supports The first model considered is an S-S-S-S square moderately thick plate with all edges elastically
Shock and Vibration 11
Table 3 Frequency parameters Ω = 1205961198872
(120588ℎ119863)12
1205872 for S-S-S-S moderately thick plates (119886119887 = 1) with uniform thickness and elastic
rotation support (1198701199090= 119870119909119886= 1198701199100= 119870119910119887= Γ119909)
Γ119909
aℎ0119886
Model sequence1 2 3 4 5 6
0
005 1945 4834 4834 7591 9515 9519(1942b) (4836) (4836) (7584) (9547) (9552)
01 1856 4500 4500 6829 8499 8513(1858b) (4510) (4510) (6843) (8539) (8554)
015 1758 4114 4114 6040 7421 7445(1761b) (4126) (4126) (6060) (7462) (7486)
02 1656 3733 3733 5335 6468 6499(1660b) (3747) (3747) (5357) (6508) (6538)
05
005 3541 7071 7071 10227 12318 12399(3546b) (7091) (7091) (10253) (12386) (12468)
01 3274 6242 6242 8719 10302 10421(3282b) (6267) (6267) (8754) (10370) (10488)
015 2947 5361 5361 7290 8465 8606(2956b) (5387) (5387) (7327) (8524) (8666)
02 2624 4599 4599 6148 7057 7199(2634b) (4624) (4624) (6182) (7107) (7250)
1
005 3541 7072 7072 10228 12320 12401(3547b) (7093) (7093) (10254) (12388) (12470)
01 3278 6248 6248 8727 10311 10429(3286b) (6274) (6274) (8762) (10378) (10497)
015 2957 5374 5374 7305 8478 8620(2966b) (5400) (5400) (7341) (8537) (8680)
02 2641 4617 4617 6166 7070 7215(2651b) (4643) (4643) (6200) (7121) (7268)
15
005 3542 7073 7073 10229 12320 12401(3547b) (7093) (7093) (10254) (12388) (12471)
01 3280 6251 6251 8729 10313 10432(3287b) (6276) (6276) (8765) (10381) (10500)
015 2960 5379 5379 7309 8482 8625(2969b) (5405) (5405) (7346) (8541) (8685)
02 2647 4623 4623 6172 7074 7221(2657b) (4649) (4649) (6206) (7125) (7274)
2
005 3542 7073 7073 10229 12321 12402(3547b) (7093) (7093) (10255) (12389) (12471)
01 3280 6252 6252 8730 10315 10434(3288b) (6277) (6277) (8766) (10382) (10501)
015 2962 5381 5381 7312 8484 8628(2971b) (5407) (5407) (7348) (8543) (8688)
02 2650 4627 4627 6175 7076 7224(2660b) (4652) (4652) (6210) (7128 (7277)
aΓ119909= 1198701198941198700 (1198700= 1 times 10
9 Nmrad 119894 = 1199090 119909119886 1199100 119910119887)bResults in parentheses are taken from FEM
12 Shock and Vibration
Table 4 Frequency parameters Ω = 1205961198872
(120588ℎ119863)12
1205872 for moderately thick plates (119886119887 = 1) with uniform thickness and all edge elastic
restraints (1198701199090= 119870119909119886= 1198701199100= 119870119910119887= Γ119909and 119896
1199090= 119896119909119886= 1198961199100= 119896119910119887= Γ119908)
Γ119909
aΓ119908
aℎ0119886
Model sequence1 2 3 4 5 6
05 0
005 0986 1967 3893 3893 4850 4850(0988b) (1971) (3897) (3897) (4852) (4852)
01 0916 1835 3515 3522 4375 4375(0917b) (1837) (3519) (3528) (4341) (4341)
015 0789 1635 2983 3027 3771 3771(0792b) (1638) (2988) (3029) (3775) (3775)
02 0646 1444 2497 2601 3249 3249(0652b) (1448) (2508) (2621) (3254) (3254)
1 05
005 3417 3417 4431 5294 5461 6448(3419b) (3419) (4435) (5303) (5478) (6456)
01 1593 1593 2401 3849 3861 4729(1611b) (1611) (2408) (3852) (3866) (4767)
015 1122 1122 1913 3272 3290 4038(1127b) (1227) (1924) (3284) (3314) (4069)
02 0890 0890 1640 2781 2829 3476(0899b) (0899) (1654) (2787) (2834) (3497)
15 1
005 4325 4325 5619 6273 6571 7619(4334b) (4334) (5681) (6299) (6642) (7642)
01 2013 2013 2812 4088 4116 4993(2024b) (2024) (2827) (4122) (4187) (4994)
015 1338 1338 2108 3430 3443 4196(1342b) (1342) (2112) (3467) (3482) (4200)
02 1038 1038 1772 2936 2965 3613(1042b) (1042) (1786) (2937) (2977) (3624)
2 2
005 5231 5231 6919 7630 8002 9220(5238b) (5238) (6931) (7637) (8011) (9241)
01 2594 2594 3439 4470 4544 5428(2597b) (2597) (3442) (4495) (4557) (5438)
015 1653 1653 2403 3613 3632 4392(1662b) (1662) (2414) (3627) (3646) (4404)
02 1238 1238 1949 3077 3099 3751(1244b) (1244) (1964) (3094) (3116) (3788)
aΓ119909= 1198701198941198700 (1198700= 1 times 10
9 Nmrad 119894 = 1199090 119909119886 1199100 119910119887) Γ119908= 1198961198941198960 (1198960= 1 times 10
9 Nm 119894 = 1199090 119909119886 1199100 119910119887)bResults in parentheses are taken from FEM
05 10 15 20 25 300
2
4
6
8
10
12
14
16
ab
The 1st orderThe 2nd orderThe 3rd order
The 4th orderThe 5th orderThe 6th order
Ω
(a)
05 10 15 20 25 30
30
60
90
120
150
180
ab
The 1st orderThe 2nd orderThe 3rd order
The 4th orderThe 5th orderThe 6th order
Ω
(b)
Figure 6 The effect of aspect ratio 119886119887 on the natural frequenciesΩ for S-S-S-S boundary condition (a) uniform thickness and (b) nonuni-form thickness
Shock and Vibration 13
Table 5 Frequency parameter Ω = 1205961198862
(120588ℎ01198630)12 for moderately thick plates with linearly varying thickness in different boundary
conditions (ℎ(119909 119910) = ℎ0(1 + 05119909)(1 + 05119910))
Boundary condition 119886119887 ℎ0119886
Model sequence1 2 3 4 5 6
S-S-S-S
101 27261 62397 62926 92423 111537 111732
(27263a) (62410) (63842) (92445) (111867) (111921)
02 23103 47010 47191 65018 75831 75865(23489a) (47209) (47349) (65057) (76605) (76473)
3201 59338 99101 138798 151102 166710 204855
(59874a) (99249) (138177) (151362) (166644) (205381)
02 41175 63765 63765 73598 83956 89084(41586a) (63313) (63669) (73661) (83918) (89547)
C-F-F-F
101 4173 10331 27407 37826 38809 64095
(4856a) (10680) (27513) (37203) (39135) (64858)
02 3983 8702 21403 29443 30286 46705(4118a) (9306) (21990) (29670) (30602) (47156)
3201 4868 15488 35620 63523 89969 100089
(4887a) (15320) (35175) (63353) (89159) (100783)
02 4469 9763 22338 42188) 50104 56331(4944a) (9636) (22087) (42177) (50580) (56559)
S-S-F-F
101 4117 24795 26216 48567) 68195 69667
(4789a) (24149) (26751) (49324) (68842) (70575)
02 3820 20516 21924 37523 48980 50327(4103a) (20041) (22125) (37221) (48894) (50622)
3201 7935 39454 65063 92877 92877 136176
(7856a) (40145) (65162) (92462) (93022) (136336)
02 6389 26149 35627 49539 56844 65170(6406a) (26536) (35626) (49831) (57144) (65981)
C-C-C-C
101 44471 80299 80681 109422 126871 128207
(45102a) (80271) (80656) (110015) (126328) (128888)
02 32162 52781 52933 69410 78474 79482(32240a) (52629) (53166) (70063) (79059) (80266)
3201 82975 115587 151992 161579 176601 212330
(83139a) (115838) (152200) (162193) (177058) (212849)
02 48577 66022 83468 90830 97474 98157(49187a) (66259) (84065) (91749) (97827) (98777)
S-C-S-C
101 36847 66664 76965 101045 113594 125551
(37901a) (68568) (78356) (101200) (113978) (125651)
02 27673 48469 51220 67027 76358 78241(27897a) (48685) (51491) (67447) (77530) (79573)
3201 77378 106848 150079 154569 173120 208391
(78003a) (107058) (150991) (155171) (173099) (208402)
02 45895 64732 73393 83282 90325 97089(45803a) (64131) 73656 83801 90518 97015
aResults in parentheses are taken from FEM
restrained That is 1198701199090
= 119870119909119886
= 1198701199100
= 119870119910119887
= Γ119909 The six
frequency parameters Ω = 1205961198862
(120588ℎ01198630)12 are given in
Table 6 for several different restraining coefficient valuesthe finite element method (FEM) results calculated usingABAQUS are also presented as a referenceThe secondmodelis a complete square moderately thick plate with all edgeselastically restrainedThat is 119896
1199090= 119896119909119886= 1198961199100= 119896119910119887= Γ119908and
1198701199090= 119870119909119886
= 1198701199100= 119870119910119887= Γ119909 The six frequency parameters
Ω = 1205961198862
(120588ℎ01198630)12 are given in Table 7 for several different
restraining coefficient values the finite element method(FEM) results are also listed as a reference It can be clearlyseen that the comparison is extremely good which impliesthat the current method is able to make correct predictionsfor the modal characteristics of linearly varying thickness
14 Shock and Vibration
Table 6 Frequency parameter Ω = 1205961198862
(120588ℎ01198630)12 for S-S-S-S moderately thick plates (119886119887 = 1) with linearly varying thickness and elastic
rotation support (ℎ(119909 119910) = ℎ0(1 + 05119909)(1 + 05119910) 119870
1199090= 119870119909119886= 1198701199100= 119870119910119887= Γ119909)
Γ119909
aℎ0119886
Model sequence1 2 3 4 5 6
0
005 28809 70080 70891 108967 134884 135295(28897b) (70334) (71129) (109112) (135254) (135772)
01 27261 62398 62926 92423 111540 111735(26606b) (61784) (62303) (91193) (111323) (111648)
015 25231 54192 54505 77118 91228 91312(25204b) (55297) (55595) (77503) (91871) (91941)
02 23103 47010 47191 65018 75832 75866(23947b) (47071) (47208) (65404) (76531) (76589)
05
005 23103 47010 47191 65018 75832 75866(23210b) (47245) (47358) (65241) (76122) (76345)
01 48377 94650 95597 135484 162635 163501(48524b) (94891) (95992) (135845) (167012) (164024)
015 36239 70939 71298 99997 118584 119009(36647b) (71214) (71687) (100024) (119114) (119457)
02 28749 56937 57037 79214 93009 93162(29142b) (57034) (57544) (79654) (93668) (93754)
1
005 49651 96759 97619 138089 165368 166304(49755b) (97245) (97964) (138544) (165864) (166878)
01 38951 73735 74251 102748 121079 121709(39021b) (74145) (74664) (102874) (121613) (121817)
015 30630 58406 58533 80432 93984 94255(30638b) (58684) (58644) (80874) (94121) (94478)
02 25548 48636 48674 66165 76708 76824(25687b) (48788) (48992) (66544) (76824) (76944)
15
005 50118 97561 98374 139096 166421 167382(50247b) (97877) (98661) (139247) (166875) (167662)
01 40310 75216 75783 104229 122402 123164(40366b) (75542) (75921) (104557) (122874) (123385)
015 31837 59352 59530 81245 94611 94983(31902b) (59451) (59871) (81375) (94784) (95105)
02 26302 49119 49156 66528 76954 77130(26418b) (49354) (49334) (66921) (77010) (77470)
2
005 50360 97983 98769 139631 166979 167954(50472b) (98005) (99114) (140154) (167188) (167986)
01 41130 76146 76724 105165 123229 124082(41421b) (76642) (78114) (105661) (123278) (124974)
015 32687 60025 60245 81832 95052 95508(32778b) (60244) (60549) (82146) (95114) (95842)
02 26893 49492 49539 66815 77138 77370(26922b) (49874) (49924) (66987) (77367) (77588)
aΓ119909= 1198701198941198700 (1198700= 1 times 10
9 Nmrad 119894 = 1199090 119909119886 1199100 119910119887)bResults in parentheses are taken from FEM
Shock and Vibration 15
Table 7 Frequency parameterΩ = 1205961198862
(120588ℎ01198630)12 for moderately thick plates (119886119887 = 1) with linearly varying thickness and elastic rotation
and translation support (ℎ(119909 119910) = ℎ0(1 + 05119909)(1 + 05119910) 119870
1199090= 119870119909119886= 1198701199100= 119870119910119887= Γ119909 and 119896
1199090= 119896119909119886= 1198961199100= 119896119910119887= Γ119908)
Γ119909
aΓ119908
aℎ0119886
Model sequence1 2 3 4 5 6
05 0
005 14751 14917 28232 55251 55397 69512(14780b) (14822) (29252) (55476) (55505) (70904)
01 11811 11910 23857 44462 45037 55453(11818b) (11897) (24711) (44376) (45257) (56511)
015 8487 8514 20011 34783 36335 44970(8500b) (8544) (20738) (34404) (36663) (45625)
02 6073 6073 17431 28679 30805 38010(6096b) (6106) (18039) (28129) (31165) (38311)
1 05
005 21225 32173 32173 44381 64061 65222(20708b) (32097) (32285) (45110) (64182) (65355)
01 8140 16878 16941 28100 48663 49015(7943b) (16838) (16970) (28888) (48700) (49171)
015 4493 11859 11878 22429 38352 39247(4383b) (11867) (11891) (23125) (38177) (39488)
02 2932 8690 8691 18919 31108 32600(3540b) (8791) (8791) (16229) (27520) (27889)
15 1
005 27819 40919 41430 54595 70792 72896(27146b) (41008) (41434) (55256) (70895) (73051)
01 11285 20068 20230 31101 51046 51429(11012b) (20025) (20270) (31849) (51126) (51569)
015 6292 13912 13922 24132 40500 41120(6139b) (13912) (13934) (24808) (40426) (41316)
02 4121 10321 10331 20051 32796 33921(4021b) (10329) (10357) (20635) (32540) (34178)
2 2
005 34615 51972 53088 68255 81344 84510(33806b) (52100) (53157) (68891) (81433) (84761)
01 15416 24659 24829 35615 53879 54494(15045b) (24625) (24888) (36303) (53970) (54626)
015 8740 16420 16481 26353 42417 42946(8529b) (16414) (16499) (26996) (42398) (43114)
02 5761 12127 12127 21434 34286 35192(5622b) (12134) (12148) (21998) (34110) (35415)
aΓ119909= 1198701198941198700 (1198700= 1 times 10
9 Nmrad 119894 = 1199090 119909119886 1199100 119910119887) Γ119908= 1198961198941198960 (1198960= 1 times 10
9 Nm 119894 = 1199090 119909119886 1199100 119910119887)bResults in parentheses are taken from FEM
moderately thick plate with not only classical boundaryconditions but also elastic edge restraints
As the last case of this subsection the influence of thegradient 120572 and 120573 on the fundamental frequency parametersfor a linearly varying thickness moderately thick plate isinvestigated The model is a square moderately thick platewith all edges elastically restrainedThat is 119896
1199090= 119896119909119886= 1198961199100=
119896119910119887= Γ119908= 2 and 119870
1199090= 119870119909119886
= 1198701199100= 119870119910119887= Γ119909= 2 The six
frequency parameters Ω = 1205961198862
(120588ℎ01198630)12 are given in
Table 8 for several different 120572 and 120573 values the finite elementmethod (FEM) results are also listed as a reference Againgood agreement can be observed Through Table 8 it is alsofound that the frequency parameter increases with increasinggradient parameters
The above studies are given as linearly varying thicknessmoderately thick plates with several of boundary condition
and different structure parameters In the next section vibra-tion results for the plates subjected to nonlinear variationthickness will be presented
34 Nonlinearly Varying Thickness Moderately Thick Plateswith Classical and Elastic Boundary Conditions In Sec-tion 33 the linearly varying thickness moderately thickplates were studied However in the practical engineeringapplications the varying thickness of amoderately thick platemay not always be linear variation in nature A variety ofpossible thickness varying cases may be encountered in prac-tice Therefore the moderately thick plates with nonlinearvariation thickness subjected to general elastic edge restraintsare examined in this subsection For the sake of brevity theindexes 119904 and 119905 will be chosen as 2 to imitate the nonlinearlyvarying thickness moderately thick plates structure in this
16 Shock and Vibration
Table 8 Frequency parameterΩ = 1205961198862
(120588ℎ01198630)12 for elastic support moderately thick plates (119886119887 = 1) with different gradient (119896
1199090= 119896119909119886=
1198961199100= 119896119910119887= Γ119908= 2 119870
1199090= 119870119909119886= 1198701199100= 119870119910119887= Γ119909= 2)
120572 120573 ℎ0119886
Model sequence1 2 3 4 5 6
120572 = 120573 = 01
005 31275 52615 52657 68954 76146 79850(31303a) (52664) (52675) (69033) (76174) (79948)
01 17270 25244 25250 33949 46050 46625(17271a) (25255) (25258) (34089) (46122) (46708)
015 10190 16266 16274 24164 37327 37526(10189a) (16269) (16280) (24323) (37401) (37622)
02 6780 12258 12260 19685 31452 31754(6780a) (12265) (12270) (19851) (31514) (31864)
120572 = 120573 = 02
005 32530 52947 53141 68954 77034 80589(32577a) (53014) (53203) (69172) (77088) (80773)
01 16882 24979 24979 34154 48057 48575(16883a) (24983) (25001) (34433) (48134) (48667)
015 9804 16284 16309 24683 38844 39085(9803a) (16283) (16322) (24980) (38909) (39191)
02 6499 12290 12295 20133 32354 32770(6498a) (12296) (12307) (20420) (32387) (32895)
120572 = 120573 = 03
005 33516 52845 53308 68704 78203 81561(33563a) (52939) (53389) (69055) (78271) (81789)
01 16419 24799 24830 34527 50061 50583(16419a) (24790) (24869) (34947) (50143) (50688)
015 9430 16334 16378 25236 40193 40506(9429a) (16329) (16395) (25660) (40243) (40625)
02 6235 12288 12293 20572 33102 33663(6235a) (12296) (12306) (20965) (33090) (33811)
120572 = 120573 = 04
005 34215 52463 53264 68430 79654 82865(34252a) (52575) (53346) (68921) (79732) (83113)
01 15921 24695 24793 35027 52011 52571(15921a) (24674) (24842) (35583) (52098) (52688)
015 9074 16388 16444 25797 41379 41789(9074a) (16381) (16464) (26336) (41404) (41928)
02 5990 12236 12238 21002 33735 34461(5989a) (12249) (12250) (21487) (33653) (34641)
120572 = 120573 = 05
005 34615 51972 53088 68255 81344 84510(33806a) (52100) (53157) (68891) (81433) (84761)
01 15416 24659 24829 35615 53879 54494(15045a) (24625) (24888) (36303) (53970) (54626)
015 8740 16420 16481 26353 42417 42946(8529a) (16414) (16499) (26996) (42398) (43114)
02 5761 12127 12127 21434 34286 35192(5622a) (12134) (12148) (21998) (34110) (35415)
aResults in parentheses are taken from FEM
section that is 119904 = 119905 = 2 Also the thickness varying functionsare also be expressed as Fourier cosine series according to (18)and (19)
In order to validate the accuracy and reliability of theproposed method for predicting the vibration behavior ofnonlinearly varying thickness moderately thick plates with
arbitrary boundary conditions the typical classical boundaryconditions viewed as the special cases of elastically restrainededges will be considered The comparison of the first six fre-quency parameters Ω = 120596119886
2
(120588ℎ01198630)12 for the moderately
thick plates with nonlinearly varying thickness is presentedin Table 9 The S-S-S-S C-F-F-F S-S-F-F C-C-C-C and
Shock and Vibration 17
Table 9 Frequency parameter Ω = 1205961198862
(120588ℎ01198630)12 for moderately thick plates with linearly varying thickness in different boundary
conditions (ℎ(119909 119910) = ℎ0(1 + 05119909
2
)(1 + 051199102
))
Boundary condition 119886119887 ℎ0119886
Model sequence1 2 3 4 5 6
S-S-S-S
101 23462 54999 55573 82950 100900 101187
(23741a) (55114) (55983) (83146) (101004) (101347)
02 18606 42725 42971 60040 71167 71761(18936a) (43014) (43148) (60247) (71642) (71993)
3201 53055 65251 85404 106700 114880 143531
(53467a) (65472) (85984) (107211) (115687) (144007)
02 17439 21531 32170 44987 59438 64179(17874a) (21748) (32651) (45264) (60188) (64879)
C-F-F-F
101 3658 9317 24135 32816 35433 59179
(3749a) (9547) (24034) (33627) (36308) (60142)
02 3520 7986 19591 26959 27491 43882(3611a) (8176) (20075) (27125) (27625) (43984)
3201 3901 9963 28984 69187 73686 89360
(3930a) (10095) (29381) (69578) (73994) (90242)
02 3292 4439 13385 23947 40850 44758(3386a) (4498) (14012) (24068) (41221) (45142)
S-S-F-F
101 3557 21268 22762 43524 60253 62429
(3649a) (21441) (22796) (43916) (60622) (62630)
02 3171 17822 19512 33616 45042 46833(3489a) (18006) (20430) (33815) (45247) (49004)
3201 6738 31658 52071 75625 86604 94903
(6957a) (31882) (52518) (75823) (86939) (95610)
02 3653 12930 20955 26517 44097 48058(6807a) (13160) (21312) (26864) (44422) (48873)
C-C-C-C
101 41020 75196 75614 104142 119683 120949
(41405a) (75452) (75882) (104711) (119637) (121137)
02 31237 51686 51868 68411 76868 77864(32010a) (52960) (52145) (69335) (77481) (78099)
3201 95823 128727 163657 173751 190890 223595
(95364a) (128932) (163588) (172350) (190350) (222948)
02 53306 71372 89384 102404 104660 120908(53587a) (71759) (90103) (102503) (105174) (121275)
S-C-S-C
101 33715 60298 71446 94187 103802 117928
(34148a) (60786) (71828) (94515) (104263) (119326)
02 26143 45499 49458 64253 72941 76191(26789a) (46622) (50680) (64329) (73474) (76843)
3201 73228 115845 137445 148166 168342 182542
(73761a) (116121) (137788) (148530) (168698) (182921)
02 41302 53648 65630 72202 81632 88105(41304a) (53757) (65842) (72794) (82033) (88474)
aResults in parentheses are taken from FEM
S-C-S-C boundary conditions are performed in the compar-ison The results adequately demonstrated the great accuracyof the modified Fourier method
We now turn to elastically restrained moderately thickplates The first one involves an S-S-S-S moderately thicksquare plate with a uniform rotational restraint along eachedge that is 119870
1199090= 119870119909119886
= 1198701199100
= 119870119910119887
= Γ119909 The calculated
frequency parameters Ω = 1205961198862
(120588ℎ01198630)12 are given in
Table 10 together with the FEM results Since the elasticallyrestrained plates with nonlinear variation thickness are rarelyinvestigated the FEMresults are used as the referenceA goodagreement is observed between the current and FEM resultsThe second example concerns amoderately thick square plateelastically supported along all edgesThe stiffness of the linear
18 Shock and Vibration
Table 10 Frequency parameterΩ = 1205961198862
(120588ℎ01198630)12 for S-S-S-S moderately thick plates (119886119887 = 1) with linearly varying thickness and elastic
rotation support (ℎ(119909 119910) = ℎ0(1 + 05119909
2
)(1 + 051199102
) 1198701199090= 119870119909119886= 1198701199100= 119870119910119887= Γ119909)
Γ119909
aℎ0119886
Model sequence1 2 3 4 5 6
0
005 23462 54999 55573 82950 100900 101187(23741b) (55114) (55983) (83146) (101004) (101347)
01 18606 42725 42971 60040 71167 71761(18936b) (43014) (43148) (60247) (71642) (71993)
015 53055 65251 85404 106700 114880 143531(53467b) (65472) (85984) (107211) (115687) (144007)
02 17439 21531 32170 44987 59438 64179(17874b) (21748) (32651) (45264) (60188) (64879)
05
005 6904 13374 13514 19400 22904 23056(9909b) (13548) (13651) (19564) (23088) (23379)
01 5316 10237 10314 14499 17155 17264(5374b) (10448) (10651) (14764) (17188) (17879)
015 4136 8201 8220 11474 13578 13671(4174b) (8348) (8651) (11764) (13688) (13879)
02 3449 6915 6924 9549 11225 11327(3474b) (6948) (6951) (9764) (11388) (11579)
1
005 7040 13617 13745 19726 23233 23402(7074b) (13748) (13851) (19964) (23488) (23879)
01 5712 10696 10793 14991 17594 17744(5774b) (10748) (10851) (15054) (17788) (17879)
015 4489 8507 8540 11759 13803 13922(4514b) (8548) (8611) (11864) (13988) (14119)
02 3677 7084 7090 9692 11330 11445(3684b) (7148) (7191) (9764) (11588) (11879)
15
005 7089 13708 13828 19847 23355 23530(7124b) (13748) (13951) (19967) (23588) (23879)
01 5895 10923 11024 15239 17813 17988(5874b) (10748) (11651) (15264) (17818) (17979)
015 4697 8694 8737 11938 13790 14083(4774b) (8748) (8951) (11864) (13688) (14879)
02 3836 7201 7210 9792 11403 11530(3874b) (7748) (7651) (9864) (11588) (11879)
2
005 7114 13754 13871 19911 23418 23597(7174b) (12848) (13851) (19864) (23588) (23879)
01 6001 11060 11159 15391 17944 18136(6004b) (11148) (11151) (15464) (17988) (18379)
015 4836 8821 8872 12061 14037 14189(4874b) (8848) (8951) (12164) (14188) (14879)
02 3954 7289 7301 9871 11457 11594(3974b) (7348) (7651) (9964) (11488) (11679)
aΓ119909= 1198701198941198700 (1198700= 1 times 10
9 Nmrad 119894 = 1199090 119909119886 1199100 119910119887)bResults in parentheses are taken from FEM
and rotational restraints is set equal to 1198961199090
= 119896119909119886
= 1198961199100
=
119896119910119887
= Γ119908and 119870
1199090= 119870119909119886
= 1198701199100
= 119870119910119887
= Γ119909 respectively
The six frequency parametersΩ = 1205961198862
(120588ℎ01198630)12 are shown
in Table 11 for several different restraining coefficient valuesIt can also be noticed that the Fourier series method is able
to predict the modal characteristics of nonlinearly varyingthickness moderately thick plate with not only classicalboundary conditions but also elastic edge restraints correctly
Finally the influence of the gradient on the fundamentalfrequency parameters of a nonlinearly varying thickness
Shock and Vibration 19
Table 11 Frequency parameterΩ = 1205961198862
(120588ℎ01198630)12 for moderately thick plates (119886119887 = 1) with linearly varying thickness and elastic rotation
and translation support (ℎ(119909 119910) = ℎ0(1 + 05119909
2
)(1 + 051199102
) 1198701199090= 119870119909119886= 1198701199100= 119870119910119887= Γ119909 and 119896
1199090= 119896119909119886= 1198961199100= 119896119910119887= Γ119908)
Γ119909
aΓ119908
aℎ0119886
Model sequence1 2 3 4 5 6
05 0
005 12926 13041 25090 47935 47943 61761(12931b) (13053) 25099 (47947) (47952) (61788)
01 10900 10988 21545 39980 40147 50322(10904b) (10992) (21557) (39984) (40166) (50334)
015 8245 8298 18081 31503 32617 40950(8247b) (8230) (18092) (31508) (32622) (40955)
02 6106 6124 15639 25631 27601 34458(6110b) (6125) (15642) (25637) (27617) (34462)
1 05
005 21575 31615 31644 42573 57889 58607(21580b) (31617) (31644) (42581) (57893) (58610)
01 8451 16024 16210 25969 43839 43867(8451b) (16025) (16217) (25972) (43844) (43870)
015 4682 11479 11480 20642 35172 35598(4684b) (11480) (11481) (20644) (35178) (35604)
02 3059 8621 8648 17320 28487 29638(3062b) (8619) (8644) (17327) (28492) (29641)
15 1
005 27713 40203 40978 52951 65352 66815(27714b) (40207) (40988) (52972) (65368) (66822)
01 11667 19322 19615 29122 46115 46183(11674b) (19327) (19619) (29137) (46120) (46188)
015 6544 13416 13481 22405 37238 37430(6547b) (13420) (13488) (22410) (37241) (37434)
02 4295 10175 10190 18543 30330 31056(4296b) (10177) (10194) (18544) (30336) (31060)
2 2
005 33453 50854 52088 66311 76610 78947(33455b) (50852) (50097) (66321) (76627) (78971)
01 15832 24130 24372 33924 49118 49362(15830b) (24127) (24374) (33931) (49147) (49382)
015 9060 15956 16099 24754 39134 39249(9064b) (15961) (16102) (24755) (39138) (39452)
02 5992 11955 11971 20034 31908 32392(5991b) (11957) (11978) (20042) (31910) (32398)
aΓ119909= 1198701198941198700 (1198700= 1 times 10
9 Nmrad 119894 = 1199090 119909119886 1199100 119910119887) Γ119908= 1198961198941198960 (1198960= 1 times 10
9 Nm 119894 = 1199090 119909119886 1199100 119910119887)bResults in parentheses are taken from FEM
moderately thick plate is investigated The plate is elasticallyrestrained in which the stiffness of the boundary springsis taken as 119896
1199090= 119896119909119886
= 1198961199100
= 119896119910119887
= Γ119908
= 2 and1198701199090
= 119870119909119886
= 1198701199100
= 119870119910119887
= Γ119909= 2 respectively The six
frequency parameters Ω = 1205961198862
(120588ℎ01198630)12 are presented in
Table 12 for several different slop values It can be seen thatthe fundamental frequency parameters will decrease with theincrease of the parameter ℎ
0119886
In the above examples it has been demonstrated thatthe presented method can be universally applied to nonlin-early varying thickness moderately thick plates with severalboundary conditions and different structure parametersNew results are obtained for plates with nonlinear variationthickness in both directions subjected to general elastic
boundary restraints which may be used for benchmarkingof researchers in the field In addition it is interesting to seethat the nature frequency decreases with the increase of theindex for thickness function
4 Conclusions
In this paper a modified Fourier method has been pre-sented to study the free vibration behaviors of moderatelythick rectangular plates with variable thickness and arbi-trary boundary conditionsThe first-order shear deformationplate theory is adopted to formulate the theoretical modelThe displacements and rotation components of the plateregardless of boundary conditions are invariantly expressedas the superposition of a 2D Fourier cosine series and four
20 Shock and Vibration
Table 12 Frequency parameterΩ = 1205961198862
(120588ℎ01198630)12 for elastic support moderately thick plates (119886119887 = 1) with different gradient (119896
1199090= 119896119909119886=
1198961199100= 119896119910119887= Γ119908= 2 119870
1199090= 119870119909119886= 1198701199100= 119870119910119887= Γ119909= 2)
120572 120573 ℎ0119886
Model sequence1 2 3 4 5 6
120572 = 120573 = 01
005 30187 51102 51141 67066 73954 77412(30188a) (51105) (51144) (67068) (73961) (77477)
01 16995 24719 24724 33071 44343 44819(16996a) (24725) (24729) (33084) (44347) (44822)
015 10077 15872 15883 23443 35937 36065(10081a) (15877) (15892) (23447) (35941) (36084)
02 6713 11953 11958 19086 30372 30604(6713a) (11957) (11965) (19092) (30378) (30612)
120572 = 120573 = 02
005 31206 51436 51614 67058 74398 77595(31208a) (51438) (51619) (67064) (74402) (77596)
01 16786 24503 24508 33144 45583 45923(16790a) (24510) (24513) (33147) (45587) (45932)
015 9821 15865 15906 23766 36897 36990(9824a) (15870) (15907) (23768) (36894) (36997)
02 6523 11978 11992 19356 30912 31162(6524a) (11981) (11997) (19362) (30916) (31174)
120572 = 120573 = 03
005 32097 51462 51900 66832 74991 77863(32098a) (51466) (51907) (66849) (74996) (77872)
01 16507 24332 24383 33330 46806 47074(16511a) (24327) (24380) (33339) (46810) (47086)
015 9562 15885 15964 24103 37748 37828(9562a) (15892) (15977) (21105) (37751) (37832)
02 6338 11992 12013 19603 31338 31634(6340a) (11994) (12011) (19612) (31344) (31639)
120572 = 120573 = 04
005 32851 51239 52046 66547 75735 78303(33455a) (50852) (50097) (66321) (76627) (78971)
01 16182 24209 24343 33598 47990 48230(33455a) (50852) (50097) (66321) (76627) (78971)
015 9307 15919 16034 24436 38491 38579(33455a) (50852) (50097) (66321) (76627) (78971)
02 6161 11986 12008 19828 31666 32040(33455a) (50852) (50097) (66321) (76627) (78971)
120572 = 120573 = 05
005 33453 50854 52088 66311 76610 78947(33455a) (50852) (50097) (66321) (76627) (78971)
01 15832 24130 24372 33924 49118 49362(15830a) (24127) (24374) (33931) (49147) (49382)
015 9060 15956 16099 24754 39134 39249(9064a) (15961) (16102) (24755) (39138) (39452)
02 5992 11955 11971 20034 31908 32392(5991a) (11957) (11978) (20042) (31910) (32398)
aResults in parentheses are taken from FEM
supplementary functions in the form of the product of apolynomial function and a single cosine series expansionto ensure and accelerate the convergence of the solutionAt each edge of the plate the general restraint conditionsare implemented by introducing one group of linear springsand two groups of rotational springs which are continuouslydistributed and determined by the stiffness of these springs
Instead of seeking a solution in strong forms in the previousstudies all the Fourier coefficients will be treated equallyand independently as the generalized coordinates and solveddirectly from the Rayleigh-Ritz technique The change of theboundary conditions can be easily achieved by only varyingthe stiffness of the three sets of boundary springs along alledges of the rectangular plates without involving any change
Shock and Vibration 21
to the solution procedure The convergence of the presentsolution is examined and the excellent accuracy is validatedby comparison with existing results published in the lit-erature and FEM data Excellent agreements are obtainedfrom these comparisons The proposed method providesa unified means for extracting the modal parameters andpredicting the vibration behaviors of moderately thick plateswith variable thickness variation functions and arbitraryelastic edge restraints A variety of free vibration results formoderately thick rectangular plates with different thicknessvariation functions and boundary conditions are presentedNew results for free vibration of moderately thick rectangularplates with various thickness variation functions and edgeconditions are presented which may be used for benchmark-ing of researchers in the field
Appendices
A Representative Calculation forStiffness and Mass Matrices
To illuminate the particular expression of the mass andstiffness matrixes clearly and tersely four new variables aredefined as 119904 = 119898 lowast (119873 + 1) + 119899 + 1 119905 = 119898
1015840
lowast (119873 + 1) + 1198991015840
+ 1119901 = (119897minus1)lowast (119872+1)+119898
1015840
+1 and 119902 = (119897minus1)lowast (119873+1)+1198991015840
+1The first row elements of K andM are given below
K1-1119904119905
=
1
2
int
119886
0
int
119887
0
[119863 (119909 119910)
times 21205821198861198981205821198861198981015840 sin 120582
119886119898119909 sin 120582
1198861198981015840119909 cos 120582
119887119899119910
times cos 1205821198871198991015840119910 + (1 minus 120583) 120582
1198871198991205821198871198991015840 cos 120582
119886119898119909
times cos 1205821198861198981015840 sin 120582
119887119899119910 sin 120582
1198871198991015840119910
+ 120581119866ℎ (119909 119910) cos 120582119886119898119909 cos 120582
1198861198981015840119909 cos 120582
119887119899
times119910 cos 1205821198871198991015840119910119889119909119889119910] 119889119909 119889119910
+ int
119886
0
1198701199091199100
cos 120582119886119898119909 cos 120582
1198861198981015840119909
+(minus1)119898+1198981015840
119870119909119910119887
cos 120582119886119898119909 cos 120582
1198861198981015840119909 119889119909
+ int
119887
0
1198701199090cos 120582119887119899119910 cos 120582
1198871198991015840119910
+(minus1)119899+1198991015840
119870119909119886cos 120582119887119899119910 cos 120582
1198871198991015840119910 119889119910
K1-2119904119901
=
1
2
int
119886
0
int
119887
0
[119863 (119909 119910)21205821198861198981205821198861198981015840 sin 120582
119886119898119909 sin 120582
1198861198981015840119909
times
2
sum
119897=1
120577119897
119887(119910) cos 120582
1198871198991015840119910
minus (1 minus 120583) 1205821198871198991015840 cos 120582
119886119898119909 cos 120582
1198861198981015840119909
times
2
sum
119897=1
1205771198971015840
119887(119910) sin 120582
1198871198991015840119910 + 120581119866ℎ (119909 119910)
times ( cos 1205821198861198981015840119909 cos 120582
1198871198991015840119910
times
2
sum
119897=1
120577119897
119887(119910) cos 120582
119886119898119909)]119889119909119889119910
+ int
119886
0
1198701199091199100
cos 120582119886119898119909 cos 120582
1198861198981015840119909
2
sum
119897=1
120577119897
119887(0)
+(minus1)1198991015840
119870119909119910119887
cos 120582119886119898119909 cos 120582
1198861198981015840119909
2
sum
119897=1
120577119897
119887(119887) 119889119909
+ int
119887
0
1198701199090
2
sum
119897=1
120577119897
119887(119910) cos 120582
1198871198991015840119910
+(minus1)119898+1198981015840
119870119909119886
2
sum
119897=1
120577119897
119887(119910) cos 120582
1198871198991015840119910119889119910
K1-3119904119902
=
1
2
int
119886
0
int
119887
0
[119863 (119909 119910) minus 21205821198861198981015840 sin 120582
1198861198981015840119909 cos 120582
1198871198991015840119910
times
2
sum
119897=1
1205771198971015840
119886(119909) cos 120582
119887119899119910
+ (1 minus 120583) 1205821198871198991205821198871198991015840 cos 120582
1198861198981015840119909
times sin 1205821198871198991015840119910
2
sum
119897=1
120577119897
119886(119909) sin 120582
119887119899119910
+ 120581119866ℎ (119909 119910) cos 1205821198861198981015840119909 cos 120582
1198871198991015840119910
times
2
sum
119897=1
120577119897
119886(119909) cos 120582
119887119899119910]119889119909119889119910
+ int
119886
0
1198701199091199100
cos 1205821198861198981015840119909
2
sum
119897=1
120577119897
119886(119909)
+(minus1)119899+1198991015840
119870119909119910119887
cos 1205821198861198981015840119909
2
sum
119897=1
120577119897
119886(119909) 119889119909
+ int
119887
0
1198701199090
2
sum
119897=1
120577119897
119886(0) cos 120582
1198871198991015840119910
+(minus1)119898+1198981015840
119870119909119886
2
sum
119897=1
120577119897
119886(0) cos 120582
1198871198991015840119910119889119910
K1-4119904119905
=
1
2
int
119886
0
int
119887
0
119863(119909 119910) 21205821198871198991205821198861198981015840 sin 120582
1198861198981015840119909 cos 120582
1198871198991015840119910
times cos 120582119886119898119909 sin 120582
119887119899119910 minus 2(1 minus 120583)
119887119899
times 1205821198861198981015840 sin 120582
1198861198981015840119909 cos 120582
1198871198991015840119910 cos 120582
119886119898119909
times sin 120582119887119899119910 + (1 minus 120583) 120582
1198861198981205821198871198991015840 cos 120582
1198861198981015840119909
times sin 1205821198871198991015840119910 sin 120582
119886119898119909 cos 120582
119887119899119910 119889119909 119889119910
22 Shock and Vibration
K1-5119904119901
=
1
2
int
119886
0
int
119887
0
119863(119909 119910) minus 21205821198861198981015840 sin 120582
1198861198981015840119909 cos 120582
1198871198991015840119910
times
2
sum
119897=1
1205771198971015840
119887(119910) cos 120582
119886119898119909
+ 2 (1 minus 120583) 1205821198861198981015840 sin 120582
1198861198981015840119909 cos 120582
1198871198991015840119910
times
2
sum
119897=1
1205771198971015840
119887(119910) cos 120582
119886119898119909
+ (1 minus 120583) 1205821198861198981205821198871198991015840 cos 120582
1198861198981015840119909 sin 120582
1198871198991015840119910
times
2
sum
119897=1
120577119897
119887(119910) sin 120582
119886119898119909119889119909119889119910
K1-6119904119902
=
1
2
int
119886
0
int
119887
0
119863(119909 119910)1205821198871198991205821198861198981015840 sin 120582
1198861198981015840119909 cos 120582
1198861198991015840119910
times
2
sum
119897=1
120577119897
119886(119909) sin 120582
1198871198991015840119910
minus 2 (1 minus 120583) 1205821198871198991205821198861198981015840 sin 120582
1198861198981015840119909 cos 120582
1198861198991015840119910
times
2
sum
119897=1
120577119897
119886(119909) sin 120582
1198871198991015840119910
+ 1205821198871198991015840 cos 120582
1198861198981015840119909 sin 120582
1198871198991015840119910
times
2
sum
119897=1
1205771198971015840
119886(119909) cos 120582
119887119899119910119889119909119889119910
K1-7119904119905
= minus
1
2
int
119886
0
int
119887
0
120581119866ℎ (119909 119910) 120582119886119898
cos 1205821198861198981015840119909 cos 120582
1198871198991015840119910
times sin 120582119886119898119909 cos 120582
119887119899119910119889119909119889119910
K1-8119904119901
= minus
1
2
int
119886
0
int
119887
0
120581119866ℎ (119909 119910) 1205821198861198981015840 cos 120582
1198861198981015840119909 cos 120582
1198871198991015840119910
times
2
sum
119897=1
120577119897
119887(119910) sin 120582
1198861198981015840119909 119889119909 119889119910
K1-9119904119902
=
1
2
int
119886
0
int
119887
0
120581119866ℎ (119909 119910) cos 1205821198861198981015840119909 cos 120582
1198871198991015840119910
times
2
sum
119897=1
1205771198971015840
119886(119909) cos 120582
119887119899119910119889119909119889119910
M1-1119904119905
=
1
24
120588int
119886
0
int
119887
0
ℎ3
(119909 119910) cos 120582119886119898119909 cos 120582
119887119899119910
times cos 1205821198861198981015840119909 cos 120582
1198871198991015840119910119889119909119889119910
M1-2119904119901
=
1
24
120588int
119886
0
int
119887
0
ℎ3
(119909 119910)
2
sum
119897=1
120577119897
119887(119910) cos 120582
119886119898119909
times cos 1205821198861198981015840119909 cos 120582
1198871198991015840119910119889119909119889119910
M1-3119904119902
=
1
24
120588int
119886
0
int
119887
0
ℎ3
(119909 119910)
2
sum
119897=1
120577119897
119886(119909) cos 120582
119887119899119910
times cos 1205821198861198981015840119909 cos 120582
1198871198991015840119910119889119909119889119910
M1-4119904119905 = M
1-5119904119901 = M1-6119904119902
= M1-7119904119905 = M
1-8119904119901 = M1-9119904119902 = 0
(A1)
B Useful Integrals for the Moderately ThickRectangular Plates with Variable Thicknessand Arbitrary Boundary Conditions
Consider
int
119871
0
cos 120582119898119909 cos 120582
119899119909 119889119909 =
0 119898 = 119899
119871
2
119898 = 119899
119871 119898 = 119899 = 0
int
119871
0
sin 120582119898119909 cos 120582
119899119909119889119909
=
minus
(minus1 + (minus1)119898+119899
) 119871119898
(1198982minus 1198992) 120587
119898 = 119899
0 119898 = 119899
int
119871
0
sin 120582119898119909 sin 120582
119899119909 119889119909 =
0 119898 = 119899
119871
2
119898 = 119899
0 119898 = 119899 = 0
int
119871
0
cos 120582119898119909120577119897
119886(119909) 119889119909
=
41198712
31205872 119897 = 1 119898 = 0
minus
41198712
(41198982
minus 3)
(9 minus 401198982+ 16119898
4) 1205872
119897 = 1 119898 = 0
minus
41198712
31205872 119897 = 2 119898 = 0
41198712
(41198982
minus 3) (minus1)119898
(9 minus 401198982+ 16119898
4) 1205872
119897 = 2 119898 = 0
Shock and Vibration 23
int
119871
0
sin 120582119898119909120577119897
119886(119909) 119889119909
=
0 119897 = 1 119898 = 0
161198712
119898(minus1)119898
(9 minus 401198982+ 16119898
4) 1205872
119897 = 1 119898 = 0
0 119897 = 2 119898 = 0
161198981198712
(9 minus 401198982+ 16119898
4) 1205872
119897 = 2 119898 = 0
int
119871
0
cos 1205821198981199091205771198971015840
119886(119909) 119889119909
=
0 119897 = 1 119898 = 0
161198711198982
(minus1)119898
(9 minus 401198982+ 16119898
4) 120587
119897 = 1 119898 = 0
0 119897 = 2 119898 = 0
161198711198982
(9 minus 401198982+ 16119898
4) 120587
119897 = 2 119898 = 0
int
119871
0
sin 1205821198981199091205771198971015840
119886(119909) 119889119909
=
0 119897 = 1 119898 = 0
4119871119898 (minus3 + 41198982
)
(9 minus 401198982+ 16119898
4) 120587
119897 = 1 119898 = 0
0 119897 = 2 119898 = 0
119871119898(
1
120587 minus 41198982120587
+
3
9120587 minus 41198982120587
) (minus1)119898
119897 = 2 119898 = 0
int
119871
0
1205771198971
119886(119909) 1205771198972
119886(119909) 119889119909 =
1198713
41205872 1198971= 1 1198972= 1
minus
21198713
31205873 1198971= 1 1198972= 2
minus
21198713
31205873 1198971= 2 1198972= 1
1198713
41205872 1198971= 2 1198972= 2
int
119871
0
1205771198971
119886(119909) 1205771198971015840
2
119886(119909) 119889119909 =
0 1198971= 1 1198972= 1
minus
1198712
8120587
1198971= 1 1198972= 2
1198712
8120587
1198971= 2 1198972= 1
0 1198971= 2 1198972= 2
int
119871
0
1205771198971015840
1
119886(119909) 1205771198971015840
2
119886(119909) 119889119909 =
5119871
16
1198971= 1 1198972= 1
minus
119871
2120587
1198971= 1 1198972= 2
minus
119871
2120587
1198971= 2 1198972= 1
5119871
16
1198971= 2 1198972= 2
int
119871
0
119909119899 cos 120582
119898119909 119889119909
=
1198711+119899
1 + 119899
119898 = 0
(1198861+119899HypergeometricPFQtimes [ 12 + 1198992 12 32 + 1198992
minus (14)1198982
1205872
]) times (1 + 119899)minus1
119898 gt 0
(B1)
Conflict of Interests
The authors declare that there is no conflict of interestsregarding the publication of this paper
Acknowledgments
The authors would like to thank the anonymous reviewersfor their very valuable comments The authors also gratefullyacknowledge the financial support from the National NaturalScience Foundation of China (no 51209052) HeilongjiangProvince Youth Science Fund Project (no QC2011C013) andHarbin Science and Technology Development InnovationFoundation of Youth (no 2011RFQXG021)
References
[1] K Liew Y Xiang and S Kitipornchai ldquoResearch on thick platevibration a literature surveyrdquo Journal of Sound and Vibrationvol 180 no 1 pp 163ndash176 1995
[2] J H Chung T Y Chung and K C Kim ldquoVibration analysisof orthotropic mindlin plates with edges elastically restrainedagainst rotationrdquo Journal of Sound and Vibration vol 163 no 1pp 151ndash163 1993
[3] Y K Cheung and D Zhou ldquoVibrations of moderately thickrectangular plates in terms of a set of static Timoshenko beamfunctionsrdquoComputers and Structures vol 78 no 6 pp 757ndash7682000
[4] CMWang ldquoNatural frequencies formula for simply supportedMindlin platesrdquo Journal of Vibration and Acoustics vol 116 no4 pp 536ndash540 1994
[5] K N Saha R C Kar and P K Datta ldquoFree vibration analysisof rectangular Mindlin plates with elastic restraints uniformlydistributed along the edgesrdquo Journal of Sound andVibration vol192 no 4 pp 885ndash902 1996
24 Shock and Vibration
[6] D J Gorman ldquoAccurate free vibration analysis of shear-deformable plates with torsional elastic edge supportrdquo Journalof Sound and Vibration vol 203 no 2 pp 209ndash218 1997
[7] D J Gorman ldquoFree vibration analysis of Mindlin plates withuniform elastic edge support by the superposition methodrdquoJournal of Sound andVibration vol 207 no 3 pp 335ndash350 1997
[8] D J Gorman ldquoAccurate free vibration analysis of point sup-ported mindlin plates by the superposition methodrdquo Journal ofSound and Vibration vol 219 no 2 pp 265ndash277 1999
[9] Y Xiang ldquoVibration of rectangular Mindlin plates resting onnon-homogenous elastic foundationsrdquo International Journal ofMechanical Sciences vol 45 no 6-7 pp 1229ndash1244 2003
[10] Y Xiang and G W Wei ldquoExact solutions for buckling andvibration of stepped rectangular Mindlin platesrdquo InternationalJournal of Solids and Structures vol 41 no 1 pp 279ndash294 2004
[11] Y L Yeh M J Jang and C C Wang ldquoAnalyzing the freevibrations of a plate using finite difference and differential trans-formationmethodrdquoAppliedMathematics andComputation vol178 no 2 pp 493ndash501 2006
[12] Y Xiang S K Lai L Zhou and C W Lim ldquoDSC-Ritz elementmethod for vibration analysis of rectangular Mindlin plateswith mixed edge supportsrdquo European Journal of MechanicsmdashASolids vol 29 no 4 pp 619ndash628 2010
[13] H Nguyen-Xuan C H Thai and T Nguyen-Thoi ldquoIsogeo-metric finite element analysis of composite sandwich platesusing a higher order shear deformation theoryrdquo Composites BEngineering vol 55 pp 558ndash574 2013
[14] C H Thai A J M Ferreira E Carrera and H Nguyen-XuanldquoIsogeometric analysis of laminated composite and sandwichplates using a layerwise deformation theoryrdquo Composite Struc-tures vol 104 pp 196ndash214 2013
[15] C H Thai A Ferreira S Bordas T Rabczuk and HNguyen-Xuan ldquoIsogeometric analysis of laminated compositeand sandwich plates using a new inverse trigonometric sheardeformation theoryrdquo European Journal of Mechanics A Solidsvol 43 pp 89ndash108 2014
[16] H Luong-Van T Nguyen-Thoi G R Liu and P Phung-VanldquoA cell-based smoothed finite elementmethod using three-nodeshear-locking free Mindlin plate element (CS-FEM-MIN3) fordynamic response of laminated composite plates on viscoelasticfoundationrdquo Engineering Analysis with Boundary Elements vol42 pp 8ndash19 2014
[17] T Nguyen-Thoi T Rabczuk T Lam-Phat V Ho-Huu andP Phung-Van ldquoFree vibration analysis of cracked Mindlinplate using an extended cell-based smoothed discrete sheargap method (XCS-DSG3)rdquo Theoretical and Applied FractureMechanics 2014
[18] H Nguyen-Xuan G R Liu and C a Thai-Hoang ldquoAnedge-based smoothed finite element method (ES-FEM) withstabilized discrete shear gap technique for analysis of Reissner-Mindlin platesrdquo Computer Methods in Applied Mechanics andEngineering vol 199 no 9ndash12 pp 471ndash489 2010
[19] P Phung-Van T Nguyen-Thoi T Bui-Xuan and Q Lieu-XuanldquoA cell-based smoothed three-node Mindlin plate element(CS-FEM-MIN3) based on the 119862
0 -type higher-order sheardeformation for geometrically nonlinear analysis of laminatedcomposite platesrdquo Computational Materials Science 2014
[20] NNguyen-Thanh T RabczukHNguyen-Xuan and S BordasldquoAn alternative alpha finite element method with discrete sheargap technique for analysis of isotropicMindlin-Reissner platesrdquoFinite Elements inAnalysis andDesign vol 47 no 5 pp 519ndash5352011
[21] KM Liew Y Xiang and S Kitipornchai ldquoTransverse vibrationof thick rectangular plates-I Comprehensive sets of boundaryconditionsrdquo Computers and Structures vol 49 no 1 pp 1ndash291993
[22] D Zhou ldquoVibrations of Mindlin rectangular plates with elasti-cally restrained edges using static Timoshenko beam functionswith the Rayleigh-Ritz methodrdquo International Journal of Solidsand Structures vol 38 no 32-33 pp 5565ndash5580 2001
[23] D Zhou SH Lo F T KAu andYK Cheung ldquoVibration anal-ysis of rectangular Mindlin plates with internal line supportsusing static Timoshenko beam functionsrdquo International Journalof Mechanical Sciences vol 44 no 12 pp 2503ndash2522 2002
[24] H S Shen J Yang and L Zhang ldquoFree and forced vibrationof Reissner-Mindlin plates with free edges resting on elasticfoundationsrdquo Journal of Sound and Vibration vol 244 no 2 pp299ndash320 2001
[25] Y Xing and B Liu ldquoClosed form solutions for free vibrations ofrectangular Mindlin platesrdquo Acta Mechanica Sinica vol 25 no5 pp 689ndash698 2009
[26] P Gagnon C Gosselin and L Cloutier ldquoA finite strip elementfor the analysis of variable thickness rectangular thick platesrdquoComputers amp Structures vol 63 no 2 pp 349ndash362 1997
[27] T Mlzusawa ldquoVibration of rectangular mindlin plates withtapered thickness by the spline strip methodrdquo Computers andStructures vol 46 no 3 pp 451ndash463 1993
[28] H Nguyen-Xuan and T Nguyen-Thoi ldquoA stabilized smoothedfinite element method for free vibration analysis of Mindlin-Reissner platesrdquo Communications in Numerical Methods inEngineering with Biomedical Applications vol 25 no 8 pp 882ndash906 2009
[29] K M Liew L X Peng and S Kitipornchai ldquoVibration anal-ysis of corrugated Reissner-Mindlin plates using a mesh-freeGalerkin methodrdquo International Journal of Mechanical Sciencesvol 51 no 9-10 pp 642ndash652 2009
[30] Y Hou GWWei and Y Xiang ldquoDSC-Ritz method for the freevibration analysis of Mindlin platesrdquo International Journal forNumerical Methods in Engineering vol 62 no 2 pp 262ndash2882005
[31] F- Liu and K M Liew ldquoVibration analysis of discontinuousMindlin plates by differential quadrature element methodrdquoJournal of Vibration and Acoustics Transactions of the ASMEvol 121 no 2 pp 204ndash208 1999
[32] P Malekzadeh G Karami and M Farid ldquoA semi-analyticalDQEM for free vibration analysis of thick plates with twoopposite edges simply supportedrdquoComputerMethods inAppliedMechanics and Engineering vol 193 no 45ndash47 pp 4781ndash47962004
[33] P Malekzadeh and S A Shahpari ldquoFree vibration analysisof variable thickness thin and moderately thick plates withelastically restrained edges by DQMrdquo Thin-Walled Structuresvol 43 no 7 pp 1037ndash1050 2005
[34] R E Diaz-Contreras and S Nomura ldquoGreenrsquos function appliedto solution ofMindlin platesrdquoComputers and Structures vol 60no 1 pp 41ndash48 1996
[35] T Sakiyama and M Huang ldquoFree vibration analysis of rect-angular plates with variable thicknessrdquo Journal of Sound andVibration vol 216 no 3 pp 379ndash397 1998
[36] M Huang X Q Ma T Sakiyama H Matuda and C MoritaldquoFree vibration analysis of orthotropic rectangular plates withvariable thickness and general boundary conditionsrdquo Journal ofSound and Vibration vol 288 no 4-5 pp 931ndash955 2005
Shock and Vibration 25
[37] I Shufrin and M Eisenberger ldquoVibration of shear deformableplates with variable thicknessmdashfirst-order and higher-orderanalysesrdquo Journal of Sound and Vibration vol 290 no 1-2 pp465ndash489 2006
[38] S A Eftekhari and A A Jafari ldquoAccurate variational approachfor free vibration of variable thickness thin and thick plates withedges elastically restrained against translation and rotationrdquoInternational Journal of Mechanical Sciences vol 68 pp 35ndash462013
[39] W L Li X Zhang J Du and Z Liu ldquoAn exact series solutionfor the transverse vibration of rectangular plates with generalelastic boundary supportsrdquo Journal of Sound amp Vibration vol321 no 1-2 pp 254ndash269 2009
[40] X Shi D Shi W L Li and Q Wang ldquoA unified method forfree vibration analysis of circular annular and sector plateswith arbitrary boundary conditionsrdquo Journal of Vibration andControl 2014
[41] D Y Shi X J Shi W L Li and Q S Wang ldquoFree transversevibrations of orthotropic thin rectangular plates with arbitraryelastic edge supportsrdquo Journal of Vibroengineering vol 16 no 1pp 389ndash398 2014
[42] Y Chen G Jin and Z Liu ldquoFree vibration analysis of circu-lar cylindrical shell with non-uniform elastic boundary con-straintsrdquo International Journal of Mechanical Sciences vol 74pp 120ndash132 2013
[43] L Dai T Yang J Du W L Li and M J Brennan ldquoAn exactseries solution for the vibration analysis of cylindrical shellswith arbitrary boundary conditionsrdquo Applied Acoustics vol 74no 3 pp 440ndash449 2013
International Journal of
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Submit your manuscripts athttpwwwhindawicom
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Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
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Shock and Vibration
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
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International Journal of
4 Shock and Vibration
on 119910 = 0
1198961199100119908 = minus119876
119910 1198701199100120595119910= minus119872
119910119910 1198701199091199100
120595119909= minus119872
119909119910
on 119910 = 119887
119896119910119887119908 = 119876
119910 119870119910119887120595119910= 119872119910119910 119870119909119910119887
120595119909= 119872119909119910
(4)
where 1198961199090
and 119896119909119886
(1198961199100
and 119896119910119887) are linear spring constants
1198701199090
and 119870119909119886
(1198701199100
and 119870119910119887) are the rotational spring con-
stants and 1198701199101199090
and 119870119910119909119886
(1198701199091199100
and 119870119909119910119887
) are the torsionalspring constants at 119909 = 0 and 119886 (119910 = 0 and 119887) respectivelyTherefore arbitrary boundary conditions of the plate can begenerated by assigning the linear springs rotational springsand torsional springs at proper stiffness For instance aclamped boundary (C) is achieved by simply setting thestiffness of the entire springs equal to infinity (which is repre-sented by a very large number 1014) Inversely a free bound-ary (F) is gained by setting the stiffness of the entire springsequal to zero
Thus the total potential energy of the spring restrainedplate which is composed of two parts namely the strainenergy of the moderately thick rectangular plates and thepotential energy stored in the boundary springs can beexpressed as
119880 =
1
2
int
119886
0
int
119887
0
119863[(
120597120595119909
120597119909
+
120597120595119910
120597119910
)
2
minus 2 (1 minus 120583)
times (
120597120595119909
120597119909
120597120595119910
120597119910
minus
1
4
(
120597120595119909
120597119910
+
120597120595119910
120597119909
)
2
)]
+ 120581119866ℎ (119909 119910)
times [(120595119909+
120597119908
120597119909
)
2
+ (120595119910+
120597119908
120597119910
)
2
] 119889119909119889119910
+
1
2
int
119886
0
[(11989611991001199082
+ 11987011991001205952
119910+ 1198701199091199100
1205952
119909)
10038161003816100381610038161003816119910=0
+ (1198961199101198871199082
+ 1198701199101198871205952
119910+ 119870119909119910119887
1205952
119909)
10038161003816100381610038161003816119910=119887
] 119889119909
+
1
2
int
119887
0
[(11989611990901199082
+ 11987011990901205952
119909+ 1198701199101199090
1205952
119910)
10038161003816100381610038161003816119909=0
+ (1198961199091198861199082
+ 1198701199091198861205952
119909+ 119870119910119909119886
1205952
119910)
10038161003816100381610038161003816119909=119886
] 119889119910
(5)
As the springs are considered with no mass while retain-ing certain stiffness the total kinetic energy of themoderatelythick rectangular plates is
119879 =
1205881205962
2
int
119887
0
int
119886
0
[
[
ℎ (119909 119910)1199082
+ ℎ3
(119909 119910) (1205952
119909+ 1205952
119910)
12
]
]
119889119909119889119910
(6)
where 120588 is the mass density and 120596 denotes the natural fre-quency of the plate
In view of satisfying arbitrarily supported boundary con-ditions of the moderately thick rectangular plate the admis-sible functions expressed in the form of the improved Fourierseries are introduced to remove the potential discontinuitieswith the functions and their derivatives Thus the moder-ately thick rectangular plate displacements and rotation areexpressed as
120595119909(119909 119910) =
infin
sum
119898=0
infin
sum
119899=0
119860119898119899
cos 120582119886119898119909 cos 120582
119887119899119910
+
2
sum
119897=1
120577119897
119887(119910)
infin
sum
119898=0
119886119897
119898cos 120582119886119898119909
+
2
sum
119897=1
120577119897
119886(119909)
infin
sum
119899=0
119887119897
119899cos 120582119887119899119910
(7)
120595119910(119909 119910) =
infin
sum
119898=0
infin
sum
119899=0
119861119898119899
cos 120582119886119898119909 cos 120582
119887119899119910
+
2
sum
119897=1
120577119897
119887(119910)
infin
sum
119898=0
119888119897
119898cos 120582119886119898119909
+
2
sum
119897=1
120577119897
119886(119909)
infin
sum
119899=0
119889119897
119899cos 120582119887119899119910
(8)
119908 (119909 119910) =
infin
sum
119898=0
infin
sum
119899=0
119862119898119899
cos 120582119886119898119909 cos 120582
119887119899119910
+
2
sum
119897=1
120577119897
119887(119910)
infin
sum
119898=0
119890119897
119898cos 120582119886119898119909
+
2
sum
119897=1
120577119897
119886(119909)
infin
sum
119899=0
119891119897
119899cos 120582119887119899119910
(9)
where 120582119886119898
= 119898120587119886 120582119887119899= 119899120587119887 and 119860
119898119899 119861119898119899 and 119862
119898119899are
the Fourier coefficients of two-dimensional Fourier seriesexpansions for the displacements and rotation functionsrespectively 119886119897
119898 119887119897119898 119888119897119898 119889119897119898 119890119897119898 and 119891119897
119898are the supplemented
coefficients of the auxiliary functions where 119897 = 1 2 Thespecific expressions of the auxiliary functions 120577119897
119886and 120577
119897
119887are
defined as
1205771
119886(119909) =
119886
2120587
sin(1205871199092119886
) +
119886
2120587
sin(31205871199092119886
)
1205772
119886(119909) = minus
119886
2120587
cos(1205871199092119886
) +
119886
2120587
cos(31205871199092119886
)
1205771
119887(119910) =
119887
2120587
sin(120587119910
2119887
) +
119887
2120587
sin(3120587119910
2119887
)
1205772
119887(119910) = minus
119887
2120587
cos(120587119910
2119887
) +
119887
2120587
cos(3120587119910
2119887
)
(10)
As shown in (7)ndash(9) the supplementary functions 1205771119886(119909)
1205772
119886(119909) 1205771
119887(119910) and 120577
2
119887(119910) are used for the displacement and
rotation expressions The theoretical meaning of introduc-ing these terms into the Fourier series is to remove the
Shock and Vibration 5
potential discontinuities and their derivatives throughout thewhole plate structure including the boundaries and then toeffectively enhance the convergence of the results To ensurethis continuity of selection expressions and correspondingderivatives at any point on the plate the first-order derivativesof the 119909 and119910 directions should exist as indicated by (4)Thisrequirement is guaranteed by the selected supplementaryfunctions because it is easy to verify that
1205771
119886(0) = 120577
1
119886(119886) = 120577
11015840
119886(119886) = 0 120577
11015840
119886(0) = 1
1205772
119886(0) = 120577
2
119886(119886) = 120577
21015840
119886(0) = 0 120577
21015840
119886(119886) = 1
(11)
Similar conditions exist for the 119910-related polynomials1205771
119887(119910) and 120577
2
119887(119910) It has to be mentioned that although the
solution is theoretically exact for the superposition of infinitenumbers of Fourier terms in actual calculation we truncatethe infinite series to 119872 and 119873 to obtain the results withacceptable accuracy
Since the energy expressions and admissible function ofthe plate have been established the remaining task is to deter-mine the Fourier expanded coefficients and supplemented
coefficients in (7)ndash(9) The Lagrangian energy functional (119871)of the plate is written as
119871 = 119879 minus 119880 (12)
Then the Lagrangian expression is minimized by takingits derivatives with respect to these coefficients
120597119871
120597120599
= 0 120599 =
119860119898119899
119886119897
119898119887119897
119899
119861119898119899
119888119897
119898119889119897
119899
119862119898119899
119890119897
119898119891119897
119899
(13)
Since the displacements and rotation components of theplate are chosen as 119872 and 119873 to obtain the results withacceptable accuracy a total of 3 lowast (119872 + 1) lowast (119873 + 1) + 6 lowast
(119872 +119873 + 2) equations are obtainedThey can be summed up in a matrix form
(K minus 1205962M)E = 0 (14)
The unknown coefficients in the displacement expres-sions can be expressed in the vector form as E where
E =
11986000 11986001 119860
11989810158400 11986011989810158401 119860
11989810158401198991015840 119860
119872119873 1198861
0 119886
1
119872 1198862
0 119886
2
119872 1198871
0 119887
1
119873 1198872
0 119887
2
119873
11986100 11986101 119861
11989810158400 11986111989810158401 119861
11989810158401198991015840 119861
119872119873 1198881
0 119888
1
119872 1198882
0 119888
2
119872 1198891
0 119889
1
119873 1198892
0 119889
2
119873
11986200 11986201 119862
11989810158400 11986211989810158401 119862
11989810158401198991015840 119862
119872119873 1198901
0 119890
1
119872 1198902
0 119890
2
119872 1198911
0 119891
1
119873 1198912
0 119891
2
119873
119879
(15)
In (14)K is the stiffness matrix for the plate andM is themass matrix They can be expressed separately as
K =
[
[
[
[
[
[
[
[
[
K1-1 K
1-2 K1-3 sdot sdot sdot K
1-9
K1198791-2 K
2-1 K2-3 sdot sdot sdot K
2-9
K1198791-3 K119879
2-3 K3-3 sdot sdot sdot K
3-9
d
K1198791-9 K119879
2-9 K1198793-9 sdot sdot sdot K
9-9
]
]
]
]
]
]
]
]
]
M =
[
[
[
[
[
[
[
[
[
M1-1 M
1-2 M1-3 sdot sdot sdot M
1-9
M1198791-2 M
2-1 M2-3 sdot sdot sdot M
2-9
M1198791-3 M119879
2-3 M3-3 sdot sdot sdot M
3-9
d
M1198791-9 M119879
2-9 M1198793-9 sdot sdot sdot M
9-9
]
]
]
]
]
]
]
]
]
(16)
The specific expressions for the elements in (16) aregiven in Appendix AMoreover all the necessary expressionswhich will be used in the calculations of the eigenvalues andeigenvectors are given in Appendix B
Obviously the natural frequencies and eigenvectors cannow be readily obtained by solving a standard matrix eigen-problem Since the components of each eigenvector are actu-ally the expansion coefficients of the modified Fourier seriesthe corresponding mode shape can be directly determinedfrom (14) In other words once the coefficient eigenvectorE is
determined for a given frequency the displacement functionsof the plate can be determined by substituting the coefficientsinto (9) When the forced vibration is involved by addingthe work done by external force in the Lagrangian energyfunction and summing the loading vector F on the right sideof (14) the characteristic equation for the forced vibration ofthe moderately thick rectangular plates is readily obtained
3 Numerical Examples and Discussion
In this section a systematic comparison between the cur-rent solutions and theoretical results published by otherresearchers or finite element method (FEM) results is car-ried out to validate the excellent accuracy reliability andfeasibility of the modified Fourier method A comprehensivestudy on the effects of elastic restraint parameters andvarying thickness in two directions is also reported Newresults are obtained for plates subjected to general elasticboundary restraints with nonlinear variable thickness in bothdirections The discussion is arranged as follows Firstly theconvergence of the modified Fourier solution is checked Inaddition the influence of the stiffness of boundary springcomponents is studied Secondly the uniform thicknessplates with various combinations of classical boundary con-ditions elastic boundary conditions and different structureparameters are examined Thirdly the nonuniform thicknessplate with linear variation in both directions various com-binations of classical boundary conditions conditions and
6 Shock and Vibration
Table 1 Frequency parameters Ω = 1205961198872
(120588ℎ119863)12
1205872 for square moderately thick plates with uniform thickness and different boundary
conditions
Boundary conditions M N Model sequence1 2 3 4 5 6 7 8
C-C-C-C
119872 = 119873 = 2 3375 7191 7191 9856 1605 1628 1747 1747119872 = 119873 = 4 3299 6308 6308 8853 1046 1056 1263 1263119872 = 119873 = 6 3294 6282 6282 8801 1037 1047 1254 1254119872 = 119873 = 8 3292 6278 6278 8797 1036 1046 1253 1253119872 = 119873 = 10 3292 6276 6276 8794 1036 1046 1253 1253119872 = 119873 = 12 3292 6276 6276 8793 1036 1046 1252 1252119872 = 119873 = 14 3292 6276 6276 8793 1036 1046 1252 1252119872 = 119873 = 16 3292 6275 6275 8792 1036 1045 1252 1252Reference [4] 3292 6276 6276 8792 1036 1046 1252 1252
E-E-E-E
119872 = 119873 = 2 06153 10861 10861 17898 27709 28417 35010 35010119872 = 119873 = 4 06150 10799 10799 17785 27695 28405 34914 34914119872 = 119873 = 6 06149 10796 10796 17773 27694 28403 34898 34898119872 = 119873 = 8 06149 10796 10796 17771 27693 28403 34893 34893119872 = 119873 = 10 06149 10796 10796 17770 27693 28403 34892 34892119872 = 119873 = 12 06149 10796 10796 17770 27693 28403 34892 34892119872 = 119873 = 14 06149 10796 10796 17770 27693 28403 34892 34892119872 = 119873 = 16 06149 10796 10796 17770 27693 28403 34891 34891Reference [33] 06153 10861 10861 17898 27709 28417 35010 35010
FEM 06150 17097 17097 17774 27694 28406 34898 34898
different structure parameters are examined Then numer-ical solutions for moderately thick rectangular plates withelastic boundary conditions are presented The effects of theelastic restraint parameters are also investigated Finally thenonuniform thickness plate with nonlinear variation in bothdirections and arbitrary boundary conditions is also studied
31 Convergence and Stiffness Value Study Since the Fourierseries is numerically truncated and only finite terms arecounted in actual calculations the proposed solution shouldbe understood as a solution with arbitrary precision Inthis subsection a uniform thickness square moderately thickrectangular plate with completely clamped boundary condi-tion (C-C-C-C) and four edges equally elastically restrainedagainst linear spring constants and rotation spring constantssupports (E-E-E-E 119870
119879= 119896119894(1198873
119863) 119870119877
= 119870119894(119887119863)
119870119879
= 10 and 119870119877
= 5) has been selected to demon-strate the convergence and accuracy of the modified Fouriermethod In Table 1 the first eight frequency parametersΩ = 120596119887
2
(120588ℎ119863)12
1205872 for the considered uniform thickness
square moderately thick rectangular plate with C-C-C-C andE-E-E-E boundary conditions are examinedThe table showsthat the proposedmethod has fast convergence behaviorThemaximum discrepancy in the worst case between the 6 times 6
truncated configuration and the 8times8 one is less than 0064In order to fully illustrate the convergence of the presentmethod Figures 3 4 and 5 present the 1st and 8th frequencyparameters Ω = 120596119887
2
(120588ℎ119863)12
1205872 with various truncated
numbers 119872 = 119873 subjected to different boundary conditionand aspect ratios A highly desired convergence characteristicis observed such that (a) sufficiently accurate results can beobtained with only a small number of terms in the seriesexpansions and (b) the solution is consistently refined asmore
terms are included in the expansions However this shouldnot constitute a problem in practice because one can alwaysverify the accuracy of the solution by increasing the trunca-tion number until a desired numerical precision is achievedAs a matter of fact this ldquoquality controlrdquo scheme can be easilyimplemented automatically In modal analysis the naturalfrequencies for higher-order modes tend to converge slower(see Table 1) Thus an adequate truncation number shouldbe dictated by the desired accuracy of the largest naturalfrequencies of interest In view of the excellent numericalbehavior of the current solution the truncation numbers willbe simply set as119872 = 119873 = 12 in the following calculations
As far as the accuracy of the present method is con-cerned the converged solutions of the present method arein excellent agreement with both the results reported byreference data and the finite element results For C-C-C-C boundary conditions the max discrepancy between thepresent results and the reference data does not exceed 0011for the worst case and in most cases is 0 Comparing theresults with exact solutions [4] it is observed that eight termsare sufficient to obtain accurate resultsMoreover with regardto the E-E-E-E boundary condition the max discrepancybetween the present results and the reference data does notexceed 031 for the worst case and in most cases is 01Regarding the results with DQM solutions [33] it can be seenthat the six terms are sufficient to obtain enough accurateresults In addition it is clear that the results of the presentapproach with just 663 DOFs (119872 times 119873 = 12 times 12) canpredict the vibration characteristics accurately Most of themare identical to those obtained from finite element method(FEM) with 10201 DOFs (S4R 001m times 001m) That isto say it needs only 662 DOFs compared with FEM toobtain the same precision solutions for the considered case
Shock and Vibration 7
1
2
3
4
5
6
The 1st orderThe 3rd order
101 103 105 107 109 1011 1013
Ω
Ki (Nm)
(a)
2
3
4
5
6
101 103 105 107 109 1011 1013
The 1st orderThe 3rd order
Ω
Ki (Nmrad)
(b)
3
4
5
6
101 103 105 107 109 1011 1013
Kij (Nmrad)
The 1st orderThe 3rd order
Ω
(c)
Figure 2 The effect of boundary spring stiffness on the natural frequencies Ω (a) translation spring (b) rotation spring and (c) torsionalspring
On the same hardware (Intel i7-39GHz) the computing timeof the present formulation for the solution (119872times119873 = 12times12)implemented in optimized MATLAB scripts is about 2125 swhereas the finite element solution consumes 34578 s that isat least 16 times more CPU time than the present method forthe same problem
As mentioned earlier in the current modeling frame-work all the classical boundary conditions and their com-binations can be conveniently viewed as special cases whenthe stiffness for the normal and tangential boundary springsbecomes zero andor infinitely large Thus the effects of thestiffness of the translation (119896
119894) rotation (119870
119894) and torsional
springs (119870119894119895) on the modal characteristics should be inves-
tigated As shown in the Figures 2(a)ndash2(c) the first and thethird frequency parameters are separately obtained by vary-ing the stiffness of one group of the boundary springs from
extremely large (1014) to extremely small (100) while assigningthe other group of the springs infinite stiffness (1014) It canbe found in Figure 2(a) that the frequency parameter almostkeeps at a level when the stiffness of the translation springs islarger than 1012 or smaller than 107 In Figure 2(b) the influ-ences of the rotation springs on frequency parameters aregiven It is shown that the frequency curves change greatlywithin the stiffness range (106 to 1010) while out of this rangethe frequency curves separately keep level In Figure 2(c) theinfluences of the torsional springs on frequency parametersare given It is shown that the frequency curves almost changewhen the stiffness changes in the whole range
Based on the analysis it can be found that the torsionalsprings almost have no effect on the structure Also the rela-tionship between the rotation springs and twisting momentcan be seen from the boundary condition expression Then
8 Shock and Vibration
32
36
40
44
48
52
2
3
4
5
6
7
8
Ω Ω
0 2 4 6 8 10 12 14 16 18M = N
ab = 1
ab = 32
(a) The 1st order
0 2 4 6 8 10 12 14 16 180
20
40
60
80
0
50
100
150
200
250
300
M = N
Ω Ω
ab = 1
ab = 32
(b) The 8th order
Figure 3 The effect of numerically truncated numbers119872 = 119873 on the natural frequenciesΩ for C-C-C-C boundary condition
0 2 4 6 8 10 12 14 16 18125
130
135
140
145
150
088
090
092
094
096
098
Ω Ω
M = N
ab = 1
ab = 32
(a) The 1st order
0 2 4 6 8 10 12 14 16 18
55
60
65
70
75
80
40
45
50
55
60
65
70
75
80Ω Ω
M = N
ab = 1
ab = 32
(b) The 8th order
Figure 4 The effect of numerically truncated numbers119872 = 119873 on the natural frequenciesΩ for F-F-F-F boundary condition
the twisting moments have small effect on the vibration char-acteristics of the structure However in this paper in order toget a more accurate prediction of the vibration characteristicsof the structure the twisting moment on boundary edges istaken into account In the latter study in addition to the factthat free boundary is not unexpected considering the torsionspring the other boundary conditions are introduced into atorsion spring and the spring stiffness takes infinity FormFigure 2 analysis it also concluded that the translation springhas a wider influence range than the rotation spring on thefrequency parameters that is for the translation springs thestable frequency parameters appearwhen the stiffness ismorethan 1012 or less 107 while for the rotation springs when thestiffness value is assigned more than 1010 the frequency para-meters become smoothThus it is suitable to use 1014 to sim-ulate the infinite stiffness value in the model validation partsand in the following examples
32 Uniform Thickness Moderately Thick Plates with Classicaland Elastic Boundary Conditions In this subsection themodified Fourier solution is applied to deal with vibrationproblems of uniform thickness moderately thick rectangu-lar plates subject to the classical boundary conditions andarbitrary elastic boundary conditions In present work threegroups of continuously distributed boundary springs areintroduced to simulate the given or typical boundary condi-tions As mentioned earlier the stiffness of these boundarysprings can take any value from zero to infinity to bettermodel many real-world restraint conditions Taking edge 119909 =0 for example the corresponding spring stiffness for the threetypes of classical boundaries and elastic boundaries is
completely free 119876119909= 0119872
119909119909= 0 and119872
119909119910= 0
1198961199090= 0 119870
1199090= 0 119870
1199101199090= 0 (17a)
Shock and Vibration 9
0 2 4 6 8 10 12 14 16 18
166
168
170
172
152
156
160
164
168
Ω Ω
ab = 1
ab = 32
M = N
(a) The 1st order
0 2 4 6 8 10 12 14 16 1850
55
60
65
70
4
5
6
7
8
9
10
Ω Ω
M = N
ab = 1
ab = 32
(b) The 8th order
Figure 5 The effect of numerically truncated numbers 119872 = 119873 on the natural frequencies Ω with four edges elastically restrained againsttranslation and rotation (119870
119879= 119896119894(1198873
119863) 119870119877= 119870119894(119887119863) 119870
119879= 100 and 119870
119877= 100)
simply supported 119908 = 0119872119909119909= 0 and 120595
119910= 0
1198961199090= 1014
1198701199090= 0 119870
1199101199090= 1014
(17b)
completely clamped 119908 = 0 120595119909= 0 and 120595
119910= 0
1198961199090= 1014
1198701199090= 1014
1198701199101199090
= 1014
(17c)
arbitrarily elastic boundary 119908 = 0 120595119909
= 0 120595119910
= 0119876119909
= 0119872119909119909
= 0 and119872119909119910
= 0
1198961199090= Γ1199081199090
1198701199090= Γ1199091199090
1198701199101199090
= 1014
(17d)
The appropriateness of the three classical boundariesdefined in (17a)ndash(17c) will be proved by several examplesgiven in following the arbitrary elastic boundaries are alsodefined in (17d) and the Γ
119908119894(Γ119908119894= 1198961198941198960 1198960= 1 times 10
9Nm2119894 = 1199090 1199091198861199100119910119887) and Γ
119909119894(Γ119909119894= 11987011989411987001198700= 1times10
9Nmrad119894 = 1199090 119909119886 1199100 119910119887) elastic restraint parameters representcorresponding spring stiffness For the sake of simplicitya four-letter string is employed to represent the restraintcondition of a plate such that F-C-S-E identifies the platewithedges 119909 = 0 119910 = 0 119909 = 119886 and 119910 = 119887 having free clampedshear-supported restrained and elastic boundary conditionsrespectively
As for the first case a uniform thickness moderately thickplate with different classical boundaries and structure param-eters is investigated here In Table 2 the comparison of thefirst eight frequency parametersΩ = 120596119887
2
(120588ℎ119863)12
1205872 of the
considered plate is presented The S-S-S-S C-F-F-F S-S-F-FF-F-F-F and S-C-S-C boundary conditions are performed inthe comparison Excellent agreements are observed betweenthe solutions obtained by the modified Fourier method thereferential data and finite element method (FEM) results forthe uniform thickness moderately thick rectangular platesIt is also verified that the definition of the three types of
classical boundaries in (17a)ndash(17c) is appropriate In additionthe elastic boundary conditions (17d) are also verified Inthe next two examples we will account for the vibrationof moderately thick plate with elastic edge supports Thefirst model considered is an S-S-S-S square moderately thickplate with all edges elastically rotationally restrained Thatis 1198701199090
= 119870119909119886
= 1198701199100
= 119870119910119887
= Γ119909 and all the
other restraining springs are set to have an infinite stiffness(namely represented by 1014 in numerical calculation) Thesix frequency parameters Ω = 120596119887
2
(120588ℎ119863)12
1205872 are given
in Table 3 for several different restraining coefficient valuesthe finite element method (FEM) results calculated usingABAQUSare also listed in this table as a referenceThe secondmodel concerns a complete square moderately thick platewith all edges elastically restrained That is 119896
1199090= 119896119909119886
=
1198961199100
= 119896119910119887
= Γ119908and 119870
1199090= 119870119909119886
= 1198701199100
= 119870119910119887
= Γ119909 The
six frequency parameters Ω = 1205961198872
(120588ℎ119863)12
1205872 are given in
Table 4 for several different restraining coefficient values thefinite element method (FEM) results are also listed in Table 4as a reference It can be clearly seen that the comparison isextremely good which implies that the current method isable tomake correct predictions for themodal characteristicsof moderately thick plate with not only classical boundaryconditions but also elastic edge restraints
The excellent agreements between the solution obtainedby the modified Fourier method and the referential datafor the moderately thick plate subjected to the combina-tions of classical boundary conditions and elastic boundaryconditions given in Tables 2ndash4 indicate that the proposedmethod is sufficiently accurate to deal with uniform thicknessmoderately thick plate with arbitrary boundary conditions
33 Linearly VariationThickness Moderately Thick Plates withClassical and Elastic Boundary Conditions In the theoreticalformulations this paper concerns the varying thicknessmoderately thick plates with classical and elastic boundary
10 Shock and Vibration
Table 2 Frequency parameters Ω = 1205961198872
(120588ℎ119863)12
1205872 for uniform thickness moderately thick plates with different classical boundaries and
structure parameters
Boundary condition ab Model sequence1 2 3 4 5 6 7 8
S-S-S-S
11931 4605 4605 7064 8605 8605 10792 10792
(1933a) (4611) (4611) (7067) (8607) (8607) (10799) (10780)(1931b) (4605) (4605) (7064) (8605) (8605) (10792) (10792)
321408 2648 4127 4605 5261 7064 7153 8184(1410a) (2651) (4128) (4608) (5267) (7074) (7155) (8188)(1408b) (2648) (4127) (4605) (5261) (7064) (7151) (8184)
C-F-F-F
10348 0817 2035 2583 2862 4816 5478 5774
(0348a) (0817) (2038) (2586) (2865) (4821) (5504) (5793)(0348b) (0816) (2034) (2582) (2860) (4811) (5477) (5772)
320155 0502 0941 1665 2292 2613 3303 3498
(0154a) (0502) (0941) (1663) (2295) (2618) (3305) (3500)(0154b) (0501) (0940) (1662) (2292) (2612) (3298) (3494)
S-S-F-F
10333 1678 1875 3559 4720 4946 6474 6632
(0335a) (1680) (1878) (3600) (4721) (4949) (6478) (6642)(0333b) (1677) (1874) (3557) (4718) (4945) (6472) (6631)
320222 0938 1631 2363 2573 4082 4482 4753
(0223a) (0940) (1632) (2364) (2575) (4092) (4483) (4759)(0222b) (0938) (1631) (2361) (2571) (4077) (4479) (4752)
F-F-F-F1 1290 1919 2363 3235 3235 5605 5605 5646
(1291a) (1920) (2365) (3239) (3239) (5607) (5607) (6648)
32 0865 0948 1961 2166 2461 2843 3551 4026(0865a) (0949) (1963) (2168) (2465) (2844) (3556) (4034)
S-C-S-C1 2700 4971 5990 7973 8787 10250 11334 12024
(2701a) (4971) (5993) (7975) (8788) (10253) (11335) (12028)
32 2348 3253 4972 5646 6489 7381 7974 9930(2348a) (3255) (4974) (5647) (6590) (7384) (7801) (9932)
aResults in parentheses are taken from FEMbResults in parentheses are taken from [11]
conditions The varying thickness function ℎ(119909 119910) can beexpressed as ℎ
0(1 + 120572119909
119904
)(1 + 120573119910119905
) in which the ℎ0 120572 and
120573 represent the initial thickness gradient in 119909 direction andgradient in 119910 direction When the indexes 119904 and 119905 take thevalue 119904 = 119905 = 1 the analyticalmodel imitates the linearly vari-ation thickness moderately thick plates structure In order tounify the description and facilitate the analytical calculationsof the involved integrals all the thickness variation functionscan be expanded into either 1D or 2D Fourier cosine seriesresulting in
ℎ (119909 119910) =
infin
sum
119898=0
infin
sum
119899=0
ℎ0119902119898119899
cos 119898120587119909119886
cos119898120587119910
119887
(18)
where119902119898119899
=
4
119886119887
int
119887
0
int
119886
0
(1 + 120572119909119904
) (1 + 120573119910119905
) cos 119898120587119909119886
cos119898120587119910
119887
119889119909 119889119910
(19)
In order to prove the validity of the present methodfor the vibration of linearly variation thickness moderately
thick plates with arbitrary boundary conditions the typicalclassical boundary conditions as the first case will be con-sidered In Table 5 the comparison of the first six frequencyparameters Ω = 120596119886
2
(120588ℎ01198630)12 of the moderately thick
plates with linearly varying thickness is presentedThe S-S-S-S C-F-F-F S-S-F-F C-C-C-C and S-C-S-C boundary condi-tions are performed in the comparison Excellent agreementsare observed between the solutions obtained by the modifiedFourier method and finite element method (FEM) results forthemoderately thick plates with linear variation thickness Toinvestigate the influence of the aspect ratio on the uniformthickness and nonuniform thickness moderately thick platesthe effect on the frequency parameters for plates with S-S-S-Sboundary conditions is presented in Figure 6 The thicknessfunctions are ℎ
0and ℎ0(1+05times119909)(1+05times119910) respectively It
is seen from Figure 6 that the influence of aspect ratios on thefrequency parameters for nonuniform thickness moderatelythick plates is more complicated
In the next two examples we also account for the vibra-tions of moderately thick plate with linear variation thicknessand elastic edge supports The first model considered is an S-S-S-S square moderately thick plate with all edges elastically
Shock and Vibration 11
Table 3 Frequency parameters Ω = 1205961198872
(120588ℎ119863)12
1205872 for S-S-S-S moderately thick plates (119886119887 = 1) with uniform thickness and elastic
rotation support (1198701199090= 119870119909119886= 1198701199100= 119870119910119887= Γ119909)
Γ119909
aℎ0119886
Model sequence1 2 3 4 5 6
0
005 1945 4834 4834 7591 9515 9519(1942b) (4836) (4836) (7584) (9547) (9552)
01 1856 4500 4500 6829 8499 8513(1858b) (4510) (4510) (6843) (8539) (8554)
015 1758 4114 4114 6040 7421 7445(1761b) (4126) (4126) (6060) (7462) (7486)
02 1656 3733 3733 5335 6468 6499(1660b) (3747) (3747) (5357) (6508) (6538)
05
005 3541 7071 7071 10227 12318 12399(3546b) (7091) (7091) (10253) (12386) (12468)
01 3274 6242 6242 8719 10302 10421(3282b) (6267) (6267) (8754) (10370) (10488)
015 2947 5361 5361 7290 8465 8606(2956b) (5387) (5387) (7327) (8524) (8666)
02 2624 4599 4599 6148 7057 7199(2634b) (4624) (4624) (6182) (7107) (7250)
1
005 3541 7072 7072 10228 12320 12401(3547b) (7093) (7093) (10254) (12388) (12470)
01 3278 6248 6248 8727 10311 10429(3286b) (6274) (6274) (8762) (10378) (10497)
015 2957 5374 5374 7305 8478 8620(2966b) (5400) (5400) (7341) (8537) (8680)
02 2641 4617 4617 6166 7070 7215(2651b) (4643) (4643) (6200) (7121) (7268)
15
005 3542 7073 7073 10229 12320 12401(3547b) (7093) (7093) (10254) (12388) (12471)
01 3280 6251 6251 8729 10313 10432(3287b) (6276) (6276) (8765) (10381) (10500)
015 2960 5379 5379 7309 8482 8625(2969b) (5405) (5405) (7346) (8541) (8685)
02 2647 4623 4623 6172 7074 7221(2657b) (4649) (4649) (6206) (7125) (7274)
2
005 3542 7073 7073 10229 12321 12402(3547b) (7093) (7093) (10255) (12389) (12471)
01 3280 6252 6252 8730 10315 10434(3288b) (6277) (6277) (8766) (10382) (10501)
015 2962 5381 5381 7312 8484 8628(2971b) (5407) (5407) (7348) (8543) (8688)
02 2650 4627 4627 6175 7076 7224(2660b) (4652) (4652) (6210) (7128 (7277)
aΓ119909= 1198701198941198700 (1198700= 1 times 10
9 Nmrad 119894 = 1199090 119909119886 1199100 119910119887)bResults in parentheses are taken from FEM
12 Shock and Vibration
Table 4 Frequency parameters Ω = 1205961198872
(120588ℎ119863)12
1205872 for moderately thick plates (119886119887 = 1) with uniform thickness and all edge elastic
restraints (1198701199090= 119870119909119886= 1198701199100= 119870119910119887= Γ119909and 119896
1199090= 119896119909119886= 1198961199100= 119896119910119887= Γ119908)
Γ119909
aΓ119908
aℎ0119886
Model sequence1 2 3 4 5 6
05 0
005 0986 1967 3893 3893 4850 4850(0988b) (1971) (3897) (3897) (4852) (4852)
01 0916 1835 3515 3522 4375 4375(0917b) (1837) (3519) (3528) (4341) (4341)
015 0789 1635 2983 3027 3771 3771(0792b) (1638) (2988) (3029) (3775) (3775)
02 0646 1444 2497 2601 3249 3249(0652b) (1448) (2508) (2621) (3254) (3254)
1 05
005 3417 3417 4431 5294 5461 6448(3419b) (3419) (4435) (5303) (5478) (6456)
01 1593 1593 2401 3849 3861 4729(1611b) (1611) (2408) (3852) (3866) (4767)
015 1122 1122 1913 3272 3290 4038(1127b) (1227) (1924) (3284) (3314) (4069)
02 0890 0890 1640 2781 2829 3476(0899b) (0899) (1654) (2787) (2834) (3497)
15 1
005 4325 4325 5619 6273 6571 7619(4334b) (4334) (5681) (6299) (6642) (7642)
01 2013 2013 2812 4088 4116 4993(2024b) (2024) (2827) (4122) (4187) (4994)
015 1338 1338 2108 3430 3443 4196(1342b) (1342) (2112) (3467) (3482) (4200)
02 1038 1038 1772 2936 2965 3613(1042b) (1042) (1786) (2937) (2977) (3624)
2 2
005 5231 5231 6919 7630 8002 9220(5238b) (5238) (6931) (7637) (8011) (9241)
01 2594 2594 3439 4470 4544 5428(2597b) (2597) (3442) (4495) (4557) (5438)
015 1653 1653 2403 3613 3632 4392(1662b) (1662) (2414) (3627) (3646) (4404)
02 1238 1238 1949 3077 3099 3751(1244b) (1244) (1964) (3094) (3116) (3788)
aΓ119909= 1198701198941198700 (1198700= 1 times 10
9 Nmrad 119894 = 1199090 119909119886 1199100 119910119887) Γ119908= 1198961198941198960 (1198960= 1 times 10
9 Nm 119894 = 1199090 119909119886 1199100 119910119887)bResults in parentheses are taken from FEM
05 10 15 20 25 300
2
4
6
8
10
12
14
16
ab
The 1st orderThe 2nd orderThe 3rd order
The 4th orderThe 5th orderThe 6th order
Ω
(a)
05 10 15 20 25 30
30
60
90
120
150
180
ab
The 1st orderThe 2nd orderThe 3rd order
The 4th orderThe 5th orderThe 6th order
Ω
(b)
Figure 6 The effect of aspect ratio 119886119887 on the natural frequenciesΩ for S-S-S-S boundary condition (a) uniform thickness and (b) nonuni-form thickness
Shock and Vibration 13
Table 5 Frequency parameter Ω = 1205961198862
(120588ℎ01198630)12 for moderately thick plates with linearly varying thickness in different boundary
conditions (ℎ(119909 119910) = ℎ0(1 + 05119909)(1 + 05119910))
Boundary condition 119886119887 ℎ0119886
Model sequence1 2 3 4 5 6
S-S-S-S
101 27261 62397 62926 92423 111537 111732
(27263a) (62410) (63842) (92445) (111867) (111921)
02 23103 47010 47191 65018 75831 75865(23489a) (47209) (47349) (65057) (76605) (76473)
3201 59338 99101 138798 151102 166710 204855
(59874a) (99249) (138177) (151362) (166644) (205381)
02 41175 63765 63765 73598 83956 89084(41586a) (63313) (63669) (73661) (83918) (89547)
C-F-F-F
101 4173 10331 27407 37826 38809 64095
(4856a) (10680) (27513) (37203) (39135) (64858)
02 3983 8702 21403 29443 30286 46705(4118a) (9306) (21990) (29670) (30602) (47156)
3201 4868 15488 35620 63523 89969 100089
(4887a) (15320) (35175) (63353) (89159) (100783)
02 4469 9763 22338 42188) 50104 56331(4944a) (9636) (22087) (42177) (50580) (56559)
S-S-F-F
101 4117 24795 26216 48567) 68195 69667
(4789a) (24149) (26751) (49324) (68842) (70575)
02 3820 20516 21924 37523 48980 50327(4103a) (20041) (22125) (37221) (48894) (50622)
3201 7935 39454 65063 92877 92877 136176
(7856a) (40145) (65162) (92462) (93022) (136336)
02 6389 26149 35627 49539 56844 65170(6406a) (26536) (35626) (49831) (57144) (65981)
C-C-C-C
101 44471 80299 80681 109422 126871 128207
(45102a) (80271) (80656) (110015) (126328) (128888)
02 32162 52781 52933 69410 78474 79482(32240a) (52629) (53166) (70063) (79059) (80266)
3201 82975 115587 151992 161579 176601 212330
(83139a) (115838) (152200) (162193) (177058) (212849)
02 48577 66022 83468 90830 97474 98157(49187a) (66259) (84065) (91749) (97827) (98777)
S-C-S-C
101 36847 66664 76965 101045 113594 125551
(37901a) (68568) (78356) (101200) (113978) (125651)
02 27673 48469 51220 67027 76358 78241(27897a) (48685) (51491) (67447) (77530) (79573)
3201 77378 106848 150079 154569 173120 208391
(78003a) (107058) (150991) (155171) (173099) (208402)
02 45895 64732 73393 83282 90325 97089(45803a) (64131) 73656 83801 90518 97015
aResults in parentheses are taken from FEM
restrained That is 1198701199090
= 119870119909119886
= 1198701199100
= 119870119910119887
= Γ119909 The six
frequency parameters Ω = 1205961198862
(120588ℎ01198630)12 are given in
Table 6 for several different restraining coefficient valuesthe finite element method (FEM) results calculated usingABAQUS are also presented as a referenceThe secondmodelis a complete square moderately thick plate with all edgeselastically restrainedThat is 119896
1199090= 119896119909119886= 1198961199100= 119896119910119887= Γ119908and
1198701199090= 119870119909119886
= 1198701199100= 119870119910119887= Γ119909 The six frequency parameters
Ω = 1205961198862
(120588ℎ01198630)12 are given in Table 7 for several different
restraining coefficient values the finite element method(FEM) results are also listed as a reference It can be clearlyseen that the comparison is extremely good which impliesthat the current method is able to make correct predictionsfor the modal characteristics of linearly varying thickness
14 Shock and Vibration
Table 6 Frequency parameter Ω = 1205961198862
(120588ℎ01198630)12 for S-S-S-S moderately thick plates (119886119887 = 1) with linearly varying thickness and elastic
rotation support (ℎ(119909 119910) = ℎ0(1 + 05119909)(1 + 05119910) 119870
1199090= 119870119909119886= 1198701199100= 119870119910119887= Γ119909)
Γ119909
aℎ0119886
Model sequence1 2 3 4 5 6
0
005 28809 70080 70891 108967 134884 135295(28897b) (70334) (71129) (109112) (135254) (135772)
01 27261 62398 62926 92423 111540 111735(26606b) (61784) (62303) (91193) (111323) (111648)
015 25231 54192 54505 77118 91228 91312(25204b) (55297) (55595) (77503) (91871) (91941)
02 23103 47010 47191 65018 75832 75866(23947b) (47071) (47208) (65404) (76531) (76589)
05
005 23103 47010 47191 65018 75832 75866(23210b) (47245) (47358) (65241) (76122) (76345)
01 48377 94650 95597 135484 162635 163501(48524b) (94891) (95992) (135845) (167012) (164024)
015 36239 70939 71298 99997 118584 119009(36647b) (71214) (71687) (100024) (119114) (119457)
02 28749 56937 57037 79214 93009 93162(29142b) (57034) (57544) (79654) (93668) (93754)
1
005 49651 96759 97619 138089 165368 166304(49755b) (97245) (97964) (138544) (165864) (166878)
01 38951 73735 74251 102748 121079 121709(39021b) (74145) (74664) (102874) (121613) (121817)
015 30630 58406 58533 80432 93984 94255(30638b) (58684) (58644) (80874) (94121) (94478)
02 25548 48636 48674 66165 76708 76824(25687b) (48788) (48992) (66544) (76824) (76944)
15
005 50118 97561 98374 139096 166421 167382(50247b) (97877) (98661) (139247) (166875) (167662)
01 40310 75216 75783 104229 122402 123164(40366b) (75542) (75921) (104557) (122874) (123385)
015 31837 59352 59530 81245 94611 94983(31902b) (59451) (59871) (81375) (94784) (95105)
02 26302 49119 49156 66528 76954 77130(26418b) (49354) (49334) (66921) (77010) (77470)
2
005 50360 97983 98769 139631 166979 167954(50472b) (98005) (99114) (140154) (167188) (167986)
01 41130 76146 76724 105165 123229 124082(41421b) (76642) (78114) (105661) (123278) (124974)
015 32687 60025 60245 81832 95052 95508(32778b) (60244) (60549) (82146) (95114) (95842)
02 26893 49492 49539 66815 77138 77370(26922b) (49874) (49924) (66987) (77367) (77588)
aΓ119909= 1198701198941198700 (1198700= 1 times 10
9 Nmrad 119894 = 1199090 119909119886 1199100 119910119887)bResults in parentheses are taken from FEM
Shock and Vibration 15
Table 7 Frequency parameterΩ = 1205961198862
(120588ℎ01198630)12 for moderately thick plates (119886119887 = 1) with linearly varying thickness and elastic rotation
and translation support (ℎ(119909 119910) = ℎ0(1 + 05119909)(1 + 05119910) 119870
1199090= 119870119909119886= 1198701199100= 119870119910119887= Γ119909 and 119896
1199090= 119896119909119886= 1198961199100= 119896119910119887= Γ119908)
Γ119909
aΓ119908
aℎ0119886
Model sequence1 2 3 4 5 6
05 0
005 14751 14917 28232 55251 55397 69512(14780b) (14822) (29252) (55476) (55505) (70904)
01 11811 11910 23857 44462 45037 55453(11818b) (11897) (24711) (44376) (45257) (56511)
015 8487 8514 20011 34783 36335 44970(8500b) (8544) (20738) (34404) (36663) (45625)
02 6073 6073 17431 28679 30805 38010(6096b) (6106) (18039) (28129) (31165) (38311)
1 05
005 21225 32173 32173 44381 64061 65222(20708b) (32097) (32285) (45110) (64182) (65355)
01 8140 16878 16941 28100 48663 49015(7943b) (16838) (16970) (28888) (48700) (49171)
015 4493 11859 11878 22429 38352 39247(4383b) (11867) (11891) (23125) (38177) (39488)
02 2932 8690 8691 18919 31108 32600(3540b) (8791) (8791) (16229) (27520) (27889)
15 1
005 27819 40919 41430 54595 70792 72896(27146b) (41008) (41434) (55256) (70895) (73051)
01 11285 20068 20230 31101 51046 51429(11012b) (20025) (20270) (31849) (51126) (51569)
015 6292 13912 13922 24132 40500 41120(6139b) (13912) (13934) (24808) (40426) (41316)
02 4121 10321 10331 20051 32796 33921(4021b) (10329) (10357) (20635) (32540) (34178)
2 2
005 34615 51972 53088 68255 81344 84510(33806b) (52100) (53157) (68891) (81433) (84761)
01 15416 24659 24829 35615 53879 54494(15045b) (24625) (24888) (36303) (53970) (54626)
015 8740 16420 16481 26353 42417 42946(8529b) (16414) (16499) (26996) (42398) (43114)
02 5761 12127 12127 21434 34286 35192(5622b) (12134) (12148) (21998) (34110) (35415)
aΓ119909= 1198701198941198700 (1198700= 1 times 10
9 Nmrad 119894 = 1199090 119909119886 1199100 119910119887) Γ119908= 1198961198941198960 (1198960= 1 times 10
9 Nm 119894 = 1199090 119909119886 1199100 119910119887)bResults in parentheses are taken from FEM
moderately thick plate with not only classical boundaryconditions but also elastic edge restraints
As the last case of this subsection the influence of thegradient 120572 and 120573 on the fundamental frequency parametersfor a linearly varying thickness moderately thick plate isinvestigated The model is a square moderately thick platewith all edges elastically restrainedThat is 119896
1199090= 119896119909119886= 1198961199100=
119896119910119887= Γ119908= 2 and 119870
1199090= 119870119909119886
= 1198701199100= 119870119910119887= Γ119909= 2 The six
frequency parameters Ω = 1205961198862
(120588ℎ01198630)12 are given in
Table 8 for several different 120572 and 120573 values the finite elementmethod (FEM) results are also listed as a reference Againgood agreement can be observed Through Table 8 it is alsofound that the frequency parameter increases with increasinggradient parameters
The above studies are given as linearly varying thicknessmoderately thick plates with several of boundary condition
and different structure parameters In the next section vibra-tion results for the plates subjected to nonlinear variationthickness will be presented
34 Nonlinearly Varying Thickness Moderately Thick Plateswith Classical and Elastic Boundary Conditions In Sec-tion 33 the linearly varying thickness moderately thickplates were studied However in the practical engineeringapplications the varying thickness of amoderately thick platemay not always be linear variation in nature A variety ofpossible thickness varying cases may be encountered in prac-tice Therefore the moderately thick plates with nonlinearvariation thickness subjected to general elastic edge restraintsare examined in this subsection For the sake of brevity theindexes 119904 and 119905 will be chosen as 2 to imitate the nonlinearlyvarying thickness moderately thick plates structure in this
16 Shock and Vibration
Table 8 Frequency parameterΩ = 1205961198862
(120588ℎ01198630)12 for elastic support moderately thick plates (119886119887 = 1) with different gradient (119896
1199090= 119896119909119886=
1198961199100= 119896119910119887= Γ119908= 2 119870
1199090= 119870119909119886= 1198701199100= 119870119910119887= Γ119909= 2)
120572 120573 ℎ0119886
Model sequence1 2 3 4 5 6
120572 = 120573 = 01
005 31275 52615 52657 68954 76146 79850(31303a) (52664) (52675) (69033) (76174) (79948)
01 17270 25244 25250 33949 46050 46625(17271a) (25255) (25258) (34089) (46122) (46708)
015 10190 16266 16274 24164 37327 37526(10189a) (16269) (16280) (24323) (37401) (37622)
02 6780 12258 12260 19685 31452 31754(6780a) (12265) (12270) (19851) (31514) (31864)
120572 = 120573 = 02
005 32530 52947 53141 68954 77034 80589(32577a) (53014) (53203) (69172) (77088) (80773)
01 16882 24979 24979 34154 48057 48575(16883a) (24983) (25001) (34433) (48134) (48667)
015 9804 16284 16309 24683 38844 39085(9803a) (16283) (16322) (24980) (38909) (39191)
02 6499 12290 12295 20133 32354 32770(6498a) (12296) (12307) (20420) (32387) (32895)
120572 = 120573 = 03
005 33516 52845 53308 68704 78203 81561(33563a) (52939) (53389) (69055) (78271) (81789)
01 16419 24799 24830 34527 50061 50583(16419a) (24790) (24869) (34947) (50143) (50688)
015 9430 16334 16378 25236 40193 40506(9429a) (16329) (16395) (25660) (40243) (40625)
02 6235 12288 12293 20572 33102 33663(6235a) (12296) (12306) (20965) (33090) (33811)
120572 = 120573 = 04
005 34215 52463 53264 68430 79654 82865(34252a) (52575) (53346) (68921) (79732) (83113)
01 15921 24695 24793 35027 52011 52571(15921a) (24674) (24842) (35583) (52098) (52688)
015 9074 16388 16444 25797 41379 41789(9074a) (16381) (16464) (26336) (41404) (41928)
02 5990 12236 12238 21002 33735 34461(5989a) (12249) (12250) (21487) (33653) (34641)
120572 = 120573 = 05
005 34615 51972 53088 68255 81344 84510(33806a) (52100) (53157) (68891) (81433) (84761)
01 15416 24659 24829 35615 53879 54494(15045a) (24625) (24888) (36303) (53970) (54626)
015 8740 16420 16481 26353 42417 42946(8529a) (16414) (16499) (26996) (42398) (43114)
02 5761 12127 12127 21434 34286 35192(5622a) (12134) (12148) (21998) (34110) (35415)
aResults in parentheses are taken from FEM
section that is 119904 = 119905 = 2 Also the thickness varying functionsare also be expressed as Fourier cosine series according to (18)and (19)
In order to validate the accuracy and reliability of theproposed method for predicting the vibration behavior ofnonlinearly varying thickness moderately thick plates with
arbitrary boundary conditions the typical classical boundaryconditions viewed as the special cases of elastically restrainededges will be considered The comparison of the first six fre-quency parameters Ω = 120596119886
2
(120588ℎ01198630)12 for the moderately
thick plates with nonlinearly varying thickness is presentedin Table 9 The S-S-S-S C-F-F-F S-S-F-F C-C-C-C and
Shock and Vibration 17
Table 9 Frequency parameter Ω = 1205961198862
(120588ℎ01198630)12 for moderately thick plates with linearly varying thickness in different boundary
conditions (ℎ(119909 119910) = ℎ0(1 + 05119909
2
)(1 + 051199102
))
Boundary condition 119886119887 ℎ0119886
Model sequence1 2 3 4 5 6
S-S-S-S
101 23462 54999 55573 82950 100900 101187
(23741a) (55114) (55983) (83146) (101004) (101347)
02 18606 42725 42971 60040 71167 71761(18936a) (43014) (43148) (60247) (71642) (71993)
3201 53055 65251 85404 106700 114880 143531
(53467a) (65472) (85984) (107211) (115687) (144007)
02 17439 21531 32170 44987 59438 64179(17874a) (21748) (32651) (45264) (60188) (64879)
C-F-F-F
101 3658 9317 24135 32816 35433 59179
(3749a) (9547) (24034) (33627) (36308) (60142)
02 3520 7986 19591 26959 27491 43882(3611a) (8176) (20075) (27125) (27625) (43984)
3201 3901 9963 28984 69187 73686 89360
(3930a) (10095) (29381) (69578) (73994) (90242)
02 3292 4439 13385 23947 40850 44758(3386a) (4498) (14012) (24068) (41221) (45142)
S-S-F-F
101 3557 21268 22762 43524 60253 62429
(3649a) (21441) (22796) (43916) (60622) (62630)
02 3171 17822 19512 33616 45042 46833(3489a) (18006) (20430) (33815) (45247) (49004)
3201 6738 31658 52071 75625 86604 94903
(6957a) (31882) (52518) (75823) (86939) (95610)
02 3653 12930 20955 26517 44097 48058(6807a) (13160) (21312) (26864) (44422) (48873)
C-C-C-C
101 41020 75196 75614 104142 119683 120949
(41405a) (75452) (75882) (104711) (119637) (121137)
02 31237 51686 51868 68411 76868 77864(32010a) (52960) (52145) (69335) (77481) (78099)
3201 95823 128727 163657 173751 190890 223595
(95364a) (128932) (163588) (172350) (190350) (222948)
02 53306 71372 89384 102404 104660 120908(53587a) (71759) (90103) (102503) (105174) (121275)
S-C-S-C
101 33715 60298 71446 94187 103802 117928
(34148a) (60786) (71828) (94515) (104263) (119326)
02 26143 45499 49458 64253 72941 76191(26789a) (46622) (50680) (64329) (73474) (76843)
3201 73228 115845 137445 148166 168342 182542
(73761a) (116121) (137788) (148530) (168698) (182921)
02 41302 53648 65630 72202 81632 88105(41304a) (53757) (65842) (72794) (82033) (88474)
aResults in parentheses are taken from FEM
S-C-S-C boundary conditions are performed in the compar-ison The results adequately demonstrated the great accuracyof the modified Fourier method
We now turn to elastically restrained moderately thickplates The first one involves an S-S-S-S moderately thicksquare plate with a uniform rotational restraint along eachedge that is 119870
1199090= 119870119909119886
= 1198701199100
= 119870119910119887
= Γ119909 The calculated
frequency parameters Ω = 1205961198862
(120588ℎ01198630)12 are given in
Table 10 together with the FEM results Since the elasticallyrestrained plates with nonlinear variation thickness are rarelyinvestigated the FEMresults are used as the referenceA goodagreement is observed between the current and FEM resultsThe second example concerns amoderately thick square plateelastically supported along all edgesThe stiffness of the linear
18 Shock and Vibration
Table 10 Frequency parameterΩ = 1205961198862
(120588ℎ01198630)12 for S-S-S-S moderately thick plates (119886119887 = 1) with linearly varying thickness and elastic
rotation support (ℎ(119909 119910) = ℎ0(1 + 05119909
2
)(1 + 051199102
) 1198701199090= 119870119909119886= 1198701199100= 119870119910119887= Γ119909)
Γ119909
aℎ0119886
Model sequence1 2 3 4 5 6
0
005 23462 54999 55573 82950 100900 101187(23741b) (55114) (55983) (83146) (101004) (101347)
01 18606 42725 42971 60040 71167 71761(18936b) (43014) (43148) (60247) (71642) (71993)
015 53055 65251 85404 106700 114880 143531(53467b) (65472) (85984) (107211) (115687) (144007)
02 17439 21531 32170 44987 59438 64179(17874b) (21748) (32651) (45264) (60188) (64879)
05
005 6904 13374 13514 19400 22904 23056(9909b) (13548) (13651) (19564) (23088) (23379)
01 5316 10237 10314 14499 17155 17264(5374b) (10448) (10651) (14764) (17188) (17879)
015 4136 8201 8220 11474 13578 13671(4174b) (8348) (8651) (11764) (13688) (13879)
02 3449 6915 6924 9549 11225 11327(3474b) (6948) (6951) (9764) (11388) (11579)
1
005 7040 13617 13745 19726 23233 23402(7074b) (13748) (13851) (19964) (23488) (23879)
01 5712 10696 10793 14991 17594 17744(5774b) (10748) (10851) (15054) (17788) (17879)
015 4489 8507 8540 11759 13803 13922(4514b) (8548) (8611) (11864) (13988) (14119)
02 3677 7084 7090 9692 11330 11445(3684b) (7148) (7191) (9764) (11588) (11879)
15
005 7089 13708 13828 19847 23355 23530(7124b) (13748) (13951) (19967) (23588) (23879)
01 5895 10923 11024 15239 17813 17988(5874b) (10748) (11651) (15264) (17818) (17979)
015 4697 8694 8737 11938 13790 14083(4774b) (8748) (8951) (11864) (13688) (14879)
02 3836 7201 7210 9792 11403 11530(3874b) (7748) (7651) (9864) (11588) (11879)
2
005 7114 13754 13871 19911 23418 23597(7174b) (12848) (13851) (19864) (23588) (23879)
01 6001 11060 11159 15391 17944 18136(6004b) (11148) (11151) (15464) (17988) (18379)
015 4836 8821 8872 12061 14037 14189(4874b) (8848) (8951) (12164) (14188) (14879)
02 3954 7289 7301 9871 11457 11594(3974b) (7348) (7651) (9964) (11488) (11679)
aΓ119909= 1198701198941198700 (1198700= 1 times 10
9 Nmrad 119894 = 1199090 119909119886 1199100 119910119887)bResults in parentheses are taken from FEM
and rotational restraints is set equal to 1198961199090
= 119896119909119886
= 1198961199100
=
119896119910119887
= Γ119908and 119870
1199090= 119870119909119886
= 1198701199100
= 119870119910119887
= Γ119909 respectively
The six frequency parametersΩ = 1205961198862
(120588ℎ01198630)12 are shown
in Table 11 for several different restraining coefficient valuesIt can also be noticed that the Fourier series method is able
to predict the modal characteristics of nonlinearly varyingthickness moderately thick plate with not only classicalboundary conditions but also elastic edge restraints correctly
Finally the influence of the gradient on the fundamentalfrequency parameters of a nonlinearly varying thickness
Shock and Vibration 19
Table 11 Frequency parameterΩ = 1205961198862
(120588ℎ01198630)12 for moderately thick plates (119886119887 = 1) with linearly varying thickness and elastic rotation
and translation support (ℎ(119909 119910) = ℎ0(1 + 05119909
2
)(1 + 051199102
) 1198701199090= 119870119909119886= 1198701199100= 119870119910119887= Γ119909 and 119896
1199090= 119896119909119886= 1198961199100= 119896119910119887= Γ119908)
Γ119909
aΓ119908
aℎ0119886
Model sequence1 2 3 4 5 6
05 0
005 12926 13041 25090 47935 47943 61761(12931b) (13053) 25099 (47947) (47952) (61788)
01 10900 10988 21545 39980 40147 50322(10904b) (10992) (21557) (39984) (40166) (50334)
015 8245 8298 18081 31503 32617 40950(8247b) (8230) (18092) (31508) (32622) (40955)
02 6106 6124 15639 25631 27601 34458(6110b) (6125) (15642) (25637) (27617) (34462)
1 05
005 21575 31615 31644 42573 57889 58607(21580b) (31617) (31644) (42581) (57893) (58610)
01 8451 16024 16210 25969 43839 43867(8451b) (16025) (16217) (25972) (43844) (43870)
015 4682 11479 11480 20642 35172 35598(4684b) (11480) (11481) (20644) (35178) (35604)
02 3059 8621 8648 17320 28487 29638(3062b) (8619) (8644) (17327) (28492) (29641)
15 1
005 27713 40203 40978 52951 65352 66815(27714b) (40207) (40988) (52972) (65368) (66822)
01 11667 19322 19615 29122 46115 46183(11674b) (19327) (19619) (29137) (46120) (46188)
015 6544 13416 13481 22405 37238 37430(6547b) (13420) (13488) (22410) (37241) (37434)
02 4295 10175 10190 18543 30330 31056(4296b) (10177) (10194) (18544) (30336) (31060)
2 2
005 33453 50854 52088 66311 76610 78947(33455b) (50852) (50097) (66321) (76627) (78971)
01 15832 24130 24372 33924 49118 49362(15830b) (24127) (24374) (33931) (49147) (49382)
015 9060 15956 16099 24754 39134 39249(9064b) (15961) (16102) (24755) (39138) (39452)
02 5992 11955 11971 20034 31908 32392(5991b) (11957) (11978) (20042) (31910) (32398)
aΓ119909= 1198701198941198700 (1198700= 1 times 10
9 Nmrad 119894 = 1199090 119909119886 1199100 119910119887) Γ119908= 1198961198941198960 (1198960= 1 times 10
9 Nm 119894 = 1199090 119909119886 1199100 119910119887)bResults in parentheses are taken from FEM
moderately thick plate is investigated The plate is elasticallyrestrained in which the stiffness of the boundary springsis taken as 119896
1199090= 119896119909119886
= 1198961199100
= 119896119910119887
= Γ119908
= 2 and1198701199090
= 119870119909119886
= 1198701199100
= 119870119910119887
= Γ119909= 2 respectively The six
frequency parameters Ω = 1205961198862
(120588ℎ01198630)12 are presented in
Table 12 for several different slop values It can be seen thatthe fundamental frequency parameters will decrease with theincrease of the parameter ℎ
0119886
In the above examples it has been demonstrated thatthe presented method can be universally applied to nonlin-early varying thickness moderately thick plates with severalboundary conditions and different structure parametersNew results are obtained for plates with nonlinear variationthickness in both directions subjected to general elastic
boundary restraints which may be used for benchmarkingof researchers in the field In addition it is interesting to seethat the nature frequency decreases with the increase of theindex for thickness function
4 Conclusions
In this paper a modified Fourier method has been pre-sented to study the free vibration behaviors of moderatelythick rectangular plates with variable thickness and arbi-trary boundary conditionsThe first-order shear deformationplate theory is adopted to formulate the theoretical modelThe displacements and rotation components of the plateregardless of boundary conditions are invariantly expressedas the superposition of a 2D Fourier cosine series and four
20 Shock and Vibration
Table 12 Frequency parameterΩ = 1205961198862
(120588ℎ01198630)12 for elastic support moderately thick plates (119886119887 = 1) with different gradient (119896
1199090= 119896119909119886=
1198961199100= 119896119910119887= Γ119908= 2 119870
1199090= 119870119909119886= 1198701199100= 119870119910119887= Γ119909= 2)
120572 120573 ℎ0119886
Model sequence1 2 3 4 5 6
120572 = 120573 = 01
005 30187 51102 51141 67066 73954 77412(30188a) (51105) (51144) (67068) (73961) (77477)
01 16995 24719 24724 33071 44343 44819(16996a) (24725) (24729) (33084) (44347) (44822)
015 10077 15872 15883 23443 35937 36065(10081a) (15877) (15892) (23447) (35941) (36084)
02 6713 11953 11958 19086 30372 30604(6713a) (11957) (11965) (19092) (30378) (30612)
120572 = 120573 = 02
005 31206 51436 51614 67058 74398 77595(31208a) (51438) (51619) (67064) (74402) (77596)
01 16786 24503 24508 33144 45583 45923(16790a) (24510) (24513) (33147) (45587) (45932)
015 9821 15865 15906 23766 36897 36990(9824a) (15870) (15907) (23768) (36894) (36997)
02 6523 11978 11992 19356 30912 31162(6524a) (11981) (11997) (19362) (30916) (31174)
120572 = 120573 = 03
005 32097 51462 51900 66832 74991 77863(32098a) (51466) (51907) (66849) (74996) (77872)
01 16507 24332 24383 33330 46806 47074(16511a) (24327) (24380) (33339) (46810) (47086)
015 9562 15885 15964 24103 37748 37828(9562a) (15892) (15977) (21105) (37751) (37832)
02 6338 11992 12013 19603 31338 31634(6340a) (11994) (12011) (19612) (31344) (31639)
120572 = 120573 = 04
005 32851 51239 52046 66547 75735 78303(33455a) (50852) (50097) (66321) (76627) (78971)
01 16182 24209 24343 33598 47990 48230(33455a) (50852) (50097) (66321) (76627) (78971)
015 9307 15919 16034 24436 38491 38579(33455a) (50852) (50097) (66321) (76627) (78971)
02 6161 11986 12008 19828 31666 32040(33455a) (50852) (50097) (66321) (76627) (78971)
120572 = 120573 = 05
005 33453 50854 52088 66311 76610 78947(33455a) (50852) (50097) (66321) (76627) (78971)
01 15832 24130 24372 33924 49118 49362(15830a) (24127) (24374) (33931) (49147) (49382)
015 9060 15956 16099 24754 39134 39249(9064a) (15961) (16102) (24755) (39138) (39452)
02 5992 11955 11971 20034 31908 32392(5991a) (11957) (11978) (20042) (31910) (32398)
aResults in parentheses are taken from FEM
supplementary functions in the form of the product of apolynomial function and a single cosine series expansionto ensure and accelerate the convergence of the solutionAt each edge of the plate the general restraint conditionsare implemented by introducing one group of linear springsand two groups of rotational springs which are continuouslydistributed and determined by the stiffness of these springs
Instead of seeking a solution in strong forms in the previousstudies all the Fourier coefficients will be treated equallyand independently as the generalized coordinates and solveddirectly from the Rayleigh-Ritz technique The change of theboundary conditions can be easily achieved by only varyingthe stiffness of the three sets of boundary springs along alledges of the rectangular plates without involving any change
Shock and Vibration 21
to the solution procedure The convergence of the presentsolution is examined and the excellent accuracy is validatedby comparison with existing results published in the lit-erature and FEM data Excellent agreements are obtainedfrom these comparisons The proposed method providesa unified means for extracting the modal parameters andpredicting the vibration behaviors of moderately thick plateswith variable thickness variation functions and arbitraryelastic edge restraints A variety of free vibration results formoderately thick rectangular plates with different thicknessvariation functions and boundary conditions are presentedNew results for free vibration of moderately thick rectangularplates with various thickness variation functions and edgeconditions are presented which may be used for benchmark-ing of researchers in the field
Appendices
A Representative Calculation forStiffness and Mass Matrices
To illuminate the particular expression of the mass andstiffness matrixes clearly and tersely four new variables aredefined as 119904 = 119898 lowast (119873 + 1) + 119899 + 1 119905 = 119898
1015840
lowast (119873 + 1) + 1198991015840
+ 1119901 = (119897minus1)lowast (119872+1)+119898
1015840
+1 and 119902 = (119897minus1)lowast (119873+1)+1198991015840
+1The first row elements of K andM are given below
K1-1119904119905
=
1
2
int
119886
0
int
119887
0
[119863 (119909 119910)
times 21205821198861198981205821198861198981015840 sin 120582
119886119898119909 sin 120582
1198861198981015840119909 cos 120582
119887119899119910
times cos 1205821198871198991015840119910 + (1 minus 120583) 120582
1198871198991205821198871198991015840 cos 120582
119886119898119909
times cos 1205821198861198981015840 sin 120582
119887119899119910 sin 120582
1198871198991015840119910
+ 120581119866ℎ (119909 119910) cos 120582119886119898119909 cos 120582
1198861198981015840119909 cos 120582
119887119899
times119910 cos 1205821198871198991015840119910119889119909119889119910] 119889119909 119889119910
+ int
119886
0
1198701199091199100
cos 120582119886119898119909 cos 120582
1198861198981015840119909
+(minus1)119898+1198981015840
119870119909119910119887
cos 120582119886119898119909 cos 120582
1198861198981015840119909 119889119909
+ int
119887
0
1198701199090cos 120582119887119899119910 cos 120582
1198871198991015840119910
+(minus1)119899+1198991015840
119870119909119886cos 120582119887119899119910 cos 120582
1198871198991015840119910 119889119910
K1-2119904119901
=
1
2
int
119886
0
int
119887
0
[119863 (119909 119910)21205821198861198981205821198861198981015840 sin 120582
119886119898119909 sin 120582
1198861198981015840119909
times
2
sum
119897=1
120577119897
119887(119910) cos 120582
1198871198991015840119910
minus (1 minus 120583) 1205821198871198991015840 cos 120582
119886119898119909 cos 120582
1198861198981015840119909
times
2
sum
119897=1
1205771198971015840
119887(119910) sin 120582
1198871198991015840119910 + 120581119866ℎ (119909 119910)
times ( cos 1205821198861198981015840119909 cos 120582
1198871198991015840119910
times
2
sum
119897=1
120577119897
119887(119910) cos 120582
119886119898119909)]119889119909119889119910
+ int
119886
0
1198701199091199100
cos 120582119886119898119909 cos 120582
1198861198981015840119909
2
sum
119897=1
120577119897
119887(0)
+(minus1)1198991015840
119870119909119910119887
cos 120582119886119898119909 cos 120582
1198861198981015840119909
2
sum
119897=1
120577119897
119887(119887) 119889119909
+ int
119887
0
1198701199090
2
sum
119897=1
120577119897
119887(119910) cos 120582
1198871198991015840119910
+(minus1)119898+1198981015840
119870119909119886
2
sum
119897=1
120577119897
119887(119910) cos 120582
1198871198991015840119910119889119910
K1-3119904119902
=
1
2
int
119886
0
int
119887
0
[119863 (119909 119910) minus 21205821198861198981015840 sin 120582
1198861198981015840119909 cos 120582
1198871198991015840119910
times
2
sum
119897=1
1205771198971015840
119886(119909) cos 120582
119887119899119910
+ (1 minus 120583) 1205821198871198991205821198871198991015840 cos 120582
1198861198981015840119909
times sin 1205821198871198991015840119910
2
sum
119897=1
120577119897
119886(119909) sin 120582
119887119899119910
+ 120581119866ℎ (119909 119910) cos 1205821198861198981015840119909 cos 120582
1198871198991015840119910
times
2
sum
119897=1
120577119897
119886(119909) cos 120582
119887119899119910]119889119909119889119910
+ int
119886
0
1198701199091199100
cos 1205821198861198981015840119909
2
sum
119897=1
120577119897
119886(119909)
+(minus1)119899+1198991015840
119870119909119910119887
cos 1205821198861198981015840119909
2
sum
119897=1
120577119897
119886(119909) 119889119909
+ int
119887
0
1198701199090
2
sum
119897=1
120577119897
119886(0) cos 120582
1198871198991015840119910
+(minus1)119898+1198981015840
119870119909119886
2
sum
119897=1
120577119897
119886(0) cos 120582
1198871198991015840119910119889119910
K1-4119904119905
=
1
2
int
119886
0
int
119887
0
119863(119909 119910) 21205821198871198991205821198861198981015840 sin 120582
1198861198981015840119909 cos 120582
1198871198991015840119910
times cos 120582119886119898119909 sin 120582
119887119899119910 minus 2(1 minus 120583)
119887119899
times 1205821198861198981015840 sin 120582
1198861198981015840119909 cos 120582
1198871198991015840119910 cos 120582
119886119898119909
times sin 120582119887119899119910 + (1 minus 120583) 120582
1198861198981205821198871198991015840 cos 120582
1198861198981015840119909
times sin 1205821198871198991015840119910 sin 120582
119886119898119909 cos 120582
119887119899119910 119889119909 119889119910
22 Shock and Vibration
K1-5119904119901
=
1
2
int
119886
0
int
119887
0
119863(119909 119910) minus 21205821198861198981015840 sin 120582
1198861198981015840119909 cos 120582
1198871198991015840119910
times
2
sum
119897=1
1205771198971015840
119887(119910) cos 120582
119886119898119909
+ 2 (1 minus 120583) 1205821198861198981015840 sin 120582
1198861198981015840119909 cos 120582
1198871198991015840119910
times
2
sum
119897=1
1205771198971015840
119887(119910) cos 120582
119886119898119909
+ (1 minus 120583) 1205821198861198981205821198871198991015840 cos 120582
1198861198981015840119909 sin 120582
1198871198991015840119910
times
2
sum
119897=1
120577119897
119887(119910) sin 120582
119886119898119909119889119909119889119910
K1-6119904119902
=
1
2
int
119886
0
int
119887
0
119863(119909 119910)1205821198871198991205821198861198981015840 sin 120582
1198861198981015840119909 cos 120582
1198861198991015840119910
times
2
sum
119897=1
120577119897
119886(119909) sin 120582
1198871198991015840119910
minus 2 (1 minus 120583) 1205821198871198991205821198861198981015840 sin 120582
1198861198981015840119909 cos 120582
1198861198991015840119910
times
2
sum
119897=1
120577119897
119886(119909) sin 120582
1198871198991015840119910
+ 1205821198871198991015840 cos 120582
1198861198981015840119909 sin 120582
1198871198991015840119910
times
2
sum
119897=1
1205771198971015840
119886(119909) cos 120582
119887119899119910119889119909119889119910
K1-7119904119905
= minus
1
2
int
119886
0
int
119887
0
120581119866ℎ (119909 119910) 120582119886119898
cos 1205821198861198981015840119909 cos 120582
1198871198991015840119910
times sin 120582119886119898119909 cos 120582
119887119899119910119889119909119889119910
K1-8119904119901
= minus
1
2
int
119886
0
int
119887
0
120581119866ℎ (119909 119910) 1205821198861198981015840 cos 120582
1198861198981015840119909 cos 120582
1198871198991015840119910
times
2
sum
119897=1
120577119897
119887(119910) sin 120582
1198861198981015840119909 119889119909 119889119910
K1-9119904119902
=
1
2
int
119886
0
int
119887
0
120581119866ℎ (119909 119910) cos 1205821198861198981015840119909 cos 120582
1198871198991015840119910
times
2
sum
119897=1
1205771198971015840
119886(119909) cos 120582
119887119899119910119889119909119889119910
M1-1119904119905
=
1
24
120588int
119886
0
int
119887
0
ℎ3
(119909 119910) cos 120582119886119898119909 cos 120582
119887119899119910
times cos 1205821198861198981015840119909 cos 120582
1198871198991015840119910119889119909119889119910
M1-2119904119901
=
1
24
120588int
119886
0
int
119887
0
ℎ3
(119909 119910)
2
sum
119897=1
120577119897
119887(119910) cos 120582
119886119898119909
times cos 1205821198861198981015840119909 cos 120582
1198871198991015840119910119889119909119889119910
M1-3119904119902
=
1
24
120588int
119886
0
int
119887
0
ℎ3
(119909 119910)
2
sum
119897=1
120577119897
119886(119909) cos 120582
119887119899119910
times cos 1205821198861198981015840119909 cos 120582
1198871198991015840119910119889119909119889119910
M1-4119904119905 = M
1-5119904119901 = M1-6119904119902
= M1-7119904119905 = M
1-8119904119901 = M1-9119904119902 = 0
(A1)
B Useful Integrals for the Moderately ThickRectangular Plates with Variable Thicknessand Arbitrary Boundary Conditions
Consider
int
119871
0
cos 120582119898119909 cos 120582
119899119909 119889119909 =
0 119898 = 119899
119871
2
119898 = 119899
119871 119898 = 119899 = 0
int
119871
0
sin 120582119898119909 cos 120582
119899119909119889119909
=
minus
(minus1 + (minus1)119898+119899
) 119871119898
(1198982minus 1198992) 120587
119898 = 119899
0 119898 = 119899
int
119871
0
sin 120582119898119909 sin 120582
119899119909 119889119909 =
0 119898 = 119899
119871
2
119898 = 119899
0 119898 = 119899 = 0
int
119871
0
cos 120582119898119909120577119897
119886(119909) 119889119909
=
41198712
31205872 119897 = 1 119898 = 0
minus
41198712
(41198982
minus 3)
(9 minus 401198982+ 16119898
4) 1205872
119897 = 1 119898 = 0
minus
41198712
31205872 119897 = 2 119898 = 0
41198712
(41198982
minus 3) (minus1)119898
(9 minus 401198982+ 16119898
4) 1205872
119897 = 2 119898 = 0
Shock and Vibration 23
int
119871
0
sin 120582119898119909120577119897
119886(119909) 119889119909
=
0 119897 = 1 119898 = 0
161198712
119898(minus1)119898
(9 minus 401198982+ 16119898
4) 1205872
119897 = 1 119898 = 0
0 119897 = 2 119898 = 0
161198981198712
(9 minus 401198982+ 16119898
4) 1205872
119897 = 2 119898 = 0
int
119871
0
cos 1205821198981199091205771198971015840
119886(119909) 119889119909
=
0 119897 = 1 119898 = 0
161198711198982
(minus1)119898
(9 minus 401198982+ 16119898
4) 120587
119897 = 1 119898 = 0
0 119897 = 2 119898 = 0
161198711198982
(9 minus 401198982+ 16119898
4) 120587
119897 = 2 119898 = 0
int
119871
0
sin 1205821198981199091205771198971015840
119886(119909) 119889119909
=
0 119897 = 1 119898 = 0
4119871119898 (minus3 + 41198982
)
(9 minus 401198982+ 16119898
4) 120587
119897 = 1 119898 = 0
0 119897 = 2 119898 = 0
119871119898(
1
120587 minus 41198982120587
+
3
9120587 minus 41198982120587
) (minus1)119898
119897 = 2 119898 = 0
int
119871
0
1205771198971
119886(119909) 1205771198972
119886(119909) 119889119909 =
1198713
41205872 1198971= 1 1198972= 1
minus
21198713
31205873 1198971= 1 1198972= 2
minus
21198713
31205873 1198971= 2 1198972= 1
1198713
41205872 1198971= 2 1198972= 2
int
119871
0
1205771198971
119886(119909) 1205771198971015840
2
119886(119909) 119889119909 =
0 1198971= 1 1198972= 1
minus
1198712
8120587
1198971= 1 1198972= 2
1198712
8120587
1198971= 2 1198972= 1
0 1198971= 2 1198972= 2
int
119871
0
1205771198971015840
1
119886(119909) 1205771198971015840
2
119886(119909) 119889119909 =
5119871
16
1198971= 1 1198972= 1
minus
119871
2120587
1198971= 1 1198972= 2
minus
119871
2120587
1198971= 2 1198972= 1
5119871
16
1198971= 2 1198972= 2
int
119871
0
119909119899 cos 120582
119898119909 119889119909
=
1198711+119899
1 + 119899
119898 = 0
(1198861+119899HypergeometricPFQtimes [ 12 + 1198992 12 32 + 1198992
minus (14)1198982
1205872
]) times (1 + 119899)minus1
119898 gt 0
(B1)
Conflict of Interests
The authors declare that there is no conflict of interestsregarding the publication of this paper
Acknowledgments
The authors would like to thank the anonymous reviewersfor their very valuable comments The authors also gratefullyacknowledge the financial support from the National NaturalScience Foundation of China (no 51209052) HeilongjiangProvince Youth Science Fund Project (no QC2011C013) andHarbin Science and Technology Development InnovationFoundation of Youth (no 2011RFQXG021)
References
[1] K Liew Y Xiang and S Kitipornchai ldquoResearch on thick platevibration a literature surveyrdquo Journal of Sound and Vibrationvol 180 no 1 pp 163ndash176 1995
[2] J H Chung T Y Chung and K C Kim ldquoVibration analysisof orthotropic mindlin plates with edges elastically restrainedagainst rotationrdquo Journal of Sound and Vibration vol 163 no 1pp 151ndash163 1993
[3] Y K Cheung and D Zhou ldquoVibrations of moderately thickrectangular plates in terms of a set of static Timoshenko beamfunctionsrdquoComputers and Structures vol 78 no 6 pp 757ndash7682000
[4] CMWang ldquoNatural frequencies formula for simply supportedMindlin platesrdquo Journal of Vibration and Acoustics vol 116 no4 pp 536ndash540 1994
[5] K N Saha R C Kar and P K Datta ldquoFree vibration analysisof rectangular Mindlin plates with elastic restraints uniformlydistributed along the edgesrdquo Journal of Sound andVibration vol192 no 4 pp 885ndash902 1996
24 Shock and Vibration
[6] D J Gorman ldquoAccurate free vibration analysis of shear-deformable plates with torsional elastic edge supportrdquo Journalof Sound and Vibration vol 203 no 2 pp 209ndash218 1997
[7] D J Gorman ldquoFree vibration analysis of Mindlin plates withuniform elastic edge support by the superposition methodrdquoJournal of Sound andVibration vol 207 no 3 pp 335ndash350 1997
[8] D J Gorman ldquoAccurate free vibration analysis of point sup-ported mindlin plates by the superposition methodrdquo Journal ofSound and Vibration vol 219 no 2 pp 265ndash277 1999
[9] Y Xiang ldquoVibration of rectangular Mindlin plates resting onnon-homogenous elastic foundationsrdquo International Journal ofMechanical Sciences vol 45 no 6-7 pp 1229ndash1244 2003
[10] Y Xiang and G W Wei ldquoExact solutions for buckling andvibration of stepped rectangular Mindlin platesrdquo InternationalJournal of Solids and Structures vol 41 no 1 pp 279ndash294 2004
[11] Y L Yeh M J Jang and C C Wang ldquoAnalyzing the freevibrations of a plate using finite difference and differential trans-formationmethodrdquoAppliedMathematics andComputation vol178 no 2 pp 493ndash501 2006
[12] Y Xiang S K Lai L Zhou and C W Lim ldquoDSC-Ritz elementmethod for vibration analysis of rectangular Mindlin plateswith mixed edge supportsrdquo European Journal of MechanicsmdashASolids vol 29 no 4 pp 619ndash628 2010
[13] H Nguyen-Xuan C H Thai and T Nguyen-Thoi ldquoIsogeo-metric finite element analysis of composite sandwich platesusing a higher order shear deformation theoryrdquo Composites BEngineering vol 55 pp 558ndash574 2013
[14] C H Thai A J M Ferreira E Carrera and H Nguyen-XuanldquoIsogeometric analysis of laminated composite and sandwichplates using a layerwise deformation theoryrdquo Composite Struc-tures vol 104 pp 196ndash214 2013
[15] C H Thai A Ferreira S Bordas T Rabczuk and HNguyen-Xuan ldquoIsogeometric analysis of laminated compositeand sandwich plates using a new inverse trigonometric sheardeformation theoryrdquo European Journal of Mechanics A Solidsvol 43 pp 89ndash108 2014
[16] H Luong-Van T Nguyen-Thoi G R Liu and P Phung-VanldquoA cell-based smoothed finite elementmethod using three-nodeshear-locking free Mindlin plate element (CS-FEM-MIN3) fordynamic response of laminated composite plates on viscoelasticfoundationrdquo Engineering Analysis with Boundary Elements vol42 pp 8ndash19 2014
[17] T Nguyen-Thoi T Rabczuk T Lam-Phat V Ho-Huu andP Phung-Van ldquoFree vibration analysis of cracked Mindlinplate using an extended cell-based smoothed discrete sheargap method (XCS-DSG3)rdquo Theoretical and Applied FractureMechanics 2014
[18] H Nguyen-Xuan G R Liu and C a Thai-Hoang ldquoAnedge-based smoothed finite element method (ES-FEM) withstabilized discrete shear gap technique for analysis of Reissner-Mindlin platesrdquo Computer Methods in Applied Mechanics andEngineering vol 199 no 9ndash12 pp 471ndash489 2010
[19] P Phung-Van T Nguyen-Thoi T Bui-Xuan and Q Lieu-XuanldquoA cell-based smoothed three-node Mindlin plate element(CS-FEM-MIN3) based on the 119862
0 -type higher-order sheardeformation for geometrically nonlinear analysis of laminatedcomposite platesrdquo Computational Materials Science 2014
[20] NNguyen-Thanh T RabczukHNguyen-Xuan and S BordasldquoAn alternative alpha finite element method with discrete sheargap technique for analysis of isotropicMindlin-Reissner platesrdquoFinite Elements inAnalysis andDesign vol 47 no 5 pp 519ndash5352011
[21] KM Liew Y Xiang and S Kitipornchai ldquoTransverse vibrationof thick rectangular plates-I Comprehensive sets of boundaryconditionsrdquo Computers and Structures vol 49 no 1 pp 1ndash291993
[22] D Zhou ldquoVibrations of Mindlin rectangular plates with elasti-cally restrained edges using static Timoshenko beam functionswith the Rayleigh-Ritz methodrdquo International Journal of Solidsand Structures vol 38 no 32-33 pp 5565ndash5580 2001
[23] D Zhou SH Lo F T KAu andYK Cheung ldquoVibration anal-ysis of rectangular Mindlin plates with internal line supportsusing static Timoshenko beam functionsrdquo International Journalof Mechanical Sciences vol 44 no 12 pp 2503ndash2522 2002
[24] H S Shen J Yang and L Zhang ldquoFree and forced vibrationof Reissner-Mindlin plates with free edges resting on elasticfoundationsrdquo Journal of Sound and Vibration vol 244 no 2 pp299ndash320 2001
[25] Y Xing and B Liu ldquoClosed form solutions for free vibrations ofrectangular Mindlin platesrdquo Acta Mechanica Sinica vol 25 no5 pp 689ndash698 2009
[26] P Gagnon C Gosselin and L Cloutier ldquoA finite strip elementfor the analysis of variable thickness rectangular thick platesrdquoComputers amp Structures vol 63 no 2 pp 349ndash362 1997
[27] T Mlzusawa ldquoVibration of rectangular mindlin plates withtapered thickness by the spline strip methodrdquo Computers andStructures vol 46 no 3 pp 451ndash463 1993
[28] H Nguyen-Xuan and T Nguyen-Thoi ldquoA stabilized smoothedfinite element method for free vibration analysis of Mindlin-Reissner platesrdquo Communications in Numerical Methods inEngineering with Biomedical Applications vol 25 no 8 pp 882ndash906 2009
[29] K M Liew L X Peng and S Kitipornchai ldquoVibration anal-ysis of corrugated Reissner-Mindlin plates using a mesh-freeGalerkin methodrdquo International Journal of Mechanical Sciencesvol 51 no 9-10 pp 642ndash652 2009
[30] Y Hou GWWei and Y Xiang ldquoDSC-Ritz method for the freevibration analysis of Mindlin platesrdquo International Journal forNumerical Methods in Engineering vol 62 no 2 pp 262ndash2882005
[31] F- Liu and K M Liew ldquoVibration analysis of discontinuousMindlin plates by differential quadrature element methodrdquoJournal of Vibration and Acoustics Transactions of the ASMEvol 121 no 2 pp 204ndash208 1999
[32] P Malekzadeh G Karami and M Farid ldquoA semi-analyticalDQEM for free vibration analysis of thick plates with twoopposite edges simply supportedrdquoComputerMethods inAppliedMechanics and Engineering vol 193 no 45ndash47 pp 4781ndash47962004
[33] P Malekzadeh and S A Shahpari ldquoFree vibration analysisof variable thickness thin and moderately thick plates withelastically restrained edges by DQMrdquo Thin-Walled Structuresvol 43 no 7 pp 1037ndash1050 2005
[34] R E Diaz-Contreras and S Nomura ldquoGreenrsquos function appliedto solution ofMindlin platesrdquoComputers and Structures vol 60no 1 pp 41ndash48 1996
[35] T Sakiyama and M Huang ldquoFree vibration analysis of rect-angular plates with variable thicknessrdquo Journal of Sound andVibration vol 216 no 3 pp 379ndash397 1998
[36] M Huang X Q Ma T Sakiyama H Matuda and C MoritaldquoFree vibration analysis of orthotropic rectangular plates withvariable thickness and general boundary conditionsrdquo Journal ofSound and Vibration vol 288 no 4-5 pp 931ndash955 2005
Shock and Vibration 25
[37] I Shufrin and M Eisenberger ldquoVibration of shear deformableplates with variable thicknessmdashfirst-order and higher-orderanalysesrdquo Journal of Sound and Vibration vol 290 no 1-2 pp465ndash489 2006
[38] S A Eftekhari and A A Jafari ldquoAccurate variational approachfor free vibration of variable thickness thin and thick plates withedges elastically restrained against translation and rotationrdquoInternational Journal of Mechanical Sciences vol 68 pp 35ndash462013
[39] W L Li X Zhang J Du and Z Liu ldquoAn exact series solutionfor the transverse vibration of rectangular plates with generalelastic boundary supportsrdquo Journal of Sound amp Vibration vol321 no 1-2 pp 254ndash269 2009
[40] X Shi D Shi W L Li and Q Wang ldquoA unified method forfree vibration analysis of circular annular and sector plateswith arbitrary boundary conditionsrdquo Journal of Vibration andControl 2014
[41] D Y Shi X J Shi W L Li and Q S Wang ldquoFree transversevibrations of orthotropic thin rectangular plates with arbitraryelastic edge supportsrdquo Journal of Vibroengineering vol 16 no 1pp 389ndash398 2014
[42] Y Chen G Jin and Z Liu ldquoFree vibration analysis of circu-lar cylindrical shell with non-uniform elastic boundary con-straintsrdquo International Journal of Mechanical Sciences vol 74pp 120ndash132 2013
[43] L Dai T Yang J Du W L Li and M J Brennan ldquoAn exactseries solution for the vibration analysis of cylindrical shellswith arbitrary boundary conditionsrdquo Applied Acoustics vol 74no 3 pp 440ndash449 2013
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Shock and Vibration
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Shock and Vibration 5
potential discontinuities and their derivatives throughout thewhole plate structure including the boundaries and then toeffectively enhance the convergence of the results To ensurethis continuity of selection expressions and correspondingderivatives at any point on the plate the first-order derivativesof the 119909 and119910 directions should exist as indicated by (4)Thisrequirement is guaranteed by the selected supplementaryfunctions because it is easy to verify that
1205771
119886(0) = 120577
1
119886(119886) = 120577
11015840
119886(119886) = 0 120577
11015840
119886(0) = 1
1205772
119886(0) = 120577
2
119886(119886) = 120577
21015840
119886(0) = 0 120577
21015840
119886(119886) = 1
(11)
Similar conditions exist for the 119910-related polynomials1205771
119887(119910) and 120577
2
119887(119910) It has to be mentioned that although the
solution is theoretically exact for the superposition of infinitenumbers of Fourier terms in actual calculation we truncatethe infinite series to 119872 and 119873 to obtain the results withacceptable accuracy
Since the energy expressions and admissible function ofthe plate have been established the remaining task is to deter-mine the Fourier expanded coefficients and supplemented
coefficients in (7)ndash(9) The Lagrangian energy functional (119871)of the plate is written as
119871 = 119879 minus 119880 (12)
Then the Lagrangian expression is minimized by takingits derivatives with respect to these coefficients
120597119871
120597120599
= 0 120599 =
119860119898119899
119886119897
119898119887119897
119899
119861119898119899
119888119897
119898119889119897
119899
119862119898119899
119890119897
119898119891119897
119899
(13)
Since the displacements and rotation components of theplate are chosen as 119872 and 119873 to obtain the results withacceptable accuracy a total of 3 lowast (119872 + 1) lowast (119873 + 1) + 6 lowast
(119872 +119873 + 2) equations are obtainedThey can be summed up in a matrix form
(K minus 1205962M)E = 0 (14)
The unknown coefficients in the displacement expres-sions can be expressed in the vector form as E where
E =
11986000 11986001 119860
11989810158400 11986011989810158401 119860
11989810158401198991015840 119860
119872119873 1198861
0 119886
1
119872 1198862
0 119886
2
119872 1198871
0 119887
1
119873 1198872
0 119887
2
119873
11986100 11986101 119861
11989810158400 11986111989810158401 119861
11989810158401198991015840 119861
119872119873 1198881
0 119888
1
119872 1198882
0 119888
2
119872 1198891
0 119889
1
119873 1198892
0 119889
2
119873
11986200 11986201 119862
11989810158400 11986211989810158401 119862
11989810158401198991015840 119862
119872119873 1198901
0 119890
1
119872 1198902
0 119890
2
119872 1198911
0 119891
1
119873 1198912
0 119891
2
119873
119879
(15)
In (14)K is the stiffness matrix for the plate andM is themass matrix They can be expressed separately as
K =
[
[
[
[
[
[
[
[
[
K1-1 K
1-2 K1-3 sdot sdot sdot K
1-9
K1198791-2 K
2-1 K2-3 sdot sdot sdot K
2-9
K1198791-3 K119879
2-3 K3-3 sdot sdot sdot K
3-9
d
K1198791-9 K119879
2-9 K1198793-9 sdot sdot sdot K
9-9
]
]
]
]
]
]
]
]
]
M =
[
[
[
[
[
[
[
[
[
M1-1 M
1-2 M1-3 sdot sdot sdot M
1-9
M1198791-2 M
2-1 M2-3 sdot sdot sdot M
2-9
M1198791-3 M119879
2-3 M3-3 sdot sdot sdot M
3-9
d
M1198791-9 M119879
2-9 M1198793-9 sdot sdot sdot M
9-9
]
]
]
]
]
]
]
]
]
(16)
The specific expressions for the elements in (16) aregiven in Appendix AMoreover all the necessary expressionswhich will be used in the calculations of the eigenvalues andeigenvectors are given in Appendix B
Obviously the natural frequencies and eigenvectors cannow be readily obtained by solving a standard matrix eigen-problem Since the components of each eigenvector are actu-ally the expansion coefficients of the modified Fourier seriesthe corresponding mode shape can be directly determinedfrom (14) In other words once the coefficient eigenvectorE is
determined for a given frequency the displacement functionsof the plate can be determined by substituting the coefficientsinto (9) When the forced vibration is involved by addingthe work done by external force in the Lagrangian energyfunction and summing the loading vector F on the right sideof (14) the characteristic equation for the forced vibration ofthe moderately thick rectangular plates is readily obtained
3 Numerical Examples and Discussion
In this section a systematic comparison between the cur-rent solutions and theoretical results published by otherresearchers or finite element method (FEM) results is car-ried out to validate the excellent accuracy reliability andfeasibility of the modified Fourier method A comprehensivestudy on the effects of elastic restraint parameters andvarying thickness in two directions is also reported Newresults are obtained for plates subjected to general elasticboundary restraints with nonlinear variable thickness in bothdirections The discussion is arranged as follows Firstly theconvergence of the modified Fourier solution is checked Inaddition the influence of the stiffness of boundary springcomponents is studied Secondly the uniform thicknessplates with various combinations of classical boundary con-ditions elastic boundary conditions and different structureparameters are examined Thirdly the nonuniform thicknessplate with linear variation in both directions various com-binations of classical boundary conditions conditions and
6 Shock and Vibration
Table 1 Frequency parameters Ω = 1205961198872
(120588ℎ119863)12
1205872 for square moderately thick plates with uniform thickness and different boundary
conditions
Boundary conditions M N Model sequence1 2 3 4 5 6 7 8
C-C-C-C
119872 = 119873 = 2 3375 7191 7191 9856 1605 1628 1747 1747119872 = 119873 = 4 3299 6308 6308 8853 1046 1056 1263 1263119872 = 119873 = 6 3294 6282 6282 8801 1037 1047 1254 1254119872 = 119873 = 8 3292 6278 6278 8797 1036 1046 1253 1253119872 = 119873 = 10 3292 6276 6276 8794 1036 1046 1253 1253119872 = 119873 = 12 3292 6276 6276 8793 1036 1046 1252 1252119872 = 119873 = 14 3292 6276 6276 8793 1036 1046 1252 1252119872 = 119873 = 16 3292 6275 6275 8792 1036 1045 1252 1252Reference [4] 3292 6276 6276 8792 1036 1046 1252 1252
E-E-E-E
119872 = 119873 = 2 06153 10861 10861 17898 27709 28417 35010 35010119872 = 119873 = 4 06150 10799 10799 17785 27695 28405 34914 34914119872 = 119873 = 6 06149 10796 10796 17773 27694 28403 34898 34898119872 = 119873 = 8 06149 10796 10796 17771 27693 28403 34893 34893119872 = 119873 = 10 06149 10796 10796 17770 27693 28403 34892 34892119872 = 119873 = 12 06149 10796 10796 17770 27693 28403 34892 34892119872 = 119873 = 14 06149 10796 10796 17770 27693 28403 34892 34892119872 = 119873 = 16 06149 10796 10796 17770 27693 28403 34891 34891Reference [33] 06153 10861 10861 17898 27709 28417 35010 35010
FEM 06150 17097 17097 17774 27694 28406 34898 34898
different structure parameters are examined Then numer-ical solutions for moderately thick rectangular plates withelastic boundary conditions are presented The effects of theelastic restraint parameters are also investigated Finally thenonuniform thickness plate with nonlinear variation in bothdirections and arbitrary boundary conditions is also studied
31 Convergence and Stiffness Value Study Since the Fourierseries is numerically truncated and only finite terms arecounted in actual calculations the proposed solution shouldbe understood as a solution with arbitrary precision Inthis subsection a uniform thickness square moderately thickrectangular plate with completely clamped boundary condi-tion (C-C-C-C) and four edges equally elastically restrainedagainst linear spring constants and rotation spring constantssupports (E-E-E-E 119870
119879= 119896119894(1198873
119863) 119870119877
= 119870119894(119887119863)
119870119879
= 10 and 119870119877
= 5) has been selected to demon-strate the convergence and accuracy of the modified Fouriermethod In Table 1 the first eight frequency parametersΩ = 120596119887
2
(120588ℎ119863)12
1205872 for the considered uniform thickness
square moderately thick rectangular plate with C-C-C-C andE-E-E-E boundary conditions are examinedThe table showsthat the proposedmethod has fast convergence behaviorThemaximum discrepancy in the worst case between the 6 times 6
truncated configuration and the 8times8 one is less than 0064In order to fully illustrate the convergence of the presentmethod Figures 3 4 and 5 present the 1st and 8th frequencyparameters Ω = 120596119887
2
(120588ℎ119863)12
1205872 with various truncated
numbers 119872 = 119873 subjected to different boundary conditionand aspect ratios A highly desired convergence characteristicis observed such that (a) sufficiently accurate results can beobtained with only a small number of terms in the seriesexpansions and (b) the solution is consistently refined asmore
terms are included in the expansions However this shouldnot constitute a problem in practice because one can alwaysverify the accuracy of the solution by increasing the trunca-tion number until a desired numerical precision is achievedAs a matter of fact this ldquoquality controlrdquo scheme can be easilyimplemented automatically In modal analysis the naturalfrequencies for higher-order modes tend to converge slower(see Table 1) Thus an adequate truncation number shouldbe dictated by the desired accuracy of the largest naturalfrequencies of interest In view of the excellent numericalbehavior of the current solution the truncation numbers willbe simply set as119872 = 119873 = 12 in the following calculations
As far as the accuracy of the present method is con-cerned the converged solutions of the present method arein excellent agreement with both the results reported byreference data and the finite element results For C-C-C-C boundary conditions the max discrepancy between thepresent results and the reference data does not exceed 0011for the worst case and in most cases is 0 Comparing theresults with exact solutions [4] it is observed that eight termsare sufficient to obtain accurate resultsMoreover with regardto the E-E-E-E boundary condition the max discrepancybetween the present results and the reference data does notexceed 031 for the worst case and in most cases is 01Regarding the results with DQM solutions [33] it can be seenthat the six terms are sufficient to obtain enough accurateresults In addition it is clear that the results of the presentapproach with just 663 DOFs (119872 times 119873 = 12 times 12) canpredict the vibration characteristics accurately Most of themare identical to those obtained from finite element method(FEM) with 10201 DOFs (S4R 001m times 001m) That isto say it needs only 662 DOFs compared with FEM toobtain the same precision solutions for the considered case
Shock and Vibration 7
1
2
3
4
5
6
The 1st orderThe 3rd order
101 103 105 107 109 1011 1013
Ω
Ki (Nm)
(a)
2
3
4
5
6
101 103 105 107 109 1011 1013
The 1st orderThe 3rd order
Ω
Ki (Nmrad)
(b)
3
4
5
6
101 103 105 107 109 1011 1013
Kij (Nmrad)
The 1st orderThe 3rd order
Ω
(c)
Figure 2 The effect of boundary spring stiffness on the natural frequencies Ω (a) translation spring (b) rotation spring and (c) torsionalspring
On the same hardware (Intel i7-39GHz) the computing timeof the present formulation for the solution (119872times119873 = 12times12)implemented in optimized MATLAB scripts is about 2125 swhereas the finite element solution consumes 34578 s that isat least 16 times more CPU time than the present method forthe same problem
As mentioned earlier in the current modeling frame-work all the classical boundary conditions and their com-binations can be conveniently viewed as special cases whenthe stiffness for the normal and tangential boundary springsbecomes zero andor infinitely large Thus the effects of thestiffness of the translation (119896
119894) rotation (119870
119894) and torsional
springs (119870119894119895) on the modal characteristics should be inves-
tigated As shown in the Figures 2(a)ndash2(c) the first and thethird frequency parameters are separately obtained by vary-ing the stiffness of one group of the boundary springs from
extremely large (1014) to extremely small (100) while assigningthe other group of the springs infinite stiffness (1014) It canbe found in Figure 2(a) that the frequency parameter almostkeeps at a level when the stiffness of the translation springs islarger than 1012 or smaller than 107 In Figure 2(b) the influ-ences of the rotation springs on frequency parameters aregiven It is shown that the frequency curves change greatlywithin the stiffness range (106 to 1010) while out of this rangethe frequency curves separately keep level In Figure 2(c) theinfluences of the torsional springs on frequency parametersare given It is shown that the frequency curves almost changewhen the stiffness changes in the whole range
Based on the analysis it can be found that the torsionalsprings almost have no effect on the structure Also the rela-tionship between the rotation springs and twisting momentcan be seen from the boundary condition expression Then
8 Shock and Vibration
32
36
40
44
48
52
2
3
4
5
6
7
8
Ω Ω
0 2 4 6 8 10 12 14 16 18M = N
ab = 1
ab = 32
(a) The 1st order
0 2 4 6 8 10 12 14 16 180
20
40
60
80
0
50
100
150
200
250
300
M = N
Ω Ω
ab = 1
ab = 32
(b) The 8th order
Figure 3 The effect of numerically truncated numbers119872 = 119873 on the natural frequenciesΩ for C-C-C-C boundary condition
0 2 4 6 8 10 12 14 16 18125
130
135
140
145
150
088
090
092
094
096
098
Ω Ω
M = N
ab = 1
ab = 32
(a) The 1st order
0 2 4 6 8 10 12 14 16 18
55
60
65
70
75
80
40
45
50
55
60
65
70
75
80Ω Ω
M = N
ab = 1
ab = 32
(b) The 8th order
Figure 4 The effect of numerically truncated numbers119872 = 119873 on the natural frequenciesΩ for F-F-F-F boundary condition
the twisting moments have small effect on the vibration char-acteristics of the structure However in this paper in order toget a more accurate prediction of the vibration characteristicsof the structure the twisting moment on boundary edges istaken into account In the latter study in addition to the factthat free boundary is not unexpected considering the torsionspring the other boundary conditions are introduced into atorsion spring and the spring stiffness takes infinity FormFigure 2 analysis it also concluded that the translation springhas a wider influence range than the rotation spring on thefrequency parameters that is for the translation springs thestable frequency parameters appearwhen the stiffness ismorethan 1012 or less 107 while for the rotation springs when thestiffness value is assigned more than 1010 the frequency para-meters become smoothThus it is suitable to use 1014 to sim-ulate the infinite stiffness value in the model validation partsand in the following examples
32 Uniform Thickness Moderately Thick Plates with Classicaland Elastic Boundary Conditions In this subsection themodified Fourier solution is applied to deal with vibrationproblems of uniform thickness moderately thick rectangu-lar plates subject to the classical boundary conditions andarbitrary elastic boundary conditions In present work threegroups of continuously distributed boundary springs areintroduced to simulate the given or typical boundary condi-tions As mentioned earlier the stiffness of these boundarysprings can take any value from zero to infinity to bettermodel many real-world restraint conditions Taking edge 119909 =0 for example the corresponding spring stiffness for the threetypes of classical boundaries and elastic boundaries is
completely free 119876119909= 0119872
119909119909= 0 and119872
119909119910= 0
1198961199090= 0 119870
1199090= 0 119870
1199101199090= 0 (17a)
Shock and Vibration 9
0 2 4 6 8 10 12 14 16 18
166
168
170
172
152
156
160
164
168
Ω Ω
ab = 1
ab = 32
M = N
(a) The 1st order
0 2 4 6 8 10 12 14 16 1850
55
60
65
70
4
5
6
7
8
9
10
Ω Ω
M = N
ab = 1
ab = 32
(b) The 8th order
Figure 5 The effect of numerically truncated numbers 119872 = 119873 on the natural frequencies Ω with four edges elastically restrained againsttranslation and rotation (119870
119879= 119896119894(1198873
119863) 119870119877= 119870119894(119887119863) 119870
119879= 100 and 119870
119877= 100)
simply supported 119908 = 0119872119909119909= 0 and 120595
119910= 0
1198961199090= 1014
1198701199090= 0 119870
1199101199090= 1014
(17b)
completely clamped 119908 = 0 120595119909= 0 and 120595
119910= 0
1198961199090= 1014
1198701199090= 1014
1198701199101199090
= 1014
(17c)
arbitrarily elastic boundary 119908 = 0 120595119909
= 0 120595119910
= 0119876119909
= 0119872119909119909
= 0 and119872119909119910
= 0
1198961199090= Γ1199081199090
1198701199090= Γ1199091199090
1198701199101199090
= 1014
(17d)
The appropriateness of the three classical boundariesdefined in (17a)ndash(17c) will be proved by several examplesgiven in following the arbitrary elastic boundaries are alsodefined in (17d) and the Γ
119908119894(Γ119908119894= 1198961198941198960 1198960= 1 times 10
9Nm2119894 = 1199090 1199091198861199100119910119887) and Γ
119909119894(Γ119909119894= 11987011989411987001198700= 1times10
9Nmrad119894 = 1199090 119909119886 1199100 119910119887) elastic restraint parameters representcorresponding spring stiffness For the sake of simplicitya four-letter string is employed to represent the restraintcondition of a plate such that F-C-S-E identifies the platewithedges 119909 = 0 119910 = 0 119909 = 119886 and 119910 = 119887 having free clampedshear-supported restrained and elastic boundary conditionsrespectively
As for the first case a uniform thickness moderately thickplate with different classical boundaries and structure param-eters is investigated here In Table 2 the comparison of thefirst eight frequency parametersΩ = 120596119887
2
(120588ℎ119863)12
1205872 of the
considered plate is presented The S-S-S-S C-F-F-F S-S-F-FF-F-F-F and S-C-S-C boundary conditions are performed inthe comparison Excellent agreements are observed betweenthe solutions obtained by the modified Fourier method thereferential data and finite element method (FEM) results forthe uniform thickness moderately thick rectangular platesIt is also verified that the definition of the three types of
classical boundaries in (17a)ndash(17c) is appropriate In additionthe elastic boundary conditions (17d) are also verified Inthe next two examples we will account for the vibrationof moderately thick plate with elastic edge supports Thefirst model considered is an S-S-S-S square moderately thickplate with all edges elastically rotationally restrained Thatis 1198701199090
= 119870119909119886
= 1198701199100
= 119870119910119887
= Γ119909 and all the
other restraining springs are set to have an infinite stiffness(namely represented by 1014 in numerical calculation) Thesix frequency parameters Ω = 120596119887
2
(120588ℎ119863)12
1205872 are given
in Table 3 for several different restraining coefficient valuesthe finite element method (FEM) results calculated usingABAQUSare also listed in this table as a referenceThe secondmodel concerns a complete square moderately thick platewith all edges elastically restrained That is 119896
1199090= 119896119909119886
=
1198961199100
= 119896119910119887
= Γ119908and 119870
1199090= 119870119909119886
= 1198701199100
= 119870119910119887
= Γ119909 The
six frequency parameters Ω = 1205961198872
(120588ℎ119863)12
1205872 are given in
Table 4 for several different restraining coefficient values thefinite element method (FEM) results are also listed in Table 4as a reference It can be clearly seen that the comparison isextremely good which implies that the current method isable tomake correct predictions for themodal characteristicsof moderately thick plate with not only classical boundaryconditions but also elastic edge restraints
The excellent agreements between the solution obtainedby the modified Fourier method and the referential datafor the moderately thick plate subjected to the combina-tions of classical boundary conditions and elastic boundaryconditions given in Tables 2ndash4 indicate that the proposedmethod is sufficiently accurate to deal with uniform thicknessmoderately thick plate with arbitrary boundary conditions
33 Linearly VariationThickness Moderately Thick Plates withClassical and Elastic Boundary Conditions In the theoreticalformulations this paper concerns the varying thicknessmoderately thick plates with classical and elastic boundary
10 Shock and Vibration
Table 2 Frequency parameters Ω = 1205961198872
(120588ℎ119863)12
1205872 for uniform thickness moderately thick plates with different classical boundaries and
structure parameters
Boundary condition ab Model sequence1 2 3 4 5 6 7 8
S-S-S-S
11931 4605 4605 7064 8605 8605 10792 10792
(1933a) (4611) (4611) (7067) (8607) (8607) (10799) (10780)(1931b) (4605) (4605) (7064) (8605) (8605) (10792) (10792)
321408 2648 4127 4605 5261 7064 7153 8184(1410a) (2651) (4128) (4608) (5267) (7074) (7155) (8188)(1408b) (2648) (4127) (4605) (5261) (7064) (7151) (8184)
C-F-F-F
10348 0817 2035 2583 2862 4816 5478 5774
(0348a) (0817) (2038) (2586) (2865) (4821) (5504) (5793)(0348b) (0816) (2034) (2582) (2860) (4811) (5477) (5772)
320155 0502 0941 1665 2292 2613 3303 3498
(0154a) (0502) (0941) (1663) (2295) (2618) (3305) (3500)(0154b) (0501) (0940) (1662) (2292) (2612) (3298) (3494)
S-S-F-F
10333 1678 1875 3559 4720 4946 6474 6632
(0335a) (1680) (1878) (3600) (4721) (4949) (6478) (6642)(0333b) (1677) (1874) (3557) (4718) (4945) (6472) (6631)
320222 0938 1631 2363 2573 4082 4482 4753
(0223a) (0940) (1632) (2364) (2575) (4092) (4483) (4759)(0222b) (0938) (1631) (2361) (2571) (4077) (4479) (4752)
F-F-F-F1 1290 1919 2363 3235 3235 5605 5605 5646
(1291a) (1920) (2365) (3239) (3239) (5607) (5607) (6648)
32 0865 0948 1961 2166 2461 2843 3551 4026(0865a) (0949) (1963) (2168) (2465) (2844) (3556) (4034)
S-C-S-C1 2700 4971 5990 7973 8787 10250 11334 12024
(2701a) (4971) (5993) (7975) (8788) (10253) (11335) (12028)
32 2348 3253 4972 5646 6489 7381 7974 9930(2348a) (3255) (4974) (5647) (6590) (7384) (7801) (9932)
aResults in parentheses are taken from FEMbResults in parentheses are taken from [11]
conditions The varying thickness function ℎ(119909 119910) can beexpressed as ℎ
0(1 + 120572119909
119904
)(1 + 120573119910119905
) in which the ℎ0 120572 and
120573 represent the initial thickness gradient in 119909 direction andgradient in 119910 direction When the indexes 119904 and 119905 take thevalue 119904 = 119905 = 1 the analyticalmodel imitates the linearly vari-ation thickness moderately thick plates structure In order tounify the description and facilitate the analytical calculationsof the involved integrals all the thickness variation functionscan be expanded into either 1D or 2D Fourier cosine seriesresulting in
ℎ (119909 119910) =
infin
sum
119898=0
infin
sum
119899=0
ℎ0119902119898119899
cos 119898120587119909119886
cos119898120587119910
119887
(18)
where119902119898119899
=
4
119886119887
int
119887
0
int
119886
0
(1 + 120572119909119904
) (1 + 120573119910119905
) cos 119898120587119909119886
cos119898120587119910
119887
119889119909 119889119910
(19)
In order to prove the validity of the present methodfor the vibration of linearly variation thickness moderately
thick plates with arbitrary boundary conditions the typicalclassical boundary conditions as the first case will be con-sidered In Table 5 the comparison of the first six frequencyparameters Ω = 120596119886
2
(120588ℎ01198630)12 of the moderately thick
plates with linearly varying thickness is presentedThe S-S-S-S C-F-F-F S-S-F-F C-C-C-C and S-C-S-C boundary condi-tions are performed in the comparison Excellent agreementsare observed between the solutions obtained by the modifiedFourier method and finite element method (FEM) results forthemoderately thick plates with linear variation thickness Toinvestigate the influence of the aspect ratio on the uniformthickness and nonuniform thickness moderately thick platesthe effect on the frequency parameters for plates with S-S-S-Sboundary conditions is presented in Figure 6 The thicknessfunctions are ℎ
0and ℎ0(1+05times119909)(1+05times119910) respectively It
is seen from Figure 6 that the influence of aspect ratios on thefrequency parameters for nonuniform thickness moderatelythick plates is more complicated
In the next two examples we also account for the vibra-tions of moderately thick plate with linear variation thicknessand elastic edge supports The first model considered is an S-S-S-S square moderately thick plate with all edges elastically
Shock and Vibration 11
Table 3 Frequency parameters Ω = 1205961198872
(120588ℎ119863)12
1205872 for S-S-S-S moderately thick plates (119886119887 = 1) with uniform thickness and elastic
rotation support (1198701199090= 119870119909119886= 1198701199100= 119870119910119887= Γ119909)
Γ119909
aℎ0119886
Model sequence1 2 3 4 5 6
0
005 1945 4834 4834 7591 9515 9519(1942b) (4836) (4836) (7584) (9547) (9552)
01 1856 4500 4500 6829 8499 8513(1858b) (4510) (4510) (6843) (8539) (8554)
015 1758 4114 4114 6040 7421 7445(1761b) (4126) (4126) (6060) (7462) (7486)
02 1656 3733 3733 5335 6468 6499(1660b) (3747) (3747) (5357) (6508) (6538)
05
005 3541 7071 7071 10227 12318 12399(3546b) (7091) (7091) (10253) (12386) (12468)
01 3274 6242 6242 8719 10302 10421(3282b) (6267) (6267) (8754) (10370) (10488)
015 2947 5361 5361 7290 8465 8606(2956b) (5387) (5387) (7327) (8524) (8666)
02 2624 4599 4599 6148 7057 7199(2634b) (4624) (4624) (6182) (7107) (7250)
1
005 3541 7072 7072 10228 12320 12401(3547b) (7093) (7093) (10254) (12388) (12470)
01 3278 6248 6248 8727 10311 10429(3286b) (6274) (6274) (8762) (10378) (10497)
015 2957 5374 5374 7305 8478 8620(2966b) (5400) (5400) (7341) (8537) (8680)
02 2641 4617 4617 6166 7070 7215(2651b) (4643) (4643) (6200) (7121) (7268)
15
005 3542 7073 7073 10229 12320 12401(3547b) (7093) (7093) (10254) (12388) (12471)
01 3280 6251 6251 8729 10313 10432(3287b) (6276) (6276) (8765) (10381) (10500)
015 2960 5379 5379 7309 8482 8625(2969b) (5405) (5405) (7346) (8541) (8685)
02 2647 4623 4623 6172 7074 7221(2657b) (4649) (4649) (6206) (7125) (7274)
2
005 3542 7073 7073 10229 12321 12402(3547b) (7093) (7093) (10255) (12389) (12471)
01 3280 6252 6252 8730 10315 10434(3288b) (6277) (6277) (8766) (10382) (10501)
015 2962 5381 5381 7312 8484 8628(2971b) (5407) (5407) (7348) (8543) (8688)
02 2650 4627 4627 6175 7076 7224(2660b) (4652) (4652) (6210) (7128 (7277)
aΓ119909= 1198701198941198700 (1198700= 1 times 10
9 Nmrad 119894 = 1199090 119909119886 1199100 119910119887)bResults in parentheses are taken from FEM
12 Shock and Vibration
Table 4 Frequency parameters Ω = 1205961198872
(120588ℎ119863)12
1205872 for moderately thick plates (119886119887 = 1) with uniform thickness and all edge elastic
restraints (1198701199090= 119870119909119886= 1198701199100= 119870119910119887= Γ119909and 119896
1199090= 119896119909119886= 1198961199100= 119896119910119887= Γ119908)
Γ119909
aΓ119908
aℎ0119886
Model sequence1 2 3 4 5 6
05 0
005 0986 1967 3893 3893 4850 4850(0988b) (1971) (3897) (3897) (4852) (4852)
01 0916 1835 3515 3522 4375 4375(0917b) (1837) (3519) (3528) (4341) (4341)
015 0789 1635 2983 3027 3771 3771(0792b) (1638) (2988) (3029) (3775) (3775)
02 0646 1444 2497 2601 3249 3249(0652b) (1448) (2508) (2621) (3254) (3254)
1 05
005 3417 3417 4431 5294 5461 6448(3419b) (3419) (4435) (5303) (5478) (6456)
01 1593 1593 2401 3849 3861 4729(1611b) (1611) (2408) (3852) (3866) (4767)
015 1122 1122 1913 3272 3290 4038(1127b) (1227) (1924) (3284) (3314) (4069)
02 0890 0890 1640 2781 2829 3476(0899b) (0899) (1654) (2787) (2834) (3497)
15 1
005 4325 4325 5619 6273 6571 7619(4334b) (4334) (5681) (6299) (6642) (7642)
01 2013 2013 2812 4088 4116 4993(2024b) (2024) (2827) (4122) (4187) (4994)
015 1338 1338 2108 3430 3443 4196(1342b) (1342) (2112) (3467) (3482) (4200)
02 1038 1038 1772 2936 2965 3613(1042b) (1042) (1786) (2937) (2977) (3624)
2 2
005 5231 5231 6919 7630 8002 9220(5238b) (5238) (6931) (7637) (8011) (9241)
01 2594 2594 3439 4470 4544 5428(2597b) (2597) (3442) (4495) (4557) (5438)
015 1653 1653 2403 3613 3632 4392(1662b) (1662) (2414) (3627) (3646) (4404)
02 1238 1238 1949 3077 3099 3751(1244b) (1244) (1964) (3094) (3116) (3788)
aΓ119909= 1198701198941198700 (1198700= 1 times 10
9 Nmrad 119894 = 1199090 119909119886 1199100 119910119887) Γ119908= 1198961198941198960 (1198960= 1 times 10
9 Nm 119894 = 1199090 119909119886 1199100 119910119887)bResults in parentheses are taken from FEM
05 10 15 20 25 300
2
4
6
8
10
12
14
16
ab
The 1st orderThe 2nd orderThe 3rd order
The 4th orderThe 5th orderThe 6th order
Ω
(a)
05 10 15 20 25 30
30
60
90
120
150
180
ab
The 1st orderThe 2nd orderThe 3rd order
The 4th orderThe 5th orderThe 6th order
Ω
(b)
Figure 6 The effect of aspect ratio 119886119887 on the natural frequenciesΩ for S-S-S-S boundary condition (a) uniform thickness and (b) nonuni-form thickness
Shock and Vibration 13
Table 5 Frequency parameter Ω = 1205961198862
(120588ℎ01198630)12 for moderately thick plates with linearly varying thickness in different boundary
conditions (ℎ(119909 119910) = ℎ0(1 + 05119909)(1 + 05119910))
Boundary condition 119886119887 ℎ0119886
Model sequence1 2 3 4 5 6
S-S-S-S
101 27261 62397 62926 92423 111537 111732
(27263a) (62410) (63842) (92445) (111867) (111921)
02 23103 47010 47191 65018 75831 75865(23489a) (47209) (47349) (65057) (76605) (76473)
3201 59338 99101 138798 151102 166710 204855
(59874a) (99249) (138177) (151362) (166644) (205381)
02 41175 63765 63765 73598 83956 89084(41586a) (63313) (63669) (73661) (83918) (89547)
C-F-F-F
101 4173 10331 27407 37826 38809 64095
(4856a) (10680) (27513) (37203) (39135) (64858)
02 3983 8702 21403 29443 30286 46705(4118a) (9306) (21990) (29670) (30602) (47156)
3201 4868 15488 35620 63523 89969 100089
(4887a) (15320) (35175) (63353) (89159) (100783)
02 4469 9763 22338 42188) 50104 56331(4944a) (9636) (22087) (42177) (50580) (56559)
S-S-F-F
101 4117 24795 26216 48567) 68195 69667
(4789a) (24149) (26751) (49324) (68842) (70575)
02 3820 20516 21924 37523 48980 50327(4103a) (20041) (22125) (37221) (48894) (50622)
3201 7935 39454 65063 92877 92877 136176
(7856a) (40145) (65162) (92462) (93022) (136336)
02 6389 26149 35627 49539 56844 65170(6406a) (26536) (35626) (49831) (57144) (65981)
C-C-C-C
101 44471 80299 80681 109422 126871 128207
(45102a) (80271) (80656) (110015) (126328) (128888)
02 32162 52781 52933 69410 78474 79482(32240a) (52629) (53166) (70063) (79059) (80266)
3201 82975 115587 151992 161579 176601 212330
(83139a) (115838) (152200) (162193) (177058) (212849)
02 48577 66022 83468 90830 97474 98157(49187a) (66259) (84065) (91749) (97827) (98777)
S-C-S-C
101 36847 66664 76965 101045 113594 125551
(37901a) (68568) (78356) (101200) (113978) (125651)
02 27673 48469 51220 67027 76358 78241(27897a) (48685) (51491) (67447) (77530) (79573)
3201 77378 106848 150079 154569 173120 208391
(78003a) (107058) (150991) (155171) (173099) (208402)
02 45895 64732 73393 83282 90325 97089(45803a) (64131) 73656 83801 90518 97015
aResults in parentheses are taken from FEM
restrained That is 1198701199090
= 119870119909119886
= 1198701199100
= 119870119910119887
= Γ119909 The six
frequency parameters Ω = 1205961198862
(120588ℎ01198630)12 are given in
Table 6 for several different restraining coefficient valuesthe finite element method (FEM) results calculated usingABAQUS are also presented as a referenceThe secondmodelis a complete square moderately thick plate with all edgeselastically restrainedThat is 119896
1199090= 119896119909119886= 1198961199100= 119896119910119887= Γ119908and
1198701199090= 119870119909119886
= 1198701199100= 119870119910119887= Γ119909 The six frequency parameters
Ω = 1205961198862
(120588ℎ01198630)12 are given in Table 7 for several different
restraining coefficient values the finite element method(FEM) results are also listed as a reference It can be clearlyseen that the comparison is extremely good which impliesthat the current method is able to make correct predictionsfor the modal characteristics of linearly varying thickness
14 Shock and Vibration
Table 6 Frequency parameter Ω = 1205961198862
(120588ℎ01198630)12 for S-S-S-S moderately thick plates (119886119887 = 1) with linearly varying thickness and elastic
rotation support (ℎ(119909 119910) = ℎ0(1 + 05119909)(1 + 05119910) 119870
1199090= 119870119909119886= 1198701199100= 119870119910119887= Γ119909)
Γ119909
aℎ0119886
Model sequence1 2 3 4 5 6
0
005 28809 70080 70891 108967 134884 135295(28897b) (70334) (71129) (109112) (135254) (135772)
01 27261 62398 62926 92423 111540 111735(26606b) (61784) (62303) (91193) (111323) (111648)
015 25231 54192 54505 77118 91228 91312(25204b) (55297) (55595) (77503) (91871) (91941)
02 23103 47010 47191 65018 75832 75866(23947b) (47071) (47208) (65404) (76531) (76589)
05
005 23103 47010 47191 65018 75832 75866(23210b) (47245) (47358) (65241) (76122) (76345)
01 48377 94650 95597 135484 162635 163501(48524b) (94891) (95992) (135845) (167012) (164024)
015 36239 70939 71298 99997 118584 119009(36647b) (71214) (71687) (100024) (119114) (119457)
02 28749 56937 57037 79214 93009 93162(29142b) (57034) (57544) (79654) (93668) (93754)
1
005 49651 96759 97619 138089 165368 166304(49755b) (97245) (97964) (138544) (165864) (166878)
01 38951 73735 74251 102748 121079 121709(39021b) (74145) (74664) (102874) (121613) (121817)
015 30630 58406 58533 80432 93984 94255(30638b) (58684) (58644) (80874) (94121) (94478)
02 25548 48636 48674 66165 76708 76824(25687b) (48788) (48992) (66544) (76824) (76944)
15
005 50118 97561 98374 139096 166421 167382(50247b) (97877) (98661) (139247) (166875) (167662)
01 40310 75216 75783 104229 122402 123164(40366b) (75542) (75921) (104557) (122874) (123385)
015 31837 59352 59530 81245 94611 94983(31902b) (59451) (59871) (81375) (94784) (95105)
02 26302 49119 49156 66528 76954 77130(26418b) (49354) (49334) (66921) (77010) (77470)
2
005 50360 97983 98769 139631 166979 167954(50472b) (98005) (99114) (140154) (167188) (167986)
01 41130 76146 76724 105165 123229 124082(41421b) (76642) (78114) (105661) (123278) (124974)
015 32687 60025 60245 81832 95052 95508(32778b) (60244) (60549) (82146) (95114) (95842)
02 26893 49492 49539 66815 77138 77370(26922b) (49874) (49924) (66987) (77367) (77588)
aΓ119909= 1198701198941198700 (1198700= 1 times 10
9 Nmrad 119894 = 1199090 119909119886 1199100 119910119887)bResults in parentheses are taken from FEM
Shock and Vibration 15
Table 7 Frequency parameterΩ = 1205961198862
(120588ℎ01198630)12 for moderately thick plates (119886119887 = 1) with linearly varying thickness and elastic rotation
and translation support (ℎ(119909 119910) = ℎ0(1 + 05119909)(1 + 05119910) 119870
1199090= 119870119909119886= 1198701199100= 119870119910119887= Γ119909 and 119896
1199090= 119896119909119886= 1198961199100= 119896119910119887= Γ119908)
Γ119909
aΓ119908
aℎ0119886
Model sequence1 2 3 4 5 6
05 0
005 14751 14917 28232 55251 55397 69512(14780b) (14822) (29252) (55476) (55505) (70904)
01 11811 11910 23857 44462 45037 55453(11818b) (11897) (24711) (44376) (45257) (56511)
015 8487 8514 20011 34783 36335 44970(8500b) (8544) (20738) (34404) (36663) (45625)
02 6073 6073 17431 28679 30805 38010(6096b) (6106) (18039) (28129) (31165) (38311)
1 05
005 21225 32173 32173 44381 64061 65222(20708b) (32097) (32285) (45110) (64182) (65355)
01 8140 16878 16941 28100 48663 49015(7943b) (16838) (16970) (28888) (48700) (49171)
015 4493 11859 11878 22429 38352 39247(4383b) (11867) (11891) (23125) (38177) (39488)
02 2932 8690 8691 18919 31108 32600(3540b) (8791) (8791) (16229) (27520) (27889)
15 1
005 27819 40919 41430 54595 70792 72896(27146b) (41008) (41434) (55256) (70895) (73051)
01 11285 20068 20230 31101 51046 51429(11012b) (20025) (20270) (31849) (51126) (51569)
015 6292 13912 13922 24132 40500 41120(6139b) (13912) (13934) (24808) (40426) (41316)
02 4121 10321 10331 20051 32796 33921(4021b) (10329) (10357) (20635) (32540) (34178)
2 2
005 34615 51972 53088 68255 81344 84510(33806b) (52100) (53157) (68891) (81433) (84761)
01 15416 24659 24829 35615 53879 54494(15045b) (24625) (24888) (36303) (53970) (54626)
015 8740 16420 16481 26353 42417 42946(8529b) (16414) (16499) (26996) (42398) (43114)
02 5761 12127 12127 21434 34286 35192(5622b) (12134) (12148) (21998) (34110) (35415)
aΓ119909= 1198701198941198700 (1198700= 1 times 10
9 Nmrad 119894 = 1199090 119909119886 1199100 119910119887) Γ119908= 1198961198941198960 (1198960= 1 times 10
9 Nm 119894 = 1199090 119909119886 1199100 119910119887)bResults in parentheses are taken from FEM
moderately thick plate with not only classical boundaryconditions but also elastic edge restraints
As the last case of this subsection the influence of thegradient 120572 and 120573 on the fundamental frequency parametersfor a linearly varying thickness moderately thick plate isinvestigated The model is a square moderately thick platewith all edges elastically restrainedThat is 119896
1199090= 119896119909119886= 1198961199100=
119896119910119887= Γ119908= 2 and 119870
1199090= 119870119909119886
= 1198701199100= 119870119910119887= Γ119909= 2 The six
frequency parameters Ω = 1205961198862
(120588ℎ01198630)12 are given in
Table 8 for several different 120572 and 120573 values the finite elementmethod (FEM) results are also listed as a reference Againgood agreement can be observed Through Table 8 it is alsofound that the frequency parameter increases with increasinggradient parameters
The above studies are given as linearly varying thicknessmoderately thick plates with several of boundary condition
and different structure parameters In the next section vibra-tion results for the plates subjected to nonlinear variationthickness will be presented
34 Nonlinearly Varying Thickness Moderately Thick Plateswith Classical and Elastic Boundary Conditions In Sec-tion 33 the linearly varying thickness moderately thickplates were studied However in the practical engineeringapplications the varying thickness of amoderately thick platemay not always be linear variation in nature A variety ofpossible thickness varying cases may be encountered in prac-tice Therefore the moderately thick plates with nonlinearvariation thickness subjected to general elastic edge restraintsare examined in this subsection For the sake of brevity theindexes 119904 and 119905 will be chosen as 2 to imitate the nonlinearlyvarying thickness moderately thick plates structure in this
16 Shock and Vibration
Table 8 Frequency parameterΩ = 1205961198862
(120588ℎ01198630)12 for elastic support moderately thick plates (119886119887 = 1) with different gradient (119896
1199090= 119896119909119886=
1198961199100= 119896119910119887= Γ119908= 2 119870
1199090= 119870119909119886= 1198701199100= 119870119910119887= Γ119909= 2)
120572 120573 ℎ0119886
Model sequence1 2 3 4 5 6
120572 = 120573 = 01
005 31275 52615 52657 68954 76146 79850(31303a) (52664) (52675) (69033) (76174) (79948)
01 17270 25244 25250 33949 46050 46625(17271a) (25255) (25258) (34089) (46122) (46708)
015 10190 16266 16274 24164 37327 37526(10189a) (16269) (16280) (24323) (37401) (37622)
02 6780 12258 12260 19685 31452 31754(6780a) (12265) (12270) (19851) (31514) (31864)
120572 = 120573 = 02
005 32530 52947 53141 68954 77034 80589(32577a) (53014) (53203) (69172) (77088) (80773)
01 16882 24979 24979 34154 48057 48575(16883a) (24983) (25001) (34433) (48134) (48667)
015 9804 16284 16309 24683 38844 39085(9803a) (16283) (16322) (24980) (38909) (39191)
02 6499 12290 12295 20133 32354 32770(6498a) (12296) (12307) (20420) (32387) (32895)
120572 = 120573 = 03
005 33516 52845 53308 68704 78203 81561(33563a) (52939) (53389) (69055) (78271) (81789)
01 16419 24799 24830 34527 50061 50583(16419a) (24790) (24869) (34947) (50143) (50688)
015 9430 16334 16378 25236 40193 40506(9429a) (16329) (16395) (25660) (40243) (40625)
02 6235 12288 12293 20572 33102 33663(6235a) (12296) (12306) (20965) (33090) (33811)
120572 = 120573 = 04
005 34215 52463 53264 68430 79654 82865(34252a) (52575) (53346) (68921) (79732) (83113)
01 15921 24695 24793 35027 52011 52571(15921a) (24674) (24842) (35583) (52098) (52688)
015 9074 16388 16444 25797 41379 41789(9074a) (16381) (16464) (26336) (41404) (41928)
02 5990 12236 12238 21002 33735 34461(5989a) (12249) (12250) (21487) (33653) (34641)
120572 = 120573 = 05
005 34615 51972 53088 68255 81344 84510(33806a) (52100) (53157) (68891) (81433) (84761)
01 15416 24659 24829 35615 53879 54494(15045a) (24625) (24888) (36303) (53970) (54626)
015 8740 16420 16481 26353 42417 42946(8529a) (16414) (16499) (26996) (42398) (43114)
02 5761 12127 12127 21434 34286 35192(5622a) (12134) (12148) (21998) (34110) (35415)
aResults in parentheses are taken from FEM
section that is 119904 = 119905 = 2 Also the thickness varying functionsare also be expressed as Fourier cosine series according to (18)and (19)
In order to validate the accuracy and reliability of theproposed method for predicting the vibration behavior ofnonlinearly varying thickness moderately thick plates with
arbitrary boundary conditions the typical classical boundaryconditions viewed as the special cases of elastically restrainededges will be considered The comparison of the first six fre-quency parameters Ω = 120596119886
2
(120588ℎ01198630)12 for the moderately
thick plates with nonlinearly varying thickness is presentedin Table 9 The S-S-S-S C-F-F-F S-S-F-F C-C-C-C and
Shock and Vibration 17
Table 9 Frequency parameter Ω = 1205961198862
(120588ℎ01198630)12 for moderately thick plates with linearly varying thickness in different boundary
conditions (ℎ(119909 119910) = ℎ0(1 + 05119909
2
)(1 + 051199102
))
Boundary condition 119886119887 ℎ0119886
Model sequence1 2 3 4 5 6
S-S-S-S
101 23462 54999 55573 82950 100900 101187
(23741a) (55114) (55983) (83146) (101004) (101347)
02 18606 42725 42971 60040 71167 71761(18936a) (43014) (43148) (60247) (71642) (71993)
3201 53055 65251 85404 106700 114880 143531
(53467a) (65472) (85984) (107211) (115687) (144007)
02 17439 21531 32170 44987 59438 64179(17874a) (21748) (32651) (45264) (60188) (64879)
C-F-F-F
101 3658 9317 24135 32816 35433 59179
(3749a) (9547) (24034) (33627) (36308) (60142)
02 3520 7986 19591 26959 27491 43882(3611a) (8176) (20075) (27125) (27625) (43984)
3201 3901 9963 28984 69187 73686 89360
(3930a) (10095) (29381) (69578) (73994) (90242)
02 3292 4439 13385 23947 40850 44758(3386a) (4498) (14012) (24068) (41221) (45142)
S-S-F-F
101 3557 21268 22762 43524 60253 62429
(3649a) (21441) (22796) (43916) (60622) (62630)
02 3171 17822 19512 33616 45042 46833(3489a) (18006) (20430) (33815) (45247) (49004)
3201 6738 31658 52071 75625 86604 94903
(6957a) (31882) (52518) (75823) (86939) (95610)
02 3653 12930 20955 26517 44097 48058(6807a) (13160) (21312) (26864) (44422) (48873)
C-C-C-C
101 41020 75196 75614 104142 119683 120949
(41405a) (75452) (75882) (104711) (119637) (121137)
02 31237 51686 51868 68411 76868 77864(32010a) (52960) (52145) (69335) (77481) (78099)
3201 95823 128727 163657 173751 190890 223595
(95364a) (128932) (163588) (172350) (190350) (222948)
02 53306 71372 89384 102404 104660 120908(53587a) (71759) (90103) (102503) (105174) (121275)
S-C-S-C
101 33715 60298 71446 94187 103802 117928
(34148a) (60786) (71828) (94515) (104263) (119326)
02 26143 45499 49458 64253 72941 76191(26789a) (46622) (50680) (64329) (73474) (76843)
3201 73228 115845 137445 148166 168342 182542
(73761a) (116121) (137788) (148530) (168698) (182921)
02 41302 53648 65630 72202 81632 88105(41304a) (53757) (65842) (72794) (82033) (88474)
aResults in parentheses are taken from FEM
S-C-S-C boundary conditions are performed in the compar-ison The results adequately demonstrated the great accuracyof the modified Fourier method
We now turn to elastically restrained moderately thickplates The first one involves an S-S-S-S moderately thicksquare plate with a uniform rotational restraint along eachedge that is 119870
1199090= 119870119909119886
= 1198701199100
= 119870119910119887
= Γ119909 The calculated
frequency parameters Ω = 1205961198862
(120588ℎ01198630)12 are given in
Table 10 together with the FEM results Since the elasticallyrestrained plates with nonlinear variation thickness are rarelyinvestigated the FEMresults are used as the referenceA goodagreement is observed between the current and FEM resultsThe second example concerns amoderately thick square plateelastically supported along all edgesThe stiffness of the linear
18 Shock and Vibration
Table 10 Frequency parameterΩ = 1205961198862
(120588ℎ01198630)12 for S-S-S-S moderately thick plates (119886119887 = 1) with linearly varying thickness and elastic
rotation support (ℎ(119909 119910) = ℎ0(1 + 05119909
2
)(1 + 051199102
) 1198701199090= 119870119909119886= 1198701199100= 119870119910119887= Γ119909)
Γ119909
aℎ0119886
Model sequence1 2 3 4 5 6
0
005 23462 54999 55573 82950 100900 101187(23741b) (55114) (55983) (83146) (101004) (101347)
01 18606 42725 42971 60040 71167 71761(18936b) (43014) (43148) (60247) (71642) (71993)
015 53055 65251 85404 106700 114880 143531(53467b) (65472) (85984) (107211) (115687) (144007)
02 17439 21531 32170 44987 59438 64179(17874b) (21748) (32651) (45264) (60188) (64879)
05
005 6904 13374 13514 19400 22904 23056(9909b) (13548) (13651) (19564) (23088) (23379)
01 5316 10237 10314 14499 17155 17264(5374b) (10448) (10651) (14764) (17188) (17879)
015 4136 8201 8220 11474 13578 13671(4174b) (8348) (8651) (11764) (13688) (13879)
02 3449 6915 6924 9549 11225 11327(3474b) (6948) (6951) (9764) (11388) (11579)
1
005 7040 13617 13745 19726 23233 23402(7074b) (13748) (13851) (19964) (23488) (23879)
01 5712 10696 10793 14991 17594 17744(5774b) (10748) (10851) (15054) (17788) (17879)
015 4489 8507 8540 11759 13803 13922(4514b) (8548) (8611) (11864) (13988) (14119)
02 3677 7084 7090 9692 11330 11445(3684b) (7148) (7191) (9764) (11588) (11879)
15
005 7089 13708 13828 19847 23355 23530(7124b) (13748) (13951) (19967) (23588) (23879)
01 5895 10923 11024 15239 17813 17988(5874b) (10748) (11651) (15264) (17818) (17979)
015 4697 8694 8737 11938 13790 14083(4774b) (8748) (8951) (11864) (13688) (14879)
02 3836 7201 7210 9792 11403 11530(3874b) (7748) (7651) (9864) (11588) (11879)
2
005 7114 13754 13871 19911 23418 23597(7174b) (12848) (13851) (19864) (23588) (23879)
01 6001 11060 11159 15391 17944 18136(6004b) (11148) (11151) (15464) (17988) (18379)
015 4836 8821 8872 12061 14037 14189(4874b) (8848) (8951) (12164) (14188) (14879)
02 3954 7289 7301 9871 11457 11594(3974b) (7348) (7651) (9964) (11488) (11679)
aΓ119909= 1198701198941198700 (1198700= 1 times 10
9 Nmrad 119894 = 1199090 119909119886 1199100 119910119887)bResults in parentheses are taken from FEM
and rotational restraints is set equal to 1198961199090
= 119896119909119886
= 1198961199100
=
119896119910119887
= Γ119908and 119870
1199090= 119870119909119886
= 1198701199100
= 119870119910119887
= Γ119909 respectively
The six frequency parametersΩ = 1205961198862
(120588ℎ01198630)12 are shown
in Table 11 for several different restraining coefficient valuesIt can also be noticed that the Fourier series method is able
to predict the modal characteristics of nonlinearly varyingthickness moderately thick plate with not only classicalboundary conditions but also elastic edge restraints correctly
Finally the influence of the gradient on the fundamentalfrequency parameters of a nonlinearly varying thickness
Shock and Vibration 19
Table 11 Frequency parameterΩ = 1205961198862
(120588ℎ01198630)12 for moderately thick plates (119886119887 = 1) with linearly varying thickness and elastic rotation
and translation support (ℎ(119909 119910) = ℎ0(1 + 05119909
2
)(1 + 051199102
) 1198701199090= 119870119909119886= 1198701199100= 119870119910119887= Γ119909 and 119896
1199090= 119896119909119886= 1198961199100= 119896119910119887= Γ119908)
Γ119909
aΓ119908
aℎ0119886
Model sequence1 2 3 4 5 6
05 0
005 12926 13041 25090 47935 47943 61761(12931b) (13053) 25099 (47947) (47952) (61788)
01 10900 10988 21545 39980 40147 50322(10904b) (10992) (21557) (39984) (40166) (50334)
015 8245 8298 18081 31503 32617 40950(8247b) (8230) (18092) (31508) (32622) (40955)
02 6106 6124 15639 25631 27601 34458(6110b) (6125) (15642) (25637) (27617) (34462)
1 05
005 21575 31615 31644 42573 57889 58607(21580b) (31617) (31644) (42581) (57893) (58610)
01 8451 16024 16210 25969 43839 43867(8451b) (16025) (16217) (25972) (43844) (43870)
015 4682 11479 11480 20642 35172 35598(4684b) (11480) (11481) (20644) (35178) (35604)
02 3059 8621 8648 17320 28487 29638(3062b) (8619) (8644) (17327) (28492) (29641)
15 1
005 27713 40203 40978 52951 65352 66815(27714b) (40207) (40988) (52972) (65368) (66822)
01 11667 19322 19615 29122 46115 46183(11674b) (19327) (19619) (29137) (46120) (46188)
015 6544 13416 13481 22405 37238 37430(6547b) (13420) (13488) (22410) (37241) (37434)
02 4295 10175 10190 18543 30330 31056(4296b) (10177) (10194) (18544) (30336) (31060)
2 2
005 33453 50854 52088 66311 76610 78947(33455b) (50852) (50097) (66321) (76627) (78971)
01 15832 24130 24372 33924 49118 49362(15830b) (24127) (24374) (33931) (49147) (49382)
015 9060 15956 16099 24754 39134 39249(9064b) (15961) (16102) (24755) (39138) (39452)
02 5992 11955 11971 20034 31908 32392(5991b) (11957) (11978) (20042) (31910) (32398)
aΓ119909= 1198701198941198700 (1198700= 1 times 10
9 Nmrad 119894 = 1199090 119909119886 1199100 119910119887) Γ119908= 1198961198941198960 (1198960= 1 times 10
9 Nm 119894 = 1199090 119909119886 1199100 119910119887)bResults in parentheses are taken from FEM
moderately thick plate is investigated The plate is elasticallyrestrained in which the stiffness of the boundary springsis taken as 119896
1199090= 119896119909119886
= 1198961199100
= 119896119910119887
= Γ119908
= 2 and1198701199090
= 119870119909119886
= 1198701199100
= 119870119910119887
= Γ119909= 2 respectively The six
frequency parameters Ω = 1205961198862
(120588ℎ01198630)12 are presented in
Table 12 for several different slop values It can be seen thatthe fundamental frequency parameters will decrease with theincrease of the parameter ℎ
0119886
In the above examples it has been demonstrated thatthe presented method can be universally applied to nonlin-early varying thickness moderately thick plates with severalboundary conditions and different structure parametersNew results are obtained for plates with nonlinear variationthickness in both directions subjected to general elastic
boundary restraints which may be used for benchmarkingof researchers in the field In addition it is interesting to seethat the nature frequency decreases with the increase of theindex for thickness function
4 Conclusions
In this paper a modified Fourier method has been pre-sented to study the free vibration behaviors of moderatelythick rectangular plates with variable thickness and arbi-trary boundary conditionsThe first-order shear deformationplate theory is adopted to formulate the theoretical modelThe displacements and rotation components of the plateregardless of boundary conditions are invariantly expressedas the superposition of a 2D Fourier cosine series and four
20 Shock and Vibration
Table 12 Frequency parameterΩ = 1205961198862
(120588ℎ01198630)12 for elastic support moderately thick plates (119886119887 = 1) with different gradient (119896
1199090= 119896119909119886=
1198961199100= 119896119910119887= Γ119908= 2 119870
1199090= 119870119909119886= 1198701199100= 119870119910119887= Γ119909= 2)
120572 120573 ℎ0119886
Model sequence1 2 3 4 5 6
120572 = 120573 = 01
005 30187 51102 51141 67066 73954 77412(30188a) (51105) (51144) (67068) (73961) (77477)
01 16995 24719 24724 33071 44343 44819(16996a) (24725) (24729) (33084) (44347) (44822)
015 10077 15872 15883 23443 35937 36065(10081a) (15877) (15892) (23447) (35941) (36084)
02 6713 11953 11958 19086 30372 30604(6713a) (11957) (11965) (19092) (30378) (30612)
120572 = 120573 = 02
005 31206 51436 51614 67058 74398 77595(31208a) (51438) (51619) (67064) (74402) (77596)
01 16786 24503 24508 33144 45583 45923(16790a) (24510) (24513) (33147) (45587) (45932)
015 9821 15865 15906 23766 36897 36990(9824a) (15870) (15907) (23768) (36894) (36997)
02 6523 11978 11992 19356 30912 31162(6524a) (11981) (11997) (19362) (30916) (31174)
120572 = 120573 = 03
005 32097 51462 51900 66832 74991 77863(32098a) (51466) (51907) (66849) (74996) (77872)
01 16507 24332 24383 33330 46806 47074(16511a) (24327) (24380) (33339) (46810) (47086)
015 9562 15885 15964 24103 37748 37828(9562a) (15892) (15977) (21105) (37751) (37832)
02 6338 11992 12013 19603 31338 31634(6340a) (11994) (12011) (19612) (31344) (31639)
120572 = 120573 = 04
005 32851 51239 52046 66547 75735 78303(33455a) (50852) (50097) (66321) (76627) (78971)
01 16182 24209 24343 33598 47990 48230(33455a) (50852) (50097) (66321) (76627) (78971)
015 9307 15919 16034 24436 38491 38579(33455a) (50852) (50097) (66321) (76627) (78971)
02 6161 11986 12008 19828 31666 32040(33455a) (50852) (50097) (66321) (76627) (78971)
120572 = 120573 = 05
005 33453 50854 52088 66311 76610 78947(33455a) (50852) (50097) (66321) (76627) (78971)
01 15832 24130 24372 33924 49118 49362(15830a) (24127) (24374) (33931) (49147) (49382)
015 9060 15956 16099 24754 39134 39249(9064a) (15961) (16102) (24755) (39138) (39452)
02 5992 11955 11971 20034 31908 32392(5991a) (11957) (11978) (20042) (31910) (32398)
aResults in parentheses are taken from FEM
supplementary functions in the form of the product of apolynomial function and a single cosine series expansionto ensure and accelerate the convergence of the solutionAt each edge of the plate the general restraint conditionsare implemented by introducing one group of linear springsand two groups of rotational springs which are continuouslydistributed and determined by the stiffness of these springs
Instead of seeking a solution in strong forms in the previousstudies all the Fourier coefficients will be treated equallyand independently as the generalized coordinates and solveddirectly from the Rayleigh-Ritz technique The change of theboundary conditions can be easily achieved by only varyingthe stiffness of the three sets of boundary springs along alledges of the rectangular plates without involving any change
Shock and Vibration 21
to the solution procedure The convergence of the presentsolution is examined and the excellent accuracy is validatedby comparison with existing results published in the lit-erature and FEM data Excellent agreements are obtainedfrom these comparisons The proposed method providesa unified means for extracting the modal parameters andpredicting the vibration behaviors of moderately thick plateswith variable thickness variation functions and arbitraryelastic edge restraints A variety of free vibration results formoderately thick rectangular plates with different thicknessvariation functions and boundary conditions are presentedNew results for free vibration of moderately thick rectangularplates with various thickness variation functions and edgeconditions are presented which may be used for benchmark-ing of researchers in the field
Appendices
A Representative Calculation forStiffness and Mass Matrices
To illuminate the particular expression of the mass andstiffness matrixes clearly and tersely four new variables aredefined as 119904 = 119898 lowast (119873 + 1) + 119899 + 1 119905 = 119898
1015840
lowast (119873 + 1) + 1198991015840
+ 1119901 = (119897minus1)lowast (119872+1)+119898
1015840
+1 and 119902 = (119897minus1)lowast (119873+1)+1198991015840
+1The first row elements of K andM are given below
K1-1119904119905
=
1
2
int
119886
0
int
119887
0
[119863 (119909 119910)
times 21205821198861198981205821198861198981015840 sin 120582
119886119898119909 sin 120582
1198861198981015840119909 cos 120582
119887119899119910
times cos 1205821198871198991015840119910 + (1 minus 120583) 120582
1198871198991205821198871198991015840 cos 120582
119886119898119909
times cos 1205821198861198981015840 sin 120582
119887119899119910 sin 120582
1198871198991015840119910
+ 120581119866ℎ (119909 119910) cos 120582119886119898119909 cos 120582
1198861198981015840119909 cos 120582
119887119899
times119910 cos 1205821198871198991015840119910119889119909119889119910] 119889119909 119889119910
+ int
119886
0
1198701199091199100
cos 120582119886119898119909 cos 120582
1198861198981015840119909
+(minus1)119898+1198981015840
119870119909119910119887
cos 120582119886119898119909 cos 120582
1198861198981015840119909 119889119909
+ int
119887
0
1198701199090cos 120582119887119899119910 cos 120582
1198871198991015840119910
+(minus1)119899+1198991015840
119870119909119886cos 120582119887119899119910 cos 120582
1198871198991015840119910 119889119910
K1-2119904119901
=
1
2
int
119886
0
int
119887
0
[119863 (119909 119910)21205821198861198981205821198861198981015840 sin 120582
119886119898119909 sin 120582
1198861198981015840119909
times
2
sum
119897=1
120577119897
119887(119910) cos 120582
1198871198991015840119910
minus (1 minus 120583) 1205821198871198991015840 cos 120582
119886119898119909 cos 120582
1198861198981015840119909
times
2
sum
119897=1
1205771198971015840
119887(119910) sin 120582
1198871198991015840119910 + 120581119866ℎ (119909 119910)
times ( cos 1205821198861198981015840119909 cos 120582
1198871198991015840119910
times
2
sum
119897=1
120577119897
119887(119910) cos 120582
119886119898119909)]119889119909119889119910
+ int
119886
0
1198701199091199100
cos 120582119886119898119909 cos 120582
1198861198981015840119909
2
sum
119897=1
120577119897
119887(0)
+(minus1)1198991015840
119870119909119910119887
cos 120582119886119898119909 cos 120582
1198861198981015840119909
2
sum
119897=1
120577119897
119887(119887) 119889119909
+ int
119887
0
1198701199090
2
sum
119897=1
120577119897
119887(119910) cos 120582
1198871198991015840119910
+(minus1)119898+1198981015840
119870119909119886
2
sum
119897=1
120577119897
119887(119910) cos 120582
1198871198991015840119910119889119910
K1-3119904119902
=
1
2
int
119886
0
int
119887
0
[119863 (119909 119910) minus 21205821198861198981015840 sin 120582
1198861198981015840119909 cos 120582
1198871198991015840119910
times
2
sum
119897=1
1205771198971015840
119886(119909) cos 120582
119887119899119910
+ (1 minus 120583) 1205821198871198991205821198871198991015840 cos 120582
1198861198981015840119909
times sin 1205821198871198991015840119910
2
sum
119897=1
120577119897
119886(119909) sin 120582
119887119899119910
+ 120581119866ℎ (119909 119910) cos 1205821198861198981015840119909 cos 120582
1198871198991015840119910
times
2
sum
119897=1
120577119897
119886(119909) cos 120582
119887119899119910]119889119909119889119910
+ int
119886
0
1198701199091199100
cos 1205821198861198981015840119909
2
sum
119897=1
120577119897
119886(119909)
+(minus1)119899+1198991015840
119870119909119910119887
cos 1205821198861198981015840119909
2
sum
119897=1
120577119897
119886(119909) 119889119909
+ int
119887
0
1198701199090
2
sum
119897=1
120577119897
119886(0) cos 120582
1198871198991015840119910
+(minus1)119898+1198981015840
119870119909119886
2
sum
119897=1
120577119897
119886(0) cos 120582
1198871198991015840119910119889119910
K1-4119904119905
=
1
2
int
119886
0
int
119887
0
119863(119909 119910) 21205821198871198991205821198861198981015840 sin 120582
1198861198981015840119909 cos 120582
1198871198991015840119910
times cos 120582119886119898119909 sin 120582
119887119899119910 minus 2(1 minus 120583)
119887119899
times 1205821198861198981015840 sin 120582
1198861198981015840119909 cos 120582
1198871198991015840119910 cos 120582
119886119898119909
times sin 120582119887119899119910 + (1 minus 120583) 120582
1198861198981205821198871198991015840 cos 120582
1198861198981015840119909
times sin 1205821198871198991015840119910 sin 120582
119886119898119909 cos 120582
119887119899119910 119889119909 119889119910
22 Shock and Vibration
K1-5119904119901
=
1
2
int
119886
0
int
119887
0
119863(119909 119910) minus 21205821198861198981015840 sin 120582
1198861198981015840119909 cos 120582
1198871198991015840119910
times
2
sum
119897=1
1205771198971015840
119887(119910) cos 120582
119886119898119909
+ 2 (1 minus 120583) 1205821198861198981015840 sin 120582
1198861198981015840119909 cos 120582
1198871198991015840119910
times
2
sum
119897=1
1205771198971015840
119887(119910) cos 120582
119886119898119909
+ (1 minus 120583) 1205821198861198981205821198871198991015840 cos 120582
1198861198981015840119909 sin 120582
1198871198991015840119910
times
2
sum
119897=1
120577119897
119887(119910) sin 120582
119886119898119909119889119909119889119910
K1-6119904119902
=
1
2
int
119886
0
int
119887
0
119863(119909 119910)1205821198871198991205821198861198981015840 sin 120582
1198861198981015840119909 cos 120582
1198861198991015840119910
times
2
sum
119897=1
120577119897
119886(119909) sin 120582
1198871198991015840119910
minus 2 (1 minus 120583) 1205821198871198991205821198861198981015840 sin 120582
1198861198981015840119909 cos 120582
1198861198991015840119910
times
2
sum
119897=1
120577119897
119886(119909) sin 120582
1198871198991015840119910
+ 1205821198871198991015840 cos 120582
1198861198981015840119909 sin 120582
1198871198991015840119910
times
2
sum
119897=1
1205771198971015840
119886(119909) cos 120582
119887119899119910119889119909119889119910
K1-7119904119905
= minus
1
2
int
119886
0
int
119887
0
120581119866ℎ (119909 119910) 120582119886119898
cos 1205821198861198981015840119909 cos 120582
1198871198991015840119910
times sin 120582119886119898119909 cos 120582
119887119899119910119889119909119889119910
K1-8119904119901
= minus
1
2
int
119886
0
int
119887
0
120581119866ℎ (119909 119910) 1205821198861198981015840 cos 120582
1198861198981015840119909 cos 120582
1198871198991015840119910
times
2
sum
119897=1
120577119897
119887(119910) sin 120582
1198861198981015840119909 119889119909 119889119910
K1-9119904119902
=
1
2
int
119886
0
int
119887
0
120581119866ℎ (119909 119910) cos 1205821198861198981015840119909 cos 120582
1198871198991015840119910
times
2
sum
119897=1
1205771198971015840
119886(119909) cos 120582
119887119899119910119889119909119889119910
M1-1119904119905
=
1
24
120588int
119886
0
int
119887
0
ℎ3
(119909 119910) cos 120582119886119898119909 cos 120582
119887119899119910
times cos 1205821198861198981015840119909 cos 120582
1198871198991015840119910119889119909119889119910
M1-2119904119901
=
1
24
120588int
119886
0
int
119887
0
ℎ3
(119909 119910)
2
sum
119897=1
120577119897
119887(119910) cos 120582
119886119898119909
times cos 1205821198861198981015840119909 cos 120582
1198871198991015840119910119889119909119889119910
M1-3119904119902
=
1
24
120588int
119886
0
int
119887
0
ℎ3
(119909 119910)
2
sum
119897=1
120577119897
119886(119909) cos 120582
119887119899119910
times cos 1205821198861198981015840119909 cos 120582
1198871198991015840119910119889119909119889119910
M1-4119904119905 = M
1-5119904119901 = M1-6119904119902
= M1-7119904119905 = M
1-8119904119901 = M1-9119904119902 = 0
(A1)
B Useful Integrals for the Moderately ThickRectangular Plates with Variable Thicknessand Arbitrary Boundary Conditions
Consider
int
119871
0
cos 120582119898119909 cos 120582
119899119909 119889119909 =
0 119898 = 119899
119871
2
119898 = 119899
119871 119898 = 119899 = 0
int
119871
0
sin 120582119898119909 cos 120582
119899119909119889119909
=
minus
(minus1 + (minus1)119898+119899
) 119871119898
(1198982minus 1198992) 120587
119898 = 119899
0 119898 = 119899
int
119871
0
sin 120582119898119909 sin 120582
119899119909 119889119909 =
0 119898 = 119899
119871
2
119898 = 119899
0 119898 = 119899 = 0
int
119871
0
cos 120582119898119909120577119897
119886(119909) 119889119909
=
41198712
31205872 119897 = 1 119898 = 0
minus
41198712
(41198982
minus 3)
(9 minus 401198982+ 16119898
4) 1205872
119897 = 1 119898 = 0
minus
41198712
31205872 119897 = 2 119898 = 0
41198712
(41198982
minus 3) (minus1)119898
(9 minus 401198982+ 16119898
4) 1205872
119897 = 2 119898 = 0
Shock and Vibration 23
int
119871
0
sin 120582119898119909120577119897
119886(119909) 119889119909
=
0 119897 = 1 119898 = 0
161198712
119898(minus1)119898
(9 minus 401198982+ 16119898
4) 1205872
119897 = 1 119898 = 0
0 119897 = 2 119898 = 0
161198981198712
(9 minus 401198982+ 16119898
4) 1205872
119897 = 2 119898 = 0
int
119871
0
cos 1205821198981199091205771198971015840
119886(119909) 119889119909
=
0 119897 = 1 119898 = 0
161198711198982
(minus1)119898
(9 minus 401198982+ 16119898
4) 120587
119897 = 1 119898 = 0
0 119897 = 2 119898 = 0
161198711198982
(9 minus 401198982+ 16119898
4) 120587
119897 = 2 119898 = 0
int
119871
0
sin 1205821198981199091205771198971015840
119886(119909) 119889119909
=
0 119897 = 1 119898 = 0
4119871119898 (minus3 + 41198982
)
(9 minus 401198982+ 16119898
4) 120587
119897 = 1 119898 = 0
0 119897 = 2 119898 = 0
119871119898(
1
120587 minus 41198982120587
+
3
9120587 minus 41198982120587
) (minus1)119898
119897 = 2 119898 = 0
int
119871
0
1205771198971
119886(119909) 1205771198972
119886(119909) 119889119909 =
1198713
41205872 1198971= 1 1198972= 1
minus
21198713
31205873 1198971= 1 1198972= 2
minus
21198713
31205873 1198971= 2 1198972= 1
1198713
41205872 1198971= 2 1198972= 2
int
119871
0
1205771198971
119886(119909) 1205771198971015840
2
119886(119909) 119889119909 =
0 1198971= 1 1198972= 1
minus
1198712
8120587
1198971= 1 1198972= 2
1198712
8120587
1198971= 2 1198972= 1
0 1198971= 2 1198972= 2
int
119871
0
1205771198971015840
1
119886(119909) 1205771198971015840
2
119886(119909) 119889119909 =
5119871
16
1198971= 1 1198972= 1
minus
119871
2120587
1198971= 1 1198972= 2
minus
119871
2120587
1198971= 2 1198972= 1
5119871
16
1198971= 2 1198972= 2
int
119871
0
119909119899 cos 120582
119898119909 119889119909
=
1198711+119899
1 + 119899
119898 = 0
(1198861+119899HypergeometricPFQtimes [ 12 + 1198992 12 32 + 1198992
minus (14)1198982
1205872
]) times (1 + 119899)minus1
119898 gt 0
(B1)
Conflict of Interests
The authors declare that there is no conflict of interestsregarding the publication of this paper
Acknowledgments
The authors would like to thank the anonymous reviewersfor their very valuable comments The authors also gratefullyacknowledge the financial support from the National NaturalScience Foundation of China (no 51209052) HeilongjiangProvince Youth Science Fund Project (no QC2011C013) andHarbin Science and Technology Development InnovationFoundation of Youth (no 2011RFQXG021)
References
[1] K Liew Y Xiang and S Kitipornchai ldquoResearch on thick platevibration a literature surveyrdquo Journal of Sound and Vibrationvol 180 no 1 pp 163ndash176 1995
[2] J H Chung T Y Chung and K C Kim ldquoVibration analysisof orthotropic mindlin plates with edges elastically restrainedagainst rotationrdquo Journal of Sound and Vibration vol 163 no 1pp 151ndash163 1993
[3] Y K Cheung and D Zhou ldquoVibrations of moderately thickrectangular plates in terms of a set of static Timoshenko beamfunctionsrdquoComputers and Structures vol 78 no 6 pp 757ndash7682000
[4] CMWang ldquoNatural frequencies formula for simply supportedMindlin platesrdquo Journal of Vibration and Acoustics vol 116 no4 pp 536ndash540 1994
[5] K N Saha R C Kar and P K Datta ldquoFree vibration analysisof rectangular Mindlin plates with elastic restraints uniformlydistributed along the edgesrdquo Journal of Sound andVibration vol192 no 4 pp 885ndash902 1996
24 Shock and Vibration
[6] D J Gorman ldquoAccurate free vibration analysis of shear-deformable plates with torsional elastic edge supportrdquo Journalof Sound and Vibration vol 203 no 2 pp 209ndash218 1997
[7] D J Gorman ldquoFree vibration analysis of Mindlin plates withuniform elastic edge support by the superposition methodrdquoJournal of Sound andVibration vol 207 no 3 pp 335ndash350 1997
[8] D J Gorman ldquoAccurate free vibration analysis of point sup-ported mindlin plates by the superposition methodrdquo Journal ofSound and Vibration vol 219 no 2 pp 265ndash277 1999
[9] Y Xiang ldquoVibration of rectangular Mindlin plates resting onnon-homogenous elastic foundationsrdquo International Journal ofMechanical Sciences vol 45 no 6-7 pp 1229ndash1244 2003
[10] Y Xiang and G W Wei ldquoExact solutions for buckling andvibration of stepped rectangular Mindlin platesrdquo InternationalJournal of Solids and Structures vol 41 no 1 pp 279ndash294 2004
[11] Y L Yeh M J Jang and C C Wang ldquoAnalyzing the freevibrations of a plate using finite difference and differential trans-formationmethodrdquoAppliedMathematics andComputation vol178 no 2 pp 493ndash501 2006
[12] Y Xiang S K Lai L Zhou and C W Lim ldquoDSC-Ritz elementmethod for vibration analysis of rectangular Mindlin plateswith mixed edge supportsrdquo European Journal of MechanicsmdashASolids vol 29 no 4 pp 619ndash628 2010
[13] H Nguyen-Xuan C H Thai and T Nguyen-Thoi ldquoIsogeo-metric finite element analysis of composite sandwich platesusing a higher order shear deformation theoryrdquo Composites BEngineering vol 55 pp 558ndash574 2013
[14] C H Thai A J M Ferreira E Carrera and H Nguyen-XuanldquoIsogeometric analysis of laminated composite and sandwichplates using a layerwise deformation theoryrdquo Composite Struc-tures vol 104 pp 196ndash214 2013
[15] C H Thai A Ferreira S Bordas T Rabczuk and HNguyen-Xuan ldquoIsogeometric analysis of laminated compositeand sandwich plates using a new inverse trigonometric sheardeformation theoryrdquo European Journal of Mechanics A Solidsvol 43 pp 89ndash108 2014
[16] H Luong-Van T Nguyen-Thoi G R Liu and P Phung-VanldquoA cell-based smoothed finite elementmethod using three-nodeshear-locking free Mindlin plate element (CS-FEM-MIN3) fordynamic response of laminated composite plates on viscoelasticfoundationrdquo Engineering Analysis with Boundary Elements vol42 pp 8ndash19 2014
[17] T Nguyen-Thoi T Rabczuk T Lam-Phat V Ho-Huu andP Phung-Van ldquoFree vibration analysis of cracked Mindlinplate using an extended cell-based smoothed discrete sheargap method (XCS-DSG3)rdquo Theoretical and Applied FractureMechanics 2014
[18] H Nguyen-Xuan G R Liu and C a Thai-Hoang ldquoAnedge-based smoothed finite element method (ES-FEM) withstabilized discrete shear gap technique for analysis of Reissner-Mindlin platesrdquo Computer Methods in Applied Mechanics andEngineering vol 199 no 9ndash12 pp 471ndash489 2010
[19] P Phung-Van T Nguyen-Thoi T Bui-Xuan and Q Lieu-XuanldquoA cell-based smoothed three-node Mindlin plate element(CS-FEM-MIN3) based on the 119862
0 -type higher-order sheardeformation for geometrically nonlinear analysis of laminatedcomposite platesrdquo Computational Materials Science 2014
[20] NNguyen-Thanh T RabczukHNguyen-Xuan and S BordasldquoAn alternative alpha finite element method with discrete sheargap technique for analysis of isotropicMindlin-Reissner platesrdquoFinite Elements inAnalysis andDesign vol 47 no 5 pp 519ndash5352011
[21] KM Liew Y Xiang and S Kitipornchai ldquoTransverse vibrationof thick rectangular plates-I Comprehensive sets of boundaryconditionsrdquo Computers and Structures vol 49 no 1 pp 1ndash291993
[22] D Zhou ldquoVibrations of Mindlin rectangular plates with elasti-cally restrained edges using static Timoshenko beam functionswith the Rayleigh-Ritz methodrdquo International Journal of Solidsand Structures vol 38 no 32-33 pp 5565ndash5580 2001
[23] D Zhou SH Lo F T KAu andYK Cheung ldquoVibration anal-ysis of rectangular Mindlin plates with internal line supportsusing static Timoshenko beam functionsrdquo International Journalof Mechanical Sciences vol 44 no 12 pp 2503ndash2522 2002
[24] H S Shen J Yang and L Zhang ldquoFree and forced vibrationof Reissner-Mindlin plates with free edges resting on elasticfoundationsrdquo Journal of Sound and Vibration vol 244 no 2 pp299ndash320 2001
[25] Y Xing and B Liu ldquoClosed form solutions for free vibrations ofrectangular Mindlin platesrdquo Acta Mechanica Sinica vol 25 no5 pp 689ndash698 2009
[26] P Gagnon C Gosselin and L Cloutier ldquoA finite strip elementfor the analysis of variable thickness rectangular thick platesrdquoComputers amp Structures vol 63 no 2 pp 349ndash362 1997
[27] T Mlzusawa ldquoVibration of rectangular mindlin plates withtapered thickness by the spline strip methodrdquo Computers andStructures vol 46 no 3 pp 451ndash463 1993
[28] H Nguyen-Xuan and T Nguyen-Thoi ldquoA stabilized smoothedfinite element method for free vibration analysis of Mindlin-Reissner platesrdquo Communications in Numerical Methods inEngineering with Biomedical Applications vol 25 no 8 pp 882ndash906 2009
[29] K M Liew L X Peng and S Kitipornchai ldquoVibration anal-ysis of corrugated Reissner-Mindlin plates using a mesh-freeGalerkin methodrdquo International Journal of Mechanical Sciencesvol 51 no 9-10 pp 642ndash652 2009
[30] Y Hou GWWei and Y Xiang ldquoDSC-Ritz method for the freevibration analysis of Mindlin platesrdquo International Journal forNumerical Methods in Engineering vol 62 no 2 pp 262ndash2882005
[31] F- Liu and K M Liew ldquoVibration analysis of discontinuousMindlin plates by differential quadrature element methodrdquoJournal of Vibration and Acoustics Transactions of the ASMEvol 121 no 2 pp 204ndash208 1999
[32] P Malekzadeh G Karami and M Farid ldquoA semi-analyticalDQEM for free vibration analysis of thick plates with twoopposite edges simply supportedrdquoComputerMethods inAppliedMechanics and Engineering vol 193 no 45ndash47 pp 4781ndash47962004
[33] P Malekzadeh and S A Shahpari ldquoFree vibration analysisof variable thickness thin and moderately thick plates withelastically restrained edges by DQMrdquo Thin-Walled Structuresvol 43 no 7 pp 1037ndash1050 2005
[34] R E Diaz-Contreras and S Nomura ldquoGreenrsquos function appliedto solution ofMindlin platesrdquoComputers and Structures vol 60no 1 pp 41ndash48 1996
[35] T Sakiyama and M Huang ldquoFree vibration analysis of rect-angular plates with variable thicknessrdquo Journal of Sound andVibration vol 216 no 3 pp 379ndash397 1998
[36] M Huang X Q Ma T Sakiyama H Matuda and C MoritaldquoFree vibration analysis of orthotropic rectangular plates withvariable thickness and general boundary conditionsrdquo Journal ofSound and Vibration vol 288 no 4-5 pp 931ndash955 2005
Shock and Vibration 25
[37] I Shufrin and M Eisenberger ldquoVibration of shear deformableplates with variable thicknessmdashfirst-order and higher-orderanalysesrdquo Journal of Sound and Vibration vol 290 no 1-2 pp465ndash489 2006
[38] S A Eftekhari and A A Jafari ldquoAccurate variational approachfor free vibration of variable thickness thin and thick plates withedges elastically restrained against translation and rotationrdquoInternational Journal of Mechanical Sciences vol 68 pp 35ndash462013
[39] W L Li X Zhang J Du and Z Liu ldquoAn exact series solutionfor the transverse vibration of rectangular plates with generalelastic boundary supportsrdquo Journal of Sound amp Vibration vol321 no 1-2 pp 254ndash269 2009
[40] X Shi D Shi W L Li and Q Wang ldquoA unified method forfree vibration analysis of circular annular and sector plateswith arbitrary boundary conditionsrdquo Journal of Vibration andControl 2014
[41] D Y Shi X J Shi W L Li and Q S Wang ldquoFree transversevibrations of orthotropic thin rectangular plates with arbitraryelastic edge supportsrdquo Journal of Vibroengineering vol 16 no 1pp 389ndash398 2014
[42] Y Chen G Jin and Z Liu ldquoFree vibration analysis of circu-lar cylindrical shell with non-uniform elastic boundary con-straintsrdquo International Journal of Mechanical Sciences vol 74pp 120ndash132 2013
[43] L Dai T Yang J Du W L Li and M J Brennan ldquoAn exactseries solution for the vibration analysis of cylindrical shellswith arbitrary boundary conditionsrdquo Applied Acoustics vol 74no 3 pp 440ndash449 2013
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Shock and Vibration
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
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DistributedSensor Networks
International Journal of
6 Shock and Vibration
Table 1 Frequency parameters Ω = 1205961198872
(120588ℎ119863)12
1205872 for square moderately thick plates with uniform thickness and different boundary
conditions
Boundary conditions M N Model sequence1 2 3 4 5 6 7 8
C-C-C-C
119872 = 119873 = 2 3375 7191 7191 9856 1605 1628 1747 1747119872 = 119873 = 4 3299 6308 6308 8853 1046 1056 1263 1263119872 = 119873 = 6 3294 6282 6282 8801 1037 1047 1254 1254119872 = 119873 = 8 3292 6278 6278 8797 1036 1046 1253 1253119872 = 119873 = 10 3292 6276 6276 8794 1036 1046 1253 1253119872 = 119873 = 12 3292 6276 6276 8793 1036 1046 1252 1252119872 = 119873 = 14 3292 6276 6276 8793 1036 1046 1252 1252119872 = 119873 = 16 3292 6275 6275 8792 1036 1045 1252 1252Reference [4] 3292 6276 6276 8792 1036 1046 1252 1252
E-E-E-E
119872 = 119873 = 2 06153 10861 10861 17898 27709 28417 35010 35010119872 = 119873 = 4 06150 10799 10799 17785 27695 28405 34914 34914119872 = 119873 = 6 06149 10796 10796 17773 27694 28403 34898 34898119872 = 119873 = 8 06149 10796 10796 17771 27693 28403 34893 34893119872 = 119873 = 10 06149 10796 10796 17770 27693 28403 34892 34892119872 = 119873 = 12 06149 10796 10796 17770 27693 28403 34892 34892119872 = 119873 = 14 06149 10796 10796 17770 27693 28403 34892 34892119872 = 119873 = 16 06149 10796 10796 17770 27693 28403 34891 34891Reference [33] 06153 10861 10861 17898 27709 28417 35010 35010
FEM 06150 17097 17097 17774 27694 28406 34898 34898
different structure parameters are examined Then numer-ical solutions for moderately thick rectangular plates withelastic boundary conditions are presented The effects of theelastic restraint parameters are also investigated Finally thenonuniform thickness plate with nonlinear variation in bothdirections and arbitrary boundary conditions is also studied
31 Convergence and Stiffness Value Study Since the Fourierseries is numerically truncated and only finite terms arecounted in actual calculations the proposed solution shouldbe understood as a solution with arbitrary precision Inthis subsection a uniform thickness square moderately thickrectangular plate with completely clamped boundary condi-tion (C-C-C-C) and four edges equally elastically restrainedagainst linear spring constants and rotation spring constantssupports (E-E-E-E 119870
119879= 119896119894(1198873
119863) 119870119877
= 119870119894(119887119863)
119870119879
= 10 and 119870119877
= 5) has been selected to demon-strate the convergence and accuracy of the modified Fouriermethod In Table 1 the first eight frequency parametersΩ = 120596119887
2
(120588ℎ119863)12
1205872 for the considered uniform thickness
square moderately thick rectangular plate with C-C-C-C andE-E-E-E boundary conditions are examinedThe table showsthat the proposedmethod has fast convergence behaviorThemaximum discrepancy in the worst case between the 6 times 6
truncated configuration and the 8times8 one is less than 0064In order to fully illustrate the convergence of the presentmethod Figures 3 4 and 5 present the 1st and 8th frequencyparameters Ω = 120596119887
2
(120588ℎ119863)12
1205872 with various truncated
numbers 119872 = 119873 subjected to different boundary conditionand aspect ratios A highly desired convergence characteristicis observed such that (a) sufficiently accurate results can beobtained with only a small number of terms in the seriesexpansions and (b) the solution is consistently refined asmore
terms are included in the expansions However this shouldnot constitute a problem in practice because one can alwaysverify the accuracy of the solution by increasing the trunca-tion number until a desired numerical precision is achievedAs a matter of fact this ldquoquality controlrdquo scheme can be easilyimplemented automatically In modal analysis the naturalfrequencies for higher-order modes tend to converge slower(see Table 1) Thus an adequate truncation number shouldbe dictated by the desired accuracy of the largest naturalfrequencies of interest In view of the excellent numericalbehavior of the current solution the truncation numbers willbe simply set as119872 = 119873 = 12 in the following calculations
As far as the accuracy of the present method is con-cerned the converged solutions of the present method arein excellent agreement with both the results reported byreference data and the finite element results For C-C-C-C boundary conditions the max discrepancy between thepresent results and the reference data does not exceed 0011for the worst case and in most cases is 0 Comparing theresults with exact solutions [4] it is observed that eight termsare sufficient to obtain accurate resultsMoreover with regardto the E-E-E-E boundary condition the max discrepancybetween the present results and the reference data does notexceed 031 for the worst case and in most cases is 01Regarding the results with DQM solutions [33] it can be seenthat the six terms are sufficient to obtain enough accurateresults In addition it is clear that the results of the presentapproach with just 663 DOFs (119872 times 119873 = 12 times 12) canpredict the vibration characteristics accurately Most of themare identical to those obtained from finite element method(FEM) with 10201 DOFs (S4R 001m times 001m) That isto say it needs only 662 DOFs compared with FEM toobtain the same precision solutions for the considered case
Shock and Vibration 7
1
2
3
4
5
6
The 1st orderThe 3rd order
101 103 105 107 109 1011 1013
Ω
Ki (Nm)
(a)
2
3
4
5
6
101 103 105 107 109 1011 1013
The 1st orderThe 3rd order
Ω
Ki (Nmrad)
(b)
3
4
5
6
101 103 105 107 109 1011 1013
Kij (Nmrad)
The 1st orderThe 3rd order
Ω
(c)
Figure 2 The effect of boundary spring stiffness on the natural frequencies Ω (a) translation spring (b) rotation spring and (c) torsionalspring
On the same hardware (Intel i7-39GHz) the computing timeof the present formulation for the solution (119872times119873 = 12times12)implemented in optimized MATLAB scripts is about 2125 swhereas the finite element solution consumes 34578 s that isat least 16 times more CPU time than the present method forthe same problem
As mentioned earlier in the current modeling frame-work all the classical boundary conditions and their com-binations can be conveniently viewed as special cases whenthe stiffness for the normal and tangential boundary springsbecomes zero andor infinitely large Thus the effects of thestiffness of the translation (119896
119894) rotation (119870
119894) and torsional
springs (119870119894119895) on the modal characteristics should be inves-
tigated As shown in the Figures 2(a)ndash2(c) the first and thethird frequency parameters are separately obtained by vary-ing the stiffness of one group of the boundary springs from
extremely large (1014) to extremely small (100) while assigningthe other group of the springs infinite stiffness (1014) It canbe found in Figure 2(a) that the frequency parameter almostkeeps at a level when the stiffness of the translation springs islarger than 1012 or smaller than 107 In Figure 2(b) the influ-ences of the rotation springs on frequency parameters aregiven It is shown that the frequency curves change greatlywithin the stiffness range (106 to 1010) while out of this rangethe frequency curves separately keep level In Figure 2(c) theinfluences of the torsional springs on frequency parametersare given It is shown that the frequency curves almost changewhen the stiffness changes in the whole range
Based on the analysis it can be found that the torsionalsprings almost have no effect on the structure Also the rela-tionship between the rotation springs and twisting momentcan be seen from the boundary condition expression Then
8 Shock and Vibration
32
36
40
44
48
52
2
3
4
5
6
7
8
Ω Ω
0 2 4 6 8 10 12 14 16 18M = N
ab = 1
ab = 32
(a) The 1st order
0 2 4 6 8 10 12 14 16 180
20
40
60
80
0
50
100
150
200
250
300
M = N
Ω Ω
ab = 1
ab = 32
(b) The 8th order
Figure 3 The effect of numerically truncated numbers119872 = 119873 on the natural frequenciesΩ for C-C-C-C boundary condition
0 2 4 6 8 10 12 14 16 18125
130
135
140
145
150
088
090
092
094
096
098
Ω Ω
M = N
ab = 1
ab = 32
(a) The 1st order
0 2 4 6 8 10 12 14 16 18
55
60
65
70
75
80
40
45
50
55
60
65
70
75
80Ω Ω
M = N
ab = 1
ab = 32
(b) The 8th order
Figure 4 The effect of numerically truncated numbers119872 = 119873 on the natural frequenciesΩ for F-F-F-F boundary condition
the twisting moments have small effect on the vibration char-acteristics of the structure However in this paper in order toget a more accurate prediction of the vibration characteristicsof the structure the twisting moment on boundary edges istaken into account In the latter study in addition to the factthat free boundary is not unexpected considering the torsionspring the other boundary conditions are introduced into atorsion spring and the spring stiffness takes infinity FormFigure 2 analysis it also concluded that the translation springhas a wider influence range than the rotation spring on thefrequency parameters that is for the translation springs thestable frequency parameters appearwhen the stiffness ismorethan 1012 or less 107 while for the rotation springs when thestiffness value is assigned more than 1010 the frequency para-meters become smoothThus it is suitable to use 1014 to sim-ulate the infinite stiffness value in the model validation partsand in the following examples
32 Uniform Thickness Moderately Thick Plates with Classicaland Elastic Boundary Conditions In this subsection themodified Fourier solution is applied to deal with vibrationproblems of uniform thickness moderately thick rectangu-lar plates subject to the classical boundary conditions andarbitrary elastic boundary conditions In present work threegroups of continuously distributed boundary springs areintroduced to simulate the given or typical boundary condi-tions As mentioned earlier the stiffness of these boundarysprings can take any value from zero to infinity to bettermodel many real-world restraint conditions Taking edge 119909 =0 for example the corresponding spring stiffness for the threetypes of classical boundaries and elastic boundaries is
completely free 119876119909= 0119872
119909119909= 0 and119872
119909119910= 0
1198961199090= 0 119870
1199090= 0 119870
1199101199090= 0 (17a)
Shock and Vibration 9
0 2 4 6 8 10 12 14 16 18
166
168
170
172
152
156
160
164
168
Ω Ω
ab = 1
ab = 32
M = N
(a) The 1st order
0 2 4 6 8 10 12 14 16 1850
55
60
65
70
4
5
6
7
8
9
10
Ω Ω
M = N
ab = 1
ab = 32
(b) The 8th order
Figure 5 The effect of numerically truncated numbers 119872 = 119873 on the natural frequencies Ω with four edges elastically restrained againsttranslation and rotation (119870
119879= 119896119894(1198873
119863) 119870119877= 119870119894(119887119863) 119870
119879= 100 and 119870
119877= 100)
simply supported 119908 = 0119872119909119909= 0 and 120595
119910= 0
1198961199090= 1014
1198701199090= 0 119870
1199101199090= 1014
(17b)
completely clamped 119908 = 0 120595119909= 0 and 120595
119910= 0
1198961199090= 1014
1198701199090= 1014
1198701199101199090
= 1014
(17c)
arbitrarily elastic boundary 119908 = 0 120595119909
= 0 120595119910
= 0119876119909
= 0119872119909119909
= 0 and119872119909119910
= 0
1198961199090= Γ1199081199090
1198701199090= Γ1199091199090
1198701199101199090
= 1014
(17d)
The appropriateness of the three classical boundariesdefined in (17a)ndash(17c) will be proved by several examplesgiven in following the arbitrary elastic boundaries are alsodefined in (17d) and the Γ
119908119894(Γ119908119894= 1198961198941198960 1198960= 1 times 10
9Nm2119894 = 1199090 1199091198861199100119910119887) and Γ
119909119894(Γ119909119894= 11987011989411987001198700= 1times10
9Nmrad119894 = 1199090 119909119886 1199100 119910119887) elastic restraint parameters representcorresponding spring stiffness For the sake of simplicitya four-letter string is employed to represent the restraintcondition of a plate such that F-C-S-E identifies the platewithedges 119909 = 0 119910 = 0 119909 = 119886 and 119910 = 119887 having free clampedshear-supported restrained and elastic boundary conditionsrespectively
As for the first case a uniform thickness moderately thickplate with different classical boundaries and structure param-eters is investigated here In Table 2 the comparison of thefirst eight frequency parametersΩ = 120596119887
2
(120588ℎ119863)12
1205872 of the
considered plate is presented The S-S-S-S C-F-F-F S-S-F-FF-F-F-F and S-C-S-C boundary conditions are performed inthe comparison Excellent agreements are observed betweenthe solutions obtained by the modified Fourier method thereferential data and finite element method (FEM) results forthe uniform thickness moderately thick rectangular platesIt is also verified that the definition of the three types of
classical boundaries in (17a)ndash(17c) is appropriate In additionthe elastic boundary conditions (17d) are also verified Inthe next two examples we will account for the vibrationof moderately thick plate with elastic edge supports Thefirst model considered is an S-S-S-S square moderately thickplate with all edges elastically rotationally restrained Thatis 1198701199090
= 119870119909119886
= 1198701199100
= 119870119910119887
= Γ119909 and all the
other restraining springs are set to have an infinite stiffness(namely represented by 1014 in numerical calculation) Thesix frequency parameters Ω = 120596119887
2
(120588ℎ119863)12
1205872 are given
in Table 3 for several different restraining coefficient valuesthe finite element method (FEM) results calculated usingABAQUSare also listed in this table as a referenceThe secondmodel concerns a complete square moderately thick platewith all edges elastically restrained That is 119896
1199090= 119896119909119886
=
1198961199100
= 119896119910119887
= Γ119908and 119870
1199090= 119870119909119886
= 1198701199100
= 119870119910119887
= Γ119909 The
six frequency parameters Ω = 1205961198872
(120588ℎ119863)12
1205872 are given in
Table 4 for several different restraining coefficient values thefinite element method (FEM) results are also listed in Table 4as a reference It can be clearly seen that the comparison isextremely good which implies that the current method isable tomake correct predictions for themodal characteristicsof moderately thick plate with not only classical boundaryconditions but also elastic edge restraints
The excellent agreements between the solution obtainedby the modified Fourier method and the referential datafor the moderately thick plate subjected to the combina-tions of classical boundary conditions and elastic boundaryconditions given in Tables 2ndash4 indicate that the proposedmethod is sufficiently accurate to deal with uniform thicknessmoderately thick plate with arbitrary boundary conditions
33 Linearly VariationThickness Moderately Thick Plates withClassical and Elastic Boundary Conditions In the theoreticalformulations this paper concerns the varying thicknessmoderately thick plates with classical and elastic boundary
10 Shock and Vibration
Table 2 Frequency parameters Ω = 1205961198872
(120588ℎ119863)12
1205872 for uniform thickness moderately thick plates with different classical boundaries and
structure parameters
Boundary condition ab Model sequence1 2 3 4 5 6 7 8
S-S-S-S
11931 4605 4605 7064 8605 8605 10792 10792
(1933a) (4611) (4611) (7067) (8607) (8607) (10799) (10780)(1931b) (4605) (4605) (7064) (8605) (8605) (10792) (10792)
321408 2648 4127 4605 5261 7064 7153 8184(1410a) (2651) (4128) (4608) (5267) (7074) (7155) (8188)(1408b) (2648) (4127) (4605) (5261) (7064) (7151) (8184)
C-F-F-F
10348 0817 2035 2583 2862 4816 5478 5774
(0348a) (0817) (2038) (2586) (2865) (4821) (5504) (5793)(0348b) (0816) (2034) (2582) (2860) (4811) (5477) (5772)
320155 0502 0941 1665 2292 2613 3303 3498
(0154a) (0502) (0941) (1663) (2295) (2618) (3305) (3500)(0154b) (0501) (0940) (1662) (2292) (2612) (3298) (3494)
S-S-F-F
10333 1678 1875 3559 4720 4946 6474 6632
(0335a) (1680) (1878) (3600) (4721) (4949) (6478) (6642)(0333b) (1677) (1874) (3557) (4718) (4945) (6472) (6631)
320222 0938 1631 2363 2573 4082 4482 4753
(0223a) (0940) (1632) (2364) (2575) (4092) (4483) (4759)(0222b) (0938) (1631) (2361) (2571) (4077) (4479) (4752)
F-F-F-F1 1290 1919 2363 3235 3235 5605 5605 5646
(1291a) (1920) (2365) (3239) (3239) (5607) (5607) (6648)
32 0865 0948 1961 2166 2461 2843 3551 4026(0865a) (0949) (1963) (2168) (2465) (2844) (3556) (4034)
S-C-S-C1 2700 4971 5990 7973 8787 10250 11334 12024
(2701a) (4971) (5993) (7975) (8788) (10253) (11335) (12028)
32 2348 3253 4972 5646 6489 7381 7974 9930(2348a) (3255) (4974) (5647) (6590) (7384) (7801) (9932)
aResults in parentheses are taken from FEMbResults in parentheses are taken from [11]
conditions The varying thickness function ℎ(119909 119910) can beexpressed as ℎ
0(1 + 120572119909
119904
)(1 + 120573119910119905
) in which the ℎ0 120572 and
120573 represent the initial thickness gradient in 119909 direction andgradient in 119910 direction When the indexes 119904 and 119905 take thevalue 119904 = 119905 = 1 the analyticalmodel imitates the linearly vari-ation thickness moderately thick plates structure In order tounify the description and facilitate the analytical calculationsof the involved integrals all the thickness variation functionscan be expanded into either 1D or 2D Fourier cosine seriesresulting in
ℎ (119909 119910) =
infin
sum
119898=0
infin
sum
119899=0
ℎ0119902119898119899
cos 119898120587119909119886
cos119898120587119910
119887
(18)
where119902119898119899
=
4
119886119887
int
119887
0
int
119886
0
(1 + 120572119909119904
) (1 + 120573119910119905
) cos 119898120587119909119886
cos119898120587119910
119887
119889119909 119889119910
(19)
In order to prove the validity of the present methodfor the vibration of linearly variation thickness moderately
thick plates with arbitrary boundary conditions the typicalclassical boundary conditions as the first case will be con-sidered In Table 5 the comparison of the first six frequencyparameters Ω = 120596119886
2
(120588ℎ01198630)12 of the moderately thick
plates with linearly varying thickness is presentedThe S-S-S-S C-F-F-F S-S-F-F C-C-C-C and S-C-S-C boundary condi-tions are performed in the comparison Excellent agreementsare observed between the solutions obtained by the modifiedFourier method and finite element method (FEM) results forthemoderately thick plates with linear variation thickness Toinvestigate the influence of the aspect ratio on the uniformthickness and nonuniform thickness moderately thick platesthe effect on the frequency parameters for plates with S-S-S-Sboundary conditions is presented in Figure 6 The thicknessfunctions are ℎ
0and ℎ0(1+05times119909)(1+05times119910) respectively It
is seen from Figure 6 that the influence of aspect ratios on thefrequency parameters for nonuniform thickness moderatelythick plates is more complicated
In the next two examples we also account for the vibra-tions of moderately thick plate with linear variation thicknessand elastic edge supports The first model considered is an S-S-S-S square moderately thick plate with all edges elastically
Shock and Vibration 11
Table 3 Frequency parameters Ω = 1205961198872
(120588ℎ119863)12
1205872 for S-S-S-S moderately thick plates (119886119887 = 1) with uniform thickness and elastic
rotation support (1198701199090= 119870119909119886= 1198701199100= 119870119910119887= Γ119909)
Γ119909
aℎ0119886
Model sequence1 2 3 4 5 6
0
005 1945 4834 4834 7591 9515 9519(1942b) (4836) (4836) (7584) (9547) (9552)
01 1856 4500 4500 6829 8499 8513(1858b) (4510) (4510) (6843) (8539) (8554)
015 1758 4114 4114 6040 7421 7445(1761b) (4126) (4126) (6060) (7462) (7486)
02 1656 3733 3733 5335 6468 6499(1660b) (3747) (3747) (5357) (6508) (6538)
05
005 3541 7071 7071 10227 12318 12399(3546b) (7091) (7091) (10253) (12386) (12468)
01 3274 6242 6242 8719 10302 10421(3282b) (6267) (6267) (8754) (10370) (10488)
015 2947 5361 5361 7290 8465 8606(2956b) (5387) (5387) (7327) (8524) (8666)
02 2624 4599 4599 6148 7057 7199(2634b) (4624) (4624) (6182) (7107) (7250)
1
005 3541 7072 7072 10228 12320 12401(3547b) (7093) (7093) (10254) (12388) (12470)
01 3278 6248 6248 8727 10311 10429(3286b) (6274) (6274) (8762) (10378) (10497)
015 2957 5374 5374 7305 8478 8620(2966b) (5400) (5400) (7341) (8537) (8680)
02 2641 4617 4617 6166 7070 7215(2651b) (4643) (4643) (6200) (7121) (7268)
15
005 3542 7073 7073 10229 12320 12401(3547b) (7093) (7093) (10254) (12388) (12471)
01 3280 6251 6251 8729 10313 10432(3287b) (6276) (6276) (8765) (10381) (10500)
015 2960 5379 5379 7309 8482 8625(2969b) (5405) (5405) (7346) (8541) (8685)
02 2647 4623 4623 6172 7074 7221(2657b) (4649) (4649) (6206) (7125) (7274)
2
005 3542 7073 7073 10229 12321 12402(3547b) (7093) (7093) (10255) (12389) (12471)
01 3280 6252 6252 8730 10315 10434(3288b) (6277) (6277) (8766) (10382) (10501)
015 2962 5381 5381 7312 8484 8628(2971b) (5407) (5407) (7348) (8543) (8688)
02 2650 4627 4627 6175 7076 7224(2660b) (4652) (4652) (6210) (7128 (7277)
aΓ119909= 1198701198941198700 (1198700= 1 times 10
9 Nmrad 119894 = 1199090 119909119886 1199100 119910119887)bResults in parentheses are taken from FEM
12 Shock and Vibration
Table 4 Frequency parameters Ω = 1205961198872
(120588ℎ119863)12
1205872 for moderately thick plates (119886119887 = 1) with uniform thickness and all edge elastic
restraints (1198701199090= 119870119909119886= 1198701199100= 119870119910119887= Γ119909and 119896
1199090= 119896119909119886= 1198961199100= 119896119910119887= Γ119908)
Γ119909
aΓ119908
aℎ0119886
Model sequence1 2 3 4 5 6
05 0
005 0986 1967 3893 3893 4850 4850(0988b) (1971) (3897) (3897) (4852) (4852)
01 0916 1835 3515 3522 4375 4375(0917b) (1837) (3519) (3528) (4341) (4341)
015 0789 1635 2983 3027 3771 3771(0792b) (1638) (2988) (3029) (3775) (3775)
02 0646 1444 2497 2601 3249 3249(0652b) (1448) (2508) (2621) (3254) (3254)
1 05
005 3417 3417 4431 5294 5461 6448(3419b) (3419) (4435) (5303) (5478) (6456)
01 1593 1593 2401 3849 3861 4729(1611b) (1611) (2408) (3852) (3866) (4767)
015 1122 1122 1913 3272 3290 4038(1127b) (1227) (1924) (3284) (3314) (4069)
02 0890 0890 1640 2781 2829 3476(0899b) (0899) (1654) (2787) (2834) (3497)
15 1
005 4325 4325 5619 6273 6571 7619(4334b) (4334) (5681) (6299) (6642) (7642)
01 2013 2013 2812 4088 4116 4993(2024b) (2024) (2827) (4122) (4187) (4994)
015 1338 1338 2108 3430 3443 4196(1342b) (1342) (2112) (3467) (3482) (4200)
02 1038 1038 1772 2936 2965 3613(1042b) (1042) (1786) (2937) (2977) (3624)
2 2
005 5231 5231 6919 7630 8002 9220(5238b) (5238) (6931) (7637) (8011) (9241)
01 2594 2594 3439 4470 4544 5428(2597b) (2597) (3442) (4495) (4557) (5438)
015 1653 1653 2403 3613 3632 4392(1662b) (1662) (2414) (3627) (3646) (4404)
02 1238 1238 1949 3077 3099 3751(1244b) (1244) (1964) (3094) (3116) (3788)
aΓ119909= 1198701198941198700 (1198700= 1 times 10
9 Nmrad 119894 = 1199090 119909119886 1199100 119910119887) Γ119908= 1198961198941198960 (1198960= 1 times 10
9 Nm 119894 = 1199090 119909119886 1199100 119910119887)bResults in parentheses are taken from FEM
05 10 15 20 25 300
2
4
6
8
10
12
14
16
ab
The 1st orderThe 2nd orderThe 3rd order
The 4th orderThe 5th orderThe 6th order
Ω
(a)
05 10 15 20 25 30
30
60
90
120
150
180
ab
The 1st orderThe 2nd orderThe 3rd order
The 4th orderThe 5th orderThe 6th order
Ω
(b)
Figure 6 The effect of aspect ratio 119886119887 on the natural frequenciesΩ for S-S-S-S boundary condition (a) uniform thickness and (b) nonuni-form thickness
Shock and Vibration 13
Table 5 Frequency parameter Ω = 1205961198862
(120588ℎ01198630)12 for moderately thick plates with linearly varying thickness in different boundary
conditions (ℎ(119909 119910) = ℎ0(1 + 05119909)(1 + 05119910))
Boundary condition 119886119887 ℎ0119886
Model sequence1 2 3 4 5 6
S-S-S-S
101 27261 62397 62926 92423 111537 111732
(27263a) (62410) (63842) (92445) (111867) (111921)
02 23103 47010 47191 65018 75831 75865(23489a) (47209) (47349) (65057) (76605) (76473)
3201 59338 99101 138798 151102 166710 204855
(59874a) (99249) (138177) (151362) (166644) (205381)
02 41175 63765 63765 73598 83956 89084(41586a) (63313) (63669) (73661) (83918) (89547)
C-F-F-F
101 4173 10331 27407 37826 38809 64095
(4856a) (10680) (27513) (37203) (39135) (64858)
02 3983 8702 21403 29443 30286 46705(4118a) (9306) (21990) (29670) (30602) (47156)
3201 4868 15488 35620 63523 89969 100089
(4887a) (15320) (35175) (63353) (89159) (100783)
02 4469 9763 22338 42188) 50104 56331(4944a) (9636) (22087) (42177) (50580) (56559)
S-S-F-F
101 4117 24795 26216 48567) 68195 69667
(4789a) (24149) (26751) (49324) (68842) (70575)
02 3820 20516 21924 37523 48980 50327(4103a) (20041) (22125) (37221) (48894) (50622)
3201 7935 39454 65063 92877 92877 136176
(7856a) (40145) (65162) (92462) (93022) (136336)
02 6389 26149 35627 49539 56844 65170(6406a) (26536) (35626) (49831) (57144) (65981)
C-C-C-C
101 44471 80299 80681 109422 126871 128207
(45102a) (80271) (80656) (110015) (126328) (128888)
02 32162 52781 52933 69410 78474 79482(32240a) (52629) (53166) (70063) (79059) (80266)
3201 82975 115587 151992 161579 176601 212330
(83139a) (115838) (152200) (162193) (177058) (212849)
02 48577 66022 83468 90830 97474 98157(49187a) (66259) (84065) (91749) (97827) (98777)
S-C-S-C
101 36847 66664 76965 101045 113594 125551
(37901a) (68568) (78356) (101200) (113978) (125651)
02 27673 48469 51220 67027 76358 78241(27897a) (48685) (51491) (67447) (77530) (79573)
3201 77378 106848 150079 154569 173120 208391
(78003a) (107058) (150991) (155171) (173099) (208402)
02 45895 64732 73393 83282 90325 97089(45803a) (64131) 73656 83801 90518 97015
aResults in parentheses are taken from FEM
restrained That is 1198701199090
= 119870119909119886
= 1198701199100
= 119870119910119887
= Γ119909 The six
frequency parameters Ω = 1205961198862
(120588ℎ01198630)12 are given in
Table 6 for several different restraining coefficient valuesthe finite element method (FEM) results calculated usingABAQUS are also presented as a referenceThe secondmodelis a complete square moderately thick plate with all edgeselastically restrainedThat is 119896
1199090= 119896119909119886= 1198961199100= 119896119910119887= Γ119908and
1198701199090= 119870119909119886
= 1198701199100= 119870119910119887= Γ119909 The six frequency parameters
Ω = 1205961198862
(120588ℎ01198630)12 are given in Table 7 for several different
restraining coefficient values the finite element method(FEM) results are also listed as a reference It can be clearlyseen that the comparison is extremely good which impliesthat the current method is able to make correct predictionsfor the modal characteristics of linearly varying thickness
14 Shock and Vibration
Table 6 Frequency parameter Ω = 1205961198862
(120588ℎ01198630)12 for S-S-S-S moderately thick plates (119886119887 = 1) with linearly varying thickness and elastic
rotation support (ℎ(119909 119910) = ℎ0(1 + 05119909)(1 + 05119910) 119870
1199090= 119870119909119886= 1198701199100= 119870119910119887= Γ119909)
Γ119909
aℎ0119886
Model sequence1 2 3 4 5 6
0
005 28809 70080 70891 108967 134884 135295(28897b) (70334) (71129) (109112) (135254) (135772)
01 27261 62398 62926 92423 111540 111735(26606b) (61784) (62303) (91193) (111323) (111648)
015 25231 54192 54505 77118 91228 91312(25204b) (55297) (55595) (77503) (91871) (91941)
02 23103 47010 47191 65018 75832 75866(23947b) (47071) (47208) (65404) (76531) (76589)
05
005 23103 47010 47191 65018 75832 75866(23210b) (47245) (47358) (65241) (76122) (76345)
01 48377 94650 95597 135484 162635 163501(48524b) (94891) (95992) (135845) (167012) (164024)
015 36239 70939 71298 99997 118584 119009(36647b) (71214) (71687) (100024) (119114) (119457)
02 28749 56937 57037 79214 93009 93162(29142b) (57034) (57544) (79654) (93668) (93754)
1
005 49651 96759 97619 138089 165368 166304(49755b) (97245) (97964) (138544) (165864) (166878)
01 38951 73735 74251 102748 121079 121709(39021b) (74145) (74664) (102874) (121613) (121817)
015 30630 58406 58533 80432 93984 94255(30638b) (58684) (58644) (80874) (94121) (94478)
02 25548 48636 48674 66165 76708 76824(25687b) (48788) (48992) (66544) (76824) (76944)
15
005 50118 97561 98374 139096 166421 167382(50247b) (97877) (98661) (139247) (166875) (167662)
01 40310 75216 75783 104229 122402 123164(40366b) (75542) (75921) (104557) (122874) (123385)
015 31837 59352 59530 81245 94611 94983(31902b) (59451) (59871) (81375) (94784) (95105)
02 26302 49119 49156 66528 76954 77130(26418b) (49354) (49334) (66921) (77010) (77470)
2
005 50360 97983 98769 139631 166979 167954(50472b) (98005) (99114) (140154) (167188) (167986)
01 41130 76146 76724 105165 123229 124082(41421b) (76642) (78114) (105661) (123278) (124974)
015 32687 60025 60245 81832 95052 95508(32778b) (60244) (60549) (82146) (95114) (95842)
02 26893 49492 49539 66815 77138 77370(26922b) (49874) (49924) (66987) (77367) (77588)
aΓ119909= 1198701198941198700 (1198700= 1 times 10
9 Nmrad 119894 = 1199090 119909119886 1199100 119910119887)bResults in parentheses are taken from FEM
Shock and Vibration 15
Table 7 Frequency parameterΩ = 1205961198862
(120588ℎ01198630)12 for moderately thick plates (119886119887 = 1) with linearly varying thickness and elastic rotation
and translation support (ℎ(119909 119910) = ℎ0(1 + 05119909)(1 + 05119910) 119870
1199090= 119870119909119886= 1198701199100= 119870119910119887= Γ119909 and 119896
1199090= 119896119909119886= 1198961199100= 119896119910119887= Γ119908)
Γ119909
aΓ119908
aℎ0119886
Model sequence1 2 3 4 5 6
05 0
005 14751 14917 28232 55251 55397 69512(14780b) (14822) (29252) (55476) (55505) (70904)
01 11811 11910 23857 44462 45037 55453(11818b) (11897) (24711) (44376) (45257) (56511)
015 8487 8514 20011 34783 36335 44970(8500b) (8544) (20738) (34404) (36663) (45625)
02 6073 6073 17431 28679 30805 38010(6096b) (6106) (18039) (28129) (31165) (38311)
1 05
005 21225 32173 32173 44381 64061 65222(20708b) (32097) (32285) (45110) (64182) (65355)
01 8140 16878 16941 28100 48663 49015(7943b) (16838) (16970) (28888) (48700) (49171)
015 4493 11859 11878 22429 38352 39247(4383b) (11867) (11891) (23125) (38177) (39488)
02 2932 8690 8691 18919 31108 32600(3540b) (8791) (8791) (16229) (27520) (27889)
15 1
005 27819 40919 41430 54595 70792 72896(27146b) (41008) (41434) (55256) (70895) (73051)
01 11285 20068 20230 31101 51046 51429(11012b) (20025) (20270) (31849) (51126) (51569)
015 6292 13912 13922 24132 40500 41120(6139b) (13912) (13934) (24808) (40426) (41316)
02 4121 10321 10331 20051 32796 33921(4021b) (10329) (10357) (20635) (32540) (34178)
2 2
005 34615 51972 53088 68255 81344 84510(33806b) (52100) (53157) (68891) (81433) (84761)
01 15416 24659 24829 35615 53879 54494(15045b) (24625) (24888) (36303) (53970) (54626)
015 8740 16420 16481 26353 42417 42946(8529b) (16414) (16499) (26996) (42398) (43114)
02 5761 12127 12127 21434 34286 35192(5622b) (12134) (12148) (21998) (34110) (35415)
aΓ119909= 1198701198941198700 (1198700= 1 times 10
9 Nmrad 119894 = 1199090 119909119886 1199100 119910119887) Γ119908= 1198961198941198960 (1198960= 1 times 10
9 Nm 119894 = 1199090 119909119886 1199100 119910119887)bResults in parentheses are taken from FEM
moderately thick plate with not only classical boundaryconditions but also elastic edge restraints
As the last case of this subsection the influence of thegradient 120572 and 120573 on the fundamental frequency parametersfor a linearly varying thickness moderately thick plate isinvestigated The model is a square moderately thick platewith all edges elastically restrainedThat is 119896
1199090= 119896119909119886= 1198961199100=
119896119910119887= Γ119908= 2 and 119870
1199090= 119870119909119886
= 1198701199100= 119870119910119887= Γ119909= 2 The six
frequency parameters Ω = 1205961198862
(120588ℎ01198630)12 are given in
Table 8 for several different 120572 and 120573 values the finite elementmethod (FEM) results are also listed as a reference Againgood agreement can be observed Through Table 8 it is alsofound that the frequency parameter increases with increasinggradient parameters
The above studies are given as linearly varying thicknessmoderately thick plates with several of boundary condition
and different structure parameters In the next section vibra-tion results for the plates subjected to nonlinear variationthickness will be presented
34 Nonlinearly Varying Thickness Moderately Thick Plateswith Classical and Elastic Boundary Conditions In Sec-tion 33 the linearly varying thickness moderately thickplates were studied However in the practical engineeringapplications the varying thickness of amoderately thick platemay not always be linear variation in nature A variety ofpossible thickness varying cases may be encountered in prac-tice Therefore the moderately thick plates with nonlinearvariation thickness subjected to general elastic edge restraintsare examined in this subsection For the sake of brevity theindexes 119904 and 119905 will be chosen as 2 to imitate the nonlinearlyvarying thickness moderately thick plates structure in this
16 Shock and Vibration
Table 8 Frequency parameterΩ = 1205961198862
(120588ℎ01198630)12 for elastic support moderately thick plates (119886119887 = 1) with different gradient (119896
1199090= 119896119909119886=
1198961199100= 119896119910119887= Γ119908= 2 119870
1199090= 119870119909119886= 1198701199100= 119870119910119887= Γ119909= 2)
120572 120573 ℎ0119886
Model sequence1 2 3 4 5 6
120572 = 120573 = 01
005 31275 52615 52657 68954 76146 79850(31303a) (52664) (52675) (69033) (76174) (79948)
01 17270 25244 25250 33949 46050 46625(17271a) (25255) (25258) (34089) (46122) (46708)
015 10190 16266 16274 24164 37327 37526(10189a) (16269) (16280) (24323) (37401) (37622)
02 6780 12258 12260 19685 31452 31754(6780a) (12265) (12270) (19851) (31514) (31864)
120572 = 120573 = 02
005 32530 52947 53141 68954 77034 80589(32577a) (53014) (53203) (69172) (77088) (80773)
01 16882 24979 24979 34154 48057 48575(16883a) (24983) (25001) (34433) (48134) (48667)
015 9804 16284 16309 24683 38844 39085(9803a) (16283) (16322) (24980) (38909) (39191)
02 6499 12290 12295 20133 32354 32770(6498a) (12296) (12307) (20420) (32387) (32895)
120572 = 120573 = 03
005 33516 52845 53308 68704 78203 81561(33563a) (52939) (53389) (69055) (78271) (81789)
01 16419 24799 24830 34527 50061 50583(16419a) (24790) (24869) (34947) (50143) (50688)
015 9430 16334 16378 25236 40193 40506(9429a) (16329) (16395) (25660) (40243) (40625)
02 6235 12288 12293 20572 33102 33663(6235a) (12296) (12306) (20965) (33090) (33811)
120572 = 120573 = 04
005 34215 52463 53264 68430 79654 82865(34252a) (52575) (53346) (68921) (79732) (83113)
01 15921 24695 24793 35027 52011 52571(15921a) (24674) (24842) (35583) (52098) (52688)
015 9074 16388 16444 25797 41379 41789(9074a) (16381) (16464) (26336) (41404) (41928)
02 5990 12236 12238 21002 33735 34461(5989a) (12249) (12250) (21487) (33653) (34641)
120572 = 120573 = 05
005 34615 51972 53088 68255 81344 84510(33806a) (52100) (53157) (68891) (81433) (84761)
01 15416 24659 24829 35615 53879 54494(15045a) (24625) (24888) (36303) (53970) (54626)
015 8740 16420 16481 26353 42417 42946(8529a) (16414) (16499) (26996) (42398) (43114)
02 5761 12127 12127 21434 34286 35192(5622a) (12134) (12148) (21998) (34110) (35415)
aResults in parentheses are taken from FEM
section that is 119904 = 119905 = 2 Also the thickness varying functionsare also be expressed as Fourier cosine series according to (18)and (19)
In order to validate the accuracy and reliability of theproposed method for predicting the vibration behavior ofnonlinearly varying thickness moderately thick plates with
arbitrary boundary conditions the typical classical boundaryconditions viewed as the special cases of elastically restrainededges will be considered The comparison of the first six fre-quency parameters Ω = 120596119886
2
(120588ℎ01198630)12 for the moderately
thick plates with nonlinearly varying thickness is presentedin Table 9 The S-S-S-S C-F-F-F S-S-F-F C-C-C-C and
Shock and Vibration 17
Table 9 Frequency parameter Ω = 1205961198862
(120588ℎ01198630)12 for moderately thick plates with linearly varying thickness in different boundary
conditions (ℎ(119909 119910) = ℎ0(1 + 05119909
2
)(1 + 051199102
))
Boundary condition 119886119887 ℎ0119886
Model sequence1 2 3 4 5 6
S-S-S-S
101 23462 54999 55573 82950 100900 101187
(23741a) (55114) (55983) (83146) (101004) (101347)
02 18606 42725 42971 60040 71167 71761(18936a) (43014) (43148) (60247) (71642) (71993)
3201 53055 65251 85404 106700 114880 143531
(53467a) (65472) (85984) (107211) (115687) (144007)
02 17439 21531 32170 44987 59438 64179(17874a) (21748) (32651) (45264) (60188) (64879)
C-F-F-F
101 3658 9317 24135 32816 35433 59179
(3749a) (9547) (24034) (33627) (36308) (60142)
02 3520 7986 19591 26959 27491 43882(3611a) (8176) (20075) (27125) (27625) (43984)
3201 3901 9963 28984 69187 73686 89360
(3930a) (10095) (29381) (69578) (73994) (90242)
02 3292 4439 13385 23947 40850 44758(3386a) (4498) (14012) (24068) (41221) (45142)
S-S-F-F
101 3557 21268 22762 43524 60253 62429
(3649a) (21441) (22796) (43916) (60622) (62630)
02 3171 17822 19512 33616 45042 46833(3489a) (18006) (20430) (33815) (45247) (49004)
3201 6738 31658 52071 75625 86604 94903
(6957a) (31882) (52518) (75823) (86939) (95610)
02 3653 12930 20955 26517 44097 48058(6807a) (13160) (21312) (26864) (44422) (48873)
C-C-C-C
101 41020 75196 75614 104142 119683 120949
(41405a) (75452) (75882) (104711) (119637) (121137)
02 31237 51686 51868 68411 76868 77864(32010a) (52960) (52145) (69335) (77481) (78099)
3201 95823 128727 163657 173751 190890 223595
(95364a) (128932) (163588) (172350) (190350) (222948)
02 53306 71372 89384 102404 104660 120908(53587a) (71759) (90103) (102503) (105174) (121275)
S-C-S-C
101 33715 60298 71446 94187 103802 117928
(34148a) (60786) (71828) (94515) (104263) (119326)
02 26143 45499 49458 64253 72941 76191(26789a) (46622) (50680) (64329) (73474) (76843)
3201 73228 115845 137445 148166 168342 182542
(73761a) (116121) (137788) (148530) (168698) (182921)
02 41302 53648 65630 72202 81632 88105(41304a) (53757) (65842) (72794) (82033) (88474)
aResults in parentheses are taken from FEM
S-C-S-C boundary conditions are performed in the compar-ison The results adequately demonstrated the great accuracyof the modified Fourier method
We now turn to elastically restrained moderately thickplates The first one involves an S-S-S-S moderately thicksquare plate with a uniform rotational restraint along eachedge that is 119870
1199090= 119870119909119886
= 1198701199100
= 119870119910119887
= Γ119909 The calculated
frequency parameters Ω = 1205961198862
(120588ℎ01198630)12 are given in
Table 10 together with the FEM results Since the elasticallyrestrained plates with nonlinear variation thickness are rarelyinvestigated the FEMresults are used as the referenceA goodagreement is observed between the current and FEM resultsThe second example concerns amoderately thick square plateelastically supported along all edgesThe stiffness of the linear
18 Shock and Vibration
Table 10 Frequency parameterΩ = 1205961198862
(120588ℎ01198630)12 for S-S-S-S moderately thick plates (119886119887 = 1) with linearly varying thickness and elastic
rotation support (ℎ(119909 119910) = ℎ0(1 + 05119909
2
)(1 + 051199102
) 1198701199090= 119870119909119886= 1198701199100= 119870119910119887= Γ119909)
Γ119909
aℎ0119886
Model sequence1 2 3 4 5 6
0
005 23462 54999 55573 82950 100900 101187(23741b) (55114) (55983) (83146) (101004) (101347)
01 18606 42725 42971 60040 71167 71761(18936b) (43014) (43148) (60247) (71642) (71993)
015 53055 65251 85404 106700 114880 143531(53467b) (65472) (85984) (107211) (115687) (144007)
02 17439 21531 32170 44987 59438 64179(17874b) (21748) (32651) (45264) (60188) (64879)
05
005 6904 13374 13514 19400 22904 23056(9909b) (13548) (13651) (19564) (23088) (23379)
01 5316 10237 10314 14499 17155 17264(5374b) (10448) (10651) (14764) (17188) (17879)
015 4136 8201 8220 11474 13578 13671(4174b) (8348) (8651) (11764) (13688) (13879)
02 3449 6915 6924 9549 11225 11327(3474b) (6948) (6951) (9764) (11388) (11579)
1
005 7040 13617 13745 19726 23233 23402(7074b) (13748) (13851) (19964) (23488) (23879)
01 5712 10696 10793 14991 17594 17744(5774b) (10748) (10851) (15054) (17788) (17879)
015 4489 8507 8540 11759 13803 13922(4514b) (8548) (8611) (11864) (13988) (14119)
02 3677 7084 7090 9692 11330 11445(3684b) (7148) (7191) (9764) (11588) (11879)
15
005 7089 13708 13828 19847 23355 23530(7124b) (13748) (13951) (19967) (23588) (23879)
01 5895 10923 11024 15239 17813 17988(5874b) (10748) (11651) (15264) (17818) (17979)
015 4697 8694 8737 11938 13790 14083(4774b) (8748) (8951) (11864) (13688) (14879)
02 3836 7201 7210 9792 11403 11530(3874b) (7748) (7651) (9864) (11588) (11879)
2
005 7114 13754 13871 19911 23418 23597(7174b) (12848) (13851) (19864) (23588) (23879)
01 6001 11060 11159 15391 17944 18136(6004b) (11148) (11151) (15464) (17988) (18379)
015 4836 8821 8872 12061 14037 14189(4874b) (8848) (8951) (12164) (14188) (14879)
02 3954 7289 7301 9871 11457 11594(3974b) (7348) (7651) (9964) (11488) (11679)
aΓ119909= 1198701198941198700 (1198700= 1 times 10
9 Nmrad 119894 = 1199090 119909119886 1199100 119910119887)bResults in parentheses are taken from FEM
and rotational restraints is set equal to 1198961199090
= 119896119909119886
= 1198961199100
=
119896119910119887
= Γ119908and 119870
1199090= 119870119909119886
= 1198701199100
= 119870119910119887
= Γ119909 respectively
The six frequency parametersΩ = 1205961198862
(120588ℎ01198630)12 are shown
in Table 11 for several different restraining coefficient valuesIt can also be noticed that the Fourier series method is able
to predict the modal characteristics of nonlinearly varyingthickness moderately thick plate with not only classicalboundary conditions but also elastic edge restraints correctly
Finally the influence of the gradient on the fundamentalfrequency parameters of a nonlinearly varying thickness
Shock and Vibration 19
Table 11 Frequency parameterΩ = 1205961198862
(120588ℎ01198630)12 for moderately thick plates (119886119887 = 1) with linearly varying thickness and elastic rotation
and translation support (ℎ(119909 119910) = ℎ0(1 + 05119909
2
)(1 + 051199102
) 1198701199090= 119870119909119886= 1198701199100= 119870119910119887= Γ119909 and 119896
1199090= 119896119909119886= 1198961199100= 119896119910119887= Γ119908)
Γ119909
aΓ119908
aℎ0119886
Model sequence1 2 3 4 5 6
05 0
005 12926 13041 25090 47935 47943 61761(12931b) (13053) 25099 (47947) (47952) (61788)
01 10900 10988 21545 39980 40147 50322(10904b) (10992) (21557) (39984) (40166) (50334)
015 8245 8298 18081 31503 32617 40950(8247b) (8230) (18092) (31508) (32622) (40955)
02 6106 6124 15639 25631 27601 34458(6110b) (6125) (15642) (25637) (27617) (34462)
1 05
005 21575 31615 31644 42573 57889 58607(21580b) (31617) (31644) (42581) (57893) (58610)
01 8451 16024 16210 25969 43839 43867(8451b) (16025) (16217) (25972) (43844) (43870)
015 4682 11479 11480 20642 35172 35598(4684b) (11480) (11481) (20644) (35178) (35604)
02 3059 8621 8648 17320 28487 29638(3062b) (8619) (8644) (17327) (28492) (29641)
15 1
005 27713 40203 40978 52951 65352 66815(27714b) (40207) (40988) (52972) (65368) (66822)
01 11667 19322 19615 29122 46115 46183(11674b) (19327) (19619) (29137) (46120) (46188)
015 6544 13416 13481 22405 37238 37430(6547b) (13420) (13488) (22410) (37241) (37434)
02 4295 10175 10190 18543 30330 31056(4296b) (10177) (10194) (18544) (30336) (31060)
2 2
005 33453 50854 52088 66311 76610 78947(33455b) (50852) (50097) (66321) (76627) (78971)
01 15832 24130 24372 33924 49118 49362(15830b) (24127) (24374) (33931) (49147) (49382)
015 9060 15956 16099 24754 39134 39249(9064b) (15961) (16102) (24755) (39138) (39452)
02 5992 11955 11971 20034 31908 32392(5991b) (11957) (11978) (20042) (31910) (32398)
aΓ119909= 1198701198941198700 (1198700= 1 times 10
9 Nmrad 119894 = 1199090 119909119886 1199100 119910119887) Γ119908= 1198961198941198960 (1198960= 1 times 10
9 Nm 119894 = 1199090 119909119886 1199100 119910119887)bResults in parentheses are taken from FEM
moderately thick plate is investigated The plate is elasticallyrestrained in which the stiffness of the boundary springsis taken as 119896
1199090= 119896119909119886
= 1198961199100
= 119896119910119887
= Γ119908
= 2 and1198701199090
= 119870119909119886
= 1198701199100
= 119870119910119887
= Γ119909= 2 respectively The six
frequency parameters Ω = 1205961198862
(120588ℎ01198630)12 are presented in
Table 12 for several different slop values It can be seen thatthe fundamental frequency parameters will decrease with theincrease of the parameter ℎ
0119886
In the above examples it has been demonstrated thatthe presented method can be universally applied to nonlin-early varying thickness moderately thick plates with severalboundary conditions and different structure parametersNew results are obtained for plates with nonlinear variationthickness in both directions subjected to general elastic
boundary restraints which may be used for benchmarkingof researchers in the field In addition it is interesting to seethat the nature frequency decreases with the increase of theindex for thickness function
4 Conclusions
In this paper a modified Fourier method has been pre-sented to study the free vibration behaviors of moderatelythick rectangular plates with variable thickness and arbi-trary boundary conditionsThe first-order shear deformationplate theory is adopted to formulate the theoretical modelThe displacements and rotation components of the plateregardless of boundary conditions are invariantly expressedas the superposition of a 2D Fourier cosine series and four
20 Shock and Vibration
Table 12 Frequency parameterΩ = 1205961198862
(120588ℎ01198630)12 for elastic support moderately thick plates (119886119887 = 1) with different gradient (119896
1199090= 119896119909119886=
1198961199100= 119896119910119887= Γ119908= 2 119870
1199090= 119870119909119886= 1198701199100= 119870119910119887= Γ119909= 2)
120572 120573 ℎ0119886
Model sequence1 2 3 4 5 6
120572 = 120573 = 01
005 30187 51102 51141 67066 73954 77412(30188a) (51105) (51144) (67068) (73961) (77477)
01 16995 24719 24724 33071 44343 44819(16996a) (24725) (24729) (33084) (44347) (44822)
015 10077 15872 15883 23443 35937 36065(10081a) (15877) (15892) (23447) (35941) (36084)
02 6713 11953 11958 19086 30372 30604(6713a) (11957) (11965) (19092) (30378) (30612)
120572 = 120573 = 02
005 31206 51436 51614 67058 74398 77595(31208a) (51438) (51619) (67064) (74402) (77596)
01 16786 24503 24508 33144 45583 45923(16790a) (24510) (24513) (33147) (45587) (45932)
015 9821 15865 15906 23766 36897 36990(9824a) (15870) (15907) (23768) (36894) (36997)
02 6523 11978 11992 19356 30912 31162(6524a) (11981) (11997) (19362) (30916) (31174)
120572 = 120573 = 03
005 32097 51462 51900 66832 74991 77863(32098a) (51466) (51907) (66849) (74996) (77872)
01 16507 24332 24383 33330 46806 47074(16511a) (24327) (24380) (33339) (46810) (47086)
015 9562 15885 15964 24103 37748 37828(9562a) (15892) (15977) (21105) (37751) (37832)
02 6338 11992 12013 19603 31338 31634(6340a) (11994) (12011) (19612) (31344) (31639)
120572 = 120573 = 04
005 32851 51239 52046 66547 75735 78303(33455a) (50852) (50097) (66321) (76627) (78971)
01 16182 24209 24343 33598 47990 48230(33455a) (50852) (50097) (66321) (76627) (78971)
015 9307 15919 16034 24436 38491 38579(33455a) (50852) (50097) (66321) (76627) (78971)
02 6161 11986 12008 19828 31666 32040(33455a) (50852) (50097) (66321) (76627) (78971)
120572 = 120573 = 05
005 33453 50854 52088 66311 76610 78947(33455a) (50852) (50097) (66321) (76627) (78971)
01 15832 24130 24372 33924 49118 49362(15830a) (24127) (24374) (33931) (49147) (49382)
015 9060 15956 16099 24754 39134 39249(9064a) (15961) (16102) (24755) (39138) (39452)
02 5992 11955 11971 20034 31908 32392(5991a) (11957) (11978) (20042) (31910) (32398)
aResults in parentheses are taken from FEM
supplementary functions in the form of the product of apolynomial function and a single cosine series expansionto ensure and accelerate the convergence of the solutionAt each edge of the plate the general restraint conditionsare implemented by introducing one group of linear springsand two groups of rotational springs which are continuouslydistributed and determined by the stiffness of these springs
Instead of seeking a solution in strong forms in the previousstudies all the Fourier coefficients will be treated equallyand independently as the generalized coordinates and solveddirectly from the Rayleigh-Ritz technique The change of theboundary conditions can be easily achieved by only varyingthe stiffness of the three sets of boundary springs along alledges of the rectangular plates without involving any change
Shock and Vibration 21
to the solution procedure The convergence of the presentsolution is examined and the excellent accuracy is validatedby comparison with existing results published in the lit-erature and FEM data Excellent agreements are obtainedfrom these comparisons The proposed method providesa unified means for extracting the modal parameters andpredicting the vibration behaviors of moderately thick plateswith variable thickness variation functions and arbitraryelastic edge restraints A variety of free vibration results formoderately thick rectangular plates with different thicknessvariation functions and boundary conditions are presentedNew results for free vibration of moderately thick rectangularplates with various thickness variation functions and edgeconditions are presented which may be used for benchmark-ing of researchers in the field
Appendices
A Representative Calculation forStiffness and Mass Matrices
To illuminate the particular expression of the mass andstiffness matrixes clearly and tersely four new variables aredefined as 119904 = 119898 lowast (119873 + 1) + 119899 + 1 119905 = 119898
1015840
lowast (119873 + 1) + 1198991015840
+ 1119901 = (119897minus1)lowast (119872+1)+119898
1015840
+1 and 119902 = (119897minus1)lowast (119873+1)+1198991015840
+1The first row elements of K andM are given below
K1-1119904119905
=
1
2
int
119886
0
int
119887
0
[119863 (119909 119910)
times 21205821198861198981205821198861198981015840 sin 120582
119886119898119909 sin 120582
1198861198981015840119909 cos 120582
119887119899119910
times cos 1205821198871198991015840119910 + (1 minus 120583) 120582
1198871198991205821198871198991015840 cos 120582
119886119898119909
times cos 1205821198861198981015840 sin 120582
119887119899119910 sin 120582
1198871198991015840119910
+ 120581119866ℎ (119909 119910) cos 120582119886119898119909 cos 120582
1198861198981015840119909 cos 120582
119887119899
times119910 cos 1205821198871198991015840119910119889119909119889119910] 119889119909 119889119910
+ int
119886
0
1198701199091199100
cos 120582119886119898119909 cos 120582
1198861198981015840119909
+(minus1)119898+1198981015840
119870119909119910119887
cos 120582119886119898119909 cos 120582
1198861198981015840119909 119889119909
+ int
119887
0
1198701199090cos 120582119887119899119910 cos 120582
1198871198991015840119910
+(minus1)119899+1198991015840
119870119909119886cos 120582119887119899119910 cos 120582
1198871198991015840119910 119889119910
K1-2119904119901
=
1
2
int
119886
0
int
119887
0
[119863 (119909 119910)21205821198861198981205821198861198981015840 sin 120582
119886119898119909 sin 120582
1198861198981015840119909
times
2
sum
119897=1
120577119897
119887(119910) cos 120582
1198871198991015840119910
minus (1 minus 120583) 1205821198871198991015840 cos 120582
119886119898119909 cos 120582
1198861198981015840119909
times
2
sum
119897=1
1205771198971015840
119887(119910) sin 120582
1198871198991015840119910 + 120581119866ℎ (119909 119910)
times ( cos 1205821198861198981015840119909 cos 120582
1198871198991015840119910
times
2
sum
119897=1
120577119897
119887(119910) cos 120582
119886119898119909)]119889119909119889119910
+ int
119886
0
1198701199091199100
cos 120582119886119898119909 cos 120582
1198861198981015840119909
2
sum
119897=1
120577119897
119887(0)
+(minus1)1198991015840
119870119909119910119887
cos 120582119886119898119909 cos 120582
1198861198981015840119909
2
sum
119897=1
120577119897
119887(119887) 119889119909
+ int
119887
0
1198701199090
2
sum
119897=1
120577119897
119887(119910) cos 120582
1198871198991015840119910
+(minus1)119898+1198981015840
119870119909119886
2
sum
119897=1
120577119897
119887(119910) cos 120582
1198871198991015840119910119889119910
K1-3119904119902
=
1
2
int
119886
0
int
119887
0
[119863 (119909 119910) minus 21205821198861198981015840 sin 120582
1198861198981015840119909 cos 120582
1198871198991015840119910
times
2
sum
119897=1
1205771198971015840
119886(119909) cos 120582
119887119899119910
+ (1 minus 120583) 1205821198871198991205821198871198991015840 cos 120582
1198861198981015840119909
times sin 1205821198871198991015840119910
2
sum
119897=1
120577119897
119886(119909) sin 120582
119887119899119910
+ 120581119866ℎ (119909 119910) cos 1205821198861198981015840119909 cos 120582
1198871198991015840119910
times
2
sum
119897=1
120577119897
119886(119909) cos 120582
119887119899119910]119889119909119889119910
+ int
119886
0
1198701199091199100
cos 1205821198861198981015840119909
2
sum
119897=1
120577119897
119886(119909)
+(minus1)119899+1198991015840
119870119909119910119887
cos 1205821198861198981015840119909
2
sum
119897=1
120577119897
119886(119909) 119889119909
+ int
119887
0
1198701199090
2
sum
119897=1
120577119897
119886(0) cos 120582
1198871198991015840119910
+(minus1)119898+1198981015840
119870119909119886
2
sum
119897=1
120577119897
119886(0) cos 120582
1198871198991015840119910119889119910
K1-4119904119905
=
1
2
int
119886
0
int
119887
0
119863(119909 119910) 21205821198871198991205821198861198981015840 sin 120582
1198861198981015840119909 cos 120582
1198871198991015840119910
times cos 120582119886119898119909 sin 120582
119887119899119910 minus 2(1 minus 120583)
119887119899
times 1205821198861198981015840 sin 120582
1198861198981015840119909 cos 120582
1198871198991015840119910 cos 120582
119886119898119909
times sin 120582119887119899119910 + (1 minus 120583) 120582
1198861198981205821198871198991015840 cos 120582
1198861198981015840119909
times sin 1205821198871198991015840119910 sin 120582
119886119898119909 cos 120582
119887119899119910 119889119909 119889119910
22 Shock and Vibration
K1-5119904119901
=
1
2
int
119886
0
int
119887
0
119863(119909 119910) minus 21205821198861198981015840 sin 120582
1198861198981015840119909 cos 120582
1198871198991015840119910
times
2
sum
119897=1
1205771198971015840
119887(119910) cos 120582
119886119898119909
+ 2 (1 minus 120583) 1205821198861198981015840 sin 120582
1198861198981015840119909 cos 120582
1198871198991015840119910
times
2
sum
119897=1
1205771198971015840
119887(119910) cos 120582
119886119898119909
+ (1 minus 120583) 1205821198861198981205821198871198991015840 cos 120582
1198861198981015840119909 sin 120582
1198871198991015840119910
times
2
sum
119897=1
120577119897
119887(119910) sin 120582
119886119898119909119889119909119889119910
K1-6119904119902
=
1
2
int
119886
0
int
119887
0
119863(119909 119910)1205821198871198991205821198861198981015840 sin 120582
1198861198981015840119909 cos 120582
1198861198991015840119910
times
2
sum
119897=1
120577119897
119886(119909) sin 120582
1198871198991015840119910
minus 2 (1 minus 120583) 1205821198871198991205821198861198981015840 sin 120582
1198861198981015840119909 cos 120582
1198861198991015840119910
times
2
sum
119897=1
120577119897
119886(119909) sin 120582
1198871198991015840119910
+ 1205821198871198991015840 cos 120582
1198861198981015840119909 sin 120582
1198871198991015840119910
times
2
sum
119897=1
1205771198971015840
119886(119909) cos 120582
119887119899119910119889119909119889119910
K1-7119904119905
= minus
1
2
int
119886
0
int
119887
0
120581119866ℎ (119909 119910) 120582119886119898
cos 1205821198861198981015840119909 cos 120582
1198871198991015840119910
times sin 120582119886119898119909 cos 120582
119887119899119910119889119909119889119910
K1-8119904119901
= minus
1
2
int
119886
0
int
119887
0
120581119866ℎ (119909 119910) 1205821198861198981015840 cos 120582
1198861198981015840119909 cos 120582
1198871198991015840119910
times
2
sum
119897=1
120577119897
119887(119910) sin 120582
1198861198981015840119909 119889119909 119889119910
K1-9119904119902
=
1
2
int
119886
0
int
119887
0
120581119866ℎ (119909 119910) cos 1205821198861198981015840119909 cos 120582
1198871198991015840119910
times
2
sum
119897=1
1205771198971015840
119886(119909) cos 120582
119887119899119910119889119909119889119910
M1-1119904119905
=
1
24
120588int
119886
0
int
119887
0
ℎ3
(119909 119910) cos 120582119886119898119909 cos 120582
119887119899119910
times cos 1205821198861198981015840119909 cos 120582
1198871198991015840119910119889119909119889119910
M1-2119904119901
=
1
24
120588int
119886
0
int
119887
0
ℎ3
(119909 119910)
2
sum
119897=1
120577119897
119887(119910) cos 120582
119886119898119909
times cos 1205821198861198981015840119909 cos 120582
1198871198991015840119910119889119909119889119910
M1-3119904119902
=
1
24
120588int
119886
0
int
119887
0
ℎ3
(119909 119910)
2
sum
119897=1
120577119897
119886(119909) cos 120582
119887119899119910
times cos 1205821198861198981015840119909 cos 120582
1198871198991015840119910119889119909119889119910
M1-4119904119905 = M
1-5119904119901 = M1-6119904119902
= M1-7119904119905 = M
1-8119904119901 = M1-9119904119902 = 0
(A1)
B Useful Integrals for the Moderately ThickRectangular Plates with Variable Thicknessand Arbitrary Boundary Conditions
Consider
int
119871
0
cos 120582119898119909 cos 120582
119899119909 119889119909 =
0 119898 = 119899
119871
2
119898 = 119899
119871 119898 = 119899 = 0
int
119871
0
sin 120582119898119909 cos 120582
119899119909119889119909
=
minus
(minus1 + (minus1)119898+119899
) 119871119898
(1198982minus 1198992) 120587
119898 = 119899
0 119898 = 119899
int
119871
0
sin 120582119898119909 sin 120582
119899119909 119889119909 =
0 119898 = 119899
119871
2
119898 = 119899
0 119898 = 119899 = 0
int
119871
0
cos 120582119898119909120577119897
119886(119909) 119889119909
=
41198712
31205872 119897 = 1 119898 = 0
minus
41198712
(41198982
minus 3)
(9 minus 401198982+ 16119898
4) 1205872
119897 = 1 119898 = 0
minus
41198712
31205872 119897 = 2 119898 = 0
41198712
(41198982
minus 3) (minus1)119898
(9 minus 401198982+ 16119898
4) 1205872
119897 = 2 119898 = 0
Shock and Vibration 23
int
119871
0
sin 120582119898119909120577119897
119886(119909) 119889119909
=
0 119897 = 1 119898 = 0
161198712
119898(minus1)119898
(9 minus 401198982+ 16119898
4) 1205872
119897 = 1 119898 = 0
0 119897 = 2 119898 = 0
161198981198712
(9 minus 401198982+ 16119898
4) 1205872
119897 = 2 119898 = 0
int
119871
0
cos 1205821198981199091205771198971015840
119886(119909) 119889119909
=
0 119897 = 1 119898 = 0
161198711198982
(minus1)119898
(9 minus 401198982+ 16119898
4) 120587
119897 = 1 119898 = 0
0 119897 = 2 119898 = 0
161198711198982
(9 minus 401198982+ 16119898
4) 120587
119897 = 2 119898 = 0
int
119871
0
sin 1205821198981199091205771198971015840
119886(119909) 119889119909
=
0 119897 = 1 119898 = 0
4119871119898 (minus3 + 41198982
)
(9 minus 401198982+ 16119898
4) 120587
119897 = 1 119898 = 0
0 119897 = 2 119898 = 0
119871119898(
1
120587 minus 41198982120587
+
3
9120587 minus 41198982120587
) (minus1)119898
119897 = 2 119898 = 0
int
119871
0
1205771198971
119886(119909) 1205771198972
119886(119909) 119889119909 =
1198713
41205872 1198971= 1 1198972= 1
minus
21198713
31205873 1198971= 1 1198972= 2
minus
21198713
31205873 1198971= 2 1198972= 1
1198713
41205872 1198971= 2 1198972= 2
int
119871
0
1205771198971
119886(119909) 1205771198971015840
2
119886(119909) 119889119909 =
0 1198971= 1 1198972= 1
minus
1198712
8120587
1198971= 1 1198972= 2
1198712
8120587
1198971= 2 1198972= 1
0 1198971= 2 1198972= 2
int
119871
0
1205771198971015840
1
119886(119909) 1205771198971015840
2
119886(119909) 119889119909 =
5119871
16
1198971= 1 1198972= 1
minus
119871
2120587
1198971= 1 1198972= 2
minus
119871
2120587
1198971= 2 1198972= 1
5119871
16
1198971= 2 1198972= 2
int
119871
0
119909119899 cos 120582
119898119909 119889119909
=
1198711+119899
1 + 119899
119898 = 0
(1198861+119899HypergeometricPFQtimes [ 12 + 1198992 12 32 + 1198992
minus (14)1198982
1205872
]) times (1 + 119899)minus1
119898 gt 0
(B1)
Conflict of Interests
The authors declare that there is no conflict of interestsregarding the publication of this paper
Acknowledgments
The authors would like to thank the anonymous reviewersfor their very valuable comments The authors also gratefullyacknowledge the financial support from the National NaturalScience Foundation of China (no 51209052) HeilongjiangProvince Youth Science Fund Project (no QC2011C013) andHarbin Science and Technology Development InnovationFoundation of Youth (no 2011RFQXG021)
References
[1] K Liew Y Xiang and S Kitipornchai ldquoResearch on thick platevibration a literature surveyrdquo Journal of Sound and Vibrationvol 180 no 1 pp 163ndash176 1995
[2] J H Chung T Y Chung and K C Kim ldquoVibration analysisof orthotropic mindlin plates with edges elastically restrainedagainst rotationrdquo Journal of Sound and Vibration vol 163 no 1pp 151ndash163 1993
[3] Y K Cheung and D Zhou ldquoVibrations of moderately thickrectangular plates in terms of a set of static Timoshenko beamfunctionsrdquoComputers and Structures vol 78 no 6 pp 757ndash7682000
[4] CMWang ldquoNatural frequencies formula for simply supportedMindlin platesrdquo Journal of Vibration and Acoustics vol 116 no4 pp 536ndash540 1994
[5] K N Saha R C Kar and P K Datta ldquoFree vibration analysisof rectangular Mindlin plates with elastic restraints uniformlydistributed along the edgesrdquo Journal of Sound andVibration vol192 no 4 pp 885ndash902 1996
24 Shock and Vibration
[6] D J Gorman ldquoAccurate free vibration analysis of shear-deformable plates with torsional elastic edge supportrdquo Journalof Sound and Vibration vol 203 no 2 pp 209ndash218 1997
[7] D J Gorman ldquoFree vibration analysis of Mindlin plates withuniform elastic edge support by the superposition methodrdquoJournal of Sound andVibration vol 207 no 3 pp 335ndash350 1997
[8] D J Gorman ldquoAccurate free vibration analysis of point sup-ported mindlin plates by the superposition methodrdquo Journal ofSound and Vibration vol 219 no 2 pp 265ndash277 1999
[9] Y Xiang ldquoVibration of rectangular Mindlin plates resting onnon-homogenous elastic foundationsrdquo International Journal ofMechanical Sciences vol 45 no 6-7 pp 1229ndash1244 2003
[10] Y Xiang and G W Wei ldquoExact solutions for buckling andvibration of stepped rectangular Mindlin platesrdquo InternationalJournal of Solids and Structures vol 41 no 1 pp 279ndash294 2004
[11] Y L Yeh M J Jang and C C Wang ldquoAnalyzing the freevibrations of a plate using finite difference and differential trans-formationmethodrdquoAppliedMathematics andComputation vol178 no 2 pp 493ndash501 2006
[12] Y Xiang S K Lai L Zhou and C W Lim ldquoDSC-Ritz elementmethod for vibration analysis of rectangular Mindlin plateswith mixed edge supportsrdquo European Journal of MechanicsmdashASolids vol 29 no 4 pp 619ndash628 2010
[13] H Nguyen-Xuan C H Thai and T Nguyen-Thoi ldquoIsogeo-metric finite element analysis of composite sandwich platesusing a higher order shear deformation theoryrdquo Composites BEngineering vol 55 pp 558ndash574 2013
[14] C H Thai A J M Ferreira E Carrera and H Nguyen-XuanldquoIsogeometric analysis of laminated composite and sandwichplates using a layerwise deformation theoryrdquo Composite Struc-tures vol 104 pp 196ndash214 2013
[15] C H Thai A Ferreira S Bordas T Rabczuk and HNguyen-Xuan ldquoIsogeometric analysis of laminated compositeand sandwich plates using a new inverse trigonometric sheardeformation theoryrdquo European Journal of Mechanics A Solidsvol 43 pp 89ndash108 2014
[16] H Luong-Van T Nguyen-Thoi G R Liu and P Phung-VanldquoA cell-based smoothed finite elementmethod using three-nodeshear-locking free Mindlin plate element (CS-FEM-MIN3) fordynamic response of laminated composite plates on viscoelasticfoundationrdquo Engineering Analysis with Boundary Elements vol42 pp 8ndash19 2014
[17] T Nguyen-Thoi T Rabczuk T Lam-Phat V Ho-Huu andP Phung-Van ldquoFree vibration analysis of cracked Mindlinplate using an extended cell-based smoothed discrete sheargap method (XCS-DSG3)rdquo Theoretical and Applied FractureMechanics 2014
[18] H Nguyen-Xuan G R Liu and C a Thai-Hoang ldquoAnedge-based smoothed finite element method (ES-FEM) withstabilized discrete shear gap technique for analysis of Reissner-Mindlin platesrdquo Computer Methods in Applied Mechanics andEngineering vol 199 no 9ndash12 pp 471ndash489 2010
[19] P Phung-Van T Nguyen-Thoi T Bui-Xuan and Q Lieu-XuanldquoA cell-based smoothed three-node Mindlin plate element(CS-FEM-MIN3) based on the 119862
0 -type higher-order sheardeformation for geometrically nonlinear analysis of laminatedcomposite platesrdquo Computational Materials Science 2014
[20] NNguyen-Thanh T RabczukHNguyen-Xuan and S BordasldquoAn alternative alpha finite element method with discrete sheargap technique for analysis of isotropicMindlin-Reissner platesrdquoFinite Elements inAnalysis andDesign vol 47 no 5 pp 519ndash5352011
[21] KM Liew Y Xiang and S Kitipornchai ldquoTransverse vibrationof thick rectangular plates-I Comprehensive sets of boundaryconditionsrdquo Computers and Structures vol 49 no 1 pp 1ndash291993
[22] D Zhou ldquoVibrations of Mindlin rectangular plates with elasti-cally restrained edges using static Timoshenko beam functionswith the Rayleigh-Ritz methodrdquo International Journal of Solidsand Structures vol 38 no 32-33 pp 5565ndash5580 2001
[23] D Zhou SH Lo F T KAu andYK Cheung ldquoVibration anal-ysis of rectangular Mindlin plates with internal line supportsusing static Timoshenko beam functionsrdquo International Journalof Mechanical Sciences vol 44 no 12 pp 2503ndash2522 2002
[24] H S Shen J Yang and L Zhang ldquoFree and forced vibrationof Reissner-Mindlin plates with free edges resting on elasticfoundationsrdquo Journal of Sound and Vibration vol 244 no 2 pp299ndash320 2001
[25] Y Xing and B Liu ldquoClosed form solutions for free vibrations ofrectangular Mindlin platesrdquo Acta Mechanica Sinica vol 25 no5 pp 689ndash698 2009
[26] P Gagnon C Gosselin and L Cloutier ldquoA finite strip elementfor the analysis of variable thickness rectangular thick platesrdquoComputers amp Structures vol 63 no 2 pp 349ndash362 1997
[27] T Mlzusawa ldquoVibration of rectangular mindlin plates withtapered thickness by the spline strip methodrdquo Computers andStructures vol 46 no 3 pp 451ndash463 1993
[28] H Nguyen-Xuan and T Nguyen-Thoi ldquoA stabilized smoothedfinite element method for free vibration analysis of Mindlin-Reissner platesrdquo Communications in Numerical Methods inEngineering with Biomedical Applications vol 25 no 8 pp 882ndash906 2009
[29] K M Liew L X Peng and S Kitipornchai ldquoVibration anal-ysis of corrugated Reissner-Mindlin plates using a mesh-freeGalerkin methodrdquo International Journal of Mechanical Sciencesvol 51 no 9-10 pp 642ndash652 2009
[30] Y Hou GWWei and Y Xiang ldquoDSC-Ritz method for the freevibration analysis of Mindlin platesrdquo International Journal forNumerical Methods in Engineering vol 62 no 2 pp 262ndash2882005
[31] F- Liu and K M Liew ldquoVibration analysis of discontinuousMindlin plates by differential quadrature element methodrdquoJournal of Vibration and Acoustics Transactions of the ASMEvol 121 no 2 pp 204ndash208 1999
[32] P Malekzadeh G Karami and M Farid ldquoA semi-analyticalDQEM for free vibration analysis of thick plates with twoopposite edges simply supportedrdquoComputerMethods inAppliedMechanics and Engineering vol 193 no 45ndash47 pp 4781ndash47962004
[33] P Malekzadeh and S A Shahpari ldquoFree vibration analysisof variable thickness thin and moderately thick plates withelastically restrained edges by DQMrdquo Thin-Walled Structuresvol 43 no 7 pp 1037ndash1050 2005
[34] R E Diaz-Contreras and S Nomura ldquoGreenrsquos function appliedto solution ofMindlin platesrdquoComputers and Structures vol 60no 1 pp 41ndash48 1996
[35] T Sakiyama and M Huang ldquoFree vibration analysis of rect-angular plates with variable thicknessrdquo Journal of Sound andVibration vol 216 no 3 pp 379ndash397 1998
[36] M Huang X Q Ma T Sakiyama H Matuda and C MoritaldquoFree vibration analysis of orthotropic rectangular plates withvariable thickness and general boundary conditionsrdquo Journal ofSound and Vibration vol 288 no 4-5 pp 931ndash955 2005
Shock and Vibration 25
[37] I Shufrin and M Eisenberger ldquoVibration of shear deformableplates with variable thicknessmdashfirst-order and higher-orderanalysesrdquo Journal of Sound and Vibration vol 290 no 1-2 pp465ndash489 2006
[38] S A Eftekhari and A A Jafari ldquoAccurate variational approachfor free vibration of variable thickness thin and thick plates withedges elastically restrained against translation and rotationrdquoInternational Journal of Mechanical Sciences vol 68 pp 35ndash462013
[39] W L Li X Zhang J Du and Z Liu ldquoAn exact series solutionfor the transverse vibration of rectangular plates with generalelastic boundary supportsrdquo Journal of Sound amp Vibration vol321 no 1-2 pp 254ndash269 2009
[40] X Shi D Shi W L Li and Q Wang ldquoA unified method forfree vibration analysis of circular annular and sector plateswith arbitrary boundary conditionsrdquo Journal of Vibration andControl 2014
[41] D Y Shi X J Shi W L Li and Q S Wang ldquoFree transversevibrations of orthotropic thin rectangular plates with arbitraryelastic edge supportsrdquo Journal of Vibroengineering vol 16 no 1pp 389ndash398 2014
[42] Y Chen G Jin and Z Liu ldquoFree vibration analysis of circu-lar cylindrical shell with non-uniform elastic boundary con-straintsrdquo International Journal of Mechanical Sciences vol 74pp 120ndash132 2013
[43] L Dai T Yang J Du W L Li and M J Brennan ldquoAn exactseries solution for the vibration analysis of cylindrical shellswith arbitrary boundary conditionsrdquo Applied Acoustics vol 74no 3 pp 440ndash449 2013
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Shock and Vibration
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Modelling amp Simulation in EngineeringHindawi Publishing Corporation httpwwwhindawicom Volume 2014
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Chemical EngineeringInternational Journal of Antennas and
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DistributedSensor Networks
International Journal of
Shock and Vibration 7
1
2
3
4
5
6
The 1st orderThe 3rd order
101 103 105 107 109 1011 1013
Ω
Ki (Nm)
(a)
2
3
4
5
6
101 103 105 107 109 1011 1013
The 1st orderThe 3rd order
Ω
Ki (Nmrad)
(b)
3
4
5
6
101 103 105 107 109 1011 1013
Kij (Nmrad)
The 1st orderThe 3rd order
Ω
(c)
Figure 2 The effect of boundary spring stiffness on the natural frequencies Ω (a) translation spring (b) rotation spring and (c) torsionalspring
On the same hardware (Intel i7-39GHz) the computing timeof the present formulation for the solution (119872times119873 = 12times12)implemented in optimized MATLAB scripts is about 2125 swhereas the finite element solution consumes 34578 s that isat least 16 times more CPU time than the present method forthe same problem
As mentioned earlier in the current modeling frame-work all the classical boundary conditions and their com-binations can be conveniently viewed as special cases whenthe stiffness for the normal and tangential boundary springsbecomes zero andor infinitely large Thus the effects of thestiffness of the translation (119896
119894) rotation (119870
119894) and torsional
springs (119870119894119895) on the modal characteristics should be inves-
tigated As shown in the Figures 2(a)ndash2(c) the first and thethird frequency parameters are separately obtained by vary-ing the stiffness of one group of the boundary springs from
extremely large (1014) to extremely small (100) while assigningthe other group of the springs infinite stiffness (1014) It canbe found in Figure 2(a) that the frequency parameter almostkeeps at a level when the stiffness of the translation springs islarger than 1012 or smaller than 107 In Figure 2(b) the influ-ences of the rotation springs on frequency parameters aregiven It is shown that the frequency curves change greatlywithin the stiffness range (106 to 1010) while out of this rangethe frequency curves separately keep level In Figure 2(c) theinfluences of the torsional springs on frequency parametersare given It is shown that the frequency curves almost changewhen the stiffness changes in the whole range
Based on the analysis it can be found that the torsionalsprings almost have no effect on the structure Also the rela-tionship between the rotation springs and twisting momentcan be seen from the boundary condition expression Then
8 Shock and Vibration
32
36
40
44
48
52
2
3
4
5
6
7
8
Ω Ω
0 2 4 6 8 10 12 14 16 18M = N
ab = 1
ab = 32
(a) The 1st order
0 2 4 6 8 10 12 14 16 180
20
40
60
80
0
50
100
150
200
250
300
M = N
Ω Ω
ab = 1
ab = 32
(b) The 8th order
Figure 3 The effect of numerically truncated numbers119872 = 119873 on the natural frequenciesΩ for C-C-C-C boundary condition
0 2 4 6 8 10 12 14 16 18125
130
135
140
145
150
088
090
092
094
096
098
Ω Ω
M = N
ab = 1
ab = 32
(a) The 1st order
0 2 4 6 8 10 12 14 16 18
55
60
65
70
75
80
40
45
50
55
60
65
70
75
80Ω Ω
M = N
ab = 1
ab = 32
(b) The 8th order
Figure 4 The effect of numerically truncated numbers119872 = 119873 on the natural frequenciesΩ for F-F-F-F boundary condition
the twisting moments have small effect on the vibration char-acteristics of the structure However in this paper in order toget a more accurate prediction of the vibration characteristicsof the structure the twisting moment on boundary edges istaken into account In the latter study in addition to the factthat free boundary is not unexpected considering the torsionspring the other boundary conditions are introduced into atorsion spring and the spring stiffness takes infinity FormFigure 2 analysis it also concluded that the translation springhas a wider influence range than the rotation spring on thefrequency parameters that is for the translation springs thestable frequency parameters appearwhen the stiffness ismorethan 1012 or less 107 while for the rotation springs when thestiffness value is assigned more than 1010 the frequency para-meters become smoothThus it is suitable to use 1014 to sim-ulate the infinite stiffness value in the model validation partsand in the following examples
32 Uniform Thickness Moderately Thick Plates with Classicaland Elastic Boundary Conditions In this subsection themodified Fourier solution is applied to deal with vibrationproblems of uniform thickness moderately thick rectangu-lar plates subject to the classical boundary conditions andarbitrary elastic boundary conditions In present work threegroups of continuously distributed boundary springs areintroduced to simulate the given or typical boundary condi-tions As mentioned earlier the stiffness of these boundarysprings can take any value from zero to infinity to bettermodel many real-world restraint conditions Taking edge 119909 =0 for example the corresponding spring stiffness for the threetypes of classical boundaries and elastic boundaries is
completely free 119876119909= 0119872
119909119909= 0 and119872
119909119910= 0
1198961199090= 0 119870
1199090= 0 119870
1199101199090= 0 (17a)
Shock and Vibration 9
0 2 4 6 8 10 12 14 16 18
166
168
170
172
152
156
160
164
168
Ω Ω
ab = 1
ab = 32
M = N
(a) The 1st order
0 2 4 6 8 10 12 14 16 1850
55
60
65
70
4
5
6
7
8
9
10
Ω Ω
M = N
ab = 1
ab = 32
(b) The 8th order
Figure 5 The effect of numerically truncated numbers 119872 = 119873 on the natural frequencies Ω with four edges elastically restrained againsttranslation and rotation (119870
119879= 119896119894(1198873
119863) 119870119877= 119870119894(119887119863) 119870
119879= 100 and 119870
119877= 100)
simply supported 119908 = 0119872119909119909= 0 and 120595
119910= 0
1198961199090= 1014
1198701199090= 0 119870
1199101199090= 1014
(17b)
completely clamped 119908 = 0 120595119909= 0 and 120595
119910= 0
1198961199090= 1014
1198701199090= 1014
1198701199101199090
= 1014
(17c)
arbitrarily elastic boundary 119908 = 0 120595119909
= 0 120595119910
= 0119876119909
= 0119872119909119909
= 0 and119872119909119910
= 0
1198961199090= Γ1199081199090
1198701199090= Γ1199091199090
1198701199101199090
= 1014
(17d)
The appropriateness of the three classical boundariesdefined in (17a)ndash(17c) will be proved by several examplesgiven in following the arbitrary elastic boundaries are alsodefined in (17d) and the Γ
119908119894(Γ119908119894= 1198961198941198960 1198960= 1 times 10
9Nm2119894 = 1199090 1199091198861199100119910119887) and Γ
119909119894(Γ119909119894= 11987011989411987001198700= 1times10
9Nmrad119894 = 1199090 119909119886 1199100 119910119887) elastic restraint parameters representcorresponding spring stiffness For the sake of simplicitya four-letter string is employed to represent the restraintcondition of a plate such that F-C-S-E identifies the platewithedges 119909 = 0 119910 = 0 119909 = 119886 and 119910 = 119887 having free clampedshear-supported restrained and elastic boundary conditionsrespectively
As for the first case a uniform thickness moderately thickplate with different classical boundaries and structure param-eters is investigated here In Table 2 the comparison of thefirst eight frequency parametersΩ = 120596119887
2
(120588ℎ119863)12
1205872 of the
considered plate is presented The S-S-S-S C-F-F-F S-S-F-FF-F-F-F and S-C-S-C boundary conditions are performed inthe comparison Excellent agreements are observed betweenthe solutions obtained by the modified Fourier method thereferential data and finite element method (FEM) results forthe uniform thickness moderately thick rectangular platesIt is also verified that the definition of the three types of
classical boundaries in (17a)ndash(17c) is appropriate In additionthe elastic boundary conditions (17d) are also verified Inthe next two examples we will account for the vibrationof moderately thick plate with elastic edge supports Thefirst model considered is an S-S-S-S square moderately thickplate with all edges elastically rotationally restrained Thatis 1198701199090
= 119870119909119886
= 1198701199100
= 119870119910119887
= Γ119909 and all the
other restraining springs are set to have an infinite stiffness(namely represented by 1014 in numerical calculation) Thesix frequency parameters Ω = 120596119887
2
(120588ℎ119863)12
1205872 are given
in Table 3 for several different restraining coefficient valuesthe finite element method (FEM) results calculated usingABAQUSare also listed in this table as a referenceThe secondmodel concerns a complete square moderately thick platewith all edges elastically restrained That is 119896
1199090= 119896119909119886
=
1198961199100
= 119896119910119887
= Γ119908and 119870
1199090= 119870119909119886
= 1198701199100
= 119870119910119887
= Γ119909 The
six frequency parameters Ω = 1205961198872
(120588ℎ119863)12
1205872 are given in
Table 4 for several different restraining coefficient values thefinite element method (FEM) results are also listed in Table 4as a reference It can be clearly seen that the comparison isextremely good which implies that the current method isable tomake correct predictions for themodal characteristicsof moderately thick plate with not only classical boundaryconditions but also elastic edge restraints
The excellent agreements between the solution obtainedby the modified Fourier method and the referential datafor the moderately thick plate subjected to the combina-tions of classical boundary conditions and elastic boundaryconditions given in Tables 2ndash4 indicate that the proposedmethod is sufficiently accurate to deal with uniform thicknessmoderately thick plate with arbitrary boundary conditions
33 Linearly VariationThickness Moderately Thick Plates withClassical and Elastic Boundary Conditions In the theoreticalformulations this paper concerns the varying thicknessmoderately thick plates with classical and elastic boundary
10 Shock and Vibration
Table 2 Frequency parameters Ω = 1205961198872
(120588ℎ119863)12
1205872 for uniform thickness moderately thick plates with different classical boundaries and
structure parameters
Boundary condition ab Model sequence1 2 3 4 5 6 7 8
S-S-S-S
11931 4605 4605 7064 8605 8605 10792 10792
(1933a) (4611) (4611) (7067) (8607) (8607) (10799) (10780)(1931b) (4605) (4605) (7064) (8605) (8605) (10792) (10792)
321408 2648 4127 4605 5261 7064 7153 8184(1410a) (2651) (4128) (4608) (5267) (7074) (7155) (8188)(1408b) (2648) (4127) (4605) (5261) (7064) (7151) (8184)
C-F-F-F
10348 0817 2035 2583 2862 4816 5478 5774
(0348a) (0817) (2038) (2586) (2865) (4821) (5504) (5793)(0348b) (0816) (2034) (2582) (2860) (4811) (5477) (5772)
320155 0502 0941 1665 2292 2613 3303 3498
(0154a) (0502) (0941) (1663) (2295) (2618) (3305) (3500)(0154b) (0501) (0940) (1662) (2292) (2612) (3298) (3494)
S-S-F-F
10333 1678 1875 3559 4720 4946 6474 6632
(0335a) (1680) (1878) (3600) (4721) (4949) (6478) (6642)(0333b) (1677) (1874) (3557) (4718) (4945) (6472) (6631)
320222 0938 1631 2363 2573 4082 4482 4753
(0223a) (0940) (1632) (2364) (2575) (4092) (4483) (4759)(0222b) (0938) (1631) (2361) (2571) (4077) (4479) (4752)
F-F-F-F1 1290 1919 2363 3235 3235 5605 5605 5646
(1291a) (1920) (2365) (3239) (3239) (5607) (5607) (6648)
32 0865 0948 1961 2166 2461 2843 3551 4026(0865a) (0949) (1963) (2168) (2465) (2844) (3556) (4034)
S-C-S-C1 2700 4971 5990 7973 8787 10250 11334 12024
(2701a) (4971) (5993) (7975) (8788) (10253) (11335) (12028)
32 2348 3253 4972 5646 6489 7381 7974 9930(2348a) (3255) (4974) (5647) (6590) (7384) (7801) (9932)
aResults in parentheses are taken from FEMbResults in parentheses are taken from [11]
conditions The varying thickness function ℎ(119909 119910) can beexpressed as ℎ
0(1 + 120572119909
119904
)(1 + 120573119910119905
) in which the ℎ0 120572 and
120573 represent the initial thickness gradient in 119909 direction andgradient in 119910 direction When the indexes 119904 and 119905 take thevalue 119904 = 119905 = 1 the analyticalmodel imitates the linearly vari-ation thickness moderately thick plates structure In order tounify the description and facilitate the analytical calculationsof the involved integrals all the thickness variation functionscan be expanded into either 1D or 2D Fourier cosine seriesresulting in
ℎ (119909 119910) =
infin
sum
119898=0
infin
sum
119899=0
ℎ0119902119898119899
cos 119898120587119909119886
cos119898120587119910
119887
(18)
where119902119898119899
=
4
119886119887
int
119887
0
int
119886
0
(1 + 120572119909119904
) (1 + 120573119910119905
) cos 119898120587119909119886
cos119898120587119910
119887
119889119909 119889119910
(19)
In order to prove the validity of the present methodfor the vibration of linearly variation thickness moderately
thick plates with arbitrary boundary conditions the typicalclassical boundary conditions as the first case will be con-sidered In Table 5 the comparison of the first six frequencyparameters Ω = 120596119886
2
(120588ℎ01198630)12 of the moderately thick
plates with linearly varying thickness is presentedThe S-S-S-S C-F-F-F S-S-F-F C-C-C-C and S-C-S-C boundary condi-tions are performed in the comparison Excellent agreementsare observed between the solutions obtained by the modifiedFourier method and finite element method (FEM) results forthemoderately thick plates with linear variation thickness Toinvestigate the influence of the aspect ratio on the uniformthickness and nonuniform thickness moderately thick platesthe effect on the frequency parameters for plates with S-S-S-Sboundary conditions is presented in Figure 6 The thicknessfunctions are ℎ
0and ℎ0(1+05times119909)(1+05times119910) respectively It
is seen from Figure 6 that the influence of aspect ratios on thefrequency parameters for nonuniform thickness moderatelythick plates is more complicated
In the next two examples we also account for the vibra-tions of moderately thick plate with linear variation thicknessand elastic edge supports The first model considered is an S-S-S-S square moderately thick plate with all edges elastically
Shock and Vibration 11
Table 3 Frequency parameters Ω = 1205961198872
(120588ℎ119863)12
1205872 for S-S-S-S moderately thick plates (119886119887 = 1) with uniform thickness and elastic
rotation support (1198701199090= 119870119909119886= 1198701199100= 119870119910119887= Γ119909)
Γ119909
aℎ0119886
Model sequence1 2 3 4 5 6
0
005 1945 4834 4834 7591 9515 9519(1942b) (4836) (4836) (7584) (9547) (9552)
01 1856 4500 4500 6829 8499 8513(1858b) (4510) (4510) (6843) (8539) (8554)
015 1758 4114 4114 6040 7421 7445(1761b) (4126) (4126) (6060) (7462) (7486)
02 1656 3733 3733 5335 6468 6499(1660b) (3747) (3747) (5357) (6508) (6538)
05
005 3541 7071 7071 10227 12318 12399(3546b) (7091) (7091) (10253) (12386) (12468)
01 3274 6242 6242 8719 10302 10421(3282b) (6267) (6267) (8754) (10370) (10488)
015 2947 5361 5361 7290 8465 8606(2956b) (5387) (5387) (7327) (8524) (8666)
02 2624 4599 4599 6148 7057 7199(2634b) (4624) (4624) (6182) (7107) (7250)
1
005 3541 7072 7072 10228 12320 12401(3547b) (7093) (7093) (10254) (12388) (12470)
01 3278 6248 6248 8727 10311 10429(3286b) (6274) (6274) (8762) (10378) (10497)
015 2957 5374 5374 7305 8478 8620(2966b) (5400) (5400) (7341) (8537) (8680)
02 2641 4617 4617 6166 7070 7215(2651b) (4643) (4643) (6200) (7121) (7268)
15
005 3542 7073 7073 10229 12320 12401(3547b) (7093) (7093) (10254) (12388) (12471)
01 3280 6251 6251 8729 10313 10432(3287b) (6276) (6276) (8765) (10381) (10500)
015 2960 5379 5379 7309 8482 8625(2969b) (5405) (5405) (7346) (8541) (8685)
02 2647 4623 4623 6172 7074 7221(2657b) (4649) (4649) (6206) (7125) (7274)
2
005 3542 7073 7073 10229 12321 12402(3547b) (7093) (7093) (10255) (12389) (12471)
01 3280 6252 6252 8730 10315 10434(3288b) (6277) (6277) (8766) (10382) (10501)
015 2962 5381 5381 7312 8484 8628(2971b) (5407) (5407) (7348) (8543) (8688)
02 2650 4627 4627 6175 7076 7224(2660b) (4652) (4652) (6210) (7128 (7277)
aΓ119909= 1198701198941198700 (1198700= 1 times 10
9 Nmrad 119894 = 1199090 119909119886 1199100 119910119887)bResults in parentheses are taken from FEM
12 Shock and Vibration
Table 4 Frequency parameters Ω = 1205961198872
(120588ℎ119863)12
1205872 for moderately thick plates (119886119887 = 1) with uniform thickness and all edge elastic
restraints (1198701199090= 119870119909119886= 1198701199100= 119870119910119887= Γ119909and 119896
1199090= 119896119909119886= 1198961199100= 119896119910119887= Γ119908)
Γ119909
aΓ119908
aℎ0119886
Model sequence1 2 3 4 5 6
05 0
005 0986 1967 3893 3893 4850 4850(0988b) (1971) (3897) (3897) (4852) (4852)
01 0916 1835 3515 3522 4375 4375(0917b) (1837) (3519) (3528) (4341) (4341)
015 0789 1635 2983 3027 3771 3771(0792b) (1638) (2988) (3029) (3775) (3775)
02 0646 1444 2497 2601 3249 3249(0652b) (1448) (2508) (2621) (3254) (3254)
1 05
005 3417 3417 4431 5294 5461 6448(3419b) (3419) (4435) (5303) (5478) (6456)
01 1593 1593 2401 3849 3861 4729(1611b) (1611) (2408) (3852) (3866) (4767)
015 1122 1122 1913 3272 3290 4038(1127b) (1227) (1924) (3284) (3314) (4069)
02 0890 0890 1640 2781 2829 3476(0899b) (0899) (1654) (2787) (2834) (3497)
15 1
005 4325 4325 5619 6273 6571 7619(4334b) (4334) (5681) (6299) (6642) (7642)
01 2013 2013 2812 4088 4116 4993(2024b) (2024) (2827) (4122) (4187) (4994)
015 1338 1338 2108 3430 3443 4196(1342b) (1342) (2112) (3467) (3482) (4200)
02 1038 1038 1772 2936 2965 3613(1042b) (1042) (1786) (2937) (2977) (3624)
2 2
005 5231 5231 6919 7630 8002 9220(5238b) (5238) (6931) (7637) (8011) (9241)
01 2594 2594 3439 4470 4544 5428(2597b) (2597) (3442) (4495) (4557) (5438)
015 1653 1653 2403 3613 3632 4392(1662b) (1662) (2414) (3627) (3646) (4404)
02 1238 1238 1949 3077 3099 3751(1244b) (1244) (1964) (3094) (3116) (3788)
aΓ119909= 1198701198941198700 (1198700= 1 times 10
9 Nmrad 119894 = 1199090 119909119886 1199100 119910119887) Γ119908= 1198961198941198960 (1198960= 1 times 10
9 Nm 119894 = 1199090 119909119886 1199100 119910119887)bResults in parentheses are taken from FEM
05 10 15 20 25 300
2
4
6
8
10
12
14
16
ab
The 1st orderThe 2nd orderThe 3rd order
The 4th orderThe 5th orderThe 6th order
Ω
(a)
05 10 15 20 25 30
30
60
90
120
150
180
ab
The 1st orderThe 2nd orderThe 3rd order
The 4th orderThe 5th orderThe 6th order
Ω
(b)
Figure 6 The effect of aspect ratio 119886119887 on the natural frequenciesΩ for S-S-S-S boundary condition (a) uniform thickness and (b) nonuni-form thickness
Shock and Vibration 13
Table 5 Frequency parameter Ω = 1205961198862
(120588ℎ01198630)12 for moderately thick plates with linearly varying thickness in different boundary
conditions (ℎ(119909 119910) = ℎ0(1 + 05119909)(1 + 05119910))
Boundary condition 119886119887 ℎ0119886
Model sequence1 2 3 4 5 6
S-S-S-S
101 27261 62397 62926 92423 111537 111732
(27263a) (62410) (63842) (92445) (111867) (111921)
02 23103 47010 47191 65018 75831 75865(23489a) (47209) (47349) (65057) (76605) (76473)
3201 59338 99101 138798 151102 166710 204855
(59874a) (99249) (138177) (151362) (166644) (205381)
02 41175 63765 63765 73598 83956 89084(41586a) (63313) (63669) (73661) (83918) (89547)
C-F-F-F
101 4173 10331 27407 37826 38809 64095
(4856a) (10680) (27513) (37203) (39135) (64858)
02 3983 8702 21403 29443 30286 46705(4118a) (9306) (21990) (29670) (30602) (47156)
3201 4868 15488 35620 63523 89969 100089
(4887a) (15320) (35175) (63353) (89159) (100783)
02 4469 9763 22338 42188) 50104 56331(4944a) (9636) (22087) (42177) (50580) (56559)
S-S-F-F
101 4117 24795 26216 48567) 68195 69667
(4789a) (24149) (26751) (49324) (68842) (70575)
02 3820 20516 21924 37523 48980 50327(4103a) (20041) (22125) (37221) (48894) (50622)
3201 7935 39454 65063 92877 92877 136176
(7856a) (40145) (65162) (92462) (93022) (136336)
02 6389 26149 35627 49539 56844 65170(6406a) (26536) (35626) (49831) (57144) (65981)
C-C-C-C
101 44471 80299 80681 109422 126871 128207
(45102a) (80271) (80656) (110015) (126328) (128888)
02 32162 52781 52933 69410 78474 79482(32240a) (52629) (53166) (70063) (79059) (80266)
3201 82975 115587 151992 161579 176601 212330
(83139a) (115838) (152200) (162193) (177058) (212849)
02 48577 66022 83468 90830 97474 98157(49187a) (66259) (84065) (91749) (97827) (98777)
S-C-S-C
101 36847 66664 76965 101045 113594 125551
(37901a) (68568) (78356) (101200) (113978) (125651)
02 27673 48469 51220 67027 76358 78241(27897a) (48685) (51491) (67447) (77530) (79573)
3201 77378 106848 150079 154569 173120 208391
(78003a) (107058) (150991) (155171) (173099) (208402)
02 45895 64732 73393 83282 90325 97089(45803a) (64131) 73656 83801 90518 97015
aResults in parentheses are taken from FEM
restrained That is 1198701199090
= 119870119909119886
= 1198701199100
= 119870119910119887
= Γ119909 The six
frequency parameters Ω = 1205961198862
(120588ℎ01198630)12 are given in
Table 6 for several different restraining coefficient valuesthe finite element method (FEM) results calculated usingABAQUS are also presented as a referenceThe secondmodelis a complete square moderately thick plate with all edgeselastically restrainedThat is 119896
1199090= 119896119909119886= 1198961199100= 119896119910119887= Γ119908and
1198701199090= 119870119909119886
= 1198701199100= 119870119910119887= Γ119909 The six frequency parameters
Ω = 1205961198862
(120588ℎ01198630)12 are given in Table 7 for several different
restraining coefficient values the finite element method(FEM) results are also listed as a reference It can be clearlyseen that the comparison is extremely good which impliesthat the current method is able to make correct predictionsfor the modal characteristics of linearly varying thickness
14 Shock and Vibration
Table 6 Frequency parameter Ω = 1205961198862
(120588ℎ01198630)12 for S-S-S-S moderately thick plates (119886119887 = 1) with linearly varying thickness and elastic
rotation support (ℎ(119909 119910) = ℎ0(1 + 05119909)(1 + 05119910) 119870
1199090= 119870119909119886= 1198701199100= 119870119910119887= Γ119909)
Γ119909
aℎ0119886
Model sequence1 2 3 4 5 6
0
005 28809 70080 70891 108967 134884 135295(28897b) (70334) (71129) (109112) (135254) (135772)
01 27261 62398 62926 92423 111540 111735(26606b) (61784) (62303) (91193) (111323) (111648)
015 25231 54192 54505 77118 91228 91312(25204b) (55297) (55595) (77503) (91871) (91941)
02 23103 47010 47191 65018 75832 75866(23947b) (47071) (47208) (65404) (76531) (76589)
05
005 23103 47010 47191 65018 75832 75866(23210b) (47245) (47358) (65241) (76122) (76345)
01 48377 94650 95597 135484 162635 163501(48524b) (94891) (95992) (135845) (167012) (164024)
015 36239 70939 71298 99997 118584 119009(36647b) (71214) (71687) (100024) (119114) (119457)
02 28749 56937 57037 79214 93009 93162(29142b) (57034) (57544) (79654) (93668) (93754)
1
005 49651 96759 97619 138089 165368 166304(49755b) (97245) (97964) (138544) (165864) (166878)
01 38951 73735 74251 102748 121079 121709(39021b) (74145) (74664) (102874) (121613) (121817)
015 30630 58406 58533 80432 93984 94255(30638b) (58684) (58644) (80874) (94121) (94478)
02 25548 48636 48674 66165 76708 76824(25687b) (48788) (48992) (66544) (76824) (76944)
15
005 50118 97561 98374 139096 166421 167382(50247b) (97877) (98661) (139247) (166875) (167662)
01 40310 75216 75783 104229 122402 123164(40366b) (75542) (75921) (104557) (122874) (123385)
015 31837 59352 59530 81245 94611 94983(31902b) (59451) (59871) (81375) (94784) (95105)
02 26302 49119 49156 66528 76954 77130(26418b) (49354) (49334) (66921) (77010) (77470)
2
005 50360 97983 98769 139631 166979 167954(50472b) (98005) (99114) (140154) (167188) (167986)
01 41130 76146 76724 105165 123229 124082(41421b) (76642) (78114) (105661) (123278) (124974)
015 32687 60025 60245 81832 95052 95508(32778b) (60244) (60549) (82146) (95114) (95842)
02 26893 49492 49539 66815 77138 77370(26922b) (49874) (49924) (66987) (77367) (77588)
aΓ119909= 1198701198941198700 (1198700= 1 times 10
9 Nmrad 119894 = 1199090 119909119886 1199100 119910119887)bResults in parentheses are taken from FEM
Shock and Vibration 15
Table 7 Frequency parameterΩ = 1205961198862
(120588ℎ01198630)12 for moderately thick plates (119886119887 = 1) with linearly varying thickness and elastic rotation
and translation support (ℎ(119909 119910) = ℎ0(1 + 05119909)(1 + 05119910) 119870
1199090= 119870119909119886= 1198701199100= 119870119910119887= Γ119909 and 119896
1199090= 119896119909119886= 1198961199100= 119896119910119887= Γ119908)
Γ119909
aΓ119908
aℎ0119886
Model sequence1 2 3 4 5 6
05 0
005 14751 14917 28232 55251 55397 69512(14780b) (14822) (29252) (55476) (55505) (70904)
01 11811 11910 23857 44462 45037 55453(11818b) (11897) (24711) (44376) (45257) (56511)
015 8487 8514 20011 34783 36335 44970(8500b) (8544) (20738) (34404) (36663) (45625)
02 6073 6073 17431 28679 30805 38010(6096b) (6106) (18039) (28129) (31165) (38311)
1 05
005 21225 32173 32173 44381 64061 65222(20708b) (32097) (32285) (45110) (64182) (65355)
01 8140 16878 16941 28100 48663 49015(7943b) (16838) (16970) (28888) (48700) (49171)
015 4493 11859 11878 22429 38352 39247(4383b) (11867) (11891) (23125) (38177) (39488)
02 2932 8690 8691 18919 31108 32600(3540b) (8791) (8791) (16229) (27520) (27889)
15 1
005 27819 40919 41430 54595 70792 72896(27146b) (41008) (41434) (55256) (70895) (73051)
01 11285 20068 20230 31101 51046 51429(11012b) (20025) (20270) (31849) (51126) (51569)
015 6292 13912 13922 24132 40500 41120(6139b) (13912) (13934) (24808) (40426) (41316)
02 4121 10321 10331 20051 32796 33921(4021b) (10329) (10357) (20635) (32540) (34178)
2 2
005 34615 51972 53088 68255 81344 84510(33806b) (52100) (53157) (68891) (81433) (84761)
01 15416 24659 24829 35615 53879 54494(15045b) (24625) (24888) (36303) (53970) (54626)
015 8740 16420 16481 26353 42417 42946(8529b) (16414) (16499) (26996) (42398) (43114)
02 5761 12127 12127 21434 34286 35192(5622b) (12134) (12148) (21998) (34110) (35415)
aΓ119909= 1198701198941198700 (1198700= 1 times 10
9 Nmrad 119894 = 1199090 119909119886 1199100 119910119887) Γ119908= 1198961198941198960 (1198960= 1 times 10
9 Nm 119894 = 1199090 119909119886 1199100 119910119887)bResults in parentheses are taken from FEM
moderately thick plate with not only classical boundaryconditions but also elastic edge restraints
As the last case of this subsection the influence of thegradient 120572 and 120573 on the fundamental frequency parametersfor a linearly varying thickness moderately thick plate isinvestigated The model is a square moderately thick platewith all edges elastically restrainedThat is 119896
1199090= 119896119909119886= 1198961199100=
119896119910119887= Γ119908= 2 and 119870
1199090= 119870119909119886
= 1198701199100= 119870119910119887= Γ119909= 2 The six
frequency parameters Ω = 1205961198862
(120588ℎ01198630)12 are given in
Table 8 for several different 120572 and 120573 values the finite elementmethod (FEM) results are also listed as a reference Againgood agreement can be observed Through Table 8 it is alsofound that the frequency parameter increases with increasinggradient parameters
The above studies are given as linearly varying thicknessmoderately thick plates with several of boundary condition
and different structure parameters In the next section vibra-tion results for the plates subjected to nonlinear variationthickness will be presented
34 Nonlinearly Varying Thickness Moderately Thick Plateswith Classical and Elastic Boundary Conditions In Sec-tion 33 the linearly varying thickness moderately thickplates were studied However in the practical engineeringapplications the varying thickness of amoderately thick platemay not always be linear variation in nature A variety ofpossible thickness varying cases may be encountered in prac-tice Therefore the moderately thick plates with nonlinearvariation thickness subjected to general elastic edge restraintsare examined in this subsection For the sake of brevity theindexes 119904 and 119905 will be chosen as 2 to imitate the nonlinearlyvarying thickness moderately thick plates structure in this
16 Shock and Vibration
Table 8 Frequency parameterΩ = 1205961198862
(120588ℎ01198630)12 for elastic support moderately thick plates (119886119887 = 1) with different gradient (119896
1199090= 119896119909119886=
1198961199100= 119896119910119887= Γ119908= 2 119870
1199090= 119870119909119886= 1198701199100= 119870119910119887= Γ119909= 2)
120572 120573 ℎ0119886
Model sequence1 2 3 4 5 6
120572 = 120573 = 01
005 31275 52615 52657 68954 76146 79850(31303a) (52664) (52675) (69033) (76174) (79948)
01 17270 25244 25250 33949 46050 46625(17271a) (25255) (25258) (34089) (46122) (46708)
015 10190 16266 16274 24164 37327 37526(10189a) (16269) (16280) (24323) (37401) (37622)
02 6780 12258 12260 19685 31452 31754(6780a) (12265) (12270) (19851) (31514) (31864)
120572 = 120573 = 02
005 32530 52947 53141 68954 77034 80589(32577a) (53014) (53203) (69172) (77088) (80773)
01 16882 24979 24979 34154 48057 48575(16883a) (24983) (25001) (34433) (48134) (48667)
015 9804 16284 16309 24683 38844 39085(9803a) (16283) (16322) (24980) (38909) (39191)
02 6499 12290 12295 20133 32354 32770(6498a) (12296) (12307) (20420) (32387) (32895)
120572 = 120573 = 03
005 33516 52845 53308 68704 78203 81561(33563a) (52939) (53389) (69055) (78271) (81789)
01 16419 24799 24830 34527 50061 50583(16419a) (24790) (24869) (34947) (50143) (50688)
015 9430 16334 16378 25236 40193 40506(9429a) (16329) (16395) (25660) (40243) (40625)
02 6235 12288 12293 20572 33102 33663(6235a) (12296) (12306) (20965) (33090) (33811)
120572 = 120573 = 04
005 34215 52463 53264 68430 79654 82865(34252a) (52575) (53346) (68921) (79732) (83113)
01 15921 24695 24793 35027 52011 52571(15921a) (24674) (24842) (35583) (52098) (52688)
015 9074 16388 16444 25797 41379 41789(9074a) (16381) (16464) (26336) (41404) (41928)
02 5990 12236 12238 21002 33735 34461(5989a) (12249) (12250) (21487) (33653) (34641)
120572 = 120573 = 05
005 34615 51972 53088 68255 81344 84510(33806a) (52100) (53157) (68891) (81433) (84761)
01 15416 24659 24829 35615 53879 54494(15045a) (24625) (24888) (36303) (53970) (54626)
015 8740 16420 16481 26353 42417 42946(8529a) (16414) (16499) (26996) (42398) (43114)
02 5761 12127 12127 21434 34286 35192(5622a) (12134) (12148) (21998) (34110) (35415)
aResults in parentheses are taken from FEM
section that is 119904 = 119905 = 2 Also the thickness varying functionsare also be expressed as Fourier cosine series according to (18)and (19)
In order to validate the accuracy and reliability of theproposed method for predicting the vibration behavior ofnonlinearly varying thickness moderately thick plates with
arbitrary boundary conditions the typical classical boundaryconditions viewed as the special cases of elastically restrainededges will be considered The comparison of the first six fre-quency parameters Ω = 120596119886
2
(120588ℎ01198630)12 for the moderately
thick plates with nonlinearly varying thickness is presentedin Table 9 The S-S-S-S C-F-F-F S-S-F-F C-C-C-C and
Shock and Vibration 17
Table 9 Frequency parameter Ω = 1205961198862
(120588ℎ01198630)12 for moderately thick plates with linearly varying thickness in different boundary
conditions (ℎ(119909 119910) = ℎ0(1 + 05119909
2
)(1 + 051199102
))
Boundary condition 119886119887 ℎ0119886
Model sequence1 2 3 4 5 6
S-S-S-S
101 23462 54999 55573 82950 100900 101187
(23741a) (55114) (55983) (83146) (101004) (101347)
02 18606 42725 42971 60040 71167 71761(18936a) (43014) (43148) (60247) (71642) (71993)
3201 53055 65251 85404 106700 114880 143531
(53467a) (65472) (85984) (107211) (115687) (144007)
02 17439 21531 32170 44987 59438 64179(17874a) (21748) (32651) (45264) (60188) (64879)
C-F-F-F
101 3658 9317 24135 32816 35433 59179
(3749a) (9547) (24034) (33627) (36308) (60142)
02 3520 7986 19591 26959 27491 43882(3611a) (8176) (20075) (27125) (27625) (43984)
3201 3901 9963 28984 69187 73686 89360
(3930a) (10095) (29381) (69578) (73994) (90242)
02 3292 4439 13385 23947 40850 44758(3386a) (4498) (14012) (24068) (41221) (45142)
S-S-F-F
101 3557 21268 22762 43524 60253 62429
(3649a) (21441) (22796) (43916) (60622) (62630)
02 3171 17822 19512 33616 45042 46833(3489a) (18006) (20430) (33815) (45247) (49004)
3201 6738 31658 52071 75625 86604 94903
(6957a) (31882) (52518) (75823) (86939) (95610)
02 3653 12930 20955 26517 44097 48058(6807a) (13160) (21312) (26864) (44422) (48873)
C-C-C-C
101 41020 75196 75614 104142 119683 120949
(41405a) (75452) (75882) (104711) (119637) (121137)
02 31237 51686 51868 68411 76868 77864(32010a) (52960) (52145) (69335) (77481) (78099)
3201 95823 128727 163657 173751 190890 223595
(95364a) (128932) (163588) (172350) (190350) (222948)
02 53306 71372 89384 102404 104660 120908(53587a) (71759) (90103) (102503) (105174) (121275)
S-C-S-C
101 33715 60298 71446 94187 103802 117928
(34148a) (60786) (71828) (94515) (104263) (119326)
02 26143 45499 49458 64253 72941 76191(26789a) (46622) (50680) (64329) (73474) (76843)
3201 73228 115845 137445 148166 168342 182542
(73761a) (116121) (137788) (148530) (168698) (182921)
02 41302 53648 65630 72202 81632 88105(41304a) (53757) (65842) (72794) (82033) (88474)
aResults in parentheses are taken from FEM
S-C-S-C boundary conditions are performed in the compar-ison The results adequately demonstrated the great accuracyof the modified Fourier method
We now turn to elastically restrained moderately thickplates The first one involves an S-S-S-S moderately thicksquare plate with a uniform rotational restraint along eachedge that is 119870
1199090= 119870119909119886
= 1198701199100
= 119870119910119887
= Γ119909 The calculated
frequency parameters Ω = 1205961198862
(120588ℎ01198630)12 are given in
Table 10 together with the FEM results Since the elasticallyrestrained plates with nonlinear variation thickness are rarelyinvestigated the FEMresults are used as the referenceA goodagreement is observed between the current and FEM resultsThe second example concerns amoderately thick square plateelastically supported along all edgesThe stiffness of the linear
18 Shock and Vibration
Table 10 Frequency parameterΩ = 1205961198862
(120588ℎ01198630)12 for S-S-S-S moderately thick plates (119886119887 = 1) with linearly varying thickness and elastic
rotation support (ℎ(119909 119910) = ℎ0(1 + 05119909
2
)(1 + 051199102
) 1198701199090= 119870119909119886= 1198701199100= 119870119910119887= Γ119909)
Γ119909
aℎ0119886
Model sequence1 2 3 4 5 6
0
005 23462 54999 55573 82950 100900 101187(23741b) (55114) (55983) (83146) (101004) (101347)
01 18606 42725 42971 60040 71167 71761(18936b) (43014) (43148) (60247) (71642) (71993)
015 53055 65251 85404 106700 114880 143531(53467b) (65472) (85984) (107211) (115687) (144007)
02 17439 21531 32170 44987 59438 64179(17874b) (21748) (32651) (45264) (60188) (64879)
05
005 6904 13374 13514 19400 22904 23056(9909b) (13548) (13651) (19564) (23088) (23379)
01 5316 10237 10314 14499 17155 17264(5374b) (10448) (10651) (14764) (17188) (17879)
015 4136 8201 8220 11474 13578 13671(4174b) (8348) (8651) (11764) (13688) (13879)
02 3449 6915 6924 9549 11225 11327(3474b) (6948) (6951) (9764) (11388) (11579)
1
005 7040 13617 13745 19726 23233 23402(7074b) (13748) (13851) (19964) (23488) (23879)
01 5712 10696 10793 14991 17594 17744(5774b) (10748) (10851) (15054) (17788) (17879)
015 4489 8507 8540 11759 13803 13922(4514b) (8548) (8611) (11864) (13988) (14119)
02 3677 7084 7090 9692 11330 11445(3684b) (7148) (7191) (9764) (11588) (11879)
15
005 7089 13708 13828 19847 23355 23530(7124b) (13748) (13951) (19967) (23588) (23879)
01 5895 10923 11024 15239 17813 17988(5874b) (10748) (11651) (15264) (17818) (17979)
015 4697 8694 8737 11938 13790 14083(4774b) (8748) (8951) (11864) (13688) (14879)
02 3836 7201 7210 9792 11403 11530(3874b) (7748) (7651) (9864) (11588) (11879)
2
005 7114 13754 13871 19911 23418 23597(7174b) (12848) (13851) (19864) (23588) (23879)
01 6001 11060 11159 15391 17944 18136(6004b) (11148) (11151) (15464) (17988) (18379)
015 4836 8821 8872 12061 14037 14189(4874b) (8848) (8951) (12164) (14188) (14879)
02 3954 7289 7301 9871 11457 11594(3974b) (7348) (7651) (9964) (11488) (11679)
aΓ119909= 1198701198941198700 (1198700= 1 times 10
9 Nmrad 119894 = 1199090 119909119886 1199100 119910119887)bResults in parentheses are taken from FEM
and rotational restraints is set equal to 1198961199090
= 119896119909119886
= 1198961199100
=
119896119910119887
= Γ119908and 119870
1199090= 119870119909119886
= 1198701199100
= 119870119910119887
= Γ119909 respectively
The six frequency parametersΩ = 1205961198862
(120588ℎ01198630)12 are shown
in Table 11 for several different restraining coefficient valuesIt can also be noticed that the Fourier series method is able
to predict the modal characteristics of nonlinearly varyingthickness moderately thick plate with not only classicalboundary conditions but also elastic edge restraints correctly
Finally the influence of the gradient on the fundamentalfrequency parameters of a nonlinearly varying thickness
Shock and Vibration 19
Table 11 Frequency parameterΩ = 1205961198862
(120588ℎ01198630)12 for moderately thick plates (119886119887 = 1) with linearly varying thickness and elastic rotation
and translation support (ℎ(119909 119910) = ℎ0(1 + 05119909
2
)(1 + 051199102
) 1198701199090= 119870119909119886= 1198701199100= 119870119910119887= Γ119909 and 119896
1199090= 119896119909119886= 1198961199100= 119896119910119887= Γ119908)
Γ119909
aΓ119908
aℎ0119886
Model sequence1 2 3 4 5 6
05 0
005 12926 13041 25090 47935 47943 61761(12931b) (13053) 25099 (47947) (47952) (61788)
01 10900 10988 21545 39980 40147 50322(10904b) (10992) (21557) (39984) (40166) (50334)
015 8245 8298 18081 31503 32617 40950(8247b) (8230) (18092) (31508) (32622) (40955)
02 6106 6124 15639 25631 27601 34458(6110b) (6125) (15642) (25637) (27617) (34462)
1 05
005 21575 31615 31644 42573 57889 58607(21580b) (31617) (31644) (42581) (57893) (58610)
01 8451 16024 16210 25969 43839 43867(8451b) (16025) (16217) (25972) (43844) (43870)
015 4682 11479 11480 20642 35172 35598(4684b) (11480) (11481) (20644) (35178) (35604)
02 3059 8621 8648 17320 28487 29638(3062b) (8619) (8644) (17327) (28492) (29641)
15 1
005 27713 40203 40978 52951 65352 66815(27714b) (40207) (40988) (52972) (65368) (66822)
01 11667 19322 19615 29122 46115 46183(11674b) (19327) (19619) (29137) (46120) (46188)
015 6544 13416 13481 22405 37238 37430(6547b) (13420) (13488) (22410) (37241) (37434)
02 4295 10175 10190 18543 30330 31056(4296b) (10177) (10194) (18544) (30336) (31060)
2 2
005 33453 50854 52088 66311 76610 78947(33455b) (50852) (50097) (66321) (76627) (78971)
01 15832 24130 24372 33924 49118 49362(15830b) (24127) (24374) (33931) (49147) (49382)
015 9060 15956 16099 24754 39134 39249(9064b) (15961) (16102) (24755) (39138) (39452)
02 5992 11955 11971 20034 31908 32392(5991b) (11957) (11978) (20042) (31910) (32398)
aΓ119909= 1198701198941198700 (1198700= 1 times 10
9 Nmrad 119894 = 1199090 119909119886 1199100 119910119887) Γ119908= 1198961198941198960 (1198960= 1 times 10
9 Nm 119894 = 1199090 119909119886 1199100 119910119887)bResults in parentheses are taken from FEM
moderately thick plate is investigated The plate is elasticallyrestrained in which the stiffness of the boundary springsis taken as 119896
1199090= 119896119909119886
= 1198961199100
= 119896119910119887
= Γ119908
= 2 and1198701199090
= 119870119909119886
= 1198701199100
= 119870119910119887
= Γ119909= 2 respectively The six
frequency parameters Ω = 1205961198862
(120588ℎ01198630)12 are presented in
Table 12 for several different slop values It can be seen thatthe fundamental frequency parameters will decrease with theincrease of the parameter ℎ
0119886
In the above examples it has been demonstrated thatthe presented method can be universally applied to nonlin-early varying thickness moderately thick plates with severalboundary conditions and different structure parametersNew results are obtained for plates with nonlinear variationthickness in both directions subjected to general elastic
boundary restraints which may be used for benchmarkingof researchers in the field In addition it is interesting to seethat the nature frequency decreases with the increase of theindex for thickness function
4 Conclusions
In this paper a modified Fourier method has been pre-sented to study the free vibration behaviors of moderatelythick rectangular plates with variable thickness and arbi-trary boundary conditionsThe first-order shear deformationplate theory is adopted to formulate the theoretical modelThe displacements and rotation components of the plateregardless of boundary conditions are invariantly expressedas the superposition of a 2D Fourier cosine series and four
20 Shock and Vibration
Table 12 Frequency parameterΩ = 1205961198862
(120588ℎ01198630)12 for elastic support moderately thick plates (119886119887 = 1) with different gradient (119896
1199090= 119896119909119886=
1198961199100= 119896119910119887= Γ119908= 2 119870
1199090= 119870119909119886= 1198701199100= 119870119910119887= Γ119909= 2)
120572 120573 ℎ0119886
Model sequence1 2 3 4 5 6
120572 = 120573 = 01
005 30187 51102 51141 67066 73954 77412(30188a) (51105) (51144) (67068) (73961) (77477)
01 16995 24719 24724 33071 44343 44819(16996a) (24725) (24729) (33084) (44347) (44822)
015 10077 15872 15883 23443 35937 36065(10081a) (15877) (15892) (23447) (35941) (36084)
02 6713 11953 11958 19086 30372 30604(6713a) (11957) (11965) (19092) (30378) (30612)
120572 = 120573 = 02
005 31206 51436 51614 67058 74398 77595(31208a) (51438) (51619) (67064) (74402) (77596)
01 16786 24503 24508 33144 45583 45923(16790a) (24510) (24513) (33147) (45587) (45932)
015 9821 15865 15906 23766 36897 36990(9824a) (15870) (15907) (23768) (36894) (36997)
02 6523 11978 11992 19356 30912 31162(6524a) (11981) (11997) (19362) (30916) (31174)
120572 = 120573 = 03
005 32097 51462 51900 66832 74991 77863(32098a) (51466) (51907) (66849) (74996) (77872)
01 16507 24332 24383 33330 46806 47074(16511a) (24327) (24380) (33339) (46810) (47086)
015 9562 15885 15964 24103 37748 37828(9562a) (15892) (15977) (21105) (37751) (37832)
02 6338 11992 12013 19603 31338 31634(6340a) (11994) (12011) (19612) (31344) (31639)
120572 = 120573 = 04
005 32851 51239 52046 66547 75735 78303(33455a) (50852) (50097) (66321) (76627) (78971)
01 16182 24209 24343 33598 47990 48230(33455a) (50852) (50097) (66321) (76627) (78971)
015 9307 15919 16034 24436 38491 38579(33455a) (50852) (50097) (66321) (76627) (78971)
02 6161 11986 12008 19828 31666 32040(33455a) (50852) (50097) (66321) (76627) (78971)
120572 = 120573 = 05
005 33453 50854 52088 66311 76610 78947(33455a) (50852) (50097) (66321) (76627) (78971)
01 15832 24130 24372 33924 49118 49362(15830a) (24127) (24374) (33931) (49147) (49382)
015 9060 15956 16099 24754 39134 39249(9064a) (15961) (16102) (24755) (39138) (39452)
02 5992 11955 11971 20034 31908 32392(5991a) (11957) (11978) (20042) (31910) (32398)
aResults in parentheses are taken from FEM
supplementary functions in the form of the product of apolynomial function and a single cosine series expansionto ensure and accelerate the convergence of the solutionAt each edge of the plate the general restraint conditionsare implemented by introducing one group of linear springsand two groups of rotational springs which are continuouslydistributed and determined by the stiffness of these springs
Instead of seeking a solution in strong forms in the previousstudies all the Fourier coefficients will be treated equallyand independently as the generalized coordinates and solveddirectly from the Rayleigh-Ritz technique The change of theboundary conditions can be easily achieved by only varyingthe stiffness of the three sets of boundary springs along alledges of the rectangular plates without involving any change
Shock and Vibration 21
to the solution procedure The convergence of the presentsolution is examined and the excellent accuracy is validatedby comparison with existing results published in the lit-erature and FEM data Excellent agreements are obtainedfrom these comparisons The proposed method providesa unified means for extracting the modal parameters andpredicting the vibration behaviors of moderately thick plateswith variable thickness variation functions and arbitraryelastic edge restraints A variety of free vibration results formoderately thick rectangular plates with different thicknessvariation functions and boundary conditions are presentedNew results for free vibration of moderately thick rectangularplates with various thickness variation functions and edgeconditions are presented which may be used for benchmark-ing of researchers in the field
Appendices
A Representative Calculation forStiffness and Mass Matrices
To illuminate the particular expression of the mass andstiffness matrixes clearly and tersely four new variables aredefined as 119904 = 119898 lowast (119873 + 1) + 119899 + 1 119905 = 119898
1015840
lowast (119873 + 1) + 1198991015840
+ 1119901 = (119897minus1)lowast (119872+1)+119898
1015840
+1 and 119902 = (119897minus1)lowast (119873+1)+1198991015840
+1The first row elements of K andM are given below
K1-1119904119905
=
1
2
int
119886
0
int
119887
0
[119863 (119909 119910)
times 21205821198861198981205821198861198981015840 sin 120582
119886119898119909 sin 120582
1198861198981015840119909 cos 120582
119887119899119910
times cos 1205821198871198991015840119910 + (1 minus 120583) 120582
1198871198991205821198871198991015840 cos 120582
119886119898119909
times cos 1205821198861198981015840 sin 120582
119887119899119910 sin 120582
1198871198991015840119910
+ 120581119866ℎ (119909 119910) cos 120582119886119898119909 cos 120582
1198861198981015840119909 cos 120582
119887119899
times119910 cos 1205821198871198991015840119910119889119909119889119910] 119889119909 119889119910
+ int
119886
0
1198701199091199100
cos 120582119886119898119909 cos 120582
1198861198981015840119909
+(minus1)119898+1198981015840
119870119909119910119887
cos 120582119886119898119909 cos 120582
1198861198981015840119909 119889119909
+ int
119887
0
1198701199090cos 120582119887119899119910 cos 120582
1198871198991015840119910
+(minus1)119899+1198991015840
119870119909119886cos 120582119887119899119910 cos 120582
1198871198991015840119910 119889119910
K1-2119904119901
=
1
2
int
119886
0
int
119887
0
[119863 (119909 119910)21205821198861198981205821198861198981015840 sin 120582
119886119898119909 sin 120582
1198861198981015840119909
times
2
sum
119897=1
120577119897
119887(119910) cos 120582
1198871198991015840119910
minus (1 minus 120583) 1205821198871198991015840 cos 120582
119886119898119909 cos 120582
1198861198981015840119909
times
2
sum
119897=1
1205771198971015840
119887(119910) sin 120582
1198871198991015840119910 + 120581119866ℎ (119909 119910)
times ( cos 1205821198861198981015840119909 cos 120582
1198871198991015840119910
times
2
sum
119897=1
120577119897
119887(119910) cos 120582
119886119898119909)]119889119909119889119910
+ int
119886
0
1198701199091199100
cos 120582119886119898119909 cos 120582
1198861198981015840119909
2
sum
119897=1
120577119897
119887(0)
+(minus1)1198991015840
119870119909119910119887
cos 120582119886119898119909 cos 120582
1198861198981015840119909
2
sum
119897=1
120577119897
119887(119887) 119889119909
+ int
119887
0
1198701199090
2
sum
119897=1
120577119897
119887(119910) cos 120582
1198871198991015840119910
+(minus1)119898+1198981015840
119870119909119886
2
sum
119897=1
120577119897
119887(119910) cos 120582
1198871198991015840119910119889119910
K1-3119904119902
=
1
2
int
119886
0
int
119887
0
[119863 (119909 119910) minus 21205821198861198981015840 sin 120582
1198861198981015840119909 cos 120582
1198871198991015840119910
times
2
sum
119897=1
1205771198971015840
119886(119909) cos 120582
119887119899119910
+ (1 minus 120583) 1205821198871198991205821198871198991015840 cos 120582
1198861198981015840119909
times sin 1205821198871198991015840119910
2
sum
119897=1
120577119897
119886(119909) sin 120582
119887119899119910
+ 120581119866ℎ (119909 119910) cos 1205821198861198981015840119909 cos 120582
1198871198991015840119910
times
2
sum
119897=1
120577119897
119886(119909) cos 120582
119887119899119910]119889119909119889119910
+ int
119886
0
1198701199091199100
cos 1205821198861198981015840119909
2
sum
119897=1
120577119897
119886(119909)
+(minus1)119899+1198991015840
119870119909119910119887
cos 1205821198861198981015840119909
2
sum
119897=1
120577119897
119886(119909) 119889119909
+ int
119887
0
1198701199090
2
sum
119897=1
120577119897
119886(0) cos 120582
1198871198991015840119910
+(minus1)119898+1198981015840
119870119909119886
2
sum
119897=1
120577119897
119886(0) cos 120582
1198871198991015840119910119889119910
K1-4119904119905
=
1
2
int
119886
0
int
119887
0
119863(119909 119910) 21205821198871198991205821198861198981015840 sin 120582
1198861198981015840119909 cos 120582
1198871198991015840119910
times cos 120582119886119898119909 sin 120582
119887119899119910 minus 2(1 minus 120583)
119887119899
times 1205821198861198981015840 sin 120582
1198861198981015840119909 cos 120582
1198871198991015840119910 cos 120582
119886119898119909
times sin 120582119887119899119910 + (1 minus 120583) 120582
1198861198981205821198871198991015840 cos 120582
1198861198981015840119909
times sin 1205821198871198991015840119910 sin 120582
119886119898119909 cos 120582
119887119899119910 119889119909 119889119910
22 Shock and Vibration
K1-5119904119901
=
1
2
int
119886
0
int
119887
0
119863(119909 119910) minus 21205821198861198981015840 sin 120582
1198861198981015840119909 cos 120582
1198871198991015840119910
times
2
sum
119897=1
1205771198971015840
119887(119910) cos 120582
119886119898119909
+ 2 (1 minus 120583) 1205821198861198981015840 sin 120582
1198861198981015840119909 cos 120582
1198871198991015840119910
times
2
sum
119897=1
1205771198971015840
119887(119910) cos 120582
119886119898119909
+ (1 minus 120583) 1205821198861198981205821198871198991015840 cos 120582
1198861198981015840119909 sin 120582
1198871198991015840119910
times
2
sum
119897=1
120577119897
119887(119910) sin 120582
119886119898119909119889119909119889119910
K1-6119904119902
=
1
2
int
119886
0
int
119887
0
119863(119909 119910)1205821198871198991205821198861198981015840 sin 120582
1198861198981015840119909 cos 120582
1198861198991015840119910
times
2
sum
119897=1
120577119897
119886(119909) sin 120582
1198871198991015840119910
minus 2 (1 minus 120583) 1205821198871198991205821198861198981015840 sin 120582
1198861198981015840119909 cos 120582
1198861198991015840119910
times
2
sum
119897=1
120577119897
119886(119909) sin 120582
1198871198991015840119910
+ 1205821198871198991015840 cos 120582
1198861198981015840119909 sin 120582
1198871198991015840119910
times
2
sum
119897=1
1205771198971015840
119886(119909) cos 120582
119887119899119910119889119909119889119910
K1-7119904119905
= minus
1
2
int
119886
0
int
119887
0
120581119866ℎ (119909 119910) 120582119886119898
cos 1205821198861198981015840119909 cos 120582
1198871198991015840119910
times sin 120582119886119898119909 cos 120582
119887119899119910119889119909119889119910
K1-8119904119901
= minus
1
2
int
119886
0
int
119887
0
120581119866ℎ (119909 119910) 1205821198861198981015840 cos 120582
1198861198981015840119909 cos 120582
1198871198991015840119910
times
2
sum
119897=1
120577119897
119887(119910) sin 120582
1198861198981015840119909 119889119909 119889119910
K1-9119904119902
=
1
2
int
119886
0
int
119887
0
120581119866ℎ (119909 119910) cos 1205821198861198981015840119909 cos 120582
1198871198991015840119910
times
2
sum
119897=1
1205771198971015840
119886(119909) cos 120582
119887119899119910119889119909119889119910
M1-1119904119905
=
1
24
120588int
119886
0
int
119887
0
ℎ3
(119909 119910) cos 120582119886119898119909 cos 120582
119887119899119910
times cos 1205821198861198981015840119909 cos 120582
1198871198991015840119910119889119909119889119910
M1-2119904119901
=
1
24
120588int
119886
0
int
119887
0
ℎ3
(119909 119910)
2
sum
119897=1
120577119897
119887(119910) cos 120582
119886119898119909
times cos 1205821198861198981015840119909 cos 120582
1198871198991015840119910119889119909119889119910
M1-3119904119902
=
1
24
120588int
119886
0
int
119887
0
ℎ3
(119909 119910)
2
sum
119897=1
120577119897
119886(119909) cos 120582
119887119899119910
times cos 1205821198861198981015840119909 cos 120582
1198871198991015840119910119889119909119889119910
M1-4119904119905 = M
1-5119904119901 = M1-6119904119902
= M1-7119904119905 = M
1-8119904119901 = M1-9119904119902 = 0
(A1)
B Useful Integrals for the Moderately ThickRectangular Plates with Variable Thicknessand Arbitrary Boundary Conditions
Consider
int
119871
0
cos 120582119898119909 cos 120582
119899119909 119889119909 =
0 119898 = 119899
119871
2
119898 = 119899
119871 119898 = 119899 = 0
int
119871
0
sin 120582119898119909 cos 120582
119899119909119889119909
=
minus
(minus1 + (minus1)119898+119899
) 119871119898
(1198982minus 1198992) 120587
119898 = 119899
0 119898 = 119899
int
119871
0
sin 120582119898119909 sin 120582
119899119909 119889119909 =
0 119898 = 119899
119871
2
119898 = 119899
0 119898 = 119899 = 0
int
119871
0
cos 120582119898119909120577119897
119886(119909) 119889119909
=
41198712
31205872 119897 = 1 119898 = 0
minus
41198712
(41198982
minus 3)
(9 minus 401198982+ 16119898
4) 1205872
119897 = 1 119898 = 0
minus
41198712
31205872 119897 = 2 119898 = 0
41198712
(41198982
minus 3) (minus1)119898
(9 minus 401198982+ 16119898
4) 1205872
119897 = 2 119898 = 0
Shock and Vibration 23
int
119871
0
sin 120582119898119909120577119897
119886(119909) 119889119909
=
0 119897 = 1 119898 = 0
161198712
119898(minus1)119898
(9 minus 401198982+ 16119898
4) 1205872
119897 = 1 119898 = 0
0 119897 = 2 119898 = 0
161198981198712
(9 minus 401198982+ 16119898
4) 1205872
119897 = 2 119898 = 0
int
119871
0
cos 1205821198981199091205771198971015840
119886(119909) 119889119909
=
0 119897 = 1 119898 = 0
161198711198982
(minus1)119898
(9 minus 401198982+ 16119898
4) 120587
119897 = 1 119898 = 0
0 119897 = 2 119898 = 0
161198711198982
(9 minus 401198982+ 16119898
4) 120587
119897 = 2 119898 = 0
int
119871
0
sin 1205821198981199091205771198971015840
119886(119909) 119889119909
=
0 119897 = 1 119898 = 0
4119871119898 (minus3 + 41198982
)
(9 minus 401198982+ 16119898
4) 120587
119897 = 1 119898 = 0
0 119897 = 2 119898 = 0
119871119898(
1
120587 minus 41198982120587
+
3
9120587 minus 41198982120587
) (minus1)119898
119897 = 2 119898 = 0
int
119871
0
1205771198971
119886(119909) 1205771198972
119886(119909) 119889119909 =
1198713
41205872 1198971= 1 1198972= 1
minus
21198713
31205873 1198971= 1 1198972= 2
minus
21198713
31205873 1198971= 2 1198972= 1
1198713
41205872 1198971= 2 1198972= 2
int
119871
0
1205771198971
119886(119909) 1205771198971015840
2
119886(119909) 119889119909 =
0 1198971= 1 1198972= 1
minus
1198712
8120587
1198971= 1 1198972= 2
1198712
8120587
1198971= 2 1198972= 1
0 1198971= 2 1198972= 2
int
119871
0
1205771198971015840
1
119886(119909) 1205771198971015840
2
119886(119909) 119889119909 =
5119871
16
1198971= 1 1198972= 1
minus
119871
2120587
1198971= 1 1198972= 2
minus
119871
2120587
1198971= 2 1198972= 1
5119871
16
1198971= 2 1198972= 2
int
119871
0
119909119899 cos 120582
119898119909 119889119909
=
1198711+119899
1 + 119899
119898 = 0
(1198861+119899HypergeometricPFQtimes [ 12 + 1198992 12 32 + 1198992
minus (14)1198982
1205872
]) times (1 + 119899)minus1
119898 gt 0
(B1)
Conflict of Interests
The authors declare that there is no conflict of interestsregarding the publication of this paper
Acknowledgments
The authors would like to thank the anonymous reviewersfor their very valuable comments The authors also gratefullyacknowledge the financial support from the National NaturalScience Foundation of China (no 51209052) HeilongjiangProvince Youth Science Fund Project (no QC2011C013) andHarbin Science and Technology Development InnovationFoundation of Youth (no 2011RFQXG021)
References
[1] K Liew Y Xiang and S Kitipornchai ldquoResearch on thick platevibration a literature surveyrdquo Journal of Sound and Vibrationvol 180 no 1 pp 163ndash176 1995
[2] J H Chung T Y Chung and K C Kim ldquoVibration analysisof orthotropic mindlin plates with edges elastically restrainedagainst rotationrdquo Journal of Sound and Vibration vol 163 no 1pp 151ndash163 1993
[3] Y K Cheung and D Zhou ldquoVibrations of moderately thickrectangular plates in terms of a set of static Timoshenko beamfunctionsrdquoComputers and Structures vol 78 no 6 pp 757ndash7682000
[4] CMWang ldquoNatural frequencies formula for simply supportedMindlin platesrdquo Journal of Vibration and Acoustics vol 116 no4 pp 536ndash540 1994
[5] K N Saha R C Kar and P K Datta ldquoFree vibration analysisof rectangular Mindlin plates with elastic restraints uniformlydistributed along the edgesrdquo Journal of Sound andVibration vol192 no 4 pp 885ndash902 1996
24 Shock and Vibration
[6] D J Gorman ldquoAccurate free vibration analysis of shear-deformable plates with torsional elastic edge supportrdquo Journalof Sound and Vibration vol 203 no 2 pp 209ndash218 1997
[7] D J Gorman ldquoFree vibration analysis of Mindlin plates withuniform elastic edge support by the superposition methodrdquoJournal of Sound andVibration vol 207 no 3 pp 335ndash350 1997
[8] D J Gorman ldquoAccurate free vibration analysis of point sup-ported mindlin plates by the superposition methodrdquo Journal ofSound and Vibration vol 219 no 2 pp 265ndash277 1999
[9] Y Xiang ldquoVibration of rectangular Mindlin plates resting onnon-homogenous elastic foundationsrdquo International Journal ofMechanical Sciences vol 45 no 6-7 pp 1229ndash1244 2003
[10] Y Xiang and G W Wei ldquoExact solutions for buckling andvibration of stepped rectangular Mindlin platesrdquo InternationalJournal of Solids and Structures vol 41 no 1 pp 279ndash294 2004
[11] Y L Yeh M J Jang and C C Wang ldquoAnalyzing the freevibrations of a plate using finite difference and differential trans-formationmethodrdquoAppliedMathematics andComputation vol178 no 2 pp 493ndash501 2006
[12] Y Xiang S K Lai L Zhou and C W Lim ldquoDSC-Ritz elementmethod for vibration analysis of rectangular Mindlin plateswith mixed edge supportsrdquo European Journal of MechanicsmdashASolids vol 29 no 4 pp 619ndash628 2010
[13] H Nguyen-Xuan C H Thai and T Nguyen-Thoi ldquoIsogeo-metric finite element analysis of composite sandwich platesusing a higher order shear deformation theoryrdquo Composites BEngineering vol 55 pp 558ndash574 2013
[14] C H Thai A J M Ferreira E Carrera and H Nguyen-XuanldquoIsogeometric analysis of laminated composite and sandwichplates using a layerwise deformation theoryrdquo Composite Struc-tures vol 104 pp 196ndash214 2013
[15] C H Thai A Ferreira S Bordas T Rabczuk and HNguyen-Xuan ldquoIsogeometric analysis of laminated compositeand sandwich plates using a new inverse trigonometric sheardeformation theoryrdquo European Journal of Mechanics A Solidsvol 43 pp 89ndash108 2014
[16] H Luong-Van T Nguyen-Thoi G R Liu and P Phung-VanldquoA cell-based smoothed finite elementmethod using three-nodeshear-locking free Mindlin plate element (CS-FEM-MIN3) fordynamic response of laminated composite plates on viscoelasticfoundationrdquo Engineering Analysis with Boundary Elements vol42 pp 8ndash19 2014
[17] T Nguyen-Thoi T Rabczuk T Lam-Phat V Ho-Huu andP Phung-Van ldquoFree vibration analysis of cracked Mindlinplate using an extended cell-based smoothed discrete sheargap method (XCS-DSG3)rdquo Theoretical and Applied FractureMechanics 2014
[18] H Nguyen-Xuan G R Liu and C a Thai-Hoang ldquoAnedge-based smoothed finite element method (ES-FEM) withstabilized discrete shear gap technique for analysis of Reissner-Mindlin platesrdquo Computer Methods in Applied Mechanics andEngineering vol 199 no 9ndash12 pp 471ndash489 2010
[19] P Phung-Van T Nguyen-Thoi T Bui-Xuan and Q Lieu-XuanldquoA cell-based smoothed three-node Mindlin plate element(CS-FEM-MIN3) based on the 119862
0 -type higher-order sheardeformation for geometrically nonlinear analysis of laminatedcomposite platesrdquo Computational Materials Science 2014
[20] NNguyen-Thanh T RabczukHNguyen-Xuan and S BordasldquoAn alternative alpha finite element method with discrete sheargap technique for analysis of isotropicMindlin-Reissner platesrdquoFinite Elements inAnalysis andDesign vol 47 no 5 pp 519ndash5352011
[21] KM Liew Y Xiang and S Kitipornchai ldquoTransverse vibrationof thick rectangular plates-I Comprehensive sets of boundaryconditionsrdquo Computers and Structures vol 49 no 1 pp 1ndash291993
[22] D Zhou ldquoVibrations of Mindlin rectangular plates with elasti-cally restrained edges using static Timoshenko beam functionswith the Rayleigh-Ritz methodrdquo International Journal of Solidsand Structures vol 38 no 32-33 pp 5565ndash5580 2001
[23] D Zhou SH Lo F T KAu andYK Cheung ldquoVibration anal-ysis of rectangular Mindlin plates with internal line supportsusing static Timoshenko beam functionsrdquo International Journalof Mechanical Sciences vol 44 no 12 pp 2503ndash2522 2002
[24] H S Shen J Yang and L Zhang ldquoFree and forced vibrationof Reissner-Mindlin plates with free edges resting on elasticfoundationsrdquo Journal of Sound and Vibration vol 244 no 2 pp299ndash320 2001
[25] Y Xing and B Liu ldquoClosed form solutions for free vibrations ofrectangular Mindlin platesrdquo Acta Mechanica Sinica vol 25 no5 pp 689ndash698 2009
[26] P Gagnon C Gosselin and L Cloutier ldquoA finite strip elementfor the analysis of variable thickness rectangular thick platesrdquoComputers amp Structures vol 63 no 2 pp 349ndash362 1997
[27] T Mlzusawa ldquoVibration of rectangular mindlin plates withtapered thickness by the spline strip methodrdquo Computers andStructures vol 46 no 3 pp 451ndash463 1993
[28] H Nguyen-Xuan and T Nguyen-Thoi ldquoA stabilized smoothedfinite element method for free vibration analysis of Mindlin-Reissner platesrdquo Communications in Numerical Methods inEngineering with Biomedical Applications vol 25 no 8 pp 882ndash906 2009
[29] K M Liew L X Peng and S Kitipornchai ldquoVibration anal-ysis of corrugated Reissner-Mindlin plates using a mesh-freeGalerkin methodrdquo International Journal of Mechanical Sciencesvol 51 no 9-10 pp 642ndash652 2009
[30] Y Hou GWWei and Y Xiang ldquoDSC-Ritz method for the freevibration analysis of Mindlin platesrdquo International Journal forNumerical Methods in Engineering vol 62 no 2 pp 262ndash2882005
[31] F- Liu and K M Liew ldquoVibration analysis of discontinuousMindlin plates by differential quadrature element methodrdquoJournal of Vibration and Acoustics Transactions of the ASMEvol 121 no 2 pp 204ndash208 1999
[32] P Malekzadeh G Karami and M Farid ldquoA semi-analyticalDQEM for free vibration analysis of thick plates with twoopposite edges simply supportedrdquoComputerMethods inAppliedMechanics and Engineering vol 193 no 45ndash47 pp 4781ndash47962004
[33] P Malekzadeh and S A Shahpari ldquoFree vibration analysisof variable thickness thin and moderately thick plates withelastically restrained edges by DQMrdquo Thin-Walled Structuresvol 43 no 7 pp 1037ndash1050 2005
[34] R E Diaz-Contreras and S Nomura ldquoGreenrsquos function appliedto solution ofMindlin platesrdquoComputers and Structures vol 60no 1 pp 41ndash48 1996
[35] T Sakiyama and M Huang ldquoFree vibration analysis of rect-angular plates with variable thicknessrdquo Journal of Sound andVibration vol 216 no 3 pp 379ndash397 1998
[36] M Huang X Q Ma T Sakiyama H Matuda and C MoritaldquoFree vibration analysis of orthotropic rectangular plates withvariable thickness and general boundary conditionsrdquo Journal ofSound and Vibration vol 288 no 4-5 pp 931ndash955 2005
Shock and Vibration 25
[37] I Shufrin and M Eisenberger ldquoVibration of shear deformableplates with variable thicknessmdashfirst-order and higher-orderanalysesrdquo Journal of Sound and Vibration vol 290 no 1-2 pp465ndash489 2006
[38] S A Eftekhari and A A Jafari ldquoAccurate variational approachfor free vibration of variable thickness thin and thick plates withedges elastically restrained against translation and rotationrdquoInternational Journal of Mechanical Sciences vol 68 pp 35ndash462013
[39] W L Li X Zhang J Du and Z Liu ldquoAn exact series solutionfor the transverse vibration of rectangular plates with generalelastic boundary supportsrdquo Journal of Sound amp Vibration vol321 no 1-2 pp 254ndash269 2009
[40] X Shi D Shi W L Li and Q Wang ldquoA unified method forfree vibration analysis of circular annular and sector plateswith arbitrary boundary conditionsrdquo Journal of Vibration andControl 2014
[41] D Y Shi X J Shi W L Li and Q S Wang ldquoFree transversevibrations of orthotropic thin rectangular plates with arbitraryelastic edge supportsrdquo Journal of Vibroengineering vol 16 no 1pp 389ndash398 2014
[42] Y Chen G Jin and Z Liu ldquoFree vibration analysis of circu-lar cylindrical shell with non-uniform elastic boundary con-straintsrdquo International Journal of Mechanical Sciences vol 74pp 120ndash132 2013
[43] L Dai T Yang J Du W L Li and M J Brennan ldquoAn exactseries solution for the vibration analysis of cylindrical shellswith arbitrary boundary conditionsrdquo Applied Acoustics vol 74no 3 pp 440ndash449 2013
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Shock and Vibration
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DistributedSensor Networks
International Journal of
8 Shock and Vibration
32
36
40
44
48
52
2
3
4
5
6
7
8
Ω Ω
0 2 4 6 8 10 12 14 16 18M = N
ab = 1
ab = 32
(a) The 1st order
0 2 4 6 8 10 12 14 16 180
20
40
60
80
0
50
100
150
200
250
300
M = N
Ω Ω
ab = 1
ab = 32
(b) The 8th order
Figure 3 The effect of numerically truncated numbers119872 = 119873 on the natural frequenciesΩ for C-C-C-C boundary condition
0 2 4 6 8 10 12 14 16 18125
130
135
140
145
150
088
090
092
094
096
098
Ω Ω
M = N
ab = 1
ab = 32
(a) The 1st order
0 2 4 6 8 10 12 14 16 18
55
60
65
70
75
80
40
45
50
55
60
65
70
75
80Ω Ω
M = N
ab = 1
ab = 32
(b) The 8th order
Figure 4 The effect of numerically truncated numbers119872 = 119873 on the natural frequenciesΩ for F-F-F-F boundary condition
the twisting moments have small effect on the vibration char-acteristics of the structure However in this paper in order toget a more accurate prediction of the vibration characteristicsof the structure the twisting moment on boundary edges istaken into account In the latter study in addition to the factthat free boundary is not unexpected considering the torsionspring the other boundary conditions are introduced into atorsion spring and the spring stiffness takes infinity FormFigure 2 analysis it also concluded that the translation springhas a wider influence range than the rotation spring on thefrequency parameters that is for the translation springs thestable frequency parameters appearwhen the stiffness ismorethan 1012 or less 107 while for the rotation springs when thestiffness value is assigned more than 1010 the frequency para-meters become smoothThus it is suitable to use 1014 to sim-ulate the infinite stiffness value in the model validation partsand in the following examples
32 Uniform Thickness Moderately Thick Plates with Classicaland Elastic Boundary Conditions In this subsection themodified Fourier solution is applied to deal with vibrationproblems of uniform thickness moderately thick rectangu-lar plates subject to the classical boundary conditions andarbitrary elastic boundary conditions In present work threegroups of continuously distributed boundary springs areintroduced to simulate the given or typical boundary condi-tions As mentioned earlier the stiffness of these boundarysprings can take any value from zero to infinity to bettermodel many real-world restraint conditions Taking edge 119909 =0 for example the corresponding spring stiffness for the threetypes of classical boundaries and elastic boundaries is
completely free 119876119909= 0119872
119909119909= 0 and119872
119909119910= 0
1198961199090= 0 119870
1199090= 0 119870
1199101199090= 0 (17a)
Shock and Vibration 9
0 2 4 6 8 10 12 14 16 18
166
168
170
172
152
156
160
164
168
Ω Ω
ab = 1
ab = 32
M = N
(a) The 1st order
0 2 4 6 8 10 12 14 16 1850
55
60
65
70
4
5
6
7
8
9
10
Ω Ω
M = N
ab = 1
ab = 32
(b) The 8th order
Figure 5 The effect of numerically truncated numbers 119872 = 119873 on the natural frequencies Ω with four edges elastically restrained againsttranslation and rotation (119870
119879= 119896119894(1198873
119863) 119870119877= 119870119894(119887119863) 119870
119879= 100 and 119870
119877= 100)
simply supported 119908 = 0119872119909119909= 0 and 120595
119910= 0
1198961199090= 1014
1198701199090= 0 119870
1199101199090= 1014
(17b)
completely clamped 119908 = 0 120595119909= 0 and 120595
119910= 0
1198961199090= 1014
1198701199090= 1014
1198701199101199090
= 1014
(17c)
arbitrarily elastic boundary 119908 = 0 120595119909
= 0 120595119910
= 0119876119909
= 0119872119909119909
= 0 and119872119909119910
= 0
1198961199090= Γ1199081199090
1198701199090= Γ1199091199090
1198701199101199090
= 1014
(17d)
The appropriateness of the three classical boundariesdefined in (17a)ndash(17c) will be proved by several examplesgiven in following the arbitrary elastic boundaries are alsodefined in (17d) and the Γ
119908119894(Γ119908119894= 1198961198941198960 1198960= 1 times 10
9Nm2119894 = 1199090 1199091198861199100119910119887) and Γ
119909119894(Γ119909119894= 11987011989411987001198700= 1times10
9Nmrad119894 = 1199090 119909119886 1199100 119910119887) elastic restraint parameters representcorresponding spring stiffness For the sake of simplicitya four-letter string is employed to represent the restraintcondition of a plate such that F-C-S-E identifies the platewithedges 119909 = 0 119910 = 0 119909 = 119886 and 119910 = 119887 having free clampedshear-supported restrained and elastic boundary conditionsrespectively
As for the first case a uniform thickness moderately thickplate with different classical boundaries and structure param-eters is investigated here In Table 2 the comparison of thefirst eight frequency parametersΩ = 120596119887
2
(120588ℎ119863)12
1205872 of the
considered plate is presented The S-S-S-S C-F-F-F S-S-F-FF-F-F-F and S-C-S-C boundary conditions are performed inthe comparison Excellent agreements are observed betweenthe solutions obtained by the modified Fourier method thereferential data and finite element method (FEM) results forthe uniform thickness moderately thick rectangular platesIt is also verified that the definition of the three types of
classical boundaries in (17a)ndash(17c) is appropriate In additionthe elastic boundary conditions (17d) are also verified Inthe next two examples we will account for the vibrationof moderately thick plate with elastic edge supports Thefirst model considered is an S-S-S-S square moderately thickplate with all edges elastically rotationally restrained Thatis 1198701199090
= 119870119909119886
= 1198701199100
= 119870119910119887
= Γ119909 and all the
other restraining springs are set to have an infinite stiffness(namely represented by 1014 in numerical calculation) Thesix frequency parameters Ω = 120596119887
2
(120588ℎ119863)12
1205872 are given
in Table 3 for several different restraining coefficient valuesthe finite element method (FEM) results calculated usingABAQUSare also listed in this table as a referenceThe secondmodel concerns a complete square moderately thick platewith all edges elastically restrained That is 119896
1199090= 119896119909119886
=
1198961199100
= 119896119910119887
= Γ119908and 119870
1199090= 119870119909119886
= 1198701199100
= 119870119910119887
= Γ119909 The
six frequency parameters Ω = 1205961198872
(120588ℎ119863)12
1205872 are given in
Table 4 for several different restraining coefficient values thefinite element method (FEM) results are also listed in Table 4as a reference It can be clearly seen that the comparison isextremely good which implies that the current method isable tomake correct predictions for themodal characteristicsof moderately thick plate with not only classical boundaryconditions but also elastic edge restraints
The excellent agreements between the solution obtainedby the modified Fourier method and the referential datafor the moderately thick plate subjected to the combina-tions of classical boundary conditions and elastic boundaryconditions given in Tables 2ndash4 indicate that the proposedmethod is sufficiently accurate to deal with uniform thicknessmoderately thick plate with arbitrary boundary conditions
33 Linearly VariationThickness Moderately Thick Plates withClassical and Elastic Boundary Conditions In the theoreticalformulations this paper concerns the varying thicknessmoderately thick plates with classical and elastic boundary
10 Shock and Vibration
Table 2 Frequency parameters Ω = 1205961198872
(120588ℎ119863)12
1205872 for uniform thickness moderately thick plates with different classical boundaries and
structure parameters
Boundary condition ab Model sequence1 2 3 4 5 6 7 8
S-S-S-S
11931 4605 4605 7064 8605 8605 10792 10792
(1933a) (4611) (4611) (7067) (8607) (8607) (10799) (10780)(1931b) (4605) (4605) (7064) (8605) (8605) (10792) (10792)
321408 2648 4127 4605 5261 7064 7153 8184(1410a) (2651) (4128) (4608) (5267) (7074) (7155) (8188)(1408b) (2648) (4127) (4605) (5261) (7064) (7151) (8184)
C-F-F-F
10348 0817 2035 2583 2862 4816 5478 5774
(0348a) (0817) (2038) (2586) (2865) (4821) (5504) (5793)(0348b) (0816) (2034) (2582) (2860) (4811) (5477) (5772)
320155 0502 0941 1665 2292 2613 3303 3498
(0154a) (0502) (0941) (1663) (2295) (2618) (3305) (3500)(0154b) (0501) (0940) (1662) (2292) (2612) (3298) (3494)
S-S-F-F
10333 1678 1875 3559 4720 4946 6474 6632
(0335a) (1680) (1878) (3600) (4721) (4949) (6478) (6642)(0333b) (1677) (1874) (3557) (4718) (4945) (6472) (6631)
320222 0938 1631 2363 2573 4082 4482 4753
(0223a) (0940) (1632) (2364) (2575) (4092) (4483) (4759)(0222b) (0938) (1631) (2361) (2571) (4077) (4479) (4752)
F-F-F-F1 1290 1919 2363 3235 3235 5605 5605 5646
(1291a) (1920) (2365) (3239) (3239) (5607) (5607) (6648)
32 0865 0948 1961 2166 2461 2843 3551 4026(0865a) (0949) (1963) (2168) (2465) (2844) (3556) (4034)
S-C-S-C1 2700 4971 5990 7973 8787 10250 11334 12024
(2701a) (4971) (5993) (7975) (8788) (10253) (11335) (12028)
32 2348 3253 4972 5646 6489 7381 7974 9930(2348a) (3255) (4974) (5647) (6590) (7384) (7801) (9932)
aResults in parentheses are taken from FEMbResults in parentheses are taken from [11]
conditions The varying thickness function ℎ(119909 119910) can beexpressed as ℎ
0(1 + 120572119909
119904
)(1 + 120573119910119905
) in which the ℎ0 120572 and
120573 represent the initial thickness gradient in 119909 direction andgradient in 119910 direction When the indexes 119904 and 119905 take thevalue 119904 = 119905 = 1 the analyticalmodel imitates the linearly vari-ation thickness moderately thick plates structure In order tounify the description and facilitate the analytical calculationsof the involved integrals all the thickness variation functionscan be expanded into either 1D or 2D Fourier cosine seriesresulting in
ℎ (119909 119910) =
infin
sum
119898=0
infin
sum
119899=0
ℎ0119902119898119899
cos 119898120587119909119886
cos119898120587119910
119887
(18)
where119902119898119899
=
4
119886119887
int
119887
0
int
119886
0
(1 + 120572119909119904
) (1 + 120573119910119905
) cos 119898120587119909119886
cos119898120587119910
119887
119889119909 119889119910
(19)
In order to prove the validity of the present methodfor the vibration of linearly variation thickness moderately
thick plates with arbitrary boundary conditions the typicalclassical boundary conditions as the first case will be con-sidered In Table 5 the comparison of the first six frequencyparameters Ω = 120596119886
2
(120588ℎ01198630)12 of the moderately thick
plates with linearly varying thickness is presentedThe S-S-S-S C-F-F-F S-S-F-F C-C-C-C and S-C-S-C boundary condi-tions are performed in the comparison Excellent agreementsare observed between the solutions obtained by the modifiedFourier method and finite element method (FEM) results forthemoderately thick plates with linear variation thickness Toinvestigate the influence of the aspect ratio on the uniformthickness and nonuniform thickness moderately thick platesthe effect on the frequency parameters for plates with S-S-S-Sboundary conditions is presented in Figure 6 The thicknessfunctions are ℎ
0and ℎ0(1+05times119909)(1+05times119910) respectively It
is seen from Figure 6 that the influence of aspect ratios on thefrequency parameters for nonuniform thickness moderatelythick plates is more complicated
In the next two examples we also account for the vibra-tions of moderately thick plate with linear variation thicknessand elastic edge supports The first model considered is an S-S-S-S square moderately thick plate with all edges elastically
Shock and Vibration 11
Table 3 Frequency parameters Ω = 1205961198872
(120588ℎ119863)12
1205872 for S-S-S-S moderately thick plates (119886119887 = 1) with uniform thickness and elastic
rotation support (1198701199090= 119870119909119886= 1198701199100= 119870119910119887= Γ119909)
Γ119909
aℎ0119886
Model sequence1 2 3 4 5 6
0
005 1945 4834 4834 7591 9515 9519(1942b) (4836) (4836) (7584) (9547) (9552)
01 1856 4500 4500 6829 8499 8513(1858b) (4510) (4510) (6843) (8539) (8554)
015 1758 4114 4114 6040 7421 7445(1761b) (4126) (4126) (6060) (7462) (7486)
02 1656 3733 3733 5335 6468 6499(1660b) (3747) (3747) (5357) (6508) (6538)
05
005 3541 7071 7071 10227 12318 12399(3546b) (7091) (7091) (10253) (12386) (12468)
01 3274 6242 6242 8719 10302 10421(3282b) (6267) (6267) (8754) (10370) (10488)
015 2947 5361 5361 7290 8465 8606(2956b) (5387) (5387) (7327) (8524) (8666)
02 2624 4599 4599 6148 7057 7199(2634b) (4624) (4624) (6182) (7107) (7250)
1
005 3541 7072 7072 10228 12320 12401(3547b) (7093) (7093) (10254) (12388) (12470)
01 3278 6248 6248 8727 10311 10429(3286b) (6274) (6274) (8762) (10378) (10497)
015 2957 5374 5374 7305 8478 8620(2966b) (5400) (5400) (7341) (8537) (8680)
02 2641 4617 4617 6166 7070 7215(2651b) (4643) (4643) (6200) (7121) (7268)
15
005 3542 7073 7073 10229 12320 12401(3547b) (7093) (7093) (10254) (12388) (12471)
01 3280 6251 6251 8729 10313 10432(3287b) (6276) (6276) (8765) (10381) (10500)
015 2960 5379 5379 7309 8482 8625(2969b) (5405) (5405) (7346) (8541) (8685)
02 2647 4623 4623 6172 7074 7221(2657b) (4649) (4649) (6206) (7125) (7274)
2
005 3542 7073 7073 10229 12321 12402(3547b) (7093) (7093) (10255) (12389) (12471)
01 3280 6252 6252 8730 10315 10434(3288b) (6277) (6277) (8766) (10382) (10501)
015 2962 5381 5381 7312 8484 8628(2971b) (5407) (5407) (7348) (8543) (8688)
02 2650 4627 4627 6175 7076 7224(2660b) (4652) (4652) (6210) (7128 (7277)
aΓ119909= 1198701198941198700 (1198700= 1 times 10
9 Nmrad 119894 = 1199090 119909119886 1199100 119910119887)bResults in parentheses are taken from FEM
12 Shock and Vibration
Table 4 Frequency parameters Ω = 1205961198872
(120588ℎ119863)12
1205872 for moderately thick plates (119886119887 = 1) with uniform thickness and all edge elastic
restraints (1198701199090= 119870119909119886= 1198701199100= 119870119910119887= Γ119909and 119896
1199090= 119896119909119886= 1198961199100= 119896119910119887= Γ119908)
Γ119909
aΓ119908
aℎ0119886
Model sequence1 2 3 4 5 6
05 0
005 0986 1967 3893 3893 4850 4850(0988b) (1971) (3897) (3897) (4852) (4852)
01 0916 1835 3515 3522 4375 4375(0917b) (1837) (3519) (3528) (4341) (4341)
015 0789 1635 2983 3027 3771 3771(0792b) (1638) (2988) (3029) (3775) (3775)
02 0646 1444 2497 2601 3249 3249(0652b) (1448) (2508) (2621) (3254) (3254)
1 05
005 3417 3417 4431 5294 5461 6448(3419b) (3419) (4435) (5303) (5478) (6456)
01 1593 1593 2401 3849 3861 4729(1611b) (1611) (2408) (3852) (3866) (4767)
015 1122 1122 1913 3272 3290 4038(1127b) (1227) (1924) (3284) (3314) (4069)
02 0890 0890 1640 2781 2829 3476(0899b) (0899) (1654) (2787) (2834) (3497)
15 1
005 4325 4325 5619 6273 6571 7619(4334b) (4334) (5681) (6299) (6642) (7642)
01 2013 2013 2812 4088 4116 4993(2024b) (2024) (2827) (4122) (4187) (4994)
015 1338 1338 2108 3430 3443 4196(1342b) (1342) (2112) (3467) (3482) (4200)
02 1038 1038 1772 2936 2965 3613(1042b) (1042) (1786) (2937) (2977) (3624)
2 2
005 5231 5231 6919 7630 8002 9220(5238b) (5238) (6931) (7637) (8011) (9241)
01 2594 2594 3439 4470 4544 5428(2597b) (2597) (3442) (4495) (4557) (5438)
015 1653 1653 2403 3613 3632 4392(1662b) (1662) (2414) (3627) (3646) (4404)
02 1238 1238 1949 3077 3099 3751(1244b) (1244) (1964) (3094) (3116) (3788)
aΓ119909= 1198701198941198700 (1198700= 1 times 10
9 Nmrad 119894 = 1199090 119909119886 1199100 119910119887) Γ119908= 1198961198941198960 (1198960= 1 times 10
9 Nm 119894 = 1199090 119909119886 1199100 119910119887)bResults in parentheses are taken from FEM
05 10 15 20 25 300
2
4
6
8
10
12
14
16
ab
The 1st orderThe 2nd orderThe 3rd order
The 4th orderThe 5th orderThe 6th order
Ω
(a)
05 10 15 20 25 30
30
60
90
120
150
180
ab
The 1st orderThe 2nd orderThe 3rd order
The 4th orderThe 5th orderThe 6th order
Ω
(b)
Figure 6 The effect of aspect ratio 119886119887 on the natural frequenciesΩ for S-S-S-S boundary condition (a) uniform thickness and (b) nonuni-form thickness
Shock and Vibration 13
Table 5 Frequency parameter Ω = 1205961198862
(120588ℎ01198630)12 for moderately thick plates with linearly varying thickness in different boundary
conditions (ℎ(119909 119910) = ℎ0(1 + 05119909)(1 + 05119910))
Boundary condition 119886119887 ℎ0119886
Model sequence1 2 3 4 5 6
S-S-S-S
101 27261 62397 62926 92423 111537 111732
(27263a) (62410) (63842) (92445) (111867) (111921)
02 23103 47010 47191 65018 75831 75865(23489a) (47209) (47349) (65057) (76605) (76473)
3201 59338 99101 138798 151102 166710 204855
(59874a) (99249) (138177) (151362) (166644) (205381)
02 41175 63765 63765 73598 83956 89084(41586a) (63313) (63669) (73661) (83918) (89547)
C-F-F-F
101 4173 10331 27407 37826 38809 64095
(4856a) (10680) (27513) (37203) (39135) (64858)
02 3983 8702 21403 29443 30286 46705(4118a) (9306) (21990) (29670) (30602) (47156)
3201 4868 15488 35620 63523 89969 100089
(4887a) (15320) (35175) (63353) (89159) (100783)
02 4469 9763 22338 42188) 50104 56331(4944a) (9636) (22087) (42177) (50580) (56559)
S-S-F-F
101 4117 24795 26216 48567) 68195 69667
(4789a) (24149) (26751) (49324) (68842) (70575)
02 3820 20516 21924 37523 48980 50327(4103a) (20041) (22125) (37221) (48894) (50622)
3201 7935 39454 65063 92877 92877 136176
(7856a) (40145) (65162) (92462) (93022) (136336)
02 6389 26149 35627 49539 56844 65170(6406a) (26536) (35626) (49831) (57144) (65981)
C-C-C-C
101 44471 80299 80681 109422 126871 128207
(45102a) (80271) (80656) (110015) (126328) (128888)
02 32162 52781 52933 69410 78474 79482(32240a) (52629) (53166) (70063) (79059) (80266)
3201 82975 115587 151992 161579 176601 212330
(83139a) (115838) (152200) (162193) (177058) (212849)
02 48577 66022 83468 90830 97474 98157(49187a) (66259) (84065) (91749) (97827) (98777)
S-C-S-C
101 36847 66664 76965 101045 113594 125551
(37901a) (68568) (78356) (101200) (113978) (125651)
02 27673 48469 51220 67027 76358 78241(27897a) (48685) (51491) (67447) (77530) (79573)
3201 77378 106848 150079 154569 173120 208391
(78003a) (107058) (150991) (155171) (173099) (208402)
02 45895 64732 73393 83282 90325 97089(45803a) (64131) 73656 83801 90518 97015
aResults in parentheses are taken from FEM
restrained That is 1198701199090
= 119870119909119886
= 1198701199100
= 119870119910119887
= Γ119909 The six
frequency parameters Ω = 1205961198862
(120588ℎ01198630)12 are given in
Table 6 for several different restraining coefficient valuesthe finite element method (FEM) results calculated usingABAQUS are also presented as a referenceThe secondmodelis a complete square moderately thick plate with all edgeselastically restrainedThat is 119896
1199090= 119896119909119886= 1198961199100= 119896119910119887= Γ119908and
1198701199090= 119870119909119886
= 1198701199100= 119870119910119887= Γ119909 The six frequency parameters
Ω = 1205961198862
(120588ℎ01198630)12 are given in Table 7 for several different
restraining coefficient values the finite element method(FEM) results are also listed as a reference It can be clearlyseen that the comparison is extremely good which impliesthat the current method is able to make correct predictionsfor the modal characteristics of linearly varying thickness
14 Shock and Vibration
Table 6 Frequency parameter Ω = 1205961198862
(120588ℎ01198630)12 for S-S-S-S moderately thick plates (119886119887 = 1) with linearly varying thickness and elastic
rotation support (ℎ(119909 119910) = ℎ0(1 + 05119909)(1 + 05119910) 119870
1199090= 119870119909119886= 1198701199100= 119870119910119887= Γ119909)
Γ119909
aℎ0119886
Model sequence1 2 3 4 5 6
0
005 28809 70080 70891 108967 134884 135295(28897b) (70334) (71129) (109112) (135254) (135772)
01 27261 62398 62926 92423 111540 111735(26606b) (61784) (62303) (91193) (111323) (111648)
015 25231 54192 54505 77118 91228 91312(25204b) (55297) (55595) (77503) (91871) (91941)
02 23103 47010 47191 65018 75832 75866(23947b) (47071) (47208) (65404) (76531) (76589)
05
005 23103 47010 47191 65018 75832 75866(23210b) (47245) (47358) (65241) (76122) (76345)
01 48377 94650 95597 135484 162635 163501(48524b) (94891) (95992) (135845) (167012) (164024)
015 36239 70939 71298 99997 118584 119009(36647b) (71214) (71687) (100024) (119114) (119457)
02 28749 56937 57037 79214 93009 93162(29142b) (57034) (57544) (79654) (93668) (93754)
1
005 49651 96759 97619 138089 165368 166304(49755b) (97245) (97964) (138544) (165864) (166878)
01 38951 73735 74251 102748 121079 121709(39021b) (74145) (74664) (102874) (121613) (121817)
015 30630 58406 58533 80432 93984 94255(30638b) (58684) (58644) (80874) (94121) (94478)
02 25548 48636 48674 66165 76708 76824(25687b) (48788) (48992) (66544) (76824) (76944)
15
005 50118 97561 98374 139096 166421 167382(50247b) (97877) (98661) (139247) (166875) (167662)
01 40310 75216 75783 104229 122402 123164(40366b) (75542) (75921) (104557) (122874) (123385)
015 31837 59352 59530 81245 94611 94983(31902b) (59451) (59871) (81375) (94784) (95105)
02 26302 49119 49156 66528 76954 77130(26418b) (49354) (49334) (66921) (77010) (77470)
2
005 50360 97983 98769 139631 166979 167954(50472b) (98005) (99114) (140154) (167188) (167986)
01 41130 76146 76724 105165 123229 124082(41421b) (76642) (78114) (105661) (123278) (124974)
015 32687 60025 60245 81832 95052 95508(32778b) (60244) (60549) (82146) (95114) (95842)
02 26893 49492 49539 66815 77138 77370(26922b) (49874) (49924) (66987) (77367) (77588)
aΓ119909= 1198701198941198700 (1198700= 1 times 10
9 Nmrad 119894 = 1199090 119909119886 1199100 119910119887)bResults in parentheses are taken from FEM
Shock and Vibration 15
Table 7 Frequency parameterΩ = 1205961198862
(120588ℎ01198630)12 for moderately thick plates (119886119887 = 1) with linearly varying thickness and elastic rotation
and translation support (ℎ(119909 119910) = ℎ0(1 + 05119909)(1 + 05119910) 119870
1199090= 119870119909119886= 1198701199100= 119870119910119887= Γ119909 and 119896
1199090= 119896119909119886= 1198961199100= 119896119910119887= Γ119908)
Γ119909
aΓ119908
aℎ0119886
Model sequence1 2 3 4 5 6
05 0
005 14751 14917 28232 55251 55397 69512(14780b) (14822) (29252) (55476) (55505) (70904)
01 11811 11910 23857 44462 45037 55453(11818b) (11897) (24711) (44376) (45257) (56511)
015 8487 8514 20011 34783 36335 44970(8500b) (8544) (20738) (34404) (36663) (45625)
02 6073 6073 17431 28679 30805 38010(6096b) (6106) (18039) (28129) (31165) (38311)
1 05
005 21225 32173 32173 44381 64061 65222(20708b) (32097) (32285) (45110) (64182) (65355)
01 8140 16878 16941 28100 48663 49015(7943b) (16838) (16970) (28888) (48700) (49171)
015 4493 11859 11878 22429 38352 39247(4383b) (11867) (11891) (23125) (38177) (39488)
02 2932 8690 8691 18919 31108 32600(3540b) (8791) (8791) (16229) (27520) (27889)
15 1
005 27819 40919 41430 54595 70792 72896(27146b) (41008) (41434) (55256) (70895) (73051)
01 11285 20068 20230 31101 51046 51429(11012b) (20025) (20270) (31849) (51126) (51569)
015 6292 13912 13922 24132 40500 41120(6139b) (13912) (13934) (24808) (40426) (41316)
02 4121 10321 10331 20051 32796 33921(4021b) (10329) (10357) (20635) (32540) (34178)
2 2
005 34615 51972 53088 68255 81344 84510(33806b) (52100) (53157) (68891) (81433) (84761)
01 15416 24659 24829 35615 53879 54494(15045b) (24625) (24888) (36303) (53970) (54626)
015 8740 16420 16481 26353 42417 42946(8529b) (16414) (16499) (26996) (42398) (43114)
02 5761 12127 12127 21434 34286 35192(5622b) (12134) (12148) (21998) (34110) (35415)
aΓ119909= 1198701198941198700 (1198700= 1 times 10
9 Nmrad 119894 = 1199090 119909119886 1199100 119910119887) Γ119908= 1198961198941198960 (1198960= 1 times 10
9 Nm 119894 = 1199090 119909119886 1199100 119910119887)bResults in parentheses are taken from FEM
moderately thick plate with not only classical boundaryconditions but also elastic edge restraints
As the last case of this subsection the influence of thegradient 120572 and 120573 on the fundamental frequency parametersfor a linearly varying thickness moderately thick plate isinvestigated The model is a square moderately thick platewith all edges elastically restrainedThat is 119896
1199090= 119896119909119886= 1198961199100=
119896119910119887= Γ119908= 2 and 119870
1199090= 119870119909119886
= 1198701199100= 119870119910119887= Γ119909= 2 The six
frequency parameters Ω = 1205961198862
(120588ℎ01198630)12 are given in
Table 8 for several different 120572 and 120573 values the finite elementmethod (FEM) results are also listed as a reference Againgood agreement can be observed Through Table 8 it is alsofound that the frequency parameter increases with increasinggradient parameters
The above studies are given as linearly varying thicknessmoderately thick plates with several of boundary condition
and different structure parameters In the next section vibra-tion results for the plates subjected to nonlinear variationthickness will be presented
34 Nonlinearly Varying Thickness Moderately Thick Plateswith Classical and Elastic Boundary Conditions In Sec-tion 33 the linearly varying thickness moderately thickplates were studied However in the practical engineeringapplications the varying thickness of amoderately thick platemay not always be linear variation in nature A variety ofpossible thickness varying cases may be encountered in prac-tice Therefore the moderately thick plates with nonlinearvariation thickness subjected to general elastic edge restraintsare examined in this subsection For the sake of brevity theindexes 119904 and 119905 will be chosen as 2 to imitate the nonlinearlyvarying thickness moderately thick plates structure in this
16 Shock and Vibration
Table 8 Frequency parameterΩ = 1205961198862
(120588ℎ01198630)12 for elastic support moderately thick plates (119886119887 = 1) with different gradient (119896
1199090= 119896119909119886=
1198961199100= 119896119910119887= Γ119908= 2 119870
1199090= 119870119909119886= 1198701199100= 119870119910119887= Γ119909= 2)
120572 120573 ℎ0119886
Model sequence1 2 3 4 5 6
120572 = 120573 = 01
005 31275 52615 52657 68954 76146 79850(31303a) (52664) (52675) (69033) (76174) (79948)
01 17270 25244 25250 33949 46050 46625(17271a) (25255) (25258) (34089) (46122) (46708)
015 10190 16266 16274 24164 37327 37526(10189a) (16269) (16280) (24323) (37401) (37622)
02 6780 12258 12260 19685 31452 31754(6780a) (12265) (12270) (19851) (31514) (31864)
120572 = 120573 = 02
005 32530 52947 53141 68954 77034 80589(32577a) (53014) (53203) (69172) (77088) (80773)
01 16882 24979 24979 34154 48057 48575(16883a) (24983) (25001) (34433) (48134) (48667)
015 9804 16284 16309 24683 38844 39085(9803a) (16283) (16322) (24980) (38909) (39191)
02 6499 12290 12295 20133 32354 32770(6498a) (12296) (12307) (20420) (32387) (32895)
120572 = 120573 = 03
005 33516 52845 53308 68704 78203 81561(33563a) (52939) (53389) (69055) (78271) (81789)
01 16419 24799 24830 34527 50061 50583(16419a) (24790) (24869) (34947) (50143) (50688)
015 9430 16334 16378 25236 40193 40506(9429a) (16329) (16395) (25660) (40243) (40625)
02 6235 12288 12293 20572 33102 33663(6235a) (12296) (12306) (20965) (33090) (33811)
120572 = 120573 = 04
005 34215 52463 53264 68430 79654 82865(34252a) (52575) (53346) (68921) (79732) (83113)
01 15921 24695 24793 35027 52011 52571(15921a) (24674) (24842) (35583) (52098) (52688)
015 9074 16388 16444 25797 41379 41789(9074a) (16381) (16464) (26336) (41404) (41928)
02 5990 12236 12238 21002 33735 34461(5989a) (12249) (12250) (21487) (33653) (34641)
120572 = 120573 = 05
005 34615 51972 53088 68255 81344 84510(33806a) (52100) (53157) (68891) (81433) (84761)
01 15416 24659 24829 35615 53879 54494(15045a) (24625) (24888) (36303) (53970) (54626)
015 8740 16420 16481 26353 42417 42946(8529a) (16414) (16499) (26996) (42398) (43114)
02 5761 12127 12127 21434 34286 35192(5622a) (12134) (12148) (21998) (34110) (35415)
aResults in parentheses are taken from FEM
section that is 119904 = 119905 = 2 Also the thickness varying functionsare also be expressed as Fourier cosine series according to (18)and (19)
In order to validate the accuracy and reliability of theproposed method for predicting the vibration behavior ofnonlinearly varying thickness moderately thick plates with
arbitrary boundary conditions the typical classical boundaryconditions viewed as the special cases of elastically restrainededges will be considered The comparison of the first six fre-quency parameters Ω = 120596119886
2
(120588ℎ01198630)12 for the moderately
thick plates with nonlinearly varying thickness is presentedin Table 9 The S-S-S-S C-F-F-F S-S-F-F C-C-C-C and
Shock and Vibration 17
Table 9 Frequency parameter Ω = 1205961198862
(120588ℎ01198630)12 for moderately thick plates with linearly varying thickness in different boundary
conditions (ℎ(119909 119910) = ℎ0(1 + 05119909
2
)(1 + 051199102
))
Boundary condition 119886119887 ℎ0119886
Model sequence1 2 3 4 5 6
S-S-S-S
101 23462 54999 55573 82950 100900 101187
(23741a) (55114) (55983) (83146) (101004) (101347)
02 18606 42725 42971 60040 71167 71761(18936a) (43014) (43148) (60247) (71642) (71993)
3201 53055 65251 85404 106700 114880 143531
(53467a) (65472) (85984) (107211) (115687) (144007)
02 17439 21531 32170 44987 59438 64179(17874a) (21748) (32651) (45264) (60188) (64879)
C-F-F-F
101 3658 9317 24135 32816 35433 59179
(3749a) (9547) (24034) (33627) (36308) (60142)
02 3520 7986 19591 26959 27491 43882(3611a) (8176) (20075) (27125) (27625) (43984)
3201 3901 9963 28984 69187 73686 89360
(3930a) (10095) (29381) (69578) (73994) (90242)
02 3292 4439 13385 23947 40850 44758(3386a) (4498) (14012) (24068) (41221) (45142)
S-S-F-F
101 3557 21268 22762 43524 60253 62429
(3649a) (21441) (22796) (43916) (60622) (62630)
02 3171 17822 19512 33616 45042 46833(3489a) (18006) (20430) (33815) (45247) (49004)
3201 6738 31658 52071 75625 86604 94903
(6957a) (31882) (52518) (75823) (86939) (95610)
02 3653 12930 20955 26517 44097 48058(6807a) (13160) (21312) (26864) (44422) (48873)
C-C-C-C
101 41020 75196 75614 104142 119683 120949
(41405a) (75452) (75882) (104711) (119637) (121137)
02 31237 51686 51868 68411 76868 77864(32010a) (52960) (52145) (69335) (77481) (78099)
3201 95823 128727 163657 173751 190890 223595
(95364a) (128932) (163588) (172350) (190350) (222948)
02 53306 71372 89384 102404 104660 120908(53587a) (71759) (90103) (102503) (105174) (121275)
S-C-S-C
101 33715 60298 71446 94187 103802 117928
(34148a) (60786) (71828) (94515) (104263) (119326)
02 26143 45499 49458 64253 72941 76191(26789a) (46622) (50680) (64329) (73474) (76843)
3201 73228 115845 137445 148166 168342 182542
(73761a) (116121) (137788) (148530) (168698) (182921)
02 41302 53648 65630 72202 81632 88105(41304a) (53757) (65842) (72794) (82033) (88474)
aResults in parentheses are taken from FEM
S-C-S-C boundary conditions are performed in the compar-ison The results adequately demonstrated the great accuracyof the modified Fourier method
We now turn to elastically restrained moderately thickplates The first one involves an S-S-S-S moderately thicksquare plate with a uniform rotational restraint along eachedge that is 119870
1199090= 119870119909119886
= 1198701199100
= 119870119910119887
= Γ119909 The calculated
frequency parameters Ω = 1205961198862
(120588ℎ01198630)12 are given in
Table 10 together with the FEM results Since the elasticallyrestrained plates with nonlinear variation thickness are rarelyinvestigated the FEMresults are used as the referenceA goodagreement is observed between the current and FEM resultsThe second example concerns amoderately thick square plateelastically supported along all edgesThe stiffness of the linear
18 Shock and Vibration
Table 10 Frequency parameterΩ = 1205961198862
(120588ℎ01198630)12 for S-S-S-S moderately thick plates (119886119887 = 1) with linearly varying thickness and elastic
rotation support (ℎ(119909 119910) = ℎ0(1 + 05119909
2
)(1 + 051199102
) 1198701199090= 119870119909119886= 1198701199100= 119870119910119887= Γ119909)
Γ119909
aℎ0119886
Model sequence1 2 3 4 5 6
0
005 23462 54999 55573 82950 100900 101187(23741b) (55114) (55983) (83146) (101004) (101347)
01 18606 42725 42971 60040 71167 71761(18936b) (43014) (43148) (60247) (71642) (71993)
015 53055 65251 85404 106700 114880 143531(53467b) (65472) (85984) (107211) (115687) (144007)
02 17439 21531 32170 44987 59438 64179(17874b) (21748) (32651) (45264) (60188) (64879)
05
005 6904 13374 13514 19400 22904 23056(9909b) (13548) (13651) (19564) (23088) (23379)
01 5316 10237 10314 14499 17155 17264(5374b) (10448) (10651) (14764) (17188) (17879)
015 4136 8201 8220 11474 13578 13671(4174b) (8348) (8651) (11764) (13688) (13879)
02 3449 6915 6924 9549 11225 11327(3474b) (6948) (6951) (9764) (11388) (11579)
1
005 7040 13617 13745 19726 23233 23402(7074b) (13748) (13851) (19964) (23488) (23879)
01 5712 10696 10793 14991 17594 17744(5774b) (10748) (10851) (15054) (17788) (17879)
015 4489 8507 8540 11759 13803 13922(4514b) (8548) (8611) (11864) (13988) (14119)
02 3677 7084 7090 9692 11330 11445(3684b) (7148) (7191) (9764) (11588) (11879)
15
005 7089 13708 13828 19847 23355 23530(7124b) (13748) (13951) (19967) (23588) (23879)
01 5895 10923 11024 15239 17813 17988(5874b) (10748) (11651) (15264) (17818) (17979)
015 4697 8694 8737 11938 13790 14083(4774b) (8748) (8951) (11864) (13688) (14879)
02 3836 7201 7210 9792 11403 11530(3874b) (7748) (7651) (9864) (11588) (11879)
2
005 7114 13754 13871 19911 23418 23597(7174b) (12848) (13851) (19864) (23588) (23879)
01 6001 11060 11159 15391 17944 18136(6004b) (11148) (11151) (15464) (17988) (18379)
015 4836 8821 8872 12061 14037 14189(4874b) (8848) (8951) (12164) (14188) (14879)
02 3954 7289 7301 9871 11457 11594(3974b) (7348) (7651) (9964) (11488) (11679)
aΓ119909= 1198701198941198700 (1198700= 1 times 10
9 Nmrad 119894 = 1199090 119909119886 1199100 119910119887)bResults in parentheses are taken from FEM
and rotational restraints is set equal to 1198961199090
= 119896119909119886
= 1198961199100
=
119896119910119887
= Γ119908and 119870
1199090= 119870119909119886
= 1198701199100
= 119870119910119887
= Γ119909 respectively
The six frequency parametersΩ = 1205961198862
(120588ℎ01198630)12 are shown
in Table 11 for several different restraining coefficient valuesIt can also be noticed that the Fourier series method is able
to predict the modal characteristics of nonlinearly varyingthickness moderately thick plate with not only classicalboundary conditions but also elastic edge restraints correctly
Finally the influence of the gradient on the fundamentalfrequency parameters of a nonlinearly varying thickness
Shock and Vibration 19
Table 11 Frequency parameterΩ = 1205961198862
(120588ℎ01198630)12 for moderately thick plates (119886119887 = 1) with linearly varying thickness and elastic rotation
and translation support (ℎ(119909 119910) = ℎ0(1 + 05119909
2
)(1 + 051199102
) 1198701199090= 119870119909119886= 1198701199100= 119870119910119887= Γ119909 and 119896
1199090= 119896119909119886= 1198961199100= 119896119910119887= Γ119908)
Γ119909
aΓ119908
aℎ0119886
Model sequence1 2 3 4 5 6
05 0
005 12926 13041 25090 47935 47943 61761(12931b) (13053) 25099 (47947) (47952) (61788)
01 10900 10988 21545 39980 40147 50322(10904b) (10992) (21557) (39984) (40166) (50334)
015 8245 8298 18081 31503 32617 40950(8247b) (8230) (18092) (31508) (32622) (40955)
02 6106 6124 15639 25631 27601 34458(6110b) (6125) (15642) (25637) (27617) (34462)
1 05
005 21575 31615 31644 42573 57889 58607(21580b) (31617) (31644) (42581) (57893) (58610)
01 8451 16024 16210 25969 43839 43867(8451b) (16025) (16217) (25972) (43844) (43870)
015 4682 11479 11480 20642 35172 35598(4684b) (11480) (11481) (20644) (35178) (35604)
02 3059 8621 8648 17320 28487 29638(3062b) (8619) (8644) (17327) (28492) (29641)
15 1
005 27713 40203 40978 52951 65352 66815(27714b) (40207) (40988) (52972) (65368) (66822)
01 11667 19322 19615 29122 46115 46183(11674b) (19327) (19619) (29137) (46120) (46188)
015 6544 13416 13481 22405 37238 37430(6547b) (13420) (13488) (22410) (37241) (37434)
02 4295 10175 10190 18543 30330 31056(4296b) (10177) (10194) (18544) (30336) (31060)
2 2
005 33453 50854 52088 66311 76610 78947(33455b) (50852) (50097) (66321) (76627) (78971)
01 15832 24130 24372 33924 49118 49362(15830b) (24127) (24374) (33931) (49147) (49382)
015 9060 15956 16099 24754 39134 39249(9064b) (15961) (16102) (24755) (39138) (39452)
02 5992 11955 11971 20034 31908 32392(5991b) (11957) (11978) (20042) (31910) (32398)
aΓ119909= 1198701198941198700 (1198700= 1 times 10
9 Nmrad 119894 = 1199090 119909119886 1199100 119910119887) Γ119908= 1198961198941198960 (1198960= 1 times 10
9 Nm 119894 = 1199090 119909119886 1199100 119910119887)bResults in parentheses are taken from FEM
moderately thick plate is investigated The plate is elasticallyrestrained in which the stiffness of the boundary springsis taken as 119896
1199090= 119896119909119886
= 1198961199100
= 119896119910119887
= Γ119908
= 2 and1198701199090
= 119870119909119886
= 1198701199100
= 119870119910119887
= Γ119909= 2 respectively The six
frequency parameters Ω = 1205961198862
(120588ℎ01198630)12 are presented in
Table 12 for several different slop values It can be seen thatthe fundamental frequency parameters will decrease with theincrease of the parameter ℎ
0119886
In the above examples it has been demonstrated thatthe presented method can be universally applied to nonlin-early varying thickness moderately thick plates with severalboundary conditions and different structure parametersNew results are obtained for plates with nonlinear variationthickness in both directions subjected to general elastic
boundary restraints which may be used for benchmarkingof researchers in the field In addition it is interesting to seethat the nature frequency decreases with the increase of theindex for thickness function
4 Conclusions
In this paper a modified Fourier method has been pre-sented to study the free vibration behaviors of moderatelythick rectangular plates with variable thickness and arbi-trary boundary conditionsThe first-order shear deformationplate theory is adopted to formulate the theoretical modelThe displacements and rotation components of the plateregardless of boundary conditions are invariantly expressedas the superposition of a 2D Fourier cosine series and four
20 Shock and Vibration
Table 12 Frequency parameterΩ = 1205961198862
(120588ℎ01198630)12 for elastic support moderately thick plates (119886119887 = 1) with different gradient (119896
1199090= 119896119909119886=
1198961199100= 119896119910119887= Γ119908= 2 119870
1199090= 119870119909119886= 1198701199100= 119870119910119887= Γ119909= 2)
120572 120573 ℎ0119886
Model sequence1 2 3 4 5 6
120572 = 120573 = 01
005 30187 51102 51141 67066 73954 77412(30188a) (51105) (51144) (67068) (73961) (77477)
01 16995 24719 24724 33071 44343 44819(16996a) (24725) (24729) (33084) (44347) (44822)
015 10077 15872 15883 23443 35937 36065(10081a) (15877) (15892) (23447) (35941) (36084)
02 6713 11953 11958 19086 30372 30604(6713a) (11957) (11965) (19092) (30378) (30612)
120572 = 120573 = 02
005 31206 51436 51614 67058 74398 77595(31208a) (51438) (51619) (67064) (74402) (77596)
01 16786 24503 24508 33144 45583 45923(16790a) (24510) (24513) (33147) (45587) (45932)
015 9821 15865 15906 23766 36897 36990(9824a) (15870) (15907) (23768) (36894) (36997)
02 6523 11978 11992 19356 30912 31162(6524a) (11981) (11997) (19362) (30916) (31174)
120572 = 120573 = 03
005 32097 51462 51900 66832 74991 77863(32098a) (51466) (51907) (66849) (74996) (77872)
01 16507 24332 24383 33330 46806 47074(16511a) (24327) (24380) (33339) (46810) (47086)
015 9562 15885 15964 24103 37748 37828(9562a) (15892) (15977) (21105) (37751) (37832)
02 6338 11992 12013 19603 31338 31634(6340a) (11994) (12011) (19612) (31344) (31639)
120572 = 120573 = 04
005 32851 51239 52046 66547 75735 78303(33455a) (50852) (50097) (66321) (76627) (78971)
01 16182 24209 24343 33598 47990 48230(33455a) (50852) (50097) (66321) (76627) (78971)
015 9307 15919 16034 24436 38491 38579(33455a) (50852) (50097) (66321) (76627) (78971)
02 6161 11986 12008 19828 31666 32040(33455a) (50852) (50097) (66321) (76627) (78971)
120572 = 120573 = 05
005 33453 50854 52088 66311 76610 78947(33455a) (50852) (50097) (66321) (76627) (78971)
01 15832 24130 24372 33924 49118 49362(15830a) (24127) (24374) (33931) (49147) (49382)
015 9060 15956 16099 24754 39134 39249(9064a) (15961) (16102) (24755) (39138) (39452)
02 5992 11955 11971 20034 31908 32392(5991a) (11957) (11978) (20042) (31910) (32398)
aResults in parentheses are taken from FEM
supplementary functions in the form of the product of apolynomial function and a single cosine series expansionto ensure and accelerate the convergence of the solutionAt each edge of the plate the general restraint conditionsare implemented by introducing one group of linear springsand two groups of rotational springs which are continuouslydistributed and determined by the stiffness of these springs
Instead of seeking a solution in strong forms in the previousstudies all the Fourier coefficients will be treated equallyand independently as the generalized coordinates and solveddirectly from the Rayleigh-Ritz technique The change of theboundary conditions can be easily achieved by only varyingthe stiffness of the three sets of boundary springs along alledges of the rectangular plates without involving any change
Shock and Vibration 21
to the solution procedure The convergence of the presentsolution is examined and the excellent accuracy is validatedby comparison with existing results published in the lit-erature and FEM data Excellent agreements are obtainedfrom these comparisons The proposed method providesa unified means for extracting the modal parameters andpredicting the vibration behaviors of moderately thick plateswith variable thickness variation functions and arbitraryelastic edge restraints A variety of free vibration results formoderately thick rectangular plates with different thicknessvariation functions and boundary conditions are presentedNew results for free vibration of moderately thick rectangularplates with various thickness variation functions and edgeconditions are presented which may be used for benchmark-ing of researchers in the field
Appendices
A Representative Calculation forStiffness and Mass Matrices
To illuminate the particular expression of the mass andstiffness matrixes clearly and tersely four new variables aredefined as 119904 = 119898 lowast (119873 + 1) + 119899 + 1 119905 = 119898
1015840
lowast (119873 + 1) + 1198991015840
+ 1119901 = (119897minus1)lowast (119872+1)+119898
1015840
+1 and 119902 = (119897minus1)lowast (119873+1)+1198991015840
+1The first row elements of K andM are given below
K1-1119904119905
=
1
2
int
119886
0
int
119887
0
[119863 (119909 119910)
times 21205821198861198981205821198861198981015840 sin 120582
119886119898119909 sin 120582
1198861198981015840119909 cos 120582
119887119899119910
times cos 1205821198871198991015840119910 + (1 minus 120583) 120582
1198871198991205821198871198991015840 cos 120582
119886119898119909
times cos 1205821198861198981015840 sin 120582
119887119899119910 sin 120582
1198871198991015840119910
+ 120581119866ℎ (119909 119910) cos 120582119886119898119909 cos 120582
1198861198981015840119909 cos 120582
119887119899
times119910 cos 1205821198871198991015840119910119889119909119889119910] 119889119909 119889119910
+ int
119886
0
1198701199091199100
cos 120582119886119898119909 cos 120582
1198861198981015840119909
+(minus1)119898+1198981015840
119870119909119910119887
cos 120582119886119898119909 cos 120582
1198861198981015840119909 119889119909
+ int
119887
0
1198701199090cos 120582119887119899119910 cos 120582
1198871198991015840119910
+(minus1)119899+1198991015840
119870119909119886cos 120582119887119899119910 cos 120582
1198871198991015840119910 119889119910
K1-2119904119901
=
1
2
int
119886
0
int
119887
0
[119863 (119909 119910)21205821198861198981205821198861198981015840 sin 120582
119886119898119909 sin 120582
1198861198981015840119909
times
2
sum
119897=1
120577119897
119887(119910) cos 120582
1198871198991015840119910
minus (1 minus 120583) 1205821198871198991015840 cos 120582
119886119898119909 cos 120582
1198861198981015840119909
times
2
sum
119897=1
1205771198971015840
119887(119910) sin 120582
1198871198991015840119910 + 120581119866ℎ (119909 119910)
times ( cos 1205821198861198981015840119909 cos 120582
1198871198991015840119910
times
2
sum
119897=1
120577119897
119887(119910) cos 120582
119886119898119909)]119889119909119889119910
+ int
119886
0
1198701199091199100
cos 120582119886119898119909 cos 120582
1198861198981015840119909
2
sum
119897=1
120577119897
119887(0)
+(minus1)1198991015840
119870119909119910119887
cos 120582119886119898119909 cos 120582
1198861198981015840119909
2
sum
119897=1
120577119897
119887(119887) 119889119909
+ int
119887
0
1198701199090
2
sum
119897=1
120577119897
119887(119910) cos 120582
1198871198991015840119910
+(minus1)119898+1198981015840
119870119909119886
2
sum
119897=1
120577119897
119887(119910) cos 120582
1198871198991015840119910119889119910
K1-3119904119902
=
1
2
int
119886
0
int
119887
0
[119863 (119909 119910) minus 21205821198861198981015840 sin 120582
1198861198981015840119909 cos 120582
1198871198991015840119910
times
2
sum
119897=1
1205771198971015840
119886(119909) cos 120582
119887119899119910
+ (1 minus 120583) 1205821198871198991205821198871198991015840 cos 120582
1198861198981015840119909
times sin 1205821198871198991015840119910
2
sum
119897=1
120577119897
119886(119909) sin 120582
119887119899119910
+ 120581119866ℎ (119909 119910) cos 1205821198861198981015840119909 cos 120582
1198871198991015840119910
times
2
sum
119897=1
120577119897
119886(119909) cos 120582
119887119899119910]119889119909119889119910
+ int
119886
0
1198701199091199100
cos 1205821198861198981015840119909
2
sum
119897=1
120577119897
119886(119909)
+(minus1)119899+1198991015840
119870119909119910119887
cos 1205821198861198981015840119909
2
sum
119897=1
120577119897
119886(119909) 119889119909
+ int
119887
0
1198701199090
2
sum
119897=1
120577119897
119886(0) cos 120582
1198871198991015840119910
+(minus1)119898+1198981015840
119870119909119886
2
sum
119897=1
120577119897
119886(0) cos 120582
1198871198991015840119910119889119910
K1-4119904119905
=
1
2
int
119886
0
int
119887
0
119863(119909 119910) 21205821198871198991205821198861198981015840 sin 120582
1198861198981015840119909 cos 120582
1198871198991015840119910
times cos 120582119886119898119909 sin 120582
119887119899119910 minus 2(1 minus 120583)
119887119899
times 1205821198861198981015840 sin 120582
1198861198981015840119909 cos 120582
1198871198991015840119910 cos 120582
119886119898119909
times sin 120582119887119899119910 + (1 minus 120583) 120582
1198861198981205821198871198991015840 cos 120582
1198861198981015840119909
times sin 1205821198871198991015840119910 sin 120582
119886119898119909 cos 120582
119887119899119910 119889119909 119889119910
22 Shock and Vibration
K1-5119904119901
=
1
2
int
119886
0
int
119887
0
119863(119909 119910) minus 21205821198861198981015840 sin 120582
1198861198981015840119909 cos 120582
1198871198991015840119910
times
2
sum
119897=1
1205771198971015840
119887(119910) cos 120582
119886119898119909
+ 2 (1 minus 120583) 1205821198861198981015840 sin 120582
1198861198981015840119909 cos 120582
1198871198991015840119910
times
2
sum
119897=1
1205771198971015840
119887(119910) cos 120582
119886119898119909
+ (1 minus 120583) 1205821198861198981205821198871198991015840 cos 120582
1198861198981015840119909 sin 120582
1198871198991015840119910
times
2
sum
119897=1
120577119897
119887(119910) sin 120582
119886119898119909119889119909119889119910
K1-6119904119902
=
1
2
int
119886
0
int
119887
0
119863(119909 119910)1205821198871198991205821198861198981015840 sin 120582
1198861198981015840119909 cos 120582
1198861198991015840119910
times
2
sum
119897=1
120577119897
119886(119909) sin 120582
1198871198991015840119910
minus 2 (1 minus 120583) 1205821198871198991205821198861198981015840 sin 120582
1198861198981015840119909 cos 120582
1198861198991015840119910
times
2
sum
119897=1
120577119897
119886(119909) sin 120582
1198871198991015840119910
+ 1205821198871198991015840 cos 120582
1198861198981015840119909 sin 120582
1198871198991015840119910
times
2
sum
119897=1
1205771198971015840
119886(119909) cos 120582
119887119899119910119889119909119889119910
K1-7119904119905
= minus
1
2
int
119886
0
int
119887
0
120581119866ℎ (119909 119910) 120582119886119898
cos 1205821198861198981015840119909 cos 120582
1198871198991015840119910
times sin 120582119886119898119909 cos 120582
119887119899119910119889119909119889119910
K1-8119904119901
= minus
1
2
int
119886
0
int
119887
0
120581119866ℎ (119909 119910) 1205821198861198981015840 cos 120582
1198861198981015840119909 cos 120582
1198871198991015840119910
times
2
sum
119897=1
120577119897
119887(119910) sin 120582
1198861198981015840119909 119889119909 119889119910
K1-9119904119902
=
1
2
int
119886
0
int
119887
0
120581119866ℎ (119909 119910) cos 1205821198861198981015840119909 cos 120582
1198871198991015840119910
times
2
sum
119897=1
1205771198971015840
119886(119909) cos 120582
119887119899119910119889119909119889119910
M1-1119904119905
=
1
24
120588int
119886
0
int
119887
0
ℎ3
(119909 119910) cos 120582119886119898119909 cos 120582
119887119899119910
times cos 1205821198861198981015840119909 cos 120582
1198871198991015840119910119889119909119889119910
M1-2119904119901
=
1
24
120588int
119886
0
int
119887
0
ℎ3
(119909 119910)
2
sum
119897=1
120577119897
119887(119910) cos 120582
119886119898119909
times cos 1205821198861198981015840119909 cos 120582
1198871198991015840119910119889119909119889119910
M1-3119904119902
=
1
24
120588int
119886
0
int
119887
0
ℎ3
(119909 119910)
2
sum
119897=1
120577119897
119886(119909) cos 120582
119887119899119910
times cos 1205821198861198981015840119909 cos 120582
1198871198991015840119910119889119909119889119910
M1-4119904119905 = M
1-5119904119901 = M1-6119904119902
= M1-7119904119905 = M
1-8119904119901 = M1-9119904119902 = 0
(A1)
B Useful Integrals for the Moderately ThickRectangular Plates with Variable Thicknessand Arbitrary Boundary Conditions
Consider
int
119871
0
cos 120582119898119909 cos 120582
119899119909 119889119909 =
0 119898 = 119899
119871
2
119898 = 119899
119871 119898 = 119899 = 0
int
119871
0
sin 120582119898119909 cos 120582
119899119909119889119909
=
minus
(minus1 + (minus1)119898+119899
) 119871119898
(1198982minus 1198992) 120587
119898 = 119899
0 119898 = 119899
int
119871
0
sin 120582119898119909 sin 120582
119899119909 119889119909 =
0 119898 = 119899
119871
2
119898 = 119899
0 119898 = 119899 = 0
int
119871
0
cos 120582119898119909120577119897
119886(119909) 119889119909
=
41198712
31205872 119897 = 1 119898 = 0
minus
41198712
(41198982
minus 3)
(9 minus 401198982+ 16119898
4) 1205872
119897 = 1 119898 = 0
minus
41198712
31205872 119897 = 2 119898 = 0
41198712
(41198982
minus 3) (minus1)119898
(9 minus 401198982+ 16119898
4) 1205872
119897 = 2 119898 = 0
Shock and Vibration 23
int
119871
0
sin 120582119898119909120577119897
119886(119909) 119889119909
=
0 119897 = 1 119898 = 0
161198712
119898(minus1)119898
(9 minus 401198982+ 16119898
4) 1205872
119897 = 1 119898 = 0
0 119897 = 2 119898 = 0
161198981198712
(9 minus 401198982+ 16119898
4) 1205872
119897 = 2 119898 = 0
int
119871
0
cos 1205821198981199091205771198971015840
119886(119909) 119889119909
=
0 119897 = 1 119898 = 0
161198711198982
(minus1)119898
(9 minus 401198982+ 16119898
4) 120587
119897 = 1 119898 = 0
0 119897 = 2 119898 = 0
161198711198982
(9 minus 401198982+ 16119898
4) 120587
119897 = 2 119898 = 0
int
119871
0
sin 1205821198981199091205771198971015840
119886(119909) 119889119909
=
0 119897 = 1 119898 = 0
4119871119898 (minus3 + 41198982
)
(9 minus 401198982+ 16119898
4) 120587
119897 = 1 119898 = 0
0 119897 = 2 119898 = 0
119871119898(
1
120587 minus 41198982120587
+
3
9120587 minus 41198982120587
) (minus1)119898
119897 = 2 119898 = 0
int
119871
0
1205771198971
119886(119909) 1205771198972
119886(119909) 119889119909 =
1198713
41205872 1198971= 1 1198972= 1
minus
21198713
31205873 1198971= 1 1198972= 2
minus
21198713
31205873 1198971= 2 1198972= 1
1198713
41205872 1198971= 2 1198972= 2
int
119871
0
1205771198971
119886(119909) 1205771198971015840
2
119886(119909) 119889119909 =
0 1198971= 1 1198972= 1
minus
1198712
8120587
1198971= 1 1198972= 2
1198712
8120587
1198971= 2 1198972= 1
0 1198971= 2 1198972= 2
int
119871
0
1205771198971015840
1
119886(119909) 1205771198971015840
2
119886(119909) 119889119909 =
5119871
16
1198971= 1 1198972= 1
minus
119871
2120587
1198971= 1 1198972= 2
minus
119871
2120587
1198971= 2 1198972= 1
5119871
16
1198971= 2 1198972= 2
int
119871
0
119909119899 cos 120582
119898119909 119889119909
=
1198711+119899
1 + 119899
119898 = 0
(1198861+119899HypergeometricPFQtimes [ 12 + 1198992 12 32 + 1198992
minus (14)1198982
1205872
]) times (1 + 119899)minus1
119898 gt 0
(B1)
Conflict of Interests
The authors declare that there is no conflict of interestsregarding the publication of this paper
Acknowledgments
The authors would like to thank the anonymous reviewersfor their very valuable comments The authors also gratefullyacknowledge the financial support from the National NaturalScience Foundation of China (no 51209052) HeilongjiangProvince Youth Science Fund Project (no QC2011C013) andHarbin Science and Technology Development InnovationFoundation of Youth (no 2011RFQXG021)
References
[1] K Liew Y Xiang and S Kitipornchai ldquoResearch on thick platevibration a literature surveyrdquo Journal of Sound and Vibrationvol 180 no 1 pp 163ndash176 1995
[2] J H Chung T Y Chung and K C Kim ldquoVibration analysisof orthotropic mindlin plates with edges elastically restrainedagainst rotationrdquo Journal of Sound and Vibration vol 163 no 1pp 151ndash163 1993
[3] Y K Cheung and D Zhou ldquoVibrations of moderately thickrectangular plates in terms of a set of static Timoshenko beamfunctionsrdquoComputers and Structures vol 78 no 6 pp 757ndash7682000
[4] CMWang ldquoNatural frequencies formula for simply supportedMindlin platesrdquo Journal of Vibration and Acoustics vol 116 no4 pp 536ndash540 1994
[5] K N Saha R C Kar and P K Datta ldquoFree vibration analysisof rectangular Mindlin plates with elastic restraints uniformlydistributed along the edgesrdquo Journal of Sound andVibration vol192 no 4 pp 885ndash902 1996
24 Shock and Vibration
[6] D J Gorman ldquoAccurate free vibration analysis of shear-deformable plates with torsional elastic edge supportrdquo Journalof Sound and Vibration vol 203 no 2 pp 209ndash218 1997
[7] D J Gorman ldquoFree vibration analysis of Mindlin plates withuniform elastic edge support by the superposition methodrdquoJournal of Sound andVibration vol 207 no 3 pp 335ndash350 1997
[8] D J Gorman ldquoAccurate free vibration analysis of point sup-ported mindlin plates by the superposition methodrdquo Journal ofSound and Vibration vol 219 no 2 pp 265ndash277 1999
[9] Y Xiang ldquoVibration of rectangular Mindlin plates resting onnon-homogenous elastic foundationsrdquo International Journal ofMechanical Sciences vol 45 no 6-7 pp 1229ndash1244 2003
[10] Y Xiang and G W Wei ldquoExact solutions for buckling andvibration of stepped rectangular Mindlin platesrdquo InternationalJournal of Solids and Structures vol 41 no 1 pp 279ndash294 2004
[11] Y L Yeh M J Jang and C C Wang ldquoAnalyzing the freevibrations of a plate using finite difference and differential trans-formationmethodrdquoAppliedMathematics andComputation vol178 no 2 pp 493ndash501 2006
[12] Y Xiang S K Lai L Zhou and C W Lim ldquoDSC-Ritz elementmethod for vibration analysis of rectangular Mindlin plateswith mixed edge supportsrdquo European Journal of MechanicsmdashASolids vol 29 no 4 pp 619ndash628 2010
[13] H Nguyen-Xuan C H Thai and T Nguyen-Thoi ldquoIsogeo-metric finite element analysis of composite sandwich platesusing a higher order shear deformation theoryrdquo Composites BEngineering vol 55 pp 558ndash574 2013
[14] C H Thai A J M Ferreira E Carrera and H Nguyen-XuanldquoIsogeometric analysis of laminated composite and sandwichplates using a layerwise deformation theoryrdquo Composite Struc-tures vol 104 pp 196ndash214 2013
[15] C H Thai A Ferreira S Bordas T Rabczuk and HNguyen-Xuan ldquoIsogeometric analysis of laminated compositeand sandwich plates using a new inverse trigonometric sheardeformation theoryrdquo European Journal of Mechanics A Solidsvol 43 pp 89ndash108 2014
[16] H Luong-Van T Nguyen-Thoi G R Liu and P Phung-VanldquoA cell-based smoothed finite elementmethod using three-nodeshear-locking free Mindlin plate element (CS-FEM-MIN3) fordynamic response of laminated composite plates on viscoelasticfoundationrdquo Engineering Analysis with Boundary Elements vol42 pp 8ndash19 2014
[17] T Nguyen-Thoi T Rabczuk T Lam-Phat V Ho-Huu andP Phung-Van ldquoFree vibration analysis of cracked Mindlinplate using an extended cell-based smoothed discrete sheargap method (XCS-DSG3)rdquo Theoretical and Applied FractureMechanics 2014
[18] H Nguyen-Xuan G R Liu and C a Thai-Hoang ldquoAnedge-based smoothed finite element method (ES-FEM) withstabilized discrete shear gap technique for analysis of Reissner-Mindlin platesrdquo Computer Methods in Applied Mechanics andEngineering vol 199 no 9ndash12 pp 471ndash489 2010
[19] P Phung-Van T Nguyen-Thoi T Bui-Xuan and Q Lieu-XuanldquoA cell-based smoothed three-node Mindlin plate element(CS-FEM-MIN3) based on the 119862
0 -type higher-order sheardeformation for geometrically nonlinear analysis of laminatedcomposite platesrdquo Computational Materials Science 2014
[20] NNguyen-Thanh T RabczukHNguyen-Xuan and S BordasldquoAn alternative alpha finite element method with discrete sheargap technique for analysis of isotropicMindlin-Reissner platesrdquoFinite Elements inAnalysis andDesign vol 47 no 5 pp 519ndash5352011
[21] KM Liew Y Xiang and S Kitipornchai ldquoTransverse vibrationof thick rectangular plates-I Comprehensive sets of boundaryconditionsrdquo Computers and Structures vol 49 no 1 pp 1ndash291993
[22] D Zhou ldquoVibrations of Mindlin rectangular plates with elasti-cally restrained edges using static Timoshenko beam functionswith the Rayleigh-Ritz methodrdquo International Journal of Solidsand Structures vol 38 no 32-33 pp 5565ndash5580 2001
[23] D Zhou SH Lo F T KAu andYK Cheung ldquoVibration anal-ysis of rectangular Mindlin plates with internal line supportsusing static Timoshenko beam functionsrdquo International Journalof Mechanical Sciences vol 44 no 12 pp 2503ndash2522 2002
[24] H S Shen J Yang and L Zhang ldquoFree and forced vibrationof Reissner-Mindlin plates with free edges resting on elasticfoundationsrdquo Journal of Sound and Vibration vol 244 no 2 pp299ndash320 2001
[25] Y Xing and B Liu ldquoClosed form solutions for free vibrations ofrectangular Mindlin platesrdquo Acta Mechanica Sinica vol 25 no5 pp 689ndash698 2009
[26] P Gagnon C Gosselin and L Cloutier ldquoA finite strip elementfor the analysis of variable thickness rectangular thick platesrdquoComputers amp Structures vol 63 no 2 pp 349ndash362 1997
[27] T Mlzusawa ldquoVibration of rectangular mindlin plates withtapered thickness by the spline strip methodrdquo Computers andStructures vol 46 no 3 pp 451ndash463 1993
[28] H Nguyen-Xuan and T Nguyen-Thoi ldquoA stabilized smoothedfinite element method for free vibration analysis of Mindlin-Reissner platesrdquo Communications in Numerical Methods inEngineering with Biomedical Applications vol 25 no 8 pp 882ndash906 2009
[29] K M Liew L X Peng and S Kitipornchai ldquoVibration anal-ysis of corrugated Reissner-Mindlin plates using a mesh-freeGalerkin methodrdquo International Journal of Mechanical Sciencesvol 51 no 9-10 pp 642ndash652 2009
[30] Y Hou GWWei and Y Xiang ldquoDSC-Ritz method for the freevibration analysis of Mindlin platesrdquo International Journal forNumerical Methods in Engineering vol 62 no 2 pp 262ndash2882005
[31] F- Liu and K M Liew ldquoVibration analysis of discontinuousMindlin plates by differential quadrature element methodrdquoJournal of Vibration and Acoustics Transactions of the ASMEvol 121 no 2 pp 204ndash208 1999
[32] P Malekzadeh G Karami and M Farid ldquoA semi-analyticalDQEM for free vibration analysis of thick plates with twoopposite edges simply supportedrdquoComputerMethods inAppliedMechanics and Engineering vol 193 no 45ndash47 pp 4781ndash47962004
[33] P Malekzadeh and S A Shahpari ldquoFree vibration analysisof variable thickness thin and moderately thick plates withelastically restrained edges by DQMrdquo Thin-Walled Structuresvol 43 no 7 pp 1037ndash1050 2005
[34] R E Diaz-Contreras and S Nomura ldquoGreenrsquos function appliedto solution ofMindlin platesrdquoComputers and Structures vol 60no 1 pp 41ndash48 1996
[35] T Sakiyama and M Huang ldquoFree vibration analysis of rect-angular plates with variable thicknessrdquo Journal of Sound andVibration vol 216 no 3 pp 379ndash397 1998
[36] M Huang X Q Ma T Sakiyama H Matuda and C MoritaldquoFree vibration analysis of orthotropic rectangular plates withvariable thickness and general boundary conditionsrdquo Journal ofSound and Vibration vol 288 no 4-5 pp 931ndash955 2005
Shock and Vibration 25
[37] I Shufrin and M Eisenberger ldquoVibration of shear deformableplates with variable thicknessmdashfirst-order and higher-orderanalysesrdquo Journal of Sound and Vibration vol 290 no 1-2 pp465ndash489 2006
[38] S A Eftekhari and A A Jafari ldquoAccurate variational approachfor free vibration of variable thickness thin and thick plates withedges elastically restrained against translation and rotationrdquoInternational Journal of Mechanical Sciences vol 68 pp 35ndash462013
[39] W L Li X Zhang J Du and Z Liu ldquoAn exact series solutionfor the transverse vibration of rectangular plates with generalelastic boundary supportsrdquo Journal of Sound amp Vibration vol321 no 1-2 pp 254ndash269 2009
[40] X Shi D Shi W L Li and Q Wang ldquoA unified method forfree vibration analysis of circular annular and sector plateswith arbitrary boundary conditionsrdquo Journal of Vibration andControl 2014
[41] D Y Shi X J Shi W L Li and Q S Wang ldquoFree transversevibrations of orthotropic thin rectangular plates with arbitraryelastic edge supportsrdquo Journal of Vibroengineering vol 16 no 1pp 389ndash398 2014
[42] Y Chen G Jin and Z Liu ldquoFree vibration analysis of circu-lar cylindrical shell with non-uniform elastic boundary con-straintsrdquo International Journal of Mechanical Sciences vol 74pp 120ndash132 2013
[43] L Dai T Yang J Du W L Li and M J Brennan ldquoAn exactseries solution for the vibration analysis of cylindrical shellswith arbitrary boundary conditionsrdquo Applied Acoustics vol 74no 3 pp 440ndash449 2013
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Shock and Vibration
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Chemical EngineeringInternational Journal of Antennas and
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DistributedSensor Networks
International Journal of
Shock and Vibration 9
0 2 4 6 8 10 12 14 16 18
166
168
170
172
152
156
160
164
168
Ω Ω
ab = 1
ab = 32
M = N
(a) The 1st order
0 2 4 6 8 10 12 14 16 1850
55
60
65
70
4
5
6
7
8
9
10
Ω Ω
M = N
ab = 1
ab = 32
(b) The 8th order
Figure 5 The effect of numerically truncated numbers 119872 = 119873 on the natural frequencies Ω with four edges elastically restrained againsttranslation and rotation (119870
119879= 119896119894(1198873
119863) 119870119877= 119870119894(119887119863) 119870
119879= 100 and 119870
119877= 100)
simply supported 119908 = 0119872119909119909= 0 and 120595
119910= 0
1198961199090= 1014
1198701199090= 0 119870
1199101199090= 1014
(17b)
completely clamped 119908 = 0 120595119909= 0 and 120595
119910= 0
1198961199090= 1014
1198701199090= 1014
1198701199101199090
= 1014
(17c)
arbitrarily elastic boundary 119908 = 0 120595119909
= 0 120595119910
= 0119876119909
= 0119872119909119909
= 0 and119872119909119910
= 0
1198961199090= Γ1199081199090
1198701199090= Γ1199091199090
1198701199101199090
= 1014
(17d)
The appropriateness of the three classical boundariesdefined in (17a)ndash(17c) will be proved by several examplesgiven in following the arbitrary elastic boundaries are alsodefined in (17d) and the Γ
119908119894(Γ119908119894= 1198961198941198960 1198960= 1 times 10
9Nm2119894 = 1199090 1199091198861199100119910119887) and Γ
119909119894(Γ119909119894= 11987011989411987001198700= 1times10
9Nmrad119894 = 1199090 119909119886 1199100 119910119887) elastic restraint parameters representcorresponding spring stiffness For the sake of simplicitya four-letter string is employed to represent the restraintcondition of a plate such that F-C-S-E identifies the platewithedges 119909 = 0 119910 = 0 119909 = 119886 and 119910 = 119887 having free clampedshear-supported restrained and elastic boundary conditionsrespectively
As for the first case a uniform thickness moderately thickplate with different classical boundaries and structure param-eters is investigated here In Table 2 the comparison of thefirst eight frequency parametersΩ = 120596119887
2
(120588ℎ119863)12
1205872 of the
considered plate is presented The S-S-S-S C-F-F-F S-S-F-FF-F-F-F and S-C-S-C boundary conditions are performed inthe comparison Excellent agreements are observed betweenthe solutions obtained by the modified Fourier method thereferential data and finite element method (FEM) results forthe uniform thickness moderately thick rectangular platesIt is also verified that the definition of the three types of
classical boundaries in (17a)ndash(17c) is appropriate In additionthe elastic boundary conditions (17d) are also verified Inthe next two examples we will account for the vibrationof moderately thick plate with elastic edge supports Thefirst model considered is an S-S-S-S square moderately thickplate with all edges elastically rotationally restrained Thatis 1198701199090
= 119870119909119886
= 1198701199100
= 119870119910119887
= Γ119909 and all the
other restraining springs are set to have an infinite stiffness(namely represented by 1014 in numerical calculation) Thesix frequency parameters Ω = 120596119887
2
(120588ℎ119863)12
1205872 are given
in Table 3 for several different restraining coefficient valuesthe finite element method (FEM) results calculated usingABAQUSare also listed in this table as a referenceThe secondmodel concerns a complete square moderately thick platewith all edges elastically restrained That is 119896
1199090= 119896119909119886
=
1198961199100
= 119896119910119887
= Γ119908and 119870
1199090= 119870119909119886
= 1198701199100
= 119870119910119887
= Γ119909 The
six frequency parameters Ω = 1205961198872
(120588ℎ119863)12
1205872 are given in
Table 4 for several different restraining coefficient values thefinite element method (FEM) results are also listed in Table 4as a reference It can be clearly seen that the comparison isextremely good which implies that the current method isable tomake correct predictions for themodal characteristicsof moderately thick plate with not only classical boundaryconditions but also elastic edge restraints
The excellent agreements between the solution obtainedby the modified Fourier method and the referential datafor the moderately thick plate subjected to the combina-tions of classical boundary conditions and elastic boundaryconditions given in Tables 2ndash4 indicate that the proposedmethod is sufficiently accurate to deal with uniform thicknessmoderately thick plate with arbitrary boundary conditions
33 Linearly VariationThickness Moderately Thick Plates withClassical and Elastic Boundary Conditions In the theoreticalformulations this paper concerns the varying thicknessmoderately thick plates with classical and elastic boundary
10 Shock and Vibration
Table 2 Frequency parameters Ω = 1205961198872
(120588ℎ119863)12
1205872 for uniform thickness moderately thick plates with different classical boundaries and
structure parameters
Boundary condition ab Model sequence1 2 3 4 5 6 7 8
S-S-S-S
11931 4605 4605 7064 8605 8605 10792 10792
(1933a) (4611) (4611) (7067) (8607) (8607) (10799) (10780)(1931b) (4605) (4605) (7064) (8605) (8605) (10792) (10792)
321408 2648 4127 4605 5261 7064 7153 8184(1410a) (2651) (4128) (4608) (5267) (7074) (7155) (8188)(1408b) (2648) (4127) (4605) (5261) (7064) (7151) (8184)
C-F-F-F
10348 0817 2035 2583 2862 4816 5478 5774
(0348a) (0817) (2038) (2586) (2865) (4821) (5504) (5793)(0348b) (0816) (2034) (2582) (2860) (4811) (5477) (5772)
320155 0502 0941 1665 2292 2613 3303 3498
(0154a) (0502) (0941) (1663) (2295) (2618) (3305) (3500)(0154b) (0501) (0940) (1662) (2292) (2612) (3298) (3494)
S-S-F-F
10333 1678 1875 3559 4720 4946 6474 6632
(0335a) (1680) (1878) (3600) (4721) (4949) (6478) (6642)(0333b) (1677) (1874) (3557) (4718) (4945) (6472) (6631)
320222 0938 1631 2363 2573 4082 4482 4753
(0223a) (0940) (1632) (2364) (2575) (4092) (4483) (4759)(0222b) (0938) (1631) (2361) (2571) (4077) (4479) (4752)
F-F-F-F1 1290 1919 2363 3235 3235 5605 5605 5646
(1291a) (1920) (2365) (3239) (3239) (5607) (5607) (6648)
32 0865 0948 1961 2166 2461 2843 3551 4026(0865a) (0949) (1963) (2168) (2465) (2844) (3556) (4034)
S-C-S-C1 2700 4971 5990 7973 8787 10250 11334 12024
(2701a) (4971) (5993) (7975) (8788) (10253) (11335) (12028)
32 2348 3253 4972 5646 6489 7381 7974 9930(2348a) (3255) (4974) (5647) (6590) (7384) (7801) (9932)
aResults in parentheses are taken from FEMbResults in parentheses are taken from [11]
conditions The varying thickness function ℎ(119909 119910) can beexpressed as ℎ
0(1 + 120572119909
119904
)(1 + 120573119910119905
) in which the ℎ0 120572 and
120573 represent the initial thickness gradient in 119909 direction andgradient in 119910 direction When the indexes 119904 and 119905 take thevalue 119904 = 119905 = 1 the analyticalmodel imitates the linearly vari-ation thickness moderately thick plates structure In order tounify the description and facilitate the analytical calculationsof the involved integrals all the thickness variation functionscan be expanded into either 1D or 2D Fourier cosine seriesresulting in
ℎ (119909 119910) =
infin
sum
119898=0
infin
sum
119899=0
ℎ0119902119898119899
cos 119898120587119909119886
cos119898120587119910
119887
(18)
where119902119898119899
=
4
119886119887
int
119887
0
int
119886
0
(1 + 120572119909119904
) (1 + 120573119910119905
) cos 119898120587119909119886
cos119898120587119910
119887
119889119909 119889119910
(19)
In order to prove the validity of the present methodfor the vibration of linearly variation thickness moderately
thick plates with arbitrary boundary conditions the typicalclassical boundary conditions as the first case will be con-sidered In Table 5 the comparison of the first six frequencyparameters Ω = 120596119886
2
(120588ℎ01198630)12 of the moderately thick
plates with linearly varying thickness is presentedThe S-S-S-S C-F-F-F S-S-F-F C-C-C-C and S-C-S-C boundary condi-tions are performed in the comparison Excellent agreementsare observed between the solutions obtained by the modifiedFourier method and finite element method (FEM) results forthemoderately thick plates with linear variation thickness Toinvestigate the influence of the aspect ratio on the uniformthickness and nonuniform thickness moderately thick platesthe effect on the frequency parameters for plates with S-S-S-Sboundary conditions is presented in Figure 6 The thicknessfunctions are ℎ
0and ℎ0(1+05times119909)(1+05times119910) respectively It
is seen from Figure 6 that the influence of aspect ratios on thefrequency parameters for nonuniform thickness moderatelythick plates is more complicated
In the next two examples we also account for the vibra-tions of moderately thick plate with linear variation thicknessand elastic edge supports The first model considered is an S-S-S-S square moderately thick plate with all edges elastically
Shock and Vibration 11
Table 3 Frequency parameters Ω = 1205961198872
(120588ℎ119863)12
1205872 for S-S-S-S moderately thick plates (119886119887 = 1) with uniform thickness and elastic
rotation support (1198701199090= 119870119909119886= 1198701199100= 119870119910119887= Γ119909)
Γ119909
aℎ0119886
Model sequence1 2 3 4 5 6
0
005 1945 4834 4834 7591 9515 9519(1942b) (4836) (4836) (7584) (9547) (9552)
01 1856 4500 4500 6829 8499 8513(1858b) (4510) (4510) (6843) (8539) (8554)
015 1758 4114 4114 6040 7421 7445(1761b) (4126) (4126) (6060) (7462) (7486)
02 1656 3733 3733 5335 6468 6499(1660b) (3747) (3747) (5357) (6508) (6538)
05
005 3541 7071 7071 10227 12318 12399(3546b) (7091) (7091) (10253) (12386) (12468)
01 3274 6242 6242 8719 10302 10421(3282b) (6267) (6267) (8754) (10370) (10488)
015 2947 5361 5361 7290 8465 8606(2956b) (5387) (5387) (7327) (8524) (8666)
02 2624 4599 4599 6148 7057 7199(2634b) (4624) (4624) (6182) (7107) (7250)
1
005 3541 7072 7072 10228 12320 12401(3547b) (7093) (7093) (10254) (12388) (12470)
01 3278 6248 6248 8727 10311 10429(3286b) (6274) (6274) (8762) (10378) (10497)
015 2957 5374 5374 7305 8478 8620(2966b) (5400) (5400) (7341) (8537) (8680)
02 2641 4617 4617 6166 7070 7215(2651b) (4643) (4643) (6200) (7121) (7268)
15
005 3542 7073 7073 10229 12320 12401(3547b) (7093) (7093) (10254) (12388) (12471)
01 3280 6251 6251 8729 10313 10432(3287b) (6276) (6276) (8765) (10381) (10500)
015 2960 5379 5379 7309 8482 8625(2969b) (5405) (5405) (7346) (8541) (8685)
02 2647 4623 4623 6172 7074 7221(2657b) (4649) (4649) (6206) (7125) (7274)
2
005 3542 7073 7073 10229 12321 12402(3547b) (7093) (7093) (10255) (12389) (12471)
01 3280 6252 6252 8730 10315 10434(3288b) (6277) (6277) (8766) (10382) (10501)
015 2962 5381 5381 7312 8484 8628(2971b) (5407) (5407) (7348) (8543) (8688)
02 2650 4627 4627 6175 7076 7224(2660b) (4652) (4652) (6210) (7128 (7277)
aΓ119909= 1198701198941198700 (1198700= 1 times 10
9 Nmrad 119894 = 1199090 119909119886 1199100 119910119887)bResults in parentheses are taken from FEM
12 Shock and Vibration
Table 4 Frequency parameters Ω = 1205961198872
(120588ℎ119863)12
1205872 for moderately thick plates (119886119887 = 1) with uniform thickness and all edge elastic
restraints (1198701199090= 119870119909119886= 1198701199100= 119870119910119887= Γ119909and 119896
1199090= 119896119909119886= 1198961199100= 119896119910119887= Γ119908)
Γ119909
aΓ119908
aℎ0119886
Model sequence1 2 3 4 5 6
05 0
005 0986 1967 3893 3893 4850 4850(0988b) (1971) (3897) (3897) (4852) (4852)
01 0916 1835 3515 3522 4375 4375(0917b) (1837) (3519) (3528) (4341) (4341)
015 0789 1635 2983 3027 3771 3771(0792b) (1638) (2988) (3029) (3775) (3775)
02 0646 1444 2497 2601 3249 3249(0652b) (1448) (2508) (2621) (3254) (3254)
1 05
005 3417 3417 4431 5294 5461 6448(3419b) (3419) (4435) (5303) (5478) (6456)
01 1593 1593 2401 3849 3861 4729(1611b) (1611) (2408) (3852) (3866) (4767)
015 1122 1122 1913 3272 3290 4038(1127b) (1227) (1924) (3284) (3314) (4069)
02 0890 0890 1640 2781 2829 3476(0899b) (0899) (1654) (2787) (2834) (3497)
15 1
005 4325 4325 5619 6273 6571 7619(4334b) (4334) (5681) (6299) (6642) (7642)
01 2013 2013 2812 4088 4116 4993(2024b) (2024) (2827) (4122) (4187) (4994)
015 1338 1338 2108 3430 3443 4196(1342b) (1342) (2112) (3467) (3482) (4200)
02 1038 1038 1772 2936 2965 3613(1042b) (1042) (1786) (2937) (2977) (3624)
2 2
005 5231 5231 6919 7630 8002 9220(5238b) (5238) (6931) (7637) (8011) (9241)
01 2594 2594 3439 4470 4544 5428(2597b) (2597) (3442) (4495) (4557) (5438)
015 1653 1653 2403 3613 3632 4392(1662b) (1662) (2414) (3627) (3646) (4404)
02 1238 1238 1949 3077 3099 3751(1244b) (1244) (1964) (3094) (3116) (3788)
aΓ119909= 1198701198941198700 (1198700= 1 times 10
9 Nmrad 119894 = 1199090 119909119886 1199100 119910119887) Γ119908= 1198961198941198960 (1198960= 1 times 10
9 Nm 119894 = 1199090 119909119886 1199100 119910119887)bResults in parentheses are taken from FEM
05 10 15 20 25 300
2
4
6
8
10
12
14
16
ab
The 1st orderThe 2nd orderThe 3rd order
The 4th orderThe 5th orderThe 6th order
Ω
(a)
05 10 15 20 25 30
30
60
90
120
150
180
ab
The 1st orderThe 2nd orderThe 3rd order
The 4th orderThe 5th orderThe 6th order
Ω
(b)
Figure 6 The effect of aspect ratio 119886119887 on the natural frequenciesΩ for S-S-S-S boundary condition (a) uniform thickness and (b) nonuni-form thickness
Shock and Vibration 13
Table 5 Frequency parameter Ω = 1205961198862
(120588ℎ01198630)12 for moderately thick plates with linearly varying thickness in different boundary
conditions (ℎ(119909 119910) = ℎ0(1 + 05119909)(1 + 05119910))
Boundary condition 119886119887 ℎ0119886
Model sequence1 2 3 4 5 6
S-S-S-S
101 27261 62397 62926 92423 111537 111732
(27263a) (62410) (63842) (92445) (111867) (111921)
02 23103 47010 47191 65018 75831 75865(23489a) (47209) (47349) (65057) (76605) (76473)
3201 59338 99101 138798 151102 166710 204855
(59874a) (99249) (138177) (151362) (166644) (205381)
02 41175 63765 63765 73598 83956 89084(41586a) (63313) (63669) (73661) (83918) (89547)
C-F-F-F
101 4173 10331 27407 37826 38809 64095
(4856a) (10680) (27513) (37203) (39135) (64858)
02 3983 8702 21403 29443 30286 46705(4118a) (9306) (21990) (29670) (30602) (47156)
3201 4868 15488 35620 63523 89969 100089
(4887a) (15320) (35175) (63353) (89159) (100783)
02 4469 9763 22338 42188) 50104 56331(4944a) (9636) (22087) (42177) (50580) (56559)
S-S-F-F
101 4117 24795 26216 48567) 68195 69667
(4789a) (24149) (26751) (49324) (68842) (70575)
02 3820 20516 21924 37523 48980 50327(4103a) (20041) (22125) (37221) (48894) (50622)
3201 7935 39454 65063 92877 92877 136176
(7856a) (40145) (65162) (92462) (93022) (136336)
02 6389 26149 35627 49539 56844 65170(6406a) (26536) (35626) (49831) (57144) (65981)
C-C-C-C
101 44471 80299 80681 109422 126871 128207
(45102a) (80271) (80656) (110015) (126328) (128888)
02 32162 52781 52933 69410 78474 79482(32240a) (52629) (53166) (70063) (79059) (80266)
3201 82975 115587 151992 161579 176601 212330
(83139a) (115838) (152200) (162193) (177058) (212849)
02 48577 66022 83468 90830 97474 98157(49187a) (66259) (84065) (91749) (97827) (98777)
S-C-S-C
101 36847 66664 76965 101045 113594 125551
(37901a) (68568) (78356) (101200) (113978) (125651)
02 27673 48469 51220 67027 76358 78241(27897a) (48685) (51491) (67447) (77530) (79573)
3201 77378 106848 150079 154569 173120 208391
(78003a) (107058) (150991) (155171) (173099) (208402)
02 45895 64732 73393 83282 90325 97089(45803a) (64131) 73656 83801 90518 97015
aResults in parentheses are taken from FEM
restrained That is 1198701199090
= 119870119909119886
= 1198701199100
= 119870119910119887
= Γ119909 The six
frequency parameters Ω = 1205961198862
(120588ℎ01198630)12 are given in
Table 6 for several different restraining coefficient valuesthe finite element method (FEM) results calculated usingABAQUS are also presented as a referenceThe secondmodelis a complete square moderately thick plate with all edgeselastically restrainedThat is 119896
1199090= 119896119909119886= 1198961199100= 119896119910119887= Γ119908and
1198701199090= 119870119909119886
= 1198701199100= 119870119910119887= Γ119909 The six frequency parameters
Ω = 1205961198862
(120588ℎ01198630)12 are given in Table 7 for several different
restraining coefficient values the finite element method(FEM) results are also listed as a reference It can be clearlyseen that the comparison is extremely good which impliesthat the current method is able to make correct predictionsfor the modal characteristics of linearly varying thickness
14 Shock and Vibration
Table 6 Frequency parameter Ω = 1205961198862
(120588ℎ01198630)12 for S-S-S-S moderately thick plates (119886119887 = 1) with linearly varying thickness and elastic
rotation support (ℎ(119909 119910) = ℎ0(1 + 05119909)(1 + 05119910) 119870
1199090= 119870119909119886= 1198701199100= 119870119910119887= Γ119909)
Γ119909
aℎ0119886
Model sequence1 2 3 4 5 6
0
005 28809 70080 70891 108967 134884 135295(28897b) (70334) (71129) (109112) (135254) (135772)
01 27261 62398 62926 92423 111540 111735(26606b) (61784) (62303) (91193) (111323) (111648)
015 25231 54192 54505 77118 91228 91312(25204b) (55297) (55595) (77503) (91871) (91941)
02 23103 47010 47191 65018 75832 75866(23947b) (47071) (47208) (65404) (76531) (76589)
05
005 23103 47010 47191 65018 75832 75866(23210b) (47245) (47358) (65241) (76122) (76345)
01 48377 94650 95597 135484 162635 163501(48524b) (94891) (95992) (135845) (167012) (164024)
015 36239 70939 71298 99997 118584 119009(36647b) (71214) (71687) (100024) (119114) (119457)
02 28749 56937 57037 79214 93009 93162(29142b) (57034) (57544) (79654) (93668) (93754)
1
005 49651 96759 97619 138089 165368 166304(49755b) (97245) (97964) (138544) (165864) (166878)
01 38951 73735 74251 102748 121079 121709(39021b) (74145) (74664) (102874) (121613) (121817)
015 30630 58406 58533 80432 93984 94255(30638b) (58684) (58644) (80874) (94121) (94478)
02 25548 48636 48674 66165 76708 76824(25687b) (48788) (48992) (66544) (76824) (76944)
15
005 50118 97561 98374 139096 166421 167382(50247b) (97877) (98661) (139247) (166875) (167662)
01 40310 75216 75783 104229 122402 123164(40366b) (75542) (75921) (104557) (122874) (123385)
015 31837 59352 59530 81245 94611 94983(31902b) (59451) (59871) (81375) (94784) (95105)
02 26302 49119 49156 66528 76954 77130(26418b) (49354) (49334) (66921) (77010) (77470)
2
005 50360 97983 98769 139631 166979 167954(50472b) (98005) (99114) (140154) (167188) (167986)
01 41130 76146 76724 105165 123229 124082(41421b) (76642) (78114) (105661) (123278) (124974)
015 32687 60025 60245 81832 95052 95508(32778b) (60244) (60549) (82146) (95114) (95842)
02 26893 49492 49539 66815 77138 77370(26922b) (49874) (49924) (66987) (77367) (77588)
aΓ119909= 1198701198941198700 (1198700= 1 times 10
9 Nmrad 119894 = 1199090 119909119886 1199100 119910119887)bResults in parentheses are taken from FEM
Shock and Vibration 15
Table 7 Frequency parameterΩ = 1205961198862
(120588ℎ01198630)12 for moderately thick plates (119886119887 = 1) with linearly varying thickness and elastic rotation
and translation support (ℎ(119909 119910) = ℎ0(1 + 05119909)(1 + 05119910) 119870
1199090= 119870119909119886= 1198701199100= 119870119910119887= Γ119909 and 119896
1199090= 119896119909119886= 1198961199100= 119896119910119887= Γ119908)
Γ119909
aΓ119908
aℎ0119886
Model sequence1 2 3 4 5 6
05 0
005 14751 14917 28232 55251 55397 69512(14780b) (14822) (29252) (55476) (55505) (70904)
01 11811 11910 23857 44462 45037 55453(11818b) (11897) (24711) (44376) (45257) (56511)
015 8487 8514 20011 34783 36335 44970(8500b) (8544) (20738) (34404) (36663) (45625)
02 6073 6073 17431 28679 30805 38010(6096b) (6106) (18039) (28129) (31165) (38311)
1 05
005 21225 32173 32173 44381 64061 65222(20708b) (32097) (32285) (45110) (64182) (65355)
01 8140 16878 16941 28100 48663 49015(7943b) (16838) (16970) (28888) (48700) (49171)
015 4493 11859 11878 22429 38352 39247(4383b) (11867) (11891) (23125) (38177) (39488)
02 2932 8690 8691 18919 31108 32600(3540b) (8791) (8791) (16229) (27520) (27889)
15 1
005 27819 40919 41430 54595 70792 72896(27146b) (41008) (41434) (55256) (70895) (73051)
01 11285 20068 20230 31101 51046 51429(11012b) (20025) (20270) (31849) (51126) (51569)
015 6292 13912 13922 24132 40500 41120(6139b) (13912) (13934) (24808) (40426) (41316)
02 4121 10321 10331 20051 32796 33921(4021b) (10329) (10357) (20635) (32540) (34178)
2 2
005 34615 51972 53088 68255 81344 84510(33806b) (52100) (53157) (68891) (81433) (84761)
01 15416 24659 24829 35615 53879 54494(15045b) (24625) (24888) (36303) (53970) (54626)
015 8740 16420 16481 26353 42417 42946(8529b) (16414) (16499) (26996) (42398) (43114)
02 5761 12127 12127 21434 34286 35192(5622b) (12134) (12148) (21998) (34110) (35415)
aΓ119909= 1198701198941198700 (1198700= 1 times 10
9 Nmrad 119894 = 1199090 119909119886 1199100 119910119887) Γ119908= 1198961198941198960 (1198960= 1 times 10
9 Nm 119894 = 1199090 119909119886 1199100 119910119887)bResults in parentheses are taken from FEM
moderately thick plate with not only classical boundaryconditions but also elastic edge restraints
As the last case of this subsection the influence of thegradient 120572 and 120573 on the fundamental frequency parametersfor a linearly varying thickness moderately thick plate isinvestigated The model is a square moderately thick platewith all edges elastically restrainedThat is 119896
1199090= 119896119909119886= 1198961199100=
119896119910119887= Γ119908= 2 and 119870
1199090= 119870119909119886
= 1198701199100= 119870119910119887= Γ119909= 2 The six
frequency parameters Ω = 1205961198862
(120588ℎ01198630)12 are given in
Table 8 for several different 120572 and 120573 values the finite elementmethod (FEM) results are also listed as a reference Againgood agreement can be observed Through Table 8 it is alsofound that the frequency parameter increases with increasinggradient parameters
The above studies are given as linearly varying thicknessmoderately thick plates with several of boundary condition
and different structure parameters In the next section vibra-tion results for the plates subjected to nonlinear variationthickness will be presented
34 Nonlinearly Varying Thickness Moderately Thick Plateswith Classical and Elastic Boundary Conditions In Sec-tion 33 the linearly varying thickness moderately thickplates were studied However in the practical engineeringapplications the varying thickness of amoderately thick platemay not always be linear variation in nature A variety ofpossible thickness varying cases may be encountered in prac-tice Therefore the moderately thick plates with nonlinearvariation thickness subjected to general elastic edge restraintsare examined in this subsection For the sake of brevity theindexes 119904 and 119905 will be chosen as 2 to imitate the nonlinearlyvarying thickness moderately thick plates structure in this
16 Shock and Vibration
Table 8 Frequency parameterΩ = 1205961198862
(120588ℎ01198630)12 for elastic support moderately thick plates (119886119887 = 1) with different gradient (119896
1199090= 119896119909119886=
1198961199100= 119896119910119887= Γ119908= 2 119870
1199090= 119870119909119886= 1198701199100= 119870119910119887= Γ119909= 2)
120572 120573 ℎ0119886
Model sequence1 2 3 4 5 6
120572 = 120573 = 01
005 31275 52615 52657 68954 76146 79850(31303a) (52664) (52675) (69033) (76174) (79948)
01 17270 25244 25250 33949 46050 46625(17271a) (25255) (25258) (34089) (46122) (46708)
015 10190 16266 16274 24164 37327 37526(10189a) (16269) (16280) (24323) (37401) (37622)
02 6780 12258 12260 19685 31452 31754(6780a) (12265) (12270) (19851) (31514) (31864)
120572 = 120573 = 02
005 32530 52947 53141 68954 77034 80589(32577a) (53014) (53203) (69172) (77088) (80773)
01 16882 24979 24979 34154 48057 48575(16883a) (24983) (25001) (34433) (48134) (48667)
015 9804 16284 16309 24683 38844 39085(9803a) (16283) (16322) (24980) (38909) (39191)
02 6499 12290 12295 20133 32354 32770(6498a) (12296) (12307) (20420) (32387) (32895)
120572 = 120573 = 03
005 33516 52845 53308 68704 78203 81561(33563a) (52939) (53389) (69055) (78271) (81789)
01 16419 24799 24830 34527 50061 50583(16419a) (24790) (24869) (34947) (50143) (50688)
015 9430 16334 16378 25236 40193 40506(9429a) (16329) (16395) (25660) (40243) (40625)
02 6235 12288 12293 20572 33102 33663(6235a) (12296) (12306) (20965) (33090) (33811)
120572 = 120573 = 04
005 34215 52463 53264 68430 79654 82865(34252a) (52575) (53346) (68921) (79732) (83113)
01 15921 24695 24793 35027 52011 52571(15921a) (24674) (24842) (35583) (52098) (52688)
015 9074 16388 16444 25797 41379 41789(9074a) (16381) (16464) (26336) (41404) (41928)
02 5990 12236 12238 21002 33735 34461(5989a) (12249) (12250) (21487) (33653) (34641)
120572 = 120573 = 05
005 34615 51972 53088 68255 81344 84510(33806a) (52100) (53157) (68891) (81433) (84761)
01 15416 24659 24829 35615 53879 54494(15045a) (24625) (24888) (36303) (53970) (54626)
015 8740 16420 16481 26353 42417 42946(8529a) (16414) (16499) (26996) (42398) (43114)
02 5761 12127 12127 21434 34286 35192(5622a) (12134) (12148) (21998) (34110) (35415)
aResults in parentheses are taken from FEM
section that is 119904 = 119905 = 2 Also the thickness varying functionsare also be expressed as Fourier cosine series according to (18)and (19)
In order to validate the accuracy and reliability of theproposed method for predicting the vibration behavior ofnonlinearly varying thickness moderately thick plates with
arbitrary boundary conditions the typical classical boundaryconditions viewed as the special cases of elastically restrainededges will be considered The comparison of the first six fre-quency parameters Ω = 120596119886
2
(120588ℎ01198630)12 for the moderately
thick plates with nonlinearly varying thickness is presentedin Table 9 The S-S-S-S C-F-F-F S-S-F-F C-C-C-C and
Shock and Vibration 17
Table 9 Frequency parameter Ω = 1205961198862
(120588ℎ01198630)12 for moderately thick plates with linearly varying thickness in different boundary
conditions (ℎ(119909 119910) = ℎ0(1 + 05119909
2
)(1 + 051199102
))
Boundary condition 119886119887 ℎ0119886
Model sequence1 2 3 4 5 6
S-S-S-S
101 23462 54999 55573 82950 100900 101187
(23741a) (55114) (55983) (83146) (101004) (101347)
02 18606 42725 42971 60040 71167 71761(18936a) (43014) (43148) (60247) (71642) (71993)
3201 53055 65251 85404 106700 114880 143531
(53467a) (65472) (85984) (107211) (115687) (144007)
02 17439 21531 32170 44987 59438 64179(17874a) (21748) (32651) (45264) (60188) (64879)
C-F-F-F
101 3658 9317 24135 32816 35433 59179
(3749a) (9547) (24034) (33627) (36308) (60142)
02 3520 7986 19591 26959 27491 43882(3611a) (8176) (20075) (27125) (27625) (43984)
3201 3901 9963 28984 69187 73686 89360
(3930a) (10095) (29381) (69578) (73994) (90242)
02 3292 4439 13385 23947 40850 44758(3386a) (4498) (14012) (24068) (41221) (45142)
S-S-F-F
101 3557 21268 22762 43524 60253 62429
(3649a) (21441) (22796) (43916) (60622) (62630)
02 3171 17822 19512 33616 45042 46833(3489a) (18006) (20430) (33815) (45247) (49004)
3201 6738 31658 52071 75625 86604 94903
(6957a) (31882) (52518) (75823) (86939) (95610)
02 3653 12930 20955 26517 44097 48058(6807a) (13160) (21312) (26864) (44422) (48873)
C-C-C-C
101 41020 75196 75614 104142 119683 120949
(41405a) (75452) (75882) (104711) (119637) (121137)
02 31237 51686 51868 68411 76868 77864(32010a) (52960) (52145) (69335) (77481) (78099)
3201 95823 128727 163657 173751 190890 223595
(95364a) (128932) (163588) (172350) (190350) (222948)
02 53306 71372 89384 102404 104660 120908(53587a) (71759) (90103) (102503) (105174) (121275)
S-C-S-C
101 33715 60298 71446 94187 103802 117928
(34148a) (60786) (71828) (94515) (104263) (119326)
02 26143 45499 49458 64253 72941 76191(26789a) (46622) (50680) (64329) (73474) (76843)
3201 73228 115845 137445 148166 168342 182542
(73761a) (116121) (137788) (148530) (168698) (182921)
02 41302 53648 65630 72202 81632 88105(41304a) (53757) (65842) (72794) (82033) (88474)
aResults in parentheses are taken from FEM
S-C-S-C boundary conditions are performed in the compar-ison The results adequately demonstrated the great accuracyof the modified Fourier method
We now turn to elastically restrained moderately thickplates The first one involves an S-S-S-S moderately thicksquare plate with a uniform rotational restraint along eachedge that is 119870
1199090= 119870119909119886
= 1198701199100
= 119870119910119887
= Γ119909 The calculated
frequency parameters Ω = 1205961198862
(120588ℎ01198630)12 are given in
Table 10 together with the FEM results Since the elasticallyrestrained plates with nonlinear variation thickness are rarelyinvestigated the FEMresults are used as the referenceA goodagreement is observed between the current and FEM resultsThe second example concerns amoderately thick square plateelastically supported along all edgesThe stiffness of the linear
18 Shock and Vibration
Table 10 Frequency parameterΩ = 1205961198862
(120588ℎ01198630)12 for S-S-S-S moderately thick plates (119886119887 = 1) with linearly varying thickness and elastic
rotation support (ℎ(119909 119910) = ℎ0(1 + 05119909
2
)(1 + 051199102
) 1198701199090= 119870119909119886= 1198701199100= 119870119910119887= Γ119909)
Γ119909
aℎ0119886
Model sequence1 2 3 4 5 6
0
005 23462 54999 55573 82950 100900 101187(23741b) (55114) (55983) (83146) (101004) (101347)
01 18606 42725 42971 60040 71167 71761(18936b) (43014) (43148) (60247) (71642) (71993)
015 53055 65251 85404 106700 114880 143531(53467b) (65472) (85984) (107211) (115687) (144007)
02 17439 21531 32170 44987 59438 64179(17874b) (21748) (32651) (45264) (60188) (64879)
05
005 6904 13374 13514 19400 22904 23056(9909b) (13548) (13651) (19564) (23088) (23379)
01 5316 10237 10314 14499 17155 17264(5374b) (10448) (10651) (14764) (17188) (17879)
015 4136 8201 8220 11474 13578 13671(4174b) (8348) (8651) (11764) (13688) (13879)
02 3449 6915 6924 9549 11225 11327(3474b) (6948) (6951) (9764) (11388) (11579)
1
005 7040 13617 13745 19726 23233 23402(7074b) (13748) (13851) (19964) (23488) (23879)
01 5712 10696 10793 14991 17594 17744(5774b) (10748) (10851) (15054) (17788) (17879)
015 4489 8507 8540 11759 13803 13922(4514b) (8548) (8611) (11864) (13988) (14119)
02 3677 7084 7090 9692 11330 11445(3684b) (7148) (7191) (9764) (11588) (11879)
15
005 7089 13708 13828 19847 23355 23530(7124b) (13748) (13951) (19967) (23588) (23879)
01 5895 10923 11024 15239 17813 17988(5874b) (10748) (11651) (15264) (17818) (17979)
015 4697 8694 8737 11938 13790 14083(4774b) (8748) (8951) (11864) (13688) (14879)
02 3836 7201 7210 9792 11403 11530(3874b) (7748) (7651) (9864) (11588) (11879)
2
005 7114 13754 13871 19911 23418 23597(7174b) (12848) (13851) (19864) (23588) (23879)
01 6001 11060 11159 15391 17944 18136(6004b) (11148) (11151) (15464) (17988) (18379)
015 4836 8821 8872 12061 14037 14189(4874b) (8848) (8951) (12164) (14188) (14879)
02 3954 7289 7301 9871 11457 11594(3974b) (7348) (7651) (9964) (11488) (11679)
aΓ119909= 1198701198941198700 (1198700= 1 times 10
9 Nmrad 119894 = 1199090 119909119886 1199100 119910119887)bResults in parentheses are taken from FEM
and rotational restraints is set equal to 1198961199090
= 119896119909119886
= 1198961199100
=
119896119910119887
= Γ119908and 119870
1199090= 119870119909119886
= 1198701199100
= 119870119910119887
= Γ119909 respectively
The six frequency parametersΩ = 1205961198862
(120588ℎ01198630)12 are shown
in Table 11 for several different restraining coefficient valuesIt can also be noticed that the Fourier series method is able
to predict the modal characteristics of nonlinearly varyingthickness moderately thick plate with not only classicalboundary conditions but also elastic edge restraints correctly
Finally the influence of the gradient on the fundamentalfrequency parameters of a nonlinearly varying thickness
Shock and Vibration 19
Table 11 Frequency parameterΩ = 1205961198862
(120588ℎ01198630)12 for moderately thick plates (119886119887 = 1) with linearly varying thickness and elastic rotation
and translation support (ℎ(119909 119910) = ℎ0(1 + 05119909
2
)(1 + 051199102
) 1198701199090= 119870119909119886= 1198701199100= 119870119910119887= Γ119909 and 119896
1199090= 119896119909119886= 1198961199100= 119896119910119887= Γ119908)
Γ119909
aΓ119908
aℎ0119886
Model sequence1 2 3 4 5 6
05 0
005 12926 13041 25090 47935 47943 61761(12931b) (13053) 25099 (47947) (47952) (61788)
01 10900 10988 21545 39980 40147 50322(10904b) (10992) (21557) (39984) (40166) (50334)
015 8245 8298 18081 31503 32617 40950(8247b) (8230) (18092) (31508) (32622) (40955)
02 6106 6124 15639 25631 27601 34458(6110b) (6125) (15642) (25637) (27617) (34462)
1 05
005 21575 31615 31644 42573 57889 58607(21580b) (31617) (31644) (42581) (57893) (58610)
01 8451 16024 16210 25969 43839 43867(8451b) (16025) (16217) (25972) (43844) (43870)
015 4682 11479 11480 20642 35172 35598(4684b) (11480) (11481) (20644) (35178) (35604)
02 3059 8621 8648 17320 28487 29638(3062b) (8619) (8644) (17327) (28492) (29641)
15 1
005 27713 40203 40978 52951 65352 66815(27714b) (40207) (40988) (52972) (65368) (66822)
01 11667 19322 19615 29122 46115 46183(11674b) (19327) (19619) (29137) (46120) (46188)
015 6544 13416 13481 22405 37238 37430(6547b) (13420) (13488) (22410) (37241) (37434)
02 4295 10175 10190 18543 30330 31056(4296b) (10177) (10194) (18544) (30336) (31060)
2 2
005 33453 50854 52088 66311 76610 78947(33455b) (50852) (50097) (66321) (76627) (78971)
01 15832 24130 24372 33924 49118 49362(15830b) (24127) (24374) (33931) (49147) (49382)
015 9060 15956 16099 24754 39134 39249(9064b) (15961) (16102) (24755) (39138) (39452)
02 5992 11955 11971 20034 31908 32392(5991b) (11957) (11978) (20042) (31910) (32398)
aΓ119909= 1198701198941198700 (1198700= 1 times 10
9 Nmrad 119894 = 1199090 119909119886 1199100 119910119887) Γ119908= 1198961198941198960 (1198960= 1 times 10
9 Nm 119894 = 1199090 119909119886 1199100 119910119887)bResults in parentheses are taken from FEM
moderately thick plate is investigated The plate is elasticallyrestrained in which the stiffness of the boundary springsis taken as 119896
1199090= 119896119909119886
= 1198961199100
= 119896119910119887
= Γ119908
= 2 and1198701199090
= 119870119909119886
= 1198701199100
= 119870119910119887
= Γ119909= 2 respectively The six
frequency parameters Ω = 1205961198862
(120588ℎ01198630)12 are presented in
Table 12 for several different slop values It can be seen thatthe fundamental frequency parameters will decrease with theincrease of the parameter ℎ
0119886
In the above examples it has been demonstrated thatthe presented method can be universally applied to nonlin-early varying thickness moderately thick plates with severalboundary conditions and different structure parametersNew results are obtained for plates with nonlinear variationthickness in both directions subjected to general elastic
boundary restraints which may be used for benchmarkingof researchers in the field In addition it is interesting to seethat the nature frequency decreases with the increase of theindex for thickness function
4 Conclusions
In this paper a modified Fourier method has been pre-sented to study the free vibration behaviors of moderatelythick rectangular plates with variable thickness and arbi-trary boundary conditionsThe first-order shear deformationplate theory is adopted to formulate the theoretical modelThe displacements and rotation components of the plateregardless of boundary conditions are invariantly expressedas the superposition of a 2D Fourier cosine series and four
20 Shock and Vibration
Table 12 Frequency parameterΩ = 1205961198862
(120588ℎ01198630)12 for elastic support moderately thick plates (119886119887 = 1) with different gradient (119896
1199090= 119896119909119886=
1198961199100= 119896119910119887= Γ119908= 2 119870
1199090= 119870119909119886= 1198701199100= 119870119910119887= Γ119909= 2)
120572 120573 ℎ0119886
Model sequence1 2 3 4 5 6
120572 = 120573 = 01
005 30187 51102 51141 67066 73954 77412(30188a) (51105) (51144) (67068) (73961) (77477)
01 16995 24719 24724 33071 44343 44819(16996a) (24725) (24729) (33084) (44347) (44822)
015 10077 15872 15883 23443 35937 36065(10081a) (15877) (15892) (23447) (35941) (36084)
02 6713 11953 11958 19086 30372 30604(6713a) (11957) (11965) (19092) (30378) (30612)
120572 = 120573 = 02
005 31206 51436 51614 67058 74398 77595(31208a) (51438) (51619) (67064) (74402) (77596)
01 16786 24503 24508 33144 45583 45923(16790a) (24510) (24513) (33147) (45587) (45932)
015 9821 15865 15906 23766 36897 36990(9824a) (15870) (15907) (23768) (36894) (36997)
02 6523 11978 11992 19356 30912 31162(6524a) (11981) (11997) (19362) (30916) (31174)
120572 = 120573 = 03
005 32097 51462 51900 66832 74991 77863(32098a) (51466) (51907) (66849) (74996) (77872)
01 16507 24332 24383 33330 46806 47074(16511a) (24327) (24380) (33339) (46810) (47086)
015 9562 15885 15964 24103 37748 37828(9562a) (15892) (15977) (21105) (37751) (37832)
02 6338 11992 12013 19603 31338 31634(6340a) (11994) (12011) (19612) (31344) (31639)
120572 = 120573 = 04
005 32851 51239 52046 66547 75735 78303(33455a) (50852) (50097) (66321) (76627) (78971)
01 16182 24209 24343 33598 47990 48230(33455a) (50852) (50097) (66321) (76627) (78971)
015 9307 15919 16034 24436 38491 38579(33455a) (50852) (50097) (66321) (76627) (78971)
02 6161 11986 12008 19828 31666 32040(33455a) (50852) (50097) (66321) (76627) (78971)
120572 = 120573 = 05
005 33453 50854 52088 66311 76610 78947(33455a) (50852) (50097) (66321) (76627) (78971)
01 15832 24130 24372 33924 49118 49362(15830a) (24127) (24374) (33931) (49147) (49382)
015 9060 15956 16099 24754 39134 39249(9064a) (15961) (16102) (24755) (39138) (39452)
02 5992 11955 11971 20034 31908 32392(5991a) (11957) (11978) (20042) (31910) (32398)
aResults in parentheses are taken from FEM
supplementary functions in the form of the product of apolynomial function and a single cosine series expansionto ensure and accelerate the convergence of the solutionAt each edge of the plate the general restraint conditionsare implemented by introducing one group of linear springsand two groups of rotational springs which are continuouslydistributed and determined by the stiffness of these springs
Instead of seeking a solution in strong forms in the previousstudies all the Fourier coefficients will be treated equallyand independently as the generalized coordinates and solveddirectly from the Rayleigh-Ritz technique The change of theboundary conditions can be easily achieved by only varyingthe stiffness of the three sets of boundary springs along alledges of the rectangular plates without involving any change
Shock and Vibration 21
to the solution procedure The convergence of the presentsolution is examined and the excellent accuracy is validatedby comparison with existing results published in the lit-erature and FEM data Excellent agreements are obtainedfrom these comparisons The proposed method providesa unified means for extracting the modal parameters andpredicting the vibration behaviors of moderately thick plateswith variable thickness variation functions and arbitraryelastic edge restraints A variety of free vibration results formoderately thick rectangular plates with different thicknessvariation functions and boundary conditions are presentedNew results for free vibration of moderately thick rectangularplates with various thickness variation functions and edgeconditions are presented which may be used for benchmark-ing of researchers in the field
Appendices
A Representative Calculation forStiffness and Mass Matrices
To illuminate the particular expression of the mass andstiffness matrixes clearly and tersely four new variables aredefined as 119904 = 119898 lowast (119873 + 1) + 119899 + 1 119905 = 119898
1015840
lowast (119873 + 1) + 1198991015840
+ 1119901 = (119897minus1)lowast (119872+1)+119898
1015840
+1 and 119902 = (119897minus1)lowast (119873+1)+1198991015840
+1The first row elements of K andM are given below
K1-1119904119905
=
1
2
int
119886
0
int
119887
0
[119863 (119909 119910)
times 21205821198861198981205821198861198981015840 sin 120582
119886119898119909 sin 120582
1198861198981015840119909 cos 120582
119887119899119910
times cos 1205821198871198991015840119910 + (1 minus 120583) 120582
1198871198991205821198871198991015840 cos 120582
119886119898119909
times cos 1205821198861198981015840 sin 120582
119887119899119910 sin 120582
1198871198991015840119910
+ 120581119866ℎ (119909 119910) cos 120582119886119898119909 cos 120582
1198861198981015840119909 cos 120582
119887119899
times119910 cos 1205821198871198991015840119910119889119909119889119910] 119889119909 119889119910
+ int
119886
0
1198701199091199100
cos 120582119886119898119909 cos 120582
1198861198981015840119909
+(minus1)119898+1198981015840
119870119909119910119887
cos 120582119886119898119909 cos 120582
1198861198981015840119909 119889119909
+ int
119887
0
1198701199090cos 120582119887119899119910 cos 120582
1198871198991015840119910
+(minus1)119899+1198991015840
119870119909119886cos 120582119887119899119910 cos 120582
1198871198991015840119910 119889119910
K1-2119904119901
=
1
2
int
119886
0
int
119887
0
[119863 (119909 119910)21205821198861198981205821198861198981015840 sin 120582
119886119898119909 sin 120582
1198861198981015840119909
times
2
sum
119897=1
120577119897
119887(119910) cos 120582
1198871198991015840119910
minus (1 minus 120583) 1205821198871198991015840 cos 120582
119886119898119909 cos 120582
1198861198981015840119909
times
2
sum
119897=1
1205771198971015840
119887(119910) sin 120582
1198871198991015840119910 + 120581119866ℎ (119909 119910)
times ( cos 1205821198861198981015840119909 cos 120582
1198871198991015840119910
times
2
sum
119897=1
120577119897
119887(119910) cos 120582
119886119898119909)]119889119909119889119910
+ int
119886
0
1198701199091199100
cos 120582119886119898119909 cos 120582
1198861198981015840119909
2
sum
119897=1
120577119897
119887(0)
+(minus1)1198991015840
119870119909119910119887
cos 120582119886119898119909 cos 120582
1198861198981015840119909
2
sum
119897=1
120577119897
119887(119887) 119889119909
+ int
119887
0
1198701199090
2
sum
119897=1
120577119897
119887(119910) cos 120582
1198871198991015840119910
+(minus1)119898+1198981015840
119870119909119886
2
sum
119897=1
120577119897
119887(119910) cos 120582
1198871198991015840119910119889119910
K1-3119904119902
=
1
2
int
119886
0
int
119887
0
[119863 (119909 119910) minus 21205821198861198981015840 sin 120582
1198861198981015840119909 cos 120582
1198871198991015840119910
times
2
sum
119897=1
1205771198971015840
119886(119909) cos 120582
119887119899119910
+ (1 minus 120583) 1205821198871198991205821198871198991015840 cos 120582
1198861198981015840119909
times sin 1205821198871198991015840119910
2
sum
119897=1
120577119897
119886(119909) sin 120582
119887119899119910
+ 120581119866ℎ (119909 119910) cos 1205821198861198981015840119909 cos 120582
1198871198991015840119910
times
2
sum
119897=1
120577119897
119886(119909) cos 120582
119887119899119910]119889119909119889119910
+ int
119886
0
1198701199091199100
cos 1205821198861198981015840119909
2
sum
119897=1
120577119897
119886(119909)
+(minus1)119899+1198991015840
119870119909119910119887
cos 1205821198861198981015840119909
2
sum
119897=1
120577119897
119886(119909) 119889119909
+ int
119887
0
1198701199090
2
sum
119897=1
120577119897
119886(0) cos 120582
1198871198991015840119910
+(minus1)119898+1198981015840
119870119909119886
2
sum
119897=1
120577119897
119886(0) cos 120582
1198871198991015840119910119889119910
K1-4119904119905
=
1
2
int
119886
0
int
119887
0
119863(119909 119910) 21205821198871198991205821198861198981015840 sin 120582
1198861198981015840119909 cos 120582
1198871198991015840119910
times cos 120582119886119898119909 sin 120582
119887119899119910 minus 2(1 minus 120583)
119887119899
times 1205821198861198981015840 sin 120582
1198861198981015840119909 cos 120582
1198871198991015840119910 cos 120582
119886119898119909
times sin 120582119887119899119910 + (1 minus 120583) 120582
1198861198981205821198871198991015840 cos 120582
1198861198981015840119909
times sin 1205821198871198991015840119910 sin 120582
119886119898119909 cos 120582
119887119899119910 119889119909 119889119910
22 Shock and Vibration
K1-5119904119901
=
1
2
int
119886
0
int
119887
0
119863(119909 119910) minus 21205821198861198981015840 sin 120582
1198861198981015840119909 cos 120582
1198871198991015840119910
times
2
sum
119897=1
1205771198971015840
119887(119910) cos 120582
119886119898119909
+ 2 (1 minus 120583) 1205821198861198981015840 sin 120582
1198861198981015840119909 cos 120582
1198871198991015840119910
times
2
sum
119897=1
1205771198971015840
119887(119910) cos 120582
119886119898119909
+ (1 minus 120583) 1205821198861198981205821198871198991015840 cos 120582
1198861198981015840119909 sin 120582
1198871198991015840119910
times
2
sum
119897=1
120577119897
119887(119910) sin 120582
119886119898119909119889119909119889119910
K1-6119904119902
=
1
2
int
119886
0
int
119887
0
119863(119909 119910)1205821198871198991205821198861198981015840 sin 120582
1198861198981015840119909 cos 120582
1198861198991015840119910
times
2
sum
119897=1
120577119897
119886(119909) sin 120582
1198871198991015840119910
minus 2 (1 minus 120583) 1205821198871198991205821198861198981015840 sin 120582
1198861198981015840119909 cos 120582
1198861198991015840119910
times
2
sum
119897=1
120577119897
119886(119909) sin 120582
1198871198991015840119910
+ 1205821198871198991015840 cos 120582
1198861198981015840119909 sin 120582
1198871198991015840119910
times
2
sum
119897=1
1205771198971015840
119886(119909) cos 120582
119887119899119910119889119909119889119910
K1-7119904119905
= minus
1
2
int
119886
0
int
119887
0
120581119866ℎ (119909 119910) 120582119886119898
cos 1205821198861198981015840119909 cos 120582
1198871198991015840119910
times sin 120582119886119898119909 cos 120582
119887119899119910119889119909119889119910
K1-8119904119901
= minus
1
2
int
119886
0
int
119887
0
120581119866ℎ (119909 119910) 1205821198861198981015840 cos 120582
1198861198981015840119909 cos 120582
1198871198991015840119910
times
2
sum
119897=1
120577119897
119887(119910) sin 120582
1198861198981015840119909 119889119909 119889119910
K1-9119904119902
=
1
2
int
119886
0
int
119887
0
120581119866ℎ (119909 119910) cos 1205821198861198981015840119909 cos 120582
1198871198991015840119910
times
2
sum
119897=1
1205771198971015840
119886(119909) cos 120582
119887119899119910119889119909119889119910
M1-1119904119905
=
1
24
120588int
119886
0
int
119887
0
ℎ3
(119909 119910) cos 120582119886119898119909 cos 120582
119887119899119910
times cos 1205821198861198981015840119909 cos 120582
1198871198991015840119910119889119909119889119910
M1-2119904119901
=
1
24
120588int
119886
0
int
119887
0
ℎ3
(119909 119910)
2
sum
119897=1
120577119897
119887(119910) cos 120582
119886119898119909
times cos 1205821198861198981015840119909 cos 120582
1198871198991015840119910119889119909119889119910
M1-3119904119902
=
1
24
120588int
119886
0
int
119887
0
ℎ3
(119909 119910)
2
sum
119897=1
120577119897
119886(119909) cos 120582
119887119899119910
times cos 1205821198861198981015840119909 cos 120582
1198871198991015840119910119889119909119889119910
M1-4119904119905 = M
1-5119904119901 = M1-6119904119902
= M1-7119904119905 = M
1-8119904119901 = M1-9119904119902 = 0
(A1)
B Useful Integrals for the Moderately ThickRectangular Plates with Variable Thicknessand Arbitrary Boundary Conditions
Consider
int
119871
0
cos 120582119898119909 cos 120582
119899119909 119889119909 =
0 119898 = 119899
119871
2
119898 = 119899
119871 119898 = 119899 = 0
int
119871
0
sin 120582119898119909 cos 120582
119899119909119889119909
=
minus
(minus1 + (minus1)119898+119899
) 119871119898
(1198982minus 1198992) 120587
119898 = 119899
0 119898 = 119899
int
119871
0
sin 120582119898119909 sin 120582
119899119909 119889119909 =
0 119898 = 119899
119871
2
119898 = 119899
0 119898 = 119899 = 0
int
119871
0
cos 120582119898119909120577119897
119886(119909) 119889119909
=
41198712
31205872 119897 = 1 119898 = 0
minus
41198712
(41198982
minus 3)
(9 minus 401198982+ 16119898
4) 1205872
119897 = 1 119898 = 0
minus
41198712
31205872 119897 = 2 119898 = 0
41198712
(41198982
minus 3) (minus1)119898
(9 minus 401198982+ 16119898
4) 1205872
119897 = 2 119898 = 0
Shock and Vibration 23
int
119871
0
sin 120582119898119909120577119897
119886(119909) 119889119909
=
0 119897 = 1 119898 = 0
161198712
119898(minus1)119898
(9 minus 401198982+ 16119898
4) 1205872
119897 = 1 119898 = 0
0 119897 = 2 119898 = 0
161198981198712
(9 minus 401198982+ 16119898
4) 1205872
119897 = 2 119898 = 0
int
119871
0
cos 1205821198981199091205771198971015840
119886(119909) 119889119909
=
0 119897 = 1 119898 = 0
161198711198982
(minus1)119898
(9 minus 401198982+ 16119898
4) 120587
119897 = 1 119898 = 0
0 119897 = 2 119898 = 0
161198711198982
(9 minus 401198982+ 16119898
4) 120587
119897 = 2 119898 = 0
int
119871
0
sin 1205821198981199091205771198971015840
119886(119909) 119889119909
=
0 119897 = 1 119898 = 0
4119871119898 (minus3 + 41198982
)
(9 minus 401198982+ 16119898
4) 120587
119897 = 1 119898 = 0
0 119897 = 2 119898 = 0
119871119898(
1
120587 minus 41198982120587
+
3
9120587 minus 41198982120587
) (minus1)119898
119897 = 2 119898 = 0
int
119871
0
1205771198971
119886(119909) 1205771198972
119886(119909) 119889119909 =
1198713
41205872 1198971= 1 1198972= 1
minus
21198713
31205873 1198971= 1 1198972= 2
minus
21198713
31205873 1198971= 2 1198972= 1
1198713
41205872 1198971= 2 1198972= 2
int
119871
0
1205771198971
119886(119909) 1205771198971015840
2
119886(119909) 119889119909 =
0 1198971= 1 1198972= 1
minus
1198712
8120587
1198971= 1 1198972= 2
1198712
8120587
1198971= 2 1198972= 1
0 1198971= 2 1198972= 2
int
119871
0
1205771198971015840
1
119886(119909) 1205771198971015840
2
119886(119909) 119889119909 =
5119871
16
1198971= 1 1198972= 1
minus
119871
2120587
1198971= 1 1198972= 2
minus
119871
2120587
1198971= 2 1198972= 1
5119871
16
1198971= 2 1198972= 2
int
119871
0
119909119899 cos 120582
119898119909 119889119909
=
1198711+119899
1 + 119899
119898 = 0
(1198861+119899HypergeometricPFQtimes [ 12 + 1198992 12 32 + 1198992
minus (14)1198982
1205872
]) times (1 + 119899)minus1
119898 gt 0
(B1)
Conflict of Interests
The authors declare that there is no conflict of interestsregarding the publication of this paper
Acknowledgments
The authors would like to thank the anonymous reviewersfor their very valuable comments The authors also gratefullyacknowledge the financial support from the National NaturalScience Foundation of China (no 51209052) HeilongjiangProvince Youth Science Fund Project (no QC2011C013) andHarbin Science and Technology Development InnovationFoundation of Youth (no 2011RFQXG021)
References
[1] K Liew Y Xiang and S Kitipornchai ldquoResearch on thick platevibration a literature surveyrdquo Journal of Sound and Vibrationvol 180 no 1 pp 163ndash176 1995
[2] J H Chung T Y Chung and K C Kim ldquoVibration analysisof orthotropic mindlin plates with edges elastically restrainedagainst rotationrdquo Journal of Sound and Vibration vol 163 no 1pp 151ndash163 1993
[3] Y K Cheung and D Zhou ldquoVibrations of moderately thickrectangular plates in terms of a set of static Timoshenko beamfunctionsrdquoComputers and Structures vol 78 no 6 pp 757ndash7682000
[4] CMWang ldquoNatural frequencies formula for simply supportedMindlin platesrdquo Journal of Vibration and Acoustics vol 116 no4 pp 536ndash540 1994
[5] K N Saha R C Kar and P K Datta ldquoFree vibration analysisof rectangular Mindlin plates with elastic restraints uniformlydistributed along the edgesrdquo Journal of Sound andVibration vol192 no 4 pp 885ndash902 1996
24 Shock and Vibration
[6] D J Gorman ldquoAccurate free vibration analysis of shear-deformable plates with torsional elastic edge supportrdquo Journalof Sound and Vibration vol 203 no 2 pp 209ndash218 1997
[7] D J Gorman ldquoFree vibration analysis of Mindlin plates withuniform elastic edge support by the superposition methodrdquoJournal of Sound andVibration vol 207 no 3 pp 335ndash350 1997
[8] D J Gorman ldquoAccurate free vibration analysis of point sup-ported mindlin plates by the superposition methodrdquo Journal ofSound and Vibration vol 219 no 2 pp 265ndash277 1999
[9] Y Xiang ldquoVibration of rectangular Mindlin plates resting onnon-homogenous elastic foundationsrdquo International Journal ofMechanical Sciences vol 45 no 6-7 pp 1229ndash1244 2003
[10] Y Xiang and G W Wei ldquoExact solutions for buckling andvibration of stepped rectangular Mindlin platesrdquo InternationalJournal of Solids and Structures vol 41 no 1 pp 279ndash294 2004
[11] Y L Yeh M J Jang and C C Wang ldquoAnalyzing the freevibrations of a plate using finite difference and differential trans-formationmethodrdquoAppliedMathematics andComputation vol178 no 2 pp 493ndash501 2006
[12] Y Xiang S K Lai L Zhou and C W Lim ldquoDSC-Ritz elementmethod for vibration analysis of rectangular Mindlin plateswith mixed edge supportsrdquo European Journal of MechanicsmdashASolids vol 29 no 4 pp 619ndash628 2010
[13] H Nguyen-Xuan C H Thai and T Nguyen-Thoi ldquoIsogeo-metric finite element analysis of composite sandwich platesusing a higher order shear deformation theoryrdquo Composites BEngineering vol 55 pp 558ndash574 2013
[14] C H Thai A J M Ferreira E Carrera and H Nguyen-XuanldquoIsogeometric analysis of laminated composite and sandwichplates using a layerwise deformation theoryrdquo Composite Struc-tures vol 104 pp 196ndash214 2013
[15] C H Thai A Ferreira S Bordas T Rabczuk and HNguyen-Xuan ldquoIsogeometric analysis of laminated compositeand sandwich plates using a new inverse trigonometric sheardeformation theoryrdquo European Journal of Mechanics A Solidsvol 43 pp 89ndash108 2014
[16] H Luong-Van T Nguyen-Thoi G R Liu and P Phung-VanldquoA cell-based smoothed finite elementmethod using three-nodeshear-locking free Mindlin plate element (CS-FEM-MIN3) fordynamic response of laminated composite plates on viscoelasticfoundationrdquo Engineering Analysis with Boundary Elements vol42 pp 8ndash19 2014
[17] T Nguyen-Thoi T Rabczuk T Lam-Phat V Ho-Huu andP Phung-Van ldquoFree vibration analysis of cracked Mindlinplate using an extended cell-based smoothed discrete sheargap method (XCS-DSG3)rdquo Theoretical and Applied FractureMechanics 2014
[18] H Nguyen-Xuan G R Liu and C a Thai-Hoang ldquoAnedge-based smoothed finite element method (ES-FEM) withstabilized discrete shear gap technique for analysis of Reissner-Mindlin platesrdquo Computer Methods in Applied Mechanics andEngineering vol 199 no 9ndash12 pp 471ndash489 2010
[19] P Phung-Van T Nguyen-Thoi T Bui-Xuan and Q Lieu-XuanldquoA cell-based smoothed three-node Mindlin plate element(CS-FEM-MIN3) based on the 119862
0 -type higher-order sheardeformation for geometrically nonlinear analysis of laminatedcomposite platesrdquo Computational Materials Science 2014
[20] NNguyen-Thanh T RabczukHNguyen-Xuan and S BordasldquoAn alternative alpha finite element method with discrete sheargap technique for analysis of isotropicMindlin-Reissner platesrdquoFinite Elements inAnalysis andDesign vol 47 no 5 pp 519ndash5352011
[21] KM Liew Y Xiang and S Kitipornchai ldquoTransverse vibrationof thick rectangular plates-I Comprehensive sets of boundaryconditionsrdquo Computers and Structures vol 49 no 1 pp 1ndash291993
[22] D Zhou ldquoVibrations of Mindlin rectangular plates with elasti-cally restrained edges using static Timoshenko beam functionswith the Rayleigh-Ritz methodrdquo International Journal of Solidsand Structures vol 38 no 32-33 pp 5565ndash5580 2001
[23] D Zhou SH Lo F T KAu andYK Cheung ldquoVibration anal-ysis of rectangular Mindlin plates with internal line supportsusing static Timoshenko beam functionsrdquo International Journalof Mechanical Sciences vol 44 no 12 pp 2503ndash2522 2002
[24] H S Shen J Yang and L Zhang ldquoFree and forced vibrationof Reissner-Mindlin plates with free edges resting on elasticfoundationsrdquo Journal of Sound and Vibration vol 244 no 2 pp299ndash320 2001
[25] Y Xing and B Liu ldquoClosed form solutions for free vibrations ofrectangular Mindlin platesrdquo Acta Mechanica Sinica vol 25 no5 pp 689ndash698 2009
[26] P Gagnon C Gosselin and L Cloutier ldquoA finite strip elementfor the analysis of variable thickness rectangular thick platesrdquoComputers amp Structures vol 63 no 2 pp 349ndash362 1997
[27] T Mlzusawa ldquoVibration of rectangular mindlin plates withtapered thickness by the spline strip methodrdquo Computers andStructures vol 46 no 3 pp 451ndash463 1993
[28] H Nguyen-Xuan and T Nguyen-Thoi ldquoA stabilized smoothedfinite element method for free vibration analysis of Mindlin-Reissner platesrdquo Communications in Numerical Methods inEngineering with Biomedical Applications vol 25 no 8 pp 882ndash906 2009
[29] K M Liew L X Peng and S Kitipornchai ldquoVibration anal-ysis of corrugated Reissner-Mindlin plates using a mesh-freeGalerkin methodrdquo International Journal of Mechanical Sciencesvol 51 no 9-10 pp 642ndash652 2009
[30] Y Hou GWWei and Y Xiang ldquoDSC-Ritz method for the freevibration analysis of Mindlin platesrdquo International Journal forNumerical Methods in Engineering vol 62 no 2 pp 262ndash2882005
[31] F- Liu and K M Liew ldquoVibration analysis of discontinuousMindlin plates by differential quadrature element methodrdquoJournal of Vibration and Acoustics Transactions of the ASMEvol 121 no 2 pp 204ndash208 1999
[32] P Malekzadeh G Karami and M Farid ldquoA semi-analyticalDQEM for free vibration analysis of thick plates with twoopposite edges simply supportedrdquoComputerMethods inAppliedMechanics and Engineering vol 193 no 45ndash47 pp 4781ndash47962004
[33] P Malekzadeh and S A Shahpari ldquoFree vibration analysisof variable thickness thin and moderately thick plates withelastically restrained edges by DQMrdquo Thin-Walled Structuresvol 43 no 7 pp 1037ndash1050 2005
[34] R E Diaz-Contreras and S Nomura ldquoGreenrsquos function appliedto solution ofMindlin platesrdquoComputers and Structures vol 60no 1 pp 41ndash48 1996
[35] T Sakiyama and M Huang ldquoFree vibration analysis of rect-angular plates with variable thicknessrdquo Journal of Sound andVibration vol 216 no 3 pp 379ndash397 1998
[36] M Huang X Q Ma T Sakiyama H Matuda and C MoritaldquoFree vibration analysis of orthotropic rectangular plates withvariable thickness and general boundary conditionsrdquo Journal ofSound and Vibration vol 288 no 4-5 pp 931ndash955 2005
Shock and Vibration 25
[37] I Shufrin and M Eisenberger ldquoVibration of shear deformableplates with variable thicknessmdashfirst-order and higher-orderanalysesrdquo Journal of Sound and Vibration vol 290 no 1-2 pp465ndash489 2006
[38] S A Eftekhari and A A Jafari ldquoAccurate variational approachfor free vibration of variable thickness thin and thick plates withedges elastically restrained against translation and rotationrdquoInternational Journal of Mechanical Sciences vol 68 pp 35ndash462013
[39] W L Li X Zhang J Du and Z Liu ldquoAn exact series solutionfor the transverse vibration of rectangular plates with generalelastic boundary supportsrdquo Journal of Sound amp Vibration vol321 no 1-2 pp 254ndash269 2009
[40] X Shi D Shi W L Li and Q Wang ldquoA unified method forfree vibration analysis of circular annular and sector plateswith arbitrary boundary conditionsrdquo Journal of Vibration andControl 2014
[41] D Y Shi X J Shi W L Li and Q S Wang ldquoFree transversevibrations of orthotropic thin rectangular plates with arbitraryelastic edge supportsrdquo Journal of Vibroengineering vol 16 no 1pp 389ndash398 2014
[42] Y Chen G Jin and Z Liu ldquoFree vibration analysis of circu-lar cylindrical shell with non-uniform elastic boundary con-straintsrdquo International Journal of Mechanical Sciences vol 74pp 120ndash132 2013
[43] L Dai T Yang J Du W L Li and M J Brennan ldquoAn exactseries solution for the vibration analysis of cylindrical shellswith arbitrary boundary conditionsrdquo Applied Acoustics vol 74no 3 pp 440ndash449 2013
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Shock and Vibration
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
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DistributedSensor Networks
International Journal of
10 Shock and Vibration
Table 2 Frequency parameters Ω = 1205961198872
(120588ℎ119863)12
1205872 for uniform thickness moderately thick plates with different classical boundaries and
structure parameters
Boundary condition ab Model sequence1 2 3 4 5 6 7 8
S-S-S-S
11931 4605 4605 7064 8605 8605 10792 10792
(1933a) (4611) (4611) (7067) (8607) (8607) (10799) (10780)(1931b) (4605) (4605) (7064) (8605) (8605) (10792) (10792)
321408 2648 4127 4605 5261 7064 7153 8184(1410a) (2651) (4128) (4608) (5267) (7074) (7155) (8188)(1408b) (2648) (4127) (4605) (5261) (7064) (7151) (8184)
C-F-F-F
10348 0817 2035 2583 2862 4816 5478 5774
(0348a) (0817) (2038) (2586) (2865) (4821) (5504) (5793)(0348b) (0816) (2034) (2582) (2860) (4811) (5477) (5772)
320155 0502 0941 1665 2292 2613 3303 3498
(0154a) (0502) (0941) (1663) (2295) (2618) (3305) (3500)(0154b) (0501) (0940) (1662) (2292) (2612) (3298) (3494)
S-S-F-F
10333 1678 1875 3559 4720 4946 6474 6632
(0335a) (1680) (1878) (3600) (4721) (4949) (6478) (6642)(0333b) (1677) (1874) (3557) (4718) (4945) (6472) (6631)
320222 0938 1631 2363 2573 4082 4482 4753
(0223a) (0940) (1632) (2364) (2575) (4092) (4483) (4759)(0222b) (0938) (1631) (2361) (2571) (4077) (4479) (4752)
F-F-F-F1 1290 1919 2363 3235 3235 5605 5605 5646
(1291a) (1920) (2365) (3239) (3239) (5607) (5607) (6648)
32 0865 0948 1961 2166 2461 2843 3551 4026(0865a) (0949) (1963) (2168) (2465) (2844) (3556) (4034)
S-C-S-C1 2700 4971 5990 7973 8787 10250 11334 12024
(2701a) (4971) (5993) (7975) (8788) (10253) (11335) (12028)
32 2348 3253 4972 5646 6489 7381 7974 9930(2348a) (3255) (4974) (5647) (6590) (7384) (7801) (9932)
aResults in parentheses are taken from FEMbResults in parentheses are taken from [11]
conditions The varying thickness function ℎ(119909 119910) can beexpressed as ℎ
0(1 + 120572119909
119904
)(1 + 120573119910119905
) in which the ℎ0 120572 and
120573 represent the initial thickness gradient in 119909 direction andgradient in 119910 direction When the indexes 119904 and 119905 take thevalue 119904 = 119905 = 1 the analyticalmodel imitates the linearly vari-ation thickness moderately thick plates structure In order tounify the description and facilitate the analytical calculationsof the involved integrals all the thickness variation functionscan be expanded into either 1D or 2D Fourier cosine seriesresulting in
ℎ (119909 119910) =
infin
sum
119898=0
infin
sum
119899=0
ℎ0119902119898119899
cos 119898120587119909119886
cos119898120587119910
119887
(18)
where119902119898119899
=
4
119886119887
int
119887
0
int
119886
0
(1 + 120572119909119904
) (1 + 120573119910119905
) cos 119898120587119909119886
cos119898120587119910
119887
119889119909 119889119910
(19)
In order to prove the validity of the present methodfor the vibration of linearly variation thickness moderately
thick plates with arbitrary boundary conditions the typicalclassical boundary conditions as the first case will be con-sidered In Table 5 the comparison of the first six frequencyparameters Ω = 120596119886
2
(120588ℎ01198630)12 of the moderately thick
plates with linearly varying thickness is presentedThe S-S-S-S C-F-F-F S-S-F-F C-C-C-C and S-C-S-C boundary condi-tions are performed in the comparison Excellent agreementsare observed between the solutions obtained by the modifiedFourier method and finite element method (FEM) results forthemoderately thick plates with linear variation thickness Toinvestigate the influence of the aspect ratio on the uniformthickness and nonuniform thickness moderately thick platesthe effect on the frequency parameters for plates with S-S-S-Sboundary conditions is presented in Figure 6 The thicknessfunctions are ℎ
0and ℎ0(1+05times119909)(1+05times119910) respectively It
is seen from Figure 6 that the influence of aspect ratios on thefrequency parameters for nonuniform thickness moderatelythick plates is more complicated
In the next two examples we also account for the vibra-tions of moderately thick plate with linear variation thicknessand elastic edge supports The first model considered is an S-S-S-S square moderately thick plate with all edges elastically
Shock and Vibration 11
Table 3 Frequency parameters Ω = 1205961198872
(120588ℎ119863)12
1205872 for S-S-S-S moderately thick plates (119886119887 = 1) with uniform thickness and elastic
rotation support (1198701199090= 119870119909119886= 1198701199100= 119870119910119887= Γ119909)
Γ119909
aℎ0119886
Model sequence1 2 3 4 5 6
0
005 1945 4834 4834 7591 9515 9519(1942b) (4836) (4836) (7584) (9547) (9552)
01 1856 4500 4500 6829 8499 8513(1858b) (4510) (4510) (6843) (8539) (8554)
015 1758 4114 4114 6040 7421 7445(1761b) (4126) (4126) (6060) (7462) (7486)
02 1656 3733 3733 5335 6468 6499(1660b) (3747) (3747) (5357) (6508) (6538)
05
005 3541 7071 7071 10227 12318 12399(3546b) (7091) (7091) (10253) (12386) (12468)
01 3274 6242 6242 8719 10302 10421(3282b) (6267) (6267) (8754) (10370) (10488)
015 2947 5361 5361 7290 8465 8606(2956b) (5387) (5387) (7327) (8524) (8666)
02 2624 4599 4599 6148 7057 7199(2634b) (4624) (4624) (6182) (7107) (7250)
1
005 3541 7072 7072 10228 12320 12401(3547b) (7093) (7093) (10254) (12388) (12470)
01 3278 6248 6248 8727 10311 10429(3286b) (6274) (6274) (8762) (10378) (10497)
015 2957 5374 5374 7305 8478 8620(2966b) (5400) (5400) (7341) (8537) (8680)
02 2641 4617 4617 6166 7070 7215(2651b) (4643) (4643) (6200) (7121) (7268)
15
005 3542 7073 7073 10229 12320 12401(3547b) (7093) (7093) (10254) (12388) (12471)
01 3280 6251 6251 8729 10313 10432(3287b) (6276) (6276) (8765) (10381) (10500)
015 2960 5379 5379 7309 8482 8625(2969b) (5405) (5405) (7346) (8541) (8685)
02 2647 4623 4623 6172 7074 7221(2657b) (4649) (4649) (6206) (7125) (7274)
2
005 3542 7073 7073 10229 12321 12402(3547b) (7093) (7093) (10255) (12389) (12471)
01 3280 6252 6252 8730 10315 10434(3288b) (6277) (6277) (8766) (10382) (10501)
015 2962 5381 5381 7312 8484 8628(2971b) (5407) (5407) (7348) (8543) (8688)
02 2650 4627 4627 6175 7076 7224(2660b) (4652) (4652) (6210) (7128 (7277)
aΓ119909= 1198701198941198700 (1198700= 1 times 10
9 Nmrad 119894 = 1199090 119909119886 1199100 119910119887)bResults in parentheses are taken from FEM
12 Shock and Vibration
Table 4 Frequency parameters Ω = 1205961198872
(120588ℎ119863)12
1205872 for moderately thick plates (119886119887 = 1) with uniform thickness and all edge elastic
restraints (1198701199090= 119870119909119886= 1198701199100= 119870119910119887= Γ119909and 119896
1199090= 119896119909119886= 1198961199100= 119896119910119887= Γ119908)
Γ119909
aΓ119908
aℎ0119886
Model sequence1 2 3 4 5 6
05 0
005 0986 1967 3893 3893 4850 4850(0988b) (1971) (3897) (3897) (4852) (4852)
01 0916 1835 3515 3522 4375 4375(0917b) (1837) (3519) (3528) (4341) (4341)
015 0789 1635 2983 3027 3771 3771(0792b) (1638) (2988) (3029) (3775) (3775)
02 0646 1444 2497 2601 3249 3249(0652b) (1448) (2508) (2621) (3254) (3254)
1 05
005 3417 3417 4431 5294 5461 6448(3419b) (3419) (4435) (5303) (5478) (6456)
01 1593 1593 2401 3849 3861 4729(1611b) (1611) (2408) (3852) (3866) (4767)
015 1122 1122 1913 3272 3290 4038(1127b) (1227) (1924) (3284) (3314) (4069)
02 0890 0890 1640 2781 2829 3476(0899b) (0899) (1654) (2787) (2834) (3497)
15 1
005 4325 4325 5619 6273 6571 7619(4334b) (4334) (5681) (6299) (6642) (7642)
01 2013 2013 2812 4088 4116 4993(2024b) (2024) (2827) (4122) (4187) (4994)
015 1338 1338 2108 3430 3443 4196(1342b) (1342) (2112) (3467) (3482) (4200)
02 1038 1038 1772 2936 2965 3613(1042b) (1042) (1786) (2937) (2977) (3624)
2 2
005 5231 5231 6919 7630 8002 9220(5238b) (5238) (6931) (7637) (8011) (9241)
01 2594 2594 3439 4470 4544 5428(2597b) (2597) (3442) (4495) (4557) (5438)
015 1653 1653 2403 3613 3632 4392(1662b) (1662) (2414) (3627) (3646) (4404)
02 1238 1238 1949 3077 3099 3751(1244b) (1244) (1964) (3094) (3116) (3788)
aΓ119909= 1198701198941198700 (1198700= 1 times 10
9 Nmrad 119894 = 1199090 119909119886 1199100 119910119887) Γ119908= 1198961198941198960 (1198960= 1 times 10
9 Nm 119894 = 1199090 119909119886 1199100 119910119887)bResults in parentheses are taken from FEM
05 10 15 20 25 300
2
4
6
8
10
12
14
16
ab
The 1st orderThe 2nd orderThe 3rd order
The 4th orderThe 5th orderThe 6th order
Ω
(a)
05 10 15 20 25 30
30
60
90
120
150
180
ab
The 1st orderThe 2nd orderThe 3rd order
The 4th orderThe 5th orderThe 6th order
Ω
(b)
Figure 6 The effect of aspect ratio 119886119887 on the natural frequenciesΩ for S-S-S-S boundary condition (a) uniform thickness and (b) nonuni-form thickness
Shock and Vibration 13
Table 5 Frequency parameter Ω = 1205961198862
(120588ℎ01198630)12 for moderately thick plates with linearly varying thickness in different boundary
conditions (ℎ(119909 119910) = ℎ0(1 + 05119909)(1 + 05119910))
Boundary condition 119886119887 ℎ0119886
Model sequence1 2 3 4 5 6
S-S-S-S
101 27261 62397 62926 92423 111537 111732
(27263a) (62410) (63842) (92445) (111867) (111921)
02 23103 47010 47191 65018 75831 75865(23489a) (47209) (47349) (65057) (76605) (76473)
3201 59338 99101 138798 151102 166710 204855
(59874a) (99249) (138177) (151362) (166644) (205381)
02 41175 63765 63765 73598 83956 89084(41586a) (63313) (63669) (73661) (83918) (89547)
C-F-F-F
101 4173 10331 27407 37826 38809 64095
(4856a) (10680) (27513) (37203) (39135) (64858)
02 3983 8702 21403 29443 30286 46705(4118a) (9306) (21990) (29670) (30602) (47156)
3201 4868 15488 35620 63523 89969 100089
(4887a) (15320) (35175) (63353) (89159) (100783)
02 4469 9763 22338 42188) 50104 56331(4944a) (9636) (22087) (42177) (50580) (56559)
S-S-F-F
101 4117 24795 26216 48567) 68195 69667
(4789a) (24149) (26751) (49324) (68842) (70575)
02 3820 20516 21924 37523 48980 50327(4103a) (20041) (22125) (37221) (48894) (50622)
3201 7935 39454 65063 92877 92877 136176
(7856a) (40145) (65162) (92462) (93022) (136336)
02 6389 26149 35627 49539 56844 65170(6406a) (26536) (35626) (49831) (57144) (65981)
C-C-C-C
101 44471 80299 80681 109422 126871 128207
(45102a) (80271) (80656) (110015) (126328) (128888)
02 32162 52781 52933 69410 78474 79482(32240a) (52629) (53166) (70063) (79059) (80266)
3201 82975 115587 151992 161579 176601 212330
(83139a) (115838) (152200) (162193) (177058) (212849)
02 48577 66022 83468 90830 97474 98157(49187a) (66259) (84065) (91749) (97827) (98777)
S-C-S-C
101 36847 66664 76965 101045 113594 125551
(37901a) (68568) (78356) (101200) (113978) (125651)
02 27673 48469 51220 67027 76358 78241(27897a) (48685) (51491) (67447) (77530) (79573)
3201 77378 106848 150079 154569 173120 208391
(78003a) (107058) (150991) (155171) (173099) (208402)
02 45895 64732 73393 83282 90325 97089(45803a) (64131) 73656 83801 90518 97015
aResults in parentheses are taken from FEM
restrained That is 1198701199090
= 119870119909119886
= 1198701199100
= 119870119910119887
= Γ119909 The six
frequency parameters Ω = 1205961198862
(120588ℎ01198630)12 are given in
Table 6 for several different restraining coefficient valuesthe finite element method (FEM) results calculated usingABAQUS are also presented as a referenceThe secondmodelis a complete square moderately thick plate with all edgeselastically restrainedThat is 119896
1199090= 119896119909119886= 1198961199100= 119896119910119887= Γ119908and
1198701199090= 119870119909119886
= 1198701199100= 119870119910119887= Γ119909 The six frequency parameters
Ω = 1205961198862
(120588ℎ01198630)12 are given in Table 7 for several different
restraining coefficient values the finite element method(FEM) results are also listed as a reference It can be clearlyseen that the comparison is extremely good which impliesthat the current method is able to make correct predictionsfor the modal characteristics of linearly varying thickness
14 Shock and Vibration
Table 6 Frequency parameter Ω = 1205961198862
(120588ℎ01198630)12 for S-S-S-S moderately thick plates (119886119887 = 1) with linearly varying thickness and elastic
rotation support (ℎ(119909 119910) = ℎ0(1 + 05119909)(1 + 05119910) 119870
1199090= 119870119909119886= 1198701199100= 119870119910119887= Γ119909)
Γ119909
aℎ0119886
Model sequence1 2 3 4 5 6
0
005 28809 70080 70891 108967 134884 135295(28897b) (70334) (71129) (109112) (135254) (135772)
01 27261 62398 62926 92423 111540 111735(26606b) (61784) (62303) (91193) (111323) (111648)
015 25231 54192 54505 77118 91228 91312(25204b) (55297) (55595) (77503) (91871) (91941)
02 23103 47010 47191 65018 75832 75866(23947b) (47071) (47208) (65404) (76531) (76589)
05
005 23103 47010 47191 65018 75832 75866(23210b) (47245) (47358) (65241) (76122) (76345)
01 48377 94650 95597 135484 162635 163501(48524b) (94891) (95992) (135845) (167012) (164024)
015 36239 70939 71298 99997 118584 119009(36647b) (71214) (71687) (100024) (119114) (119457)
02 28749 56937 57037 79214 93009 93162(29142b) (57034) (57544) (79654) (93668) (93754)
1
005 49651 96759 97619 138089 165368 166304(49755b) (97245) (97964) (138544) (165864) (166878)
01 38951 73735 74251 102748 121079 121709(39021b) (74145) (74664) (102874) (121613) (121817)
015 30630 58406 58533 80432 93984 94255(30638b) (58684) (58644) (80874) (94121) (94478)
02 25548 48636 48674 66165 76708 76824(25687b) (48788) (48992) (66544) (76824) (76944)
15
005 50118 97561 98374 139096 166421 167382(50247b) (97877) (98661) (139247) (166875) (167662)
01 40310 75216 75783 104229 122402 123164(40366b) (75542) (75921) (104557) (122874) (123385)
015 31837 59352 59530 81245 94611 94983(31902b) (59451) (59871) (81375) (94784) (95105)
02 26302 49119 49156 66528 76954 77130(26418b) (49354) (49334) (66921) (77010) (77470)
2
005 50360 97983 98769 139631 166979 167954(50472b) (98005) (99114) (140154) (167188) (167986)
01 41130 76146 76724 105165 123229 124082(41421b) (76642) (78114) (105661) (123278) (124974)
015 32687 60025 60245 81832 95052 95508(32778b) (60244) (60549) (82146) (95114) (95842)
02 26893 49492 49539 66815 77138 77370(26922b) (49874) (49924) (66987) (77367) (77588)
aΓ119909= 1198701198941198700 (1198700= 1 times 10
9 Nmrad 119894 = 1199090 119909119886 1199100 119910119887)bResults in parentheses are taken from FEM
Shock and Vibration 15
Table 7 Frequency parameterΩ = 1205961198862
(120588ℎ01198630)12 for moderately thick plates (119886119887 = 1) with linearly varying thickness and elastic rotation
and translation support (ℎ(119909 119910) = ℎ0(1 + 05119909)(1 + 05119910) 119870
1199090= 119870119909119886= 1198701199100= 119870119910119887= Γ119909 and 119896
1199090= 119896119909119886= 1198961199100= 119896119910119887= Γ119908)
Γ119909
aΓ119908
aℎ0119886
Model sequence1 2 3 4 5 6
05 0
005 14751 14917 28232 55251 55397 69512(14780b) (14822) (29252) (55476) (55505) (70904)
01 11811 11910 23857 44462 45037 55453(11818b) (11897) (24711) (44376) (45257) (56511)
015 8487 8514 20011 34783 36335 44970(8500b) (8544) (20738) (34404) (36663) (45625)
02 6073 6073 17431 28679 30805 38010(6096b) (6106) (18039) (28129) (31165) (38311)
1 05
005 21225 32173 32173 44381 64061 65222(20708b) (32097) (32285) (45110) (64182) (65355)
01 8140 16878 16941 28100 48663 49015(7943b) (16838) (16970) (28888) (48700) (49171)
015 4493 11859 11878 22429 38352 39247(4383b) (11867) (11891) (23125) (38177) (39488)
02 2932 8690 8691 18919 31108 32600(3540b) (8791) (8791) (16229) (27520) (27889)
15 1
005 27819 40919 41430 54595 70792 72896(27146b) (41008) (41434) (55256) (70895) (73051)
01 11285 20068 20230 31101 51046 51429(11012b) (20025) (20270) (31849) (51126) (51569)
015 6292 13912 13922 24132 40500 41120(6139b) (13912) (13934) (24808) (40426) (41316)
02 4121 10321 10331 20051 32796 33921(4021b) (10329) (10357) (20635) (32540) (34178)
2 2
005 34615 51972 53088 68255 81344 84510(33806b) (52100) (53157) (68891) (81433) (84761)
01 15416 24659 24829 35615 53879 54494(15045b) (24625) (24888) (36303) (53970) (54626)
015 8740 16420 16481 26353 42417 42946(8529b) (16414) (16499) (26996) (42398) (43114)
02 5761 12127 12127 21434 34286 35192(5622b) (12134) (12148) (21998) (34110) (35415)
aΓ119909= 1198701198941198700 (1198700= 1 times 10
9 Nmrad 119894 = 1199090 119909119886 1199100 119910119887) Γ119908= 1198961198941198960 (1198960= 1 times 10
9 Nm 119894 = 1199090 119909119886 1199100 119910119887)bResults in parentheses are taken from FEM
moderately thick plate with not only classical boundaryconditions but also elastic edge restraints
As the last case of this subsection the influence of thegradient 120572 and 120573 on the fundamental frequency parametersfor a linearly varying thickness moderately thick plate isinvestigated The model is a square moderately thick platewith all edges elastically restrainedThat is 119896
1199090= 119896119909119886= 1198961199100=
119896119910119887= Γ119908= 2 and 119870
1199090= 119870119909119886
= 1198701199100= 119870119910119887= Γ119909= 2 The six
frequency parameters Ω = 1205961198862
(120588ℎ01198630)12 are given in
Table 8 for several different 120572 and 120573 values the finite elementmethod (FEM) results are also listed as a reference Againgood agreement can be observed Through Table 8 it is alsofound that the frequency parameter increases with increasinggradient parameters
The above studies are given as linearly varying thicknessmoderately thick plates with several of boundary condition
and different structure parameters In the next section vibra-tion results for the plates subjected to nonlinear variationthickness will be presented
34 Nonlinearly Varying Thickness Moderately Thick Plateswith Classical and Elastic Boundary Conditions In Sec-tion 33 the linearly varying thickness moderately thickplates were studied However in the practical engineeringapplications the varying thickness of amoderately thick platemay not always be linear variation in nature A variety ofpossible thickness varying cases may be encountered in prac-tice Therefore the moderately thick plates with nonlinearvariation thickness subjected to general elastic edge restraintsare examined in this subsection For the sake of brevity theindexes 119904 and 119905 will be chosen as 2 to imitate the nonlinearlyvarying thickness moderately thick plates structure in this
16 Shock and Vibration
Table 8 Frequency parameterΩ = 1205961198862
(120588ℎ01198630)12 for elastic support moderately thick plates (119886119887 = 1) with different gradient (119896
1199090= 119896119909119886=
1198961199100= 119896119910119887= Γ119908= 2 119870
1199090= 119870119909119886= 1198701199100= 119870119910119887= Γ119909= 2)
120572 120573 ℎ0119886
Model sequence1 2 3 4 5 6
120572 = 120573 = 01
005 31275 52615 52657 68954 76146 79850(31303a) (52664) (52675) (69033) (76174) (79948)
01 17270 25244 25250 33949 46050 46625(17271a) (25255) (25258) (34089) (46122) (46708)
015 10190 16266 16274 24164 37327 37526(10189a) (16269) (16280) (24323) (37401) (37622)
02 6780 12258 12260 19685 31452 31754(6780a) (12265) (12270) (19851) (31514) (31864)
120572 = 120573 = 02
005 32530 52947 53141 68954 77034 80589(32577a) (53014) (53203) (69172) (77088) (80773)
01 16882 24979 24979 34154 48057 48575(16883a) (24983) (25001) (34433) (48134) (48667)
015 9804 16284 16309 24683 38844 39085(9803a) (16283) (16322) (24980) (38909) (39191)
02 6499 12290 12295 20133 32354 32770(6498a) (12296) (12307) (20420) (32387) (32895)
120572 = 120573 = 03
005 33516 52845 53308 68704 78203 81561(33563a) (52939) (53389) (69055) (78271) (81789)
01 16419 24799 24830 34527 50061 50583(16419a) (24790) (24869) (34947) (50143) (50688)
015 9430 16334 16378 25236 40193 40506(9429a) (16329) (16395) (25660) (40243) (40625)
02 6235 12288 12293 20572 33102 33663(6235a) (12296) (12306) (20965) (33090) (33811)
120572 = 120573 = 04
005 34215 52463 53264 68430 79654 82865(34252a) (52575) (53346) (68921) (79732) (83113)
01 15921 24695 24793 35027 52011 52571(15921a) (24674) (24842) (35583) (52098) (52688)
015 9074 16388 16444 25797 41379 41789(9074a) (16381) (16464) (26336) (41404) (41928)
02 5990 12236 12238 21002 33735 34461(5989a) (12249) (12250) (21487) (33653) (34641)
120572 = 120573 = 05
005 34615 51972 53088 68255 81344 84510(33806a) (52100) (53157) (68891) (81433) (84761)
01 15416 24659 24829 35615 53879 54494(15045a) (24625) (24888) (36303) (53970) (54626)
015 8740 16420 16481 26353 42417 42946(8529a) (16414) (16499) (26996) (42398) (43114)
02 5761 12127 12127 21434 34286 35192(5622a) (12134) (12148) (21998) (34110) (35415)
aResults in parentheses are taken from FEM
section that is 119904 = 119905 = 2 Also the thickness varying functionsare also be expressed as Fourier cosine series according to (18)and (19)
In order to validate the accuracy and reliability of theproposed method for predicting the vibration behavior ofnonlinearly varying thickness moderately thick plates with
arbitrary boundary conditions the typical classical boundaryconditions viewed as the special cases of elastically restrainededges will be considered The comparison of the first six fre-quency parameters Ω = 120596119886
2
(120588ℎ01198630)12 for the moderately
thick plates with nonlinearly varying thickness is presentedin Table 9 The S-S-S-S C-F-F-F S-S-F-F C-C-C-C and
Shock and Vibration 17
Table 9 Frequency parameter Ω = 1205961198862
(120588ℎ01198630)12 for moderately thick plates with linearly varying thickness in different boundary
conditions (ℎ(119909 119910) = ℎ0(1 + 05119909
2
)(1 + 051199102
))
Boundary condition 119886119887 ℎ0119886
Model sequence1 2 3 4 5 6
S-S-S-S
101 23462 54999 55573 82950 100900 101187
(23741a) (55114) (55983) (83146) (101004) (101347)
02 18606 42725 42971 60040 71167 71761(18936a) (43014) (43148) (60247) (71642) (71993)
3201 53055 65251 85404 106700 114880 143531
(53467a) (65472) (85984) (107211) (115687) (144007)
02 17439 21531 32170 44987 59438 64179(17874a) (21748) (32651) (45264) (60188) (64879)
C-F-F-F
101 3658 9317 24135 32816 35433 59179
(3749a) (9547) (24034) (33627) (36308) (60142)
02 3520 7986 19591 26959 27491 43882(3611a) (8176) (20075) (27125) (27625) (43984)
3201 3901 9963 28984 69187 73686 89360
(3930a) (10095) (29381) (69578) (73994) (90242)
02 3292 4439 13385 23947 40850 44758(3386a) (4498) (14012) (24068) (41221) (45142)
S-S-F-F
101 3557 21268 22762 43524 60253 62429
(3649a) (21441) (22796) (43916) (60622) (62630)
02 3171 17822 19512 33616 45042 46833(3489a) (18006) (20430) (33815) (45247) (49004)
3201 6738 31658 52071 75625 86604 94903
(6957a) (31882) (52518) (75823) (86939) (95610)
02 3653 12930 20955 26517 44097 48058(6807a) (13160) (21312) (26864) (44422) (48873)
C-C-C-C
101 41020 75196 75614 104142 119683 120949
(41405a) (75452) (75882) (104711) (119637) (121137)
02 31237 51686 51868 68411 76868 77864(32010a) (52960) (52145) (69335) (77481) (78099)
3201 95823 128727 163657 173751 190890 223595
(95364a) (128932) (163588) (172350) (190350) (222948)
02 53306 71372 89384 102404 104660 120908(53587a) (71759) (90103) (102503) (105174) (121275)
S-C-S-C
101 33715 60298 71446 94187 103802 117928
(34148a) (60786) (71828) (94515) (104263) (119326)
02 26143 45499 49458 64253 72941 76191(26789a) (46622) (50680) (64329) (73474) (76843)
3201 73228 115845 137445 148166 168342 182542
(73761a) (116121) (137788) (148530) (168698) (182921)
02 41302 53648 65630 72202 81632 88105(41304a) (53757) (65842) (72794) (82033) (88474)
aResults in parentheses are taken from FEM
S-C-S-C boundary conditions are performed in the compar-ison The results adequately demonstrated the great accuracyof the modified Fourier method
We now turn to elastically restrained moderately thickplates The first one involves an S-S-S-S moderately thicksquare plate with a uniform rotational restraint along eachedge that is 119870
1199090= 119870119909119886
= 1198701199100
= 119870119910119887
= Γ119909 The calculated
frequency parameters Ω = 1205961198862
(120588ℎ01198630)12 are given in
Table 10 together with the FEM results Since the elasticallyrestrained plates with nonlinear variation thickness are rarelyinvestigated the FEMresults are used as the referenceA goodagreement is observed between the current and FEM resultsThe second example concerns amoderately thick square plateelastically supported along all edgesThe stiffness of the linear
18 Shock and Vibration
Table 10 Frequency parameterΩ = 1205961198862
(120588ℎ01198630)12 for S-S-S-S moderately thick plates (119886119887 = 1) with linearly varying thickness and elastic
rotation support (ℎ(119909 119910) = ℎ0(1 + 05119909
2
)(1 + 051199102
) 1198701199090= 119870119909119886= 1198701199100= 119870119910119887= Γ119909)
Γ119909
aℎ0119886
Model sequence1 2 3 4 5 6
0
005 23462 54999 55573 82950 100900 101187(23741b) (55114) (55983) (83146) (101004) (101347)
01 18606 42725 42971 60040 71167 71761(18936b) (43014) (43148) (60247) (71642) (71993)
015 53055 65251 85404 106700 114880 143531(53467b) (65472) (85984) (107211) (115687) (144007)
02 17439 21531 32170 44987 59438 64179(17874b) (21748) (32651) (45264) (60188) (64879)
05
005 6904 13374 13514 19400 22904 23056(9909b) (13548) (13651) (19564) (23088) (23379)
01 5316 10237 10314 14499 17155 17264(5374b) (10448) (10651) (14764) (17188) (17879)
015 4136 8201 8220 11474 13578 13671(4174b) (8348) (8651) (11764) (13688) (13879)
02 3449 6915 6924 9549 11225 11327(3474b) (6948) (6951) (9764) (11388) (11579)
1
005 7040 13617 13745 19726 23233 23402(7074b) (13748) (13851) (19964) (23488) (23879)
01 5712 10696 10793 14991 17594 17744(5774b) (10748) (10851) (15054) (17788) (17879)
015 4489 8507 8540 11759 13803 13922(4514b) (8548) (8611) (11864) (13988) (14119)
02 3677 7084 7090 9692 11330 11445(3684b) (7148) (7191) (9764) (11588) (11879)
15
005 7089 13708 13828 19847 23355 23530(7124b) (13748) (13951) (19967) (23588) (23879)
01 5895 10923 11024 15239 17813 17988(5874b) (10748) (11651) (15264) (17818) (17979)
015 4697 8694 8737 11938 13790 14083(4774b) (8748) (8951) (11864) (13688) (14879)
02 3836 7201 7210 9792 11403 11530(3874b) (7748) (7651) (9864) (11588) (11879)
2
005 7114 13754 13871 19911 23418 23597(7174b) (12848) (13851) (19864) (23588) (23879)
01 6001 11060 11159 15391 17944 18136(6004b) (11148) (11151) (15464) (17988) (18379)
015 4836 8821 8872 12061 14037 14189(4874b) (8848) (8951) (12164) (14188) (14879)
02 3954 7289 7301 9871 11457 11594(3974b) (7348) (7651) (9964) (11488) (11679)
aΓ119909= 1198701198941198700 (1198700= 1 times 10
9 Nmrad 119894 = 1199090 119909119886 1199100 119910119887)bResults in parentheses are taken from FEM
and rotational restraints is set equal to 1198961199090
= 119896119909119886
= 1198961199100
=
119896119910119887
= Γ119908and 119870
1199090= 119870119909119886
= 1198701199100
= 119870119910119887
= Γ119909 respectively
The six frequency parametersΩ = 1205961198862
(120588ℎ01198630)12 are shown
in Table 11 for several different restraining coefficient valuesIt can also be noticed that the Fourier series method is able
to predict the modal characteristics of nonlinearly varyingthickness moderately thick plate with not only classicalboundary conditions but also elastic edge restraints correctly
Finally the influence of the gradient on the fundamentalfrequency parameters of a nonlinearly varying thickness
Shock and Vibration 19
Table 11 Frequency parameterΩ = 1205961198862
(120588ℎ01198630)12 for moderately thick plates (119886119887 = 1) with linearly varying thickness and elastic rotation
and translation support (ℎ(119909 119910) = ℎ0(1 + 05119909
2
)(1 + 051199102
) 1198701199090= 119870119909119886= 1198701199100= 119870119910119887= Γ119909 and 119896
1199090= 119896119909119886= 1198961199100= 119896119910119887= Γ119908)
Γ119909
aΓ119908
aℎ0119886
Model sequence1 2 3 4 5 6
05 0
005 12926 13041 25090 47935 47943 61761(12931b) (13053) 25099 (47947) (47952) (61788)
01 10900 10988 21545 39980 40147 50322(10904b) (10992) (21557) (39984) (40166) (50334)
015 8245 8298 18081 31503 32617 40950(8247b) (8230) (18092) (31508) (32622) (40955)
02 6106 6124 15639 25631 27601 34458(6110b) (6125) (15642) (25637) (27617) (34462)
1 05
005 21575 31615 31644 42573 57889 58607(21580b) (31617) (31644) (42581) (57893) (58610)
01 8451 16024 16210 25969 43839 43867(8451b) (16025) (16217) (25972) (43844) (43870)
015 4682 11479 11480 20642 35172 35598(4684b) (11480) (11481) (20644) (35178) (35604)
02 3059 8621 8648 17320 28487 29638(3062b) (8619) (8644) (17327) (28492) (29641)
15 1
005 27713 40203 40978 52951 65352 66815(27714b) (40207) (40988) (52972) (65368) (66822)
01 11667 19322 19615 29122 46115 46183(11674b) (19327) (19619) (29137) (46120) (46188)
015 6544 13416 13481 22405 37238 37430(6547b) (13420) (13488) (22410) (37241) (37434)
02 4295 10175 10190 18543 30330 31056(4296b) (10177) (10194) (18544) (30336) (31060)
2 2
005 33453 50854 52088 66311 76610 78947(33455b) (50852) (50097) (66321) (76627) (78971)
01 15832 24130 24372 33924 49118 49362(15830b) (24127) (24374) (33931) (49147) (49382)
015 9060 15956 16099 24754 39134 39249(9064b) (15961) (16102) (24755) (39138) (39452)
02 5992 11955 11971 20034 31908 32392(5991b) (11957) (11978) (20042) (31910) (32398)
aΓ119909= 1198701198941198700 (1198700= 1 times 10
9 Nmrad 119894 = 1199090 119909119886 1199100 119910119887) Γ119908= 1198961198941198960 (1198960= 1 times 10
9 Nm 119894 = 1199090 119909119886 1199100 119910119887)bResults in parentheses are taken from FEM
moderately thick plate is investigated The plate is elasticallyrestrained in which the stiffness of the boundary springsis taken as 119896
1199090= 119896119909119886
= 1198961199100
= 119896119910119887
= Γ119908
= 2 and1198701199090
= 119870119909119886
= 1198701199100
= 119870119910119887
= Γ119909= 2 respectively The six
frequency parameters Ω = 1205961198862
(120588ℎ01198630)12 are presented in
Table 12 for several different slop values It can be seen thatthe fundamental frequency parameters will decrease with theincrease of the parameter ℎ
0119886
In the above examples it has been demonstrated thatthe presented method can be universally applied to nonlin-early varying thickness moderately thick plates with severalboundary conditions and different structure parametersNew results are obtained for plates with nonlinear variationthickness in both directions subjected to general elastic
boundary restraints which may be used for benchmarkingof researchers in the field In addition it is interesting to seethat the nature frequency decreases with the increase of theindex for thickness function
4 Conclusions
In this paper a modified Fourier method has been pre-sented to study the free vibration behaviors of moderatelythick rectangular plates with variable thickness and arbi-trary boundary conditionsThe first-order shear deformationplate theory is adopted to formulate the theoretical modelThe displacements and rotation components of the plateregardless of boundary conditions are invariantly expressedas the superposition of a 2D Fourier cosine series and four
20 Shock and Vibration
Table 12 Frequency parameterΩ = 1205961198862
(120588ℎ01198630)12 for elastic support moderately thick plates (119886119887 = 1) with different gradient (119896
1199090= 119896119909119886=
1198961199100= 119896119910119887= Γ119908= 2 119870
1199090= 119870119909119886= 1198701199100= 119870119910119887= Γ119909= 2)
120572 120573 ℎ0119886
Model sequence1 2 3 4 5 6
120572 = 120573 = 01
005 30187 51102 51141 67066 73954 77412(30188a) (51105) (51144) (67068) (73961) (77477)
01 16995 24719 24724 33071 44343 44819(16996a) (24725) (24729) (33084) (44347) (44822)
015 10077 15872 15883 23443 35937 36065(10081a) (15877) (15892) (23447) (35941) (36084)
02 6713 11953 11958 19086 30372 30604(6713a) (11957) (11965) (19092) (30378) (30612)
120572 = 120573 = 02
005 31206 51436 51614 67058 74398 77595(31208a) (51438) (51619) (67064) (74402) (77596)
01 16786 24503 24508 33144 45583 45923(16790a) (24510) (24513) (33147) (45587) (45932)
015 9821 15865 15906 23766 36897 36990(9824a) (15870) (15907) (23768) (36894) (36997)
02 6523 11978 11992 19356 30912 31162(6524a) (11981) (11997) (19362) (30916) (31174)
120572 = 120573 = 03
005 32097 51462 51900 66832 74991 77863(32098a) (51466) (51907) (66849) (74996) (77872)
01 16507 24332 24383 33330 46806 47074(16511a) (24327) (24380) (33339) (46810) (47086)
015 9562 15885 15964 24103 37748 37828(9562a) (15892) (15977) (21105) (37751) (37832)
02 6338 11992 12013 19603 31338 31634(6340a) (11994) (12011) (19612) (31344) (31639)
120572 = 120573 = 04
005 32851 51239 52046 66547 75735 78303(33455a) (50852) (50097) (66321) (76627) (78971)
01 16182 24209 24343 33598 47990 48230(33455a) (50852) (50097) (66321) (76627) (78971)
015 9307 15919 16034 24436 38491 38579(33455a) (50852) (50097) (66321) (76627) (78971)
02 6161 11986 12008 19828 31666 32040(33455a) (50852) (50097) (66321) (76627) (78971)
120572 = 120573 = 05
005 33453 50854 52088 66311 76610 78947(33455a) (50852) (50097) (66321) (76627) (78971)
01 15832 24130 24372 33924 49118 49362(15830a) (24127) (24374) (33931) (49147) (49382)
015 9060 15956 16099 24754 39134 39249(9064a) (15961) (16102) (24755) (39138) (39452)
02 5992 11955 11971 20034 31908 32392(5991a) (11957) (11978) (20042) (31910) (32398)
aResults in parentheses are taken from FEM
supplementary functions in the form of the product of apolynomial function and a single cosine series expansionto ensure and accelerate the convergence of the solutionAt each edge of the plate the general restraint conditionsare implemented by introducing one group of linear springsand two groups of rotational springs which are continuouslydistributed and determined by the stiffness of these springs
Instead of seeking a solution in strong forms in the previousstudies all the Fourier coefficients will be treated equallyand independently as the generalized coordinates and solveddirectly from the Rayleigh-Ritz technique The change of theboundary conditions can be easily achieved by only varyingthe stiffness of the three sets of boundary springs along alledges of the rectangular plates without involving any change
Shock and Vibration 21
to the solution procedure The convergence of the presentsolution is examined and the excellent accuracy is validatedby comparison with existing results published in the lit-erature and FEM data Excellent agreements are obtainedfrom these comparisons The proposed method providesa unified means for extracting the modal parameters andpredicting the vibration behaviors of moderately thick plateswith variable thickness variation functions and arbitraryelastic edge restraints A variety of free vibration results formoderately thick rectangular plates with different thicknessvariation functions and boundary conditions are presentedNew results for free vibration of moderately thick rectangularplates with various thickness variation functions and edgeconditions are presented which may be used for benchmark-ing of researchers in the field
Appendices
A Representative Calculation forStiffness and Mass Matrices
To illuminate the particular expression of the mass andstiffness matrixes clearly and tersely four new variables aredefined as 119904 = 119898 lowast (119873 + 1) + 119899 + 1 119905 = 119898
1015840
lowast (119873 + 1) + 1198991015840
+ 1119901 = (119897minus1)lowast (119872+1)+119898
1015840
+1 and 119902 = (119897minus1)lowast (119873+1)+1198991015840
+1The first row elements of K andM are given below
K1-1119904119905
=
1
2
int
119886
0
int
119887
0
[119863 (119909 119910)
times 21205821198861198981205821198861198981015840 sin 120582
119886119898119909 sin 120582
1198861198981015840119909 cos 120582
119887119899119910
times cos 1205821198871198991015840119910 + (1 minus 120583) 120582
1198871198991205821198871198991015840 cos 120582
119886119898119909
times cos 1205821198861198981015840 sin 120582
119887119899119910 sin 120582
1198871198991015840119910
+ 120581119866ℎ (119909 119910) cos 120582119886119898119909 cos 120582
1198861198981015840119909 cos 120582
119887119899
times119910 cos 1205821198871198991015840119910119889119909119889119910] 119889119909 119889119910
+ int
119886
0
1198701199091199100
cos 120582119886119898119909 cos 120582
1198861198981015840119909
+(minus1)119898+1198981015840
119870119909119910119887
cos 120582119886119898119909 cos 120582
1198861198981015840119909 119889119909
+ int
119887
0
1198701199090cos 120582119887119899119910 cos 120582
1198871198991015840119910
+(minus1)119899+1198991015840
119870119909119886cos 120582119887119899119910 cos 120582
1198871198991015840119910 119889119910
K1-2119904119901
=
1
2
int
119886
0
int
119887
0
[119863 (119909 119910)21205821198861198981205821198861198981015840 sin 120582
119886119898119909 sin 120582
1198861198981015840119909
times
2
sum
119897=1
120577119897
119887(119910) cos 120582
1198871198991015840119910
minus (1 minus 120583) 1205821198871198991015840 cos 120582
119886119898119909 cos 120582
1198861198981015840119909
times
2
sum
119897=1
1205771198971015840
119887(119910) sin 120582
1198871198991015840119910 + 120581119866ℎ (119909 119910)
times ( cos 1205821198861198981015840119909 cos 120582
1198871198991015840119910
times
2
sum
119897=1
120577119897
119887(119910) cos 120582
119886119898119909)]119889119909119889119910
+ int
119886
0
1198701199091199100
cos 120582119886119898119909 cos 120582
1198861198981015840119909
2
sum
119897=1
120577119897
119887(0)
+(minus1)1198991015840
119870119909119910119887
cos 120582119886119898119909 cos 120582
1198861198981015840119909
2
sum
119897=1
120577119897
119887(119887) 119889119909
+ int
119887
0
1198701199090
2
sum
119897=1
120577119897
119887(119910) cos 120582
1198871198991015840119910
+(minus1)119898+1198981015840
119870119909119886
2
sum
119897=1
120577119897
119887(119910) cos 120582
1198871198991015840119910119889119910
K1-3119904119902
=
1
2
int
119886
0
int
119887
0
[119863 (119909 119910) minus 21205821198861198981015840 sin 120582
1198861198981015840119909 cos 120582
1198871198991015840119910
times
2
sum
119897=1
1205771198971015840
119886(119909) cos 120582
119887119899119910
+ (1 minus 120583) 1205821198871198991205821198871198991015840 cos 120582
1198861198981015840119909
times sin 1205821198871198991015840119910
2
sum
119897=1
120577119897
119886(119909) sin 120582
119887119899119910
+ 120581119866ℎ (119909 119910) cos 1205821198861198981015840119909 cos 120582
1198871198991015840119910
times
2
sum
119897=1
120577119897
119886(119909) cos 120582
119887119899119910]119889119909119889119910
+ int
119886
0
1198701199091199100
cos 1205821198861198981015840119909
2
sum
119897=1
120577119897
119886(119909)
+(minus1)119899+1198991015840
119870119909119910119887
cos 1205821198861198981015840119909
2
sum
119897=1
120577119897
119886(119909) 119889119909
+ int
119887
0
1198701199090
2
sum
119897=1
120577119897
119886(0) cos 120582
1198871198991015840119910
+(minus1)119898+1198981015840
119870119909119886
2
sum
119897=1
120577119897
119886(0) cos 120582
1198871198991015840119910119889119910
K1-4119904119905
=
1
2
int
119886
0
int
119887
0
119863(119909 119910) 21205821198871198991205821198861198981015840 sin 120582
1198861198981015840119909 cos 120582
1198871198991015840119910
times cos 120582119886119898119909 sin 120582
119887119899119910 minus 2(1 minus 120583)
119887119899
times 1205821198861198981015840 sin 120582
1198861198981015840119909 cos 120582
1198871198991015840119910 cos 120582
119886119898119909
times sin 120582119887119899119910 + (1 minus 120583) 120582
1198861198981205821198871198991015840 cos 120582
1198861198981015840119909
times sin 1205821198871198991015840119910 sin 120582
119886119898119909 cos 120582
119887119899119910 119889119909 119889119910
22 Shock and Vibration
K1-5119904119901
=
1
2
int
119886
0
int
119887
0
119863(119909 119910) minus 21205821198861198981015840 sin 120582
1198861198981015840119909 cos 120582
1198871198991015840119910
times
2
sum
119897=1
1205771198971015840
119887(119910) cos 120582
119886119898119909
+ 2 (1 minus 120583) 1205821198861198981015840 sin 120582
1198861198981015840119909 cos 120582
1198871198991015840119910
times
2
sum
119897=1
1205771198971015840
119887(119910) cos 120582
119886119898119909
+ (1 minus 120583) 1205821198861198981205821198871198991015840 cos 120582
1198861198981015840119909 sin 120582
1198871198991015840119910
times
2
sum
119897=1
120577119897
119887(119910) sin 120582
119886119898119909119889119909119889119910
K1-6119904119902
=
1
2
int
119886
0
int
119887
0
119863(119909 119910)1205821198871198991205821198861198981015840 sin 120582
1198861198981015840119909 cos 120582
1198861198991015840119910
times
2
sum
119897=1
120577119897
119886(119909) sin 120582
1198871198991015840119910
minus 2 (1 minus 120583) 1205821198871198991205821198861198981015840 sin 120582
1198861198981015840119909 cos 120582
1198861198991015840119910
times
2
sum
119897=1
120577119897
119886(119909) sin 120582
1198871198991015840119910
+ 1205821198871198991015840 cos 120582
1198861198981015840119909 sin 120582
1198871198991015840119910
times
2
sum
119897=1
1205771198971015840
119886(119909) cos 120582
119887119899119910119889119909119889119910
K1-7119904119905
= minus
1
2
int
119886
0
int
119887
0
120581119866ℎ (119909 119910) 120582119886119898
cos 1205821198861198981015840119909 cos 120582
1198871198991015840119910
times sin 120582119886119898119909 cos 120582
119887119899119910119889119909119889119910
K1-8119904119901
= minus
1
2
int
119886
0
int
119887
0
120581119866ℎ (119909 119910) 1205821198861198981015840 cos 120582
1198861198981015840119909 cos 120582
1198871198991015840119910
times
2
sum
119897=1
120577119897
119887(119910) sin 120582
1198861198981015840119909 119889119909 119889119910
K1-9119904119902
=
1
2
int
119886
0
int
119887
0
120581119866ℎ (119909 119910) cos 1205821198861198981015840119909 cos 120582
1198871198991015840119910
times
2
sum
119897=1
1205771198971015840
119886(119909) cos 120582
119887119899119910119889119909119889119910
M1-1119904119905
=
1
24
120588int
119886
0
int
119887
0
ℎ3
(119909 119910) cos 120582119886119898119909 cos 120582
119887119899119910
times cos 1205821198861198981015840119909 cos 120582
1198871198991015840119910119889119909119889119910
M1-2119904119901
=
1
24
120588int
119886
0
int
119887
0
ℎ3
(119909 119910)
2
sum
119897=1
120577119897
119887(119910) cos 120582
119886119898119909
times cos 1205821198861198981015840119909 cos 120582
1198871198991015840119910119889119909119889119910
M1-3119904119902
=
1
24
120588int
119886
0
int
119887
0
ℎ3
(119909 119910)
2
sum
119897=1
120577119897
119886(119909) cos 120582
119887119899119910
times cos 1205821198861198981015840119909 cos 120582
1198871198991015840119910119889119909119889119910
M1-4119904119905 = M
1-5119904119901 = M1-6119904119902
= M1-7119904119905 = M
1-8119904119901 = M1-9119904119902 = 0
(A1)
B Useful Integrals for the Moderately ThickRectangular Plates with Variable Thicknessand Arbitrary Boundary Conditions
Consider
int
119871
0
cos 120582119898119909 cos 120582
119899119909 119889119909 =
0 119898 = 119899
119871
2
119898 = 119899
119871 119898 = 119899 = 0
int
119871
0
sin 120582119898119909 cos 120582
119899119909119889119909
=
minus
(minus1 + (minus1)119898+119899
) 119871119898
(1198982minus 1198992) 120587
119898 = 119899
0 119898 = 119899
int
119871
0
sin 120582119898119909 sin 120582
119899119909 119889119909 =
0 119898 = 119899
119871
2
119898 = 119899
0 119898 = 119899 = 0
int
119871
0
cos 120582119898119909120577119897
119886(119909) 119889119909
=
41198712
31205872 119897 = 1 119898 = 0
minus
41198712
(41198982
minus 3)
(9 minus 401198982+ 16119898
4) 1205872
119897 = 1 119898 = 0
minus
41198712
31205872 119897 = 2 119898 = 0
41198712
(41198982
minus 3) (minus1)119898
(9 minus 401198982+ 16119898
4) 1205872
119897 = 2 119898 = 0
Shock and Vibration 23
int
119871
0
sin 120582119898119909120577119897
119886(119909) 119889119909
=
0 119897 = 1 119898 = 0
161198712
119898(minus1)119898
(9 minus 401198982+ 16119898
4) 1205872
119897 = 1 119898 = 0
0 119897 = 2 119898 = 0
161198981198712
(9 minus 401198982+ 16119898
4) 1205872
119897 = 2 119898 = 0
int
119871
0
cos 1205821198981199091205771198971015840
119886(119909) 119889119909
=
0 119897 = 1 119898 = 0
161198711198982
(minus1)119898
(9 minus 401198982+ 16119898
4) 120587
119897 = 1 119898 = 0
0 119897 = 2 119898 = 0
161198711198982
(9 minus 401198982+ 16119898
4) 120587
119897 = 2 119898 = 0
int
119871
0
sin 1205821198981199091205771198971015840
119886(119909) 119889119909
=
0 119897 = 1 119898 = 0
4119871119898 (minus3 + 41198982
)
(9 minus 401198982+ 16119898
4) 120587
119897 = 1 119898 = 0
0 119897 = 2 119898 = 0
119871119898(
1
120587 minus 41198982120587
+
3
9120587 minus 41198982120587
) (minus1)119898
119897 = 2 119898 = 0
int
119871
0
1205771198971
119886(119909) 1205771198972
119886(119909) 119889119909 =
1198713
41205872 1198971= 1 1198972= 1
minus
21198713
31205873 1198971= 1 1198972= 2
minus
21198713
31205873 1198971= 2 1198972= 1
1198713
41205872 1198971= 2 1198972= 2
int
119871
0
1205771198971
119886(119909) 1205771198971015840
2
119886(119909) 119889119909 =
0 1198971= 1 1198972= 1
minus
1198712
8120587
1198971= 1 1198972= 2
1198712
8120587
1198971= 2 1198972= 1
0 1198971= 2 1198972= 2
int
119871
0
1205771198971015840
1
119886(119909) 1205771198971015840
2
119886(119909) 119889119909 =
5119871
16
1198971= 1 1198972= 1
minus
119871
2120587
1198971= 1 1198972= 2
minus
119871
2120587
1198971= 2 1198972= 1
5119871
16
1198971= 2 1198972= 2
int
119871
0
119909119899 cos 120582
119898119909 119889119909
=
1198711+119899
1 + 119899
119898 = 0
(1198861+119899HypergeometricPFQtimes [ 12 + 1198992 12 32 + 1198992
minus (14)1198982
1205872
]) times (1 + 119899)minus1
119898 gt 0
(B1)
Conflict of Interests
The authors declare that there is no conflict of interestsregarding the publication of this paper
Acknowledgments
The authors would like to thank the anonymous reviewersfor their very valuable comments The authors also gratefullyacknowledge the financial support from the National NaturalScience Foundation of China (no 51209052) HeilongjiangProvince Youth Science Fund Project (no QC2011C013) andHarbin Science and Technology Development InnovationFoundation of Youth (no 2011RFQXG021)
References
[1] K Liew Y Xiang and S Kitipornchai ldquoResearch on thick platevibration a literature surveyrdquo Journal of Sound and Vibrationvol 180 no 1 pp 163ndash176 1995
[2] J H Chung T Y Chung and K C Kim ldquoVibration analysisof orthotropic mindlin plates with edges elastically restrainedagainst rotationrdquo Journal of Sound and Vibration vol 163 no 1pp 151ndash163 1993
[3] Y K Cheung and D Zhou ldquoVibrations of moderately thickrectangular plates in terms of a set of static Timoshenko beamfunctionsrdquoComputers and Structures vol 78 no 6 pp 757ndash7682000
[4] CMWang ldquoNatural frequencies formula for simply supportedMindlin platesrdquo Journal of Vibration and Acoustics vol 116 no4 pp 536ndash540 1994
[5] K N Saha R C Kar and P K Datta ldquoFree vibration analysisof rectangular Mindlin plates with elastic restraints uniformlydistributed along the edgesrdquo Journal of Sound andVibration vol192 no 4 pp 885ndash902 1996
24 Shock and Vibration
[6] D J Gorman ldquoAccurate free vibration analysis of shear-deformable plates with torsional elastic edge supportrdquo Journalof Sound and Vibration vol 203 no 2 pp 209ndash218 1997
[7] D J Gorman ldquoFree vibration analysis of Mindlin plates withuniform elastic edge support by the superposition methodrdquoJournal of Sound andVibration vol 207 no 3 pp 335ndash350 1997
[8] D J Gorman ldquoAccurate free vibration analysis of point sup-ported mindlin plates by the superposition methodrdquo Journal ofSound and Vibration vol 219 no 2 pp 265ndash277 1999
[9] Y Xiang ldquoVibration of rectangular Mindlin plates resting onnon-homogenous elastic foundationsrdquo International Journal ofMechanical Sciences vol 45 no 6-7 pp 1229ndash1244 2003
[10] Y Xiang and G W Wei ldquoExact solutions for buckling andvibration of stepped rectangular Mindlin platesrdquo InternationalJournal of Solids and Structures vol 41 no 1 pp 279ndash294 2004
[11] Y L Yeh M J Jang and C C Wang ldquoAnalyzing the freevibrations of a plate using finite difference and differential trans-formationmethodrdquoAppliedMathematics andComputation vol178 no 2 pp 493ndash501 2006
[12] Y Xiang S K Lai L Zhou and C W Lim ldquoDSC-Ritz elementmethod for vibration analysis of rectangular Mindlin plateswith mixed edge supportsrdquo European Journal of MechanicsmdashASolids vol 29 no 4 pp 619ndash628 2010
[13] H Nguyen-Xuan C H Thai and T Nguyen-Thoi ldquoIsogeo-metric finite element analysis of composite sandwich platesusing a higher order shear deformation theoryrdquo Composites BEngineering vol 55 pp 558ndash574 2013
[14] C H Thai A J M Ferreira E Carrera and H Nguyen-XuanldquoIsogeometric analysis of laminated composite and sandwichplates using a layerwise deformation theoryrdquo Composite Struc-tures vol 104 pp 196ndash214 2013
[15] C H Thai A Ferreira S Bordas T Rabczuk and HNguyen-Xuan ldquoIsogeometric analysis of laminated compositeand sandwich plates using a new inverse trigonometric sheardeformation theoryrdquo European Journal of Mechanics A Solidsvol 43 pp 89ndash108 2014
[16] H Luong-Van T Nguyen-Thoi G R Liu and P Phung-VanldquoA cell-based smoothed finite elementmethod using three-nodeshear-locking free Mindlin plate element (CS-FEM-MIN3) fordynamic response of laminated composite plates on viscoelasticfoundationrdquo Engineering Analysis with Boundary Elements vol42 pp 8ndash19 2014
[17] T Nguyen-Thoi T Rabczuk T Lam-Phat V Ho-Huu andP Phung-Van ldquoFree vibration analysis of cracked Mindlinplate using an extended cell-based smoothed discrete sheargap method (XCS-DSG3)rdquo Theoretical and Applied FractureMechanics 2014
[18] H Nguyen-Xuan G R Liu and C a Thai-Hoang ldquoAnedge-based smoothed finite element method (ES-FEM) withstabilized discrete shear gap technique for analysis of Reissner-Mindlin platesrdquo Computer Methods in Applied Mechanics andEngineering vol 199 no 9ndash12 pp 471ndash489 2010
[19] P Phung-Van T Nguyen-Thoi T Bui-Xuan and Q Lieu-XuanldquoA cell-based smoothed three-node Mindlin plate element(CS-FEM-MIN3) based on the 119862
0 -type higher-order sheardeformation for geometrically nonlinear analysis of laminatedcomposite platesrdquo Computational Materials Science 2014
[20] NNguyen-Thanh T RabczukHNguyen-Xuan and S BordasldquoAn alternative alpha finite element method with discrete sheargap technique for analysis of isotropicMindlin-Reissner platesrdquoFinite Elements inAnalysis andDesign vol 47 no 5 pp 519ndash5352011
[21] KM Liew Y Xiang and S Kitipornchai ldquoTransverse vibrationof thick rectangular plates-I Comprehensive sets of boundaryconditionsrdquo Computers and Structures vol 49 no 1 pp 1ndash291993
[22] D Zhou ldquoVibrations of Mindlin rectangular plates with elasti-cally restrained edges using static Timoshenko beam functionswith the Rayleigh-Ritz methodrdquo International Journal of Solidsand Structures vol 38 no 32-33 pp 5565ndash5580 2001
[23] D Zhou SH Lo F T KAu andYK Cheung ldquoVibration anal-ysis of rectangular Mindlin plates with internal line supportsusing static Timoshenko beam functionsrdquo International Journalof Mechanical Sciences vol 44 no 12 pp 2503ndash2522 2002
[24] H S Shen J Yang and L Zhang ldquoFree and forced vibrationof Reissner-Mindlin plates with free edges resting on elasticfoundationsrdquo Journal of Sound and Vibration vol 244 no 2 pp299ndash320 2001
[25] Y Xing and B Liu ldquoClosed form solutions for free vibrations ofrectangular Mindlin platesrdquo Acta Mechanica Sinica vol 25 no5 pp 689ndash698 2009
[26] P Gagnon C Gosselin and L Cloutier ldquoA finite strip elementfor the analysis of variable thickness rectangular thick platesrdquoComputers amp Structures vol 63 no 2 pp 349ndash362 1997
[27] T Mlzusawa ldquoVibration of rectangular mindlin plates withtapered thickness by the spline strip methodrdquo Computers andStructures vol 46 no 3 pp 451ndash463 1993
[28] H Nguyen-Xuan and T Nguyen-Thoi ldquoA stabilized smoothedfinite element method for free vibration analysis of Mindlin-Reissner platesrdquo Communications in Numerical Methods inEngineering with Biomedical Applications vol 25 no 8 pp 882ndash906 2009
[29] K M Liew L X Peng and S Kitipornchai ldquoVibration anal-ysis of corrugated Reissner-Mindlin plates using a mesh-freeGalerkin methodrdquo International Journal of Mechanical Sciencesvol 51 no 9-10 pp 642ndash652 2009
[30] Y Hou GWWei and Y Xiang ldquoDSC-Ritz method for the freevibration analysis of Mindlin platesrdquo International Journal forNumerical Methods in Engineering vol 62 no 2 pp 262ndash2882005
[31] F- Liu and K M Liew ldquoVibration analysis of discontinuousMindlin plates by differential quadrature element methodrdquoJournal of Vibration and Acoustics Transactions of the ASMEvol 121 no 2 pp 204ndash208 1999
[32] P Malekzadeh G Karami and M Farid ldquoA semi-analyticalDQEM for free vibration analysis of thick plates with twoopposite edges simply supportedrdquoComputerMethods inAppliedMechanics and Engineering vol 193 no 45ndash47 pp 4781ndash47962004
[33] P Malekzadeh and S A Shahpari ldquoFree vibration analysisof variable thickness thin and moderately thick plates withelastically restrained edges by DQMrdquo Thin-Walled Structuresvol 43 no 7 pp 1037ndash1050 2005
[34] R E Diaz-Contreras and S Nomura ldquoGreenrsquos function appliedto solution ofMindlin platesrdquoComputers and Structures vol 60no 1 pp 41ndash48 1996
[35] T Sakiyama and M Huang ldquoFree vibration analysis of rect-angular plates with variable thicknessrdquo Journal of Sound andVibration vol 216 no 3 pp 379ndash397 1998
[36] M Huang X Q Ma T Sakiyama H Matuda and C MoritaldquoFree vibration analysis of orthotropic rectangular plates withvariable thickness and general boundary conditionsrdquo Journal ofSound and Vibration vol 288 no 4-5 pp 931ndash955 2005
Shock and Vibration 25
[37] I Shufrin and M Eisenberger ldquoVibration of shear deformableplates with variable thicknessmdashfirst-order and higher-orderanalysesrdquo Journal of Sound and Vibration vol 290 no 1-2 pp465ndash489 2006
[38] S A Eftekhari and A A Jafari ldquoAccurate variational approachfor free vibration of variable thickness thin and thick plates withedges elastically restrained against translation and rotationrdquoInternational Journal of Mechanical Sciences vol 68 pp 35ndash462013
[39] W L Li X Zhang J Du and Z Liu ldquoAn exact series solutionfor the transverse vibration of rectangular plates with generalelastic boundary supportsrdquo Journal of Sound amp Vibration vol321 no 1-2 pp 254ndash269 2009
[40] X Shi D Shi W L Li and Q Wang ldquoA unified method forfree vibration analysis of circular annular and sector plateswith arbitrary boundary conditionsrdquo Journal of Vibration andControl 2014
[41] D Y Shi X J Shi W L Li and Q S Wang ldquoFree transversevibrations of orthotropic thin rectangular plates with arbitraryelastic edge supportsrdquo Journal of Vibroengineering vol 16 no 1pp 389ndash398 2014
[42] Y Chen G Jin and Z Liu ldquoFree vibration analysis of circu-lar cylindrical shell with non-uniform elastic boundary con-straintsrdquo International Journal of Mechanical Sciences vol 74pp 120ndash132 2013
[43] L Dai T Yang J Du W L Li and M J Brennan ldquoAn exactseries solution for the vibration analysis of cylindrical shellswith arbitrary boundary conditionsrdquo Applied Acoustics vol 74no 3 pp 440ndash449 2013
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Active and Passive Electronic Components
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Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Shock and Vibration
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Civil EngineeringAdvances in
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DistributedSensor Networks
International Journal of
Shock and Vibration 11
Table 3 Frequency parameters Ω = 1205961198872
(120588ℎ119863)12
1205872 for S-S-S-S moderately thick plates (119886119887 = 1) with uniform thickness and elastic
rotation support (1198701199090= 119870119909119886= 1198701199100= 119870119910119887= Γ119909)
Γ119909
aℎ0119886
Model sequence1 2 3 4 5 6
0
005 1945 4834 4834 7591 9515 9519(1942b) (4836) (4836) (7584) (9547) (9552)
01 1856 4500 4500 6829 8499 8513(1858b) (4510) (4510) (6843) (8539) (8554)
015 1758 4114 4114 6040 7421 7445(1761b) (4126) (4126) (6060) (7462) (7486)
02 1656 3733 3733 5335 6468 6499(1660b) (3747) (3747) (5357) (6508) (6538)
05
005 3541 7071 7071 10227 12318 12399(3546b) (7091) (7091) (10253) (12386) (12468)
01 3274 6242 6242 8719 10302 10421(3282b) (6267) (6267) (8754) (10370) (10488)
015 2947 5361 5361 7290 8465 8606(2956b) (5387) (5387) (7327) (8524) (8666)
02 2624 4599 4599 6148 7057 7199(2634b) (4624) (4624) (6182) (7107) (7250)
1
005 3541 7072 7072 10228 12320 12401(3547b) (7093) (7093) (10254) (12388) (12470)
01 3278 6248 6248 8727 10311 10429(3286b) (6274) (6274) (8762) (10378) (10497)
015 2957 5374 5374 7305 8478 8620(2966b) (5400) (5400) (7341) (8537) (8680)
02 2641 4617 4617 6166 7070 7215(2651b) (4643) (4643) (6200) (7121) (7268)
15
005 3542 7073 7073 10229 12320 12401(3547b) (7093) (7093) (10254) (12388) (12471)
01 3280 6251 6251 8729 10313 10432(3287b) (6276) (6276) (8765) (10381) (10500)
015 2960 5379 5379 7309 8482 8625(2969b) (5405) (5405) (7346) (8541) (8685)
02 2647 4623 4623 6172 7074 7221(2657b) (4649) (4649) (6206) (7125) (7274)
2
005 3542 7073 7073 10229 12321 12402(3547b) (7093) (7093) (10255) (12389) (12471)
01 3280 6252 6252 8730 10315 10434(3288b) (6277) (6277) (8766) (10382) (10501)
015 2962 5381 5381 7312 8484 8628(2971b) (5407) (5407) (7348) (8543) (8688)
02 2650 4627 4627 6175 7076 7224(2660b) (4652) (4652) (6210) (7128 (7277)
aΓ119909= 1198701198941198700 (1198700= 1 times 10
9 Nmrad 119894 = 1199090 119909119886 1199100 119910119887)bResults in parentheses are taken from FEM
12 Shock and Vibration
Table 4 Frequency parameters Ω = 1205961198872
(120588ℎ119863)12
1205872 for moderately thick plates (119886119887 = 1) with uniform thickness and all edge elastic
restraints (1198701199090= 119870119909119886= 1198701199100= 119870119910119887= Γ119909and 119896
1199090= 119896119909119886= 1198961199100= 119896119910119887= Γ119908)
Γ119909
aΓ119908
aℎ0119886
Model sequence1 2 3 4 5 6
05 0
005 0986 1967 3893 3893 4850 4850(0988b) (1971) (3897) (3897) (4852) (4852)
01 0916 1835 3515 3522 4375 4375(0917b) (1837) (3519) (3528) (4341) (4341)
015 0789 1635 2983 3027 3771 3771(0792b) (1638) (2988) (3029) (3775) (3775)
02 0646 1444 2497 2601 3249 3249(0652b) (1448) (2508) (2621) (3254) (3254)
1 05
005 3417 3417 4431 5294 5461 6448(3419b) (3419) (4435) (5303) (5478) (6456)
01 1593 1593 2401 3849 3861 4729(1611b) (1611) (2408) (3852) (3866) (4767)
015 1122 1122 1913 3272 3290 4038(1127b) (1227) (1924) (3284) (3314) (4069)
02 0890 0890 1640 2781 2829 3476(0899b) (0899) (1654) (2787) (2834) (3497)
15 1
005 4325 4325 5619 6273 6571 7619(4334b) (4334) (5681) (6299) (6642) (7642)
01 2013 2013 2812 4088 4116 4993(2024b) (2024) (2827) (4122) (4187) (4994)
015 1338 1338 2108 3430 3443 4196(1342b) (1342) (2112) (3467) (3482) (4200)
02 1038 1038 1772 2936 2965 3613(1042b) (1042) (1786) (2937) (2977) (3624)
2 2
005 5231 5231 6919 7630 8002 9220(5238b) (5238) (6931) (7637) (8011) (9241)
01 2594 2594 3439 4470 4544 5428(2597b) (2597) (3442) (4495) (4557) (5438)
015 1653 1653 2403 3613 3632 4392(1662b) (1662) (2414) (3627) (3646) (4404)
02 1238 1238 1949 3077 3099 3751(1244b) (1244) (1964) (3094) (3116) (3788)
aΓ119909= 1198701198941198700 (1198700= 1 times 10
9 Nmrad 119894 = 1199090 119909119886 1199100 119910119887) Γ119908= 1198961198941198960 (1198960= 1 times 10
9 Nm 119894 = 1199090 119909119886 1199100 119910119887)bResults in parentheses are taken from FEM
05 10 15 20 25 300
2
4
6
8
10
12
14
16
ab
The 1st orderThe 2nd orderThe 3rd order
The 4th orderThe 5th orderThe 6th order
Ω
(a)
05 10 15 20 25 30
30
60
90
120
150
180
ab
The 1st orderThe 2nd orderThe 3rd order
The 4th orderThe 5th orderThe 6th order
Ω
(b)
Figure 6 The effect of aspect ratio 119886119887 on the natural frequenciesΩ for S-S-S-S boundary condition (a) uniform thickness and (b) nonuni-form thickness
Shock and Vibration 13
Table 5 Frequency parameter Ω = 1205961198862
(120588ℎ01198630)12 for moderately thick plates with linearly varying thickness in different boundary
conditions (ℎ(119909 119910) = ℎ0(1 + 05119909)(1 + 05119910))
Boundary condition 119886119887 ℎ0119886
Model sequence1 2 3 4 5 6
S-S-S-S
101 27261 62397 62926 92423 111537 111732
(27263a) (62410) (63842) (92445) (111867) (111921)
02 23103 47010 47191 65018 75831 75865(23489a) (47209) (47349) (65057) (76605) (76473)
3201 59338 99101 138798 151102 166710 204855
(59874a) (99249) (138177) (151362) (166644) (205381)
02 41175 63765 63765 73598 83956 89084(41586a) (63313) (63669) (73661) (83918) (89547)
C-F-F-F
101 4173 10331 27407 37826 38809 64095
(4856a) (10680) (27513) (37203) (39135) (64858)
02 3983 8702 21403 29443 30286 46705(4118a) (9306) (21990) (29670) (30602) (47156)
3201 4868 15488 35620 63523 89969 100089
(4887a) (15320) (35175) (63353) (89159) (100783)
02 4469 9763 22338 42188) 50104 56331(4944a) (9636) (22087) (42177) (50580) (56559)
S-S-F-F
101 4117 24795 26216 48567) 68195 69667
(4789a) (24149) (26751) (49324) (68842) (70575)
02 3820 20516 21924 37523 48980 50327(4103a) (20041) (22125) (37221) (48894) (50622)
3201 7935 39454 65063 92877 92877 136176
(7856a) (40145) (65162) (92462) (93022) (136336)
02 6389 26149 35627 49539 56844 65170(6406a) (26536) (35626) (49831) (57144) (65981)
C-C-C-C
101 44471 80299 80681 109422 126871 128207
(45102a) (80271) (80656) (110015) (126328) (128888)
02 32162 52781 52933 69410 78474 79482(32240a) (52629) (53166) (70063) (79059) (80266)
3201 82975 115587 151992 161579 176601 212330
(83139a) (115838) (152200) (162193) (177058) (212849)
02 48577 66022 83468 90830 97474 98157(49187a) (66259) (84065) (91749) (97827) (98777)
S-C-S-C
101 36847 66664 76965 101045 113594 125551
(37901a) (68568) (78356) (101200) (113978) (125651)
02 27673 48469 51220 67027 76358 78241(27897a) (48685) (51491) (67447) (77530) (79573)
3201 77378 106848 150079 154569 173120 208391
(78003a) (107058) (150991) (155171) (173099) (208402)
02 45895 64732 73393 83282 90325 97089(45803a) (64131) 73656 83801 90518 97015
aResults in parentheses are taken from FEM
restrained That is 1198701199090
= 119870119909119886
= 1198701199100
= 119870119910119887
= Γ119909 The six
frequency parameters Ω = 1205961198862
(120588ℎ01198630)12 are given in
Table 6 for several different restraining coefficient valuesthe finite element method (FEM) results calculated usingABAQUS are also presented as a referenceThe secondmodelis a complete square moderately thick plate with all edgeselastically restrainedThat is 119896
1199090= 119896119909119886= 1198961199100= 119896119910119887= Γ119908and
1198701199090= 119870119909119886
= 1198701199100= 119870119910119887= Γ119909 The six frequency parameters
Ω = 1205961198862
(120588ℎ01198630)12 are given in Table 7 for several different
restraining coefficient values the finite element method(FEM) results are also listed as a reference It can be clearlyseen that the comparison is extremely good which impliesthat the current method is able to make correct predictionsfor the modal characteristics of linearly varying thickness
14 Shock and Vibration
Table 6 Frequency parameter Ω = 1205961198862
(120588ℎ01198630)12 for S-S-S-S moderately thick plates (119886119887 = 1) with linearly varying thickness and elastic
rotation support (ℎ(119909 119910) = ℎ0(1 + 05119909)(1 + 05119910) 119870
1199090= 119870119909119886= 1198701199100= 119870119910119887= Γ119909)
Γ119909
aℎ0119886
Model sequence1 2 3 4 5 6
0
005 28809 70080 70891 108967 134884 135295(28897b) (70334) (71129) (109112) (135254) (135772)
01 27261 62398 62926 92423 111540 111735(26606b) (61784) (62303) (91193) (111323) (111648)
015 25231 54192 54505 77118 91228 91312(25204b) (55297) (55595) (77503) (91871) (91941)
02 23103 47010 47191 65018 75832 75866(23947b) (47071) (47208) (65404) (76531) (76589)
05
005 23103 47010 47191 65018 75832 75866(23210b) (47245) (47358) (65241) (76122) (76345)
01 48377 94650 95597 135484 162635 163501(48524b) (94891) (95992) (135845) (167012) (164024)
015 36239 70939 71298 99997 118584 119009(36647b) (71214) (71687) (100024) (119114) (119457)
02 28749 56937 57037 79214 93009 93162(29142b) (57034) (57544) (79654) (93668) (93754)
1
005 49651 96759 97619 138089 165368 166304(49755b) (97245) (97964) (138544) (165864) (166878)
01 38951 73735 74251 102748 121079 121709(39021b) (74145) (74664) (102874) (121613) (121817)
015 30630 58406 58533 80432 93984 94255(30638b) (58684) (58644) (80874) (94121) (94478)
02 25548 48636 48674 66165 76708 76824(25687b) (48788) (48992) (66544) (76824) (76944)
15
005 50118 97561 98374 139096 166421 167382(50247b) (97877) (98661) (139247) (166875) (167662)
01 40310 75216 75783 104229 122402 123164(40366b) (75542) (75921) (104557) (122874) (123385)
015 31837 59352 59530 81245 94611 94983(31902b) (59451) (59871) (81375) (94784) (95105)
02 26302 49119 49156 66528 76954 77130(26418b) (49354) (49334) (66921) (77010) (77470)
2
005 50360 97983 98769 139631 166979 167954(50472b) (98005) (99114) (140154) (167188) (167986)
01 41130 76146 76724 105165 123229 124082(41421b) (76642) (78114) (105661) (123278) (124974)
015 32687 60025 60245 81832 95052 95508(32778b) (60244) (60549) (82146) (95114) (95842)
02 26893 49492 49539 66815 77138 77370(26922b) (49874) (49924) (66987) (77367) (77588)
aΓ119909= 1198701198941198700 (1198700= 1 times 10
9 Nmrad 119894 = 1199090 119909119886 1199100 119910119887)bResults in parentheses are taken from FEM
Shock and Vibration 15
Table 7 Frequency parameterΩ = 1205961198862
(120588ℎ01198630)12 for moderately thick plates (119886119887 = 1) with linearly varying thickness and elastic rotation
and translation support (ℎ(119909 119910) = ℎ0(1 + 05119909)(1 + 05119910) 119870
1199090= 119870119909119886= 1198701199100= 119870119910119887= Γ119909 and 119896
1199090= 119896119909119886= 1198961199100= 119896119910119887= Γ119908)
Γ119909
aΓ119908
aℎ0119886
Model sequence1 2 3 4 5 6
05 0
005 14751 14917 28232 55251 55397 69512(14780b) (14822) (29252) (55476) (55505) (70904)
01 11811 11910 23857 44462 45037 55453(11818b) (11897) (24711) (44376) (45257) (56511)
015 8487 8514 20011 34783 36335 44970(8500b) (8544) (20738) (34404) (36663) (45625)
02 6073 6073 17431 28679 30805 38010(6096b) (6106) (18039) (28129) (31165) (38311)
1 05
005 21225 32173 32173 44381 64061 65222(20708b) (32097) (32285) (45110) (64182) (65355)
01 8140 16878 16941 28100 48663 49015(7943b) (16838) (16970) (28888) (48700) (49171)
015 4493 11859 11878 22429 38352 39247(4383b) (11867) (11891) (23125) (38177) (39488)
02 2932 8690 8691 18919 31108 32600(3540b) (8791) (8791) (16229) (27520) (27889)
15 1
005 27819 40919 41430 54595 70792 72896(27146b) (41008) (41434) (55256) (70895) (73051)
01 11285 20068 20230 31101 51046 51429(11012b) (20025) (20270) (31849) (51126) (51569)
015 6292 13912 13922 24132 40500 41120(6139b) (13912) (13934) (24808) (40426) (41316)
02 4121 10321 10331 20051 32796 33921(4021b) (10329) (10357) (20635) (32540) (34178)
2 2
005 34615 51972 53088 68255 81344 84510(33806b) (52100) (53157) (68891) (81433) (84761)
01 15416 24659 24829 35615 53879 54494(15045b) (24625) (24888) (36303) (53970) (54626)
015 8740 16420 16481 26353 42417 42946(8529b) (16414) (16499) (26996) (42398) (43114)
02 5761 12127 12127 21434 34286 35192(5622b) (12134) (12148) (21998) (34110) (35415)
aΓ119909= 1198701198941198700 (1198700= 1 times 10
9 Nmrad 119894 = 1199090 119909119886 1199100 119910119887) Γ119908= 1198961198941198960 (1198960= 1 times 10
9 Nm 119894 = 1199090 119909119886 1199100 119910119887)bResults in parentheses are taken from FEM
moderately thick plate with not only classical boundaryconditions but also elastic edge restraints
As the last case of this subsection the influence of thegradient 120572 and 120573 on the fundamental frequency parametersfor a linearly varying thickness moderately thick plate isinvestigated The model is a square moderately thick platewith all edges elastically restrainedThat is 119896
1199090= 119896119909119886= 1198961199100=
119896119910119887= Γ119908= 2 and 119870
1199090= 119870119909119886
= 1198701199100= 119870119910119887= Γ119909= 2 The six
frequency parameters Ω = 1205961198862
(120588ℎ01198630)12 are given in
Table 8 for several different 120572 and 120573 values the finite elementmethod (FEM) results are also listed as a reference Againgood agreement can be observed Through Table 8 it is alsofound that the frequency parameter increases with increasinggradient parameters
The above studies are given as linearly varying thicknessmoderately thick plates with several of boundary condition
and different structure parameters In the next section vibra-tion results for the plates subjected to nonlinear variationthickness will be presented
34 Nonlinearly Varying Thickness Moderately Thick Plateswith Classical and Elastic Boundary Conditions In Sec-tion 33 the linearly varying thickness moderately thickplates were studied However in the practical engineeringapplications the varying thickness of amoderately thick platemay not always be linear variation in nature A variety ofpossible thickness varying cases may be encountered in prac-tice Therefore the moderately thick plates with nonlinearvariation thickness subjected to general elastic edge restraintsare examined in this subsection For the sake of brevity theindexes 119904 and 119905 will be chosen as 2 to imitate the nonlinearlyvarying thickness moderately thick plates structure in this
16 Shock and Vibration
Table 8 Frequency parameterΩ = 1205961198862
(120588ℎ01198630)12 for elastic support moderately thick plates (119886119887 = 1) with different gradient (119896
1199090= 119896119909119886=
1198961199100= 119896119910119887= Γ119908= 2 119870
1199090= 119870119909119886= 1198701199100= 119870119910119887= Γ119909= 2)
120572 120573 ℎ0119886
Model sequence1 2 3 4 5 6
120572 = 120573 = 01
005 31275 52615 52657 68954 76146 79850(31303a) (52664) (52675) (69033) (76174) (79948)
01 17270 25244 25250 33949 46050 46625(17271a) (25255) (25258) (34089) (46122) (46708)
015 10190 16266 16274 24164 37327 37526(10189a) (16269) (16280) (24323) (37401) (37622)
02 6780 12258 12260 19685 31452 31754(6780a) (12265) (12270) (19851) (31514) (31864)
120572 = 120573 = 02
005 32530 52947 53141 68954 77034 80589(32577a) (53014) (53203) (69172) (77088) (80773)
01 16882 24979 24979 34154 48057 48575(16883a) (24983) (25001) (34433) (48134) (48667)
015 9804 16284 16309 24683 38844 39085(9803a) (16283) (16322) (24980) (38909) (39191)
02 6499 12290 12295 20133 32354 32770(6498a) (12296) (12307) (20420) (32387) (32895)
120572 = 120573 = 03
005 33516 52845 53308 68704 78203 81561(33563a) (52939) (53389) (69055) (78271) (81789)
01 16419 24799 24830 34527 50061 50583(16419a) (24790) (24869) (34947) (50143) (50688)
015 9430 16334 16378 25236 40193 40506(9429a) (16329) (16395) (25660) (40243) (40625)
02 6235 12288 12293 20572 33102 33663(6235a) (12296) (12306) (20965) (33090) (33811)
120572 = 120573 = 04
005 34215 52463 53264 68430 79654 82865(34252a) (52575) (53346) (68921) (79732) (83113)
01 15921 24695 24793 35027 52011 52571(15921a) (24674) (24842) (35583) (52098) (52688)
015 9074 16388 16444 25797 41379 41789(9074a) (16381) (16464) (26336) (41404) (41928)
02 5990 12236 12238 21002 33735 34461(5989a) (12249) (12250) (21487) (33653) (34641)
120572 = 120573 = 05
005 34615 51972 53088 68255 81344 84510(33806a) (52100) (53157) (68891) (81433) (84761)
01 15416 24659 24829 35615 53879 54494(15045a) (24625) (24888) (36303) (53970) (54626)
015 8740 16420 16481 26353 42417 42946(8529a) (16414) (16499) (26996) (42398) (43114)
02 5761 12127 12127 21434 34286 35192(5622a) (12134) (12148) (21998) (34110) (35415)
aResults in parentheses are taken from FEM
section that is 119904 = 119905 = 2 Also the thickness varying functionsare also be expressed as Fourier cosine series according to (18)and (19)
In order to validate the accuracy and reliability of theproposed method for predicting the vibration behavior ofnonlinearly varying thickness moderately thick plates with
arbitrary boundary conditions the typical classical boundaryconditions viewed as the special cases of elastically restrainededges will be considered The comparison of the first six fre-quency parameters Ω = 120596119886
2
(120588ℎ01198630)12 for the moderately
thick plates with nonlinearly varying thickness is presentedin Table 9 The S-S-S-S C-F-F-F S-S-F-F C-C-C-C and
Shock and Vibration 17
Table 9 Frequency parameter Ω = 1205961198862
(120588ℎ01198630)12 for moderately thick plates with linearly varying thickness in different boundary
conditions (ℎ(119909 119910) = ℎ0(1 + 05119909
2
)(1 + 051199102
))
Boundary condition 119886119887 ℎ0119886
Model sequence1 2 3 4 5 6
S-S-S-S
101 23462 54999 55573 82950 100900 101187
(23741a) (55114) (55983) (83146) (101004) (101347)
02 18606 42725 42971 60040 71167 71761(18936a) (43014) (43148) (60247) (71642) (71993)
3201 53055 65251 85404 106700 114880 143531
(53467a) (65472) (85984) (107211) (115687) (144007)
02 17439 21531 32170 44987 59438 64179(17874a) (21748) (32651) (45264) (60188) (64879)
C-F-F-F
101 3658 9317 24135 32816 35433 59179
(3749a) (9547) (24034) (33627) (36308) (60142)
02 3520 7986 19591 26959 27491 43882(3611a) (8176) (20075) (27125) (27625) (43984)
3201 3901 9963 28984 69187 73686 89360
(3930a) (10095) (29381) (69578) (73994) (90242)
02 3292 4439 13385 23947 40850 44758(3386a) (4498) (14012) (24068) (41221) (45142)
S-S-F-F
101 3557 21268 22762 43524 60253 62429
(3649a) (21441) (22796) (43916) (60622) (62630)
02 3171 17822 19512 33616 45042 46833(3489a) (18006) (20430) (33815) (45247) (49004)
3201 6738 31658 52071 75625 86604 94903
(6957a) (31882) (52518) (75823) (86939) (95610)
02 3653 12930 20955 26517 44097 48058(6807a) (13160) (21312) (26864) (44422) (48873)
C-C-C-C
101 41020 75196 75614 104142 119683 120949
(41405a) (75452) (75882) (104711) (119637) (121137)
02 31237 51686 51868 68411 76868 77864(32010a) (52960) (52145) (69335) (77481) (78099)
3201 95823 128727 163657 173751 190890 223595
(95364a) (128932) (163588) (172350) (190350) (222948)
02 53306 71372 89384 102404 104660 120908(53587a) (71759) (90103) (102503) (105174) (121275)
S-C-S-C
101 33715 60298 71446 94187 103802 117928
(34148a) (60786) (71828) (94515) (104263) (119326)
02 26143 45499 49458 64253 72941 76191(26789a) (46622) (50680) (64329) (73474) (76843)
3201 73228 115845 137445 148166 168342 182542
(73761a) (116121) (137788) (148530) (168698) (182921)
02 41302 53648 65630 72202 81632 88105(41304a) (53757) (65842) (72794) (82033) (88474)
aResults in parentheses are taken from FEM
S-C-S-C boundary conditions are performed in the compar-ison The results adequately demonstrated the great accuracyof the modified Fourier method
We now turn to elastically restrained moderately thickplates The first one involves an S-S-S-S moderately thicksquare plate with a uniform rotational restraint along eachedge that is 119870
1199090= 119870119909119886
= 1198701199100
= 119870119910119887
= Γ119909 The calculated
frequency parameters Ω = 1205961198862
(120588ℎ01198630)12 are given in
Table 10 together with the FEM results Since the elasticallyrestrained plates with nonlinear variation thickness are rarelyinvestigated the FEMresults are used as the referenceA goodagreement is observed between the current and FEM resultsThe second example concerns amoderately thick square plateelastically supported along all edgesThe stiffness of the linear
18 Shock and Vibration
Table 10 Frequency parameterΩ = 1205961198862
(120588ℎ01198630)12 for S-S-S-S moderately thick plates (119886119887 = 1) with linearly varying thickness and elastic
rotation support (ℎ(119909 119910) = ℎ0(1 + 05119909
2
)(1 + 051199102
) 1198701199090= 119870119909119886= 1198701199100= 119870119910119887= Γ119909)
Γ119909
aℎ0119886
Model sequence1 2 3 4 5 6
0
005 23462 54999 55573 82950 100900 101187(23741b) (55114) (55983) (83146) (101004) (101347)
01 18606 42725 42971 60040 71167 71761(18936b) (43014) (43148) (60247) (71642) (71993)
015 53055 65251 85404 106700 114880 143531(53467b) (65472) (85984) (107211) (115687) (144007)
02 17439 21531 32170 44987 59438 64179(17874b) (21748) (32651) (45264) (60188) (64879)
05
005 6904 13374 13514 19400 22904 23056(9909b) (13548) (13651) (19564) (23088) (23379)
01 5316 10237 10314 14499 17155 17264(5374b) (10448) (10651) (14764) (17188) (17879)
015 4136 8201 8220 11474 13578 13671(4174b) (8348) (8651) (11764) (13688) (13879)
02 3449 6915 6924 9549 11225 11327(3474b) (6948) (6951) (9764) (11388) (11579)
1
005 7040 13617 13745 19726 23233 23402(7074b) (13748) (13851) (19964) (23488) (23879)
01 5712 10696 10793 14991 17594 17744(5774b) (10748) (10851) (15054) (17788) (17879)
015 4489 8507 8540 11759 13803 13922(4514b) (8548) (8611) (11864) (13988) (14119)
02 3677 7084 7090 9692 11330 11445(3684b) (7148) (7191) (9764) (11588) (11879)
15
005 7089 13708 13828 19847 23355 23530(7124b) (13748) (13951) (19967) (23588) (23879)
01 5895 10923 11024 15239 17813 17988(5874b) (10748) (11651) (15264) (17818) (17979)
015 4697 8694 8737 11938 13790 14083(4774b) (8748) (8951) (11864) (13688) (14879)
02 3836 7201 7210 9792 11403 11530(3874b) (7748) (7651) (9864) (11588) (11879)
2
005 7114 13754 13871 19911 23418 23597(7174b) (12848) (13851) (19864) (23588) (23879)
01 6001 11060 11159 15391 17944 18136(6004b) (11148) (11151) (15464) (17988) (18379)
015 4836 8821 8872 12061 14037 14189(4874b) (8848) (8951) (12164) (14188) (14879)
02 3954 7289 7301 9871 11457 11594(3974b) (7348) (7651) (9964) (11488) (11679)
aΓ119909= 1198701198941198700 (1198700= 1 times 10
9 Nmrad 119894 = 1199090 119909119886 1199100 119910119887)bResults in parentheses are taken from FEM
and rotational restraints is set equal to 1198961199090
= 119896119909119886
= 1198961199100
=
119896119910119887
= Γ119908and 119870
1199090= 119870119909119886
= 1198701199100
= 119870119910119887
= Γ119909 respectively
The six frequency parametersΩ = 1205961198862
(120588ℎ01198630)12 are shown
in Table 11 for several different restraining coefficient valuesIt can also be noticed that the Fourier series method is able
to predict the modal characteristics of nonlinearly varyingthickness moderately thick plate with not only classicalboundary conditions but also elastic edge restraints correctly
Finally the influence of the gradient on the fundamentalfrequency parameters of a nonlinearly varying thickness
Shock and Vibration 19
Table 11 Frequency parameterΩ = 1205961198862
(120588ℎ01198630)12 for moderately thick plates (119886119887 = 1) with linearly varying thickness and elastic rotation
and translation support (ℎ(119909 119910) = ℎ0(1 + 05119909
2
)(1 + 051199102
) 1198701199090= 119870119909119886= 1198701199100= 119870119910119887= Γ119909 and 119896
1199090= 119896119909119886= 1198961199100= 119896119910119887= Γ119908)
Γ119909
aΓ119908
aℎ0119886
Model sequence1 2 3 4 5 6
05 0
005 12926 13041 25090 47935 47943 61761(12931b) (13053) 25099 (47947) (47952) (61788)
01 10900 10988 21545 39980 40147 50322(10904b) (10992) (21557) (39984) (40166) (50334)
015 8245 8298 18081 31503 32617 40950(8247b) (8230) (18092) (31508) (32622) (40955)
02 6106 6124 15639 25631 27601 34458(6110b) (6125) (15642) (25637) (27617) (34462)
1 05
005 21575 31615 31644 42573 57889 58607(21580b) (31617) (31644) (42581) (57893) (58610)
01 8451 16024 16210 25969 43839 43867(8451b) (16025) (16217) (25972) (43844) (43870)
015 4682 11479 11480 20642 35172 35598(4684b) (11480) (11481) (20644) (35178) (35604)
02 3059 8621 8648 17320 28487 29638(3062b) (8619) (8644) (17327) (28492) (29641)
15 1
005 27713 40203 40978 52951 65352 66815(27714b) (40207) (40988) (52972) (65368) (66822)
01 11667 19322 19615 29122 46115 46183(11674b) (19327) (19619) (29137) (46120) (46188)
015 6544 13416 13481 22405 37238 37430(6547b) (13420) (13488) (22410) (37241) (37434)
02 4295 10175 10190 18543 30330 31056(4296b) (10177) (10194) (18544) (30336) (31060)
2 2
005 33453 50854 52088 66311 76610 78947(33455b) (50852) (50097) (66321) (76627) (78971)
01 15832 24130 24372 33924 49118 49362(15830b) (24127) (24374) (33931) (49147) (49382)
015 9060 15956 16099 24754 39134 39249(9064b) (15961) (16102) (24755) (39138) (39452)
02 5992 11955 11971 20034 31908 32392(5991b) (11957) (11978) (20042) (31910) (32398)
aΓ119909= 1198701198941198700 (1198700= 1 times 10
9 Nmrad 119894 = 1199090 119909119886 1199100 119910119887) Γ119908= 1198961198941198960 (1198960= 1 times 10
9 Nm 119894 = 1199090 119909119886 1199100 119910119887)bResults in parentheses are taken from FEM
moderately thick plate is investigated The plate is elasticallyrestrained in which the stiffness of the boundary springsis taken as 119896
1199090= 119896119909119886
= 1198961199100
= 119896119910119887
= Γ119908
= 2 and1198701199090
= 119870119909119886
= 1198701199100
= 119870119910119887
= Γ119909= 2 respectively The six
frequency parameters Ω = 1205961198862
(120588ℎ01198630)12 are presented in
Table 12 for several different slop values It can be seen thatthe fundamental frequency parameters will decrease with theincrease of the parameter ℎ
0119886
In the above examples it has been demonstrated thatthe presented method can be universally applied to nonlin-early varying thickness moderately thick plates with severalboundary conditions and different structure parametersNew results are obtained for plates with nonlinear variationthickness in both directions subjected to general elastic
boundary restraints which may be used for benchmarkingof researchers in the field In addition it is interesting to seethat the nature frequency decreases with the increase of theindex for thickness function
4 Conclusions
In this paper a modified Fourier method has been pre-sented to study the free vibration behaviors of moderatelythick rectangular plates with variable thickness and arbi-trary boundary conditionsThe first-order shear deformationplate theory is adopted to formulate the theoretical modelThe displacements and rotation components of the plateregardless of boundary conditions are invariantly expressedas the superposition of a 2D Fourier cosine series and four
20 Shock and Vibration
Table 12 Frequency parameterΩ = 1205961198862
(120588ℎ01198630)12 for elastic support moderately thick plates (119886119887 = 1) with different gradient (119896
1199090= 119896119909119886=
1198961199100= 119896119910119887= Γ119908= 2 119870
1199090= 119870119909119886= 1198701199100= 119870119910119887= Γ119909= 2)
120572 120573 ℎ0119886
Model sequence1 2 3 4 5 6
120572 = 120573 = 01
005 30187 51102 51141 67066 73954 77412(30188a) (51105) (51144) (67068) (73961) (77477)
01 16995 24719 24724 33071 44343 44819(16996a) (24725) (24729) (33084) (44347) (44822)
015 10077 15872 15883 23443 35937 36065(10081a) (15877) (15892) (23447) (35941) (36084)
02 6713 11953 11958 19086 30372 30604(6713a) (11957) (11965) (19092) (30378) (30612)
120572 = 120573 = 02
005 31206 51436 51614 67058 74398 77595(31208a) (51438) (51619) (67064) (74402) (77596)
01 16786 24503 24508 33144 45583 45923(16790a) (24510) (24513) (33147) (45587) (45932)
015 9821 15865 15906 23766 36897 36990(9824a) (15870) (15907) (23768) (36894) (36997)
02 6523 11978 11992 19356 30912 31162(6524a) (11981) (11997) (19362) (30916) (31174)
120572 = 120573 = 03
005 32097 51462 51900 66832 74991 77863(32098a) (51466) (51907) (66849) (74996) (77872)
01 16507 24332 24383 33330 46806 47074(16511a) (24327) (24380) (33339) (46810) (47086)
015 9562 15885 15964 24103 37748 37828(9562a) (15892) (15977) (21105) (37751) (37832)
02 6338 11992 12013 19603 31338 31634(6340a) (11994) (12011) (19612) (31344) (31639)
120572 = 120573 = 04
005 32851 51239 52046 66547 75735 78303(33455a) (50852) (50097) (66321) (76627) (78971)
01 16182 24209 24343 33598 47990 48230(33455a) (50852) (50097) (66321) (76627) (78971)
015 9307 15919 16034 24436 38491 38579(33455a) (50852) (50097) (66321) (76627) (78971)
02 6161 11986 12008 19828 31666 32040(33455a) (50852) (50097) (66321) (76627) (78971)
120572 = 120573 = 05
005 33453 50854 52088 66311 76610 78947(33455a) (50852) (50097) (66321) (76627) (78971)
01 15832 24130 24372 33924 49118 49362(15830a) (24127) (24374) (33931) (49147) (49382)
015 9060 15956 16099 24754 39134 39249(9064a) (15961) (16102) (24755) (39138) (39452)
02 5992 11955 11971 20034 31908 32392(5991a) (11957) (11978) (20042) (31910) (32398)
aResults in parentheses are taken from FEM
supplementary functions in the form of the product of apolynomial function and a single cosine series expansionto ensure and accelerate the convergence of the solutionAt each edge of the plate the general restraint conditionsare implemented by introducing one group of linear springsand two groups of rotational springs which are continuouslydistributed and determined by the stiffness of these springs
Instead of seeking a solution in strong forms in the previousstudies all the Fourier coefficients will be treated equallyand independently as the generalized coordinates and solveddirectly from the Rayleigh-Ritz technique The change of theboundary conditions can be easily achieved by only varyingthe stiffness of the three sets of boundary springs along alledges of the rectangular plates without involving any change
Shock and Vibration 21
to the solution procedure The convergence of the presentsolution is examined and the excellent accuracy is validatedby comparison with existing results published in the lit-erature and FEM data Excellent agreements are obtainedfrom these comparisons The proposed method providesa unified means for extracting the modal parameters andpredicting the vibration behaviors of moderately thick plateswith variable thickness variation functions and arbitraryelastic edge restraints A variety of free vibration results formoderately thick rectangular plates with different thicknessvariation functions and boundary conditions are presentedNew results for free vibration of moderately thick rectangularplates with various thickness variation functions and edgeconditions are presented which may be used for benchmark-ing of researchers in the field
Appendices
A Representative Calculation forStiffness and Mass Matrices
To illuminate the particular expression of the mass andstiffness matrixes clearly and tersely four new variables aredefined as 119904 = 119898 lowast (119873 + 1) + 119899 + 1 119905 = 119898
1015840
lowast (119873 + 1) + 1198991015840
+ 1119901 = (119897minus1)lowast (119872+1)+119898
1015840
+1 and 119902 = (119897minus1)lowast (119873+1)+1198991015840
+1The first row elements of K andM are given below
K1-1119904119905
=
1
2
int
119886
0
int
119887
0
[119863 (119909 119910)
times 21205821198861198981205821198861198981015840 sin 120582
119886119898119909 sin 120582
1198861198981015840119909 cos 120582
119887119899119910
times cos 1205821198871198991015840119910 + (1 minus 120583) 120582
1198871198991205821198871198991015840 cos 120582
119886119898119909
times cos 1205821198861198981015840 sin 120582
119887119899119910 sin 120582
1198871198991015840119910
+ 120581119866ℎ (119909 119910) cos 120582119886119898119909 cos 120582
1198861198981015840119909 cos 120582
119887119899
times119910 cos 1205821198871198991015840119910119889119909119889119910] 119889119909 119889119910
+ int
119886
0
1198701199091199100
cos 120582119886119898119909 cos 120582
1198861198981015840119909
+(minus1)119898+1198981015840
119870119909119910119887
cos 120582119886119898119909 cos 120582
1198861198981015840119909 119889119909
+ int
119887
0
1198701199090cos 120582119887119899119910 cos 120582
1198871198991015840119910
+(minus1)119899+1198991015840
119870119909119886cos 120582119887119899119910 cos 120582
1198871198991015840119910 119889119910
K1-2119904119901
=
1
2
int
119886
0
int
119887
0
[119863 (119909 119910)21205821198861198981205821198861198981015840 sin 120582
119886119898119909 sin 120582
1198861198981015840119909
times
2
sum
119897=1
120577119897
119887(119910) cos 120582
1198871198991015840119910
minus (1 minus 120583) 1205821198871198991015840 cos 120582
119886119898119909 cos 120582
1198861198981015840119909
times
2
sum
119897=1
1205771198971015840
119887(119910) sin 120582
1198871198991015840119910 + 120581119866ℎ (119909 119910)
times ( cos 1205821198861198981015840119909 cos 120582
1198871198991015840119910
times
2
sum
119897=1
120577119897
119887(119910) cos 120582
119886119898119909)]119889119909119889119910
+ int
119886
0
1198701199091199100
cos 120582119886119898119909 cos 120582
1198861198981015840119909
2
sum
119897=1
120577119897
119887(0)
+(minus1)1198991015840
119870119909119910119887
cos 120582119886119898119909 cos 120582
1198861198981015840119909
2
sum
119897=1
120577119897
119887(119887) 119889119909
+ int
119887
0
1198701199090
2
sum
119897=1
120577119897
119887(119910) cos 120582
1198871198991015840119910
+(minus1)119898+1198981015840
119870119909119886
2
sum
119897=1
120577119897
119887(119910) cos 120582
1198871198991015840119910119889119910
K1-3119904119902
=
1
2
int
119886
0
int
119887
0
[119863 (119909 119910) minus 21205821198861198981015840 sin 120582
1198861198981015840119909 cos 120582
1198871198991015840119910
times
2
sum
119897=1
1205771198971015840
119886(119909) cos 120582
119887119899119910
+ (1 minus 120583) 1205821198871198991205821198871198991015840 cos 120582
1198861198981015840119909
times sin 1205821198871198991015840119910
2
sum
119897=1
120577119897
119886(119909) sin 120582
119887119899119910
+ 120581119866ℎ (119909 119910) cos 1205821198861198981015840119909 cos 120582
1198871198991015840119910
times
2
sum
119897=1
120577119897
119886(119909) cos 120582
119887119899119910]119889119909119889119910
+ int
119886
0
1198701199091199100
cos 1205821198861198981015840119909
2
sum
119897=1
120577119897
119886(119909)
+(minus1)119899+1198991015840
119870119909119910119887
cos 1205821198861198981015840119909
2
sum
119897=1
120577119897
119886(119909) 119889119909
+ int
119887
0
1198701199090
2
sum
119897=1
120577119897
119886(0) cos 120582
1198871198991015840119910
+(minus1)119898+1198981015840
119870119909119886
2
sum
119897=1
120577119897
119886(0) cos 120582
1198871198991015840119910119889119910
K1-4119904119905
=
1
2
int
119886
0
int
119887
0
119863(119909 119910) 21205821198871198991205821198861198981015840 sin 120582
1198861198981015840119909 cos 120582
1198871198991015840119910
times cos 120582119886119898119909 sin 120582
119887119899119910 minus 2(1 minus 120583)
119887119899
times 1205821198861198981015840 sin 120582
1198861198981015840119909 cos 120582
1198871198991015840119910 cos 120582
119886119898119909
times sin 120582119887119899119910 + (1 minus 120583) 120582
1198861198981205821198871198991015840 cos 120582
1198861198981015840119909
times sin 1205821198871198991015840119910 sin 120582
119886119898119909 cos 120582
119887119899119910 119889119909 119889119910
22 Shock and Vibration
K1-5119904119901
=
1
2
int
119886
0
int
119887
0
119863(119909 119910) minus 21205821198861198981015840 sin 120582
1198861198981015840119909 cos 120582
1198871198991015840119910
times
2
sum
119897=1
1205771198971015840
119887(119910) cos 120582
119886119898119909
+ 2 (1 minus 120583) 1205821198861198981015840 sin 120582
1198861198981015840119909 cos 120582
1198871198991015840119910
times
2
sum
119897=1
1205771198971015840
119887(119910) cos 120582
119886119898119909
+ (1 minus 120583) 1205821198861198981205821198871198991015840 cos 120582
1198861198981015840119909 sin 120582
1198871198991015840119910
times
2
sum
119897=1
120577119897
119887(119910) sin 120582
119886119898119909119889119909119889119910
K1-6119904119902
=
1
2
int
119886
0
int
119887
0
119863(119909 119910)1205821198871198991205821198861198981015840 sin 120582
1198861198981015840119909 cos 120582
1198861198991015840119910
times
2
sum
119897=1
120577119897
119886(119909) sin 120582
1198871198991015840119910
minus 2 (1 minus 120583) 1205821198871198991205821198861198981015840 sin 120582
1198861198981015840119909 cos 120582
1198861198991015840119910
times
2
sum
119897=1
120577119897
119886(119909) sin 120582
1198871198991015840119910
+ 1205821198871198991015840 cos 120582
1198861198981015840119909 sin 120582
1198871198991015840119910
times
2
sum
119897=1
1205771198971015840
119886(119909) cos 120582
119887119899119910119889119909119889119910
K1-7119904119905
= minus
1
2
int
119886
0
int
119887
0
120581119866ℎ (119909 119910) 120582119886119898
cos 1205821198861198981015840119909 cos 120582
1198871198991015840119910
times sin 120582119886119898119909 cos 120582
119887119899119910119889119909119889119910
K1-8119904119901
= minus
1
2
int
119886
0
int
119887
0
120581119866ℎ (119909 119910) 1205821198861198981015840 cos 120582
1198861198981015840119909 cos 120582
1198871198991015840119910
times
2
sum
119897=1
120577119897
119887(119910) sin 120582
1198861198981015840119909 119889119909 119889119910
K1-9119904119902
=
1
2
int
119886
0
int
119887
0
120581119866ℎ (119909 119910) cos 1205821198861198981015840119909 cos 120582
1198871198991015840119910
times
2
sum
119897=1
1205771198971015840
119886(119909) cos 120582
119887119899119910119889119909119889119910
M1-1119904119905
=
1
24
120588int
119886
0
int
119887
0
ℎ3
(119909 119910) cos 120582119886119898119909 cos 120582
119887119899119910
times cos 1205821198861198981015840119909 cos 120582
1198871198991015840119910119889119909119889119910
M1-2119904119901
=
1
24
120588int
119886
0
int
119887
0
ℎ3
(119909 119910)
2
sum
119897=1
120577119897
119887(119910) cos 120582
119886119898119909
times cos 1205821198861198981015840119909 cos 120582
1198871198991015840119910119889119909119889119910
M1-3119904119902
=
1
24
120588int
119886
0
int
119887
0
ℎ3
(119909 119910)
2
sum
119897=1
120577119897
119886(119909) cos 120582
119887119899119910
times cos 1205821198861198981015840119909 cos 120582
1198871198991015840119910119889119909119889119910
M1-4119904119905 = M
1-5119904119901 = M1-6119904119902
= M1-7119904119905 = M
1-8119904119901 = M1-9119904119902 = 0
(A1)
B Useful Integrals for the Moderately ThickRectangular Plates with Variable Thicknessand Arbitrary Boundary Conditions
Consider
int
119871
0
cos 120582119898119909 cos 120582
119899119909 119889119909 =
0 119898 = 119899
119871
2
119898 = 119899
119871 119898 = 119899 = 0
int
119871
0
sin 120582119898119909 cos 120582
119899119909119889119909
=
minus
(minus1 + (minus1)119898+119899
) 119871119898
(1198982minus 1198992) 120587
119898 = 119899
0 119898 = 119899
int
119871
0
sin 120582119898119909 sin 120582
119899119909 119889119909 =
0 119898 = 119899
119871
2
119898 = 119899
0 119898 = 119899 = 0
int
119871
0
cos 120582119898119909120577119897
119886(119909) 119889119909
=
41198712
31205872 119897 = 1 119898 = 0
minus
41198712
(41198982
minus 3)
(9 minus 401198982+ 16119898
4) 1205872
119897 = 1 119898 = 0
minus
41198712
31205872 119897 = 2 119898 = 0
41198712
(41198982
minus 3) (minus1)119898
(9 minus 401198982+ 16119898
4) 1205872
119897 = 2 119898 = 0
Shock and Vibration 23
int
119871
0
sin 120582119898119909120577119897
119886(119909) 119889119909
=
0 119897 = 1 119898 = 0
161198712
119898(minus1)119898
(9 minus 401198982+ 16119898
4) 1205872
119897 = 1 119898 = 0
0 119897 = 2 119898 = 0
161198981198712
(9 minus 401198982+ 16119898
4) 1205872
119897 = 2 119898 = 0
int
119871
0
cos 1205821198981199091205771198971015840
119886(119909) 119889119909
=
0 119897 = 1 119898 = 0
161198711198982
(minus1)119898
(9 minus 401198982+ 16119898
4) 120587
119897 = 1 119898 = 0
0 119897 = 2 119898 = 0
161198711198982
(9 minus 401198982+ 16119898
4) 120587
119897 = 2 119898 = 0
int
119871
0
sin 1205821198981199091205771198971015840
119886(119909) 119889119909
=
0 119897 = 1 119898 = 0
4119871119898 (minus3 + 41198982
)
(9 minus 401198982+ 16119898
4) 120587
119897 = 1 119898 = 0
0 119897 = 2 119898 = 0
119871119898(
1
120587 minus 41198982120587
+
3
9120587 minus 41198982120587
) (minus1)119898
119897 = 2 119898 = 0
int
119871
0
1205771198971
119886(119909) 1205771198972
119886(119909) 119889119909 =
1198713
41205872 1198971= 1 1198972= 1
minus
21198713
31205873 1198971= 1 1198972= 2
minus
21198713
31205873 1198971= 2 1198972= 1
1198713
41205872 1198971= 2 1198972= 2
int
119871
0
1205771198971
119886(119909) 1205771198971015840
2
119886(119909) 119889119909 =
0 1198971= 1 1198972= 1
minus
1198712
8120587
1198971= 1 1198972= 2
1198712
8120587
1198971= 2 1198972= 1
0 1198971= 2 1198972= 2
int
119871
0
1205771198971015840
1
119886(119909) 1205771198971015840
2
119886(119909) 119889119909 =
5119871
16
1198971= 1 1198972= 1
minus
119871
2120587
1198971= 1 1198972= 2
minus
119871
2120587
1198971= 2 1198972= 1
5119871
16
1198971= 2 1198972= 2
int
119871
0
119909119899 cos 120582
119898119909 119889119909
=
1198711+119899
1 + 119899
119898 = 0
(1198861+119899HypergeometricPFQtimes [ 12 + 1198992 12 32 + 1198992
minus (14)1198982
1205872
]) times (1 + 119899)minus1
119898 gt 0
(B1)
Conflict of Interests
The authors declare that there is no conflict of interestsregarding the publication of this paper
Acknowledgments
The authors would like to thank the anonymous reviewersfor their very valuable comments The authors also gratefullyacknowledge the financial support from the National NaturalScience Foundation of China (no 51209052) HeilongjiangProvince Youth Science Fund Project (no QC2011C013) andHarbin Science and Technology Development InnovationFoundation of Youth (no 2011RFQXG021)
References
[1] K Liew Y Xiang and S Kitipornchai ldquoResearch on thick platevibration a literature surveyrdquo Journal of Sound and Vibrationvol 180 no 1 pp 163ndash176 1995
[2] J H Chung T Y Chung and K C Kim ldquoVibration analysisof orthotropic mindlin plates with edges elastically restrainedagainst rotationrdquo Journal of Sound and Vibration vol 163 no 1pp 151ndash163 1993
[3] Y K Cheung and D Zhou ldquoVibrations of moderately thickrectangular plates in terms of a set of static Timoshenko beamfunctionsrdquoComputers and Structures vol 78 no 6 pp 757ndash7682000
[4] CMWang ldquoNatural frequencies formula for simply supportedMindlin platesrdquo Journal of Vibration and Acoustics vol 116 no4 pp 536ndash540 1994
[5] K N Saha R C Kar and P K Datta ldquoFree vibration analysisof rectangular Mindlin plates with elastic restraints uniformlydistributed along the edgesrdquo Journal of Sound andVibration vol192 no 4 pp 885ndash902 1996
24 Shock and Vibration
[6] D J Gorman ldquoAccurate free vibration analysis of shear-deformable plates with torsional elastic edge supportrdquo Journalof Sound and Vibration vol 203 no 2 pp 209ndash218 1997
[7] D J Gorman ldquoFree vibration analysis of Mindlin plates withuniform elastic edge support by the superposition methodrdquoJournal of Sound andVibration vol 207 no 3 pp 335ndash350 1997
[8] D J Gorman ldquoAccurate free vibration analysis of point sup-ported mindlin plates by the superposition methodrdquo Journal ofSound and Vibration vol 219 no 2 pp 265ndash277 1999
[9] Y Xiang ldquoVibration of rectangular Mindlin plates resting onnon-homogenous elastic foundationsrdquo International Journal ofMechanical Sciences vol 45 no 6-7 pp 1229ndash1244 2003
[10] Y Xiang and G W Wei ldquoExact solutions for buckling andvibration of stepped rectangular Mindlin platesrdquo InternationalJournal of Solids and Structures vol 41 no 1 pp 279ndash294 2004
[11] Y L Yeh M J Jang and C C Wang ldquoAnalyzing the freevibrations of a plate using finite difference and differential trans-formationmethodrdquoAppliedMathematics andComputation vol178 no 2 pp 493ndash501 2006
[12] Y Xiang S K Lai L Zhou and C W Lim ldquoDSC-Ritz elementmethod for vibration analysis of rectangular Mindlin plateswith mixed edge supportsrdquo European Journal of MechanicsmdashASolids vol 29 no 4 pp 619ndash628 2010
[13] H Nguyen-Xuan C H Thai and T Nguyen-Thoi ldquoIsogeo-metric finite element analysis of composite sandwich platesusing a higher order shear deformation theoryrdquo Composites BEngineering vol 55 pp 558ndash574 2013
[14] C H Thai A J M Ferreira E Carrera and H Nguyen-XuanldquoIsogeometric analysis of laminated composite and sandwichplates using a layerwise deformation theoryrdquo Composite Struc-tures vol 104 pp 196ndash214 2013
[15] C H Thai A Ferreira S Bordas T Rabczuk and HNguyen-Xuan ldquoIsogeometric analysis of laminated compositeand sandwich plates using a new inverse trigonometric sheardeformation theoryrdquo European Journal of Mechanics A Solidsvol 43 pp 89ndash108 2014
[16] H Luong-Van T Nguyen-Thoi G R Liu and P Phung-VanldquoA cell-based smoothed finite elementmethod using three-nodeshear-locking free Mindlin plate element (CS-FEM-MIN3) fordynamic response of laminated composite plates on viscoelasticfoundationrdquo Engineering Analysis with Boundary Elements vol42 pp 8ndash19 2014
[17] T Nguyen-Thoi T Rabczuk T Lam-Phat V Ho-Huu andP Phung-Van ldquoFree vibration analysis of cracked Mindlinplate using an extended cell-based smoothed discrete sheargap method (XCS-DSG3)rdquo Theoretical and Applied FractureMechanics 2014
[18] H Nguyen-Xuan G R Liu and C a Thai-Hoang ldquoAnedge-based smoothed finite element method (ES-FEM) withstabilized discrete shear gap technique for analysis of Reissner-Mindlin platesrdquo Computer Methods in Applied Mechanics andEngineering vol 199 no 9ndash12 pp 471ndash489 2010
[19] P Phung-Van T Nguyen-Thoi T Bui-Xuan and Q Lieu-XuanldquoA cell-based smoothed three-node Mindlin plate element(CS-FEM-MIN3) based on the 119862
0 -type higher-order sheardeformation for geometrically nonlinear analysis of laminatedcomposite platesrdquo Computational Materials Science 2014
[20] NNguyen-Thanh T RabczukHNguyen-Xuan and S BordasldquoAn alternative alpha finite element method with discrete sheargap technique for analysis of isotropicMindlin-Reissner platesrdquoFinite Elements inAnalysis andDesign vol 47 no 5 pp 519ndash5352011
[21] KM Liew Y Xiang and S Kitipornchai ldquoTransverse vibrationof thick rectangular plates-I Comprehensive sets of boundaryconditionsrdquo Computers and Structures vol 49 no 1 pp 1ndash291993
[22] D Zhou ldquoVibrations of Mindlin rectangular plates with elasti-cally restrained edges using static Timoshenko beam functionswith the Rayleigh-Ritz methodrdquo International Journal of Solidsand Structures vol 38 no 32-33 pp 5565ndash5580 2001
[23] D Zhou SH Lo F T KAu andYK Cheung ldquoVibration anal-ysis of rectangular Mindlin plates with internal line supportsusing static Timoshenko beam functionsrdquo International Journalof Mechanical Sciences vol 44 no 12 pp 2503ndash2522 2002
[24] H S Shen J Yang and L Zhang ldquoFree and forced vibrationof Reissner-Mindlin plates with free edges resting on elasticfoundationsrdquo Journal of Sound and Vibration vol 244 no 2 pp299ndash320 2001
[25] Y Xing and B Liu ldquoClosed form solutions for free vibrations ofrectangular Mindlin platesrdquo Acta Mechanica Sinica vol 25 no5 pp 689ndash698 2009
[26] P Gagnon C Gosselin and L Cloutier ldquoA finite strip elementfor the analysis of variable thickness rectangular thick platesrdquoComputers amp Structures vol 63 no 2 pp 349ndash362 1997
[27] T Mlzusawa ldquoVibration of rectangular mindlin plates withtapered thickness by the spline strip methodrdquo Computers andStructures vol 46 no 3 pp 451ndash463 1993
[28] H Nguyen-Xuan and T Nguyen-Thoi ldquoA stabilized smoothedfinite element method for free vibration analysis of Mindlin-Reissner platesrdquo Communications in Numerical Methods inEngineering with Biomedical Applications vol 25 no 8 pp 882ndash906 2009
[29] K M Liew L X Peng and S Kitipornchai ldquoVibration anal-ysis of corrugated Reissner-Mindlin plates using a mesh-freeGalerkin methodrdquo International Journal of Mechanical Sciencesvol 51 no 9-10 pp 642ndash652 2009
[30] Y Hou GWWei and Y Xiang ldquoDSC-Ritz method for the freevibration analysis of Mindlin platesrdquo International Journal forNumerical Methods in Engineering vol 62 no 2 pp 262ndash2882005
[31] F- Liu and K M Liew ldquoVibration analysis of discontinuousMindlin plates by differential quadrature element methodrdquoJournal of Vibration and Acoustics Transactions of the ASMEvol 121 no 2 pp 204ndash208 1999
[32] P Malekzadeh G Karami and M Farid ldquoA semi-analyticalDQEM for free vibration analysis of thick plates with twoopposite edges simply supportedrdquoComputerMethods inAppliedMechanics and Engineering vol 193 no 45ndash47 pp 4781ndash47962004
[33] P Malekzadeh and S A Shahpari ldquoFree vibration analysisof variable thickness thin and moderately thick plates withelastically restrained edges by DQMrdquo Thin-Walled Structuresvol 43 no 7 pp 1037ndash1050 2005
[34] R E Diaz-Contreras and S Nomura ldquoGreenrsquos function appliedto solution ofMindlin platesrdquoComputers and Structures vol 60no 1 pp 41ndash48 1996
[35] T Sakiyama and M Huang ldquoFree vibration analysis of rect-angular plates with variable thicknessrdquo Journal of Sound andVibration vol 216 no 3 pp 379ndash397 1998
[36] M Huang X Q Ma T Sakiyama H Matuda and C MoritaldquoFree vibration analysis of orthotropic rectangular plates withvariable thickness and general boundary conditionsrdquo Journal ofSound and Vibration vol 288 no 4-5 pp 931ndash955 2005
Shock and Vibration 25
[37] I Shufrin and M Eisenberger ldquoVibration of shear deformableplates with variable thicknessmdashfirst-order and higher-orderanalysesrdquo Journal of Sound and Vibration vol 290 no 1-2 pp465ndash489 2006
[38] S A Eftekhari and A A Jafari ldquoAccurate variational approachfor free vibration of variable thickness thin and thick plates withedges elastically restrained against translation and rotationrdquoInternational Journal of Mechanical Sciences vol 68 pp 35ndash462013
[39] W L Li X Zhang J Du and Z Liu ldquoAn exact series solutionfor the transverse vibration of rectangular plates with generalelastic boundary supportsrdquo Journal of Sound amp Vibration vol321 no 1-2 pp 254ndash269 2009
[40] X Shi D Shi W L Li and Q Wang ldquoA unified method forfree vibration analysis of circular annular and sector plateswith arbitrary boundary conditionsrdquo Journal of Vibration andControl 2014
[41] D Y Shi X J Shi W L Li and Q S Wang ldquoFree transversevibrations of orthotropic thin rectangular plates with arbitraryelastic edge supportsrdquo Journal of Vibroengineering vol 16 no 1pp 389ndash398 2014
[42] Y Chen G Jin and Z Liu ldquoFree vibration analysis of circu-lar cylindrical shell with non-uniform elastic boundary con-straintsrdquo International Journal of Mechanical Sciences vol 74pp 120ndash132 2013
[43] L Dai T Yang J Du W L Li and M J Brennan ldquoAn exactseries solution for the vibration analysis of cylindrical shellswith arbitrary boundary conditionsrdquo Applied Acoustics vol 74no 3 pp 440ndash449 2013
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Shock and Vibration
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Chemical EngineeringInternational Journal of Antennas and
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Navigation and Observation
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DistributedSensor Networks
International Journal of
12 Shock and Vibration
Table 4 Frequency parameters Ω = 1205961198872
(120588ℎ119863)12
1205872 for moderately thick plates (119886119887 = 1) with uniform thickness and all edge elastic
restraints (1198701199090= 119870119909119886= 1198701199100= 119870119910119887= Γ119909and 119896
1199090= 119896119909119886= 1198961199100= 119896119910119887= Γ119908)
Γ119909
aΓ119908
aℎ0119886
Model sequence1 2 3 4 5 6
05 0
005 0986 1967 3893 3893 4850 4850(0988b) (1971) (3897) (3897) (4852) (4852)
01 0916 1835 3515 3522 4375 4375(0917b) (1837) (3519) (3528) (4341) (4341)
015 0789 1635 2983 3027 3771 3771(0792b) (1638) (2988) (3029) (3775) (3775)
02 0646 1444 2497 2601 3249 3249(0652b) (1448) (2508) (2621) (3254) (3254)
1 05
005 3417 3417 4431 5294 5461 6448(3419b) (3419) (4435) (5303) (5478) (6456)
01 1593 1593 2401 3849 3861 4729(1611b) (1611) (2408) (3852) (3866) (4767)
015 1122 1122 1913 3272 3290 4038(1127b) (1227) (1924) (3284) (3314) (4069)
02 0890 0890 1640 2781 2829 3476(0899b) (0899) (1654) (2787) (2834) (3497)
15 1
005 4325 4325 5619 6273 6571 7619(4334b) (4334) (5681) (6299) (6642) (7642)
01 2013 2013 2812 4088 4116 4993(2024b) (2024) (2827) (4122) (4187) (4994)
015 1338 1338 2108 3430 3443 4196(1342b) (1342) (2112) (3467) (3482) (4200)
02 1038 1038 1772 2936 2965 3613(1042b) (1042) (1786) (2937) (2977) (3624)
2 2
005 5231 5231 6919 7630 8002 9220(5238b) (5238) (6931) (7637) (8011) (9241)
01 2594 2594 3439 4470 4544 5428(2597b) (2597) (3442) (4495) (4557) (5438)
015 1653 1653 2403 3613 3632 4392(1662b) (1662) (2414) (3627) (3646) (4404)
02 1238 1238 1949 3077 3099 3751(1244b) (1244) (1964) (3094) (3116) (3788)
aΓ119909= 1198701198941198700 (1198700= 1 times 10
9 Nmrad 119894 = 1199090 119909119886 1199100 119910119887) Γ119908= 1198961198941198960 (1198960= 1 times 10
9 Nm 119894 = 1199090 119909119886 1199100 119910119887)bResults in parentheses are taken from FEM
05 10 15 20 25 300
2
4
6
8
10
12
14
16
ab
The 1st orderThe 2nd orderThe 3rd order
The 4th orderThe 5th orderThe 6th order
Ω
(a)
05 10 15 20 25 30
30
60
90
120
150
180
ab
The 1st orderThe 2nd orderThe 3rd order
The 4th orderThe 5th orderThe 6th order
Ω
(b)
Figure 6 The effect of aspect ratio 119886119887 on the natural frequenciesΩ for S-S-S-S boundary condition (a) uniform thickness and (b) nonuni-form thickness
Shock and Vibration 13
Table 5 Frequency parameter Ω = 1205961198862
(120588ℎ01198630)12 for moderately thick plates with linearly varying thickness in different boundary
conditions (ℎ(119909 119910) = ℎ0(1 + 05119909)(1 + 05119910))
Boundary condition 119886119887 ℎ0119886
Model sequence1 2 3 4 5 6
S-S-S-S
101 27261 62397 62926 92423 111537 111732
(27263a) (62410) (63842) (92445) (111867) (111921)
02 23103 47010 47191 65018 75831 75865(23489a) (47209) (47349) (65057) (76605) (76473)
3201 59338 99101 138798 151102 166710 204855
(59874a) (99249) (138177) (151362) (166644) (205381)
02 41175 63765 63765 73598 83956 89084(41586a) (63313) (63669) (73661) (83918) (89547)
C-F-F-F
101 4173 10331 27407 37826 38809 64095
(4856a) (10680) (27513) (37203) (39135) (64858)
02 3983 8702 21403 29443 30286 46705(4118a) (9306) (21990) (29670) (30602) (47156)
3201 4868 15488 35620 63523 89969 100089
(4887a) (15320) (35175) (63353) (89159) (100783)
02 4469 9763 22338 42188) 50104 56331(4944a) (9636) (22087) (42177) (50580) (56559)
S-S-F-F
101 4117 24795 26216 48567) 68195 69667
(4789a) (24149) (26751) (49324) (68842) (70575)
02 3820 20516 21924 37523 48980 50327(4103a) (20041) (22125) (37221) (48894) (50622)
3201 7935 39454 65063 92877 92877 136176
(7856a) (40145) (65162) (92462) (93022) (136336)
02 6389 26149 35627 49539 56844 65170(6406a) (26536) (35626) (49831) (57144) (65981)
C-C-C-C
101 44471 80299 80681 109422 126871 128207
(45102a) (80271) (80656) (110015) (126328) (128888)
02 32162 52781 52933 69410 78474 79482(32240a) (52629) (53166) (70063) (79059) (80266)
3201 82975 115587 151992 161579 176601 212330
(83139a) (115838) (152200) (162193) (177058) (212849)
02 48577 66022 83468 90830 97474 98157(49187a) (66259) (84065) (91749) (97827) (98777)
S-C-S-C
101 36847 66664 76965 101045 113594 125551
(37901a) (68568) (78356) (101200) (113978) (125651)
02 27673 48469 51220 67027 76358 78241(27897a) (48685) (51491) (67447) (77530) (79573)
3201 77378 106848 150079 154569 173120 208391
(78003a) (107058) (150991) (155171) (173099) (208402)
02 45895 64732 73393 83282 90325 97089(45803a) (64131) 73656 83801 90518 97015
aResults in parentheses are taken from FEM
restrained That is 1198701199090
= 119870119909119886
= 1198701199100
= 119870119910119887
= Γ119909 The six
frequency parameters Ω = 1205961198862
(120588ℎ01198630)12 are given in
Table 6 for several different restraining coefficient valuesthe finite element method (FEM) results calculated usingABAQUS are also presented as a referenceThe secondmodelis a complete square moderately thick plate with all edgeselastically restrainedThat is 119896
1199090= 119896119909119886= 1198961199100= 119896119910119887= Γ119908and
1198701199090= 119870119909119886
= 1198701199100= 119870119910119887= Γ119909 The six frequency parameters
Ω = 1205961198862
(120588ℎ01198630)12 are given in Table 7 for several different
restraining coefficient values the finite element method(FEM) results are also listed as a reference It can be clearlyseen that the comparison is extremely good which impliesthat the current method is able to make correct predictionsfor the modal characteristics of linearly varying thickness
14 Shock and Vibration
Table 6 Frequency parameter Ω = 1205961198862
(120588ℎ01198630)12 for S-S-S-S moderately thick plates (119886119887 = 1) with linearly varying thickness and elastic
rotation support (ℎ(119909 119910) = ℎ0(1 + 05119909)(1 + 05119910) 119870
1199090= 119870119909119886= 1198701199100= 119870119910119887= Γ119909)
Γ119909
aℎ0119886
Model sequence1 2 3 4 5 6
0
005 28809 70080 70891 108967 134884 135295(28897b) (70334) (71129) (109112) (135254) (135772)
01 27261 62398 62926 92423 111540 111735(26606b) (61784) (62303) (91193) (111323) (111648)
015 25231 54192 54505 77118 91228 91312(25204b) (55297) (55595) (77503) (91871) (91941)
02 23103 47010 47191 65018 75832 75866(23947b) (47071) (47208) (65404) (76531) (76589)
05
005 23103 47010 47191 65018 75832 75866(23210b) (47245) (47358) (65241) (76122) (76345)
01 48377 94650 95597 135484 162635 163501(48524b) (94891) (95992) (135845) (167012) (164024)
015 36239 70939 71298 99997 118584 119009(36647b) (71214) (71687) (100024) (119114) (119457)
02 28749 56937 57037 79214 93009 93162(29142b) (57034) (57544) (79654) (93668) (93754)
1
005 49651 96759 97619 138089 165368 166304(49755b) (97245) (97964) (138544) (165864) (166878)
01 38951 73735 74251 102748 121079 121709(39021b) (74145) (74664) (102874) (121613) (121817)
015 30630 58406 58533 80432 93984 94255(30638b) (58684) (58644) (80874) (94121) (94478)
02 25548 48636 48674 66165 76708 76824(25687b) (48788) (48992) (66544) (76824) (76944)
15
005 50118 97561 98374 139096 166421 167382(50247b) (97877) (98661) (139247) (166875) (167662)
01 40310 75216 75783 104229 122402 123164(40366b) (75542) (75921) (104557) (122874) (123385)
015 31837 59352 59530 81245 94611 94983(31902b) (59451) (59871) (81375) (94784) (95105)
02 26302 49119 49156 66528 76954 77130(26418b) (49354) (49334) (66921) (77010) (77470)
2
005 50360 97983 98769 139631 166979 167954(50472b) (98005) (99114) (140154) (167188) (167986)
01 41130 76146 76724 105165 123229 124082(41421b) (76642) (78114) (105661) (123278) (124974)
015 32687 60025 60245 81832 95052 95508(32778b) (60244) (60549) (82146) (95114) (95842)
02 26893 49492 49539 66815 77138 77370(26922b) (49874) (49924) (66987) (77367) (77588)
aΓ119909= 1198701198941198700 (1198700= 1 times 10
9 Nmrad 119894 = 1199090 119909119886 1199100 119910119887)bResults in parentheses are taken from FEM
Shock and Vibration 15
Table 7 Frequency parameterΩ = 1205961198862
(120588ℎ01198630)12 for moderately thick plates (119886119887 = 1) with linearly varying thickness and elastic rotation
and translation support (ℎ(119909 119910) = ℎ0(1 + 05119909)(1 + 05119910) 119870
1199090= 119870119909119886= 1198701199100= 119870119910119887= Γ119909 and 119896
1199090= 119896119909119886= 1198961199100= 119896119910119887= Γ119908)
Γ119909
aΓ119908
aℎ0119886
Model sequence1 2 3 4 5 6
05 0
005 14751 14917 28232 55251 55397 69512(14780b) (14822) (29252) (55476) (55505) (70904)
01 11811 11910 23857 44462 45037 55453(11818b) (11897) (24711) (44376) (45257) (56511)
015 8487 8514 20011 34783 36335 44970(8500b) (8544) (20738) (34404) (36663) (45625)
02 6073 6073 17431 28679 30805 38010(6096b) (6106) (18039) (28129) (31165) (38311)
1 05
005 21225 32173 32173 44381 64061 65222(20708b) (32097) (32285) (45110) (64182) (65355)
01 8140 16878 16941 28100 48663 49015(7943b) (16838) (16970) (28888) (48700) (49171)
015 4493 11859 11878 22429 38352 39247(4383b) (11867) (11891) (23125) (38177) (39488)
02 2932 8690 8691 18919 31108 32600(3540b) (8791) (8791) (16229) (27520) (27889)
15 1
005 27819 40919 41430 54595 70792 72896(27146b) (41008) (41434) (55256) (70895) (73051)
01 11285 20068 20230 31101 51046 51429(11012b) (20025) (20270) (31849) (51126) (51569)
015 6292 13912 13922 24132 40500 41120(6139b) (13912) (13934) (24808) (40426) (41316)
02 4121 10321 10331 20051 32796 33921(4021b) (10329) (10357) (20635) (32540) (34178)
2 2
005 34615 51972 53088 68255 81344 84510(33806b) (52100) (53157) (68891) (81433) (84761)
01 15416 24659 24829 35615 53879 54494(15045b) (24625) (24888) (36303) (53970) (54626)
015 8740 16420 16481 26353 42417 42946(8529b) (16414) (16499) (26996) (42398) (43114)
02 5761 12127 12127 21434 34286 35192(5622b) (12134) (12148) (21998) (34110) (35415)
aΓ119909= 1198701198941198700 (1198700= 1 times 10
9 Nmrad 119894 = 1199090 119909119886 1199100 119910119887) Γ119908= 1198961198941198960 (1198960= 1 times 10
9 Nm 119894 = 1199090 119909119886 1199100 119910119887)bResults in parentheses are taken from FEM
moderately thick plate with not only classical boundaryconditions but also elastic edge restraints
As the last case of this subsection the influence of thegradient 120572 and 120573 on the fundamental frequency parametersfor a linearly varying thickness moderately thick plate isinvestigated The model is a square moderately thick platewith all edges elastically restrainedThat is 119896
1199090= 119896119909119886= 1198961199100=
119896119910119887= Γ119908= 2 and 119870
1199090= 119870119909119886
= 1198701199100= 119870119910119887= Γ119909= 2 The six
frequency parameters Ω = 1205961198862
(120588ℎ01198630)12 are given in
Table 8 for several different 120572 and 120573 values the finite elementmethod (FEM) results are also listed as a reference Againgood agreement can be observed Through Table 8 it is alsofound that the frequency parameter increases with increasinggradient parameters
The above studies are given as linearly varying thicknessmoderately thick plates with several of boundary condition
and different structure parameters In the next section vibra-tion results for the plates subjected to nonlinear variationthickness will be presented
34 Nonlinearly Varying Thickness Moderately Thick Plateswith Classical and Elastic Boundary Conditions In Sec-tion 33 the linearly varying thickness moderately thickplates were studied However in the practical engineeringapplications the varying thickness of amoderately thick platemay not always be linear variation in nature A variety ofpossible thickness varying cases may be encountered in prac-tice Therefore the moderately thick plates with nonlinearvariation thickness subjected to general elastic edge restraintsare examined in this subsection For the sake of brevity theindexes 119904 and 119905 will be chosen as 2 to imitate the nonlinearlyvarying thickness moderately thick plates structure in this
16 Shock and Vibration
Table 8 Frequency parameterΩ = 1205961198862
(120588ℎ01198630)12 for elastic support moderately thick plates (119886119887 = 1) with different gradient (119896
1199090= 119896119909119886=
1198961199100= 119896119910119887= Γ119908= 2 119870
1199090= 119870119909119886= 1198701199100= 119870119910119887= Γ119909= 2)
120572 120573 ℎ0119886
Model sequence1 2 3 4 5 6
120572 = 120573 = 01
005 31275 52615 52657 68954 76146 79850(31303a) (52664) (52675) (69033) (76174) (79948)
01 17270 25244 25250 33949 46050 46625(17271a) (25255) (25258) (34089) (46122) (46708)
015 10190 16266 16274 24164 37327 37526(10189a) (16269) (16280) (24323) (37401) (37622)
02 6780 12258 12260 19685 31452 31754(6780a) (12265) (12270) (19851) (31514) (31864)
120572 = 120573 = 02
005 32530 52947 53141 68954 77034 80589(32577a) (53014) (53203) (69172) (77088) (80773)
01 16882 24979 24979 34154 48057 48575(16883a) (24983) (25001) (34433) (48134) (48667)
015 9804 16284 16309 24683 38844 39085(9803a) (16283) (16322) (24980) (38909) (39191)
02 6499 12290 12295 20133 32354 32770(6498a) (12296) (12307) (20420) (32387) (32895)
120572 = 120573 = 03
005 33516 52845 53308 68704 78203 81561(33563a) (52939) (53389) (69055) (78271) (81789)
01 16419 24799 24830 34527 50061 50583(16419a) (24790) (24869) (34947) (50143) (50688)
015 9430 16334 16378 25236 40193 40506(9429a) (16329) (16395) (25660) (40243) (40625)
02 6235 12288 12293 20572 33102 33663(6235a) (12296) (12306) (20965) (33090) (33811)
120572 = 120573 = 04
005 34215 52463 53264 68430 79654 82865(34252a) (52575) (53346) (68921) (79732) (83113)
01 15921 24695 24793 35027 52011 52571(15921a) (24674) (24842) (35583) (52098) (52688)
015 9074 16388 16444 25797 41379 41789(9074a) (16381) (16464) (26336) (41404) (41928)
02 5990 12236 12238 21002 33735 34461(5989a) (12249) (12250) (21487) (33653) (34641)
120572 = 120573 = 05
005 34615 51972 53088 68255 81344 84510(33806a) (52100) (53157) (68891) (81433) (84761)
01 15416 24659 24829 35615 53879 54494(15045a) (24625) (24888) (36303) (53970) (54626)
015 8740 16420 16481 26353 42417 42946(8529a) (16414) (16499) (26996) (42398) (43114)
02 5761 12127 12127 21434 34286 35192(5622a) (12134) (12148) (21998) (34110) (35415)
aResults in parentheses are taken from FEM
section that is 119904 = 119905 = 2 Also the thickness varying functionsare also be expressed as Fourier cosine series according to (18)and (19)
In order to validate the accuracy and reliability of theproposed method for predicting the vibration behavior ofnonlinearly varying thickness moderately thick plates with
arbitrary boundary conditions the typical classical boundaryconditions viewed as the special cases of elastically restrainededges will be considered The comparison of the first six fre-quency parameters Ω = 120596119886
2
(120588ℎ01198630)12 for the moderately
thick plates with nonlinearly varying thickness is presentedin Table 9 The S-S-S-S C-F-F-F S-S-F-F C-C-C-C and
Shock and Vibration 17
Table 9 Frequency parameter Ω = 1205961198862
(120588ℎ01198630)12 for moderately thick plates with linearly varying thickness in different boundary
conditions (ℎ(119909 119910) = ℎ0(1 + 05119909
2
)(1 + 051199102
))
Boundary condition 119886119887 ℎ0119886
Model sequence1 2 3 4 5 6
S-S-S-S
101 23462 54999 55573 82950 100900 101187
(23741a) (55114) (55983) (83146) (101004) (101347)
02 18606 42725 42971 60040 71167 71761(18936a) (43014) (43148) (60247) (71642) (71993)
3201 53055 65251 85404 106700 114880 143531
(53467a) (65472) (85984) (107211) (115687) (144007)
02 17439 21531 32170 44987 59438 64179(17874a) (21748) (32651) (45264) (60188) (64879)
C-F-F-F
101 3658 9317 24135 32816 35433 59179
(3749a) (9547) (24034) (33627) (36308) (60142)
02 3520 7986 19591 26959 27491 43882(3611a) (8176) (20075) (27125) (27625) (43984)
3201 3901 9963 28984 69187 73686 89360
(3930a) (10095) (29381) (69578) (73994) (90242)
02 3292 4439 13385 23947 40850 44758(3386a) (4498) (14012) (24068) (41221) (45142)
S-S-F-F
101 3557 21268 22762 43524 60253 62429
(3649a) (21441) (22796) (43916) (60622) (62630)
02 3171 17822 19512 33616 45042 46833(3489a) (18006) (20430) (33815) (45247) (49004)
3201 6738 31658 52071 75625 86604 94903
(6957a) (31882) (52518) (75823) (86939) (95610)
02 3653 12930 20955 26517 44097 48058(6807a) (13160) (21312) (26864) (44422) (48873)
C-C-C-C
101 41020 75196 75614 104142 119683 120949
(41405a) (75452) (75882) (104711) (119637) (121137)
02 31237 51686 51868 68411 76868 77864(32010a) (52960) (52145) (69335) (77481) (78099)
3201 95823 128727 163657 173751 190890 223595
(95364a) (128932) (163588) (172350) (190350) (222948)
02 53306 71372 89384 102404 104660 120908(53587a) (71759) (90103) (102503) (105174) (121275)
S-C-S-C
101 33715 60298 71446 94187 103802 117928
(34148a) (60786) (71828) (94515) (104263) (119326)
02 26143 45499 49458 64253 72941 76191(26789a) (46622) (50680) (64329) (73474) (76843)
3201 73228 115845 137445 148166 168342 182542
(73761a) (116121) (137788) (148530) (168698) (182921)
02 41302 53648 65630 72202 81632 88105(41304a) (53757) (65842) (72794) (82033) (88474)
aResults in parentheses are taken from FEM
S-C-S-C boundary conditions are performed in the compar-ison The results adequately demonstrated the great accuracyof the modified Fourier method
We now turn to elastically restrained moderately thickplates The first one involves an S-S-S-S moderately thicksquare plate with a uniform rotational restraint along eachedge that is 119870
1199090= 119870119909119886
= 1198701199100
= 119870119910119887
= Γ119909 The calculated
frequency parameters Ω = 1205961198862
(120588ℎ01198630)12 are given in
Table 10 together with the FEM results Since the elasticallyrestrained plates with nonlinear variation thickness are rarelyinvestigated the FEMresults are used as the referenceA goodagreement is observed between the current and FEM resultsThe second example concerns amoderately thick square plateelastically supported along all edgesThe stiffness of the linear
18 Shock and Vibration
Table 10 Frequency parameterΩ = 1205961198862
(120588ℎ01198630)12 for S-S-S-S moderately thick plates (119886119887 = 1) with linearly varying thickness and elastic
rotation support (ℎ(119909 119910) = ℎ0(1 + 05119909
2
)(1 + 051199102
) 1198701199090= 119870119909119886= 1198701199100= 119870119910119887= Γ119909)
Γ119909
aℎ0119886
Model sequence1 2 3 4 5 6
0
005 23462 54999 55573 82950 100900 101187(23741b) (55114) (55983) (83146) (101004) (101347)
01 18606 42725 42971 60040 71167 71761(18936b) (43014) (43148) (60247) (71642) (71993)
015 53055 65251 85404 106700 114880 143531(53467b) (65472) (85984) (107211) (115687) (144007)
02 17439 21531 32170 44987 59438 64179(17874b) (21748) (32651) (45264) (60188) (64879)
05
005 6904 13374 13514 19400 22904 23056(9909b) (13548) (13651) (19564) (23088) (23379)
01 5316 10237 10314 14499 17155 17264(5374b) (10448) (10651) (14764) (17188) (17879)
015 4136 8201 8220 11474 13578 13671(4174b) (8348) (8651) (11764) (13688) (13879)
02 3449 6915 6924 9549 11225 11327(3474b) (6948) (6951) (9764) (11388) (11579)
1
005 7040 13617 13745 19726 23233 23402(7074b) (13748) (13851) (19964) (23488) (23879)
01 5712 10696 10793 14991 17594 17744(5774b) (10748) (10851) (15054) (17788) (17879)
015 4489 8507 8540 11759 13803 13922(4514b) (8548) (8611) (11864) (13988) (14119)
02 3677 7084 7090 9692 11330 11445(3684b) (7148) (7191) (9764) (11588) (11879)
15
005 7089 13708 13828 19847 23355 23530(7124b) (13748) (13951) (19967) (23588) (23879)
01 5895 10923 11024 15239 17813 17988(5874b) (10748) (11651) (15264) (17818) (17979)
015 4697 8694 8737 11938 13790 14083(4774b) (8748) (8951) (11864) (13688) (14879)
02 3836 7201 7210 9792 11403 11530(3874b) (7748) (7651) (9864) (11588) (11879)
2
005 7114 13754 13871 19911 23418 23597(7174b) (12848) (13851) (19864) (23588) (23879)
01 6001 11060 11159 15391 17944 18136(6004b) (11148) (11151) (15464) (17988) (18379)
015 4836 8821 8872 12061 14037 14189(4874b) (8848) (8951) (12164) (14188) (14879)
02 3954 7289 7301 9871 11457 11594(3974b) (7348) (7651) (9964) (11488) (11679)
aΓ119909= 1198701198941198700 (1198700= 1 times 10
9 Nmrad 119894 = 1199090 119909119886 1199100 119910119887)bResults in parentheses are taken from FEM
and rotational restraints is set equal to 1198961199090
= 119896119909119886
= 1198961199100
=
119896119910119887
= Γ119908and 119870
1199090= 119870119909119886
= 1198701199100
= 119870119910119887
= Γ119909 respectively
The six frequency parametersΩ = 1205961198862
(120588ℎ01198630)12 are shown
in Table 11 for several different restraining coefficient valuesIt can also be noticed that the Fourier series method is able
to predict the modal characteristics of nonlinearly varyingthickness moderately thick plate with not only classicalboundary conditions but also elastic edge restraints correctly
Finally the influence of the gradient on the fundamentalfrequency parameters of a nonlinearly varying thickness
Shock and Vibration 19
Table 11 Frequency parameterΩ = 1205961198862
(120588ℎ01198630)12 for moderately thick plates (119886119887 = 1) with linearly varying thickness and elastic rotation
and translation support (ℎ(119909 119910) = ℎ0(1 + 05119909
2
)(1 + 051199102
) 1198701199090= 119870119909119886= 1198701199100= 119870119910119887= Γ119909 and 119896
1199090= 119896119909119886= 1198961199100= 119896119910119887= Γ119908)
Γ119909
aΓ119908
aℎ0119886
Model sequence1 2 3 4 5 6
05 0
005 12926 13041 25090 47935 47943 61761(12931b) (13053) 25099 (47947) (47952) (61788)
01 10900 10988 21545 39980 40147 50322(10904b) (10992) (21557) (39984) (40166) (50334)
015 8245 8298 18081 31503 32617 40950(8247b) (8230) (18092) (31508) (32622) (40955)
02 6106 6124 15639 25631 27601 34458(6110b) (6125) (15642) (25637) (27617) (34462)
1 05
005 21575 31615 31644 42573 57889 58607(21580b) (31617) (31644) (42581) (57893) (58610)
01 8451 16024 16210 25969 43839 43867(8451b) (16025) (16217) (25972) (43844) (43870)
015 4682 11479 11480 20642 35172 35598(4684b) (11480) (11481) (20644) (35178) (35604)
02 3059 8621 8648 17320 28487 29638(3062b) (8619) (8644) (17327) (28492) (29641)
15 1
005 27713 40203 40978 52951 65352 66815(27714b) (40207) (40988) (52972) (65368) (66822)
01 11667 19322 19615 29122 46115 46183(11674b) (19327) (19619) (29137) (46120) (46188)
015 6544 13416 13481 22405 37238 37430(6547b) (13420) (13488) (22410) (37241) (37434)
02 4295 10175 10190 18543 30330 31056(4296b) (10177) (10194) (18544) (30336) (31060)
2 2
005 33453 50854 52088 66311 76610 78947(33455b) (50852) (50097) (66321) (76627) (78971)
01 15832 24130 24372 33924 49118 49362(15830b) (24127) (24374) (33931) (49147) (49382)
015 9060 15956 16099 24754 39134 39249(9064b) (15961) (16102) (24755) (39138) (39452)
02 5992 11955 11971 20034 31908 32392(5991b) (11957) (11978) (20042) (31910) (32398)
aΓ119909= 1198701198941198700 (1198700= 1 times 10
9 Nmrad 119894 = 1199090 119909119886 1199100 119910119887) Γ119908= 1198961198941198960 (1198960= 1 times 10
9 Nm 119894 = 1199090 119909119886 1199100 119910119887)bResults in parentheses are taken from FEM
moderately thick plate is investigated The plate is elasticallyrestrained in which the stiffness of the boundary springsis taken as 119896
1199090= 119896119909119886
= 1198961199100
= 119896119910119887
= Γ119908
= 2 and1198701199090
= 119870119909119886
= 1198701199100
= 119870119910119887
= Γ119909= 2 respectively The six
frequency parameters Ω = 1205961198862
(120588ℎ01198630)12 are presented in
Table 12 for several different slop values It can be seen thatthe fundamental frequency parameters will decrease with theincrease of the parameter ℎ
0119886
In the above examples it has been demonstrated thatthe presented method can be universally applied to nonlin-early varying thickness moderately thick plates with severalboundary conditions and different structure parametersNew results are obtained for plates with nonlinear variationthickness in both directions subjected to general elastic
boundary restraints which may be used for benchmarkingof researchers in the field In addition it is interesting to seethat the nature frequency decreases with the increase of theindex for thickness function
4 Conclusions
In this paper a modified Fourier method has been pre-sented to study the free vibration behaviors of moderatelythick rectangular plates with variable thickness and arbi-trary boundary conditionsThe first-order shear deformationplate theory is adopted to formulate the theoretical modelThe displacements and rotation components of the plateregardless of boundary conditions are invariantly expressedas the superposition of a 2D Fourier cosine series and four
20 Shock and Vibration
Table 12 Frequency parameterΩ = 1205961198862
(120588ℎ01198630)12 for elastic support moderately thick plates (119886119887 = 1) with different gradient (119896
1199090= 119896119909119886=
1198961199100= 119896119910119887= Γ119908= 2 119870
1199090= 119870119909119886= 1198701199100= 119870119910119887= Γ119909= 2)
120572 120573 ℎ0119886
Model sequence1 2 3 4 5 6
120572 = 120573 = 01
005 30187 51102 51141 67066 73954 77412(30188a) (51105) (51144) (67068) (73961) (77477)
01 16995 24719 24724 33071 44343 44819(16996a) (24725) (24729) (33084) (44347) (44822)
015 10077 15872 15883 23443 35937 36065(10081a) (15877) (15892) (23447) (35941) (36084)
02 6713 11953 11958 19086 30372 30604(6713a) (11957) (11965) (19092) (30378) (30612)
120572 = 120573 = 02
005 31206 51436 51614 67058 74398 77595(31208a) (51438) (51619) (67064) (74402) (77596)
01 16786 24503 24508 33144 45583 45923(16790a) (24510) (24513) (33147) (45587) (45932)
015 9821 15865 15906 23766 36897 36990(9824a) (15870) (15907) (23768) (36894) (36997)
02 6523 11978 11992 19356 30912 31162(6524a) (11981) (11997) (19362) (30916) (31174)
120572 = 120573 = 03
005 32097 51462 51900 66832 74991 77863(32098a) (51466) (51907) (66849) (74996) (77872)
01 16507 24332 24383 33330 46806 47074(16511a) (24327) (24380) (33339) (46810) (47086)
015 9562 15885 15964 24103 37748 37828(9562a) (15892) (15977) (21105) (37751) (37832)
02 6338 11992 12013 19603 31338 31634(6340a) (11994) (12011) (19612) (31344) (31639)
120572 = 120573 = 04
005 32851 51239 52046 66547 75735 78303(33455a) (50852) (50097) (66321) (76627) (78971)
01 16182 24209 24343 33598 47990 48230(33455a) (50852) (50097) (66321) (76627) (78971)
015 9307 15919 16034 24436 38491 38579(33455a) (50852) (50097) (66321) (76627) (78971)
02 6161 11986 12008 19828 31666 32040(33455a) (50852) (50097) (66321) (76627) (78971)
120572 = 120573 = 05
005 33453 50854 52088 66311 76610 78947(33455a) (50852) (50097) (66321) (76627) (78971)
01 15832 24130 24372 33924 49118 49362(15830a) (24127) (24374) (33931) (49147) (49382)
015 9060 15956 16099 24754 39134 39249(9064a) (15961) (16102) (24755) (39138) (39452)
02 5992 11955 11971 20034 31908 32392(5991a) (11957) (11978) (20042) (31910) (32398)
aResults in parentheses are taken from FEM
supplementary functions in the form of the product of apolynomial function and a single cosine series expansionto ensure and accelerate the convergence of the solutionAt each edge of the plate the general restraint conditionsare implemented by introducing one group of linear springsand two groups of rotational springs which are continuouslydistributed and determined by the stiffness of these springs
Instead of seeking a solution in strong forms in the previousstudies all the Fourier coefficients will be treated equallyand independently as the generalized coordinates and solveddirectly from the Rayleigh-Ritz technique The change of theboundary conditions can be easily achieved by only varyingthe stiffness of the three sets of boundary springs along alledges of the rectangular plates without involving any change
Shock and Vibration 21
to the solution procedure The convergence of the presentsolution is examined and the excellent accuracy is validatedby comparison with existing results published in the lit-erature and FEM data Excellent agreements are obtainedfrom these comparisons The proposed method providesa unified means for extracting the modal parameters andpredicting the vibration behaviors of moderately thick plateswith variable thickness variation functions and arbitraryelastic edge restraints A variety of free vibration results formoderately thick rectangular plates with different thicknessvariation functions and boundary conditions are presentedNew results for free vibration of moderately thick rectangularplates with various thickness variation functions and edgeconditions are presented which may be used for benchmark-ing of researchers in the field
Appendices
A Representative Calculation forStiffness and Mass Matrices
To illuminate the particular expression of the mass andstiffness matrixes clearly and tersely four new variables aredefined as 119904 = 119898 lowast (119873 + 1) + 119899 + 1 119905 = 119898
1015840
lowast (119873 + 1) + 1198991015840
+ 1119901 = (119897minus1)lowast (119872+1)+119898
1015840
+1 and 119902 = (119897minus1)lowast (119873+1)+1198991015840
+1The first row elements of K andM are given below
K1-1119904119905
=
1
2
int
119886
0
int
119887
0
[119863 (119909 119910)
times 21205821198861198981205821198861198981015840 sin 120582
119886119898119909 sin 120582
1198861198981015840119909 cos 120582
119887119899119910
times cos 1205821198871198991015840119910 + (1 minus 120583) 120582
1198871198991205821198871198991015840 cos 120582
119886119898119909
times cos 1205821198861198981015840 sin 120582
119887119899119910 sin 120582
1198871198991015840119910
+ 120581119866ℎ (119909 119910) cos 120582119886119898119909 cos 120582
1198861198981015840119909 cos 120582
119887119899
times119910 cos 1205821198871198991015840119910119889119909119889119910] 119889119909 119889119910
+ int
119886
0
1198701199091199100
cos 120582119886119898119909 cos 120582
1198861198981015840119909
+(minus1)119898+1198981015840
119870119909119910119887
cos 120582119886119898119909 cos 120582
1198861198981015840119909 119889119909
+ int
119887
0
1198701199090cos 120582119887119899119910 cos 120582
1198871198991015840119910
+(minus1)119899+1198991015840
119870119909119886cos 120582119887119899119910 cos 120582
1198871198991015840119910 119889119910
K1-2119904119901
=
1
2
int
119886
0
int
119887
0
[119863 (119909 119910)21205821198861198981205821198861198981015840 sin 120582
119886119898119909 sin 120582
1198861198981015840119909
times
2
sum
119897=1
120577119897
119887(119910) cos 120582
1198871198991015840119910
minus (1 minus 120583) 1205821198871198991015840 cos 120582
119886119898119909 cos 120582
1198861198981015840119909
times
2
sum
119897=1
1205771198971015840
119887(119910) sin 120582
1198871198991015840119910 + 120581119866ℎ (119909 119910)
times ( cos 1205821198861198981015840119909 cos 120582
1198871198991015840119910
times
2
sum
119897=1
120577119897
119887(119910) cos 120582
119886119898119909)]119889119909119889119910
+ int
119886
0
1198701199091199100
cos 120582119886119898119909 cos 120582
1198861198981015840119909
2
sum
119897=1
120577119897
119887(0)
+(minus1)1198991015840
119870119909119910119887
cos 120582119886119898119909 cos 120582
1198861198981015840119909
2
sum
119897=1
120577119897
119887(119887) 119889119909
+ int
119887
0
1198701199090
2
sum
119897=1
120577119897
119887(119910) cos 120582
1198871198991015840119910
+(minus1)119898+1198981015840
119870119909119886
2
sum
119897=1
120577119897
119887(119910) cos 120582
1198871198991015840119910119889119910
K1-3119904119902
=
1
2
int
119886
0
int
119887
0
[119863 (119909 119910) minus 21205821198861198981015840 sin 120582
1198861198981015840119909 cos 120582
1198871198991015840119910
times
2
sum
119897=1
1205771198971015840
119886(119909) cos 120582
119887119899119910
+ (1 minus 120583) 1205821198871198991205821198871198991015840 cos 120582
1198861198981015840119909
times sin 1205821198871198991015840119910
2
sum
119897=1
120577119897
119886(119909) sin 120582
119887119899119910
+ 120581119866ℎ (119909 119910) cos 1205821198861198981015840119909 cos 120582
1198871198991015840119910
times
2
sum
119897=1
120577119897
119886(119909) cos 120582
119887119899119910]119889119909119889119910
+ int
119886
0
1198701199091199100
cos 1205821198861198981015840119909
2
sum
119897=1
120577119897
119886(119909)
+(minus1)119899+1198991015840
119870119909119910119887
cos 1205821198861198981015840119909
2
sum
119897=1
120577119897
119886(119909) 119889119909
+ int
119887
0
1198701199090
2
sum
119897=1
120577119897
119886(0) cos 120582
1198871198991015840119910
+(minus1)119898+1198981015840
119870119909119886
2
sum
119897=1
120577119897
119886(0) cos 120582
1198871198991015840119910119889119910
K1-4119904119905
=
1
2
int
119886
0
int
119887
0
119863(119909 119910) 21205821198871198991205821198861198981015840 sin 120582
1198861198981015840119909 cos 120582
1198871198991015840119910
times cos 120582119886119898119909 sin 120582
119887119899119910 minus 2(1 minus 120583)
119887119899
times 1205821198861198981015840 sin 120582
1198861198981015840119909 cos 120582
1198871198991015840119910 cos 120582
119886119898119909
times sin 120582119887119899119910 + (1 minus 120583) 120582
1198861198981205821198871198991015840 cos 120582
1198861198981015840119909
times sin 1205821198871198991015840119910 sin 120582
119886119898119909 cos 120582
119887119899119910 119889119909 119889119910
22 Shock and Vibration
K1-5119904119901
=
1
2
int
119886
0
int
119887
0
119863(119909 119910) minus 21205821198861198981015840 sin 120582
1198861198981015840119909 cos 120582
1198871198991015840119910
times
2
sum
119897=1
1205771198971015840
119887(119910) cos 120582
119886119898119909
+ 2 (1 minus 120583) 1205821198861198981015840 sin 120582
1198861198981015840119909 cos 120582
1198871198991015840119910
times
2
sum
119897=1
1205771198971015840
119887(119910) cos 120582
119886119898119909
+ (1 minus 120583) 1205821198861198981205821198871198991015840 cos 120582
1198861198981015840119909 sin 120582
1198871198991015840119910
times
2
sum
119897=1
120577119897
119887(119910) sin 120582
119886119898119909119889119909119889119910
K1-6119904119902
=
1
2
int
119886
0
int
119887
0
119863(119909 119910)1205821198871198991205821198861198981015840 sin 120582
1198861198981015840119909 cos 120582
1198861198991015840119910
times
2
sum
119897=1
120577119897
119886(119909) sin 120582
1198871198991015840119910
minus 2 (1 minus 120583) 1205821198871198991205821198861198981015840 sin 120582
1198861198981015840119909 cos 120582
1198861198991015840119910
times
2
sum
119897=1
120577119897
119886(119909) sin 120582
1198871198991015840119910
+ 1205821198871198991015840 cos 120582
1198861198981015840119909 sin 120582
1198871198991015840119910
times
2
sum
119897=1
1205771198971015840
119886(119909) cos 120582
119887119899119910119889119909119889119910
K1-7119904119905
= minus
1
2
int
119886
0
int
119887
0
120581119866ℎ (119909 119910) 120582119886119898
cos 1205821198861198981015840119909 cos 120582
1198871198991015840119910
times sin 120582119886119898119909 cos 120582
119887119899119910119889119909119889119910
K1-8119904119901
= minus
1
2
int
119886
0
int
119887
0
120581119866ℎ (119909 119910) 1205821198861198981015840 cos 120582
1198861198981015840119909 cos 120582
1198871198991015840119910
times
2
sum
119897=1
120577119897
119887(119910) sin 120582
1198861198981015840119909 119889119909 119889119910
K1-9119904119902
=
1
2
int
119886
0
int
119887
0
120581119866ℎ (119909 119910) cos 1205821198861198981015840119909 cos 120582
1198871198991015840119910
times
2
sum
119897=1
1205771198971015840
119886(119909) cos 120582
119887119899119910119889119909119889119910
M1-1119904119905
=
1
24
120588int
119886
0
int
119887
0
ℎ3
(119909 119910) cos 120582119886119898119909 cos 120582
119887119899119910
times cos 1205821198861198981015840119909 cos 120582
1198871198991015840119910119889119909119889119910
M1-2119904119901
=
1
24
120588int
119886
0
int
119887
0
ℎ3
(119909 119910)
2
sum
119897=1
120577119897
119887(119910) cos 120582
119886119898119909
times cos 1205821198861198981015840119909 cos 120582
1198871198991015840119910119889119909119889119910
M1-3119904119902
=
1
24
120588int
119886
0
int
119887
0
ℎ3
(119909 119910)
2
sum
119897=1
120577119897
119886(119909) cos 120582
119887119899119910
times cos 1205821198861198981015840119909 cos 120582
1198871198991015840119910119889119909119889119910
M1-4119904119905 = M
1-5119904119901 = M1-6119904119902
= M1-7119904119905 = M
1-8119904119901 = M1-9119904119902 = 0
(A1)
B Useful Integrals for the Moderately ThickRectangular Plates with Variable Thicknessand Arbitrary Boundary Conditions
Consider
int
119871
0
cos 120582119898119909 cos 120582
119899119909 119889119909 =
0 119898 = 119899
119871
2
119898 = 119899
119871 119898 = 119899 = 0
int
119871
0
sin 120582119898119909 cos 120582
119899119909119889119909
=
minus
(minus1 + (minus1)119898+119899
) 119871119898
(1198982minus 1198992) 120587
119898 = 119899
0 119898 = 119899
int
119871
0
sin 120582119898119909 sin 120582
119899119909 119889119909 =
0 119898 = 119899
119871
2
119898 = 119899
0 119898 = 119899 = 0
int
119871
0
cos 120582119898119909120577119897
119886(119909) 119889119909
=
41198712
31205872 119897 = 1 119898 = 0
minus
41198712
(41198982
minus 3)
(9 minus 401198982+ 16119898
4) 1205872
119897 = 1 119898 = 0
minus
41198712
31205872 119897 = 2 119898 = 0
41198712
(41198982
minus 3) (minus1)119898
(9 minus 401198982+ 16119898
4) 1205872
119897 = 2 119898 = 0
Shock and Vibration 23
int
119871
0
sin 120582119898119909120577119897
119886(119909) 119889119909
=
0 119897 = 1 119898 = 0
161198712
119898(minus1)119898
(9 minus 401198982+ 16119898
4) 1205872
119897 = 1 119898 = 0
0 119897 = 2 119898 = 0
161198981198712
(9 minus 401198982+ 16119898
4) 1205872
119897 = 2 119898 = 0
int
119871
0
cos 1205821198981199091205771198971015840
119886(119909) 119889119909
=
0 119897 = 1 119898 = 0
161198711198982
(minus1)119898
(9 minus 401198982+ 16119898
4) 120587
119897 = 1 119898 = 0
0 119897 = 2 119898 = 0
161198711198982
(9 minus 401198982+ 16119898
4) 120587
119897 = 2 119898 = 0
int
119871
0
sin 1205821198981199091205771198971015840
119886(119909) 119889119909
=
0 119897 = 1 119898 = 0
4119871119898 (minus3 + 41198982
)
(9 minus 401198982+ 16119898
4) 120587
119897 = 1 119898 = 0
0 119897 = 2 119898 = 0
119871119898(
1
120587 minus 41198982120587
+
3
9120587 minus 41198982120587
) (minus1)119898
119897 = 2 119898 = 0
int
119871
0
1205771198971
119886(119909) 1205771198972
119886(119909) 119889119909 =
1198713
41205872 1198971= 1 1198972= 1
minus
21198713
31205873 1198971= 1 1198972= 2
minus
21198713
31205873 1198971= 2 1198972= 1
1198713
41205872 1198971= 2 1198972= 2
int
119871
0
1205771198971
119886(119909) 1205771198971015840
2
119886(119909) 119889119909 =
0 1198971= 1 1198972= 1
minus
1198712
8120587
1198971= 1 1198972= 2
1198712
8120587
1198971= 2 1198972= 1
0 1198971= 2 1198972= 2
int
119871
0
1205771198971015840
1
119886(119909) 1205771198971015840
2
119886(119909) 119889119909 =
5119871
16
1198971= 1 1198972= 1
minus
119871
2120587
1198971= 1 1198972= 2
minus
119871
2120587
1198971= 2 1198972= 1
5119871
16
1198971= 2 1198972= 2
int
119871
0
119909119899 cos 120582
119898119909 119889119909
=
1198711+119899
1 + 119899
119898 = 0
(1198861+119899HypergeometricPFQtimes [ 12 + 1198992 12 32 + 1198992
minus (14)1198982
1205872
]) times (1 + 119899)minus1
119898 gt 0
(B1)
Conflict of Interests
The authors declare that there is no conflict of interestsregarding the publication of this paper
Acknowledgments
The authors would like to thank the anonymous reviewersfor their very valuable comments The authors also gratefullyacknowledge the financial support from the National NaturalScience Foundation of China (no 51209052) HeilongjiangProvince Youth Science Fund Project (no QC2011C013) andHarbin Science and Technology Development InnovationFoundation of Youth (no 2011RFQXG021)
References
[1] K Liew Y Xiang and S Kitipornchai ldquoResearch on thick platevibration a literature surveyrdquo Journal of Sound and Vibrationvol 180 no 1 pp 163ndash176 1995
[2] J H Chung T Y Chung and K C Kim ldquoVibration analysisof orthotropic mindlin plates with edges elastically restrainedagainst rotationrdquo Journal of Sound and Vibration vol 163 no 1pp 151ndash163 1993
[3] Y K Cheung and D Zhou ldquoVibrations of moderately thickrectangular plates in terms of a set of static Timoshenko beamfunctionsrdquoComputers and Structures vol 78 no 6 pp 757ndash7682000
[4] CMWang ldquoNatural frequencies formula for simply supportedMindlin platesrdquo Journal of Vibration and Acoustics vol 116 no4 pp 536ndash540 1994
[5] K N Saha R C Kar and P K Datta ldquoFree vibration analysisof rectangular Mindlin plates with elastic restraints uniformlydistributed along the edgesrdquo Journal of Sound andVibration vol192 no 4 pp 885ndash902 1996
24 Shock and Vibration
[6] D J Gorman ldquoAccurate free vibration analysis of shear-deformable plates with torsional elastic edge supportrdquo Journalof Sound and Vibration vol 203 no 2 pp 209ndash218 1997
[7] D J Gorman ldquoFree vibration analysis of Mindlin plates withuniform elastic edge support by the superposition methodrdquoJournal of Sound andVibration vol 207 no 3 pp 335ndash350 1997
[8] D J Gorman ldquoAccurate free vibration analysis of point sup-ported mindlin plates by the superposition methodrdquo Journal ofSound and Vibration vol 219 no 2 pp 265ndash277 1999
[9] Y Xiang ldquoVibration of rectangular Mindlin plates resting onnon-homogenous elastic foundationsrdquo International Journal ofMechanical Sciences vol 45 no 6-7 pp 1229ndash1244 2003
[10] Y Xiang and G W Wei ldquoExact solutions for buckling andvibration of stepped rectangular Mindlin platesrdquo InternationalJournal of Solids and Structures vol 41 no 1 pp 279ndash294 2004
[11] Y L Yeh M J Jang and C C Wang ldquoAnalyzing the freevibrations of a plate using finite difference and differential trans-formationmethodrdquoAppliedMathematics andComputation vol178 no 2 pp 493ndash501 2006
[12] Y Xiang S K Lai L Zhou and C W Lim ldquoDSC-Ritz elementmethod for vibration analysis of rectangular Mindlin plateswith mixed edge supportsrdquo European Journal of MechanicsmdashASolids vol 29 no 4 pp 619ndash628 2010
[13] H Nguyen-Xuan C H Thai and T Nguyen-Thoi ldquoIsogeo-metric finite element analysis of composite sandwich platesusing a higher order shear deformation theoryrdquo Composites BEngineering vol 55 pp 558ndash574 2013
[14] C H Thai A J M Ferreira E Carrera and H Nguyen-XuanldquoIsogeometric analysis of laminated composite and sandwichplates using a layerwise deformation theoryrdquo Composite Struc-tures vol 104 pp 196ndash214 2013
[15] C H Thai A Ferreira S Bordas T Rabczuk and HNguyen-Xuan ldquoIsogeometric analysis of laminated compositeand sandwich plates using a new inverse trigonometric sheardeformation theoryrdquo European Journal of Mechanics A Solidsvol 43 pp 89ndash108 2014
[16] H Luong-Van T Nguyen-Thoi G R Liu and P Phung-VanldquoA cell-based smoothed finite elementmethod using three-nodeshear-locking free Mindlin plate element (CS-FEM-MIN3) fordynamic response of laminated composite plates on viscoelasticfoundationrdquo Engineering Analysis with Boundary Elements vol42 pp 8ndash19 2014
[17] T Nguyen-Thoi T Rabczuk T Lam-Phat V Ho-Huu andP Phung-Van ldquoFree vibration analysis of cracked Mindlinplate using an extended cell-based smoothed discrete sheargap method (XCS-DSG3)rdquo Theoretical and Applied FractureMechanics 2014
[18] H Nguyen-Xuan G R Liu and C a Thai-Hoang ldquoAnedge-based smoothed finite element method (ES-FEM) withstabilized discrete shear gap technique for analysis of Reissner-Mindlin platesrdquo Computer Methods in Applied Mechanics andEngineering vol 199 no 9ndash12 pp 471ndash489 2010
[19] P Phung-Van T Nguyen-Thoi T Bui-Xuan and Q Lieu-XuanldquoA cell-based smoothed three-node Mindlin plate element(CS-FEM-MIN3) based on the 119862
0 -type higher-order sheardeformation for geometrically nonlinear analysis of laminatedcomposite platesrdquo Computational Materials Science 2014
[20] NNguyen-Thanh T RabczukHNguyen-Xuan and S BordasldquoAn alternative alpha finite element method with discrete sheargap technique for analysis of isotropicMindlin-Reissner platesrdquoFinite Elements inAnalysis andDesign vol 47 no 5 pp 519ndash5352011
[21] KM Liew Y Xiang and S Kitipornchai ldquoTransverse vibrationof thick rectangular plates-I Comprehensive sets of boundaryconditionsrdquo Computers and Structures vol 49 no 1 pp 1ndash291993
[22] D Zhou ldquoVibrations of Mindlin rectangular plates with elasti-cally restrained edges using static Timoshenko beam functionswith the Rayleigh-Ritz methodrdquo International Journal of Solidsand Structures vol 38 no 32-33 pp 5565ndash5580 2001
[23] D Zhou SH Lo F T KAu andYK Cheung ldquoVibration anal-ysis of rectangular Mindlin plates with internal line supportsusing static Timoshenko beam functionsrdquo International Journalof Mechanical Sciences vol 44 no 12 pp 2503ndash2522 2002
[24] H S Shen J Yang and L Zhang ldquoFree and forced vibrationof Reissner-Mindlin plates with free edges resting on elasticfoundationsrdquo Journal of Sound and Vibration vol 244 no 2 pp299ndash320 2001
[25] Y Xing and B Liu ldquoClosed form solutions for free vibrations ofrectangular Mindlin platesrdquo Acta Mechanica Sinica vol 25 no5 pp 689ndash698 2009
[26] P Gagnon C Gosselin and L Cloutier ldquoA finite strip elementfor the analysis of variable thickness rectangular thick platesrdquoComputers amp Structures vol 63 no 2 pp 349ndash362 1997
[27] T Mlzusawa ldquoVibration of rectangular mindlin plates withtapered thickness by the spline strip methodrdquo Computers andStructures vol 46 no 3 pp 451ndash463 1993
[28] H Nguyen-Xuan and T Nguyen-Thoi ldquoA stabilized smoothedfinite element method for free vibration analysis of Mindlin-Reissner platesrdquo Communications in Numerical Methods inEngineering with Biomedical Applications vol 25 no 8 pp 882ndash906 2009
[29] K M Liew L X Peng and S Kitipornchai ldquoVibration anal-ysis of corrugated Reissner-Mindlin plates using a mesh-freeGalerkin methodrdquo International Journal of Mechanical Sciencesvol 51 no 9-10 pp 642ndash652 2009
[30] Y Hou GWWei and Y Xiang ldquoDSC-Ritz method for the freevibration analysis of Mindlin platesrdquo International Journal forNumerical Methods in Engineering vol 62 no 2 pp 262ndash2882005
[31] F- Liu and K M Liew ldquoVibration analysis of discontinuousMindlin plates by differential quadrature element methodrdquoJournal of Vibration and Acoustics Transactions of the ASMEvol 121 no 2 pp 204ndash208 1999
[32] P Malekzadeh G Karami and M Farid ldquoA semi-analyticalDQEM for free vibration analysis of thick plates with twoopposite edges simply supportedrdquoComputerMethods inAppliedMechanics and Engineering vol 193 no 45ndash47 pp 4781ndash47962004
[33] P Malekzadeh and S A Shahpari ldquoFree vibration analysisof variable thickness thin and moderately thick plates withelastically restrained edges by DQMrdquo Thin-Walled Structuresvol 43 no 7 pp 1037ndash1050 2005
[34] R E Diaz-Contreras and S Nomura ldquoGreenrsquos function appliedto solution ofMindlin platesrdquoComputers and Structures vol 60no 1 pp 41ndash48 1996
[35] T Sakiyama and M Huang ldquoFree vibration analysis of rect-angular plates with variable thicknessrdquo Journal of Sound andVibration vol 216 no 3 pp 379ndash397 1998
[36] M Huang X Q Ma T Sakiyama H Matuda and C MoritaldquoFree vibration analysis of orthotropic rectangular plates withvariable thickness and general boundary conditionsrdquo Journal ofSound and Vibration vol 288 no 4-5 pp 931ndash955 2005
Shock and Vibration 25
[37] I Shufrin and M Eisenberger ldquoVibration of shear deformableplates with variable thicknessmdashfirst-order and higher-orderanalysesrdquo Journal of Sound and Vibration vol 290 no 1-2 pp465ndash489 2006
[38] S A Eftekhari and A A Jafari ldquoAccurate variational approachfor free vibration of variable thickness thin and thick plates withedges elastically restrained against translation and rotationrdquoInternational Journal of Mechanical Sciences vol 68 pp 35ndash462013
[39] W L Li X Zhang J Du and Z Liu ldquoAn exact series solutionfor the transverse vibration of rectangular plates with generalelastic boundary supportsrdquo Journal of Sound amp Vibration vol321 no 1-2 pp 254ndash269 2009
[40] X Shi D Shi W L Li and Q Wang ldquoA unified method forfree vibration analysis of circular annular and sector plateswith arbitrary boundary conditionsrdquo Journal of Vibration andControl 2014
[41] D Y Shi X J Shi W L Li and Q S Wang ldquoFree transversevibrations of orthotropic thin rectangular plates with arbitraryelastic edge supportsrdquo Journal of Vibroengineering vol 16 no 1pp 389ndash398 2014
[42] Y Chen G Jin and Z Liu ldquoFree vibration analysis of circu-lar cylindrical shell with non-uniform elastic boundary con-straintsrdquo International Journal of Mechanical Sciences vol 74pp 120ndash132 2013
[43] L Dai T Yang J Du W L Li and M J Brennan ldquoAn exactseries solution for the vibration analysis of cylindrical shellswith arbitrary boundary conditionsrdquo Applied Acoustics vol 74no 3 pp 440ndash449 2013
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Shock and Vibration
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Shock and Vibration 13
Table 5 Frequency parameter Ω = 1205961198862
(120588ℎ01198630)12 for moderately thick plates with linearly varying thickness in different boundary
conditions (ℎ(119909 119910) = ℎ0(1 + 05119909)(1 + 05119910))
Boundary condition 119886119887 ℎ0119886
Model sequence1 2 3 4 5 6
S-S-S-S
101 27261 62397 62926 92423 111537 111732
(27263a) (62410) (63842) (92445) (111867) (111921)
02 23103 47010 47191 65018 75831 75865(23489a) (47209) (47349) (65057) (76605) (76473)
3201 59338 99101 138798 151102 166710 204855
(59874a) (99249) (138177) (151362) (166644) (205381)
02 41175 63765 63765 73598 83956 89084(41586a) (63313) (63669) (73661) (83918) (89547)
C-F-F-F
101 4173 10331 27407 37826 38809 64095
(4856a) (10680) (27513) (37203) (39135) (64858)
02 3983 8702 21403 29443 30286 46705(4118a) (9306) (21990) (29670) (30602) (47156)
3201 4868 15488 35620 63523 89969 100089
(4887a) (15320) (35175) (63353) (89159) (100783)
02 4469 9763 22338 42188) 50104 56331(4944a) (9636) (22087) (42177) (50580) (56559)
S-S-F-F
101 4117 24795 26216 48567) 68195 69667
(4789a) (24149) (26751) (49324) (68842) (70575)
02 3820 20516 21924 37523 48980 50327(4103a) (20041) (22125) (37221) (48894) (50622)
3201 7935 39454 65063 92877 92877 136176
(7856a) (40145) (65162) (92462) (93022) (136336)
02 6389 26149 35627 49539 56844 65170(6406a) (26536) (35626) (49831) (57144) (65981)
C-C-C-C
101 44471 80299 80681 109422 126871 128207
(45102a) (80271) (80656) (110015) (126328) (128888)
02 32162 52781 52933 69410 78474 79482(32240a) (52629) (53166) (70063) (79059) (80266)
3201 82975 115587 151992 161579 176601 212330
(83139a) (115838) (152200) (162193) (177058) (212849)
02 48577 66022 83468 90830 97474 98157(49187a) (66259) (84065) (91749) (97827) (98777)
S-C-S-C
101 36847 66664 76965 101045 113594 125551
(37901a) (68568) (78356) (101200) (113978) (125651)
02 27673 48469 51220 67027 76358 78241(27897a) (48685) (51491) (67447) (77530) (79573)
3201 77378 106848 150079 154569 173120 208391
(78003a) (107058) (150991) (155171) (173099) (208402)
02 45895 64732 73393 83282 90325 97089(45803a) (64131) 73656 83801 90518 97015
aResults in parentheses are taken from FEM
restrained That is 1198701199090
= 119870119909119886
= 1198701199100
= 119870119910119887
= Γ119909 The six
frequency parameters Ω = 1205961198862
(120588ℎ01198630)12 are given in
Table 6 for several different restraining coefficient valuesthe finite element method (FEM) results calculated usingABAQUS are also presented as a referenceThe secondmodelis a complete square moderately thick plate with all edgeselastically restrainedThat is 119896
1199090= 119896119909119886= 1198961199100= 119896119910119887= Γ119908and
1198701199090= 119870119909119886
= 1198701199100= 119870119910119887= Γ119909 The six frequency parameters
Ω = 1205961198862
(120588ℎ01198630)12 are given in Table 7 for several different
restraining coefficient values the finite element method(FEM) results are also listed as a reference It can be clearlyseen that the comparison is extremely good which impliesthat the current method is able to make correct predictionsfor the modal characteristics of linearly varying thickness
14 Shock and Vibration
Table 6 Frequency parameter Ω = 1205961198862
(120588ℎ01198630)12 for S-S-S-S moderately thick plates (119886119887 = 1) with linearly varying thickness and elastic
rotation support (ℎ(119909 119910) = ℎ0(1 + 05119909)(1 + 05119910) 119870
1199090= 119870119909119886= 1198701199100= 119870119910119887= Γ119909)
Γ119909
aℎ0119886
Model sequence1 2 3 4 5 6
0
005 28809 70080 70891 108967 134884 135295(28897b) (70334) (71129) (109112) (135254) (135772)
01 27261 62398 62926 92423 111540 111735(26606b) (61784) (62303) (91193) (111323) (111648)
015 25231 54192 54505 77118 91228 91312(25204b) (55297) (55595) (77503) (91871) (91941)
02 23103 47010 47191 65018 75832 75866(23947b) (47071) (47208) (65404) (76531) (76589)
05
005 23103 47010 47191 65018 75832 75866(23210b) (47245) (47358) (65241) (76122) (76345)
01 48377 94650 95597 135484 162635 163501(48524b) (94891) (95992) (135845) (167012) (164024)
015 36239 70939 71298 99997 118584 119009(36647b) (71214) (71687) (100024) (119114) (119457)
02 28749 56937 57037 79214 93009 93162(29142b) (57034) (57544) (79654) (93668) (93754)
1
005 49651 96759 97619 138089 165368 166304(49755b) (97245) (97964) (138544) (165864) (166878)
01 38951 73735 74251 102748 121079 121709(39021b) (74145) (74664) (102874) (121613) (121817)
015 30630 58406 58533 80432 93984 94255(30638b) (58684) (58644) (80874) (94121) (94478)
02 25548 48636 48674 66165 76708 76824(25687b) (48788) (48992) (66544) (76824) (76944)
15
005 50118 97561 98374 139096 166421 167382(50247b) (97877) (98661) (139247) (166875) (167662)
01 40310 75216 75783 104229 122402 123164(40366b) (75542) (75921) (104557) (122874) (123385)
015 31837 59352 59530 81245 94611 94983(31902b) (59451) (59871) (81375) (94784) (95105)
02 26302 49119 49156 66528 76954 77130(26418b) (49354) (49334) (66921) (77010) (77470)
2
005 50360 97983 98769 139631 166979 167954(50472b) (98005) (99114) (140154) (167188) (167986)
01 41130 76146 76724 105165 123229 124082(41421b) (76642) (78114) (105661) (123278) (124974)
015 32687 60025 60245 81832 95052 95508(32778b) (60244) (60549) (82146) (95114) (95842)
02 26893 49492 49539 66815 77138 77370(26922b) (49874) (49924) (66987) (77367) (77588)
aΓ119909= 1198701198941198700 (1198700= 1 times 10
9 Nmrad 119894 = 1199090 119909119886 1199100 119910119887)bResults in parentheses are taken from FEM
Shock and Vibration 15
Table 7 Frequency parameterΩ = 1205961198862
(120588ℎ01198630)12 for moderately thick plates (119886119887 = 1) with linearly varying thickness and elastic rotation
and translation support (ℎ(119909 119910) = ℎ0(1 + 05119909)(1 + 05119910) 119870
1199090= 119870119909119886= 1198701199100= 119870119910119887= Γ119909 and 119896
1199090= 119896119909119886= 1198961199100= 119896119910119887= Γ119908)
Γ119909
aΓ119908
aℎ0119886
Model sequence1 2 3 4 5 6
05 0
005 14751 14917 28232 55251 55397 69512(14780b) (14822) (29252) (55476) (55505) (70904)
01 11811 11910 23857 44462 45037 55453(11818b) (11897) (24711) (44376) (45257) (56511)
015 8487 8514 20011 34783 36335 44970(8500b) (8544) (20738) (34404) (36663) (45625)
02 6073 6073 17431 28679 30805 38010(6096b) (6106) (18039) (28129) (31165) (38311)
1 05
005 21225 32173 32173 44381 64061 65222(20708b) (32097) (32285) (45110) (64182) (65355)
01 8140 16878 16941 28100 48663 49015(7943b) (16838) (16970) (28888) (48700) (49171)
015 4493 11859 11878 22429 38352 39247(4383b) (11867) (11891) (23125) (38177) (39488)
02 2932 8690 8691 18919 31108 32600(3540b) (8791) (8791) (16229) (27520) (27889)
15 1
005 27819 40919 41430 54595 70792 72896(27146b) (41008) (41434) (55256) (70895) (73051)
01 11285 20068 20230 31101 51046 51429(11012b) (20025) (20270) (31849) (51126) (51569)
015 6292 13912 13922 24132 40500 41120(6139b) (13912) (13934) (24808) (40426) (41316)
02 4121 10321 10331 20051 32796 33921(4021b) (10329) (10357) (20635) (32540) (34178)
2 2
005 34615 51972 53088 68255 81344 84510(33806b) (52100) (53157) (68891) (81433) (84761)
01 15416 24659 24829 35615 53879 54494(15045b) (24625) (24888) (36303) (53970) (54626)
015 8740 16420 16481 26353 42417 42946(8529b) (16414) (16499) (26996) (42398) (43114)
02 5761 12127 12127 21434 34286 35192(5622b) (12134) (12148) (21998) (34110) (35415)
aΓ119909= 1198701198941198700 (1198700= 1 times 10
9 Nmrad 119894 = 1199090 119909119886 1199100 119910119887) Γ119908= 1198961198941198960 (1198960= 1 times 10
9 Nm 119894 = 1199090 119909119886 1199100 119910119887)bResults in parentheses are taken from FEM
moderately thick plate with not only classical boundaryconditions but also elastic edge restraints
As the last case of this subsection the influence of thegradient 120572 and 120573 on the fundamental frequency parametersfor a linearly varying thickness moderately thick plate isinvestigated The model is a square moderately thick platewith all edges elastically restrainedThat is 119896
1199090= 119896119909119886= 1198961199100=
119896119910119887= Γ119908= 2 and 119870
1199090= 119870119909119886
= 1198701199100= 119870119910119887= Γ119909= 2 The six
frequency parameters Ω = 1205961198862
(120588ℎ01198630)12 are given in
Table 8 for several different 120572 and 120573 values the finite elementmethod (FEM) results are also listed as a reference Againgood agreement can be observed Through Table 8 it is alsofound that the frequency parameter increases with increasinggradient parameters
The above studies are given as linearly varying thicknessmoderately thick plates with several of boundary condition
and different structure parameters In the next section vibra-tion results for the plates subjected to nonlinear variationthickness will be presented
34 Nonlinearly Varying Thickness Moderately Thick Plateswith Classical and Elastic Boundary Conditions In Sec-tion 33 the linearly varying thickness moderately thickplates were studied However in the practical engineeringapplications the varying thickness of amoderately thick platemay not always be linear variation in nature A variety ofpossible thickness varying cases may be encountered in prac-tice Therefore the moderately thick plates with nonlinearvariation thickness subjected to general elastic edge restraintsare examined in this subsection For the sake of brevity theindexes 119904 and 119905 will be chosen as 2 to imitate the nonlinearlyvarying thickness moderately thick plates structure in this
16 Shock and Vibration
Table 8 Frequency parameterΩ = 1205961198862
(120588ℎ01198630)12 for elastic support moderately thick plates (119886119887 = 1) with different gradient (119896
1199090= 119896119909119886=
1198961199100= 119896119910119887= Γ119908= 2 119870
1199090= 119870119909119886= 1198701199100= 119870119910119887= Γ119909= 2)
120572 120573 ℎ0119886
Model sequence1 2 3 4 5 6
120572 = 120573 = 01
005 31275 52615 52657 68954 76146 79850(31303a) (52664) (52675) (69033) (76174) (79948)
01 17270 25244 25250 33949 46050 46625(17271a) (25255) (25258) (34089) (46122) (46708)
015 10190 16266 16274 24164 37327 37526(10189a) (16269) (16280) (24323) (37401) (37622)
02 6780 12258 12260 19685 31452 31754(6780a) (12265) (12270) (19851) (31514) (31864)
120572 = 120573 = 02
005 32530 52947 53141 68954 77034 80589(32577a) (53014) (53203) (69172) (77088) (80773)
01 16882 24979 24979 34154 48057 48575(16883a) (24983) (25001) (34433) (48134) (48667)
015 9804 16284 16309 24683 38844 39085(9803a) (16283) (16322) (24980) (38909) (39191)
02 6499 12290 12295 20133 32354 32770(6498a) (12296) (12307) (20420) (32387) (32895)
120572 = 120573 = 03
005 33516 52845 53308 68704 78203 81561(33563a) (52939) (53389) (69055) (78271) (81789)
01 16419 24799 24830 34527 50061 50583(16419a) (24790) (24869) (34947) (50143) (50688)
015 9430 16334 16378 25236 40193 40506(9429a) (16329) (16395) (25660) (40243) (40625)
02 6235 12288 12293 20572 33102 33663(6235a) (12296) (12306) (20965) (33090) (33811)
120572 = 120573 = 04
005 34215 52463 53264 68430 79654 82865(34252a) (52575) (53346) (68921) (79732) (83113)
01 15921 24695 24793 35027 52011 52571(15921a) (24674) (24842) (35583) (52098) (52688)
015 9074 16388 16444 25797 41379 41789(9074a) (16381) (16464) (26336) (41404) (41928)
02 5990 12236 12238 21002 33735 34461(5989a) (12249) (12250) (21487) (33653) (34641)
120572 = 120573 = 05
005 34615 51972 53088 68255 81344 84510(33806a) (52100) (53157) (68891) (81433) (84761)
01 15416 24659 24829 35615 53879 54494(15045a) (24625) (24888) (36303) (53970) (54626)
015 8740 16420 16481 26353 42417 42946(8529a) (16414) (16499) (26996) (42398) (43114)
02 5761 12127 12127 21434 34286 35192(5622a) (12134) (12148) (21998) (34110) (35415)
aResults in parentheses are taken from FEM
section that is 119904 = 119905 = 2 Also the thickness varying functionsare also be expressed as Fourier cosine series according to (18)and (19)
In order to validate the accuracy and reliability of theproposed method for predicting the vibration behavior ofnonlinearly varying thickness moderately thick plates with
arbitrary boundary conditions the typical classical boundaryconditions viewed as the special cases of elastically restrainededges will be considered The comparison of the first six fre-quency parameters Ω = 120596119886
2
(120588ℎ01198630)12 for the moderately
thick plates with nonlinearly varying thickness is presentedin Table 9 The S-S-S-S C-F-F-F S-S-F-F C-C-C-C and
Shock and Vibration 17
Table 9 Frequency parameter Ω = 1205961198862
(120588ℎ01198630)12 for moderately thick plates with linearly varying thickness in different boundary
conditions (ℎ(119909 119910) = ℎ0(1 + 05119909
2
)(1 + 051199102
))
Boundary condition 119886119887 ℎ0119886
Model sequence1 2 3 4 5 6
S-S-S-S
101 23462 54999 55573 82950 100900 101187
(23741a) (55114) (55983) (83146) (101004) (101347)
02 18606 42725 42971 60040 71167 71761(18936a) (43014) (43148) (60247) (71642) (71993)
3201 53055 65251 85404 106700 114880 143531
(53467a) (65472) (85984) (107211) (115687) (144007)
02 17439 21531 32170 44987 59438 64179(17874a) (21748) (32651) (45264) (60188) (64879)
C-F-F-F
101 3658 9317 24135 32816 35433 59179
(3749a) (9547) (24034) (33627) (36308) (60142)
02 3520 7986 19591 26959 27491 43882(3611a) (8176) (20075) (27125) (27625) (43984)
3201 3901 9963 28984 69187 73686 89360
(3930a) (10095) (29381) (69578) (73994) (90242)
02 3292 4439 13385 23947 40850 44758(3386a) (4498) (14012) (24068) (41221) (45142)
S-S-F-F
101 3557 21268 22762 43524 60253 62429
(3649a) (21441) (22796) (43916) (60622) (62630)
02 3171 17822 19512 33616 45042 46833(3489a) (18006) (20430) (33815) (45247) (49004)
3201 6738 31658 52071 75625 86604 94903
(6957a) (31882) (52518) (75823) (86939) (95610)
02 3653 12930 20955 26517 44097 48058(6807a) (13160) (21312) (26864) (44422) (48873)
C-C-C-C
101 41020 75196 75614 104142 119683 120949
(41405a) (75452) (75882) (104711) (119637) (121137)
02 31237 51686 51868 68411 76868 77864(32010a) (52960) (52145) (69335) (77481) (78099)
3201 95823 128727 163657 173751 190890 223595
(95364a) (128932) (163588) (172350) (190350) (222948)
02 53306 71372 89384 102404 104660 120908(53587a) (71759) (90103) (102503) (105174) (121275)
S-C-S-C
101 33715 60298 71446 94187 103802 117928
(34148a) (60786) (71828) (94515) (104263) (119326)
02 26143 45499 49458 64253 72941 76191(26789a) (46622) (50680) (64329) (73474) (76843)
3201 73228 115845 137445 148166 168342 182542
(73761a) (116121) (137788) (148530) (168698) (182921)
02 41302 53648 65630 72202 81632 88105(41304a) (53757) (65842) (72794) (82033) (88474)
aResults in parentheses are taken from FEM
S-C-S-C boundary conditions are performed in the compar-ison The results adequately demonstrated the great accuracyof the modified Fourier method
We now turn to elastically restrained moderately thickplates The first one involves an S-S-S-S moderately thicksquare plate with a uniform rotational restraint along eachedge that is 119870
1199090= 119870119909119886
= 1198701199100
= 119870119910119887
= Γ119909 The calculated
frequency parameters Ω = 1205961198862
(120588ℎ01198630)12 are given in
Table 10 together with the FEM results Since the elasticallyrestrained plates with nonlinear variation thickness are rarelyinvestigated the FEMresults are used as the referenceA goodagreement is observed between the current and FEM resultsThe second example concerns amoderately thick square plateelastically supported along all edgesThe stiffness of the linear
18 Shock and Vibration
Table 10 Frequency parameterΩ = 1205961198862
(120588ℎ01198630)12 for S-S-S-S moderately thick plates (119886119887 = 1) with linearly varying thickness and elastic
rotation support (ℎ(119909 119910) = ℎ0(1 + 05119909
2
)(1 + 051199102
) 1198701199090= 119870119909119886= 1198701199100= 119870119910119887= Γ119909)
Γ119909
aℎ0119886
Model sequence1 2 3 4 5 6
0
005 23462 54999 55573 82950 100900 101187(23741b) (55114) (55983) (83146) (101004) (101347)
01 18606 42725 42971 60040 71167 71761(18936b) (43014) (43148) (60247) (71642) (71993)
015 53055 65251 85404 106700 114880 143531(53467b) (65472) (85984) (107211) (115687) (144007)
02 17439 21531 32170 44987 59438 64179(17874b) (21748) (32651) (45264) (60188) (64879)
05
005 6904 13374 13514 19400 22904 23056(9909b) (13548) (13651) (19564) (23088) (23379)
01 5316 10237 10314 14499 17155 17264(5374b) (10448) (10651) (14764) (17188) (17879)
015 4136 8201 8220 11474 13578 13671(4174b) (8348) (8651) (11764) (13688) (13879)
02 3449 6915 6924 9549 11225 11327(3474b) (6948) (6951) (9764) (11388) (11579)
1
005 7040 13617 13745 19726 23233 23402(7074b) (13748) (13851) (19964) (23488) (23879)
01 5712 10696 10793 14991 17594 17744(5774b) (10748) (10851) (15054) (17788) (17879)
015 4489 8507 8540 11759 13803 13922(4514b) (8548) (8611) (11864) (13988) (14119)
02 3677 7084 7090 9692 11330 11445(3684b) (7148) (7191) (9764) (11588) (11879)
15
005 7089 13708 13828 19847 23355 23530(7124b) (13748) (13951) (19967) (23588) (23879)
01 5895 10923 11024 15239 17813 17988(5874b) (10748) (11651) (15264) (17818) (17979)
015 4697 8694 8737 11938 13790 14083(4774b) (8748) (8951) (11864) (13688) (14879)
02 3836 7201 7210 9792 11403 11530(3874b) (7748) (7651) (9864) (11588) (11879)
2
005 7114 13754 13871 19911 23418 23597(7174b) (12848) (13851) (19864) (23588) (23879)
01 6001 11060 11159 15391 17944 18136(6004b) (11148) (11151) (15464) (17988) (18379)
015 4836 8821 8872 12061 14037 14189(4874b) (8848) (8951) (12164) (14188) (14879)
02 3954 7289 7301 9871 11457 11594(3974b) (7348) (7651) (9964) (11488) (11679)
aΓ119909= 1198701198941198700 (1198700= 1 times 10
9 Nmrad 119894 = 1199090 119909119886 1199100 119910119887)bResults in parentheses are taken from FEM
and rotational restraints is set equal to 1198961199090
= 119896119909119886
= 1198961199100
=
119896119910119887
= Γ119908and 119870
1199090= 119870119909119886
= 1198701199100
= 119870119910119887
= Γ119909 respectively
The six frequency parametersΩ = 1205961198862
(120588ℎ01198630)12 are shown
in Table 11 for several different restraining coefficient valuesIt can also be noticed that the Fourier series method is able
to predict the modal characteristics of nonlinearly varyingthickness moderately thick plate with not only classicalboundary conditions but also elastic edge restraints correctly
Finally the influence of the gradient on the fundamentalfrequency parameters of a nonlinearly varying thickness
Shock and Vibration 19
Table 11 Frequency parameterΩ = 1205961198862
(120588ℎ01198630)12 for moderately thick plates (119886119887 = 1) with linearly varying thickness and elastic rotation
and translation support (ℎ(119909 119910) = ℎ0(1 + 05119909
2
)(1 + 051199102
) 1198701199090= 119870119909119886= 1198701199100= 119870119910119887= Γ119909 and 119896
1199090= 119896119909119886= 1198961199100= 119896119910119887= Γ119908)
Γ119909
aΓ119908
aℎ0119886
Model sequence1 2 3 4 5 6
05 0
005 12926 13041 25090 47935 47943 61761(12931b) (13053) 25099 (47947) (47952) (61788)
01 10900 10988 21545 39980 40147 50322(10904b) (10992) (21557) (39984) (40166) (50334)
015 8245 8298 18081 31503 32617 40950(8247b) (8230) (18092) (31508) (32622) (40955)
02 6106 6124 15639 25631 27601 34458(6110b) (6125) (15642) (25637) (27617) (34462)
1 05
005 21575 31615 31644 42573 57889 58607(21580b) (31617) (31644) (42581) (57893) (58610)
01 8451 16024 16210 25969 43839 43867(8451b) (16025) (16217) (25972) (43844) (43870)
015 4682 11479 11480 20642 35172 35598(4684b) (11480) (11481) (20644) (35178) (35604)
02 3059 8621 8648 17320 28487 29638(3062b) (8619) (8644) (17327) (28492) (29641)
15 1
005 27713 40203 40978 52951 65352 66815(27714b) (40207) (40988) (52972) (65368) (66822)
01 11667 19322 19615 29122 46115 46183(11674b) (19327) (19619) (29137) (46120) (46188)
015 6544 13416 13481 22405 37238 37430(6547b) (13420) (13488) (22410) (37241) (37434)
02 4295 10175 10190 18543 30330 31056(4296b) (10177) (10194) (18544) (30336) (31060)
2 2
005 33453 50854 52088 66311 76610 78947(33455b) (50852) (50097) (66321) (76627) (78971)
01 15832 24130 24372 33924 49118 49362(15830b) (24127) (24374) (33931) (49147) (49382)
015 9060 15956 16099 24754 39134 39249(9064b) (15961) (16102) (24755) (39138) (39452)
02 5992 11955 11971 20034 31908 32392(5991b) (11957) (11978) (20042) (31910) (32398)
aΓ119909= 1198701198941198700 (1198700= 1 times 10
9 Nmrad 119894 = 1199090 119909119886 1199100 119910119887) Γ119908= 1198961198941198960 (1198960= 1 times 10
9 Nm 119894 = 1199090 119909119886 1199100 119910119887)bResults in parentheses are taken from FEM
moderately thick plate is investigated The plate is elasticallyrestrained in which the stiffness of the boundary springsis taken as 119896
1199090= 119896119909119886
= 1198961199100
= 119896119910119887
= Γ119908
= 2 and1198701199090
= 119870119909119886
= 1198701199100
= 119870119910119887
= Γ119909= 2 respectively The six
frequency parameters Ω = 1205961198862
(120588ℎ01198630)12 are presented in
Table 12 for several different slop values It can be seen thatthe fundamental frequency parameters will decrease with theincrease of the parameter ℎ
0119886
In the above examples it has been demonstrated thatthe presented method can be universally applied to nonlin-early varying thickness moderately thick plates with severalboundary conditions and different structure parametersNew results are obtained for plates with nonlinear variationthickness in both directions subjected to general elastic
boundary restraints which may be used for benchmarkingof researchers in the field In addition it is interesting to seethat the nature frequency decreases with the increase of theindex for thickness function
4 Conclusions
In this paper a modified Fourier method has been pre-sented to study the free vibration behaviors of moderatelythick rectangular plates with variable thickness and arbi-trary boundary conditionsThe first-order shear deformationplate theory is adopted to formulate the theoretical modelThe displacements and rotation components of the plateregardless of boundary conditions are invariantly expressedas the superposition of a 2D Fourier cosine series and four
20 Shock and Vibration
Table 12 Frequency parameterΩ = 1205961198862
(120588ℎ01198630)12 for elastic support moderately thick plates (119886119887 = 1) with different gradient (119896
1199090= 119896119909119886=
1198961199100= 119896119910119887= Γ119908= 2 119870
1199090= 119870119909119886= 1198701199100= 119870119910119887= Γ119909= 2)
120572 120573 ℎ0119886
Model sequence1 2 3 4 5 6
120572 = 120573 = 01
005 30187 51102 51141 67066 73954 77412(30188a) (51105) (51144) (67068) (73961) (77477)
01 16995 24719 24724 33071 44343 44819(16996a) (24725) (24729) (33084) (44347) (44822)
015 10077 15872 15883 23443 35937 36065(10081a) (15877) (15892) (23447) (35941) (36084)
02 6713 11953 11958 19086 30372 30604(6713a) (11957) (11965) (19092) (30378) (30612)
120572 = 120573 = 02
005 31206 51436 51614 67058 74398 77595(31208a) (51438) (51619) (67064) (74402) (77596)
01 16786 24503 24508 33144 45583 45923(16790a) (24510) (24513) (33147) (45587) (45932)
015 9821 15865 15906 23766 36897 36990(9824a) (15870) (15907) (23768) (36894) (36997)
02 6523 11978 11992 19356 30912 31162(6524a) (11981) (11997) (19362) (30916) (31174)
120572 = 120573 = 03
005 32097 51462 51900 66832 74991 77863(32098a) (51466) (51907) (66849) (74996) (77872)
01 16507 24332 24383 33330 46806 47074(16511a) (24327) (24380) (33339) (46810) (47086)
015 9562 15885 15964 24103 37748 37828(9562a) (15892) (15977) (21105) (37751) (37832)
02 6338 11992 12013 19603 31338 31634(6340a) (11994) (12011) (19612) (31344) (31639)
120572 = 120573 = 04
005 32851 51239 52046 66547 75735 78303(33455a) (50852) (50097) (66321) (76627) (78971)
01 16182 24209 24343 33598 47990 48230(33455a) (50852) (50097) (66321) (76627) (78971)
015 9307 15919 16034 24436 38491 38579(33455a) (50852) (50097) (66321) (76627) (78971)
02 6161 11986 12008 19828 31666 32040(33455a) (50852) (50097) (66321) (76627) (78971)
120572 = 120573 = 05
005 33453 50854 52088 66311 76610 78947(33455a) (50852) (50097) (66321) (76627) (78971)
01 15832 24130 24372 33924 49118 49362(15830a) (24127) (24374) (33931) (49147) (49382)
015 9060 15956 16099 24754 39134 39249(9064a) (15961) (16102) (24755) (39138) (39452)
02 5992 11955 11971 20034 31908 32392(5991a) (11957) (11978) (20042) (31910) (32398)
aResults in parentheses are taken from FEM
supplementary functions in the form of the product of apolynomial function and a single cosine series expansionto ensure and accelerate the convergence of the solutionAt each edge of the plate the general restraint conditionsare implemented by introducing one group of linear springsand two groups of rotational springs which are continuouslydistributed and determined by the stiffness of these springs
Instead of seeking a solution in strong forms in the previousstudies all the Fourier coefficients will be treated equallyand independently as the generalized coordinates and solveddirectly from the Rayleigh-Ritz technique The change of theboundary conditions can be easily achieved by only varyingthe stiffness of the three sets of boundary springs along alledges of the rectangular plates without involving any change
Shock and Vibration 21
to the solution procedure The convergence of the presentsolution is examined and the excellent accuracy is validatedby comparison with existing results published in the lit-erature and FEM data Excellent agreements are obtainedfrom these comparisons The proposed method providesa unified means for extracting the modal parameters andpredicting the vibration behaviors of moderately thick plateswith variable thickness variation functions and arbitraryelastic edge restraints A variety of free vibration results formoderately thick rectangular plates with different thicknessvariation functions and boundary conditions are presentedNew results for free vibration of moderately thick rectangularplates with various thickness variation functions and edgeconditions are presented which may be used for benchmark-ing of researchers in the field
Appendices
A Representative Calculation forStiffness and Mass Matrices
To illuminate the particular expression of the mass andstiffness matrixes clearly and tersely four new variables aredefined as 119904 = 119898 lowast (119873 + 1) + 119899 + 1 119905 = 119898
1015840
lowast (119873 + 1) + 1198991015840
+ 1119901 = (119897minus1)lowast (119872+1)+119898
1015840
+1 and 119902 = (119897minus1)lowast (119873+1)+1198991015840
+1The first row elements of K andM are given below
K1-1119904119905
=
1
2
int
119886
0
int
119887
0
[119863 (119909 119910)
times 21205821198861198981205821198861198981015840 sin 120582
119886119898119909 sin 120582
1198861198981015840119909 cos 120582
119887119899119910
times cos 1205821198871198991015840119910 + (1 minus 120583) 120582
1198871198991205821198871198991015840 cos 120582
119886119898119909
times cos 1205821198861198981015840 sin 120582
119887119899119910 sin 120582
1198871198991015840119910
+ 120581119866ℎ (119909 119910) cos 120582119886119898119909 cos 120582
1198861198981015840119909 cos 120582
119887119899
times119910 cos 1205821198871198991015840119910119889119909119889119910] 119889119909 119889119910
+ int
119886
0
1198701199091199100
cos 120582119886119898119909 cos 120582
1198861198981015840119909
+(minus1)119898+1198981015840
119870119909119910119887
cos 120582119886119898119909 cos 120582
1198861198981015840119909 119889119909
+ int
119887
0
1198701199090cos 120582119887119899119910 cos 120582
1198871198991015840119910
+(minus1)119899+1198991015840
119870119909119886cos 120582119887119899119910 cos 120582
1198871198991015840119910 119889119910
K1-2119904119901
=
1
2
int
119886
0
int
119887
0
[119863 (119909 119910)21205821198861198981205821198861198981015840 sin 120582
119886119898119909 sin 120582
1198861198981015840119909
times
2
sum
119897=1
120577119897
119887(119910) cos 120582
1198871198991015840119910
minus (1 minus 120583) 1205821198871198991015840 cos 120582
119886119898119909 cos 120582
1198861198981015840119909
times
2
sum
119897=1
1205771198971015840
119887(119910) sin 120582
1198871198991015840119910 + 120581119866ℎ (119909 119910)
times ( cos 1205821198861198981015840119909 cos 120582
1198871198991015840119910
times
2
sum
119897=1
120577119897
119887(119910) cos 120582
119886119898119909)]119889119909119889119910
+ int
119886
0
1198701199091199100
cos 120582119886119898119909 cos 120582
1198861198981015840119909
2
sum
119897=1
120577119897
119887(0)
+(minus1)1198991015840
119870119909119910119887
cos 120582119886119898119909 cos 120582
1198861198981015840119909
2
sum
119897=1
120577119897
119887(119887) 119889119909
+ int
119887
0
1198701199090
2
sum
119897=1
120577119897
119887(119910) cos 120582
1198871198991015840119910
+(minus1)119898+1198981015840
119870119909119886
2
sum
119897=1
120577119897
119887(119910) cos 120582
1198871198991015840119910119889119910
K1-3119904119902
=
1
2
int
119886
0
int
119887
0
[119863 (119909 119910) minus 21205821198861198981015840 sin 120582
1198861198981015840119909 cos 120582
1198871198991015840119910
times
2
sum
119897=1
1205771198971015840
119886(119909) cos 120582
119887119899119910
+ (1 minus 120583) 1205821198871198991205821198871198991015840 cos 120582
1198861198981015840119909
times sin 1205821198871198991015840119910
2
sum
119897=1
120577119897
119886(119909) sin 120582
119887119899119910
+ 120581119866ℎ (119909 119910) cos 1205821198861198981015840119909 cos 120582
1198871198991015840119910
times
2
sum
119897=1
120577119897
119886(119909) cos 120582
119887119899119910]119889119909119889119910
+ int
119886
0
1198701199091199100
cos 1205821198861198981015840119909
2
sum
119897=1
120577119897
119886(119909)
+(minus1)119899+1198991015840
119870119909119910119887
cos 1205821198861198981015840119909
2
sum
119897=1
120577119897
119886(119909) 119889119909
+ int
119887
0
1198701199090
2
sum
119897=1
120577119897
119886(0) cos 120582
1198871198991015840119910
+(minus1)119898+1198981015840
119870119909119886
2
sum
119897=1
120577119897
119886(0) cos 120582
1198871198991015840119910119889119910
K1-4119904119905
=
1
2
int
119886
0
int
119887
0
119863(119909 119910) 21205821198871198991205821198861198981015840 sin 120582
1198861198981015840119909 cos 120582
1198871198991015840119910
times cos 120582119886119898119909 sin 120582
119887119899119910 minus 2(1 minus 120583)
119887119899
times 1205821198861198981015840 sin 120582
1198861198981015840119909 cos 120582
1198871198991015840119910 cos 120582
119886119898119909
times sin 120582119887119899119910 + (1 minus 120583) 120582
1198861198981205821198871198991015840 cos 120582
1198861198981015840119909
times sin 1205821198871198991015840119910 sin 120582
119886119898119909 cos 120582
119887119899119910 119889119909 119889119910
22 Shock and Vibration
K1-5119904119901
=
1
2
int
119886
0
int
119887
0
119863(119909 119910) minus 21205821198861198981015840 sin 120582
1198861198981015840119909 cos 120582
1198871198991015840119910
times
2
sum
119897=1
1205771198971015840
119887(119910) cos 120582
119886119898119909
+ 2 (1 minus 120583) 1205821198861198981015840 sin 120582
1198861198981015840119909 cos 120582
1198871198991015840119910
times
2
sum
119897=1
1205771198971015840
119887(119910) cos 120582
119886119898119909
+ (1 minus 120583) 1205821198861198981205821198871198991015840 cos 120582
1198861198981015840119909 sin 120582
1198871198991015840119910
times
2
sum
119897=1
120577119897
119887(119910) sin 120582
119886119898119909119889119909119889119910
K1-6119904119902
=
1
2
int
119886
0
int
119887
0
119863(119909 119910)1205821198871198991205821198861198981015840 sin 120582
1198861198981015840119909 cos 120582
1198861198991015840119910
times
2
sum
119897=1
120577119897
119886(119909) sin 120582
1198871198991015840119910
minus 2 (1 minus 120583) 1205821198871198991205821198861198981015840 sin 120582
1198861198981015840119909 cos 120582
1198861198991015840119910
times
2
sum
119897=1
120577119897
119886(119909) sin 120582
1198871198991015840119910
+ 1205821198871198991015840 cos 120582
1198861198981015840119909 sin 120582
1198871198991015840119910
times
2
sum
119897=1
1205771198971015840
119886(119909) cos 120582
119887119899119910119889119909119889119910
K1-7119904119905
= minus
1
2
int
119886
0
int
119887
0
120581119866ℎ (119909 119910) 120582119886119898
cos 1205821198861198981015840119909 cos 120582
1198871198991015840119910
times sin 120582119886119898119909 cos 120582
119887119899119910119889119909119889119910
K1-8119904119901
= minus
1
2
int
119886
0
int
119887
0
120581119866ℎ (119909 119910) 1205821198861198981015840 cos 120582
1198861198981015840119909 cos 120582
1198871198991015840119910
times
2
sum
119897=1
120577119897
119887(119910) sin 120582
1198861198981015840119909 119889119909 119889119910
K1-9119904119902
=
1
2
int
119886
0
int
119887
0
120581119866ℎ (119909 119910) cos 1205821198861198981015840119909 cos 120582
1198871198991015840119910
times
2
sum
119897=1
1205771198971015840
119886(119909) cos 120582
119887119899119910119889119909119889119910
M1-1119904119905
=
1
24
120588int
119886
0
int
119887
0
ℎ3
(119909 119910) cos 120582119886119898119909 cos 120582
119887119899119910
times cos 1205821198861198981015840119909 cos 120582
1198871198991015840119910119889119909119889119910
M1-2119904119901
=
1
24
120588int
119886
0
int
119887
0
ℎ3
(119909 119910)
2
sum
119897=1
120577119897
119887(119910) cos 120582
119886119898119909
times cos 1205821198861198981015840119909 cos 120582
1198871198991015840119910119889119909119889119910
M1-3119904119902
=
1
24
120588int
119886
0
int
119887
0
ℎ3
(119909 119910)
2
sum
119897=1
120577119897
119886(119909) cos 120582
119887119899119910
times cos 1205821198861198981015840119909 cos 120582
1198871198991015840119910119889119909119889119910
M1-4119904119905 = M
1-5119904119901 = M1-6119904119902
= M1-7119904119905 = M
1-8119904119901 = M1-9119904119902 = 0
(A1)
B Useful Integrals for the Moderately ThickRectangular Plates with Variable Thicknessand Arbitrary Boundary Conditions
Consider
int
119871
0
cos 120582119898119909 cos 120582
119899119909 119889119909 =
0 119898 = 119899
119871
2
119898 = 119899
119871 119898 = 119899 = 0
int
119871
0
sin 120582119898119909 cos 120582
119899119909119889119909
=
minus
(minus1 + (minus1)119898+119899
) 119871119898
(1198982minus 1198992) 120587
119898 = 119899
0 119898 = 119899
int
119871
0
sin 120582119898119909 sin 120582
119899119909 119889119909 =
0 119898 = 119899
119871
2
119898 = 119899
0 119898 = 119899 = 0
int
119871
0
cos 120582119898119909120577119897
119886(119909) 119889119909
=
41198712
31205872 119897 = 1 119898 = 0
minus
41198712
(41198982
minus 3)
(9 minus 401198982+ 16119898
4) 1205872
119897 = 1 119898 = 0
minus
41198712
31205872 119897 = 2 119898 = 0
41198712
(41198982
minus 3) (minus1)119898
(9 minus 401198982+ 16119898
4) 1205872
119897 = 2 119898 = 0
Shock and Vibration 23
int
119871
0
sin 120582119898119909120577119897
119886(119909) 119889119909
=
0 119897 = 1 119898 = 0
161198712
119898(minus1)119898
(9 minus 401198982+ 16119898
4) 1205872
119897 = 1 119898 = 0
0 119897 = 2 119898 = 0
161198981198712
(9 minus 401198982+ 16119898
4) 1205872
119897 = 2 119898 = 0
int
119871
0
cos 1205821198981199091205771198971015840
119886(119909) 119889119909
=
0 119897 = 1 119898 = 0
161198711198982
(minus1)119898
(9 minus 401198982+ 16119898
4) 120587
119897 = 1 119898 = 0
0 119897 = 2 119898 = 0
161198711198982
(9 minus 401198982+ 16119898
4) 120587
119897 = 2 119898 = 0
int
119871
0
sin 1205821198981199091205771198971015840
119886(119909) 119889119909
=
0 119897 = 1 119898 = 0
4119871119898 (minus3 + 41198982
)
(9 minus 401198982+ 16119898
4) 120587
119897 = 1 119898 = 0
0 119897 = 2 119898 = 0
119871119898(
1
120587 minus 41198982120587
+
3
9120587 minus 41198982120587
) (minus1)119898
119897 = 2 119898 = 0
int
119871
0
1205771198971
119886(119909) 1205771198972
119886(119909) 119889119909 =
1198713
41205872 1198971= 1 1198972= 1
minus
21198713
31205873 1198971= 1 1198972= 2
minus
21198713
31205873 1198971= 2 1198972= 1
1198713
41205872 1198971= 2 1198972= 2
int
119871
0
1205771198971
119886(119909) 1205771198971015840
2
119886(119909) 119889119909 =
0 1198971= 1 1198972= 1
minus
1198712
8120587
1198971= 1 1198972= 2
1198712
8120587
1198971= 2 1198972= 1
0 1198971= 2 1198972= 2
int
119871
0
1205771198971015840
1
119886(119909) 1205771198971015840
2
119886(119909) 119889119909 =
5119871
16
1198971= 1 1198972= 1
minus
119871
2120587
1198971= 1 1198972= 2
minus
119871
2120587
1198971= 2 1198972= 1
5119871
16
1198971= 2 1198972= 2
int
119871
0
119909119899 cos 120582
119898119909 119889119909
=
1198711+119899
1 + 119899
119898 = 0
(1198861+119899HypergeometricPFQtimes [ 12 + 1198992 12 32 + 1198992
minus (14)1198982
1205872
]) times (1 + 119899)minus1
119898 gt 0
(B1)
Conflict of Interests
The authors declare that there is no conflict of interestsregarding the publication of this paper
Acknowledgments
The authors would like to thank the anonymous reviewersfor their very valuable comments The authors also gratefullyacknowledge the financial support from the National NaturalScience Foundation of China (no 51209052) HeilongjiangProvince Youth Science Fund Project (no QC2011C013) andHarbin Science and Technology Development InnovationFoundation of Youth (no 2011RFQXG021)
References
[1] K Liew Y Xiang and S Kitipornchai ldquoResearch on thick platevibration a literature surveyrdquo Journal of Sound and Vibrationvol 180 no 1 pp 163ndash176 1995
[2] J H Chung T Y Chung and K C Kim ldquoVibration analysisof orthotropic mindlin plates with edges elastically restrainedagainst rotationrdquo Journal of Sound and Vibration vol 163 no 1pp 151ndash163 1993
[3] Y K Cheung and D Zhou ldquoVibrations of moderately thickrectangular plates in terms of a set of static Timoshenko beamfunctionsrdquoComputers and Structures vol 78 no 6 pp 757ndash7682000
[4] CMWang ldquoNatural frequencies formula for simply supportedMindlin platesrdquo Journal of Vibration and Acoustics vol 116 no4 pp 536ndash540 1994
[5] K N Saha R C Kar and P K Datta ldquoFree vibration analysisof rectangular Mindlin plates with elastic restraints uniformlydistributed along the edgesrdquo Journal of Sound andVibration vol192 no 4 pp 885ndash902 1996
24 Shock and Vibration
[6] D J Gorman ldquoAccurate free vibration analysis of shear-deformable plates with torsional elastic edge supportrdquo Journalof Sound and Vibration vol 203 no 2 pp 209ndash218 1997
[7] D J Gorman ldquoFree vibration analysis of Mindlin plates withuniform elastic edge support by the superposition methodrdquoJournal of Sound andVibration vol 207 no 3 pp 335ndash350 1997
[8] D J Gorman ldquoAccurate free vibration analysis of point sup-ported mindlin plates by the superposition methodrdquo Journal ofSound and Vibration vol 219 no 2 pp 265ndash277 1999
[9] Y Xiang ldquoVibration of rectangular Mindlin plates resting onnon-homogenous elastic foundationsrdquo International Journal ofMechanical Sciences vol 45 no 6-7 pp 1229ndash1244 2003
[10] Y Xiang and G W Wei ldquoExact solutions for buckling andvibration of stepped rectangular Mindlin platesrdquo InternationalJournal of Solids and Structures vol 41 no 1 pp 279ndash294 2004
[11] Y L Yeh M J Jang and C C Wang ldquoAnalyzing the freevibrations of a plate using finite difference and differential trans-formationmethodrdquoAppliedMathematics andComputation vol178 no 2 pp 493ndash501 2006
[12] Y Xiang S K Lai L Zhou and C W Lim ldquoDSC-Ritz elementmethod for vibration analysis of rectangular Mindlin plateswith mixed edge supportsrdquo European Journal of MechanicsmdashASolids vol 29 no 4 pp 619ndash628 2010
[13] H Nguyen-Xuan C H Thai and T Nguyen-Thoi ldquoIsogeo-metric finite element analysis of composite sandwich platesusing a higher order shear deformation theoryrdquo Composites BEngineering vol 55 pp 558ndash574 2013
[14] C H Thai A J M Ferreira E Carrera and H Nguyen-XuanldquoIsogeometric analysis of laminated composite and sandwichplates using a layerwise deformation theoryrdquo Composite Struc-tures vol 104 pp 196ndash214 2013
[15] C H Thai A Ferreira S Bordas T Rabczuk and HNguyen-Xuan ldquoIsogeometric analysis of laminated compositeand sandwich plates using a new inverse trigonometric sheardeformation theoryrdquo European Journal of Mechanics A Solidsvol 43 pp 89ndash108 2014
[16] H Luong-Van T Nguyen-Thoi G R Liu and P Phung-VanldquoA cell-based smoothed finite elementmethod using three-nodeshear-locking free Mindlin plate element (CS-FEM-MIN3) fordynamic response of laminated composite plates on viscoelasticfoundationrdquo Engineering Analysis with Boundary Elements vol42 pp 8ndash19 2014
[17] T Nguyen-Thoi T Rabczuk T Lam-Phat V Ho-Huu andP Phung-Van ldquoFree vibration analysis of cracked Mindlinplate using an extended cell-based smoothed discrete sheargap method (XCS-DSG3)rdquo Theoretical and Applied FractureMechanics 2014
[18] H Nguyen-Xuan G R Liu and C a Thai-Hoang ldquoAnedge-based smoothed finite element method (ES-FEM) withstabilized discrete shear gap technique for analysis of Reissner-Mindlin platesrdquo Computer Methods in Applied Mechanics andEngineering vol 199 no 9ndash12 pp 471ndash489 2010
[19] P Phung-Van T Nguyen-Thoi T Bui-Xuan and Q Lieu-XuanldquoA cell-based smoothed three-node Mindlin plate element(CS-FEM-MIN3) based on the 119862
0 -type higher-order sheardeformation for geometrically nonlinear analysis of laminatedcomposite platesrdquo Computational Materials Science 2014
[20] NNguyen-Thanh T RabczukHNguyen-Xuan and S BordasldquoAn alternative alpha finite element method with discrete sheargap technique for analysis of isotropicMindlin-Reissner platesrdquoFinite Elements inAnalysis andDesign vol 47 no 5 pp 519ndash5352011
[21] KM Liew Y Xiang and S Kitipornchai ldquoTransverse vibrationof thick rectangular plates-I Comprehensive sets of boundaryconditionsrdquo Computers and Structures vol 49 no 1 pp 1ndash291993
[22] D Zhou ldquoVibrations of Mindlin rectangular plates with elasti-cally restrained edges using static Timoshenko beam functionswith the Rayleigh-Ritz methodrdquo International Journal of Solidsand Structures vol 38 no 32-33 pp 5565ndash5580 2001
[23] D Zhou SH Lo F T KAu andYK Cheung ldquoVibration anal-ysis of rectangular Mindlin plates with internal line supportsusing static Timoshenko beam functionsrdquo International Journalof Mechanical Sciences vol 44 no 12 pp 2503ndash2522 2002
[24] H S Shen J Yang and L Zhang ldquoFree and forced vibrationof Reissner-Mindlin plates with free edges resting on elasticfoundationsrdquo Journal of Sound and Vibration vol 244 no 2 pp299ndash320 2001
[25] Y Xing and B Liu ldquoClosed form solutions for free vibrations ofrectangular Mindlin platesrdquo Acta Mechanica Sinica vol 25 no5 pp 689ndash698 2009
[26] P Gagnon C Gosselin and L Cloutier ldquoA finite strip elementfor the analysis of variable thickness rectangular thick platesrdquoComputers amp Structures vol 63 no 2 pp 349ndash362 1997
[27] T Mlzusawa ldquoVibration of rectangular mindlin plates withtapered thickness by the spline strip methodrdquo Computers andStructures vol 46 no 3 pp 451ndash463 1993
[28] H Nguyen-Xuan and T Nguyen-Thoi ldquoA stabilized smoothedfinite element method for free vibration analysis of Mindlin-Reissner platesrdquo Communications in Numerical Methods inEngineering with Biomedical Applications vol 25 no 8 pp 882ndash906 2009
[29] K M Liew L X Peng and S Kitipornchai ldquoVibration anal-ysis of corrugated Reissner-Mindlin plates using a mesh-freeGalerkin methodrdquo International Journal of Mechanical Sciencesvol 51 no 9-10 pp 642ndash652 2009
[30] Y Hou GWWei and Y Xiang ldquoDSC-Ritz method for the freevibration analysis of Mindlin platesrdquo International Journal forNumerical Methods in Engineering vol 62 no 2 pp 262ndash2882005
[31] F- Liu and K M Liew ldquoVibration analysis of discontinuousMindlin plates by differential quadrature element methodrdquoJournal of Vibration and Acoustics Transactions of the ASMEvol 121 no 2 pp 204ndash208 1999
[32] P Malekzadeh G Karami and M Farid ldquoA semi-analyticalDQEM for free vibration analysis of thick plates with twoopposite edges simply supportedrdquoComputerMethods inAppliedMechanics and Engineering vol 193 no 45ndash47 pp 4781ndash47962004
[33] P Malekzadeh and S A Shahpari ldquoFree vibration analysisof variable thickness thin and moderately thick plates withelastically restrained edges by DQMrdquo Thin-Walled Structuresvol 43 no 7 pp 1037ndash1050 2005
[34] R E Diaz-Contreras and S Nomura ldquoGreenrsquos function appliedto solution ofMindlin platesrdquoComputers and Structures vol 60no 1 pp 41ndash48 1996
[35] T Sakiyama and M Huang ldquoFree vibration analysis of rect-angular plates with variable thicknessrdquo Journal of Sound andVibration vol 216 no 3 pp 379ndash397 1998
[36] M Huang X Q Ma T Sakiyama H Matuda and C MoritaldquoFree vibration analysis of orthotropic rectangular plates withvariable thickness and general boundary conditionsrdquo Journal ofSound and Vibration vol 288 no 4-5 pp 931ndash955 2005
Shock and Vibration 25
[37] I Shufrin and M Eisenberger ldquoVibration of shear deformableplates with variable thicknessmdashfirst-order and higher-orderanalysesrdquo Journal of Sound and Vibration vol 290 no 1-2 pp465ndash489 2006
[38] S A Eftekhari and A A Jafari ldquoAccurate variational approachfor free vibration of variable thickness thin and thick plates withedges elastically restrained against translation and rotationrdquoInternational Journal of Mechanical Sciences vol 68 pp 35ndash462013
[39] W L Li X Zhang J Du and Z Liu ldquoAn exact series solutionfor the transverse vibration of rectangular plates with generalelastic boundary supportsrdquo Journal of Sound amp Vibration vol321 no 1-2 pp 254ndash269 2009
[40] X Shi D Shi W L Li and Q Wang ldquoA unified method forfree vibration analysis of circular annular and sector plateswith arbitrary boundary conditionsrdquo Journal of Vibration andControl 2014
[41] D Y Shi X J Shi W L Li and Q S Wang ldquoFree transversevibrations of orthotropic thin rectangular plates with arbitraryelastic edge supportsrdquo Journal of Vibroengineering vol 16 no 1pp 389ndash398 2014
[42] Y Chen G Jin and Z Liu ldquoFree vibration analysis of circu-lar cylindrical shell with non-uniform elastic boundary con-straintsrdquo International Journal of Mechanical Sciences vol 74pp 120ndash132 2013
[43] L Dai T Yang J Du W L Li and M J Brennan ldquoAn exactseries solution for the vibration analysis of cylindrical shellswith arbitrary boundary conditionsrdquo Applied Acoustics vol 74no 3 pp 440ndash449 2013
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Shock and Vibration
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International Journal of
14 Shock and Vibration
Table 6 Frequency parameter Ω = 1205961198862
(120588ℎ01198630)12 for S-S-S-S moderately thick plates (119886119887 = 1) with linearly varying thickness and elastic
rotation support (ℎ(119909 119910) = ℎ0(1 + 05119909)(1 + 05119910) 119870
1199090= 119870119909119886= 1198701199100= 119870119910119887= Γ119909)
Γ119909
aℎ0119886
Model sequence1 2 3 4 5 6
0
005 28809 70080 70891 108967 134884 135295(28897b) (70334) (71129) (109112) (135254) (135772)
01 27261 62398 62926 92423 111540 111735(26606b) (61784) (62303) (91193) (111323) (111648)
015 25231 54192 54505 77118 91228 91312(25204b) (55297) (55595) (77503) (91871) (91941)
02 23103 47010 47191 65018 75832 75866(23947b) (47071) (47208) (65404) (76531) (76589)
05
005 23103 47010 47191 65018 75832 75866(23210b) (47245) (47358) (65241) (76122) (76345)
01 48377 94650 95597 135484 162635 163501(48524b) (94891) (95992) (135845) (167012) (164024)
015 36239 70939 71298 99997 118584 119009(36647b) (71214) (71687) (100024) (119114) (119457)
02 28749 56937 57037 79214 93009 93162(29142b) (57034) (57544) (79654) (93668) (93754)
1
005 49651 96759 97619 138089 165368 166304(49755b) (97245) (97964) (138544) (165864) (166878)
01 38951 73735 74251 102748 121079 121709(39021b) (74145) (74664) (102874) (121613) (121817)
015 30630 58406 58533 80432 93984 94255(30638b) (58684) (58644) (80874) (94121) (94478)
02 25548 48636 48674 66165 76708 76824(25687b) (48788) (48992) (66544) (76824) (76944)
15
005 50118 97561 98374 139096 166421 167382(50247b) (97877) (98661) (139247) (166875) (167662)
01 40310 75216 75783 104229 122402 123164(40366b) (75542) (75921) (104557) (122874) (123385)
015 31837 59352 59530 81245 94611 94983(31902b) (59451) (59871) (81375) (94784) (95105)
02 26302 49119 49156 66528 76954 77130(26418b) (49354) (49334) (66921) (77010) (77470)
2
005 50360 97983 98769 139631 166979 167954(50472b) (98005) (99114) (140154) (167188) (167986)
01 41130 76146 76724 105165 123229 124082(41421b) (76642) (78114) (105661) (123278) (124974)
015 32687 60025 60245 81832 95052 95508(32778b) (60244) (60549) (82146) (95114) (95842)
02 26893 49492 49539 66815 77138 77370(26922b) (49874) (49924) (66987) (77367) (77588)
aΓ119909= 1198701198941198700 (1198700= 1 times 10
9 Nmrad 119894 = 1199090 119909119886 1199100 119910119887)bResults in parentheses are taken from FEM
Shock and Vibration 15
Table 7 Frequency parameterΩ = 1205961198862
(120588ℎ01198630)12 for moderately thick plates (119886119887 = 1) with linearly varying thickness and elastic rotation
and translation support (ℎ(119909 119910) = ℎ0(1 + 05119909)(1 + 05119910) 119870
1199090= 119870119909119886= 1198701199100= 119870119910119887= Γ119909 and 119896
1199090= 119896119909119886= 1198961199100= 119896119910119887= Γ119908)
Γ119909
aΓ119908
aℎ0119886
Model sequence1 2 3 4 5 6
05 0
005 14751 14917 28232 55251 55397 69512(14780b) (14822) (29252) (55476) (55505) (70904)
01 11811 11910 23857 44462 45037 55453(11818b) (11897) (24711) (44376) (45257) (56511)
015 8487 8514 20011 34783 36335 44970(8500b) (8544) (20738) (34404) (36663) (45625)
02 6073 6073 17431 28679 30805 38010(6096b) (6106) (18039) (28129) (31165) (38311)
1 05
005 21225 32173 32173 44381 64061 65222(20708b) (32097) (32285) (45110) (64182) (65355)
01 8140 16878 16941 28100 48663 49015(7943b) (16838) (16970) (28888) (48700) (49171)
015 4493 11859 11878 22429 38352 39247(4383b) (11867) (11891) (23125) (38177) (39488)
02 2932 8690 8691 18919 31108 32600(3540b) (8791) (8791) (16229) (27520) (27889)
15 1
005 27819 40919 41430 54595 70792 72896(27146b) (41008) (41434) (55256) (70895) (73051)
01 11285 20068 20230 31101 51046 51429(11012b) (20025) (20270) (31849) (51126) (51569)
015 6292 13912 13922 24132 40500 41120(6139b) (13912) (13934) (24808) (40426) (41316)
02 4121 10321 10331 20051 32796 33921(4021b) (10329) (10357) (20635) (32540) (34178)
2 2
005 34615 51972 53088 68255 81344 84510(33806b) (52100) (53157) (68891) (81433) (84761)
01 15416 24659 24829 35615 53879 54494(15045b) (24625) (24888) (36303) (53970) (54626)
015 8740 16420 16481 26353 42417 42946(8529b) (16414) (16499) (26996) (42398) (43114)
02 5761 12127 12127 21434 34286 35192(5622b) (12134) (12148) (21998) (34110) (35415)
aΓ119909= 1198701198941198700 (1198700= 1 times 10
9 Nmrad 119894 = 1199090 119909119886 1199100 119910119887) Γ119908= 1198961198941198960 (1198960= 1 times 10
9 Nm 119894 = 1199090 119909119886 1199100 119910119887)bResults in parentheses are taken from FEM
moderately thick plate with not only classical boundaryconditions but also elastic edge restraints
As the last case of this subsection the influence of thegradient 120572 and 120573 on the fundamental frequency parametersfor a linearly varying thickness moderately thick plate isinvestigated The model is a square moderately thick platewith all edges elastically restrainedThat is 119896
1199090= 119896119909119886= 1198961199100=
119896119910119887= Γ119908= 2 and 119870
1199090= 119870119909119886
= 1198701199100= 119870119910119887= Γ119909= 2 The six
frequency parameters Ω = 1205961198862
(120588ℎ01198630)12 are given in
Table 8 for several different 120572 and 120573 values the finite elementmethod (FEM) results are also listed as a reference Againgood agreement can be observed Through Table 8 it is alsofound that the frequency parameter increases with increasinggradient parameters
The above studies are given as linearly varying thicknessmoderately thick plates with several of boundary condition
and different structure parameters In the next section vibra-tion results for the plates subjected to nonlinear variationthickness will be presented
34 Nonlinearly Varying Thickness Moderately Thick Plateswith Classical and Elastic Boundary Conditions In Sec-tion 33 the linearly varying thickness moderately thickplates were studied However in the practical engineeringapplications the varying thickness of amoderately thick platemay not always be linear variation in nature A variety ofpossible thickness varying cases may be encountered in prac-tice Therefore the moderately thick plates with nonlinearvariation thickness subjected to general elastic edge restraintsare examined in this subsection For the sake of brevity theindexes 119904 and 119905 will be chosen as 2 to imitate the nonlinearlyvarying thickness moderately thick plates structure in this
16 Shock and Vibration
Table 8 Frequency parameterΩ = 1205961198862
(120588ℎ01198630)12 for elastic support moderately thick plates (119886119887 = 1) with different gradient (119896
1199090= 119896119909119886=
1198961199100= 119896119910119887= Γ119908= 2 119870
1199090= 119870119909119886= 1198701199100= 119870119910119887= Γ119909= 2)
120572 120573 ℎ0119886
Model sequence1 2 3 4 5 6
120572 = 120573 = 01
005 31275 52615 52657 68954 76146 79850(31303a) (52664) (52675) (69033) (76174) (79948)
01 17270 25244 25250 33949 46050 46625(17271a) (25255) (25258) (34089) (46122) (46708)
015 10190 16266 16274 24164 37327 37526(10189a) (16269) (16280) (24323) (37401) (37622)
02 6780 12258 12260 19685 31452 31754(6780a) (12265) (12270) (19851) (31514) (31864)
120572 = 120573 = 02
005 32530 52947 53141 68954 77034 80589(32577a) (53014) (53203) (69172) (77088) (80773)
01 16882 24979 24979 34154 48057 48575(16883a) (24983) (25001) (34433) (48134) (48667)
015 9804 16284 16309 24683 38844 39085(9803a) (16283) (16322) (24980) (38909) (39191)
02 6499 12290 12295 20133 32354 32770(6498a) (12296) (12307) (20420) (32387) (32895)
120572 = 120573 = 03
005 33516 52845 53308 68704 78203 81561(33563a) (52939) (53389) (69055) (78271) (81789)
01 16419 24799 24830 34527 50061 50583(16419a) (24790) (24869) (34947) (50143) (50688)
015 9430 16334 16378 25236 40193 40506(9429a) (16329) (16395) (25660) (40243) (40625)
02 6235 12288 12293 20572 33102 33663(6235a) (12296) (12306) (20965) (33090) (33811)
120572 = 120573 = 04
005 34215 52463 53264 68430 79654 82865(34252a) (52575) (53346) (68921) (79732) (83113)
01 15921 24695 24793 35027 52011 52571(15921a) (24674) (24842) (35583) (52098) (52688)
015 9074 16388 16444 25797 41379 41789(9074a) (16381) (16464) (26336) (41404) (41928)
02 5990 12236 12238 21002 33735 34461(5989a) (12249) (12250) (21487) (33653) (34641)
120572 = 120573 = 05
005 34615 51972 53088 68255 81344 84510(33806a) (52100) (53157) (68891) (81433) (84761)
01 15416 24659 24829 35615 53879 54494(15045a) (24625) (24888) (36303) (53970) (54626)
015 8740 16420 16481 26353 42417 42946(8529a) (16414) (16499) (26996) (42398) (43114)
02 5761 12127 12127 21434 34286 35192(5622a) (12134) (12148) (21998) (34110) (35415)
aResults in parentheses are taken from FEM
section that is 119904 = 119905 = 2 Also the thickness varying functionsare also be expressed as Fourier cosine series according to (18)and (19)
In order to validate the accuracy and reliability of theproposed method for predicting the vibration behavior ofnonlinearly varying thickness moderately thick plates with
arbitrary boundary conditions the typical classical boundaryconditions viewed as the special cases of elastically restrainededges will be considered The comparison of the first six fre-quency parameters Ω = 120596119886
2
(120588ℎ01198630)12 for the moderately
thick plates with nonlinearly varying thickness is presentedin Table 9 The S-S-S-S C-F-F-F S-S-F-F C-C-C-C and
Shock and Vibration 17
Table 9 Frequency parameter Ω = 1205961198862
(120588ℎ01198630)12 for moderately thick plates with linearly varying thickness in different boundary
conditions (ℎ(119909 119910) = ℎ0(1 + 05119909
2
)(1 + 051199102
))
Boundary condition 119886119887 ℎ0119886
Model sequence1 2 3 4 5 6
S-S-S-S
101 23462 54999 55573 82950 100900 101187
(23741a) (55114) (55983) (83146) (101004) (101347)
02 18606 42725 42971 60040 71167 71761(18936a) (43014) (43148) (60247) (71642) (71993)
3201 53055 65251 85404 106700 114880 143531
(53467a) (65472) (85984) (107211) (115687) (144007)
02 17439 21531 32170 44987 59438 64179(17874a) (21748) (32651) (45264) (60188) (64879)
C-F-F-F
101 3658 9317 24135 32816 35433 59179
(3749a) (9547) (24034) (33627) (36308) (60142)
02 3520 7986 19591 26959 27491 43882(3611a) (8176) (20075) (27125) (27625) (43984)
3201 3901 9963 28984 69187 73686 89360
(3930a) (10095) (29381) (69578) (73994) (90242)
02 3292 4439 13385 23947 40850 44758(3386a) (4498) (14012) (24068) (41221) (45142)
S-S-F-F
101 3557 21268 22762 43524 60253 62429
(3649a) (21441) (22796) (43916) (60622) (62630)
02 3171 17822 19512 33616 45042 46833(3489a) (18006) (20430) (33815) (45247) (49004)
3201 6738 31658 52071 75625 86604 94903
(6957a) (31882) (52518) (75823) (86939) (95610)
02 3653 12930 20955 26517 44097 48058(6807a) (13160) (21312) (26864) (44422) (48873)
C-C-C-C
101 41020 75196 75614 104142 119683 120949
(41405a) (75452) (75882) (104711) (119637) (121137)
02 31237 51686 51868 68411 76868 77864(32010a) (52960) (52145) (69335) (77481) (78099)
3201 95823 128727 163657 173751 190890 223595
(95364a) (128932) (163588) (172350) (190350) (222948)
02 53306 71372 89384 102404 104660 120908(53587a) (71759) (90103) (102503) (105174) (121275)
S-C-S-C
101 33715 60298 71446 94187 103802 117928
(34148a) (60786) (71828) (94515) (104263) (119326)
02 26143 45499 49458 64253 72941 76191(26789a) (46622) (50680) (64329) (73474) (76843)
3201 73228 115845 137445 148166 168342 182542
(73761a) (116121) (137788) (148530) (168698) (182921)
02 41302 53648 65630 72202 81632 88105(41304a) (53757) (65842) (72794) (82033) (88474)
aResults in parentheses are taken from FEM
S-C-S-C boundary conditions are performed in the compar-ison The results adequately demonstrated the great accuracyof the modified Fourier method
We now turn to elastically restrained moderately thickplates The first one involves an S-S-S-S moderately thicksquare plate with a uniform rotational restraint along eachedge that is 119870
1199090= 119870119909119886
= 1198701199100
= 119870119910119887
= Γ119909 The calculated
frequency parameters Ω = 1205961198862
(120588ℎ01198630)12 are given in
Table 10 together with the FEM results Since the elasticallyrestrained plates with nonlinear variation thickness are rarelyinvestigated the FEMresults are used as the referenceA goodagreement is observed between the current and FEM resultsThe second example concerns amoderately thick square plateelastically supported along all edgesThe stiffness of the linear
18 Shock and Vibration
Table 10 Frequency parameterΩ = 1205961198862
(120588ℎ01198630)12 for S-S-S-S moderately thick plates (119886119887 = 1) with linearly varying thickness and elastic
rotation support (ℎ(119909 119910) = ℎ0(1 + 05119909
2
)(1 + 051199102
) 1198701199090= 119870119909119886= 1198701199100= 119870119910119887= Γ119909)
Γ119909
aℎ0119886
Model sequence1 2 3 4 5 6
0
005 23462 54999 55573 82950 100900 101187(23741b) (55114) (55983) (83146) (101004) (101347)
01 18606 42725 42971 60040 71167 71761(18936b) (43014) (43148) (60247) (71642) (71993)
015 53055 65251 85404 106700 114880 143531(53467b) (65472) (85984) (107211) (115687) (144007)
02 17439 21531 32170 44987 59438 64179(17874b) (21748) (32651) (45264) (60188) (64879)
05
005 6904 13374 13514 19400 22904 23056(9909b) (13548) (13651) (19564) (23088) (23379)
01 5316 10237 10314 14499 17155 17264(5374b) (10448) (10651) (14764) (17188) (17879)
015 4136 8201 8220 11474 13578 13671(4174b) (8348) (8651) (11764) (13688) (13879)
02 3449 6915 6924 9549 11225 11327(3474b) (6948) (6951) (9764) (11388) (11579)
1
005 7040 13617 13745 19726 23233 23402(7074b) (13748) (13851) (19964) (23488) (23879)
01 5712 10696 10793 14991 17594 17744(5774b) (10748) (10851) (15054) (17788) (17879)
015 4489 8507 8540 11759 13803 13922(4514b) (8548) (8611) (11864) (13988) (14119)
02 3677 7084 7090 9692 11330 11445(3684b) (7148) (7191) (9764) (11588) (11879)
15
005 7089 13708 13828 19847 23355 23530(7124b) (13748) (13951) (19967) (23588) (23879)
01 5895 10923 11024 15239 17813 17988(5874b) (10748) (11651) (15264) (17818) (17979)
015 4697 8694 8737 11938 13790 14083(4774b) (8748) (8951) (11864) (13688) (14879)
02 3836 7201 7210 9792 11403 11530(3874b) (7748) (7651) (9864) (11588) (11879)
2
005 7114 13754 13871 19911 23418 23597(7174b) (12848) (13851) (19864) (23588) (23879)
01 6001 11060 11159 15391 17944 18136(6004b) (11148) (11151) (15464) (17988) (18379)
015 4836 8821 8872 12061 14037 14189(4874b) (8848) (8951) (12164) (14188) (14879)
02 3954 7289 7301 9871 11457 11594(3974b) (7348) (7651) (9964) (11488) (11679)
aΓ119909= 1198701198941198700 (1198700= 1 times 10
9 Nmrad 119894 = 1199090 119909119886 1199100 119910119887)bResults in parentheses are taken from FEM
and rotational restraints is set equal to 1198961199090
= 119896119909119886
= 1198961199100
=
119896119910119887
= Γ119908and 119870
1199090= 119870119909119886
= 1198701199100
= 119870119910119887
= Γ119909 respectively
The six frequency parametersΩ = 1205961198862
(120588ℎ01198630)12 are shown
in Table 11 for several different restraining coefficient valuesIt can also be noticed that the Fourier series method is able
to predict the modal characteristics of nonlinearly varyingthickness moderately thick plate with not only classicalboundary conditions but also elastic edge restraints correctly
Finally the influence of the gradient on the fundamentalfrequency parameters of a nonlinearly varying thickness
Shock and Vibration 19
Table 11 Frequency parameterΩ = 1205961198862
(120588ℎ01198630)12 for moderately thick plates (119886119887 = 1) with linearly varying thickness and elastic rotation
and translation support (ℎ(119909 119910) = ℎ0(1 + 05119909
2
)(1 + 051199102
) 1198701199090= 119870119909119886= 1198701199100= 119870119910119887= Γ119909 and 119896
1199090= 119896119909119886= 1198961199100= 119896119910119887= Γ119908)
Γ119909
aΓ119908
aℎ0119886
Model sequence1 2 3 4 5 6
05 0
005 12926 13041 25090 47935 47943 61761(12931b) (13053) 25099 (47947) (47952) (61788)
01 10900 10988 21545 39980 40147 50322(10904b) (10992) (21557) (39984) (40166) (50334)
015 8245 8298 18081 31503 32617 40950(8247b) (8230) (18092) (31508) (32622) (40955)
02 6106 6124 15639 25631 27601 34458(6110b) (6125) (15642) (25637) (27617) (34462)
1 05
005 21575 31615 31644 42573 57889 58607(21580b) (31617) (31644) (42581) (57893) (58610)
01 8451 16024 16210 25969 43839 43867(8451b) (16025) (16217) (25972) (43844) (43870)
015 4682 11479 11480 20642 35172 35598(4684b) (11480) (11481) (20644) (35178) (35604)
02 3059 8621 8648 17320 28487 29638(3062b) (8619) (8644) (17327) (28492) (29641)
15 1
005 27713 40203 40978 52951 65352 66815(27714b) (40207) (40988) (52972) (65368) (66822)
01 11667 19322 19615 29122 46115 46183(11674b) (19327) (19619) (29137) (46120) (46188)
015 6544 13416 13481 22405 37238 37430(6547b) (13420) (13488) (22410) (37241) (37434)
02 4295 10175 10190 18543 30330 31056(4296b) (10177) (10194) (18544) (30336) (31060)
2 2
005 33453 50854 52088 66311 76610 78947(33455b) (50852) (50097) (66321) (76627) (78971)
01 15832 24130 24372 33924 49118 49362(15830b) (24127) (24374) (33931) (49147) (49382)
015 9060 15956 16099 24754 39134 39249(9064b) (15961) (16102) (24755) (39138) (39452)
02 5992 11955 11971 20034 31908 32392(5991b) (11957) (11978) (20042) (31910) (32398)
aΓ119909= 1198701198941198700 (1198700= 1 times 10
9 Nmrad 119894 = 1199090 119909119886 1199100 119910119887) Γ119908= 1198961198941198960 (1198960= 1 times 10
9 Nm 119894 = 1199090 119909119886 1199100 119910119887)bResults in parentheses are taken from FEM
moderately thick plate is investigated The plate is elasticallyrestrained in which the stiffness of the boundary springsis taken as 119896
1199090= 119896119909119886
= 1198961199100
= 119896119910119887
= Γ119908
= 2 and1198701199090
= 119870119909119886
= 1198701199100
= 119870119910119887
= Γ119909= 2 respectively The six
frequency parameters Ω = 1205961198862
(120588ℎ01198630)12 are presented in
Table 12 for several different slop values It can be seen thatthe fundamental frequency parameters will decrease with theincrease of the parameter ℎ
0119886
In the above examples it has been demonstrated thatthe presented method can be universally applied to nonlin-early varying thickness moderately thick plates with severalboundary conditions and different structure parametersNew results are obtained for plates with nonlinear variationthickness in both directions subjected to general elastic
boundary restraints which may be used for benchmarkingof researchers in the field In addition it is interesting to seethat the nature frequency decreases with the increase of theindex for thickness function
4 Conclusions
In this paper a modified Fourier method has been pre-sented to study the free vibration behaviors of moderatelythick rectangular plates with variable thickness and arbi-trary boundary conditionsThe first-order shear deformationplate theory is adopted to formulate the theoretical modelThe displacements and rotation components of the plateregardless of boundary conditions are invariantly expressedas the superposition of a 2D Fourier cosine series and four
20 Shock and Vibration
Table 12 Frequency parameterΩ = 1205961198862
(120588ℎ01198630)12 for elastic support moderately thick plates (119886119887 = 1) with different gradient (119896
1199090= 119896119909119886=
1198961199100= 119896119910119887= Γ119908= 2 119870
1199090= 119870119909119886= 1198701199100= 119870119910119887= Γ119909= 2)
120572 120573 ℎ0119886
Model sequence1 2 3 4 5 6
120572 = 120573 = 01
005 30187 51102 51141 67066 73954 77412(30188a) (51105) (51144) (67068) (73961) (77477)
01 16995 24719 24724 33071 44343 44819(16996a) (24725) (24729) (33084) (44347) (44822)
015 10077 15872 15883 23443 35937 36065(10081a) (15877) (15892) (23447) (35941) (36084)
02 6713 11953 11958 19086 30372 30604(6713a) (11957) (11965) (19092) (30378) (30612)
120572 = 120573 = 02
005 31206 51436 51614 67058 74398 77595(31208a) (51438) (51619) (67064) (74402) (77596)
01 16786 24503 24508 33144 45583 45923(16790a) (24510) (24513) (33147) (45587) (45932)
015 9821 15865 15906 23766 36897 36990(9824a) (15870) (15907) (23768) (36894) (36997)
02 6523 11978 11992 19356 30912 31162(6524a) (11981) (11997) (19362) (30916) (31174)
120572 = 120573 = 03
005 32097 51462 51900 66832 74991 77863(32098a) (51466) (51907) (66849) (74996) (77872)
01 16507 24332 24383 33330 46806 47074(16511a) (24327) (24380) (33339) (46810) (47086)
015 9562 15885 15964 24103 37748 37828(9562a) (15892) (15977) (21105) (37751) (37832)
02 6338 11992 12013 19603 31338 31634(6340a) (11994) (12011) (19612) (31344) (31639)
120572 = 120573 = 04
005 32851 51239 52046 66547 75735 78303(33455a) (50852) (50097) (66321) (76627) (78971)
01 16182 24209 24343 33598 47990 48230(33455a) (50852) (50097) (66321) (76627) (78971)
015 9307 15919 16034 24436 38491 38579(33455a) (50852) (50097) (66321) (76627) (78971)
02 6161 11986 12008 19828 31666 32040(33455a) (50852) (50097) (66321) (76627) (78971)
120572 = 120573 = 05
005 33453 50854 52088 66311 76610 78947(33455a) (50852) (50097) (66321) (76627) (78971)
01 15832 24130 24372 33924 49118 49362(15830a) (24127) (24374) (33931) (49147) (49382)
015 9060 15956 16099 24754 39134 39249(9064a) (15961) (16102) (24755) (39138) (39452)
02 5992 11955 11971 20034 31908 32392(5991a) (11957) (11978) (20042) (31910) (32398)
aResults in parentheses are taken from FEM
supplementary functions in the form of the product of apolynomial function and a single cosine series expansionto ensure and accelerate the convergence of the solutionAt each edge of the plate the general restraint conditionsare implemented by introducing one group of linear springsand two groups of rotational springs which are continuouslydistributed and determined by the stiffness of these springs
Instead of seeking a solution in strong forms in the previousstudies all the Fourier coefficients will be treated equallyand independently as the generalized coordinates and solveddirectly from the Rayleigh-Ritz technique The change of theboundary conditions can be easily achieved by only varyingthe stiffness of the three sets of boundary springs along alledges of the rectangular plates without involving any change
Shock and Vibration 21
to the solution procedure The convergence of the presentsolution is examined and the excellent accuracy is validatedby comparison with existing results published in the lit-erature and FEM data Excellent agreements are obtainedfrom these comparisons The proposed method providesa unified means for extracting the modal parameters andpredicting the vibration behaviors of moderately thick plateswith variable thickness variation functions and arbitraryelastic edge restraints A variety of free vibration results formoderately thick rectangular plates with different thicknessvariation functions and boundary conditions are presentedNew results for free vibration of moderately thick rectangularplates with various thickness variation functions and edgeconditions are presented which may be used for benchmark-ing of researchers in the field
Appendices
A Representative Calculation forStiffness and Mass Matrices
To illuminate the particular expression of the mass andstiffness matrixes clearly and tersely four new variables aredefined as 119904 = 119898 lowast (119873 + 1) + 119899 + 1 119905 = 119898
1015840
lowast (119873 + 1) + 1198991015840
+ 1119901 = (119897minus1)lowast (119872+1)+119898
1015840
+1 and 119902 = (119897minus1)lowast (119873+1)+1198991015840
+1The first row elements of K andM are given below
K1-1119904119905
=
1
2
int
119886
0
int
119887
0
[119863 (119909 119910)
times 21205821198861198981205821198861198981015840 sin 120582
119886119898119909 sin 120582
1198861198981015840119909 cos 120582
119887119899119910
times cos 1205821198871198991015840119910 + (1 minus 120583) 120582
1198871198991205821198871198991015840 cos 120582
119886119898119909
times cos 1205821198861198981015840 sin 120582
119887119899119910 sin 120582
1198871198991015840119910
+ 120581119866ℎ (119909 119910) cos 120582119886119898119909 cos 120582
1198861198981015840119909 cos 120582
119887119899
times119910 cos 1205821198871198991015840119910119889119909119889119910] 119889119909 119889119910
+ int
119886
0
1198701199091199100
cos 120582119886119898119909 cos 120582
1198861198981015840119909
+(minus1)119898+1198981015840
119870119909119910119887
cos 120582119886119898119909 cos 120582
1198861198981015840119909 119889119909
+ int
119887
0
1198701199090cos 120582119887119899119910 cos 120582
1198871198991015840119910
+(minus1)119899+1198991015840
119870119909119886cos 120582119887119899119910 cos 120582
1198871198991015840119910 119889119910
K1-2119904119901
=
1
2
int
119886
0
int
119887
0
[119863 (119909 119910)21205821198861198981205821198861198981015840 sin 120582
119886119898119909 sin 120582
1198861198981015840119909
times
2
sum
119897=1
120577119897
119887(119910) cos 120582
1198871198991015840119910
minus (1 minus 120583) 1205821198871198991015840 cos 120582
119886119898119909 cos 120582
1198861198981015840119909
times
2
sum
119897=1
1205771198971015840
119887(119910) sin 120582
1198871198991015840119910 + 120581119866ℎ (119909 119910)
times ( cos 1205821198861198981015840119909 cos 120582
1198871198991015840119910
times
2
sum
119897=1
120577119897
119887(119910) cos 120582
119886119898119909)]119889119909119889119910
+ int
119886
0
1198701199091199100
cos 120582119886119898119909 cos 120582
1198861198981015840119909
2
sum
119897=1
120577119897
119887(0)
+(minus1)1198991015840
119870119909119910119887
cos 120582119886119898119909 cos 120582
1198861198981015840119909
2
sum
119897=1
120577119897
119887(119887) 119889119909
+ int
119887
0
1198701199090
2
sum
119897=1
120577119897
119887(119910) cos 120582
1198871198991015840119910
+(minus1)119898+1198981015840
119870119909119886
2
sum
119897=1
120577119897
119887(119910) cos 120582
1198871198991015840119910119889119910
K1-3119904119902
=
1
2
int
119886
0
int
119887
0
[119863 (119909 119910) minus 21205821198861198981015840 sin 120582
1198861198981015840119909 cos 120582
1198871198991015840119910
times
2
sum
119897=1
1205771198971015840
119886(119909) cos 120582
119887119899119910
+ (1 minus 120583) 1205821198871198991205821198871198991015840 cos 120582
1198861198981015840119909
times sin 1205821198871198991015840119910
2
sum
119897=1
120577119897
119886(119909) sin 120582
119887119899119910
+ 120581119866ℎ (119909 119910) cos 1205821198861198981015840119909 cos 120582
1198871198991015840119910
times
2
sum
119897=1
120577119897
119886(119909) cos 120582
119887119899119910]119889119909119889119910
+ int
119886
0
1198701199091199100
cos 1205821198861198981015840119909
2
sum
119897=1
120577119897
119886(119909)
+(minus1)119899+1198991015840
119870119909119910119887
cos 1205821198861198981015840119909
2
sum
119897=1
120577119897
119886(119909) 119889119909
+ int
119887
0
1198701199090
2
sum
119897=1
120577119897
119886(0) cos 120582
1198871198991015840119910
+(minus1)119898+1198981015840
119870119909119886
2
sum
119897=1
120577119897
119886(0) cos 120582
1198871198991015840119910119889119910
K1-4119904119905
=
1
2
int
119886
0
int
119887
0
119863(119909 119910) 21205821198871198991205821198861198981015840 sin 120582
1198861198981015840119909 cos 120582
1198871198991015840119910
times cos 120582119886119898119909 sin 120582
119887119899119910 minus 2(1 minus 120583)
119887119899
times 1205821198861198981015840 sin 120582
1198861198981015840119909 cos 120582
1198871198991015840119910 cos 120582
119886119898119909
times sin 120582119887119899119910 + (1 minus 120583) 120582
1198861198981205821198871198991015840 cos 120582
1198861198981015840119909
times sin 1205821198871198991015840119910 sin 120582
119886119898119909 cos 120582
119887119899119910 119889119909 119889119910
22 Shock and Vibration
K1-5119904119901
=
1
2
int
119886
0
int
119887
0
119863(119909 119910) minus 21205821198861198981015840 sin 120582
1198861198981015840119909 cos 120582
1198871198991015840119910
times
2
sum
119897=1
1205771198971015840
119887(119910) cos 120582
119886119898119909
+ 2 (1 minus 120583) 1205821198861198981015840 sin 120582
1198861198981015840119909 cos 120582
1198871198991015840119910
times
2
sum
119897=1
1205771198971015840
119887(119910) cos 120582
119886119898119909
+ (1 minus 120583) 1205821198861198981205821198871198991015840 cos 120582
1198861198981015840119909 sin 120582
1198871198991015840119910
times
2
sum
119897=1
120577119897
119887(119910) sin 120582
119886119898119909119889119909119889119910
K1-6119904119902
=
1
2
int
119886
0
int
119887
0
119863(119909 119910)1205821198871198991205821198861198981015840 sin 120582
1198861198981015840119909 cos 120582
1198861198991015840119910
times
2
sum
119897=1
120577119897
119886(119909) sin 120582
1198871198991015840119910
minus 2 (1 minus 120583) 1205821198871198991205821198861198981015840 sin 120582
1198861198981015840119909 cos 120582
1198861198991015840119910
times
2
sum
119897=1
120577119897
119886(119909) sin 120582
1198871198991015840119910
+ 1205821198871198991015840 cos 120582
1198861198981015840119909 sin 120582
1198871198991015840119910
times
2
sum
119897=1
1205771198971015840
119886(119909) cos 120582
119887119899119910119889119909119889119910
K1-7119904119905
= minus
1
2
int
119886
0
int
119887
0
120581119866ℎ (119909 119910) 120582119886119898
cos 1205821198861198981015840119909 cos 120582
1198871198991015840119910
times sin 120582119886119898119909 cos 120582
119887119899119910119889119909119889119910
K1-8119904119901
= minus
1
2
int
119886
0
int
119887
0
120581119866ℎ (119909 119910) 1205821198861198981015840 cos 120582
1198861198981015840119909 cos 120582
1198871198991015840119910
times
2
sum
119897=1
120577119897
119887(119910) sin 120582
1198861198981015840119909 119889119909 119889119910
K1-9119904119902
=
1
2
int
119886
0
int
119887
0
120581119866ℎ (119909 119910) cos 1205821198861198981015840119909 cos 120582
1198871198991015840119910
times
2
sum
119897=1
1205771198971015840
119886(119909) cos 120582
119887119899119910119889119909119889119910
M1-1119904119905
=
1
24
120588int
119886
0
int
119887
0
ℎ3
(119909 119910) cos 120582119886119898119909 cos 120582
119887119899119910
times cos 1205821198861198981015840119909 cos 120582
1198871198991015840119910119889119909119889119910
M1-2119904119901
=
1
24
120588int
119886
0
int
119887
0
ℎ3
(119909 119910)
2
sum
119897=1
120577119897
119887(119910) cos 120582
119886119898119909
times cos 1205821198861198981015840119909 cos 120582
1198871198991015840119910119889119909119889119910
M1-3119904119902
=
1
24
120588int
119886
0
int
119887
0
ℎ3
(119909 119910)
2
sum
119897=1
120577119897
119886(119909) cos 120582
119887119899119910
times cos 1205821198861198981015840119909 cos 120582
1198871198991015840119910119889119909119889119910
M1-4119904119905 = M
1-5119904119901 = M1-6119904119902
= M1-7119904119905 = M
1-8119904119901 = M1-9119904119902 = 0
(A1)
B Useful Integrals for the Moderately ThickRectangular Plates with Variable Thicknessand Arbitrary Boundary Conditions
Consider
int
119871
0
cos 120582119898119909 cos 120582
119899119909 119889119909 =
0 119898 = 119899
119871
2
119898 = 119899
119871 119898 = 119899 = 0
int
119871
0
sin 120582119898119909 cos 120582
119899119909119889119909
=
minus
(minus1 + (minus1)119898+119899
) 119871119898
(1198982minus 1198992) 120587
119898 = 119899
0 119898 = 119899
int
119871
0
sin 120582119898119909 sin 120582
119899119909 119889119909 =
0 119898 = 119899
119871
2
119898 = 119899
0 119898 = 119899 = 0
int
119871
0
cos 120582119898119909120577119897
119886(119909) 119889119909
=
41198712
31205872 119897 = 1 119898 = 0
minus
41198712
(41198982
minus 3)
(9 minus 401198982+ 16119898
4) 1205872
119897 = 1 119898 = 0
minus
41198712
31205872 119897 = 2 119898 = 0
41198712
(41198982
minus 3) (minus1)119898
(9 minus 401198982+ 16119898
4) 1205872
119897 = 2 119898 = 0
Shock and Vibration 23
int
119871
0
sin 120582119898119909120577119897
119886(119909) 119889119909
=
0 119897 = 1 119898 = 0
161198712
119898(minus1)119898
(9 minus 401198982+ 16119898
4) 1205872
119897 = 1 119898 = 0
0 119897 = 2 119898 = 0
161198981198712
(9 minus 401198982+ 16119898
4) 1205872
119897 = 2 119898 = 0
int
119871
0
cos 1205821198981199091205771198971015840
119886(119909) 119889119909
=
0 119897 = 1 119898 = 0
161198711198982
(minus1)119898
(9 minus 401198982+ 16119898
4) 120587
119897 = 1 119898 = 0
0 119897 = 2 119898 = 0
161198711198982
(9 minus 401198982+ 16119898
4) 120587
119897 = 2 119898 = 0
int
119871
0
sin 1205821198981199091205771198971015840
119886(119909) 119889119909
=
0 119897 = 1 119898 = 0
4119871119898 (minus3 + 41198982
)
(9 minus 401198982+ 16119898
4) 120587
119897 = 1 119898 = 0
0 119897 = 2 119898 = 0
119871119898(
1
120587 minus 41198982120587
+
3
9120587 minus 41198982120587
) (minus1)119898
119897 = 2 119898 = 0
int
119871
0
1205771198971
119886(119909) 1205771198972
119886(119909) 119889119909 =
1198713
41205872 1198971= 1 1198972= 1
minus
21198713
31205873 1198971= 1 1198972= 2
minus
21198713
31205873 1198971= 2 1198972= 1
1198713
41205872 1198971= 2 1198972= 2
int
119871
0
1205771198971
119886(119909) 1205771198971015840
2
119886(119909) 119889119909 =
0 1198971= 1 1198972= 1
minus
1198712
8120587
1198971= 1 1198972= 2
1198712
8120587
1198971= 2 1198972= 1
0 1198971= 2 1198972= 2
int
119871
0
1205771198971015840
1
119886(119909) 1205771198971015840
2
119886(119909) 119889119909 =
5119871
16
1198971= 1 1198972= 1
minus
119871
2120587
1198971= 1 1198972= 2
minus
119871
2120587
1198971= 2 1198972= 1
5119871
16
1198971= 2 1198972= 2
int
119871
0
119909119899 cos 120582
119898119909 119889119909
=
1198711+119899
1 + 119899
119898 = 0
(1198861+119899HypergeometricPFQtimes [ 12 + 1198992 12 32 + 1198992
minus (14)1198982
1205872
]) times (1 + 119899)minus1
119898 gt 0
(B1)
Conflict of Interests
The authors declare that there is no conflict of interestsregarding the publication of this paper
Acknowledgments
The authors would like to thank the anonymous reviewersfor their very valuable comments The authors also gratefullyacknowledge the financial support from the National NaturalScience Foundation of China (no 51209052) HeilongjiangProvince Youth Science Fund Project (no QC2011C013) andHarbin Science and Technology Development InnovationFoundation of Youth (no 2011RFQXG021)
References
[1] K Liew Y Xiang and S Kitipornchai ldquoResearch on thick platevibration a literature surveyrdquo Journal of Sound and Vibrationvol 180 no 1 pp 163ndash176 1995
[2] J H Chung T Y Chung and K C Kim ldquoVibration analysisof orthotropic mindlin plates with edges elastically restrainedagainst rotationrdquo Journal of Sound and Vibration vol 163 no 1pp 151ndash163 1993
[3] Y K Cheung and D Zhou ldquoVibrations of moderately thickrectangular plates in terms of a set of static Timoshenko beamfunctionsrdquoComputers and Structures vol 78 no 6 pp 757ndash7682000
[4] CMWang ldquoNatural frequencies formula for simply supportedMindlin platesrdquo Journal of Vibration and Acoustics vol 116 no4 pp 536ndash540 1994
[5] K N Saha R C Kar and P K Datta ldquoFree vibration analysisof rectangular Mindlin plates with elastic restraints uniformlydistributed along the edgesrdquo Journal of Sound andVibration vol192 no 4 pp 885ndash902 1996
24 Shock and Vibration
[6] D J Gorman ldquoAccurate free vibration analysis of shear-deformable plates with torsional elastic edge supportrdquo Journalof Sound and Vibration vol 203 no 2 pp 209ndash218 1997
[7] D J Gorman ldquoFree vibration analysis of Mindlin plates withuniform elastic edge support by the superposition methodrdquoJournal of Sound andVibration vol 207 no 3 pp 335ndash350 1997
[8] D J Gorman ldquoAccurate free vibration analysis of point sup-ported mindlin plates by the superposition methodrdquo Journal ofSound and Vibration vol 219 no 2 pp 265ndash277 1999
[9] Y Xiang ldquoVibration of rectangular Mindlin plates resting onnon-homogenous elastic foundationsrdquo International Journal ofMechanical Sciences vol 45 no 6-7 pp 1229ndash1244 2003
[10] Y Xiang and G W Wei ldquoExact solutions for buckling andvibration of stepped rectangular Mindlin platesrdquo InternationalJournal of Solids and Structures vol 41 no 1 pp 279ndash294 2004
[11] Y L Yeh M J Jang and C C Wang ldquoAnalyzing the freevibrations of a plate using finite difference and differential trans-formationmethodrdquoAppliedMathematics andComputation vol178 no 2 pp 493ndash501 2006
[12] Y Xiang S K Lai L Zhou and C W Lim ldquoDSC-Ritz elementmethod for vibration analysis of rectangular Mindlin plateswith mixed edge supportsrdquo European Journal of MechanicsmdashASolids vol 29 no 4 pp 619ndash628 2010
[13] H Nguyen-Xuan C H Thai and T Nguyen-Thoi ldquoIsogeo-metric finite element analysis of composite sandwich platesusing a higher order shear deformation theoryrdquo Composites BEngineering vol 55 pp 558ndash574 2013
[14] C H Thai A J M Ferreira E Carrera and H Nguyen-XuanldquoIsogeometric analysis of laminated composite and sandwichplates using a layerwise deformation theoryrdquo Composite Struc-tures vol 104 pp 196ndash214 2013
[15] C H Thai A Ferreira S Bordas T Rabczuk and HNguyen-Xuan ldquoIsogeometric analysis of laminated compositeand sandwich plates using a new inverse trigonometric sheardeformation theoryrdquo European Journal of Mechanics A Solidsvol 43 pp 89ndash108 2014
[16] H Luong-Van T Nguyen-Thoi G R Liu and P Phung-VanldquoA cell-based smoothed finite elementmethod using three-nodeshear-locking free Mindlin plate element (CS-FEM-MIN3) fordynamic response of laminated composite plates on viscoelasticfoundationrdquo Engineering Analysis with Boundary Elements vol42 pp 8ndash19 2014
[17] T Nguyen-Thoi T Rabczuk T Lam-Phat V Ho-Huu andP Phung-Van ldquoFree vibration analysis of cracked Mindlinplate using an extended cell-based smoothed discrete sheargap method (XCS-DSG3)rdquo Theoretical and Applied FractureMechanics 2014
[18] H Nguyen-Xuan G R Liu and C a Thai-Hoang ldquoAnedge-based smoothed finite element method (ES-FEM) withstabilized discrete shear gap technique for analysis of Reissner-Mindlin platesrdquo Computer Methods in Applied Mechanics andEngineering vol 199 no 9ndash12 pp 471ndash489 2010
[19] P Phung-Van T Nguyen-Thoi T Bui-Xuan and Q Lieu-XuanldquoA cell-based smoothed three-node Mindlin plate element(CS-FEM-MIN3) based on the 119862
0 -type higher-order sheardeformation for geometrically nonlinear analysis of laminatedcomposite platesrdquo Computational Materials Science 2014
[20] NNguyen-Thanh T RabczukHNguyen-Xuan and S BordasldquoAn alternative alpha finite element method with discrete sheargap technique for analysis of isotropicMindlin-Reissner platesrdquoFinite Elements inAnalysis andDesign vol 47 no 5 pp 519ndash5352011
[21] KM Liew Y Xiang and S Kitipornchai ldquoTransverse vibrationof thick rectangular plates-I Comprehensive sets of boundaryconditionsrdquo Computers and Structures vol 49 no 1 pp 1ndash291993
[22] D Zhou ldquoVibrations of Mindlin rectangular plates with elasti-cally restrained edges using static Timoshenko beam functionswith the Rayleigh-Ritz methodrdquo International Journal of Solidsand Structures vol 38 no 32-33 pp 5565ndash5580 2001
[23] D Zhou SH Lo F T KAu andYK Cheung ldquoVibration anal-ysis of rectangular Mindlin plates with internal line supportsusing static Timoshenko beam functionsrdquo International Journalof Mechanical Sciences vol 44 no 12 pp 2503ndash2522 2002
[24] H S Shen J Yang and L Zhang ldquoFree and forced vibrationof Reissner-Mindlin plates with free edges resting on elasticfoundationsrdquo Journal of Sound and Vibration vol 244 no 2 pp299ndash320 2001
[25] Y Xing and B Liu ldquoClosed form solutions for free vibrations ofrectangular Mindlin platesrdquo Acta Mechanica Sinica vol 25 no5 pp 689ndash698 2009
[26] P Gagnon C Gosselin and L Cloutier ldquoA finite strip elementfor the analysis of variable thickness rectangular thick platesrdquoComputers amp Structures vol 63 no 2 pp 349ndash362 1997
[27] T Mlzusawa ldquoVibration of rectangular mindlin plates withtapered thickness by the spline strip methodrdquo Computers andStructures vol 46 no 3 pp 451ndash463 1993
[28] H Nguyen-Xuan and T Nguyen-Thoi ldquoA stabilized smoothedfinite element method for free vibration analysis of Mindlin-Reissner platesrdquo Communications in Numerical Methods inEngineering with Biomedical Applications vol 25 no 8 pp 882ndash906 2009
[29] K M Liew L X Peng and S Kitipornchai ldquoVibration anal-ysis of corrugated Reissner-Mindlin plates using a mesh-freeGalerkin methodrdquo International Journal of Mechanical Sciencesvol 51 no 9-10 pp 642ndash652 2009
[30] Y Hou GWWei and Y Xiang ldquoDSC-Ritz method for the freevibration analysis of Mindlin platesrdquo International Journal forNumerical Methods in Engineering vol 62 no 2 pp 262ndash2882005
[31] F- Liu and K M Liew ldquoVibration analysis of discontinuousMindlin plates by differential quadrature element methodrdquoJournal of Vibration and Acoustics Transactions of the ASMEvol 121 no 2 pp 204ndash208 1999
[32] P Malekzadeh G Karami and M Farid ldquoA semi-analyticalDQEM for free vibration analysis of thick plates with twoopposite edges simply supportedrdquoComputerMethods inAppliedMechanics and Engineering vol 193 no 45ndash47 pp 4781ndash47962004
[33] P Malekzadeh and S A Shahpari ldquoFree vibration analysisof variable thickness thin and moderately thick plates withelastically restrained edges by DQMrdquo Thin-Walled Structuresvol 43 no 7 pp 1037ndash1050 2005
[34] R E Diaz-Contreras and S Nomura ldquoGreenrsquos function appliedto solution ofMindlin platesrdquoComputers and Structures vol 60no 1 pp 41ndash48 1996
[35] T Sakiyama and M Huang ldquoFree vibration analysis of rect-angular plates with variable thicknessrdquo Journal of Sound andVibration vol 216 no 3 pp 379ndash397 1998
[36] M Huang X Q Ma T Sakiyama H Matuda and C MoritaldquoFree vibration analysis of orthotropic rectangular plates withvariable thickness and general boundary conditionsrdquo Journal ofSound and Vibration vol 288 no 4-5 pp 931ndash955 2005
Shock and Vibration 25
[37] I Shufrin and M Eisenberger ldquoVibration of shear deformableplates with variable thicknessmdashfirst-order and higher-orderanalysesrdquo Journal of Sound and Vibration vol 290 no 1-2 pp465ndash489 2006
[38] S A Eftekhari and A A Jafari ldquoAccurate variational approachfor free vibration of variable thickness thin and thick plates withedges elastically restrained against translation and rotationrdquoInternational Journal of Mechanical Sciences vol 68 pp 35ndash462013
[39] W L Li X Zhang J Du and Z Liu ldquoAn exact series solutionfor the transverse vibration of rectangular plates with generalelastic boundary supportsrdquo Journal of Sound amp Vibration vol321 no 1-2 pp 254ndash269 2009
[40] X Shi D Shi W L Li and Q Wang ldquoA unified method forfree vibration analysis of circular annular and sector plateswith arbitrary boundary conditionsrdquo Journal of Vibration andControl 2014
[41] D Y Shi X J Shi W L Li and Q S Wang ldquoFree transversevibrations of orthotropic thin rectangular plates with arbitraryelastic edge supportsrdquo Journal of Vibroengineering vol 16 no 1pp 389ndash398 2014
[42] Y Chen G Jin and Z Liu ldquoFree vibration analysis of circu-lar cylindrical shell with non-uniform elastic boundary con-straintsrdquo International Journal of Mechanical Sciences vol 74pp 120ndash132 2013
[43] L Dai T Yang J Du W L Li and M J Brennan ldquoAn exactseries solution for the vibration analysis of cylindrical shellswith arbitrary boundary conditionsrdquo Applied Acoustics vol 74no 3 pp 440ndash449 2013
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Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
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Shock and Vibration
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
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Chemical EngineeringInternational Journal of Antennas and
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DistributedSensor Networks
International Journal of
Shock and Vibration 15
Table 7 Frequency parameterΩ = 1205961198862
(120588ℎ01198630)12 for moderately thick plates (119886119887 = 1) with linearly varying thickness and elastic rotation
and translation support (ℎ(119909 119910) = ℎ0(1 + 05119909)(1 + 05119910) 119870
1199090= 119870119909119886= 1198701199100= 119870119910119887= Γ119909 and 119896
1199090= 119896119909119886= 1198961199100= 119896119910119887= Γ119908)
Γ119909
aΓ119908
aℎ0119886
Model sequence1 2 3 4 5 6
05 0
005 14751 14917 28232 55251 55397 69512(14780b) (14822) (29252) (55476) (55505) (70904)
01 11811 11910 23857 44462 45037 55453(11818b) (11897) (24711) (44376) (45257) (56511)
015 8487 8514 20011 34783 36335 44970(8500b) (8544) (20738) (34404) (36663) (45625)
02 6073 6073 17431 28679 30805 38010(6096b) (6106) (18039) (28129) (31165) (38311)
1 05
005 21225 32173 32173 44381 64061 65222(20708b) (32097) (32285) (45110) (64182) (65355)
01 8140 16878 16941 28100 48663 49015(7943b) (16838) (16970) (28888) (48700) (49171)
015 4493 11859 11878 22429 38352 39247(4383b) (11867) (11891) (23125) (38177) (39488)
02 2932 8690 8691 18919 31108 32600(3540b) (8791) (8791) (16229) (27520) (27889)
15 1
005 27819 40919 41430 54595 70792 72896(27146b) (41008) (41434) (55256) (70895) (73051)
01 11285 20068 20230 31101 51046 51429(11012b) (20025) (20270) (31849) (51126) (51569)
015 6292 13912 13922 24132 40500 41120(6139b) (13912) (13934) (24808) (40426) (41316)
02 4121 10321 10331 20051 32796 33921(4021b) (10329) (10357) (20635) (32540) (34178)
2 2
005 34615 51972 53088 68255 81344 84510(33806b) (52100) (53157) (68891) (81433) (84761)
01 15416 24659 24829 35615 53879 54494(15045b) (24625) (24888) (36303) (53970) (54626)
015 8740 16420 16481 26353 42417 42946(8529b) (16414) (16499) (26996) (42398) (43114)
02 5761 12127 12127 21434 34286 35192(5622b) (12134) (12148) (21998) (34110) (35415)
aΓ119909= 1198701198941198700 (1198700= 1 times 10
9 Nmrad 119894 = 1199090 119909119886 1199100 119910119887) Γ119908= 1198961198941198960 (1198960= 1 times 10
9 Nm 119894 = 1199090 119909119886 1199100 119910119887)bResults in parentheses are taken from FEM
moderately thick plate with not only classical boundaryconditions but also elastic edge restraints
As the last case of this subsection the influence of thegradient 120572 and 120573 on the fundamental frequency parametersfor a linearly varying thickness moderately thick plate isinvestigated The model is a square moderately thick platewith all edges elastically restrainedThat is 119896
1199090= 119896119909119886= 1198961199100=
119896119910119887= Γ119908= 2 and 119870
1199090= 119870119909119886
= 1198701199100= 119870119910119887= Γ119909= 2 The six
frequency parameters Ω = 1205961198862
(120588ℎ01198630)12 are given in
Table 8 for several different 120572 and 120573 values the finite elementmethod (FEM) results are also listed as a reference Againgood agreement can be observed Through Table 8 it is alsofound that the frequency parameter increases with increasinggradient parameters
The above studies are given as linearly varying thicknessmoderately thick plates with several of boundary condition
and different structure parameters In the next section vibra-tion results for the plates subjected to nonlinear variationthickness will be presented
34 Nonlinearly Varying Thickness Moderately Thick Plateswith Classical and Elastic Boundary Conditions In Sec-tion 33 the linearly varying thickness moderately thickplates were studied However in the practical engineeringapplications the varying thickness of amoderately thick platemay not always be linear variation in nature A variety ofpossible thickness varying cases may be encountered in prac-tice Therefore the moderately thick plates with nonlinearvariation thickness subjected to general elastic edge restraintsare examined in this subsection For the sake of brevity theindexes 119904 and 119905 will be chosen as 2 to imitate the nonlinearlyvarying thickness moderately thick plates structure in this
16 Shock and Vibration
Table 8 Frequency parameterΩ = 1205961198862
(120588ℎ01198630)12 for elastic support moderately thick plates (119886119887 = 1) with different gradient (119896
1199090= 119896119909119886=
1198961199100= 119896119910119887= Γ119908= 2 119870
1199090= 119870119909119886= 1198701199100= 119870119910119887= Γ119909= 2)
120572 120573 ℎ0119886
Model sequence1 2 3 4 5 6
120572 = 120573 = 01
005 31275 52615 52657 68954 76146 79850(31303a) (52664) (52675) (69033) (76174) (79948)
01 17270 25244 25250 33949 46050 46625(17271a) (25255) (25258) (34089) (46122) (46708)
015 10190 16266 16274 24164 37327 37526(10189a) (16269) (16280) (24323) (37401) (37622)
02 6780 12258 12260 19685 31452 31754(6780a) (12265) (12270) (19851) (31514) (31864)
120572 = 120573 = 02
005 32530 52947 53141 68954 77034 80589(32577a) (53014) (53203) (69172) (77088) (80773)
01 16882 24979 24979 34154 48057 48575(16883a) (24983) (25001) (34433) (48134) (48667)
015 9804 16284 16309 24683 38844 39085(9803a) (16283) (16322) (24980) (38909) (39191)
02 6499 12290 12295 20133 32354 32770(6498a) (12296) (12307) (20420) (32387) (32895)
120572 = 120573 = 03
005 33516 52845 53308 68704 78203 81561(33563a) (52939) (53389) (69055) (78271) (81789)
01 16419 24799 24830 34527 50061 50583(16419a) (24790) (24869) (34947) (50143) (50688)
015 9430 16334 16378 25236 40193 40506(9429a) (16329) (16395) (25660) (40243) (40625)
02 6235 12288 12293 20572 33102 33663(6235a) (12296) (12306) (20965) (33090) (33811)
120572 = 120573 = 04
005 34215 52463 53264 68430 79654 82865(34252a) (52575) (53346) (68921) (79732) (83113)
01 15921 24695 24793 35027 52011 52571(15921a) (24674) (24842) (35583) (52098) (52688)
015 9074 16388 16444 25797 41379 41789(9074a) (16381) (16464) (26336) (41404) (41928)
02 5990 12236 12238 21002 33735 34461(5989a) (12249) (12250) (21487) (33653) (34641)
120572 = 120573 = 05
005 34615 51972 53088 68255 81344 84510(33806a) (52100) (53157) (68891) (81433) (84761)
01 15416 24659 24829 35615 53879 54494(15045a) (24625) (24888) (36303) (53970) (54626)
015 8740 16420 16481 26353 42417 42946(8529a) (16414) (16499) (26996) (42398) (43114)
02 5761 12127 12127 21434 34286 35192(5622a) (12134) (12148) (21998) (34110) (35415)
aResults in parentheses are taken from FEM
section that is 119904 = 119905 = 2 Also the thickness varying functionsare also be expressed as Fourier cosine series according to (18)and (19)
In order to validate the accuracy and reliability of theproposed method for predicting the vibration behavior ofnonlinearly varying thickness moderately thick plates with
arbitrary boundary conditions the typical classical boundaryconditions viewed as the special cases of elastically restrainededges will be considered The comparison of the first six fre-quency parameters Ω = 120596119886
2
(120588ℎ01198630)12 for the moderately
thick plates with nonlinearly varying thickness is presentedin Table 9 The S-S-S-S C-F-F-F S-S-F-F C-C-C-C and
Shock and Vibration 17
Table 9 Frequency parameter Ω = 1205961198862
(120588ℎ01198630)12 for moderately thick plates with linearly varying thickness in different boundary
conditions (ℎ(119909 119910) = ℎ0(1 + 05119909
2
)(1 + 051199102
))
Boundary condition 119886119887 ℎ0119886
Model sequence1 2 3 4 5 6
S-S-S-S
101 23462 54999 55573 82950 100900 101187
(23741a) (55114) (55983) (83146) (101004) (101347)
02 18606 42725 42971 60040 71167 71761(18936a) (43014) (43148) (60247) (71642) (71993)
3201 53055 65251 85404 106700 114880 143531
(53467a) (65472) (85984) (107211) (115687) (144007)
02 17439 21531 32170 44987 59438 64179(17874a) (21748) (32651) (45264) (60188) (64879)
C-F-F-F
101 3658 9317 24135 32816 35433 59179
(3749a) (9547) (24034) (33627) (36308) (60142)
02 3520 7986 19591 26959 27491 43882(3611a) (8176) (20075) (27125) (27625) (43984)
3201 3901 9963 28984 69187 73686 89360
(3930a) (10095) (29381) (69578) (73994) (90242)
02 3292 4439 13385 23947 40850 44758(3386a) (4498) (14012) (24068) (41221) (45142)
S-S-F-F
101 3557 21268 22762 43524 60253 62429
(3649a) (21441) (22796) (43916) (60622) (62630)
02 3171 17822 19512 33616 45042 46833(3489a) (18006) (20430) (33815) (45247) (49004)
3201 6738 31658 52071 75625 86604 94903
(6957a) (31882) (52518) (75823) (86939) (95610)
02 3653 12930 20955 26517 44097 48058(6807a) (13160) (21312) (26864) (44422) (48873)
C-C-C-C
101 41020 75196 75614 104142 119683 120949
(41405a) (75452) (75882) (104711) (119637) (121137)
02 31237 51686 51868 68411 76868 77864(32010a) (52960) (52145) (69335) (77481) (78099)
3201 95823 128727 163657 173751 190890 223595
(95364a) (128932) (163588) (172350) (190350) (222948)
02 53306 71372 89384 102404 104660 120908(53587a) (71759) (90103) (102503) (105174) (121275)
S-C-S-C
101 33715 60298 71446 94187 103802 117928
(34148a) (60786) (71828) (94515) (104263) (119326)
02 26143 45499 49458 64253 72941 76191(26789a) (46622) (50680) (64329) (73474) (76843)
3201 73228 115845 137445 148166 168342 182542
(73761a) (116121) (137788) (148530) (168698) (182921)
02 41302 53648 65630 72202 81632 88105(41304a) (53757) (65842) (72794) (82033) (88474)
aResults in parentheses are taken from FEM
S-C-S-C boundary conditions are performed in the compar-ison The results adequately demonstrated the great accuracyof the modified Fourier method
We now turn to elastically restrained moderately thickplates The first one involves an S-S-S-S moderately thicksquare plate with a uniform rotational restraint along eachedge that is 119870
1199090= 119870119909119886
= 1198701199100
= 119870119910119887
= Γ119909 The calculated
frequency parameters Ω = 1205961198862
(120588ℎ01198630)12 are given in
Table 10 together with the FEM results Since the elasticallyrestrained plates with nonlinear variation thickness are rarelyinvestigated the FEMresults are used as the referenceA goodagreement is observed between the current and FEM resultsThe second example concerns amoderately thick square plateelastically supported along all edgesThe stiffness of the linear
18 Shock and Vibration
Table 10 Frequency parameterΩ = 1205961198862
(120588ℎ01198630)12 for S-S-S-S moderately thick plates (119886119887 = 1) with linearly varying thickness and elastic
rotation support (ℎ(119909 119910) = ℎ0(1 + 05119909
2
)(1 + 051199102
) 1198701199090= 119870119909119886= 1198701199100= 119870119910119887= Γ119909)
Γ119909
aℎ0119886
Model sequence1 2 3 4 5 6
0
005 23462 54999 55573 82950 100900 101187(23741b) (55114) (55983) (83146) (101004) (101347)
01 18606 42725 42971 60040 71167 71761(18936b) (43014) (43148) (60247) (71642) (71993)
015 53055 65251 85404 106700 114880 143531(53467b) (65472) (85984) (107211) (115687) (144007)
02 17439 21531 32170 44987 59438 64179(17874b) (21748) (32651) (45264) (60188) (64879)
05
005 6904 13374 13514 19400 22904 23056(9909b) (13548) (13651) (19564) (23088) (23379)
01 5316 10237 10314 14499 17155 17264(5374b) (10448) (10651) (14764) (17188) (17879)
015 4136 8201 8220 11474 13578 13671(4174b) (8348) (8651) (11764) (13688) (13879)
02 3449 6915 6924 9549 11225 11327(3474b) (6948) (6951) (9764) (11388) (11579)
1
005 7040 13617 13745 19726 23233 23402(7074b) (13748) (13851) (19964) (23488) (23879)
01 5712 10696 10793 14991 17594 17744(5774b) (10748) (10851) (15054) (17788) (17879)
015 4489 8507 8540 11759 13803 13922(4514b) (8548) (8611) (11864) (13988) (14119)
02 3677 7084 7090 9692 11330 11445(3684b) (7148) (7191) (9764) (11588) (11879)
15
005 7089 13708 13828 19847 23355 23530(7124b) (13748) (13951) (19967) (23588) (23879)
01 5895 10923 11024 15239 17813 17988(5874b) (10748) (11651) (15264) (17818) (17979)
015 4697 8694 8737 11938 13790 14083(4774b) (8748) (8951) (11864) (13688) (14879)
02 3836 7201 7210 9792 11403 11530(3874b) (7748) (7651) (9864) (11588) (11879)
2
005 7114 13754 13871 19911 23418 23597(7174b) (12848) (13851) (19864) (23588) (23879)
01 6001 11060 11159 15391 17944 18136(6004b) (11148) (11151) (15464) (17988) (18379)
015 4836 8821 8872 12061 14037 14189(4874b) (8848) (8951) (12164) (14188) (14879)
02 3954 7289 7301 9871 11457 11594(3974b) (7348) (7651) (9964) (11488) (11679)
aΓ119909= 1198701198941198700 (1198700= 1 times 10
9 Nmrad 119894 = 1199090 119909119886 1199100 119910119887)bResults in parentheses are taken from FEM
and rotational restraints is set equal to 1198961199090
= 119896119909119886
= 1198961199100
=
119896119910119887
= Γ119908and 119870
1199090= 119870119909119886
= 1198701199100
= 119870119910119887
= Γ119909 respectively
The six frequency parametersΩ = 1205961198862
(120588ℎ01198630)12 are shown
in Table 11 for several different restraining coefficient valuesIt can also be noticed that the Fourier series method is able
to predict the modal characteristics of nonlinearly varyingthickness moderately thick plate with not only classicalboundary conditions but also elastic edge restraints correctly
Finally the influence of the gradient on the fundamentalfrequency parameters of a nonlinearly varying thickness
Shock and Vibration 19
Table 11 Frequency parameterΩ = 1205961198862
(120588ℎ01198630)12 for moderately thick plates (119886119887 = 1) with linearly varying thickness and elastic rotation
and translation support (ℎ(119909 119910) = ℎ0(1 + 05119909
2
)(1 + 051199102
) 1198701199090= 119870119909119886= 1198701199100= 119870119910119887= Γ119909 and 119896
1199090= 119896119909119886= 1198961199100= 119896119910119887= Γ119908)
Γ119909
aΓ119908
aℎ0119886
Model sequence1 2 3 4 5 6
05 0
005 12926 13041 25090 47935 47943 61761(12931b) (13053) 25099 (47947) (47952) (61788)
01 10900 10988 21545 39980 40147 50322(10904b) (10992) (21557) (39984) (40166) (50334)
015 8245 8298 18081 31503 32617 40950(8247b) (8230) (18092) (31508) (32622) (40955)
02 6106 6124 15639 25631 27601 34458(6110b) (6125) (15642) (25637) (27617) (34462)
1 05
005 21575 31615 31644 42573 57889 58607(21580b) (31617) (31644) (42581) (57893) (58610)
01 8451 16024 16210 25969 43839 43867(8451b) (16025) (16217) (25972) (43844) (43870)
015 4682 11479 11480 20642 35172 35598(4684b) (11480) (11481) (20644) (35178) (35604)
02 3059 8621 8648 17320 28487 29638(3062b) (8619) (8644) (17327) (28492) (29641)
15 1
005 27713 40203 40978 52951 65352 66815(27714b) (40207) (40988) (52972) (65368) (66822)
01 11667 19322 19615 29122 46115 46183(11674b) (19327) (19619) (29137) (46120) (46188)
015 6544 13416 13481 22405 37238 37430(6547b) (13420) (13488) (22410) (37241) (37434)
02 4295 10175 10190 18543 30330 31056(4296b) (10177) (10194) (18544) (30336) (31060)
2 2
005 33453 50854 52088 66311 76610 78947(33455b) (50852) (50097) (66321) (76627) (78971)
01 15832 24130 24372 33924 49118 49362(15830b) (24127) (24374) (33931) (49147) (49382)
015 9060 15956 16099 24754 39134 39249(9064b) (15961) (16102) (24755) (39138) (39452)
02 5992 11955 11971 20034 31908 32392(5991b) (11957) (11978) (20042) (31910) (32398)
aΓ119909= 1198701198941198700 (1198700= 1 times 10
9 Nmrad 119894 = 1199090 119909119886 1199100 119910119887) Γ119908= 1198961198941198960 (1198960= 1 times 10
9 Nm 119894 = 1199090 119909119886 1199100 119910119887)bResults in parentheses are taken from FEM
moderately thick plate is investigated The plate is elasticallyrestrained in which the stiffness of the boundary springsis taken as 119896
1199090= 119896119909119886
= 1198961199100
= 119896119910119887
= Γ119908
= 2 and1198701199090
= 119870119909119886
= 1198701199100
= 119870119910119887
= Γ119909= 2 respectively The six
frequency parameters Ω = 1205961198862
(120588ℎ01198630)12 are presented in
Table 12 for several different slop values It can be seen thatthe fundamental frequency parameters will decrease with theincrease of the parameter ℎ
0119886
In the above examples it has been demonstrated thatthe presented method can be universally applied to nonlin-early varying thickness moderately thick plates with severalboundary conditions and different structure parametersNew results are obtained for plates with nonlinear variationthickness in both directions subjected to general elastic
boundary restraints which may be used for benchmarkingof researchers in the field In addition it is interesting to seethat the nature frequency decreases with the increase of theindex for thickness function
4 Conclusions
In this paper a modified Fourier method has been pre-sented to study the free vibration behaviors of moderatelythick rectangular plates with variable thickness and arbi-trary boundary conditionsThe first-order shear deformationplate theory is adopted to formulate the theoretical modelThe displacements and rotation components of the plateregardless of boundary conditions are invariantly expressedas the superposition of a 2D Fourier cosine series and four
20 Shock and Vibration
Table 12 Frequency parameterΩ = 1205961198862
(120588ℎ01198630)12 for elastic support moderately thick plates (119886119887 = 1) with different gradient (119896
1199090= 119896119909119886=
1198961199100= 119896119910119887= Γ119908= 2 119870
1199090= 119870119909119886= 1198701199100= 119870119910119887= Γ119909= 2)
120572 120573 ℎ0119886
Model sequence1 2 3 4 5 6
120572 = 120573 = 01
005 30187 51102 51141 67066 73954 77412(30188a) (51105) (51144) (67068) (73961) (77477)
01 16995 24719 24724 33071 44343 44819(16996a) (24725) (24729) (33084) (44347) (44822)
015 10077 15872 15883 23443 35937 36065(10081a) (15877) (15892) (23447) (35941) (36084)
02 6713 11953 11958 19086 30372 30604(6713a) (11957) (11965) (19092) (30378) (30612)
120572 = 120573 = 02
005 31206 51436 51614 67058 74398 77595(31208a) (51438) (51619) (67064) (74402) (77596)
01 16786 24503 24508 33144 45583 45923(16790a) (24510) (24513) (33147) (45587) (45932)
015 9821 15865 15906 23766 36897 36990(9824a) (15870) (15907) (23768) (36894) (36997)
02 6523 11978 11992 19356 30912 31162(6524a) (11981) (11997) (19362) (30916) (31174)
120572 = 120573 = 03
005 32097 51462 51900 66832 74991 77863(32098a) (51466) (51907) (66849) (74996) (77872)
01 16507 24332 24383 33330 46806 47074(16511a) (24327) (24380) (33339) (46810) (47086)
015 9562 15885 15964 24103 37748 37828(9562a) (15892) (15977) (21105) (37751) (37832)
02 6338 11992 12013 19603 31338 31634(6340a) (11994) (12011) (19612) (31344) (31639)
120572 = 120573 = 04
005 32851 51239 52046 66547 75735 78303(33455a) (50852) (50097) (66321) (76627) (78971)
01 16182 24209 24343 33598 47990 48230(33455a) (50852) (50097) (66321) (76627) (78971)
015 9307 15919 16034 24436 38491 38579(33455a) (50852) (50097) (66321) (76627) (78971)
02 6161 11986 12008 19828 31666 32040(33455a) (50852) (50097) (66321) (76627) (78971)
120572 = 120573 = 05
005 33453 50854 52088 66311 76610 78947(33455a) (50852) (50097) (66321) (76627) (78971)
01 15832 24130 24372 33924 49118 49362(15830a) (24127) (24374) (33931) (49147) (49382)
015 9060 15956 16099 24754 39134 39249(9064a) (15961) (16102) (24755) (39138) (39452)
02 5992 11955 11971 20034 31908 32392(5991a) (11957) (11978) (20042) (31910) (32398)
aResults in parentheses are taken from FEM
supplementary functions in the form of the product of apolynomial function and a single cosine series expansionto ensure and accelerate the convergence of the solutionAt each edge of the plate the general restraint conditionsare implemented by introducing one group of linear springsand two groups of rotational springs which are continuouslydistributed and determined by the stiffness of these springs
Instead of seeking a solution in strong forms in the previousstudies all the Fourier coefficients will be treated equallyand independently as the generalized coordinates and solveddirectly from the Rayleigh-Ritz technique The change of theboundary conditions can be easily achieved by only varyingthe stiffness of the three sets of boundary springs along alledges of the rectangular plates without involving any change
Shock and Vibration 21
to the solution procedure The convergence of the presentsolution is examined and the excellent accuracy is validatedby comparison with existing results published in the lit-erature and FEM data Excellent agreements are obtainedfrom these comparisons The proposed method providesa unified means for extracting the modal parameters andpredicting the vibration behaviors of moderately thick plateswith variable thickness variation functions and arbitraryelastic edge restraints A variety of free vibration results formoderately thick rectangular plates with different thicknessvariation functions and boundary conditions are presentedNew results for free vibration of moderately thick rectangularplates with various thickness variation functions and edgeconditions are presented which may be used for benchmark-ing of researchers in the field
Appendices
A Representative Calculation forStiffness and Mass Matrices
To illuminate the particular expression of the mass andstiffness matrixes clearly and tersely four new variables aredefined as 119904 = 119898 lowast (119873 + 1) + 119899 + 1 119905 = 119898
1015840
lowast (119873 + 1) + 1198991015840
+ 1119901 = (119897minus1)lowast (119872+1)+119898
1015840
+1 and 119902 = (119897minus1)lowast (119873+1)+1198991015840
+1The first row elements of K andM are given below
K1-1119904119905
=
1
2
int
119886
0
int
119887
0
[119863 (119909 119910)
times 21205821198861198981205821198861198981015840 sin 120582
119886119898119909 sin 120582
1198861198981015840119909 cos 120582
119887119899119910
times cos 1205821198871198991015840119910 + (1 minus 120583) 120582
1198871198991205821198871198991015840 cos 120582
119886119898119909
times cos 1205821198861198981015840 sin 120582
119887119899119910 sin 120582
1198871198991015840119910
+ 120581119866ℎ (119909 119910) cos 120582119886119898119909 cos 120582
1198861198981015840119909 cos 120582
119887119899
times119910 cos 1205821198871198991015840119910119889119909119889119910] 119889119909 119889119910
+ int
119886
0
1198701199091199100
cos 120582119886119898119909 cos 120582
1198861198981015840119909
+(minus1)119898+1198981015840
119870119909119910119887
cos 120582119886119898119909 cos 120582
1198861198981015840119909 119889119909
+ int
119887
0
1198701199090cos 120582119887119899119910 cos 120582
1198871198991015840119910
+(minus1)119899+1198991015840
119870119909119886cos 120582119887119899119910 cos 120582
1198871198991015840119910 119889119910
K1-2119904119901
=
1
2
int
119886
0
int
119887
0
[119863 (119909 119910)21205821198861198981205821198861198981015840 sin 120582
119886119898119909 sin 120582
1198861198981015840119909
times
2
sum
119897=1
120577119897
119887(119910) cos 120582
1198871198991015840119910
minus (1 minus 120583) 1205821198871198991015840 cos 120582
119886119898119909 cos 120582
1198861198981015840119909
times
2
sum
119897=1
1205771198971015840
119887(119910) sin 120582
1198871198991015840119910 + 120581119866ℎ (119909 119910)
times ( cos 1205821198861198981015840119909 cos 120582
1198871198991015840119910
times
2
sum
119897=1
120577119897
119887(119910) cos 120582
119886119898119909)]119889119909119889119910
+ int
119886
0
1198701199091199100
cos 120582119886119898119909 cos 120582
1198861198981015840119909
2
sum
119897=1
120577119897
119887(0)
+(minus1)1198991015840
119870119909119910119887
cos 120582119886119898119909 cos 120582
1198861198981015840119909
2
sum
119897=1
120577119897
119887(119887) 119889119909
+ int
119887
0
1198701199090
2
sum
119897=1
120577119897
119887(119910) cos 120582
1198871198991015840119910
+(minus1)119898+1198981015840
119870119909119886
2
sum
119897=1
120577119897
119887(119910) cos 120582
1198871198991015840119910119889119910
K1-3119904119902
=
1
2
int
119886
0
int
119887
0
[119863 (119909 119910) minus 21205821198861198981015840 sin 120582
1198861198981015840119909 cos 120582
1198871198991015840119910
times
2
sum
119897=1
1205771198971015840
119886(119909) cos 120582
119887119899119910
+ (1 minus 120583) 1205821198871198991205821198871198991015840 cos 120582
1198861198981015840119909
times sin 1205821198871198991015840119910
2
sum
119897=1
120577119897
119886(119909) sin 120582
119887119899119910
+ 120581119866ℎ (119909 119910) cos 1205821198861198981015840119909 cos 120582
1198871198991015840119910
times
2
sum
119897=1
120577119897
119886(119909) cos 120582
119887119899119910]119889119909119889119910
+ int
119886
0
1198701199091199100
cos 1205821198861198981015840119909
2
sum
119897=1
120577119897
119886(119909)
+(minus1)119899+1198991015840
119870119909119910119887
cos 1205821198861198981015840119909
2
sum
119897=1
120577119897
119886(119909) 119889119909
+ int
119887
0
1198701199090
2
sum
119897=1
120577119897
119886(0) cos 120582
1198871198991015840119910
+(minus1)119898+1198981015840
119870119909119886
2
sum
119897=1
120577119897
119886(0) cos 120582
1198871198991015840119910119889119910
K1-4119904119905
=
1
2
int
119886
0
int
119887
0
119863(119909 119910) 21205821198871198991205821198861198981015840 sin 120582
1198861198981015840119909 cos 120582
1198871198991015840119910
times cos 120582119886119898119909 sin 120582
119887119899119910 minus 2(1 minus 120583)
119887119899
times 1205821198861198981015840 sin 120582
1198861198981015840119909 cos 120582
1198871198991015840119910 cos 120582
119886119898119909
times sin 120582119887119899119910 + (1 minus 120583) 120582
1198861198981205821198871198991015840 cos 120582
1198861198981015840119909
times sin 1205821198871198991015840119910 sin 120582
119886119898119909 cos 120582
119887119899119910 119889119909 119889119910
22 Shock and Vibration
K1-5119904119901
=
1
2
int
119886
0
int
119887
0
119863(119909 119910) minus 21205821198861198981015840 sin 120582
1198861198981015840119909 cos 120582
1198871198991015840119910
times
2
sum
119897=1
1205771198971015840
119887(119910) cos 120582
119886119898119909
+ 2 (1 minus 120583) 1205821198861198981015840 sin 120582
1198861198981015840119909 cos 120582
1198871198991015840119910
times
2
sum
119897=1
1205771198971015840
119887(119910) cos 120582
119886119898119909
+ (1 minus 120583) 1205821198861198981205821198871198991015840 cos 120582
1198861198981015840119909 sin 120582
1198871198991015840119910
times
2
sum
119897=1
120577119897
119887(119910) sin 120582
119886119898119909119889119909119889119910
K1-6119904119902
=
1
2
int
119886
0
int
119887
0
119863(119909 119910)1205821198871198991205821198861198981015840 sin 120582
1198861198981015840119909 cos 120582
1198861198991015840119910
times
2
sum
119897=1
120577119897
119886(119909) sin 120582
1198871198991015840119910
minus 2 (1 minus 120583) 1205821198871198991205821198861198981015840 sin 120582
1198861198981015840119909 cos 120582
1198861198991015840119910
times
2
sum
119897=1
120577119897
119886(119909) sin 120582
1198871198991015840119910
+ 1205821198871198991015840 cos 120582
1198861198981015840119909 sin 120582
1198871198991015840119910
times
2
sum
119897=1
1205771198971015840
119886(119909) cos 120582
119887119899119910119889119909119889119910
K1-7119904119905
= minus
1
2
int
119886
0
int
119887
0
120581119866ℎ (119909 119910) 120582119886119898
cos 1205821198861198981015840119909 cos 120582
1198871198991015840119910
times sin 120582119886119898119909 cos 120582
119887119899119910119889119909119889119910
K1-8119904119901
= minus
1
2
int
119886
0
int
119887
0
120581119866ℎ (119909 119910) 1205821198861198981015840 cos 120582
1198861198981015840119909 cos 120582
1198871198991015840119910
times
2
sum
119897=1
120577119897
119887(119910) sin 120582
1198861198981015840119909 119889119909 119889119910
K1-9119904119902
=
1
2
int
119886
0
int
119887
0
120581119866ℎ (119909 119910) cos 1205821198861198981015840119909 cos 120582
1198871198991015840119910
times
2
sum
119897=1
1205771198971015840
119886(119909) cos 120582
119887119899119910119889119909119889119910
M1-1119904119905
=
1
24
120588int
119886
0
int
119887
0
ℎ3
(119909 119910) cos 120582119886119898119909 cos 120582
119887119899119910
times cos 1205821198861198981015840119909 cos 120582
1198871198991015840119910119889119909119889119910
M1-2119904119901
=
1
24
120588int
119886
0
int
119887
0
ℎ3
(119909 119910)
2
sum
119897=1
120577119897
119887(119910) cos 120582
119886119898119909
times cos 1205821198861198981015840119909 cos 120582
1198871198991015840119910119889119909119889119910
M1-3119904119902
=
1
24
120588int
119886
0
int
119887
0
ℎ3
(119909 119910)
2
sum
119897=1
120577119897
119886(119909) cos 120582
119887119899119910
times cos 1205821198861198981015840119909 cos 120582
1198871198991015840119910119889119909119889119910
M1-4119904119905 = M
1-5119904119901 = M1-6119904119902
= M1-7119904119905 = M
1-8119904119901 = M1-9119904119902 = 0
(A1)
B Useful Integrals for the Moderately ThickRectangular Plates with Variable Thicknessand Arbitrary Boundary Conditions
Consider
int
119871
0
cos 120582119898119909 cos 120582
119899119909 119889119909 =
0 119898 = 119899
119871
2
119898 = 119899
119871 119898 = 119899 = 0
int
119871
0
sin 120582119898119909 cos 120582
119899119909119889119909
=
minus
(minus1 + (minus1)119898+119899
) 119871119898
(1198982minus 1198992) 120587
119898 = 119899
0 119898 = 119899
int
119871
0
sin 120582119898119909 sin 120582
119899119909 119889119909 =
0 119898 = 119899
119871
2
119898 = 119899
0 119898 = 119899 = 0
int
119871
0
cos 120582119898119909120577119897
119886(119909) 119889119909
=
41198712
31205872 119897 = 1 119898 = 0
minus
41198712
(41198982
minus 3)
(9 minus 401198982+ 16119898
4) 1205872
119897 = 1 119898 = 0
minus
41198712
31205872 119897 = 2 119898 = 0
41198712
(41198982
minus 3) (minus1)119898
(9 minus 401198982+ 16119898
4) 1205872
119897 = 2 119898 = 0
Shock and Vibration 23
int
119871
0
sin 120582119898119909120577119897
119886(119909) 119889119909
=
0 119897 = 1 119898 = 0
161198712
119898(minus1)119898
(9 minus 401198982+ 16119898
4) 1205872
119897 = 1 119898 = 0
0 119897 = 2 119898 = 0
161198981198712
(9 minus 401198982+ 16119898
4) 1205872
119897 = 2 119898 = 0
int
119871
0
cos 1205821198981199091205771198971015840
119886(119909) 119889119909
=
0 119897 = 1 119898 = 0
161198711198982
(minus1)119898
(9 minus 401198982+ 16119898
4) 120587
119897 = 1 119898 = 0
0 119897 = 2 119898 = 0
161198711198982
(9 minus 401198982+ 16119898
4) 120587
119897 = 2 119898 = 0
int
119871
0
sin 1205821198981199091205771198971015840
119886(119909) 119889119909
=
0 119897 = 1 119898 = 0
4119871119898 (minus3 + 41198982
)
(9 minus 401198982+ 16119898
4) 120587
119897 = 1 119898 = 0
0 119897 = 2 119898 = 0
119871119898(
1
120587 minus 41198982120587
+
3
9120587 minus 41198982120587
) (minus1)119898
119897 = 2 119898 = 0
int
119871
0
1205771198971
119886(119909) 1205771198972
119886(119909) 119889119909 =
1198713
41205872 1198971= 1 1198972= 1
minus
21198713
31205873 1198971= 1 1198972= 2
minus
21198713
31205873 1198971= 2 1198972= 1
1198713
41205872 1198971= 2 1198972= 2
int
119871
0
1205771198971
119886(119909) 1205771198971015840
2
119886(119909) 119889119909 =
0 1198971= 1 1198972= 1
minus
1198712
8120587
1198971= 1 1198972= 2
1198712
8120587
1198971= 2 1198972= 1
0 1198971= 2 1198972= 2
int
119871
0
1205771198971015840
1
119886(119909) 1205771198971015840
2
119886(119909) 119889119909 =
5119871
16
1198971= 1 1198972= 1
minus
119871
2120587
1198971= 1 1198972= 2
minus
119871
2120587
1198971= 2 1198972= 1
5119871
16
1198971= 2 1198972= 2
int
119871
0
119909119899 cos 120582
119898119909 119889119909
=
1198711+119899
1 + 119899
119898 = 0
(1198861+119899HypergeometricPFQtimes [ 12 + 1198992 12 32 + 1198992
minus (14)1198982
1205872
]) times (1 + 119899)minus1
119898 gt 0
(B1)
Conflict of Interests
The authors declare that there is no conflict of interestsregarding the publication of this paper
Acknowledgments
The authors would like to thank the anonymous reviewersfor their very valuable comments The authors also gratefullyacknowledge the financial support from the National NaturalScience Foundation of China (no 51209052) HeilongjiangProvince Youth Science Fund Project (no QC2011C013) andHarbin Science and Technology Development InnovationFoundation of Youth (no 2011RFQXG021)
References
[1] K Liew Y Xiang and S Kitipornchai ldquoResearch on thick platevibration a literature surveyrdquo Journal of Sound and Vibrationvol 180 no 1 pp 163ndash176 1995
[2] J H Chung T Y Chung and K C Kim ldquoVibration analysisof orthotropic mindlin plates with edges elastically restrainedagainst rotationrdquo Journal of Sound and Vibration vol 163 no 1pp 151ndash163 1993
[3] Y K Cheung and D Zhou ldquoVibrations of moderately thickrectangular plates in terms of a set of static Timoshenko beamfunctionsrdquoComputers and Structures vol 78 no 6 pp 757ndash7682000
[4] CMWang ldquoNatural frequencies formula for simply supportedMindlin platesrdquo Journal of Vibration and Acoustics vol 116 no4 pp 536ndash540 1994
[5] K N Saha R C Kar and P K Datta ldquoFree vibration analysisof rectangular Mindlin plates with elastic restraints uniformlydistributed along the edgesrdquo Journal of Sound andVibration vol192 no 4 pp 885ndash902 1996
24 Shock and Vibration
[6] D J Gorman ldquoAccurate free vibration analysis of shear-deformable plates with torsional elastic edge supportrdquo Journalof Sound and Vibration vol 203 no 2 pp 209ndash218 1997
[7] D J Gorman ldquoFree vibration analysis of Mindlin plates withuniform elastic edge support by the superposition methodrdquoJournal of Sound andVibration vol 207 no 3 pp 335ndash350 1997
[8] D J Gorman ldquoAccurate free vibration analysis of point sup-ported mindlin plates by the superposition methodrdquo Journal ofSound and Vibration vol 219 no 2 pp 265ndash277 1999
[9] Y Xiang ldquoVibration of rectangular Mindlin plates resting onnon-homogenous elastic foundationsrdquo International Journal ofMechanical Sciences vol 45 no 6-7 pp 1229ndash1244 2003
[10] Y Xiang and G W Wei ldquoExact solutions for buckling andvibration of stepped rectangular Mindlin platesrdquo InternationalJournal of Solids and Structures vol 41 no 1 pp 279ndash294 2004
[11] Y L Yeh M J Jang and C C Wang ldquoAnalyzing the freevibrations of a plate using finite difference and differential trans-formationmethodrdquoAppliedMathematics andComputation vol178 no 2 pp 493ndash501 2006
[12] Y Xiang S K Lai L Zhou and C W Lim ldquoDSC-Ritz elementmethod for vibration analysis of rectangular Mindlin plateswith mixed edge supportsrdquo European Journal of MechanicsmdashASolids vol 29 no 4 pp 619ndash628 2010
[13] H Nguyen-Xuan C H Thai and T Nguyen-Thoi ldquoIsogeo-metric finite element analysis of composite sandwich platesusing a higher order shear deformation theoryrdquo Composites BEngineering vol 55 pp 558ndash574 2013
[14] C H Thai A J M Ferreira E Carrera and H Nguyen-XuanldquoIsogeometric analysis of laminated composite and sandwichplates using a layerwise deformation theoryrdquo Composite Struc-tures vol 104 pp 196ndash214 2013
[15] C H Thai A Ferreira S Bordas T Rabczuk and HNguyen-Xuan ldquoIsogeometric analysis of laminated compositeand sandwich plates using a new inverse trigonometric sheardeformation theoryrdquo European Journal of Mechanics A Solidsvol 43 pp 89ndash108 2014
[16] H Luong-Van T Nguyen-Thoi G R Liu and P Phung-VanldquoA cell-based smoothed finite elementmethod using three-nodeshear-locking free Mindlin plate element (CS-FEM-MIN3) fordynamic response of laminated composite plates on viscoelasticfoundationrdquo Engineering Analysis with Boundary Elements vol42 pp 8ndash19 2014
[17] T Nguyen-Thoi T Rabczuk T Lam-Phat V Ho-Huu andP Phung-Van ldquoFree vibration analysis of cracked Mindlinplate using an extended cell-based smoothed discrete sheargap method (XCS-DSG3)rdquo Theoretical and Applied FractureMechanics 2014
[18] H Nguyen-Xuan G R Liu and C a Thai-Hoang ldquoAnedge-based smoothed finite element method (ES-FEM) withstabilized discrete shear gap technique for analysis of Reissner-Mindlin platesrdquo Computer Methods in Applied Mechanics andEngineering vol 199 no 9ndash12 pp 471ndash489 2010
[19] P Phung-Van T Nguyen-Thoi T Bui-Xuan and Q Lieu-XuanldquoA cell-based smoothed three-node Mindlin plate element(CS-FEM-MIN3) based on the 119862
0 -type higher-order sheardeformation for geometrically nonlinear analysis of laminatedcomposite platesrdquo Computational Materials Science 2014
[20] NNguyen-Thanh T RabczukHNguyen-Xuan and S BordasldquoAn alternative alpha finite element method with discrete sheargap technique for analysis of isotropicMindlin-Reissner platesrdquoFinite Elements inAnalysis andDesign vol 47 no 5 pp 519ndash5352011
[21] KM Liew Y Xiang and S Kitipornchai ldquoTransverse vibrationof thick rectangular plates-I Comprehensive sets of boundaryconditionsrdquo Computers and Structures vol 49 no 1 pp 1ndash291993
[22] D Zhou ldquoVibrations of Mindlin rectangular plates with elasti-cally restrained edges using static Timoshenko beam functionswith the Rayleigh-Ritz methodrdquo International Journal of Solidsand Structures vol 38 no 32-33 pp 5565ndash5580 2001
[23] D Zhou SH Lo F T KAu andYK Cheung ldquoVibration anal-ysis of rectangular Mindlin plates with internal line supportsusing static Timoshenko beam functionsrdquo International Journalof Mechanical Sciences vol 44 no 12 pp 2503ndash2522 2002
[24] H S Shen J Yang and L Zhang ldquoFree and forced vibrationof Reissner-Mindlin plates with free edges resting on elasticfoundationsrdquo Journal of Sound and Vibration vol 244 no 2 pp299ndash320 2001
[25] Y Xing and B Liu ldquoClosed form solutions for free vibrations ofrectangular Mindlin platesrdquo Acta Mechanica Sinica vol 25 no5 pp 689ndash698 2009
[26] P Gagnon C Gosselin and L Cloutier ldquoA finite strip elementfor the analysis of variable thickness rectangular thick platesrdquoComputers amp Structures vol 63 no 2 pp 349ndash362 1997
[27] T Mlzusawa ldquoVibration of rectangular mindlin plates withtapered thickness by the spline strip methodrdquo Computers andStructures vol 46 no 3 pp 451ndash463 1993
[28] H Nguyen-Xuan and T Nguyen-Thoi ldquoA stabilized smoothedfinite element method for free vibration analysis of Mindlin-Reissner platesrdquo Communications in Numerical Methods inEngineering with Biomedical Applications vol 25 no 8 pp 882ndash906 2009
[29] K M Liew L X Peng and S Kitipornchai ldquoVibration anal-ysis of corrugated Reissner-Mindlin plates using a mesh-freeGalerkin methodrdquo International Journal of Mechanical Sciencesvol 51 no 9-10 pp 642ndash652 2009
[30] Y Hou GWWei and Y Xiang ldquoDSC-Ritz method for the freevibration analysis of Mindlin platesrdquo International Journal forNumerical Methods in Engineering vol 62 no 2 pp 262ndash2882005
[31] F- Liu and K M Liew ldquoVibration analysis of discontinuousMindlin plates by differential quadrature element methodrdquoJournal of Vibration and Acoustics Transactions of the ASMEvol 121 no 2 pp 204ndash208 1999
[32] P Malekzadeh G Karami and M Farid ldquoA semi-analyticalDQEM for free vibration analysis of thick plates with twoopposite edges simply supportedrdquoComputerMethods inAppliedMechanics and Engineering vol 193 no 45ndash47 pp 4781ndash47962004
[33] P Malekzadeh and S A Shahpari ldquoFree vibration analysisof variable thickness thin and moderately thick plates withelastically restrained edges by DQMrdquo Thin-Walled Structuresvol 43 no 7 pp 1037ndash1050 2005
[34] R E Diaz-Contreras and S Nomura ldquoGreenrsquos function appliedto solution ofMindlin platesrdquoComputers and Structures vol 60no 1 pp 41ndash48 1996
[35] T Sakiyama and M Huang ldquoFree vibration analysis of rect-angular plates with variable thicknessrdquo Journal of Sound andVibration vol 216 no 3 pp 379ndash397 1998
[36] M Huang X Q Ma T Sakiyama H Matuda and C MoritaldquoFree vibration analysis of orthotropic rectangular plates withvariable thickness and general boundary conditionsrdquo Journal ofSound and Vibration vol 288 no 4-5 pp 931ndash955 2005
Shock and Vibration 25
[37] I Shufrin and M Eisenberger ldquoVibration of shear deformableplates with variable thicknessmdashfirst-order and higher-orderanalysesrdquo Journal of Sound and Vibration vol 290 no 1-2 pp465ndash489 2006
[38] S A Eftekhari and A A Jafari ldquoAccurate variational approachfor free vibration of variable thickness thin and thick plates withedges elastically restrained against translation and rotationrdquoInternational Journal of Mechanical Sciences vol 68 pp 35ndash462013
[39] W L Li X Zhang J Du and Z Liu ldquoAn exact series solutionfor the transverse vibration of rectangular plates with generalelastic boundary supportsrdquo Journal of Sound amp Vibration vol321 no 1-2 pp 254ndash269 2009
[40] X Shi D Shi W L Li and Q Wang ldquoA unified method forfree vibration analysis of circular annular and sector plateswith arbitrary boundary conditionsrdquo Journal of Vibration andControl 2014
[41] D Y Shi X J Shi W L Li and Q S Wang ldquoFree transversevibrations of orthotropic thin rectangular plates with arbitraryelastic edge supportsrdquo Journal of Vibroengineering vol 16 no 1pp 389ndash398 2014
[42] Y Chen G Jin and Z Liu ldquoFree vibration analysis of circu-lar cylindrical shell with non-uniform elastic boundary con-straintsrdquo International Journal of Mechanical Sciences vol 74pp 120ndash132 2013
[43] L Dai T Yang J Du W L Li and M J Brennan ldquoAn exactseries solution for the vibration analysis of cylindrical shellswith arbitrary boundary conditionsrdquo Applied Acoustics vol 74no 3 pp 440ndash449 2013
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Shock and Vibration
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DistributedSensor Networks
International Journal of
16 Shock and Vibration
Table 8 Frequency parameterΩ = 1205961198862
(120588ℎ01198630)12 for elastic support moderately thick plates (119886119887 = 1) with different gradient (119896
1199090= 119896119909119886=
1198961199100= 119896119910119887= Γ119908= 2 119870
1199090= 119870119909119886= 1198701199100= 119870119910119887= Γ119909= 2)
120572 120573 ℎ0119886
Model sequence1 2 3 4 5 6
120572 = 120573 = 01
005 31275 52615 52657 68954 76146 79850(31303a) (52664) (52675) (69033) (76174) (79948)
01 17270 25244 25250 33949 46050 46625(17271a) (25255) (25258) (34089) (46122) (46708)
015 10190 16266 16274 24164 37327 37526(10189a) (16269) (16280) (24323) (37401) (37622)
02 6780 12258 12260 19685 31452 31754(6780a) (12265) (12270) (19851) (31514) (31864)
120572 = 120573 = 02
005 32530 52947 53141 68954 77034 80589(32577a) (53014) (53203) (69172) (77088) (80773)
01 16882 24979 24979 34154 48057 48575(16883a) (24983) (25001) (34433) (48134) (48667)
015 9804 16284 16309 24683 38844 39085(9803a) (16283) (16322) (24980) (38909) (39191)
02 6499 12290 12295 20133 32354 32770(6498a) (12296) (12307) (20420) (32387) (32895)
120572 = 120573 = 03
005 33516 52845 53308 68704 78203 81561(33563a) (52939) (53389) (69055) (78271) (81789)
01 16419 24799 24830 34527 50061 50583(16419a) (24790) (24869) (34947) (50143) (50688)
015 9430 16334 16378 25236 40193 40506(9429a) (16329) (16395) (25660) (40243) (40625)
02 6235 12288 12293 20572 33102 33663(6235a) (12296) (12306) (20965) (33090) (33811)
120572 = 120573 = 04
005 34215 52463 53264 68430 79654 82865(34252a) (52575) (53346) (68921) (79732) (83113)
01 15921 24695 24793 35027 52011 52571(15921a) (24674) (24842) (35583) (52098) (52688)
015 9074 16388 16444 25797 41379 41789(9074a) (16381) (16464) (26336) (41404) (41928)
02 5990 12236 12238 21002 33735 34461(5989a) (12249) (12250) (21487) (33653) (34641)
120572 = 120573 = 05
005 34615 51972 53088 68255 81344 84510(33806a) (52100) (53157) (68891) (81433) (84761)
01 15416 24659 24829 35615 53879 54494(15045a) (24625) (24888) (36303) (53970) (54626)
015 8740 16420 16481 26353 42417 42946(8529a) (16414) (16499) (26996) (42398) (43114)
02 5761 12127 12127 21434 34286 35192(5622a) (12134) (12148) (21998) (34110) (35415)
aResults in parentheses are taken from FEM
section that is 119904 = 119905 = 2 Also the thickness varying functionsare also be expressed as Fourier cosine series according to (18)and (19)
In order to validate the accuracy and reliability of theproposed method for predicting the vibration behavior ofnonlinearly varying thickness moderately thick plates with
arbitrary boundary conditions the typical classical boundaryconditions viewed as the special cases of elastically restrainededges will be considered The comparison of the first six fre-quency parameters Ω = 120596119886
2
(120588ℎ01198630)12 for the moderately
thick plates with nonlinearly varying thickness is presentedin Table 9 The S-S-S-S C-F-F-F S-S-F-F C-C-C-C and
Shock and Vibration 17
Table 9 Frequency parameter Ω = 1205961198862
(120588ℎ01198630)12 for moderately thick plates with linearly varying thickness in different boundary
conditions (ℎ(119909 119910) = ℎ0(1 + 05119909
2
)(1 + 051199102
))
Boundary condition 119886119887 ℎ0119886
Model sequence1 2 3 4 5 6
S-S-S-S
101 23462 54999 55573 82950 100900 101187
(23741a) (55114) (55983) (83146) (101004) (101347)
02 18606 42725 42971 60040 71167 71761(18936a) (43014) (43148) (60247) (71642) (71993)
3201 53055 65251 85404 106700 114880 143531
(53467a) (65472) (85984) (107211) (115687) (144007)
02 17439 21531 32170 44987 59438 64179(17874a) (21748) (32651) (45264) (60188) (64879)
C-F-F-F
101 3658 9317 24135 32816 35433 59179
(3749a) (9547) (24034) (33627) (36308) (60142)
02 3520 7986 19591 26959 27491 43882(3611a) (8176) (20075) (27125) (27625) (43984)
3201 3901 9963 28984 69187 73686 89360
(3930a) (10095) (29381) (69578) (73994) (90242)
02 3292 4439 13385 23947 40850 44758(3386a) (4498) (14012) (24068) (41221) (45142)
S-S-F-F
101 3557 21268 22762 43524 60253 62429
(3649a) (21441) (22796) (43916) (60622) (62630)
02 3171 17822 19512 33616 45042 46833(3489a) (18006) (20430) (33815) (45247) (49004)
3201 6738 31658 52071 75625 86604 94903
(6957a) (31882) (52518) (75823) (86939) (95610)
02 3653 12930 20955 26517 44097 48058(6807a) (13160) (21312) (26864) (44422) (48873)
C-C-C-C
101 41020 75196 75614 104142 119683 120949
(41405a) (75452) (75882) (104711) (119637) (121137)
02 31237 51686 51868 68411 76868 77864(32010a) (52960) (52145) (69335) (77481) (78099)
3201 95823 128727 163657 173751 190890 223595
(95364a) (128932) (163588) (172350) (190350) (222948)
02 53306 71372 89384 102404 104660 120908(53587a) (71759) (90103) (102503) (105174) (121275)
S-C-S-C
101 33715 60298 71446 94187 103802 117928
(34148a) (60786) (71828) (94515) (104263) (119326)
02 26143 45499 49458 64253 72941 76191(26789a) (46622) (50680) (64329) (73474) (76843)
3201 73228 115845 137445 148166 168342 182542
(73761a) (116121) (137788) (148530) (168698) (182921)
02 41302 53648 65630 72202 81632 88105(41304a) (53757) (65842) (72794) (82033) (88474)
aResults in parentheses are taken from FEM
S-C-S-C boundary conditions are performed in the compar-ison The results adequately demonstrated the great accuracyof the modified Fourier method
We now turn to elastically restrained moderately thickplates The first one involves an S-S-S-S moderately thicksquare plate with a uniform rotational restraint along eachedge that is 119870
1199090= 119870119909119886
= 1198701199100
= 119870119910119887
= Γ119909 The calculated
frequency parameters Ω = 1205961198862
(120588ℎ01198630)12 are given in
Table 10 together with the FEM results Since the elasticallyrestrained plates with nonlinear variation thickness are rarelyinvestigated the FEMresults are used as the referenceA goodagreement is observed between the current and FEM resultsThe second example concerns amoderately thick square plateelastically supported along all edgesThe stiffness of the linear
18 Shock and Vibration
Table 10 Frequency parameterΩ = 1205961198862
(120588ℎ01198630)12 for S-S-S-S moderately thick plates (119886119887 = 1) with linearly varying thickness and elastic
rotation support (ℎ(119909 119910) = ℎ0(1 + 05119909
2
)(1 + 051199102
) 1198701199090= 119870119909119886= 1198701199100= 119870119910119887= Γ119909)
Γ119909
aℎ0119886
Model sequence1 2 3 4 5 6
0
005 23462 54999 55573 82950 100900 101187(23741b) (55114) (55983) (83146) (101004) (101347)
01 18606 42725 42971 60040 71167 71761(18936b) (43014) (43148) (60247) (71642) (71993)
015 53055 65251 85404 106700 114880 143531(53467b) (65472) (85984) (107211) (115687) (144007)
02 17439 21531 32170 44987 59438 64179(17874b) (21748) (32651) (45264) (60188) (64879)
05
005 6904 13374 13514 19400 22904 23056(9909b) (13548) (13651) (19564) (23088) (23379)
01 5316 10237 10314 14499 17155 17264(5374b) (10448) (10651) (14764) (17188) (17879)
015 4136 8201 8220 11474 13578 13671(4174b) (8348) (8651) (11764) (13688) (13879)
02 3449 6915 6924 9549 11225 11327(3474b) (6948) (6951) (9764) (11388) (11579)
1
005 7040 13617 13745 19726 23233 23402(7074b) (13748) (13851) (19964) (23488) (23879)
01 5712 10696 10793 14991 17594 17744(5774b) (10748) (10851) (15054) (17788) (17879)
015 4489 8507 8540 11759 13803 13922(4514b) (8548) (8611) (11864) (13988) (14119)
02 3677 7084 7090 9692 11330 11445(3684b) (7148) (7191) (9764) (11588) (11879)
15
005 7089 13708 13828 19847 23355 23530(7124b) (13748) (13951) (19967) (23588) (23879)
01 5895 10923 11024 15239 17813 17988(5874b) (10748) (11651) (15264) (17818) (17979)
015 4697 8694 8737 11938 13790 14083(4774b) (8748) (8951) (11864) (13688) (14879)
02 3836 7201 7210 9792 11403 11530(3874b) (7748) (7651) (9864) (11588) (11879)
2
005 7114 13754 13871 19911 23418 23597(7174b) (12848) (13851) (19864) (23588) (23879)
01 6001 11060 11159 15391 17944 18136(6004b) (11148) (11151) (15464) (17988) (18379)
015 4836 8821 8872 12061 14037 14189(4874b) (8848) (8951) (12164) (14188) (14879)
02 3954 7289 7301 9871 11457 11594(3974b) (7348) (7651) (9964) (11488) (11679)
aΓ119909= 1198701198941198700 (1198700= 1 times 10
9 Nmrad 119894 = 1199090 119909119886 1199100 119910119887)bResults in parentheses are taken from FEM
and rotational restraints is set equal to 1198961199090
= 119896119909119886
= 1198961199100
=
119896119910119887
= Γ119908and 119870
1199090= 119870119909119886
= 1198701199100
= 119870119910119887
= Γ119909 respectively
The six frequency parametersΩ = 1205961198862
(120588ℎ01198630)12 are shown
in Table 11 for several different restraining coefficient valuesIt can also be noticed that the Fourier series method is able
to predict the modal characteristics of nonlinearly varyingthickness moderately thick plate with not only classicalboundary conditions but also elastic edge restraints correctly
Finally the influence of the gradient on the fundamentalfrequency parameters of a nonlinearly varying thickness
Shock and Vibration 19
Table 11 Frequency parameterΩ = 1205961198862
(120588ℎ01198630)12 for moderately thick plates (119886119887 = 1) with linearly varying thickness and elastic rotation
and translation support (ℎ(119909 119910) = ℎ0(1 + 05119909
2
)(1 + 051199102
) 1198701199090= 119870119909119886= 1198701199100= 119870119910119887= Γ119909 and 119896
1199090= 119896119909119886= 1198961199100= 119896119910119887= Γ119908)
Γ119909
aΓ119908
aℎ0119886
Model sequence1 2 3 4 5 6
05 0
005 12926 13041 25090 47935 47943 61761(12931b) (13053) 25099 (47947) (47952) (61788)
01 10900 10988 21545 39980 40147 50322(10904b) (10992) (21557) (39984) (40166) (50334)
015 8245 8298 18081 31503 32617 40950(8247b) (8230) (18092) (31508) (32622) (40955)
02 6106 6124 15639 25631 27601 34458(6110b) (6125) (15642) (25637) (27617) (34462)
1 05
005 21575 31615 31644 42573 57889 58607(21580b) (31617) (31644) (42581) (57893) (58610)
01 8451 16024 16210 25969 43839 43867(8451b) (16025) (16217) (25972) (43844) (43870)
015 4682 11479 11480 20642 35172 35598(4684b) (11480) (11481) (20644) (35178) (35604)
02 3059 8621 8648 17320 28487 29638(3062b) (8619) (8644) (17327) (28492) (29641)
15 1
005 27713 40203 40978 52951 65352 66815(27714b) (40207) (40988) (52972) (65368) (66822)
01 11667 19322 19615 29122 46115 46183(11674b) (19327) (19619) (29137) (46120) (46188)
015 6544 13416 13481 22405 37238 37430(6547b) (13420) (13488) (22410) (37241) (37434)
02 4295 10175 10190 18543 30330 31056(4296b) (10177) (10194) (18544) (30336) (31060)
2 2
005 33453 50854 52088 66311 76610 78947(33455b) (50852) (50097) (66321) (76627) (78971)
01 15832 24130 24372 33924 49118 49362(15830b) (24127) (24374) (33931) (49147) (49382)
015 9060 15956 16099 24754 39134 39249(9064b) (15961) (16102) (24755) (39138) (39452)
02 5992 11955 11971 20034 31908 32392(5991b) (11957) (11978) (20042) (31910) (32398)
aΓ119909= 1198701198941198700 (1198700= 1 times 10
9 Nmrad 119894 = 1199090 119909119886 1199100 119910119887) Γ119908= 1198961198941198960 (1198960= 1 times 10
9 Nm 119894 = 1199090 119909119886 1199100 119910119887)bResults in parentheses are taken from FEM
moderately thick plate is investigated The plate is elasticallyrestrained in which the stiffness of the boundary springsis taken as 119896
1199090= 119896119909119886
= 1198961199100
= 119896119910119887
= Γ119908
= 2 and1198701199090
= 119870119909119886
= 1198701199100
= 119870119910119887
= Γ119909= 2 respectively The six
frequency parameters Ω = 1205961198862
(120588ℎ01198630)12 are presented in
Table 12 for several different slop values It can be seen thatthe fundamental frequency parameters will decrease with theincrease of the parameter ℎ
0119886
In the above examples it has been demonstrated thatthe presented method can be universally applied to nonlin-early varying thickness moderately thick plates with severalboundary conditions and different structure parametersNew results are obtained for plates with nonlinear variationthickness in both directions subjected to general elastic
boundary restraints which may be used for benchmarkingof researchers in the field In addition it is interesting to seethat the nature frequency decreases with the increase of theindex for thickness function
4 Conclusions
In this paper a modified Fourier method has been pre-sented to study the free vibration behaviors of moderatelythick rectangular plates with variable thickness and arbi-trary boundary conditionsThe first-order shear deformationplate theory is adopted to formulate the theoretical modelThe displacements and rotation components of the plateregardless of boundary conditions are invariantly expressedas the superposition of a 2D Fourier cosine series and four
20 Shock and Vibration
Table 12 Frequency parameterΩ = 1205961198862
(120588ℎ01198630)12 for elastic support moderately thick plates (119886119887 = 1) with different gradient (119896
1199090= 119896119909119886=
1198961199100= 119896119910119887= Γ119908= 2 119870
1199090= 119870119909119886= 1198701199100= 119870119910119887= Γ119909= 2)
120572 120573 ℎ0119886
Model sequence1 2 3 4 5 6
120572 = 120573 = 01
005 30187 51102 51141 67066 73954 77412(30188a) (51105) (51144) (67068) (73961) (77477)
01 16995 24719 24724 33071 44343 44819(16996a) (24725) (24729) (33084) (44347) (44822)
015 10077 15872 15883 23443 35937 36065(10081a) (15877) (15892) (23447) (35941) (36084)
02 6713 11953 11958 19086 30372 30604(6713a) (11957) (11965) (19092) (30378) (30612)
120572 = 120573 = 02
005 31206 51436 51614 67058 74398 77595(31208a) (51438) (51619) (67064) (74402) (77596)
01 16786 24503 24508 33144 45583 45923(16790a) (24510) (24513) (33147) (45587) (45932)
015 9821 15865 15906 23766 36897 36990(9824a) (15870) (15907) (23768) (36894) (36997)
02 6523 11978 11992 19356 30912 31162(6524a) (11981) (11997) (19362) (30916) (31174)
120572 = 120573 = 03
005 32097 51462 51900 66832 74991 77863(32098a) (51466) (51907) (66849) (74996) (77872)
01 16507 24332 24383 33330 46806 47074(16511a) (24327) (24380) (33339) (46810) (47086)
015 9562 15885 15964 24103 37748 37828(9562a) (15892) (15977) (21105) (37751) (37832)
02 6338 11992 12013 19603 31338 31634(6340a) (11994) (12011) (19612) (31344) (31639)
120572 = 120573 = 04
005 32851 51239 52046 66547 75735 78303(33455a) (50852) (50097) (66321) (76627) (78971)
01 16182 24209 24343 33598 47990 48230(33455a) (50852) (50097) (66321) (76627) (78971)
015 9307 15919 16034 24436 38491 38579(33455a) (50852) (50097) (66321) (76627) (78971)
02 6161 11986 12008 19828 31666 32040(33455a) (50852) (50097) (66321) (76627) (78971)
120572 = 120573 = 05
005 33453 50854 52088 66311 76610 78947(33455a) (50852) (50097) (66321) (76627) (78971)
01 15832 24130 24372 33924 49118 49362(15830a) (24127) (24374) (33931) (49147) (49382)
015 9060 15956 16099 24754 39134 39249(9064a) (15961) (16102) (24755) (39138) (39452)
02 5992 11955 11971 20034 31908 32392(5991a) (11957) (11978) (20042) (31910) (32398)
aResults in parentheses are taken from FEM
supplementary functions in the form of the product of apolynomial function and a single cosine series expansionto ensure and accelerate the convergence of the solutionAt each edge of the plate the general restraint conditionsare implemented by introducing one group of linear springsand two groups of rotational springs which are continuouslydistributed and determined by the stiffness of these springs
Instead of seeking a solution in strong forms in the previousstudies all the Fourier coefficients will be treated equallyand independently as the generalized coordinates and solveddirectly from the Rayleigh-Ritz technique The change of theboundary conditions can be easily achieved by only varyingthe stiffness of the three sets of boundary springs along alledges of the rectangular plates without involving any change
Shock and Vibration 21
to the solution procedure The convergence of the presentsolution is examined and the excellent accuracy is validatedby comparison with existing results published in the lit-erature and FEM data Excellent agreements are obtainedfrom these comparisons The proposed method providesa unified means for extracting the modal parameters andpredicting the vibration behaviors of moderately thick plateswith variable thickness variation functions and arbitraryelastic edge restraints A variety of free vibration results formoderately thick rectangular plates with different thicknessvariation functions and boundary conditions are presentedNew results for free vibration of moderately thick rectangularplates with various thickness variation functions and edgeconditions are presented which may be used for benchmark-ing of researchers in the field
Appendices
A Representative Calculation forStiffness and Mass Matrices
To illuminate the particular expression of the mass andstiffness matrixes clearly and tersely four new variables aredefined as 119904 = 119898 lowast (119873 + 1) + 119899 + 1 119905 = 119898
1015840
lowast (119873 + 1) + 1198991015840
+ 1119901 = (119897minus1)lowast (119872+1)+119898
1015840
+1 and 119902 = (119897minus1)lowast (119873+1)+1198991015840
+1The first row elements of K andM are given below
K1-1119904119905
=
1
2
int
119886
0
int
119887
0
[119863 (119909 119910)
times 21205821198861198981205821198861198981015840 sin 120582
119886119898119909 sin 120582
1198861198981015840119909 cos 120582
119887119899119910
times cos 1205821198871198991015840119910 + (1 minus 120583) 120582
1198871198991205821198871198991015840 cos 120582
119886119898119909
times cos 1205821198861198981015840 sin 120582
119887119899119910 sin 120582
1198871198991015840119910
+ 120581119866ℎ (119909 119910) cos 120582119886119898119909 cos 120582
1198861198981015840119909 cos 120582
119887119899
times119910 cos 1205821198871198991015840119910119889119909119889119910] 119889119909 119889119910
+ int
119886
0
1198701199091199100
cos 120582119886119898119909 cos 120582
1198861198981015840119909
+(minus1)119898+1198981015840
119870119909119910119887
cos 120582119886119898119909 cos 120582
1198861198981015840119909 119889119909
+ int
119887
0
1198701199090cos 120582119887119899119910 cos 120582
1198871198991015840119910
+(minus1)119899+1198991015840
119870119909119886cos 120582119887119899119910 cos 120582
1198871198991015840119910 119889119910
K1-2119904119901
=
1
2
int
119886
0
int
119887
0
[119863 (119909 119910)21205821198861198981205821198861198981015840 sin 120582
119886119898119909 sin 120582
1198861198981015840119909
times
2
sum
119897=1
120577119897
119887(119910) cos 120582
1198871198991015840119910
minus (1 minus 120583) 1205821198871198991015840 cos 120582
119886119898119909 cos 120582
1198861198981015840119909
times
2
sum
119897=1
1205771198971015840
119887(119910) sin 120582
1198871198991015840119910 + 120581119866ℎ (119909 119910)
times ( cos 1205821198861198981015840119909 cos 120582
1198871198991015840119910
times
2
sum
119897=1
120577119897
119887(119910) cos 120582
119886119898119909)]119889119909119889119910
+ int
119886
0
1198701199091199100
cos 120582119886119898119909 cos 120582
1198861198981015840119909
2
sum
119897=1
120577119897
119887(0)
+(minus1)1198991015840
119870119909119910119887
cos 120582119886119898119909 cos 120582
1198861198981015840119909
2
sum
119897=1
120577119897
119887(119887) 119889119909
+ int
119887
0
1198701199090
2
sum
119897=1
120577119897
119887(119910) cos 120582
1198871198991015840119910
+(minus1)119898+1198981015840
119870119909119886
2
sum
119897=1
120577119897
119887(119910) cos 120582
1198871198991015840119910119889119910
K1-3119904119902
=
1
2
int
119886
0
int
119887
0
[119863 (119909 119910) minus 21205821198861198981015840 sin 120582
1198861198981015840119909 cos 120582
1198871198991015840119910
times
2
sum
119897=1
1205771198971015840
119886(119909) cos 120582
119887119899119910
+ (1 minus 120583) 1205821198871198991205821198871198991015840 cos 120582
1198861198981015840119909
times sin 1205821198871198991015840119910
2
sum
119897=1
120577119897
119886(119909) sin 120582
119887119899119910
+ 120581119866ℎ (119909 119910) cos 1205821198861198981015840119909 cos 120582
1198871198991015840119910
times
2
sum
119897=1
120577119897
119886(119909) cos 120582
119887119899119910]119889119909119889119910
+ int
119886
0
1198701199091199100
cos 1205821198861198981015840119909
2
sum
119897=1
120577119897
119886(119909)
+(minus1)119899+1198991015840
119870119909119910119887
cos 1205821198861198981015840119909
2
sum
119897=1
120577119897
119886(119909) 119889119909
+ int
119887
0
1198701199090
2
sum
119897=1
120577119897
119886(0) cos 120582
1198871198991015840119910
+(minus1)119898+1198981015840
119870119909119886
2
sum
119897=1
120577119897
119886(0) cos 120582
1198871198991015840119910119889119910
K1-4119904119905
=
1
2
int
119886
0
int
119887
0
119863(119909 119910) 21205821198871198991205821198861198981015840 sin 120582
1198861198981015840119909 cos 120582
1198871198991015840119910
times cos 120582119886119898119909 sin 120582
119887119899119910 minus 2(1 minus 120583)
119887119899
times 1205821198861198981015840 sin 120582
1198861198981015840119909 cos 120582
1198871198991015840119910 cos 120582
119886119898119909
times sin 120582119887119899119910 + (1 minus 120583) 120582
1198861198981205821198871198991015840 cos 120582
1198861198981015840119909
times sin 1205821198871198991015840119910 sin 120582
119886119898119909 cos 120582
119887119899119910 119889119909 119889119910
22 Shock and Vibration
K1-5119904119901
=
1
2
int
119886
0
int
119887
0
119863(119909 119910) minus 21205821198861198981015840 sin 120582
1198861198981015840119909 cos 120582
1198871198991015840119910
times
2
sum
119897=1
1205771198971015840
119887(119910) cos 120582
119886119898119909
+ 2 (1 minus 120583) 1205821198861198981015840 sin 120582
1198861198981015840119909 cos 120582
1198871198991015840119910
times
2
sum
119897=1
1205771198971015840
119887(119910) cos 120582
119886119898119909
+ (1 minus 120583) 1205821198861198981205821198871198991015840 cos 120582
1198861198981015840119909 sin 120582
1198871198991015840119910
times
2
sum
119897=1
120577119897
119887(119910) sin 120582
119886119898119909119889119909119889119910
K1-6119904119902
=
1
2
int
119886
0
int
119887
0
119863(119909 119910)1205821198871198991205821198861198981015840 sin 120582
1198861198981015840119909 cos 120582
1198861198991015840119910
times
2
sum
119897=1
120577119897
119886(119909) sin 120582
1198871198991015840119910
minus 2 (1 minus 120583) 1205821198871198991205821198861198981015840 sin 120582
1198861198981015840119909 cos 120582
1198861198991015840119910
times
2
sum
119897=1
120577119897
119886(119909) sin 120582
1198871198991015840119910
+ 1205821198871198991015840 cos 120582
1198861198981015840119909 sin 120582
1198871198991015840119910
times
2
sum
119897=1
1205771198971015840
119886(119909) cos 120582
119887119899119910119889119909119889119910
K1-7119904119905
= minus
1
2
int
119886
0
int
119887
0
120581119866ℎ (119909 119910) 120582119886119898
cos 1205821198861198981015840119909 cos 120582
1198871198991015840119910
times sin 120582119886119898119909 cos 120582
119887119899119910119889119909119889119910
K1-8119904119901
= minus
1
2
int
119886
0
int
119887
0
120581119866ℎ (119909 119910) 1205821198861198981015840 cos 120582
1198861198981015840119909 cos 120582
1198871198991015840119910
times
2
sum
119897=1
120577119897
119887(119910) sin 120582
1198861198981015840119909 119889119909 119889119910
K1-9119904119902
=
1
2
int
119886
0
int
119887
0
120581119866ℎ (119909 119910) cos 1205821198861198981015840119909 cos 120582
1198871198991015840119910
times
2
sum
119897=1
1205771198971015840
119886(119909) cos 120582
119887119899119910119889119909119889119910
M1-1119904119905
=
1
24
120588int
119886
0
int
119887
0
ℎ3
(119909 119910) cos 120582119886119898119909 cos 120582
119887119899119910
times cos 1205821198861198981015840119909 cos 120582
1198871198991015840119910119889119909119889119910
M1-2119904119901
=
1
24
120588int
119886
0
int
119887
0
ℎ3
(119909 119910)
2
sum
119897=1
120577119897
119887(119910) cos 120582
119886119898119909
times cos 1205821198861198981015840119909 cos 120582
1198871198991015840119910119889119909119889119910
M1-3119904119902
=
1
24
120588int
119886
0
int
119887
0
ℎ3
(119909 119910)
2
sum
119897=1
120577119897
119886(119909) cos 120582
119887119899119910
times cos 1205821198861198981015840119909 cos 120582
1198871198991015840119910119889119909119889119910
M1-4119904119905 = M
1-5119904119901 = M1-6119904119902
= M1-7119904119905 = M
1-8119904119901 = M1-9119904119902 = 0
(A1)
B Useful Integrals for the Moderately ThickRectangular Plates with Variable Thicknessand Arbitrary Boundary Conditions
Consider
int
119871
0
cos 120582119898119909 cos 120582
119899119909 119889119909 =
0 119898 = 119899
119871
2
119898 = 119899
119871 119898 = 119899 = 0
int
119871
0
sin 120582119898119909 cos 120582
119899119909119889119909
=
minus
(minus1 + (minus1)119898+119899
) 119871119898
(1198982minus 1198992) 120587
119898 = 119899
0 119898 = 119899
int
119871
0
sin 120582119898119909 sin 120582
119899119909 119889119909 =
0 119898 = 119899
119871
2
119898 = 119899
0 119898 = 119899 = 0
int
119871
0
cos 120582119898119909120577119897
119886(119909) 119889119909
=
41198712
31205872 119897 = 1 119898 = 0
minus
41198712
(41198982
minus 3)
(9 minus 401198982+ 16119898
4) 1205872
119897 = 1 119898 = 0
minus
41198712
31205872 119897 = 2 119898 = 0
41198712
(41198982
minus 3) (minus1)119898
(9 minus 401198982+ 16119898
4) 1205872
119897 = 2 119898 = 0
Shock and Vibration 23
int
119871
0
sin 120582119898119909120577119897
119886(119909) 119889119909
=
0 119897 = 1 119898 = 0
161198712
119898(minus1)119898
(9 minus 401198982+ 16119898
4) 1205872
119897 = 1 119898 = 0
0 119897 = 2 119898 = 0
161198981198712
(9 minus 401198982+ 16119898
4) 1205872
119897 = 2 119898 = 0
int
119871
0
cos 1205821198981199091205771198971015840
119886(119909) 119889119909
=
0 119897 = 1 119898 = 0
161198711198982
(minus1)119898
(9 minus 401198982+ 16119898
4) 120587
119897 = 1 119898 = 0
0 119897 = 2 119898 = 0
161198711198982
(9 minus 401198982+ 16119898
4) 120587
119897 = 2 119898 = 0
int
119871
0
sin 1205821198981199091205771198971015840
119886(119909) 119889119909
=
0 119897 = 1 119898 = 0
4119871119898 (minus3 + 41198982
)
(9 minus 401198982+ 16119898
4) 120587
119897 = 1 119898 = 0
0 119897 = 2 119898 = 0
119871119898(
1
120587 minus 41198982120587
+
3
9120587 minus 41198982120587
) (minus1)119898
119897 = 2 119898 = 0
int
119871
0
1205771198971
119886(119909) 1205771198972
119886(119909) 119889119909 =
1198713
41205872 1198971= 1 1198972= 1
minus
21198713
31205873 1198971= 1 1198972= 2
minus
21198713
31205873 1198971= 2 1198972= 1
1198713
41205872 1198971= 2 1198972= 2
int
119871
0
1205771198971
119886(119909) 1205771198971015840
2
119886(119909) 119889119909 =
0 1198971= 1 1198972= 1
minus
1198712
8120587
1198971= 1 1198972= 2
1198712
8120587
1198971= 2 1198972= 1
0 1198971= 2 1198972= 2
int
119871
0
1205771198971015840
1
119886(119909) 1205771198971015840
2
119886(119909) 119889119909 =
5119871
16
1198971= 1 1198972= 1
minus
119871
2120587
1198971= 1 1198972= 2
minus
119871
2120587
1198971= 2 1198972= 1
5119871
16
1198971= 2 1198972= 2
int
119871
0
119909119899 cos 120582
119898119909 119889119909
=
1198711+119899
1 + 119899
119898 = 0
(1198861+119899HypergeometricPFQtimes [ 12 + 1198992 12 32 + 1198992
minus (14)1198982
1205872
]) times (1 + 119899)minus1
119898 gt 0
(B1)
Conflict of Interests
The authors declare that there is no conflict of interestsregarding the publication of this paper
Acknowledgments
The authors would like to thank the anonymous reviewersfor their very valuable comments The authors also gratefullyacknowledge the financial support from the National NaturalScience Foundation of China (no 51209052) HeilongjiangProvince Youth Science Fund Project (no QC2011C013) andHarbin Science and Technology Development InnovationFoundation of Youth (no 2011RFQXG021)
References
[1] K Liew Y Xiang and S Kitipornchai ldquoResearch on thick platevibration a literature surveyrdquo Journal of Sound and Vibrationvol 180 no 1 pp 163ndash176 1995
[2] J H Chung T Y Chung and K C Kim ldquoVibration analysisof orthotropic mindlin plates with edges elastically restrainedagainst rotationrdquo Journal of Sound and Vibration vol 163 no 1pp 151ndash163 1993
[3] Y K Cheung and D Zhou ldquoVibrations of moderately thickrectangular plates in terms of a set of static Timoshenko beamfunctionsrdquoComputers and Structures vol 78 no 6 pp 757ndash7682000
[4] CMWang ldquoNatural frequencies formula for simply supportedMindlin platesrdquo Journal of Vibration and Acoustics vol 116 no4 pp 536ndash540 1994
[5] K N Saha R C Kar and P K Datta ldquoFree vibration analysisof rectangular Mindlin plates with elastic restraints uniformlydistributed along the edgesrdquo Journal of Sound andVibration vol192 no 4 pp 885ndash902 1996
24 Shock and Vibration
[6] D J Gorman ldquoAccurate free vibration analysis of shear-deformable plates with torsional elastic edge supportrdquo Journalof Sound and Vibration vol 203 no 2 pp 209ndash218 1997
[7] D J Gorman ldquoFree vibration analysis of Mindlin plates withuniform elastic edge support by the superposition methodrdquoJournal of Sound andVibration vol 207 no 3 pp 335ndash350 1997
[8] D J Gorman ldquoAccurate free vibration analysis of point sup-ported mindlin plates by the superposition methodrdquo Journal ofSound and Vibration vol 219 no 2 pp 265ndash277 1999
[9] Y Xiang ldquoVibration of rectangular Mindlin plates resting onnon-homogenous elastic foundationsrdquo International Journal ofMechanical Sciences vol 45 no 6-7 pp 1229ndash1244 2003
[10] Y Xiang and G W Wei ldquoExact solutions for buckling andvibration of stepped rectangular Mindlin platesrdquo InternationalJournal of Solids and Structures vol 41 no 1 pp 279ndash294 2004
[11] Y L Yeh M J Jang and C C Wang ldquoAnalyzing the freevibrations of a plate using finite difference and differential trans-formationmethodrdquoAppliedMathematics andComputation vol178 no 2 pp 493ndash501 2006
[12] Y Xiang S K Lai L Zhou and C W Lim ldquoDSC-Ritz elementmethod for vibration analysis of rectangular Mindlin plateswith mixed edge supportsrdquo European Journal of MechanicsmdashASolids vol 29 no 4 pp 619ndash628 2010
[13] H Nguyen-Xuan C H Thai and T Nguyen-Thoi ldquoIsogeo-metric finite element analysis of composite sandwich platesusing a higher order shear deformation theoryrdquo Composites BEngineering vol 55 pp 558ndash574 2013
[14] C H Thai A J M Ferreira E Carrera and H Nguyen-XuanldquoIsogeometric analysis of laminated composite and sandwichplates using a layerwise deformation theoryrdquo Composite Struc-tures vol 104 pp 196ndash214 2013
[15] C H Thai A Ferreira S Bordas T Rabczuk and HNguyen-Xuan ldquoIsogeometric analysis of laminated compositeand sandwich plates using a new inverse trigonometric sheardeformation theoryrdquo European Journal of Mechanics A Solidsvol 43 pp 89ndash108 2014
[16] H Luong-Van T Nguyen-Thoi G R Liu and P Phung-VanldquoA cell-based smoothed finite elementmethod using three-nodeshear-locking free Mindlin plate element (CS-FEM-MIN3) fordynamic response of laminated composite plates on viscoelasticfoundationrdquo Engineering Analysis with Boundary Elements vol42 pp 8ndash19 2014
[17] T Nguyen-Thoi T Rabczuk T Lam-Phat V Ho-Huu andP Phung-Van ldquoFree vibration analysis of cracked Mindlinplate using an extended cell-based smoothed discrete sheargap method (XCS-DSG3)rdquo Theoretical and Applied FractureMechanics 2014
[18] H Nguyen-Xuan G R Liu and C a Thai-Hoang ldquoAnedge-based smoothed finite element method (ES-FEM) withstabilized discrete shear gap technique for analysis of Reissner-Mindlin platesrdquo Computer Methods in Applied Mechanics andEngineering vol 199 no 9ndash12 pp 471ndash489 2010
[19] P Phung-Van T Nguyen-Thoi T Bui-Xuan and Q Lieu-XuanldquoA cell-based smoothed three-node Mindlin plate element(CS-FEM-MIN3) based on the 119862
0 -type higher-order sheardeformation for geometrically nonlinear analysis of laminatedcomposite platesrdquo Computational Materials Science 2014
[20] NNguyen-Thanh T RabczukHNguyen-Xuan and S BordasldquoAn alternative alpha finite element method with discrete sheargap technique for analysis of isotropicMindlin-Reissner platesrdquoFinite Elements inAnalysis andDesign vol 47 no 5 pp 519ndash5352011
[21] KM Liew Y Xiang and S Kitipornchai ldquoTransverse vibrationof thick rectangular plates-I Comprehensive sets of boundaryconditionsrdquo Computers and Structures vol 49 no 1 pp 1ndash291993
[22] D Zhou ldquoVibrations of Mindlin rectangular plates with elasti-cally restrained edges using static Timoshenko beam functionswith the Rayleigh-Ritz methodrdquo International Journal of Solidsand Structures vol 38 no 32-33 pp 5565ndash5580 2001
[23] D Zhou SH Lo F T KAu andYK Cheung ldquoVibration anal-ysis of rectangular Mindlin plates with internal line supportsusing static Timoshenko beam functionsrdquo International Journalof Mechanical Sciences vol 44 no 12 pp 2503ndash2522 2002
[24] H S Shen J Yang and L Zhang ldquoFree and forced vibrationof Reissner-Mindlin plates with free edges resting on elasticfoundationsrdquo Journal of Sound and Vibration vol 244 no 2 pp299ndash320 2001
[25] Y Xing and B Liu ldquoClosed form solutions for free vibrations ofrectangular Mindlin platesrdquo Acta Mechanica Sinica vol 25 no5 pp 689ndash698 2009
[26] P Gagnon C Gosselin and L Cloutier ldquoA finite strip elementfor the analysis of variable thickness rectangular thick platesrdquoComputers amp Structures vol 63 no 2 pp 349ndash362 1997
[27] T Mlzusawa ldquoVibration of rectangular mindlin plates withtapered thickness by the spline strip methodrdquo Computers andStructures vol 46 no 3 pp 451ndash463 1993
[28] H Nguyen-Xuan and T Nguyen-Thoi ldquoA stabilized smoothedfinite element method for free vibration analysis of Mindlin-Reissner platesrdquo Communications in Numerical Methods inEngineering with Biomedical Applications vol 25 no 8 pp 882ndash906 2009
[29] K M Liew L X Peng and S Kitipornchai ldquoVibration anal-ysis of corrugated Reissner-Mindlin plates using a mesh-freeGalerkin methodrdquo International Journal of Mechanical Sciencesvol 51 no 9-10 pp 642ndash652 2009
[30] Y Hou GWWei and Y Xiang ldquoDSC-Ritz method for the freevibration analysis of Mindlin platesrdquo International Journal forNumerical Methods in Engineering vol 62 no 2 pp 262ndash2882005
[31] F- Liu and K M Liew ldquoVibration analysis of discontinuousMindlin plates by differential quadrature element methodrdquoJournal of Vibration and Acoustics Transactions of the ASMEvol 121 no 2 pp 204ndash208 1999
[32] P Malekzadeh G Karami and M Farid ldquoA semi-analyticalDQEM for free vibration analysis of thick plates with twoopposite edges simply supportedrdquoComputerMethods inAppliedMechanics and Engineering vol 193 no 45ndash47 pp 4781ndash47962004
[33] P Malekzadeh and S A Shahpari ldquoFree vibration analysisof variable thickness thin and moderately thick plates withelastically restrained edges by DQMrdquo Thin-Walled Structuresvol 43 no 7 pp 1037ndash1050 2005
[34] R E Diaz-Contreras and S Nomura ldquoGreenrsquos function appliedto solution ofMindlin platesrdquoComputers and Structures vol 60no 1 pp 41ndash48 1996
[35] T Sakiyama and M Huang ldquoFree vibration analysis of rect-angular plates with variable thicknessrdquo Journal of Sound andVibration vol 216 no 3 pp 379ndash397 1998
[36] M Huang X Q Ma T Sakiyama H Matuda and C MoritaldquoFree vibration analysis of orthotropic rectangular plates withvariable thickness and general boundary conditionsrdquo Journal ofSound and Vibration vol 288 no 4-5 pp 931ndash955 2005
Shock and Vibration 25
[37] I Shufrin and M Eisenberger ldquoVibration of shear deformableplates with variable thicknessmdashfirst-order and higher-orderanalysesrdquo Journal of Sound and Vibration vol 290 no 1-2 pp465ndash489 2006
[38] S A Eftekhari and A A Jafari ldquoAccurate variational approachfor free vibration of variable thickness thin and thick plates withedges elastically restrained against translation and rotationrdquoInternational Journal of Mechanical Sciences vol 68 pp 35ndash462013
[39] W L Li X Zhang J Du and Z Liu ldquoAn exact series solutionfor the transverse vibration of rectangular plates with generalelastic boundary supportsrdquo Journal of Sound amp Vibration vol321 no 1-2 pp 254ndash269 2009
[40] X Shi D Shi W L Li and Q Wang ldquoA unified method forfree vibration analysis of circular annular and sector plateswith arbitrary boundary conditionsrdquo Journal of Vibration andControl 2014
[41] D Y Shi X J Shi W L Li and Q S Wang ldquoFree transversevibrations of orthotropic thin rectangular plates with arbitraryelastic edge supportsrdquo Journal of Vibroengineering vol 16 no 1pp 389ndash398 2014
[42] Y Chen G Jin and Z Liu ldquoFree vibration analysis of circu-lar cylindrical shell with non-uniform elastic boundary con-straintsrdquo International Journal of Mechanical Sciences vol 74pp 120ndash132 2013
[43] L Dai T Yang J Du W L Li and M J Brennan ldquoAn exactseries solution for the vibration analysis of cylindrical shellswith arbitrary boundary conditionsrdquo Applied Acoustics vol 74no 3 pp 440ndash449 2013
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Shock and Vibration
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International Journal of
Shock and Vibration 17
Table 9 Frequency parameter Ω = 1205961198862
(120588ℎ01198630)12 for moderately thick plates with linearly varying thickness in different boundary
conditions (ℎ(119909 119910) = ℎ0(1 + 05119909
2
)(1 + 051199102
))
Boundary condition 119886119887 ℎ0119886
Model sequence1 2 3 4 5 6
S-S-S-S
101 23462 54999 55573 82950 100900 101187
(23741a) (55114) (55983) (83146) (101004) (101347)
02 18606 42725 42971 60040 71167 71761(18936a) (43014) (43148) (60247) (71642) (71993)
3201 53055 65251 85404 106700 114880 143531
(53467a) (65472) (85984) (107211) (115687) (144007)
02 17439 21531 32170 44987 59438 64179(17874a) (21748) (32651) (45264) (60188) (64879)
C-F-F-F
101 3658 9317 24135 32816 35433 59179
(3749a) (9547) (24034) (33627) (36308) (60142)
02 3520 7986 19591 26959 27491 43882(3611a) (8176) (20075) (27125) (27625) (43984)
3201 3901 9963 28984 69187 73686 89360
(3930a) (10095) (29381) (69578) (73994) (90242)
02 3292 4439 13385 23947 40850 44758(3386a) (4498) (14012) (24068) (41221) (45142)
S-S-F-F
101 3557 21268 22762 43524 60253 62429
(3649a) (21441) (22796) (43916) (60622) (62630)
02 3171 17822 19512 33616 45042 46833(3489a) (18006) (20430) (33815) (45247) (49004)
3201 6738 31658 52071 75625 86604 94903
(6957a) (31882) (52518) (75823) (86939) (95610)
02 3653 12930 20955 26517 44097 48058(6807a) (13160) (21312) (26864) (44422) (48873)
C-C-C-C
101 41020 75196 75614 104142 119683 120949
(41405a) (75452) (75882) (104711) (119637) (121137)
02 31237 51686 51868 68411 76868 77864(32010a) (52960) (52145) (69335) (77481) (78099)
3201 95823 128727 163657 173751 190890 223595
(95364a) (128932) (163588) (172350) (190350) (222948)
02 53306 71372 89384 102404 104660 120908(53587a) (71759) (90103) (102503) (105174) (121275)
S-C-S-C
101 33715 60298 71446 94187 103802 117928
(34148a) (60786) (71828) (94515) (104263) (119326)
02 26143 45499 49458 64253 72941 76191(26789a) (46622) (50680) (64329) (73474) (76843)
3201 73228 115845 137445 148166 168342 182542
(73761a) (116121) (137788) (148530) (168698) (182921)
02 41302 53648 65630 72202 81632 88105(41304a) (53757) (65842) (72794) (82033) (88474)
aResults in parentheses are taken from FEM
S-C-S-C boundary conditions are performed in the compar-ison The results adequately demonstrated the great accuracyof the modified Fourier method
We now turn to elastically restrained moderately thickplates The first one involves an S-S-S-S moderately thicksquare plate with a uniform rotational restraint along eachedge that is 119870
1199090= 119870119909119886
= 1198701199100
= 119870119910119887
= Γ119909 The calculated
frequency parameters Ω = 1205961198862
(120588ℎ01198630)12 are given in
Table 10 together with the FEM results Since the elasticallyrestrained plates with nonlinear variation thickness are rarelyinvestigated the FEMresults are used as the referenceA goodagreement is observed between the current and FEM resultsThe second example concerns amoderately thick square plateelastically supported along all edgesThe stiffness of the linear
18 Shock and Vibration
Table 10 Frequency parameterΩ = 1205961198862
(120588ℎ01198630)12 for S-S-S-S moderately thick plates (119886119887 = 1) with linearly varying thickness and elastic
rotation support (ℎ(119909 119910) = ℎ0(1 + 05119909
2
)(1 + 051199102
) 1198701199090= 119870119909119886= 1198701199100= 119870119910119887= Γ119909)
Γ119909
aℎ0119886
Model sequence1 2 3 4 5 6
0
005 23462 54999 55573 82950 100900 101187(23741b) (55114) (55983) (83146) (101004) (101347)
01 18606 42725 42971 60040 71167 71761(18936b) (43014) (43148) (60247) (71642) (71993)
015 53055 65251 85404 106700 114880 143531(53467b) (65472) (85984) (107211) (115687) (144007)
02 17439 21531 32170 44987 59438 64179(17874b) (21748) (32651) (45264) (60188) (64879)
05
005 6904 13374 13514 19400 22904 23056(9909b) (13548) (13651) (19564) (23088) (23379)
01 5316 10237 10314 14499 17155 17264(5374b) (10448) (10651) (14764) (17188) (17879)
015 4136 8201 8220 11474 13578 13671(4174b) (8348) (8651) (11764) (13688) (13879)
02 3449 6915 6924 9549 11225 11327(3474b) (6948) (6951) (9764) (11388) (11579)
1
005 7040 13617 13745 19726 23233 23402(7074b) (13748) (13851) (19964) (23488) (23879)
01 5712 10696 10793 14991 17594 17744(5774b) (10748) (10851) (15054) (17788) (17879)
015 4489 8507 8540 11759 13803 13922(4514b) (8548) (8611) (11864) (13988) (14119)
02 3677 7084 7090 9692 11330 11445(3684b) (7148) (7191) (9764) (11588) (11879)
15
005 7089 13708 13828 19847 23355 23530(7124b) (13748) (13951) (19967) (23588) (23879)
01 5895 10923 11024 15239 17813 17988(5874b) (10748) (11651) (15264) (17818) (17979)
015 4697 8694 8737 11938 13790 14083(4774b) (8748) (8951) (11864) (13688) (14879)
02 3836 7201 7210 9792 11403 11530(3874b) (7748) (7651) (9864) (11588) (11879)
2
005 7114 13754 13871 19911 23418 23597(7174b) (12848) (13851) (19864) (23588) (23879)
01 6001 11060 11159 15391 17944 18136(6004b) (11148) (11151) (15464) (17988) (18379)
015 4836 8821 8872 12061 14037 14189(4874b) (8848) (8951) (12164) (14188) (14879)
02 3954 7289 7301 9871 11457 11594(3974b) (7348) (7651) (9964) (11488) (11679)
aΓ119909= 1198701198941198700 (1198700= 1 times 10
9 Nmrad 119894 = 1199090 119909119886 1199100 119910119887)bResults in parentheses are taken from FEM
and rotational restraints is set equal to 1198961199090
= 119896119909119886
= 1198961199100
=
119896119910119887
= Γ119908and 119870
1199090= 119870119909119886
= 1198701199100
= 119870119910119887
= Γ119909 respectively
The six frequency parametersΩ = 1205961198862
(120588ℎ01198630)12 are shown
in Table 11 for several different restraining coefficient valuesIt can also be noticed that the Fourier series method is able
to predict the modal characteristics of nonlinearly varyingthickness moderately thick plate with not only classicalboundary conditions but also elastic edge restraints correctly
Finally the influence of the gradient on the fundamentalfrequency parameters of a nonlinearly varying thickness
Shock and Vibration 19
Table 11 Frequency parameterΩ = 1205961198862
(120588ℎ01198630)12 for moderately thick plates (119886119887 = 1) with linearly varying thickness and elastic rotation
and translation support (ℎ(119909 119910) = ℎ0(1 + 05119909
2
)(1 + 051199102
) 1198701199090= 119870119909119886= 1198701199100= 119870119910119887= Γ119909 and 119896
1199090= 119896119909119886= 1198961199100= 119896119910119887= Γ119908)
Γ119909
aΓ119908
aℎ0119886
Model sequence1 2 3 4 5 6
05 0
005 12926 13041 25090 47935 47943 61761(12931b) (13053) 25099 (47947) (47952) (61788)
01 10900 10988 21545 39980 40147 50322(10904b) (10992) (21557) (39984) (40166) (50334)
015 8245 8298 18081 31503 32617 40950(8247b) (8230) (18092) (31508) (32622) (40955)
02 6106 6124 15639 25631 27601 34458(6110b) (6125) (15642) (25637) (27617) (34462)
1 05
005 21575 31615 31644 42573 57889 58607(21580b) (31617) (31644) (42581) (57893) (58610)
01 8451 16024 16210 25969 43839 43867(8451b) (16025) (16217) (25972) (43844) (43870)
015 4682 11479 11480 20642 35172 35598(4684b) (11480) (11481) (20644) (35178) (35604)
02 3059 8621 8648 17320 28487 29638(3062b) (8619) (8644) (17327) (28492) (29641)
15 1
005 27713 40203 40978 52951 65352 66815(27714b) (40207) (40988) (52972) (65368) (66822)
01 11667 19322 19615 29122 46115 46183(11674b) (19327) (19619) (29137) (46120) (46188)
015 6544 13416 13481 22405 37238 37430(6547b) (13420) (13488) (22410) (37241) (37434)
02 4295 10175 10190 18543 30330 31056(4296b) (10177) (10194) (18544) (30336) (31060)
2 2
005 33453 50854 52088 66311 76610 78947(33455b) (50852) (50097) (66321) (76627) (78971)
01 15832 24130 24372 33924 49118 49362(15830b) (24127) (24374) (33931) (49147) (49382)
015 9060 15956 16099 24754 39134 39249(9064b) (15961) (16102) (24755) (39138) (39452)
02 5992 11955 11971 20034 31908 32392(5991b) (11957) (11978) (20042) (31910) (32398)
aΓ119909= 1198701198941198700 (1198700= 1 times 10
9 Nmrad 119894 = 1199090 119909119886 1199100 119910119887) Γ119908= 1198961198941198960 (1198960= 1 times 10
9 Nm 119894 = 1199090 119909119886 1199100 119910119887)bResults in parentheses are taken from FEM
moderately thick plate is investigated The plate is elasticallyrestrained in which the stiffness of the boundary springsis taken as 119896
1199090= 119896119909119886
= 1198961199100
= 119896119910119887
= Γ119908
= 2 and1198701199090
= 119870119909119886
= 1198701199100
= 119870119910119887
= Γ119909= 2 respectively The six
frequency parameters Ω = 1205961198862
(120588ℎ01198630)12 are presented in
Table 12 for several different slop values It can be seen thatthe fundamental frequency parameters will decrease with theincrease of the parameter ℎ
0119886
In the above examples it has been demonstrated thatthe presented method can be universally applied to nonlin-early varying thickness moderately thick plates with severalboundary conditions and different structure parametersNew results are obtained for plates with nonlinear variationthickness in both directions subjected to general elastic
boundary restraints which may be used for benchmarkingof researchers in the field In addition it is interesting to seethat the nature frequency decreases with the increase of theindex for thickness function
4 Conclusions
In this paper a modified Fourier method has been pre-sented to study the free vibration behaviors of moderatelythick rectangular plates with variable thickness and arbi-trary boundary conditionsThe first-order shear deformationplate theory is adopted to formulate the theoretical modelThe displacements and rotation components of the plateregardless of boundary conditions are invariantly expressedas the superposition of a 2D Fourier cosine series and four
20 Shock and Vibration
Table 12 Frequency parameterΩ = 1205961198862
(120588ℎ01198630)12 for elastic support moderately thick plates (119886119887 = 1) with different gradient (119896
1199090= 119896119909119886=
1198961199100= 119896119910119887= Γ119908= 2 119870
1199090= 119870119909119886= 1198701199100= 119870119910119887= Γ119909= 2)
120572 120573 ℎ0119886
Model sequence1 2 3 4 5 6
120572 = 120573 = 01
005 30187 51102 51141 67066 73954 77412(30188a) (51105) (51144) (67068) (73961) (77477)
01 16995 24719 24724 33071 44343 44819(16996a) (24725) (24729) (33084) (44347) (44822)
015 10077 15872 15883 23443 35937 36065(10081a) (15877) (15892) (23447) (35941) (36084)
02 6713 11953 11958 19086 30372 30604(6713a) (11957) (11965) (19092) (30378) (30612)
120572 = 120573 = 02
005 31206 51436 51614 67058 74398 77595(31208a) (51438) (51619) (67064) (74402) (77596)
01 16786 24503 24508 33144 45583 45923(16790a) (24510) (24513) (33147) (45587) (45932)
015 9821 15865 15906 23766 36897 36990(9824a) (15870) (15907) (23768) (36894) (36997)
02 6523 11978 11992 19356 30912 31162(6524a) (11981) (11997) (19362) (30916) (31174)
120572 = 120573 = 03
005 32097 51462 51900 66832 74991 77863(32098a) (51466) (51907) (66849) (74996) (77872)
01 16507 24332 24383 33330 46806 47074(16511a) (24327) (24380) (33339) (46810) (47086)
015 9562 15885 15964 24103 37748 37828(9562a) (15892) (15977) (21105) (37751) (37832)
02 6338 11992 12013 19603 31338 31634(6340a) (11994) (12011) (19612) (31344) (31639)
120572 = 120573 = 04
005 32851 51239 52046 66547 75735 78303(33455a) (50852) (50097) (66321) (76627) (78971)
01 16182 24209 24343 33598 47990 48230(33455a) (50852) (50097) (66321) (76627) (78971)
015 9307 15919 16034 24436 38491 38579(33455a) (50852) (50097) (66321) (76627) (78971)
02 6161 11986 12008 19828 31666 32040(33455a) (50852) (50097) (66321) (76627) (78971)
120572 = 120573 = 05
005 33453 50854 52088 66311 76610 78947(33455a) (50852) (50097) (66321) (76627) (78971)
01 15832 24130 24372 33924 49118 49362(15830a) (24127) (24374) (33931) (49147) (49382)
015 9060 15956 16099 24754 39134 39249(9064a) (15961) (16102) (24755) (39138) (39452)
02 5992 11955 11971 20034 31908 32392(5991a) (11957) (11978) (20042) (31910) (32398)
aResults in parentheses are taken from FEM
supplementary functions in the form of the product of apolynomial function and a single cosine series expansionto ensure and accelerate the convergence of the solutionAt each edge of the plate the general restraint conditionsare implemented by introducing one group of linear springsand two groups of rotational springs which are continuouslydistributed and determined by the stiffness of these springs
Instead of seeking a solution in strong forms in the previousstudies all the Fourier coefficients will be treated equallyand independently as the generalized coordinates and solveddirectly from the Rayleigh-Ritz technique The change of theboundary conditions can be easily achieved by only varyingthe stiffness of the three sets of boundary springs along alledges of the rectangular plates without involving any change
Shock and Vibration 21
to the solution procedure The convergence of the presentsolution is examined and the excellent accuracy is validatedby comparison with existing results published in the lit-erature and FEM data Excellent agreements are obtainedfrom these comparisons The proposed method providesa unified means for extracting the modal parameters andpredicting the vibration behaviors of moderately thick plateswith variable thickness variation functions and arbitraryelastic edge restraints A variety of free vibration results formoderately thick rectangular plates with different thicknessvariation functions and boundary conditions are presentedNew results for free vibration of moderately thick rectangularplates with various thickness variation functions and edgeconditions are presented which may be used for benchmark-ing of researchers in the field
Appendices
A Representative Calculation forStiffness and Mass Matrices
To illuminate the particular expression of the mass andstiffness matrixes clearly and tersely four new variables aredefined as 119904 = 119898 lowast (119873 + 1) + 119899 + 1 119905 = 119898
1015840
lowast (119873 + 1) + 1198991015840
+ 1119901 = (119897minus1)lowast (119872+1)+119898
1015840
+1 and 119902 = (119897minus1)lowast (119873+1)+1198991015840
+1The first row elements of K andM are given below
K1-1119904119905
=
1
2
int
119886
0
int
119887
0
[119863 (119909 119910)
times 21205821198861198981205821198861198981015840 sin 120582
119886119898119909 sin 120582
1198861198981015840119909 cos 120582
119887119899119910
times cos 1205821198871198991015840119910 + (1 minus 120583) 120582
1198871198991205821198871198991015840 cos 120582
119886119898119909
times cos 1205821198861198981015840 sin 120582
119887119899119910 sin 120582
1198871198991015840119910
+ 120581119866ℎ (119909 119910) cos 120582119886119898119909 cos 120582
1198861198981015840119909 cos 120582
119887119899
times119910 cos 1205821198871198991015840119910119889119909119889119910] 119889119909 119889119910
+ int
119886
0
1198701199091199100
cos 120582119886119898119909 cos 120582
1198861198981015840119909
+(minus1)119898+1198981015840
119870119909119910119887
cos 120582119886119898119909 cos 120582
1198861198981015840119909 119889119909
+ int
119887
0
1198701199090cos 120582119887119899119910 cos 120582
1198871198991015840119910
+(minus1)119899+1198991015840
119870119909119886cos 120582119887119899119910 cos 120582
1198871198991015840119910 119889119910
K1-2119904119901
=
1
2
int
119886
0
int
119887
0
[119863 (119909 119910)21205821198861198981205821198861198981015840 sin 120582
119886119898119909 sin 120582
1198861198981015840119909
times
2
sum
119897=1
120577119897
119887(119910) cos 120582
1198871198991015840119910
minus (1 minus 120583) 1205821198871198991015840 cos 120582
119886119898119909 cos 120582
1198861198981015840119909
times
2
sum
119897=1
1205771198971015840
119887(119910) sin 120582
1198871198991015840119910 + 120581119866ℎ (119909 119910)
times ( cos 1205821198861198981015840119909 cos 120582
1198871198991015840119910
times
2
sum
119897=1
120577119897
119887(119910) cos 120582
119886119898119909)]119889119909119889119910
+ int
119886
0
1198701199091199100
cos 120582119886119898119909 cos 120582
1198861198981015840119909
2
sum
119897=1
120577119897
119887(0)
+(minus1)1198991015840
119870119909119910119887
cos 120582119886119898119909 cos 120582
1198861198981015840119909
2
sum
119897=1
120577119897
119887(119887) 119889119909
+ int
119887
0
1198701199090
2
sum
119897=1
120577119897
119887(119910) cos 120582
1198871198991015840119910
+(minus1)119898+1198981015840
119870119909119886
2
sum
119897=1
120577119897
119887(119910) cos 120582
1198871198991015840119910119889119910
K1-3119904119902
=
1
2
int
119886
0
int
119887
0
[119863 (119909 119910) minus 21205821198861198981015840 sin 120582
1198861198981015840119909 cos 120582
1198871198991015840119910
times
2
sum
119897=1
1205771198971015840
119886(119909) cos 120582
119887119899119910
+ (1 minus 120583) 1205821198871198991205821198871198991015840 cos 120582
1198861198981015840119909
times sin 1205821198871198991015840119910
2
sum
119897=1
120577119897
119886(119909) sin 120582
119887119899119910
+ 120581119866ℎ (119909 119910) cos 1205821198861198981015840119909 cos 120582
1198871198991015840119910
times
2
sum
119897=1
120577119897
119886(119909) cos 120582
119887119899119910]119889119909119889119910
+ int
119886
0
1198701199091199100
cos 1205821198861198981015840119909
2
sum
119897=1
120577119897
119886(119909)
+(minus1)119899+1198991015840
119870119909119910119887
cos 1205821198861198981015840119909
2
sum
119897=1
120577119897
119886(119909) 119889119909
+ int
119887
0
1198701199090
2
sum
119897=1
120577119897
119886(0) cos 120582
1198871198991015840119910
+(minus1)119898+1198981015840
119870119909119886
2
sum
119897=1
120577119897
119886(0) cos 120582
1198871198991015840119910119889119910
K1-4119904119905
=
1
2
int
119886
0
int
119887
0
119863(119909 119910) 21205821198871198991205821198861198981015840 sin 120582
1198861198981015840119909 cos 120582
1198871198991015840119910
times cos 120582119886119898119909 sin 120582
119887119899119910 minus 2(1 minus 120583)
119887119899
times 1205821198861198981015840 sin 120582
1198861198981015840119909 cos 120582
1198871198991015840119910 cos 120582
119886119898119909
times sin 120582119887119899119910 + (1 minus 120583) 120582
1198861198981205821198871198991015840 cos 120582
1198861198981015840119909
times sin 1205821198871198991015840119910 sin 120582
119886119898119909 cos 120582
119887119899119910 119889119909 119889119910
22 Shock and Vibration
K1-5119904119901
=
1
2
int
119886
0
int
119887
0
119863(119909 119910) minus 21205821198861198981015840 sin 120582
1198861198981015840119909 cos 120582
1198871198991015840119910
times
2
sum
119897=1
1205771198971015840
119887(119910) cos 120582
119886119898119909
+ 2 (1 minus 120583) 1205821198861198981015840 sin 120582
1198861198981015840119909 cos 120582
1198871198991015840119910
times
2
sum
119897=1
1205771198971015840
119887(119910) cos 120582
119886119898119909
+ (1 minus 120583) 1205821198861198981205821198871198991015840 cos 120582
1198861198981015840119909 sin 120582
1198871198991015840119910
times
2
sum
119897=1
120577119897
119887(119910) sin 120582
119886119898119909119889119909119889119910
K1-6119904119902
=
1
2
int
119886
0
int
119887
0
119863(119909 119910)1205821198871198991205821198861198981015840 sin 120582
1198861198981015840119909 cos 120582
1198861198991015840119910
times
2
sum
119897=1
120577119897
119886(119909) sin 120582
1198871198991015840119910
minus 2 (1 minus 120583) 1205821198871198991205821198861198981015840 sin 120582
1198861198981015840119909 cos 120582
1198861198991015840119910
times
2
sum
119897=1
120577119897
119886(119909) sin 120582
1198871198991015840119910
+ 1205821198871198991015840 cos 120582
1198861198981015840119909 sin 120582
1198871198991015840119910
times
2
sum
119897=1
1205771198971015840
119886(119909) cos 120582
119887119899119910119889119909119889119910
K1-7119904119905
= minus
1
2
int
119886
0
int
119887
0
120581119866ℎ (119909 119910) 120582119886119898
cos 1205821198861198981015840119909 cos 120582
1198871198991015840119910
times sin 120582119886119898119909 cos 120582
119887119899119910119889119909119889119910
K1-8119904119901
= minus
1
2
int
119886
0
int
119887
0
120581119866ℎ (119909 119910) 1205821198861198981015840 cos 120582
1198861198981015840119909 cos 120582
1198871198991015840119910
times
2
sum
119897=1
120577119897
119887(119910) sin 120582
1198861198981015840119909 119889119909 119889119910
K1-9119904119902
=
1
2
int
119886
0
int
119887
0
120581119866ℎ (119909 119910) cos 1205821198861198981015840119909 cos 120582
1198871198991015840119910
times
2
sum
119897=1
1205771198971015840
119886(119909) cos 120582
119887119899119910119889119909119889119910
M1-1119904119905
=
1
24
120588int
119886
0
int
119887
0
ℎ3
(119909 119910) cos 120582119886119898119909 cos 120582
119887119899119910
times cos 1205821198861198981015840119909 cos 120582
1198871198991015840119910119889119909119889119910
M1-2119904119901
=
1
24
120588int
119886
0
int
119887
0
ℎ3
(119909 119910)
2
sum
119897=1
120577119897
119887(119910) cos 120582
119886119898119909
times cos 1205821198861198981015840119909 cos 120582
1198871198991015840119910119889119909119889119910
M1-3119904119902
=
1
24
120588int
119886
0
int
119887
0
ℎ3
(119909 119910)
2
sum
119897=1
120577119897
119886(119909) cos 120582
119887119899119910
times cos 1205821198861198981015840119909 cos 120582
1198871198991015840119910119889119909119889119910
M1-4119904119905 = M
1-5119904119901 = M1-6119904119902
= M1-7119904119905 = M
1-8119904119901 = M1-9119904119902 = 0
(A1)
B Useful Integrals for the Moderately ThickRectangular Plates with Variable Thicknessand Arbitrary Boundary Conditions
Consider
int
119871
0
cos 120582119898119909 cos 120582
119899119909 119889119909 =
0 119898 = 119899
119871
2
119898 = 119899
119871 119898 = 119899 = 0
int
119871
0
sin 120582119898119909 cos 120582
119899119909119889119909
=
minus
(minus1 + (minus1)119898+119899
) 119871119898
(1198982minus 1198992) 120587
119898 = 119899
0 119898 = 119899
int
119871
0
sin 120582119898119909 sin 120582
119899119909 119889119909 =
0 119898 = 119899
119871
2
119898 = 119899
0 119898 = 119899 = 0
int
119871
0
cos 120582119898119909120577119897
119886(119909) 119889119909
=
41198712
31205872 119897 = 1 119898 = 0
minus
41198712
(41198982
minus 3)
(9 minus 401198982+ 16119898
4) 1205872
119897 = 1 119898 = 0
minus
41198712
31205872 119897 = 2 119898 = 0
41198712
(41198982
minus 3) (minus1)119898
(9 minus 401198982+ 16119898
4) 1205872
119897 = 2 119898 = 0
Shock and Vibration 23
int
119871
0
sin 120582119898119909120577119897
119886(119909) 119889119909
=
0 119897 = 1 119898 = 0
161198712
119898(minus1)119898
(9 minus 401198982+ 16119898
4) 1205872
119897 = 1 119898 = 0
0 119897 = 2 119898 = 0
161198981198712
(9 minus 401198982+ 16119898
4) 1205872
119897 = 2 119898 = 0
int
119871
0
cos 1205821198981199091205771198971015840
119886(119909) 119889119909
=
0 119897 = 1 119898 = 0
161198711198982
(minus1)119898
(9 minus 401198982+ 16119898
4) 120587
119897 = 1 119898 = 0
0 119897 = 2 119898 = 0
161198711198982
(9 minus 401198982+ 16119898
4) 120587
119897 = 2 119898 = 0
int
119871
0
sin 1205821198981199091205771198971015840
119886(119909) 119889119909
=
0 119897 = 1 119898 = 0
4119871119898 (minus3 + 41198982
)
(9 minus 401198982+ 16119898
4) 120587
119897 = 1 119898 = 0
0 119897 = 2 119898 = 0
119871119898(
1
120587 minus 41198982120587
+
3
9120587 minus 41198982120587
) (minus1)119898
119897 = 2 119898 = 0
int
119871
0
1205771198971
119886(119909) 1205771198972
119886(119909) 119889119909 =
1198713
41205872 1198971= 1 1198972= 1
minus
21198713
31205873 1198971= 1 1198972= 2
minus
21198713
31205873 1198971= 2 1198972= 1
1198713
41205872 1198971= 2 1198972= 2
int
119871
0
1205771198971
119886(119909) 1205771198971015840
2
119886(119909) 119889119909 =
0 1198971= 1 1198972= 1
minus
1198712
8120587
1198971= 1 1198972= 2
1198712
8120587
1198971= 2 1198972= 1
0 1198971= 2 1198972= 2
int
119871
0
1205771198971015840
1
119886(119909) 1205771198971015840
2
119886(119909) 119889119909 =
5119871
16
1198971= 1 1198972= 1
minus
119871
2120587
1198971= 1 1198972= 2
minus
119871
2120587
1198971= 2 1198972= 1
5119871
16
1198971= 2 1198972= 2
int
119871
0
119909119899 cos 120582
119898119909 119889119909
=
1198711+119899
1 + 119899
119898 = 0
(1198861+119899HypergeometricPFQtimes [ 12 + 1198992 12 32 + 1198992
minus (14)1198982
1205872
]) times (1 + 119899)minus1
119898 gt 0
(B1)
Conflict of Interests
The authors declare that there is no conflict of interestsregarding the publication of this paper
Acknowledgments
The authors would like to thank the anonymous reviewersfor their very valuable comments The authors also gratefullyacknowledge the financial support from the National NaturalScience Foundation of China (no 51209052) HeilongjiangProvince Youth Science Fund Project (no QC2011C013) andHarbin Science and Technology Development InnovationFoundation of Youth (no 2011RFQXG021)
References
[1] K Liew Y Xiang and S Kitipornchai ldquoResearch on thick platevibration a literature surveyrdquo Journal of Sound and Vibrationvol 180 no 1 pp 163ndash176 1995
[2] J H Chung T Y Chung and K C Kim ldquoVibration analysisof orthotropic mindlin plates with edges elastically restrainedagainst rotationrdquo Journal of Sound and Vibration vol 163 no 1pp 151ndash163 1993
[3] Y K Cheung and D Zhou ldquoVibrations of moderately thickrectangular plates in terms of a set of static Timoshenko beamfunctionsrdquoComputers and Structures vol 78 no 6 pp 757ndash7682000
[4] CMWang ldquoNatural frequencies formula for simply supportedMindlin platesrdquo Journal of Vibration and Acoustics vol 116 no4 pp 536ndash540 1994
[5] K N Saha R C Kar and P K Datta ldquoFree vibration analysisof rectangular Mindlin plates with elastic restraints uniformlydistributed along the edgesrdquo Journal of Sound andVibration vol192 no 4 pp 885ndash902 1996
24 Shock and Vibration
[6] D J Gorman ldquoAccurate free vibration analysis of shear-deformable plates with torsional elastic edge supportrdquo Journalof Sound and Vibration vol 203 no 2 pp 209ndash218 1997
[7] D J Gorman ldquoFree vibration analysis of Mindlin plates withuniform elastic edge support by the superposition methodrdquoJournal of Sound andVibration vol 207 no 3 pp 335ndash350 1997
[8] D J Gorman ldquoAccurate free vibration analysis of point sup-ported mindlin plates by the superposition methodrdquo Journal ofSound and Vibration vol 219 no 2 pp 265ndash277 1999
[9] Y Xiang ldquoVibration of rectangular Mindlin plates resting onnon-homogenous elastic foundationsrdquo International Journal ofMechanical Sciences vol 45 no 6-7 pp 1229ndash1244 2003
[10] Y Xiang and G W Wei ldquoExact solutions for buckling andvibration of stepped rectangular Mindlin platesrdquo InternationalJournal of Solids and Structures vol 41 no 1 pp 279ndash294 2004
[11] Y L Yeh M J Jang and C C Wang ldquoAnalyzing the freevibrations of a plate using finite difference and differential trans-formationmethodrdquoAppliedMathematics andComputation vol178 no 2 pp 493ndash501 2006
[12] Y Xiang S K Lai L Zhou and C W Lim ldquoDSC-Ritz elementmethod for vibration analysis of rectangular Mindlin plateswith mixed edge supportsrdquo European Journal of MechanicsmdashASolids vol 29 no 4 pp 619ndash628 2010
[13] H Nguyen-Xuan C H Thai and T Nguyen-Thoi ldquoIsogeo-metric finite element analysis of composite sandwich platesusing a higher order shear deformation theoryrdquo Composites BEngineering vol 55 pp 558ndash574 2013
[14] C H Thai A J M Ferreira E Carrera and H Nguyen-XuanldquoIsogeometric analysis of laminated composite and sandwichplates using a layerwise deformation theoryrdquo Composite Struc-tures vol 104 pp 196ndash214 2013
[15] C H Thai A Ferreira S Bordas T Rabczuk and HNguyen-Xuan ldquoIsogeometric analysis of laminated compositeand sandwich plates using a new inverse trigonometric sheardeformation theoryrdquo European Journal of Mechanics A Solidsvol 43 pp 89ndash108 2014
[16] H Luong-Van T Nguyen-Thoi G R Liu and P Phung-VanldquoA cell-based smoothed finite elementmethod using three-nodeshear-locking free Mindlin plate element (CS-FEM-MIN3) fordynamic response of laminated composite plates on viscoelasticfoundationrdquo Engineering Analysis with Boundary Elements vol42 pp 8ndash19 2014
[17] T Nguyen-Thoi T Rabczuk T Lam-Phat V Ho-Huu andP Phung-Van ldquoFree vibration analysis of cracked Mindlinplate using an extended cell-based smoothed discrete sheargap method (XCS-DSG3)rdquo Theoretical and Applied FractureMechanics 2014
[18] H Nguyen-Xuan G R Liu and C a Thai-Hoang ldquoAnedge-based smoothed finite element method (ES-FEM) withstabilized discrete shear gap technique for analysis of Reissner-Mindlin platesrdquo Computer Methods in Applied Mechanics andEngineering vol 199 no 9ndash12 pp 471ndash489 2010
[19] P Phung-Van T Nguyen-Thoi T Bui-Xuan and Q Lieu-XuanldquoA cell-based smoothed three-node Mindlin plate element(CS-FEM-MIN3) based on the 119862
0 -type higher-order sheardeformation for geometrically nonlinear analysis of laminatedcomposite platesrdquo Computational Materials Science 2014
[20] NNguyen-Thanh T RabczukHNguyen-Xuan and S BordasldquoAn alternative alpha finite element method with discrete sheargap technique for analysis of isotropicMindlin-Reissner platesrdquoFinite Elements inAnalysis andDesign vol 47 no 5 pp 519ndash5352011
[21] KM Liew Y Xiang and S Kitipornchai ldquoTransverse vibrationof thick rectangular plates-I Comprehensive sets of boundaryconditionsrdquo Computers and Structures vol 49 no 1 pp 1ndash291993
[22] D Zhou ldquoVibrations of Mindlin rectangular plates with elasti-cally restrained edges using static Timoshenko beam functionswith the Rayleigh-Ritz methodrdquo International Journal of Solidsand Structures vol 38 no 32-33 pp 5565ndash5580 2001
[23] D Zhou SH Lo F T KAu andYK Cheung ldquoVibration anal-ysis of rectangular Mindlin plates with internal line supportsusing static Timoshenko beam functionsrdquo International Journalof Mechanical Sciences vol 44 no 12 pp 2503ndash2522 2002
[24] H S Shen J Yang and L Zhang ldquoFree and forced vibrationof Reissner-Mindlin plates with free edges resting on elasticfoundationsrdquo Journal of Sound and Vibration vol 244 no 2 pp299ndash320 2001
[25] Y Xing and B Liu ldquoClosed form solutions for free vibrations ofrectangular Mindlin platesrdquo Acta Mechanica Sinica vol 25 no5 pp 689ndash698 2009
[26] P Gagnon C Gosselin and L Cloutier ldquoA finite strip elementfor the analysis of variable thickness rectangular thick platesrdquoComputers amp Structures vol 63 no 2 pp 349ndash362 1997
[27] T Mlzusawa ldquoVibration of rectangular mindlin plates withtapered thickness by the spline strip methodrdquo Computers andStructures vol 46 no 3 pp 451ndash463 1993
[28] H Nguyen-Xuan and T Nguyen-Thoi ldquoA stabilized smoothedfinite element method for free vibration analysis of Mindlin-Reissner platesrdquo Communications in Numerical Methods inEngineering with Biomedical Applications vol 25 no 8 pp 882ndash906 2009
[29] K M Liew L X Peng and S Kitipornchai ldquoVibration anal-ysis of corrugated Reissner-Mindlin plates using a mesh-freeGalerkin methodrdquo International Journal of Mechanical Sciencesvol 51 no 9-10 pp 642ndash652 2009
[30] Y Hou GWWei and Y Xiang ldquoDSC-Ritz method for the freevibration analysis of Mindlin platesrdquo International Journal forNumerical Methods in Engineering vol 62 no 2 pp 262ndash2882005
[31] F- Liu and K M Liew ldquoVibration analysis of discontinuousMindlin plates by differential quadrature element methodrdquoJournal of Vibration and Acoustics Transactions of the ASMEvol 121 no 2 pp 204ndash208 1999
[32] P Malekzadeh G Karami and M Farid ldquoA semi-analyticalDQEM for free vibration analysis of thick plates with twoopposite edges simply supportedrdquoComputerMethods inAppliedMechanics and Engineering vol 193 no 45ndash47 pp 4781ndash47962004
[33] P Malekzadeh and S A Shahpari ldquoFree vibration analysisof variable thickness thin and moderately thick plates withelastically restrained edges by DQMrdquo Thin-Walled Structuresvol 43 no 7 pp 1037ndash1050 2005
[34] R E Diaz-Contreras and S Nomura ldquoGreenrsquos function appliedto solution ofMindlin platesrdquoComputers and Structures vol 60no 1 pp 41ndash48 1996
[35] T Sakiyama and M Huang ldquoFree vibration analysis of rect-angular plates with variable thicknessrdquo Journal of Sound andVibration vol 216 no 3 pp 379ndash397 1998
[36] M Huang X Q Ma T Sakiyama H Matuda and C MoritaldquoFree vibration analysis of orthotropic rectangular plates withvariable thickness and general boundary conditionsrdquo Journal ofSound and Vibration vol 288 no 4-5 pp 931ndash955 2005
Shock and Vibration 25
[37] I Shufrin and M Eisenberger ldquoVibration of shear deformableplates with variable thicknessmdashfirst-order and higher-orderanalysesrdquo Journal of Sound and Vibration vol 290 no 1-2 pp465ndash489 2006
[38] S A Eftekhari and A A Jafari ldquoAccurate variational approachfor free vibration of variable thickness thin and thick plates withedges elastically restrained against translation and rotationrdquoInternational Journal of Mechanical Sciences vol 68 pp 35ndash462013
[39] W L Li X Zhang J Du and Z Liu ldquoAn exact series solutionfor the transverse vibration of rectangular plates with generalelastic boundary supportsrdquo Journal of Sound amp Vibration vol321 no 1-2 pp 254ndash269 2009
[40] X Shi D Shi W L Li and Q Wang ldquoA unified method forfree vibration analysis of circular annular and sector plateswith arbitrary boundary conditionsrdquo Journal of Vibration andControl 2014
[41] D Y Shi X J Shi W L Li and Q S Wang ldquoFree transversevibrations of orthotropic thin rectangular plates with arbitraryelastic edge supportsrdquo Journal of Vibroengineering vol 16 no 1pp 389ndash398 2014
[42] Y Chen G Jin and Z Liu ldquoFree vibration analysis of circu-lar cylindrical shell with non-uniform elastic boundary con-straintsrdquo International Journal of Mechanical Sciences vol 74pp 120ndash132 2013
[43] L Dai T Yang J Du W L Li and M J Brennan ldquoAn exactseries solution for the vibration analysis of cylindrical shellswith arbitrary boundary conditionsrdquo Applied Acoustics vol 74no 3 pp 440ndash449 2013
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Shock and Vibration
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18 Shock and Vibration
Table 10 Frequency parameterΩ = 1205961198862
(120588ℎ01198630)12 for S-S-S-S moderately thick plates (119886119887 = 1) with linearly varying thickness and elastic
rotation support (ℎ(119909 119910) = ℎ0(1 + 05119909
2
)(1 + 051199102
) 1198701199090= 119870119909119886= 1198701199100= 119870119910119887= Γ119909)
Γ119909
aℎ0119886
Model sequence1 2 3 4 5 6
0
005 23462 54999 55573 82950 100900 101187(23741b) (55114) (55983) (83146) (101004) (101347)
01 18606 42725 42971 60040 71167 71761(18936b) (43014) (43148) (60247) (71642) (71993)
015 53055 65251 85404 106700 114880 143531(53467b) (65472) (85984) (107211) (115687) (144007)
02 17439 21531 32170 44987 59438 64179(17874b) (21748) (32651) (45264) (60188) (64879)
05
005 6904 13374 13514 19400 22904 23056(9909b) (13548) (13651) (19564) (23088) (23379)
01 5316 10237 10314 14499 17155 17264(5374b) (10448) (10651) (14764) (17188) (17879)
015 4136 8201 8220 11474 13578 13671(4174b) (8348) (8651) (11764) (13688) (13879)
02 3449 6915 6924 9549 11225 11327(3474b) (6948) (6951) (9764) (11388) (11579)
1
005 7040 13617 13745 19726 23233 23402(7074b) (13748) (13851) (19964) (23488) (23879)
01 5712 10696 10793 14991 17594 17744(5774b) (10748) (10851) (15054) (17788) (17879)
015 4489 8507 8540 11759 13803 13922(4514b) (8548) (8611) (11864) (13988) (14119)
02 3677 7084 7090 9692 11330 11445(3684b) (7148) (7191) (9764) (11588) (11879)
15
005 7089 13708 13828 19847 23355 23530(7124b) (13748) (13951) (19967) (23588) (23879)
01 5895 10923 11024 15239 17813 17988(5874b) (10748) (11651) (15264) (17818) (17979)
015 4697 8694 8737 11938 13790 14083(4774b) (8748) (8951) (11864) (13688) (14879)
02 3836 7201 7210 9792 11403 11530(3874b) (7748) (7651) (9864) (11588) (11879)
2
005 7114 13754 13871 19911 23418 23597(7174b) (12848) (13851) (19864) (23588) (23879)
01 6001 11060 11159 15391 17944 18136(6004b) (11148) (11151) (15464) (17988) (18379)
015 4836 8821 8872 12061 14037 14189(4874b) (8848) (8951) (12164) (14188) (14879)
02 3954 7289 7301 9871 11457 11594(3974b) (7348) (7651) (9964) (11488) (11679)
aΓ119909= 1198701198941198700 (1198700= 1 times 10
9 Nmrad 119894 = 1199090 119909119886 1199100 119910119887)bResults in parentheses are taken from FEM
and rotational restraints is set equal to 1198961199090
= 119896119909119886
= 1198961199100
=
119896119910119887
= Γ119908and 119870
1199090= 119870119909119886
= 1198701199100
= 119870119910119887
= Γ119909 respectively
The six frequency parametersΩ = 1205961198862
(120588ℎ01198630)12 are shown
in Table 11 for several different restraining coefficient valuesIt can also be noticed that the Fourier series method is able
to predict the modal characteristics of nonlinearly varyingthickness moderately thick plate with not only classicalboundary conditions but also elastic edge restraints correctly
Finally the influence of the gradient on the fundamentalfrequency parameters of a nonlinearly varying thickness
Shock and Vibration 19
Table 11 Frequency parameterΩ = 1205961198862
(120588ℎ01198630)12 for moderately thick plates (119886119887 = 1) with linearly varying thickness and elastic rotation
and translation support (ℎ(119909 119910) = ℎ0(1 + 05119909
2
)(1 + 051199102
) 1198701199090= 119870119909119886= 1198701199100= 119870119910119887= Γ119909 and 119896
1199090= 119896119909119886= 1198961199100= 119896119910119887= Γ119908)
Γ119909
aΓ119908
aℎ0119886
Model sequence1 2 3 4 5 6
05 0
005 12926 13041 25090 47935 47943 61761(12931b) (13053) 25099 (47947) (47952) (61788)
01 10900 10988 21545 39980 40147 50322(10904b) (10992) (21557) (39984) (40166) (50334)
015 8245 8298 18081 31503 32617 40950(8247b) (8230) (18092) (31508) (32622) (40955)
02 6106 6124 15639 25631 27601 34458(6110b) (6125) (15642) (25637) (27617) (34462)
1 05
005 21575 31615 31644 42573 57889 58607(21580b) (31617) (31644) (42581) (57893) (58610)
01 8451 16024 16210 25969 43839 43867(8451b) (16025) (16217) (25972) (43844) (43870)
015 4682 11479 11480 20642 35172 35598(4684b) (11480) (11481) (20644) (35178) (35604)
02 3059 8621 8648 17320 28487 29638(3062b) (8619) (8644) (17327) (28492) (29641)
15 1
005 27713 40203 40978 52951 65352 66815(27714b) (40207) (40988) (52972) (65368) (66822)
01 11667 19322 19615 29122 46115 46183(11674b) (19327) (19619) (29137) (46120) (46188)
015 6544 13416 13481 22405 37238 37430(6547b) (13420) (13488) (22410) (37241) (37434)
02 4295 10175 10190 18543 30330 31056(4296b) (10177) (10194) (18544) (30336) (31060)
2 2
005 33453 50854 52088 66311 76610 78947(33455b) (50852) (50097) (66321) (76627) (78971)
01 15832 24130 24372 33924 49118 49362(15830b) (24127) (24374) (33931) (49147) (49382)
015 9060 15956 16099 24754 39134 39249(9064b) (15961) (16102) (24755) (39138) (39452)
02 5992 11955 11971 20034 31908 32392(5991b) (11957) (11978) (20042) (31910) (32398)
aΓ119909= 1198701198941198700 (1198700= 1 times 10
9 Nmrad 119894 = 1199090 119909119886 1199100 119910119887) Γ119908= 1198961198941198960 (1198960= 1 times 10
9 Nm 119894 = 1199090 119909119886 1199100 119910119887)bResults in parentheses are taken from FEM
moderately thick plate is investigated The plate is elasticallyrestrained in which the stiffness of the boundary springsis taken as 119896
1199090= 119896119909119886
= 1198961199100
= 119896119910119887
= Γ119908
= 2 and1198701199090
= 119870119909119886
= 1198701199100
= 119870119910119887
= Γ119909= 2 respectively The six
frequency parameters Ω = 1205961198862
(120588ℎ01198630)12 are presented in
Table 12 for several different slop values It can be seen thatthe fundamental frequency parameters will decrease with theincrease of the parameter ℎ
0119886
In the above examples it has been demonstrated thatthe presented method can be universally applied to nonlin-early varying thickness moderately thick plates with severalboundary conditions and different structure parametersNew results are obtained for plates with nonlinear variationthickness in both directions subjected to general elastic
boundary restraints which may be used for benchmarkingof researchers in the field In addition it is interesting to seethat the nature frequency decreases with the increase of theindex for thickness function
4 Conclusions
In this paper a modified Fourier method has been pre-sented to study the free vibration behaviors of moderatelythick rectangular plates with variable thickness and arbi-trary boundary conditionsThe first-order shear deformationplate theory is adopted to formulate the theoretical modelThe displacements and rotation components of the plateregardless of boundary conditions are invariantly expressedas the superposition of a 2D Fourier cosine series and four
20 Shock and Vibration
Table 12 Frequency parameterΩ = 1205961198862
(120588ℎ01198630)12 for elastic support moderately thick plates (119886119887 = 1) with different gradient (119896
1199090= 119896119909119886=
1198961199100= 119896119910119887= Γ119908= 2 119870
1199090= 119870119909119886= 1198701199100= 119870119910119887= Γ119909= 2)
120572 120573 ℎ0119886
Model sequence1 2 3 4 5 6
120572 = 120573 = 01
005 30187 51102 51141 67066 73954 77412(30188a) (51105) (51144) (67068) (73961) (77477)
01 16995 24719 24724 33071 44343 44819(16996a) (24725) (24729) (33084) (44347) (44822)
015 10077 15872 15883 23443 35937 36065(10081a) (15877) (15892) (23447) (35941) (36084)
02 6713 11953 11958 19086 30372 30604(6713a) (11957) (11965) (19092) (30378) (30612)
120572 = 120573 = 02
005 31206 51436 51614 67058 74398 77595(31208a) (51438) (51619) (67064) (74402) (77596)
01 16786 24503 24508 33144 45583 45923(16790a) (24510) (24513) (33147) (45587) (45932)
015 9821 15865 15906 23766 36897 36990(9824a) (15870) (15907) (23768) (36894) (36997)
02 6523 11978 11992 19356 30912 31162(6524a) (11981) (11997) (19362) (30916) (31174)
120572 = 120573 = 03
005 32097 51462 51900 66832 74991 77863(32098a) (51466) (51907) (66849) (74996) (77872)
01 16507 24332 24383 33330 46806 47074(16511a) (24327) (24380) (33339) (46810) (47086)
015 9562 15885 15964 24103 37748 37828(9562a) (15892) (15977) (21105) (37751) (37832)
02 6338 11992 12013 19603 31338 31634(6340a) (11994) (12011) (19612) (31344) (31639)
120572 = 120573 = 04
005 32851 51239 52046 66547 75735 78303(33455a) (50852) (50097) (66321) (76627) (78971)
01 16182 24209 24343 33598 47990 48230(33455a) (50852) (50097) (66321) (76627) (78971)
015 9307 15919 16034 24436 38491 38579(33455a) (50852) (50097) (66321) (76627) (78971)
02 6161 11986 12008 19828 31666 32040(33455a) (50852) (50097) (66321) (76627) (78971)
120572 = 120573 = 05
005 33453 50854 52088 66311 76610 78947(33455a) (50852) (50097) (66321) (76627) (78971)
01 15832 24130 24372 33924 49118 49362(15830a) (24127) (24374) (33931) (49147) (49382)
015 9060 15956 16099 24754 39134 39249(9064a) (15961) (16102) (24755) (39138) (39452)
02 5992 11955 11971 20034 31908 32392(5991a) (11957) (11978) (20042) (31910) (32398)
aResults in parentheses are taken from FEM
supplementary functions in the form of the product of apolynomial function and a single cosine series expansionto ensure and accelerate the convergence of the solutionAt each edge of the plate the general restraint conditionsare implemented by introducing one group of linear springsand two groups of rotational springs which are continuouslydistributed and determined by the stiffness of these springs
Instead of seeking a solution in strong forms in the previousstudies all the Fourier coefficients will be treated equallyand independently as the generalized coordinates and solveddirectly from the Rayleigh-Ritz technique The change of theboundary conditions can be easily achieved by only varyingthe stiffness of the three sets of boundary springs along alledges of the rectangular plates without involving any change
Shock and Vibration 21
to the solution procedure The convergence of the presentsolution is examined and the excellent accuracy is validatedby comparison with existing results published in the lit-erature and FEM data Excellent agreements are obtainedfrom these comparisons The proposed method providesa unified means for extracting the modal parameters andpredicting the vibration behaviors of moderately thick plateswith variable thickness variation functions and arbitraryelastic edge restraints A variety of free vibration results formoderately thick rectangular plates with different thicknessvariation functions and boundary conditions are presentedNew results for free vibration of moderately thick rectangularplates with various thickness variation functions and edgeconditions are presented which may be used for benchmark-ing of researchers in the field
Appendices
A Representative Calculation forStiffness and Mass Matrices
To illuminate the particular expression of the mass andstiffness matrixes clearly and tersely four new variables aredefined as 119904 = 119898 lowast (119873 + 1) + 119899 + 1 119905 = 119898
1015840
lowast (119873 + 1) + 1198991015840
+ 1119901 = (119897minus1)lowast (119872+1)+119898
1015840
+1 and 119902 = (119897minus1)lowast (119873+1)+1198991015840
+1The first row elements of K andM are given below
K1-1119904119905
=
1
2
int
119886
0
int
119887
0
[119863 (119909 119910)
times 21205821198861198981205821198861198981015840 sin 120582
119886119898119909 sin 120582
1198861198981015840119909 cos 120582
119887119899119910
times cos 1205821198871198991015840119910 + (1 minus 120583) 120582
1198871198991205821198871198991015840 cos 120582
119886119898119909
times cos 1205821198861198981015840 sin 120582
119887119899119910 sin 120582
1198871198991015840119910
+ 120581119866ℎ (119909 119910) cos 120582119886119898119909 cos 120582
1198861198981015840119909 cos 120582
119887119899
times119910 cos 1205821198871198991015840119910119889119909119889119910] 119889119909 119889119910
+ int
119886
0
1198701199091199100
cos 120582119886119898119909 cos 120582
1198861198981015840119909
+(minus1)119898+1198981015840
119870119909119910119887
cos 120582119886119898119909 cos 120582
1198861198981015840119909 119889119909
+ int
119887
0
1198701199090cos 120582119887119899119910 cos 120582
1198871198991015840119910
+(minus1)119899+1198991015840
119870119909119886cos 120582119887119899119910 cos 120582
1198871198991015840119910 119889119910
K1-2119904119901
=
1
2
int
119886
0
int
119887
0
[119863 (119909 119910)21205821198861198981205821198861198981015840 sin 120582
119886119898119909 sin 120582
1198861198981015840119909
times
2
sum
119897=1
120577119897
119887(119910) cos 120582
1198871198991015840119910
minus (1 minus 120583) 1205821198871198991015840 cos 120582
119886119898119909 cos 120582
1198861198981015840119909
times
2
sum
119897=1
1205771198971015840
119887(119910) sin 120582
1198871198991015840119910 + 120581119866ℎ (119909 119910)
times ( cos 1205821198861198981015840119909 cos 120582
1198871198991015840119910
times
2
sum
119897=1
120577119897
119887(119910) cos 120582
119886119898119909)]119889119909119889119910
+ int
119886
0
1198701199091199100
cos 120582119886119898119909 cos 120582
1198861198981015840119909
2
sum
119897=1
120577119897
119887(0)
+(minus1)1198991015840
119870119909119910119887
cos 120582119886119898119909 cos 120582
1198861198981015840119909
2
sum
119897=1
120577119897
119887(119887) 119889119909
+ int
119887
0
1198701199090
2
sum
119897=1
120577119897
119887(119910) cos 120582
1198871198991015840119910
+(minus1)119898+1198981015840
119870119909119886
2
sum
119897=1
120577119897
119887(119910) cos 120582
1198871198991015840119910119889119910
K1-3119904119902
=
1
2
int
119886
0
int
119887
0
[119863 (119909 119910) minus 21205821198861198981015840 sin 120582
1198861198981015840119909 cos 120582
1198871198991015840119910
times
2
sum
119897=1
1205771198971015840
119886(119909) cos 120582
119887119899119910
+ (1 minus 120583) 1205821198871198991205821198871198991015840 cos 120582
1198861198981015840119909
times sin 1205821198871198991015840119910
2
sum
119897=1
120577119897
119886(119909) sin 120582
119887119899119910
+ 120581119866ℎ (119909 119910) cos 1205821198861198981015840119909 cos 120582
1198871198991015840119910
times
2
sum
119897=1
120577119897
119886(119909) cos 120582
119887119899119910]119889119909119889119910
+ int
119886
0
1198701199091199100
cos 1205821198861198981015840119909
2
sum
119897=1
120577119897
119886(119909)
+(minus1)119899+1198991015840
119870119909119910119887
cos 1205821198861198981015840119909
2
sum
119897=1
120577119897
119886(119909) 119889119909
+ int
119887
0
1198701199090
2
sum
119897=1
120577119897
119886(0) cos 120582
1198871198991015840119910
+(minus1)119898+1198981015840
119870119909119886
2
sum
119897=1
120577119897
119886(0) cos 120582
1198871198991015840119910119889119910
K1-4119904119905
=
1
2
int
119886
0
int
119887
0
119863(119909 119910) 21205821198871198991205821198861198981015840 sin 120582
1198861198981015840119909 cos 120582
1198871198991015840119910
times cos 120582119886119898119909 sin 120582
119887119899119910 minus 2(1 minus 120583)
119887119899
times 1205821198861198981015840 sin 120582
1198861198981015840119909 cos 120582
1198871198991015840119910 cos 120582
119886119898119909
times sin 120582119887119899119910 + (1 minus 120583) 120582
1198861198981205821198871198991015840 cos 120582
1198861198981015840119909
times sin 1205821198871198991015840119910 sin 120582
119886119898119909 cos 120582
119887119899119910 119889119909 119889119910
22 Shock and Vibration
K1-5119904119901
=
1
2
int
119886
0
int
119887
0
119863(119909 119910) minus 21205821198861198981015840 sin 120582
1198861198981015840119909 cos 120582
1198871198991015840119910
times
2
sum
119897=1
1205771198971015840
119887(119910) cos 120582
119886119898119909
+ 2 (1 minus 120583) 1205821198861198981015840 sin 120582
1198861198981015840119909 cos 120582
1198871198991015840119910
times
2
sum
119897=1
1205771198971015840
119887(119910) cos 120582
119886119898119909
+ (1 minus 120583) 1205821198861198981205821198871198991015840 cos 120582
1198861198981015840119909 sin 120582
1198871198991015840119910
times
2
sum
119897=1
120577119897
119887(119910) sin 120582
119886119898119909119889119909119889119910
K1-6119904119902
=
1
2
int
119886
0
int
119887
0
119863(119909 119910)1205821198871198991205821198861198981015840 sin 120582
1198861198981015840119909 cos 120582
1198861198991015840119910
times
2
sum
119897=1
120577119897
119886(119909) sin 120582
1198871198991015840119910
minus 2 (1 minus 120583) 1205821198871198991205821198861198981015840 sin 120582
1198861198981015840119909 cos 120582
1198861198991015840119910
times
2
sum
119897=1
120577119897
119886(119909) sin 120582
1198871198991015840119910
+ 1205821198871198991015840 cos 120582
1198861198981015840119909 sin 120582
1198871198991015840119910
times
2
sum
119897=1
1205771198971015840
119886(119909) cos 120582
119887119899119910119889119909119889119910
K1-7119904119905
= minus
1
2
int
119886
0
int
119887
0
120581119866ℎ (119909 119910) 120582119886119898
cos 1205821198861198981015840119909 cos 120582
1198871198991015840119910
times sin 120582119886119898119909 cos 120582
119887119899119910119889119909119889119910
K1-8119904119901
= minus
1
2
int
119886
0
int
119887
0
120581119866ℎ (119909 119910) 1205821198861198981015840 cos 120582
1198861198981015840119909 cos 120582
1198871198991015840119910
times
2
sum
119897=1
120577119897
119887(119910) sin 120582
1198861198981015840119909 119889119909 119889119910
K1-9119904119902
=
1
2
int
119886
0
int
119887
0
120581119866ℎ (119909 119910) cos 1205821198861198981015840119909 cos 120582
1198871198991015840119910
times
2
sum
119897=1
1205771198971015840
119886(119909) cos 120582
119887119899119910119889119909119889119910
M1-1119904119905
=
1
24
120588int
119886
0
int
119887
0
ℎ3
(119909 119910) cos 120582119886119898119909 cos 120582
119887119899119910
times cos 1205821198861198981015840119909 cos 120582
1198871198991015840119910119889119909119889119910
M1-2119904119901
=
1
24
120588int
119886
0
int
119887
0
ℎ3
(119909 119910)
2
sum
119897=1
120577119897
119887(119910) cos 120582
119886119898119909
times cos 1205821198861198981015840119909 cos 120582
1198871198991015840119910119889119909119889119910
M1-3119904119902
=
1
24
120588int
119886
0
int
119887
0
ℎ3
(119909 119910)
2
sum
119897=1
120577119897
119886(119909) cos 120582
119887119899119910
times cos 1205821198861198981015840119909 cos 120582
1198871198991015840119910119889119909119889119910
M1-4119904119905 = M
1-5119904119901 = M1-6119904119902
= M1-7119904119905 = M
1-8119904119901 = M1-9119904119902 = 0
(A1)
B Useful Integrals for the Moderately ThickRectangular Plates with Variable Thicknessand Arbitrary Boundary Conditions
Consider
int
119871
0
cos 120582119898119909 cos 120582
119899119909 119889119909 =
0 119898 = 119899
119871
2
119898 = 119899
119871 119898 = 119899 = 0
int
119871
0
sin 120582119898119909 cos 120582
119899119909119889119909
=
minus
(minus1 + (minus1)119898+119899
) 119871119898
(1198982minus 1198992) 120587
119898 = 119899
0 119898 = 119899
int
119871
0
sin 120582119898119909 sin 120582
119899119909 119889119909 =
0 119898 = 119899
119871
2
119898 = 119899
0 119898 = 119899 = 0
int
119871
0
cos 120582119898119909120577119897
119886(119909) 119889119909
=
41198712
31205872 119897 = 1 119898 = 0
minus
41198712
(41198982
minus 3)
(9 minus 401198982+ 16119898
4) 1205872
119897 = 1 119898 = 0
minus
41198712
31205872 119897 = 2 119898 = 0
41198712
(41198982
minus 3) (minus1)119898
(9 minus 401198982+ 16119898
4) 1205872
119897 = 2 119898 = 0
Shock and Vibration 23
int
119871
0
sin 120582119898119909120577119897
119886(119909) 119889119909
=
0 119897 = 1 119898 = 0
161198712
119898(minus1)119898
(9 minus 401198982+ 16119898
4) 1205872
119897 = 1 119898 = 0
0 119897 = 2 119898 = 0
161198981198712
(9 minus 401198982+ 16119898
4) 1205872
119897 = 2 119898 = 0
int
119871
0
cos 1205821198981199091205771198971015840
119886(119909) 119889119909
=
0 119897 = 1 119898 = 0
161198711198982
(minus1)119898
(9 minus 401198982+ 16119898
4) 120587
119897 = 1 119898 = 0
0 119897 = 2 119898 = 0
161198711198982
(9 minus 401198982+ 16119898
4) 120587
119897 = 2 119898 = 0
int
119871
0
sin 1205821198981199091205771198971015840
119886(119909) 119889119909
=
0 119897 = 1 119898 = 0
4119871119898 (minus3 + 41198982
)
(9 minus 401198982+ 16119898
4) 120587
119897 = 1 119898 = 0
0 119897 = 2 119898 = 0
119871119898(
1
120587 minus 41198982120587
+
3
9120587 minus 41198982120587
) (minus1)119898
119897 = 2 119898 = 0
int
119871
0
1205771198971
119886(119909) 1205771198972
119886(119909) 119889119909 =
1198713
41205872 1198971= 1 1198972= 1
minus
21198713
31205873 1198971= 1 1198972= 2
minus
21198713
31205873 1198971= 2 1198972= 1
1198713
41205872 1198971= 2 1198972= 2
int
119871
0
1205771198971
119886(119909) 1205771198971015840
2
119886(119909) 119889119909 =
0 1198971= 1 1198972= 1
minus
1198712
8120587
1198971= 1 1198972= 2
1198712
8120587
1198971= 2 1198972= 1
0 1198971= 2 1198972= 2
int
119871
0
1205771198971015840
1
119886(119909) 1205771198971015840
2
119886(119909) 119889119909 =
5119871
16
1198971= 1 1198972= 1
minus
119871
2120587
1198971= 1 1198972= 2
minus
119871
2120587
1198971= 2 1198972= 1
5119871
16
1198971= 2 1198972= 2
int
119871
0
119909119899 cos 120582
119898119909 119889119909
=
1198711+119899
1 + 119899
119898 = 0
(1198861+119899HypergeometricPFQtimes [ 12 + 1198992 12 32 + 1198992
minus (14)1198982
1205872
]) times (1 + 119899)minus1
119898 gt 0
(B1)
Conflict of Interests
The authors declare that there is no conflict of interestsregarding the publication of this paper
Acknowledgments
The authors would like to thank the anonymous reviewersfor their very valuable comments The authors also gratefullyacknowledge the financial support from the National NaturalScience Foundation of China (no 51209052) HeilongjiangProvince Youth Science Fund Project (no QC2011C013) andHarbin Science and Technology Development InnovationFoundation of Youth (no 2011RFQXG021)
References
[1] K Liew Y Xiang and S Kitipornchai ldquoResearch on thick platevibration a literature surveyrdquo Journal of Sound and Vibrationvol 180 no 1 pp 163ndash176 1995
[2] J H Chung T Y Chung and K C Kim ldquoVibration analysisof orthotropic mindlin plates with edges elastically restrainedagainst rotationrdquo Journal of Sound and Vibration vol 163 no 1pp 151ndash163 1993
[3] Y K Cheung and D Zhou ldquoVibrations of moderately thickrectangular plates in terms of a set of static Timoshenko beamfunctionsrdquoComputers and Structures vol 78 no 6 pp 757ndash7682000
[4] CMWang ldquoNatural frequencies formula for simply supportedMindlin platesrdquo Journal of Vibration and Acoustics vol 116 no4 pp 536ndash540 1994
[5] K N Saha R C Kar and P K Datta ldquoFree vibration analysisof rectangular Mindlin plates with elastic restraints uniformlydistributed along the edgesrdquo Journal of Sound andVibration vol192 no 4 pp 885ndash902 1996
24 Shock and Vibration
[6] D J Gorman ldquoAccurate free vibration analysis of shear-deformable plates with torsional elastic edge supportrdquo Journalof Sound and Vibration vol 203 no 2 pp 209ndash218 1997
[7] D J Gorman ldquoFree vibration analysis of Mindlin plates withuniform elastic edge support by the superposition methodrdquoJournal of Sound andVibration vol 207 no 3 pp 335ndash350 1997
[8] D J Gorman ldquoAccurate free vibration analysis of point sup-ported mindlin plates by the superposition methodrdquo Journal ofSound and Vibration vol 219 no 2 pp 265ndash277 1999
[9] Y Xiang ldquoVibration of rectangular Mindlin plates resting onnon-homogenous elastic foundationsrdquo International Journal ofMechanical Sciences vol 45 no 6-7 pp 1229ndash1244 2003
[10] Y Xiang and G W Wei ldquoExact solutions for buckling andvibration of stepped rectangular Mindlin platesrdquo InternationalJournal of Solids and Structures vol 41 no 1 pp 279ndash294 2004
[11] Y L Yeh M J Jang and C C Wang ldquoAnalyzing the freevibrations of a plate using finite difference and differential trans-formationmethodrdquoAppliedMathematics andComputation vol178 no 2 pp 493ndash501 2006
[12] Y Xiang S K Lai L Zhou and C W Lim ldquoDSC-Ritz elementmethod for vibration analysis of rectangular Mindlin plateswith mixed edge supportsrdquo European Journal of MechanicsmdashASolids vol 29 no 4 pp 619ndash628 2010
[13] H Nguyen-Xuan C H Thai and T Nguyen-Thoi ldquoIsogeo-metric finite element analysis of composite sandwich platesusing a higher order shear deformation theoryrdquo Composites BEngineering vol 55 pp 558ndash574 2013
[14] C H Thai A J M Ferreira E Carrera and H Nguyen-XuanldquoIsogeometric analysis of laminated composite and sandwichplates using a layerwise deformation theoryrdquo Composite Struc-tures vol 104 pp 196ndash214 2013
[15] C H Thai A Ferreira S Bordas T Rabczuk and HNguyen-Xuan ldquoIsogeometric analysis of laminated compositeand sandwich plates using a new inverse trigonometric sheardeformation theoryrdquo European Journal of Mechanics A Solidsvol 43 pp 89ndash108 2014
[16] H Luong-Van T Nguyen-Thoi G R Liu and P Phung-VanldquoA cell-based smoothed finite elementmethod using three-nodeshear-locking free Mindlin plate element (CS-FEM-MIN3) fordynamic response of laminated composite plates on viscoelasticfoundationrdquo Engineering Analysis with Boundary Elements vol42 pp 8ndash19 2014
[17] T Nguyen-Thoi T Rabczuk T Lam-Phat V Ho-Huu andP Phung-Van ldquoFree vibration analysis of cracked Mindlinplate using an extended cell-based smoothed discrete sheargap method (XCS-DSG3)rdquo Theoretical and Applied FractureMechanics 2014
[18] H Nguyen-Xuan G R Liu and C a Thai-Hoang ldquoAnedge-based smoothed finite element method (ES-FEM) withstabilized discrete shear gap technique for analysis of Reissner-Mindlin platesrdquo Computer Methods in Applied Mechanics andEngineering vol 199 no 9ndash12 pp 471ndash489 2010
[19] P Phung-Van T Nguyen-Thoi T Bui-Xuan and Q Lieu-XuanldquoA cell-based smoothed three-node Mindlin plate element(CS-FEM-MIN3) based on the 119862
0 -type higher-order sheardeformation for geometrically nonlinear analysis of laminatedcomposite platesrdquo Computational Materials Science 2014
[20] NNguyen-Thanh T RabczukHNguyen-Xuan and S BordasldquoAn alternative alpha finite element method with discrete sheargap technique for analysis of isotropicMindlin-Reissner platesrdquoFinite Elements inAnalysis andDesign vol 47 no 5 pp 519ndash5352011
[21] KM Liew Y Xiang and S Kitipornchai ldquoTransverse vibrationof thick rectangular plates-I Comprehensive sets of boundaryconditionsrdquo Computers and Structures vol 49 no 1 pp 1ndash291993
[22] D Zhou ldquoVibrations of Mindlin rectangular plates with elasti-cally restrained edges using static Timoshenko beam functionswith the Rayleigh-Ritz methodrdquo International Journal of Solidsand Structures vol 38 no 32-33 pp 5565ndash5580 2001
[23] D Zhou SH Lo F T KAu andYK Cheung ldquoVibration anal-ysis of rectangular Mindlin plates with internal line supportsusing static Timoshenko beam functionsrdquo International Journalof Mechanical Sciences vol 44 no 12 pp 2503ndash2522 2002
[24] H S Shen J Yang and L Zhang ldquoFree and forced vibrationof Reissner-Mindlin plates with free edges resting on elasticfoundationsrdquo Journal of Sound and Vibration vol 244 no 2 pp299ndash320 2001
[25] Y Xing and B Liu ldquoClosed form solutions for free vibrations ofrectangular Mindlin platesrdquo Acta Mechanica Sinica vol 25 no5 pp 689ndash698 2009
[26] P Gagnon C Gosselin and L Cloutier ldquoA finite strip elementfor the analysis of variable thickness rectangular thick platesrdquoComputers amp Structures vol 63 no 2 pp 349ndash362 1997
[27] T Mlzusawa ldquoVibration of rectangular mindlin plates withtapered thickness by the spline strip methodrdquo Computers andStructures vol 46 no 3 pp 451ndash463 1993
[28] H Nguyen-Xuan and T Nguyen-Thoi ldquoA stabilized smoothedfinite element method for free vibration analysis of Mindlin-Reissner platesrdquo Communications in Numerical Methods inEngineering with Biomedical Applications vol 25 no 8 pp 882ndash906 2009
[29] K M Liew L X Peng and S Kitipornchai ldquoVibration anal-ysis of corrugated Reissner-Mindlin plates using a mesh-freeGalerkin methodrdquo International Journal of Mechanical Sciencesvol 51 no 9-10 pp 642ndash652 2009
[30] Y Hou GWWei and Y Xiang ldquoDSC-Ritz method for the freevibration analysis of Mindlin platesrdquo International Journal forNumerical Methods in Engineering vol 62 no 2 pp 262ndash2882005
[31] F- Liu and K M Liew ldquoVibration analysis of discontinuousMindlin plates by differential quadrature element methodrdquoJournal of Vibration and Acoustics Transactions of the ASMEvol 121 no 2 pp 204ndash208 1999
[32] P Malekzadeh G Karami and M Farid ldquoA semi-analyticalDQEM for free vibration analysis of thick plates with twoopposite edges simply supportedrdquoComputerMethods inAppliedMechanics and Engineering vol 193 no 45ndash47 pp 4781ndash47962004
[33] P Malekzadeh and S A Shahpari ldquoFree vibration analysisof variable thickness thin and moderately thick plates withelastically restrained edges by DQMrdquo Thin-Walled Structuresvol 43 no 7 pp 1037ndash1050 2005
[34] R E Diaz-Contreras and S Nomura ldquoGreenrsquos function appliedto solution ofMindlin platesrdquoComputers and Structures vol 60no 1 pp 41ndash48 1996
[35] T Sakiyama and M Huang ldquoFree vibration analysis of rect-angular plates with variable thicknessrdquo Journal of Sound andVibration vol 216 no 3 pp 379ndash397 1998
[36] M Huang X Q Ma T Sakiyama H Matuda and C MoritaldquoFree vibration analysis of orthotropic rectangular plates withvariable thickness and general boundary conditionsrdquo Journal ofSound and Vibration vol 288 no 4-5 pp 931ndash955 2005
Shock and Vibration 25
[37] I Shufrin and M Eisenberger ldquoVibration of shear deformableplates with variable thicknessmdashfirst-order and higher-orderanalysesrdquo Journal of Sound and Vibration vol 290 no 1-2 pp465ndash489 2006
[38] S A Eftekhari and A A Jafari ldquoAccurate variational approachfor free vibration of variable thickness thin and thick plates withedges elastically restrained against translation and rotationrdquoInternational Journal of Mechanical Sciences vol 68 pp 35ndash462013
[39] W L Li X Zhang J Du and Z Liu ldquoAn exact series solutionfor the transverse vibration of rectangular plates with generalelastic boundary supportsrdquo Journal of Sound amp Vibration vol321 no 1-2 pp 254ndash269 2009
[40] X Shi D Shi W L Li and Q Wang ldquoA unified method forfree vibration analysis of circular annular and sector plateswith arbitrary boundary conditionsrdquo Journal of Vibration andControl 2014
[41] D Y Shi X J Shi W L Li and Q S Wang ldquoFree transversevibrations of orthotropic thin rectangular plates with arbitraryelastic edge supportsrdquo Journal of Vibroengineering vol 16 no 1pp 389ndash398 2014
[42] Y Chen G Jin and Z Liu ldquoFree vibration analysis of circu-lar cylindrical shell with non-uniform elastic boundary con-straintsrdquo International Journal of Mechanical Sciences vol 74pp 120ndash132 2013
[43] L Dai T Yang J Du W L Li and M J Brennan ldquoAn exactseries solution for the vibration analysis of cylindrical shellswith arbitrary boundary conditionsrdquo Applied Acoustics vol 74no 3 pp 440ndash449 2013
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Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
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Shock and Vibration
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
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DistributedSensor Networks
International Journal of
Shock and Vibration 19
Table 11 Frequency parameterΩ = 1205961198862
(120588ℎ01198630)12 for moderately thick plates (119886119887 = 1) with linearly varying thickness and elastic rotation
and translation support (ℎ(119909 119910) = ℎ0(1 + 05119909
2
)(1 + 051199102
) 1198701199090= 119870119909119886= 1198701199100= 119870119910119887= Γ119909 and 119896
1199090= 119896119909119886= 1198961199100= 119896119910119887= Γ119908)
Γ119909
aΓ119908
aℎ0119886
Model sequence1 2 3 4 5 6
05 0
005 12926 13041 25090 47935 47943 61761(12931b) (13053) 25099 (47947) (47952) (61788)
01 10900 10988 21545 39980 40147 50322(10904b) (10992) (21557) (39984) (40166) (50334)
015 8245 8298 18081 31503 32617 40950(8247b) (8230) (18092) (31508) (32622) (40955)
02 6106 6124 15639 25631 27601 34458(6110b) (6125) (15642) (25637) (27617) (34462)
1 05
005 21575 31615 31644 42573 57889 58607(21580b) (31617) (31644) (42581) (57893) (58610)
01 8451 16024 16210 25969 43839 43867(8451b) (16025) (16217) (25972) (43844) (43870)
015 4682 11479 11480 20642 35172 35598(4684b) (11480) (11481) (20644) (35178) (35604)
02 3059 8621 8648 17320 28487 29638(3062b) (8619) (8644) (17327) (28492) (29641)
15 1
005 27713 40203 40978 52951 65352 66815(27714b) (40207) (40988) (52972) (65368) (66822)
01 11667 19322 19615 29122 46115 46183(11674b) (19327) (19619) (29137) (46120) (46188)
015 6544 13416 13481 22405 37238 37430(6547b) (13420) (13488) (22410) (37241) (37434)
02 4295 10175 10190 18543 30330 31056(4296b) (10177) (10194) (18544) (30336) (31060)
2 2
005 33453 50854 52088 66311 76610 78947(33455b) (50852) (50097) (66321) (76627) (78971)
01 15832 24130 24372 33924 49118 49362(15830b) (24127) (24374) (33931) (49147) (49382)
015 9060 15956 16099 24754 39134 39249(9064b) (15961) (16102) (24755) (39138) (39452)
02 5992 11955 11971 20034 31908 32392(5991b) (11957) (11978) (20042) (31910) (32398)
aΓ119909= 1198701198941198700 (1198700= 1 times 10
9 Nmrad 119894 = 1199090 119909119886 1199100 119910119887) Γ119908= 1198961198941198960 (1198960= 1 times 10
9 Nm 119894 = 1199090 119909119886 1199100 119910119887)bResults in parentheses are taken from FEM
moderately thick plate is investigated The plate is elasticallyrestrained in which the stiffness of the boundary springsis taken as 119896
1199090= 119896119909119886
= 1198961199100
= 119896119910119887
= Γ119908
= 2 and1198701199090
= 119870119909119886
= 1198701199100
= 119870119910119887
= Γ119909= 2 respectively The six
frequency parameters Ω = 1205961198862
(120588ℎ01198630)12 are presented in
Table 12 for several different slop values It can be seen thatthe fundamental frequency parameters will decrease with theincrease of the parameter ℎ
0119886
In the above examples it has been demonstrated thatthe presented method can be universally applied to nonlin-early varying thickness moderately thick plates with severalboundary conditions and different structure parametersNew results are obtained for plates with nonlinear variationthickness in both directions subjected to general elastic
boundary restraints which may be used for benchmarkingof researchers in the field In addition it is interesting to seethat the nature frequency decreases with the increase of theindex for thickness function
4 Conclusions
In this paper a modified Fourier method has been pre-sented to study the free vibration behaviors of moderatelythick rectangular plates with variable thickness and arbi-trary boundary conditionsThe first-order shear deformationplate theory is adopted to formulate the theoretical modelThe displacements and rotation components of the plateregardless of boundary conditions are invariantly expressedas the superposition of a 2D Fourier cosine series and four
20 Shock and Vibration
Table 12 Frequency parameterΩ = 1205961198862
(120588ℎ01198630)12 for elastic support moderately thick plates (119886119887 = 1) with different gradient (119896
1199090= 119896119909119886=
1198961199100= 119896119910119887= Γ119908= 2 119870
1199090= 119870119909119886= 1198701199100= 119870119910119887= Γ119909= 2)
120572 120573 ℎ0119886
Model sequence1 2 3 4 5 6
120572 = 120573 = 01
005 30187 51102 51141 67066 73954 77412(30188a) (51105) (51144) (67068) (73961) (77477)
01 16995 24719 24724 33071 44343 44819(16996a) (24725) (24729) (33084) (44347) (44822)
015 10077 15872 15883 23443 35937 36065(10081a) (15877) (15892) (23447) (35941) (36084)
02 6713 11953 11958 19086 30372 30604(6713a) (11957) (11965) (19092) (30378) (30612)
120572 = 120573 = 02
005 31206 51436 51614 67058 74398 77595(31208a) (51438) (51619) (67064) (74402) (77596)
01 16786 24503 24508 33144 45583 45923(16790a) (24510) (24513) (33147) (45587) (45932)
015 9821 15865 15906 23766 36897 36990(9824a) (15870) (15907) (23768) (36894) (36997)
02 6523 11978 11992 19356 30912 31162(6524a) (11981) (11997) (19362) (30916) (31174)
120572 = 120573 = 03
005 32097 51462 51900 66832 74991 77863(32098a) (51466) (51907) (66849) (74996) (77872)
01 16507 24332 24383 33330 46806 47074(16511a) (24327) (24380) (33339) (46810) (47086)
015 9562 15885 15964 24103 37748 37828(9562a) (15892) (15977) (21105) (37751) (37832)
02 6338 11992 12013 19603 31338 31634(6340a) (11994) (12011) (19612) (31344) (31639)
120572 = 120573 = 04
005 32851 51239 52046 66547 75735 78303(33455a) (50852) (50097) (66321) (76627) (78971)
01 16182 24209 24343 33598 47990 48230(33455a) (50852) (50097) (66321) (76627) (78971)
015 9307 15919 16034 24436 38491 38579(33455a) (50852) (50097) (66321) (76627) (78971)
02 6161 11986 12008 19828 31666 32040(33455a) (50852) (50097) (66321) (76627) (78971)
120572 = 120573 = 05
005 33453 50854 52088 66311 76610 78947(33455a) (50852) (50097) (66321) (76627) (78971)
01 15832 24130 24372 33924 49118 49362(15830a) (24127) (24374) (33931) (49147) (49382)
015 9060 15956 16099 24754 39134 39249(9064a) (15961) (16102) (24755) (39138) (39452)
02 5992 11955 11971 20034 31908 32392(5991a) (11957) (11978) (20042) (31910) (32398)
aResults in parentheses are taken from FEM
supplementary functions in the form of the product of apolynomial function and a single cosine series expansionto ensure and accelerate the convergence of the solutionAt each edge of the plate the general restraint conditionsare implemented by introducing one group of linear springsand two groups of rotational springs which are continuouslydistributed and determined by the stiffness of these springs
Instead of seeking a solution in strong forms in the previousstudies all the Fourier coefficients will be treated equallyand independently as the generalized coordinates and solveddirectly from the Rayleigh-Ritz technique The change of theboundary conditions can be easily achieved by only varyingthe stiffness of the three sets of boundary springs along alledges of the rectangular plates without involving any change
Shock and Vibration 21
to the solution procedure The convergence of the presentsolution is examined and the excellent accuracy is validatedby comparison with existing results published in the lit-erature and FEM data Excellent agreements are obtainedfrom these comparisons The proposed method providesa unified means for extracting the modal parameters andpredicting the vibration behaviors of moderately thick plateswith variable thickness variation functions and arbitraryelastic edge restraints A variety of free vibration results formoderately thick rectangular plates with different thicknessvariation functions and boundary conditions are presentedNew results for free vibration of moderately thick rectangularplates with various thickness variation functions and edgeconditions are presented which may be used for benchmark-ing of researchers in the field
Appendices
A Representative Calculation forStiffness and Mass Matrices
To illuminate the particular expression of the mass andstiffness matrixes clearly and tersely four new variables aredefined as 119904 = 119898 lowast (119873 + 1) + 119899 + 1 119905 = 119898
1015840
lowast (119873 + 1) + 1198991015840
+ 1119901 = (119897minus1)lowast (119872+1)+119898
1015840
+1 and 119902 = (119897minus1)lowast (119873+1)+1198991015840
+1The first row elements of K andM are given below
K1-1119904119905
=
1
2
int
119886
0
int
119887
0
[119863 (119909 119910)
times 21205821198861198981205821198861198981015840 sin 120582
119886119898119909 sin 120582
1198861198981015840119909 cos 120582
119887119899119910
times cos 1205821198871198991015840119910 + (1 minus 120583) 120582
1198871198991205821198871198991015840 cos 120582
119886119898119909
times cos 1205821198861198981015840 sin 120582
119887119899119910 sin 120582
1198871198991015840119910
+ 120581119866ℎ (119909 119910) cos 120582119886119898119909 cos 120582
1198861198981015840119909 cos 120582
119887119899
times119910 cos 1205821198871198991015840119910119889119909119889119910] 119889119909 119889119910
+ int
119886
0
1198701199091199100
cos 120582119886119898119909 cos 120582
1198861198981015840119909
+(minus1)119898+1198981015840
119870119909119910119887
cos 120582119886119898119909 cos 120582
1198861198981015840119909 119889119909
+ int
119887
0
1198701199090cos 120582119887119899119910 cos 120582
1198871198991015840119910
+(minus1)119899+1198991015840
119870119909119886cos 120582119887119899119910 cos 120582
1198871198991015840119910 119889119910
K1-2119904119901
=
1
2
int
119886
0
int
119887
0
[119863 (119909 119910)21205821198861198981205821198861198981015840 sin 120582
119886119898119909 sin 120582
1198861198981015840119909
times
2
sum
119897=1
120577119897
119887(119910) cos 120582
1198871198991015840119910
minus (1 minus 120583) 1205821198871198991015840 cos 120582
119886119898119909 cos 120582
1198861198981015840119909
times
2
sum
119897=1
1205771198971015840
119887(119910) sin 120582
1198871198991015840119910 + 120581119866ℎ (119909 119910)
times ( cos 1205821198861198981015840119909 cos 120582
1198871198991015840119910
times
2
sum
119897=1
120577119897
119887(119910) cos 120582
119886119898119909)]119889119909119889119910
+ int
119886
0
1198701199091199100
cos 120582119886119898119909 cos 120582
1198861198981015840119909
2
sum
119897=1
120577119897
119887(0)
+(minus1)1198991015840
119870119909119910119887
cos 120582119886119898119909 cos 120582
1198861198981015840119909
2
sum
119897=1
120577119897
119887(119887) 119889119909
+ int
119887
0
1198701199090
2
sum
119897=1
120577119897
119887(119910) cos 120582
1198871198991015840119910
+(minus1)119898+1198981015840
119870119909119886
2
sum
119897=1
120577119897
119887(119910) cos 120582
1198871198991015840119910119889119910
K1-3119904119902
=
1
2
int
119886
0
int
119887
0
[119863 (119909 119910) minus 21205821198861198981015840 sin 120582
1198861198981015840119909 cos 120582
1198871198991015840119910
times
2
sum
119897=1
1205771198971015840
119886(119909) cos 120582
119887119899119910
+ (1 minus 120583) 1205821198871198991205821198871198991015840 cos 120582
1198861198981015840119909
times sin 1205821198871198991015840119910
2
sum
119897=1
120577119897
119886(119909) sin 120582
119887119899119910
+ 120581119866ℎ (119909 119910) cos 1205821198861198981015840119909 cos 120582
1198871198991015840119910
times
2
sum
119897=1
120577119897
119886(119909) cos 120582
119887119899119910]119889119909119889119910
+ int
119886
0
1198701199091199100
cos 1205821198861198981015840119909
2
sum
119897=1
120577119897
119886(119909)
+(minus1)119899+1198991015840
119870119909119910119887
cos 1205821198861198981015840119909
2
sum
119897=1
120577119897
119886(119909) 119889119909
+ int
119887
0
1198701199090
2
sum
119897=1
120577119897
119886(0) cos 120582
1198871198991015840119910
+(minus1)119898+1198981015840
119870119909119886
2
sum
119897=1
120577119897
119886(0) cos 120582
1198871198991015840119910119889119910
K1-4119904119905
=
1
2
int
119886
0
int
119887
0
119863(119909 119910) 21205821198871198991205821198861198981015840 sin 120582
1198861198981015840119909 cos 120582
1198871198991015840119910
times cos 120582119886119898119909 sin 120582
119887119899119910 minus 2(1 minus 120583)
119887119899
times 1205821198861198981015840 sin 120582
1198861198981015840119909 cos 120582
1198871198991015840119910 cos 120582
119886119898119909
times sin 120582119887119899119910 + (1 minus 120583) 120582
1198861198981205821198871198991015840 cos 120582
1198861198981015840119909
times sin 1205821198871198991015840119910 sin 120582
119886119898119909 cos 120582
119887119899119910 119889119909 119889119910
22 Shock and Vibration
K1-5119904119901
=
1
2
int
119886
0
int
119887
0
119863(119909 119910) minus 21205821198861198981015840 sin 120582
1198861198981015840119909 cos 120582
1198871198991015840119910
times
2
sum
119897=1
1205771198971015840
119887(119910) cos 120582
119886119898119909
+ 2 (1 minus 120583) 1205821198861198981015840 sin 120582
1198861198981015840119909 cos 120582
1198871198991015840119910
times
2
sum
119897=1
1205771198971015840
119887(119910) cos 120582
119886119898119909
+ (1 minus 120583) 1205821198861198981205821198871198991015840 cos 120582
1198861198981015840119909 sin 120582
1198871198991015840119910
times
2
sum
119897=1
120577119897
119887(119910) sin 120582
119886119898119909119889119909119889119910
K1-6119904119902
=
1
2
int
119886
0
int
119887
0
119863(119909 119910)1205821198871198991205821198861198981015840 sin 120582
1198861198981015840119909 cos 120582
1198861198991015840119910
times
2
sum
119897=1
120577119897
119886(119909) sin 120582
1198871198991015840119910
minus 2 (1 minus 120583) 1205821198871198991205821198861198981015840 sin 120582
1198861198981015840119909 cos 120582
1198861198991015840119910
times
2
sum
119897=1
120577119897
119886(119909) sin 120582
1198871198991015840119910
+ 1205821198871198991015840 cos 120582
1198861198981015840119909 sin 120582
1198871198991015840119910
times
2
sum
119897=1
1205771198971015840
119886(119909) cos 120582
119887119899119910119889119909119889119910
K1-7119904119905
= minus
1
2
int
119886
0
int
119887
0
120581119866ℎ (119909 119910) 120582119886119898
cos 1205821198861198981015840119909 cos 120582
1198871198991015840119910
times sin 120582119886119898119909 cos 120582
119887119899119910119889119909119889119910
K1-8119904119901
= minus
1
2
int
119886
0
int
119887
0
120581119866ℎ (119909 119910) 1205821198861198981015840 cos 120582
1198861198981015840119909 cos 120582
1198871198991015840119910
times
2
sum
119897=1
120577119897
119887(119910) sin 120582
1198861198981015840119909 119889119909 119889119910
K1-9119904119902
=
1
2
int
119886
0
int
119887
0
120581119866ℎ (119909 119910) cos 1205821198861198981015840119909 cos 120582
1198871198991015840119910
times
2
sum
119897=1
1205771198971015840
119886(119909) cos 120582
119887119899119910119889119909119889119910
M1-1119904119905
=
1
24
120588int
119886
0
int
119887
0
ℎ3
(119909 119910) cos 120582119886119898119909 cos 120582
119887119899119910
times cos 1205821198861198981015840119909 cos 120582
1198871198991015840119910119889119909119889119910
M1-2119904119901
=
1
24
120588int
119886
0
int
119887
0
ℎ3
(119909 119910)
2
sum
119897=1
120577119897
119887(119910) cos 120582
119886119898119909
times cos 1205821198861198981015840119909 cos 120582
1198871198991015840119910119889119909119889119910
M1-3119904119902
=
1
24
120588int
119886
0
int
119887
0
ℎ3
(119909 119910)
2
sum
119897=1
120577119897
119886(119909) cos 120582
119887119899119910
times cos 1205821198861198981015840119909 cos 120582
1198871198991015840119910119889119909119889119910
M1-4119904119905 = M
1-5119904119901 = M1-6119904119902
= M1-7119904119905 = M
1-8119904119901 = M1-9119904119902 = 0
(A1)
B Useful Integrals for the Moderately ThickRectangular Plates with Variable Thicknessand Arbitrary Boundary Conditions
Consider
int
119871
0
cos 120582119898119909 cos 120582
119899119909 119889119909 =
0 119898 = 119899
119871
2
119898 = 119899
119871 119898 = 119899 = 0
int
119871
0
sin 120582119898119909 cos 120582
119899119909119889119909
=
minus
(minus1 + (minus1)119898+119899
) 119871119898
(1198982minus 1198992) 120587
119898 = 119899
0 119898 = 119899
int
119871
0
sin 120582119898119909 sin 120582
119899119909 119889119909 =
0 119898 = 119899
119871
2
119898 = 119899
0 119898 = 119899 = 0
int
119871
0
cos 120582119898119909120577119897
119886(119909) 119889119909
=
41198712
31205872 119897 = 1 119898 = 0
minus
41198712
(41198982
minus 3)
(9 minus 401198982+ 16119898
4) 1205872
119897 = 1 119898 = 0
minus
41198712
31205872 119897 = 2 119898 = 0
41198712
(41198982
minus 3) (minus1)119898
(9 minus 401198982+ 16119898
4) 1205872
119897 = 2 119898 = 0
Shock and Vibration 23
int
119871
0
sin 120582119898119909120577119897
119886(119909) 119889119909
=
0 119897 = 1 119898 = 0
161198712
119898(minus1)119898
(9 minus 401198982+ 16119898
4) 1205872
119897 = 1 119898 = 0
0 119897 = 2 119898 = 0
161198981198712
(9 minus 401198982+ 16119898
4) 1205872
119897 = 2 119898 = 0
int
119871
0
cos 1205821198981199091205771198971015840
119886(119909) 119889119909
=
0 119897 = 1 119898 = 0
161198711198982
(minus1)119898
(9 minus 401198982+ 16119898
4) 120587
119897 = 1 119898 = 0
0 119897 = 2 119898 = 0
161198711198982
(9 minus 401198982+ 16119898
4) 120587
119897 = 2 119898 = 0
int
119871
0
sin 1205821198981199091205771198971015840
119886(119909) 119889119909
=
0 119897 = 1 119898 = 0
4119871119898 (minus3 + 41198982
)
(9 minus 401198982+ 16119898
4) 120587
119897 = 1 119898 = 0
0 119897 = 2 119898 = 0
119871119898(
1
120587 minus 41198982120587
+
3
9120587 minus 41198982120587
) (minus1)119898
119897 = 2 119898 = 0
int
119871
0
1205771198971
119886(119909) 1205771198972
119886(119909) 119889119909 =
1198713
41205872 1198971= 1 1198972= 1
minus
21198713
31205873 1198971= 1 1198972= 2
minus
21198713
31205873 1198971= 2 1198972= 1
1198713
41205872 1198971= 2 1198972= 2
int
119871
0
1205771198971
119886(119909) 1205771198971015840
2
119886(119909) 119889119909 =
0 1198971= 1 1198972= 1
minus
1198712
8120587
1198971= 1 1198972= 2
1198712
8120587
1198971= 2 1198972= 1
0 1198971= 2 1198972= 2
int
119871
0
1205771198971015840
1
119886(119909) 1205771198971015840
2
119886(119909) 119889119909 =
5119871
16
1198971= 1 1198972= 1
minus
119871
2120587
1198971= 1 1198972= 2
minus
119871
2120587
1198971= 2 1198972= 1
5119871
16
1198971= 2 1198972= 2
int
119871
0
119909119899 cos 120582
119898119909 119889119909
=
1198711+119899
1 + 119899
119898 = 0
(1198861+119899HypergeometricPFQtimes [ 12 + 1198992 12 32 + 1198992
minus (14)1198982
1205872
]) times (1 + 119899)minus1
119898 gt 0
(B1)
Conflict of Interests
The authors declare that there is no conflict of interestsregarding the publication of this paper
Acknowledgments
The authors would like to thank the anonymous reviewersfor their very valuable comments The authors also gratefullyacknowledge the financial support from the National NaturalScience Foundation of China (no 51209052) HeilongjiangProvince Youth Science Fund Project (no QC2011C013) andHarbin Science and Technology Development InnovationFoundation of Youth (no 2011RFQXG021)
References
[1] K Liew Y Xiang and S Kitipornchai ldquoResearch on thick platevibration a literature surveyrdquo Journal of Sound and Vibrationvol 180 no 1 pp 163ndash176 1995
[2] J H Chung T Y Chung and K C Kim ldquoVibration analysisof orthotropic mindlin plates with edges elastically restrainedagainst rotationrdquo Journal of Sound and Vibration vol 163 no 1pp 151ndash163 1993
[3] Y K Cheung and D Zhou ldquoVibrations of moderately thickrectangular plates in terms of a set of static Timoshenko beamfunctionsrdquoComputers and Structures vol 78 no 6 pp 757ndash7682000
[4] CMWang ldquoNatural frequencies formula for simply supportedMindlin platesrdquo Journal of Vibration and Acoustics vol 116 no4 pp 536ndash540 1994
[5] K N Saha R C Kar and P K Datta ldquoFree vibration analysisof rectangular Mindlin plates with elastic restraints uniformlydistributed along the edgesrdquo Journal of Sound andVibration vol192 no 4 pp 885ndash902 1996
24 Shock and Vibration
[6] D J Gorman ldquoAccurate free vibration analysis of shear-deformable plates with torsional elastic edge supportrdquo Journalof Sound and Vibration vol 203 no 2 pp 209ndash218 1997
[7] D J Gorman ldquoFree vibration analysis of Mindlin plates withuniform elastic edge support by the superposition methodrdquoJournal of Sound andVibration vol 207 no 3 pp 335ndash350 1997
[8] D J Gorman ldquoAccurate free vibration analysis of point sup-ported mindlin plates by the superposition methodrdquo Journal ofSound and Vibration vol 219 no 2 pp 265ndash277 1999
[9] Y Xiang ldquoVibration of rectangular Mindlin plates resting onnon-homogenous elastic foundationsrdquo International Journal ofMechanical Sciences vol 45 no 6-7 pp 1229ndash1244 2003
[10] Y Xiang and G W Wei ldquoExact solutions for buckling andvibration of stepped rectangular Mindlin platesrdquo InternationalJournal of Solids and Structures vol 41 no 1 pp 279ndash294 2004
[11] Y L Yeh M J Jang and C C Wang ldquoAnalyzing the freevibrations of a plate using finite difference and differential trans-formationmethodrdquoAppliedMathematics andComputation vol178 no 2 pp 493ndash501 2006
[12] Y Xiang S K Lai L Zhou and C W Lim ldquoDSC-Ritz elementmethod for vibration analysis of rectangular Mindlin plateswith mixed edge supportsrdquo European Journal of MechanicsmdashASolids vol 29 no 4 pp 619ndash628 2010
[13] H Nguyen-Xuan C H Thai and T Nguyen-Thoi ldquoIsogeo-metric finite element analysis of composite sandwich platesusing a higher order shear deformation theoryrdquo Composites BEngineering vol 55 pp 558ndash574 2013
[14] C H Thai A J M Ferreira E Carrera and H Nguyen-XuanldquoIsogeometric analysis of laminated composite and sandwichplates using a layerwise deformation theoryrdquo Composite Struc-tures vol 104 pp 196ndash214 2013
[15] C H Thai A Ferreira S Bordas T Rabczuk and HNguyen-Xuan ldquoIsogeometric analysis of laminated compositeand sandwich plates using a new inverse trigonometric sheardeformation theoryrdquo European Journal of Mechanics A Solidsvol 43 pp 89ndash108 2014
[16] H Luong-Van T Nguyen-Thoi G R Liu and P Phung-VanldquoA cell-based smoothed finite elementmethod using three-nodeshear-locking free Mindlin plate element (CS-FEM-MIN3) fordynamic response of laminated composite plates on viscoelasticfoundationrdquo Engineering Analysis with Boundary Elements vol42 pp 8ndash19 2014
[17] T Nguyen-Thoi T Rabczuk T Lam-Phat V Ho-Huu andP Phung-Van ldquoFree vibration analysis of cracked Mindlinplate using an extended cell-based smoothed discrete sheargap method (XCS-DSG3)rdquo Theoretical and Applied FractureMechanics 2014
[18] H Nguyen-Xuan G R Liu and C a Thai-Hoang ldquoAnedge-based smoothed finite element method (ES-FEM) withstabilized discrete shear gap technique for analysis of Reissner-Mindlin platesrdquo Computer Methods in Applied Mechanics andEngineering vol 199 no 9ndash12 pp 471ndash489 2010
[19] P Phung-Van T Nguyen-Thoi T Bui-Xuan and Q Lieu-XuanldquoA cell-based smoothed three-node Mindlin plate element(CS-FEM-MIN3) based on the 119862
0 -type higher-order sheardeformation for geometrically nonlinear analysis of laminatedcomposite platesrdquo Computational Materials Science 2014
[20] NNguyen-Thanh T RabczukHNguyen-Xuan and S BordasldquoAn alternative alpha finite element method with discrete sheargap technique for analysis of isotropicMindlin-Reissner platesrdquoFinite Elements inAnalysis andDesign vol 47 no 5 pp 519ndash5352011
[21] KM Liew Y Xiang and S Kitipornchai ldquoTransverse vibrationof thick rectangular plates-I Comprehensive sets of boundaryconditionsrdquo Computers and Structures vol 49 no 1 pp 1ndash291993
[22] D Zhou ldquoVibrations of Mindlin rectangular plates with elasti-cally restrained edges using static Timoshenko beam functionswith the Rayleigh-Ritz methodrdquo International Journal of Solidsand Structures vol 38 no 32-33 pp 5565ndash5580 2001
[23] D Zhou SH Lo F T KAu andYK Cheung ldquoVibration anal-ysis of rectangular Mindlin plates with internal line supportsusing static Timoshenko beam functionsrdquo International Journalof Mechanical Sciences vol 44 no 12 pp 2503ndash2522 2002
[24] H S Shen J Yang and L Zhang ldquoFree and forced vibrationof Reissner-Mindlin plates with free edges resting on elasticfoundationsrdquo Journal of Sound and Vibration vol 244 no 2 pp299ndash320 2001
[25] Y Xing and B Liu ldquoClosed form solutions for free vibrations ofrectangular Mindlin platesrdquo Acta Mechanica Sinica vol 25 no5 pp 689ndash698 2009
[26] P Gagnon C Gosselin and L Cloutier ldquoA finite strip elementfor the analysis of variable thickness rectangular thick platesrdquoComputers amp Structures vol 63 no 2 pp 349ndash362 1997
[27] T Mlzusawa ldquoVibration of rectangular mindlin plates withtapered thickness by the spline strip methodrdquo Computers andStructures vol 46 no 3 pp 451ndash463 1993
[28] H Nguyen-Xuan and T Nguyen-Thoi ldquoA stabilized smoothedfinite element method for free vibration analysis of Mindlin-Reissner platesrdquo Communications in Numerical Methods inEngineering with Biomedical Applications vol 25 no 8 pp 882ndash906 2009
[29] K M Liew L X Peng and S Kitipornchai ldquoVibration anal-ysis of corrugated Reissner-Mindlin plates using a mesh-freeGalerkin methodrdquo International Journal of Mechanical Sciencesvol 51 no 9-10 pp 642ndash652 2009
[30] Y Hou GWWei and Y Xiang ldquoDSC-Ritz method for the freevibration analysis of Mindlin platesrdquo International Journal forNumerical Methods in Engineering vol 62 no 2 pp 262ndash2882005
[31] F- Liu and K M Liew ldquoVibration analysis of discontinuousMindlin plates by differential quadrature element methodrdquoJournal of Vibration and Acoustics Transactions of the ASMEvol 121 no 2 pp 204ndash208 1999
[32] P Malekzadeh G Karami and M Farid ldquoA semi-analyticalDQEM for free vibration analysis of thick plates with twoopposite edges simply supportedrdquoComputerMethods inAppliedMechanics and Engineering vol 193 no 45ndash47 pp 4781ndash47962004
[33] P Malekzadeh and S A Shahpari ldquoFree vibration analysisof variable thickness thin and moderately thick plates withelastically restrained edges by DQMrdquo Thin-Walled Structuresvol 43 no 7 pp 1037ndash1050 2005
[34] R E Diaz-Contreras and S Nomura ldquoGreenrsquos function appliedto solution ofMindlin platesrdquoComputers and Structures vol 60no 1 pp 41ndash48 1996
[35] T Sakiyama and M Huang ldquoFree vibration analysis of rect-angular plates with variable thicknessrdquo Journal of Sound andVibration vol 216 no 3 pp 379ndash397 1998
[36] M Huang X Q Ma T Sakiyama H Matuda and C MoritaldquoFree vibration analysis of orthotropic rectangular plates withvariable thickness and general boundary conditionsrdquo Journal ofSound and Vibration vol 288 no 4-5 pp 931ndash955 2005
Shock and Vibration 25
[37] I Shufrin and M Eisenberger ldquoVibration of shear deformableplates with variable thicknessmdashfirst-order and higher-orderanalysesrdquo Journal of Sound and Vibration vol 290 no 1-2 pp465ndash489 2006
[38] S A Eftekhari and A A Jafari ldquoAccurate variational approachfor free vibration of variable thickness thin and thick plates withedges elastically restrained against translation and rotationrdquoInternational Journal of Mechanical Sciences vol 68 pp 35ndash462013
[39] W L Li X Zhang J Du and Z Liu ldquoAn exact series solutionfor the transverse vibration of rectangular plates with generalelastic boundary supportsrdquo Journal of Sound amp Vibration vol321 no 1-2 pp 254ndash269 2009
[40] X Shi D Shi W L Li and Q Wang ldquoA unified method forfree vibration analysis of circular annular and sector plateswith arbitrary boundary conditionsrdquo Journal of Vibration andControl 2014
[41] D Y Shi X J Shi W L Li and Q S Wang ldquoFree transversevibrations of orthotropic thin rectangular plates with arbitraryelastic edge supportsrdquo Journal of Vibroengineering vol 16 no 1pp 389ndash398 2014
[42] Y Chen G Jin and Z Liu ldquoFree vibration analysis of circu-lar cylindrical shell with non-uniform elastic boundary con-straintsrdquo International Journal of Mechanical Sciences vol 74pp 120ndash132 2013
[43] L Dai T Yang J Du W L Li and M J Brennan ldquoAn exactseries solution for the vibration analysis of cylindrical shellswith arbitrary boundary conditionsrdquo Applied Acoustics vol 74no 3 pp 440ndash449 2013
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Shock and Vibration
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Navigation and Observation
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DistributedSensor Networks
International Journal of
20 Shock and Vibration
Table 12 Frequency parameterΩ = 1205961198862
(120588ℎ01198630)12 for elastic support moderately thick plates (119886119887 = 1) with different gradient (119896
1199090= 119896119909119886=
1198961199100= 119896119910119887= Γ119908= 2 119870
1199090= 119870119909119886= 1198701199100= 119870119910119887= Γ119909= 2)
120572 120573 ℎ0119886
Model sequence1 2 3 4 5 6
120572 = 120573 = 01
005 30187 51102 51141 67066 73954 77412(30188a) (51105) (51144) (67068) (73961) (77477)
01 16995 24719 24724 33071 44343 44819(16996a) (24725) (24729) (33084) (44347) (44822)
015 10077 15872 15883 23443 35937 36065(10081a) (15877) (15892) (23447) (35941) (36084)
02 6713 11953 11958 19086 30372 30604(6713a) (11957) (11965) (19092) (30378) (30612)
120572 = 120573 = 02
005 31206 51436 51614 67058 74398 77595(31208a) (51438) (51619) (67064) (74402) (77596)
01 16786 24503 24508 33144 45583 45923(16790a) (24510) (24513) (33147) (45587) (45932)
015 9821 15865 15906 23766 36897 36990(9824a) (15870) (15907) (23768) (36894) (36997)
02 6523 11978 11992 19356 30912 31162(6524a) (11981) (11997) (19362) (30916) (31174)
120572 = 120573 = 03
005 32097 51462 51900 66832 74991 77863(32098a) (51466) (51907) (66849) (74996) (77872)
01 16507 24332 24383 33330 46806 47074(16511a) (24327) (24380) (33339) (46810) (47086)
015 9562 15885 15964 24103 37748 37828(9562a) (15892) (15977) (21105) (37751) (37832)
02 6338 11992 12013 19603 31338 31634(6340a) (11994) (12011) (19612) (31344) (31639)
120572 = 120573 = 04
005 32851 51239 52046 66547 75735 78303(33455a) (50852) (50097) (66321) (76627) (78971)
01 16182 24209 24343 33598 47990 48230(33455a) (50852) (50097) (66321) (76627) (78971)
015 9307 15919 16034 24436 38491 38579(33455a) (50852) (50097) (66321) (76627) (78971)
02 6161 11986 12008 19828 31666 32040(33455a) (50852) (50097) (66321) (76627) (78971)
120572 = 120573 = 05
005 33453 50854 52088 66311 76610 78947(33455a) (50852) (50097) (66321) (76627) (78971)
01 15832 24130 24372 33924 49118 49362(15830a) (24127) (24374) (33931) (49147) (49382)
015 9060 15956 16099 24754 39134 39249(9064a) (15961) (16102) (24755) (39138) (39452)
02 5992 11955 11971 20034 31908 32392(5991a) (11957) (11978) (20042) (31910) (32398)
aResults in parentheses are taken from FEM
supplementary functions in the form of the product of apolynomial function and a single cosine series expansionto ensure and accelerate the convergence of the solutionAt each edge of the plate the general restraint conditionsare implemented by introducing one group of linear springsand two groups of rotational springs which are continuouslydistributed and determined by the stiffness of these springs
Instead of seeking a solution in strong forms in the previousstudies all the Fourier coefficients will be treated equallyand independently as the generalized coordinates and solveddirectly from the Rayleigh-Ritz technique The change of theboundary conditions can be easily achieved by only varyingthe stiffness of the three sets of boundary springs along alledges of the rectangular plates without involving any change
Shock and Vibration 21
to the solution procedure The convergence of the presentsolution is examined and the excellent accuracy is validatedby comparison with existing results published in the lit-erature and FEM data Excellent agreements are obtainedfrom these comparisons The proposed method providesa unified means for extracting the modal parameters andpredicting the vibration behaviors of moderately thick plateswith variable thickness variation functions and arbitraryelastic edge restraints A variety of free vibration results formoderately thick rectangular plates with different thicknessvariation functions and boundary conditions are presentedNew results for free vibration of moderately thick rectangularplates with various thickness variation functions and edgeconditions are presented which may be used for benchmark-ing of researchers in the field
Appendices
A Representative Calculation forStiffness and Mass Matrices
To illuminate the particular expression of the mass andstiffness matrixes clearly and tersely four new variables aredefined as 119904 = 119898 lowast (119873 + 1) + 119899 + 1 119905 = 119898
1015840
lowast (119873 + 1) + 1198991015840
+ 1119901 = (119897minus1)lowast (119872+1)+119898
1015840
+1 and 119902 = (119897minus1)lowast (119873+1)+1198991015840
+1The first row elements of K andM are given below
K1-1119904119905
=
1
2
int
119886
0
int
119887
0
[119863 (119909 119910)
times 21205821198861198981205821198861198981015840 sin 120582
119886119898119909 sin 120582
1198861198981015840119909 cos 120582
119887119899119910
times cos 1205821198871198991015840119910 + (1 minus 120583) 120582
1198871198991205821198871198991015840 cos 120582
119886119898119909
times cos 1205821198861198981015840 sin 120582
119887119899119910 sin 120582
1198871198991015840119910
+ 120581119866ℎ (119909 119910) cos 120582119886119898119909 cos 120582
1198861198981015840119909 cos 120582
119887119899
times119910 cos 1205821198871198991015840119910119889119909119889119910] 119889119909 119889119910
+ int
119886
0
1198701199091199100
cos 120582119886119898119909 cos 120582
1198861198981015840119909
+(minus1)119898+1198981015840
119870119909119910119887
cos 120582119886119898119909 cos 120582
1198861198981015840119909 119889119909
+ int
119887
0
1198701199090cos 120582119887119899119910 cos 120582
1198871198991015840119910
+(minus1)119899+1198991015840
119870119909119886cos 120582119887119899119910 cos 120582
1198871198991015840119910 119889119910
K1-2119904119901
=
1
2
int
119886
0
int
119887
0
[119863 (119909 119910)21205821198861198981205821198861198981015840 sin 120582
119886119898119909 sin 120582
1198861198981015840119909
times
2
sum
119897=1
120577119897
119887(119910) cos 120582
1198871198991015840119910
minus (1 minus 120583) 1205821198871198991015840 cos 120582
119886119898119909 cos 120582
1198861198981015840119909
times
2
sum
119897=1
1205771198971015840
119887(119910) sin 120582
1198871198991015840119910 + 120581119866ℎ (119909 119910)
times ( cos 1205821198861198981015840119909 cos 120582
1198871198991015840119910
times
2
sum
119897=1
120577119897
119887(119910) cos 120582
119886119898119909)]119889119909119889119910
+ int
119886
0
1198701199091199100
cos 120582119886119898119909 cos 120582
1198861198981015840119909
2
sum
119897=1
120577119897
119887(0)
+(minus1)1198991015840
119870119909119910119887
cos 120582119886119898119909 cos 120582
1198861198981015840119909
2
sum
119897=1
120577119897
119887(119887) 119889119909
+ int
119887
0
1198701199090
2
sum
119897=1
120577119897
119887(119910) cos 120582
1198871198991015840119910
+(minus1)119898+1198981015840
119870119909119886
2
sum
119897=1
120577119897
119887(119910) cos 120582
1198871198991015840119910119889119910
K1-3119904119902
=
1
2
int
119886
0
int
119887
0
[119863 (119909 119910) minus 21205821198861198981015840 sin 120582
1198861198981015840119909 cos 120582
1198871198991015840119910
times
2
sum
119897=1
1205771198971015840
119886(119909) cos 120582
119887119899119910
+ (1 minus 120583) 1205821198871198991205821198871198991015840 cos 120582
1198861198981015840119909
times sin 1205821198871198991015840119910
2
sum
119897=1
120577119897
119886(119909) sin 120582
119887119899119910
+ 120581119866ℎ (119909 119910) cos 1205821198861198981015840119909 cos 120582
1198871198991015840119910
times
2
sum
119897=1
120577119897
119886(119909) cos 120582
119887119899119910]119889119909119889119910
+ int
119886
0
1198701199091199100
cos 1205821198861198981015840119909
2
sum
119897=1
120577119897
119886(119909)
+(minus1)119899+1198991015840
119870119909119910119887
cos 1205821198861198981015840119909
2
sum
119897=1
120577119897
119886(119909) 119889119909
+ int
119887
0
1198701199090
2
sum
119897=1
120577119897
119886(0) cos 120582
1198871198991015840119910
+(minus1)119898+1198981015840
119870119909119886
2
sum
119897=1
120577119897
119886(0) cos 120582
1198871198991015840119910119889119910
K1-4119904119905
=
1
2
int
119886
0
int
119887
0
119863(119909 119910) 21205821198871198991205821198861198981015840 sin 120582
1198861198981015840119909 cos 120582
1198871198991015840119910
times cos 120582119886119898119909 sin 120582
119887119899119910 minus 2(1 minus 120583)
119887119899
times 1205821198861198981015840 sin 120582
1198861198981015840119909 cos 120582
1198871198991015840119910 cos 120582
119886119898119909
times sin 120582119887119899119910 + (1 minus 120583) 120582
1198861198981205821198871198991015840 cos 120582
1198861198981015840119909
times sin 1205821198871198991015840119910 sin 120582
119886119898119909 cos 120582
119887119899119910 119889119909 119889119910
22 Shock and Vibration
K1-5119904119901
=
1
2
int
119886
0
int
119887
0
119863(119909 119910) minus 21205821198861198981015840 sin 120582
1198861198981015840119909 cos 120582
1198871198991015840119910
times
2
sum
119897=1
1205771198971015840
119887(119910) cos 120582
119886119898119909
+ 2 (1 minus 120583) 1205821198861198981015840 sin 120582
1198861198981015840119909 cos 120582
1198871198991015840119910
times
2
sum
119897=1
1205771198971015840
119887(119910) cos 120582
119886119898119909
+ (1 minus 120583) 1205821198861198981205821198871198991015840 cos 120582
1198861198981015840119909 sin 120582
1198871198991015840119910
times
2
sum
119897=1
120577119897
119887(119910) sin 120582
119886119898119909119889119909119889119910
K1-6119904119902
=
1
2
int
119886
0
int
119887
0
119863(119909 119910)1205821198871198991205821198861198981015840 sin 120582
1198861198981015840119909 cos 120582
1198861198991015840119910
times
2
sum
119897=1
120577119897
119886(119909) sin 120582
1198871198991015840119910
minus 2 (1 minus 120583) 1205821198871198991205821198861198981015840 sin 120582
1198861198981015840119909 cos 120582
1198861198991015840119910
times
2
sum
119897=1
120577119897
119886(119909) sin 120582
1198871198991015840119910
+ 1205821198871198991015840 cos 120582
1198861198981015840119909 sin 120582
1198871198991015840119910
times
2
sum
119897=1
1205771198971015840
119886(119909) cos 120582
119887119899119910119889119909119889119910
K1-7119904119905
= minus
1
2
int
119886
0
int
119887
0
120581119866ℎ (119909 119910) 120582119886119898
cos 1205821198861198981015840119909 cos 120582
1198871198991015840119910
times sin 120582119886119898119909 cos 120582
119887119899119910119889119909119889119910
K1-8119904119901
= minus
1
2
int
119886
0
int
119887
0
120581119866ℎ (119909 119910) 1205821198861198981015840 cos 120582
1198861198981015840119909 cos 120582
1198871198991015840119910
times
2
sum
119897=1
120577119897
119887(119910) sin 120582
1198861198981015840119909 119889119909 119889119910
K1-9119904119902
=
1
2
int
119886
0
int
119887
0
120581119866ℎ (119909 119910) cos 1205821198861198981015840119909 cos 120582
1198871198991015840119910
times
2
sum
119897=1
1205771198971015840
119886(119909) cos 120582
119887119899119910119889119909119889119910
M1-1119904119905
=
1
24
120588int
119886
0
int
119887
0
ℎ3
(119909 119910) cos 120582119886119898119909 cos 120582
119887119899119910
times cos 1205821198861198981015840119909 cos 120582
1198871198991015840119910119889119909119889119910
M1-2119904119901
=
1
24
120588int
119886
0
int
119887
0
ℎ3
(119909 119910)
2
sum
119897=1
120577119897
119887(119910) cos 120582
119886119898119909
times cos 1205821198861198981015840119909 cos 120582
1198871198991015840119910119889119909119889119910
M1-3119904119902
=
1
24
120588int
119886
0
int
119887
0
ℎ3
(119909 119910)
2
sum
119897=1
120577119897
119886(119909) cos 120582
119887119899119910
times cos 1205821198861198981015840119909 cos 120582
1198871198991015840119910119889119909119889119910
M1-4119904119905 = M
1-5119904119901 = M1-6119904119902
= M1-7119904119905 = M
1-8119904119901 = M1-9119904119902 = 0
(A1)
B Useful Integrals for the Moderately ThickRectangular Plates with Variable Thicknessand Arbitrary Boundary Conditions
Consider
int
119871
0
cos 120582119898119909 cos 120582
119899119909 119889119909 =
0 119898 = 119899
119871
2
119898 = 119899
119871 119898 = 119899 = 0
int
119871
0
sin 120582119898119909 cos 120582
119899119909119889119909
=
minus
(minus1 + (minus1)119898+119899
) 119871119898
(1198982minus 1198992) 120587
119898 = 119899
0 119898 = 119899
int
119871
0
sin 120582119898119909 sin 120582
119899119909 119889119909 =
0 119898 = 119899
119871
2
119898 = 119899
0 119898 = 119899 = 0
int
119871
0
cos 120582119898119909120577119897
119886(119909) 119889119909
=
41198712
31205872 119897 = 1 119898 = 0
minus
41198712
(41198982
minus 3)
(9 minus 401198982+ 16119898
4) 1205872
119897 = 1 119898 = 0
minus
41198712
31205872 119897 = 2 119898 = 0
41198712
(41198982
minus 3) (minus1)119898
(9 minus 401198982+ 16119898
4) 1205872
119897 = 2 119898 = 0
Shock and Vibration 23
int
119871
0
sin 120582119898119909120577119897
119886(119909) 119889119909
=
0 119897 = 1 119898 = 0
161198712
119898(minus1)119898
(9 minus 401198982+ 16119898
4) 1205872
119897 = 1 119898 = 0
0 119897 = 2 119898 = 0
161198981198712
(9 minus 401198982+ 16119898
4) 1205872
119897 = 2 119898 = 0
int
119871
0
cos 1205821198981199091205771198971015840
119886(119909) 119889119909
=
0 119897 = 1 119898 = 0
161198711198982
(minus1)119898
(9 minus 401198982+ 16119898
4) 120587
119897 = 1 119898 = 0
0 119897 = 2 119898 = 0
161198711198982
(9 minus 401198982+ 16119898
4) 120587
119897 = 2 119898 = 0
int
119871
0
sin 1205821198981199091205771198971015840
119886(119909) 119889119909
=
0 119897 = 1 119898 = 0
4119871119898 (minus3 + 41198982
)
(9 minus 401198982+ 16119898
4) 120587
119897 = 1 119898 = 0
0 119897 = 2 119898 = 0
119871119898(
1
120587 minus 41198982120587
+
3
9120587 minus 41198982120587
) (minus1)119898
119897 = 2 119898 = 0
int
119871
0
1205771198971
119886(119909) 1205771198972
119886(119909) 119889119909 =
1198713
41205872 1198971= 1 1198972= 1
minus
21198713
31205873 1198971= 1 1198972= 2
minus
21198713
31205873 1198971= 2 1198972= 1
1198713
41205872 1198971= 2 1198972= 2
int
119871
0
1205771198971
119886(119909) 1205771198971015840
2
119886(119909) 119889119909 =
0 1198971= 1 1198972= 1
minus
1198712
8120587
1198971= 1 1198972= 2
1198712
8120587
1198971= 2 1198972= 1
0 1198971= 2 1198972= 2
int
119871
0
1205771198971015840
1
119886(119909) 1205771198971015840
2
119886(119909) 119889119909 =
5119871
16
1198971= 1 1198972= 1
minus
119871
2120587
1198971= 1 1198972= 2
minus
119871
2120587
1198971= 2 1198972= 1
5119871
16
1198971= 2 1198972= 2
int
119871
0
119909119899 cos 120582
119898119909 119889119909
=
1198711+119899
1 + 119899
119898 = 0
(1198861+119899HypergeometricPFQtimes [ 12 + 1198992 12 32 + 1198992
minus (14)1198982
1205872
]) times (1 + 119899)minus1
119898 gt 0
(B1)
Conflict of Interests
The authors declare that there is no conflict of interestsregarding the publication of this paper
Acknowledgments
The authors would like to thank the anonymous reviewersfor their very valuable comments The authors also gratefullyacknowledge the financial support from the National NaturalScience Foundation of China (no 51209052) HeilongjiangProvince Youth Science Fund Project (no QC2011C013) andHarbin Science and Technology Development InnovationFoundation of Youth (no 2011RFQXG021)
References
[1] K Liew Y Xiang and S Kitipornchai ldquoResearch on thick platevibration a literature surveyrdquo Journal of Sound and Vibrationvol 180 no 1 pp 163ndash176 1995
[2] J H Chung T Y Chung and K C Kim ldquoVibration analysisof orthotropic mindlin plates with edges elastically restrainedagainst rotationrdquo Journal of Sound and Vibration vol 163 no 1pp 151ndash163 1993
[3] Y K Cheung and D Zhou ldquoVibrations of moderately thickrectangular plates in terms of a set of static Timoshenko beamfunctionsrdquoComputers and Structures vol 78 no 6 pp 757ndash7682000
[4] CMWang ldquoNatural frequencies formula for simply supportedMindlin platesrdquo Journal of Vibration and Acoustics vol 116 no4 pp 536ndash540 1994
[5] K N Saha R C Kar and P K Datta ldquoFree vibration analysisof rectangular Mindlin plates with elastic restraints uniformlydistributed along the edgesrdquo Journal of Sound andVibration vol192 no 4 pp 885ndash902 1996
24 Shock and Vibration
[6] D J Gorman ldquoAccurate free vibration analysis of shear-deformable plates with torsional elastic edge supportrdquo Journalof Sound and Vibration vol 203 no 2 pp 209ndash218 1997
[7] D J Gorman ldquoFree vibration analysis of Mindlin plates withuniform elastic edge support by the superposition methodrdquoJournal of Sound andVibration vol 207 no 3 pp 335ndash350 1997
[8] D J Gorman ldquoAccurate free vibration analysis of point sup-ported mindlin plates by the superposition methodrdquo Journal ofSound and Vibration vol 219 no 2 pp 265ndash277 1999
[9] Y Xiang ldquoVibration of rectangular Mindlin plates resting onnon-homogenous elastic foundationsrdquo International Journal ofMechanical Sciences vol 45 no 6-7 pp 1229ndash1244 2003
[10] Y Xiang and G W Wei ldquoExact solutions for buckling andvibration of stepped rectangular Mindlin platesrdquo InternationalJournal of Solids and Structures vol 41 no 1 pp 279ndash294 2004
[11] Y L Yeh M J Jang and C C Wang ldquoAnalyzing the freevibrations of a plate using finite difference and differential trans-formationmethodrdquoAppliedMathematics andComputation vol178 no 2 pp 493ndash501 2006
[12] Y Xiang S K Lai L Zhou and C W Lim ldquoDSC-Ritz elementmethod for vibration analysis of rectangular Mindlin plateswith mixed edge supportsrdquo European Journal of MechanicsmdashASolids vol 29 no 4 pp 619ndash628 2010
[13] H Nguyen-Xuan C H Thai and T Nguyen-Thoi ldquoIsogeo-metric finite element analysis of composite sandwich platesusing a higher order shear deformation theoryrdquo Composites BEngineering vol 55 pp 558ndash574 2013
[14] C H Thai A J M Ferreira E Carrera and H Nguyen-XuanldquoIsogeometric analysis of laminated composite and sandwichplates using a layerwise deformation theoryrdquo Composite Struc-tures vol 104 pp 196ndash214 2013
[15] C H Thai A Ferreira S Bordas T Rabczuk and HNguyen-Xuan ldquoIsogeometric analysis of laminated compositeand sandwich plates using a new inverse trigonometric sheardeformation theoryrdquo European Journal of Mechanics A Solidsvol 43 pp 89ndash108 2014
[16] H Luong-Van T Nguyen-Thoi G R Liu and P Phung-VanldquoA cell-based smoothed finite elementmethod using three-nodeshear-locking free Mindlin plate element (CS-FEM-MIN3) fordynamic response of laminated composite plates on viscoelasticfoundationrdquo Engineering Analysis with Boundary Elements vol42 pp 8ndash19 2014
[17] T Nguyen-Thoi T Rabczuk T Lam-Phat V Ho-Huu andP Phung-Van ldquoFree vibration analysis of cracked Mindlinplate using an extended cell-based smoothed discrete sheargap method (XCS-DSG3)rdquo Theoretical and Applied FractureMechanics 2014
[18] H Nguyen-Xuan G R Liu and C a Thai-Hoang ldquoAnedge-based smoothed finite element method (ES-FEM) withstabilized discrete shear gap technique for analysis of Reissner-Mindlin platesrdquo Computer Methods in Applied Mechanics andEngineering vol 199 no 9ndash12 pp 471ndash489 2010
[19] P Phung-Van T Nguyen-Thoi T Bui-Xuan and Q Lieu-XuanldquoA cell-based smoothed three-node Mindlin plate element(CS-FEM-MIN3) based on the 119862
0 -type higher-order sheardeformation for geometrically nonlinear analysis of laminatedcomposite platesrdquo Computational Materials Science 2014
[20] NNguyen-Thanh T RabczukHNguyen-Xuan and S BordasldquoAn alternative alpha finite element method with discrete sheargap technique for analysis of isotropicMindlin-Reissner platesrdquoFinite Elements inAnalysis andDesign vol 47 no 5 pp 519ndash5352011
[21] KM Liew Y Xiang and S Kitipornchai ldquoTransverse vibrationof thick rectangular plates-I Comprehensive sets of boundaryconditionsrdquo Computers and Structures vol 49 no 1 pp 1ndash291993
[22] D Zhou ldquoVibrations of Mindlin rectangular plates with elasti-cally restrained edges using static Timoshenko beam functionswith the Rayleigh-Ritz methodrdquo International Journal of Solidsand Structures vol 38 no 32-33 pp 5565ndash5580 2001
[23] D Zhou SH Lo F T KAu andYK Cheung ldquoVibration anal-ysis of rectangular Mindlin plates with internal line supportsusing static Timoshenko beam functionsrdquo International Journalof Mechanical Sciences vol 44 no 12 pp 2503ndash2522 2002
[24] H S Shen J Yang and L Zhang ldquoFree and forced vibrationof Reissner-Mindlin plates with free edges resting on elasticfoundationsrdquo Journal of Sound and Vibration vol 244 no 2 pp299ndash320 2001
[25] Y Xing and B Liu ldquoClosed form solutions for free vibrations ofrectangular Mindlin platesrdquo Acta Mechanica Sinica vol 25 no5 pp 689ndash698 2009
[26] P Gagnon C Gosselin and L Cloutier ldquoA finite strip elementfor the analysis of variable thickness rectangular thick platesrdquoComputers amp Structures vol 63 no 2 pp 349ndash362 1997
[27] T Mlzusawa ldquoVibration of rectangular mindlin plates withtapered thickness by the spline strip methodrdquo Computers andStructures vol 46 no 3 pp 451ndash463 1993
[28] H Nguyen-Xuan and T Nguyen-Thoi ldquoA stabilized smoothedfinite element method for free vibration analysis of Mindlin-Reissner platesrdquo Communications in Numerical Methods inEngineering with Biomedical Applications vol 25 no 8 pp 882ndash906 2009
[29] K M Liew L X Peng and S Kitipornchai ldquoVibration anal-ysis of corrugated Reissner-Mindlin plates using a mesh-freeGalerkin methodrdquo International Journal of Mechanical Sciencesvol 51 no 9-10 pp 642ndash652 2009
[30] Y Hou GWWei and Y Xiang ldquoDSC-Ritz method for the freevibration analysis of Mindlin platesrdquo International Journal forNumerical Methods in Engineering vol 62 no 2 pp 262ndash2882005
[31] F- Liu and K M Liew ldquoVibration analysis of discontinuousMindlin plates by differential quadrature element methodrdquoJournal of Vibration and Acoustics Transactions of the ASMEvol 121 no 2 pp 204ndash208 1999
[32] P Malekzadeh G Karami and M Farid ldquoA semi-analyticalDQEM for free vibration analysis of thick plates with twoopposite edges simply supportedrdquoComputerMethods inAppliedMechanics and Engineering vol 193 no 45ndash47 pp 4781ndash47962004
[33] P Malekzadeh and S A Shahpari ldquoFree vibration analysisof variable thickness thin and moderately thick plates withelastically restrained edges by DQMrdquo Thin-Walled Structuresvol 43 no 7 pp 1037ndash1050 2005
[34] R E Diaz-Contreras and S Nomura ldquoGreenrsquos function appliedto solution ofMindlin platesrdquoComputers and Structures vol 60no 1 pp 41ndash48 1996
[35] T Sakiyama and M Huang ldquoFree vibration analysis of rect-angular plates with variable thicknessrdquo Journal of Sound andVibration vol 216 no 3 pp 379ndash397 1998
[36] M Huang X Q Ma T Sakiyama H Matuda and C MoritaldquoFree vibration analysis of orthotropic rectangular plates withvariable thickness and general boundary conditionsrdquo Journal ofSound and Vibration vol 288 no 4-5 pp 931ndash955 2005
Shock and Vibration 25
[37] I Shufrin and M Eisenberger ldquoVibration of shear deformableplates with variable thicknessmdashfirst-order and higher-orderanalysesrdquo Journal of Sound and Vibration vol 290 no 1-2 pp465ndash489 2006
[38] S A Eftekhari and A A Jafari ldquoAccurate variational approachfor free vibration of variable thickness thin and thick plates withedges elastically restrained against translation and rotationrdquoInternational Journal of Mechanical Sciences vol 68 pp 35ndash462013
[39] W L Li X Zhang J Du and Z Liu ldquoAn exact series solutionfor the transverse vibration of rectangular plates with generalelastic boundary supportsrdquo Journal of Sound amp Vibration vol321 no 1-2 pp 254ndash269 2009
[40] X Shi D Shi W L Li and Q Wang ldquoA unified method forfree vibration analysis of circular annular and sector plateswith arbitrary boundary conditionsrdquo Journal of Vibration andControl 2014
[41] D Y Shi X J Shi W L Li and Q S Wang ldquoFree transversevibrations of orthotropic thin rectangular plates with arbitraryelastic edge supportsrdquo Journal of Vibroengineering vol 16 no 1pp 389ndash398 2014
[42] Y Chen G Jin and Z Liu ldquoFree vibration analysis of circu-lar cylindrical shell with non-uniform elastic boundary con-straintsrdquo International Journal of Mechanical Sciences vol 74pp 120ndash132 2013
[43] L Dai T Yang J Du W L Li and M J Brennan ldquoAn exactseries solution for the vibration analysis of cylindrical shellswith arbitrary boundary conditionsrdquo Applied Acoustics vol 74no 3 pp 440ndash449 2013
International Journal of
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Shock and Vibration
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International Journal of
Shock and Vibration 21
to the solution procedure The convergence of the presentsolution is examined and the excellent accuracy is validatedby comparison with existing results published in the lit-erature and FEM data Excellent agreements are obtainedfrom these comparisons The proposed method providesa unified means for extracting the modal parameters andpredicting the vibration behaviors of moderately thick plateswith variable thickness variation functions and arbitraryelastic edge restraints A variety of free vibration results formoderately thick rectangular plates with different thicknessvariation functions and boundary conditions are presentedNew results for free vibration of moderately thick rectangularplates with various thickness variation functions and edgeconditions are presented which may be used for benchmark-ing of researchers in the field
Appendices
A Representative Calculation forStiffness and Mass Matrices
To illuminate the particular expression of the mass andstiffness matrixes clearly and tersely four new variables aredefined as 119904 = 119898 lowast (119873 + 1) + 119899 + 1 119905 = 119898
1015840
lowast (119873 + 1) + 1198991015840
+ 1119901 = (119897minus1)lowast (119872+1)+119898
1015840
+1 and 119902 = (119897minus1)lowast (119873+1)+1198991015840
+1The first row elements of K andM are given below
K1-1119904119905
=
1
2
int
119886
0
int
119887
0
[119863 (119909 119910)
times 21205821198861198981205821198861198981015840 sin 120582
119886119898119909 sin 120582
1198861198981015840119909 cos 120582
119887119899119910
times cos 1205821198871198991015840119910 + (1 minus 120583) 120582
1198871198991205821198871198991015840 cos 120582
119886119898119909
times cos 1205821198861198981015840 sin 120582
119887119899119910 sin 120582
1198871198991015840119910
+ 120581119866ℎ (119909 119910) cos 120582119886119898119909 cos 120582
1198861198981015840119909 cos 120582
119887119899
times119910 cos 1205821198871198991015840119910119889119909119889119910] 119889119909 119889119910
+ int
119886
0
1198701199091199100
cos 120582119886119898119909 cos 120582
1198861198981015840119909
+(minus1)119898+1198981015840
119870119909119910119887
cos 120582119886119898119909 cos 120582
1198861198981015840119909 119889119909
+ int
119887
0
1198701199090cos 120582119887119899119910 cos 120582
1198871198991015840119910
+(minus1)119899+1198991015840
119870119909119886cos 120582119887119899119910 cos 120582
1198871198991015840119910 119889119910
K1-2119904119901
=
1
2
int
119886
0
int
119887
0
[119863 (119909 119910)21205821198861198981205821198861198981015840 sin 120582
119886119898119909 sin 120582
1198861198981015840119909
times
2
sum
119897=1
120577119897
119887(119910) cos 120582
1198871198991015840119910
minus (1 minus 120583) 1205821198871198991015840 cos 120582
119886119898119909 cos 120582
1198861198981015840119909
times
2
sum
119897=1
1205771198971015840
119887(119910) sin 120582
1198871198991015840119910 + 120581119866ℎ (119909 119910)
times ( cos 1205821198861198981015840119909 cos 120582
1198871198991015840119910
times
2
sum
119897=1
120577119897
119887(119910) cos 120582
119886119898119909)]119889119909119889119910
+ int
119886
0
1198701199091199100
cos 120582119886119898119909 cos 120582
1198861198981015840119909
2
sum
119897=1
120577119897
119887(0)
+(minus1)1198991015840
119870119909119910119887
cos 120582119886119898119909 cos 120582
1198861198981015840119909
2
sum
119897=1
120577119897
119887(119887) 119889119909
+ int
119887
0
1198701199090
2
sum
119897=1
120577119897
119887(119910) cos 120582
1198871198991015840119910
+(minus1)119898+1198981015840
119870119909119886
2
sum
119897=1
120577119897
119887(119910) cos 120582
1198871198991015840119910119889119910
K1-3119904119902
=
1
2
int
119886
0
int
119887
0
[119863 (119909 119910) minus 21205821198861198981015840 sin 120582
1198861198981015840119909 cos 120582
1198871198991015840119910
times
2
sum
119897=1
1205771198971015840
119886(119909) cos 120582
119887119899119910
+ (1 minus 120583) 1205821198871198991205821198871198991015840 cos 120582
1198861198981015840119909
times sin 1205821198871198991015840119910
2
sum
119897=1
120577119897
119886(119909) sin 120582
119887119899119910
+ 120581119866ℎ (119909 119910) cos 1205821198861198981015840119909 cos 120582
1198871198991015840119910
times
2
sum
119897=1
120577119897
119886(119909) cos 120582
119887119899119910]119889119909119889119910
+ int
119886
0
1198701199091199100
cos 1205821198861198981015840119909
2
sum
119897=1
120577119897
119886(119909)
+(minus1)119899+1198991015840
119870119909119910119887
cos 1205821198861198981015840119909
2
sum
119897=1
120577119897
119886(119909) 119889119909
+ int
119887
0
1198701199090
2
sum
119897=1
120577119897
119886(0) cos 120582
1198871198991015840119910
+(minus1)119898+1198981015840
119870119909119886
2
sum
119897=1
120577119897
119886(0) cos 120582
1198871198991015840119910119889119910
K1-4119904119905
=
1
2
int
119886
0
int
119887
0
119863(119909 119910) 21205821198871198991205821198861198981015840 sin 120582
1198861198981015840119909 cos 120582
1198871198991015840119910
times cos 120582119886119898119909 sin 120582
119887119899119910 minus 2(1 minus 120583)
119887119899
times 1205821198861198981015840 sin 120582
1198861198981015840119909 cos 120582
1198871198991015840119910 cos 120582
119886119898119909
times sin 120582119887119899119910 + (1 minus 120583) 120582
1198861198981205821198871198991015840 cos 120582
1198861198981015840119909
times sin 1205821198871198991015840119910 sin 120582
119886119898119909 cos 120582
119887119899119910 119889119909 119889119910
22 Shock and Vibration
K1-5119904119901
=
1
2
int
119886
0
int
119887
0
119863(119909 119910) minus 21205821198861198981015840 sin 120582
1198861198981015840119909 cos 120582
1198871198991015840119910
times
2
sum
119897=1
1205771198971015840
119887(119910) cos 120582
119886119898119909
+ 2 (1 minus 120583) 1205821198861198981015840 sin 120582
1198861198981015840119909 cos 120582
1198871198991015840119910
times
2
sum
119897=1
1205771198971015840
119887(119910) cos 120582
119886119898119909
+ (1 minus 120583) 1205821198861198981205821198871198991015840 cos 120582
1198861198981015840119909 sin 120582
1198871198991015840119910
times
2
sum
119897=1
120577119897
119887(119910) sin 120582
119886119898119909119889119909119889119910
K1-6119904119902
=
1
2
int
119886
0
int
119887
0
119863(119909 119910)1205821198871198991205821198861198981015840 sin 120582
1198861198981015840119909 cos 120582
1198861198991015840119910
times
2
sum
119897=1
120577119897
119886(119909) sin 120582
1198871198991015840119910
minus 2 (1 minus 120583) 1205821198871198991205821198861198981015840 sin 120582
1198861198981015840119909 cos 120582
1198861198991015840119910
times
2
sum
119897=1
120577119897
119886(119909) sin 120582
1198871198991015840119910
+ 1205821198871198991015840 cos 120582
1198861198981015840119909 sin 120582
1198871198991015840119910
times
2
sum
119897=1
1205771198971015840
119886(119909) cos 120582
119887119899119910119889119909119889119910
K1-7119904119905
= minus
1
2
int
119886
0
int
119887
0
120581119866ℎ (119909 119910) 120582119886119898
cos 1205821198861198981015840119909 cos 120582
1198871198991015840119910
times sin 120582119886119898119909 cos 120582
119887119899119910119889119909119889119910
K1-8119904119901
= minus
1
2
int
119886
0
int
119887
0
120581119866ℎ (119909 119910) 1205821198861198981015840 cos 120582
1198861198981015840119909 cos 120582
1198871198991015840119910
times
2
sum
119897=1
120577119897
119887(119910) sin 120582
1198861198981015840119909 119889119909 119889119910
K1-9119904119902
=
1
2
int
119886
0
int
119887
0
120581119866ℎ (119909 119910) cos 1205821198861198981015840119909 cos 120582
1198871198991015840119910
times
2
sum
119897=1
1205771198971015840
119886(119909) cos 120582
119887119899119910119889119909119889119910
M1-1119904119905
=
1
24
120588int
119886
0
int
119887
0
ℎ3
(119909 119910) cos 120582119886119898119909 cos 120582
119887119899119910
times cos 1205821198861198981015840119909 cos 120582
1198871198991015840119910119889119909119889119910
M1-2119904119901
=
1
24
120588int
119886
0
int
119887
0
ℎ3
(119909 119910)
2
sum
119897=1
120577119897
119887(119910) cos 120582
119886119898119909
times cos 1205821198861198981015840119909 cos 120582
1198871198991015840119910119889119909119889119910
M1-3119904119902
=
1
24
120588int
119886
0
int
119887
0
ℎ3
(119909 119910)
2
sum
119897=1
120577119897
119886(119909) cos 120582
119887119899119910
times cos 1205821198861198981015840119909 cos 120582
1198871198991015840119910119889119909119889119910
M1-4119904119905 = M
1-5119904119901 = M1-6119904119902
= M1-7119904119905 = M
1-8119904119901 = M1-9119904119902 = 0
(A1)
B Useful Integrals for the Moderately ThickRectangular Plates with Variable Thicknessand Arbitrary Boundary Conditions
Consider
int
119871
0
cos 120582119898119909 cos 120582
119899119909 119889119909 =
0 119898 = 119899
119871
2
119898 = 119899
119871 119898 = 119899 = 0
int
119871
0
sin 120582119898119909 cos 120582
119899119909119889119909
=
minus
(minus1 + (minus1)119898+119899
) 119871119898
(1198982minus 1198992) 120587
119898 = 119899
0 119898 = 119899
int
119871
0
sin 120582119898119909 sin 120582
119899119909 119889119909 =
0 119898 = 119899
119871
2
119898 = 119899
0 119898 = 119899 = 0
int
119871
0
cos 120582119898119909120577119897
119886(119909) 119889119909
=
41198712
31205872 119897 = 1 119898 = 0
minus
41198712
(41198982
minus 3)
(9 minus 401198982+ 16119898
4) 1205872
119897 = 1 119898 = 0
minus
41198712
31205872 119897 = 2 119898 = 0
41198712
(41198982
minus 3) (minus1)119898
(9 minus 401198982+ 16119898
4) 1205872
119897 = 2 119898 = 0
Shock and Vibration 23
int
119871
0
sin 120582119898119909120577119897
119886(119909) 119889119909
=
0 119897 = 1 119898 = 0
161198712
119898(minus1)119898
(9 minus 401198982+ 16119898
4) 1205872
119897 = 1 119898 = 0
0 119897 = 2 119898 = 0
161198981198712
(9 minus 401198982+ 16119898
4) 1205872
119897 = 2 119898 = 0
int
119871
0
cos 1205821198981199091205771198971015840
119886(119909) 119889119909
=
0 119897 = 1 119898 = 0
161198711198982
(minus1)119898
(9 minus 401198982+ 16119898
4) 120587
119897 = 1 119898 = 0
0 119897 = 2 119898 = 0
161198711198982
(9 minus 401198982+ 16119898
4) 120587
119897 = 2 119898 = 0
int
119871
0
sin 1205821198981199091205771198971015840
119886(119909) 119889119909
=
0 119897 = 1 119898 = 0
4119871119898 (minus3 + 41198982
)
(9 minus 401198982+ 16119898
4) 120587
119897 = 1 119898 = 0
0 119897 = 2 119898 = 0
119871119898(
1
120587 minus 41198982120587
+
3
9120587 minus 41198982120587
) (minus1)119898
119897 = 2 119898 = 0
int
119871
0
1205771198971
119886(119909) 1205771198972
119886(119909) 119889119909 =
1198713
41205872 1198971= 1 1198972= 1
minus
21198713
31205873 1198971= 1 1198972= 2
minus
21198713
31205873 1198971= 2 1198972= 1
1198713
41205872 1198971= 2 1198972= 2
int
119871
0
1205771198971
119886(119909) 1205771198971015840
2
119886(119909) 119889119909 =
0 1198971= 1 1198972= 1
minus
1198712
8120587
1198971= 1 1198972= 2
1198712
8120587
1198971= 2 1198972= 1
0 1198971= 2 1198972= 2
int
119871
0
1205771198971015840
1
119886(119909) 1205771198971015840
2
119886(119909) 119889119909 =
5119871
16
1198971= 1 1198972= 1
minus
119871
2120587
1198971= 1 1198972= 2
minus
119871
2120587
1198971= 2 1198972= 1
5119871
16
1198971= 2 1198972= 2
int
119871
0
119909119899 cos 120582
119898119909 119889119909
=
1198711+119899
1 + 119899
119898 = 0
(1198861+119899HypergeometricPFQtimes [ 12 + 1198992 12 32 + 1198992
minus (14)1198982
1205872
]) times (1 + 119899)minus1
119898 gt 0
(B1)
Conflict of Interests
The authors declare that there is no conflict of interestsregarding the publication of this paper
Acknowledgments
The authors would like to thank the anonymous reviewersfor their very valuable comments The authors also gratefullyacknowledge the financial support from the National NaturalScience Foundation of China (no 51209052) HeilongjiangProvince Youth Science Fund Project (no QC2011C013) andHarbin Science and Technology Development InnovationFoundation of Youth (no 2011RFQXG021)
References
[1] K Liew Y Xiang and S Kitipornchai ldquoResearch on thick platevibration a literature surveyrdquo Journal of Sound and Vibrationvol 180 no 1 pp 163ndash176 1995
[2] J H Chung T Y Chung and K C Kim ldquoVibration analysisof orthotropic mindlin plates with edges elastically restrainedagainst rotationrdquo Journal of Sound and Vibration vol 163 no 1pp 151ndash163 1993
[3] Y K Cheung and D Zhou ldquoVibrations of moderately thickrectangular plates in terms of a set of static Timoshenko beamfunctionsrdquoComputers and Structures vol 78 no 6 pp 757ndash7682000
[4] CMWang ldquoNatural frequencies formula for simply supportedMindlin platesrdquo Journal of Vibration and Acoustics vol 116 no4 pp 536ndash540 1994
[5] K N Saha R C Kar and P K Datta ldquoFree vibration analysisof rectangular Mindlin plates with elastic restraints uniformlydistributed along the edgesrdquo Journal of Sound andVibration vol192 no 4 pp 885ndash902 1996
24 Shock and Vibration
[6] D J Gorman ldquoAccurate free vibration analysis of shear-deformable plates with torsional elastic edge supportrdquo Journalof Sound and Vibration vol 203 no 2 pp 209ndash218 1997
[7] D J Gorman ldquoFree vibration analysis of Mindlin plates withuniform elastic edge support by the superposition methodrdquoJournal of Sound andVibration vol 207 no 3 pp 335ndash350 1997
[8] D J Gorman ldquoAccurate free vibration analysis of point sup-ported mindlin plates by the superposition methodrdquo Journal ofSound and Vibration vol 219 no 2 pp 265ndash277 1999
[9] Y Xiang ldquoVibration of rectangular Mindlin plates resting onnon-homogenous elastic foundationsrdquo International Journal ofMechanical Sciences vol 45 no 6-7 pp 1229ndash1244 2003
[10] Y Xiang and G W Wei ldquoExact solutions for buckling andvibration of stepped rectangular Mindlin platesrdquo InternationalJournal of Solids and Structures vol 41 no 1 pp 279ndash294 2004
[11] Y L Yeh M J Jang and C C Wang ldquoAnalyzing the freevibrations of a plate using finite difference and differential trans-formationmethodrdquoAppliedMathematics andComputation vol178 no 2 pp 493ndash501 2006
[12] Y Xiang S K Lai L Zhou and C W Lim ldquoDSC-Ritz elementmethod for vibration analysis of rectangular Mindlin plateswith mixed edge supportsrdquo European Journal of MechanicsmdashASolids vol 29 no 4 pp 619ndash628 2010
[13] H Nguyen-Xuan C H Thai and T Nguyen-Thoi ldquoIsogeo-metric finite element analysis of composite sandwich platesusing a higher order shear deformation theoryrdquo Composites BEngineering vol 55 pp 558ndash574 2013
[14] C H Thai A J M Ferreira E Carrera and H Nguyen-XuanldquoIsogeometric analysis of laminated composite and sandwichplates using a layerwise deformation theoryrdquo Composite Struc-tures vol 104 pp 196ndash214 2013
[15] C H Thai A Ferreira S Bordas T Rabczuk and HNguyen-Xuan ldquoIsogeometric analysis of laminated compositeand sandwich plates using a new inverse trigonometric sheardeformation theoryrdquo European Journal of Mechanics A Solidsvol 43 pp 89ndash108 2014
[16] H Luong-Van T Nguyen-Thoi G R Liu and P Phung-VanldquoA cell-based smoothed finite elementmethod using three-nodeshear-locking free Mindlin plate element (CS-FEM-MIN3) fordynamic response of laminated composite plates on viscoelasticfoundationrdquo Engineering Analysis with Boundary Elements vol42 pp 8ndash19 2014
[17] T Nguyen-Thoi T Rabczuk T Lam-Phat V Ho-Huu andP Phung-Van ldquoFree vibration analysis of cracked Mindlinplate using an extended cell-based smoothed discrete sheargap method (XCS-DSG3)rdquo Theoretical and Applied FractureMechanics 2014
[18] H Nguyen-Xuan G R Liu and C a Thai-Hoang ldquoAnedge-based smoothed finite element method (ES-FEM) withstabilized discrete shear gap technique for analysis of Reissner-Mindlin platesrdquo Computer Methods in Applied Mechanics andEngineering vol 199 no 9ndash12 pp 471ndash489 2010
[19] P Phung-Van T Nguyen-Thoi T Bui-Xuan and Q Lieu-XuanldquoA cell-based smoothed three-node Mindlin plate element(CS-FEM-MIN3) based on the 119862
0 -type higher-order sheardeformation for geometrically nonlinear analysis of laminatedcomposite platesrdquo Computational Materials Science 2014
[20] NNguyen-Thanh T RabczukHNguyen-Xuan and S BordasldquoAn alternative alpha finite element method with discrete sheargap technique for analysis of isotropicMindlin-Reissner platesrdquoFinite Elements inAnalysis andDesign vol 47 no 5 pp 519ndash5352011
[21] KM Liew Y Xiang and S Kitipornchai ldquoTransverse vibrationof thick rectangular plates-I Comprehensive sets of boundaryconditionsrdquo Computers and Structures vol 49 no 1 pp 1ndash291993
[22] D Zhou ldquoVibrations of Mindlin rectangular plates with elasti-cally restrained edges using static Timoshenko beam functionswith the Rayleigh-Ritz methodrdquo International Journal of Solidsand Structures vol 38 no 32-33 pp 5565ndash5580 2001
[23] D Zhou SH Lo F T KAu andYK Cheung ldquoVibration anal-ysis of rectangular Mindlin plates with internal line supportsusing static Timoshenko beam functionsrdquo International Journalof Mechanical Sciences vol 44 no 12 pp 2503ndash2522 2002
[24] H S Shen J Yang and L Zhang ldquoFree and forced vibrationof Reissner-Mindlin plates with free edges resting on elasticfoundationsrdquo Journal of Sound and Vibration vol 244 no 2 pp299ndash320 2001
[25] Y Xing and B Liu ldquoClosed form solutions for free vibrations ofrectangular Mindlin platesrdquo Acta Mechanica Sinica vol 25 no5 pp 689ndash698 2009
[26] P Gagnon C Gosselin and L Cloutier ldquoA finite strip elementfor the analysis of variable thickness rectangular thick platesrdquoComputers amp Structures vol 63 no 2 pp 349ndash362 1997
[27] T Mlzusawa ldquoVibration of rectangular mindlin plates withtapered thickness by the spline strip methodrdquo Computers andStructures vol 46 no 3 pp 451ndash463 1993
[28] H Nguyen-Xuan and T Nguyen-Thoi ldquoA stabilized smoothedfinite element method for free vibration analysis of Mindlin-Reissner platesrdquo Communications in Numerical Methods inEngineering with Biomedical Applications vol 25 no 8 pp 882ndash906 2009
[29] K M Liew L X Peng and S Kitipornchai ldquoVibration anal-ysis of corrugated Reissner-Mindlin plates using a mesh-freeGalerkin methodrdquo International Journal of Mechanical Sciencesvol 51 no 9-10 pp 642ndash652 2009
[30] Y Hou GWWei and Y Xiang ldquoDSC-Ritz method for the freevibration analysis of Mindlin platesrdquo International Journal forNumerical Methods in Engineering vol 62 no 2 pp 262ndash2882005
[31] F- Liu and K M Liew ldquoVibration analysis of discontinuousMindlin plates by differential quadrature element methodrdquoJournal of Vibration and Acoustics Transactions of the ASMEvol 121 no 2 pp 204ndash208 1999
[32] P Malekzadeh G Karami and M Farid ldquoA semi-analyticalDQEM for free vibration analysis of thick plates with twoopposite edges simply supportedrdquoComputerMethods inAppliedMechanics and Engineering vol 193 no 45ndash47 pp 4781ndash47962004
[33] P Malekzadeh and S A Shahpari ldquoFree vibration analysisof variable thickness thin and moderately thick plates withelastically restrained edges by DQMrdquo Thin-Walled Structuresvol 43 no 7 pp 1037ndash1050 2005
[34] R E Diaz-Contreras and S Nomura ldquoGreenrsquos function appliedto solution ofMindlin platesrdquoComputers and Structures vol 60no 1 pp 41ndash48 1996
[35] T Sakiyama and M Huang ldquoFree vibration analysis of rect-angular plates with variable thicknessrdquo Journal of Sound andVibration vol 216 no 3 pp 379ndash397 1998
[36] M Huang X Q Ma T Sakiyama H Matuda and C MoritaldquoFree vibration analysis of orthotropic rectangular plates withvariable thickness and general boundary conditionsrdquo Journal ofSound and Vibration vol 288 no 4-5 pp 931ndash955 2005
Shock and Vibration 25
[37] I Shufrin and M Eisenberger ldquoVibration of shear deformableplates with variable thicknessmdashfirst-order and higher-orderanalysesrdquo Journal of Sound and Vibration vol 290 no 1-2 pp465ndash489 2006
[38] S A Eftekhari and A A Jafari ldquoAccurate variational approachfor free vibration of variable thickness thin and thick plates withedges elastically restrained against translation and rotationrdquoInternational Journal of Mechanical Sciences vol 68 pp 35ndash462013
[39] W L Li X Zhang J Du and Z Liu ldquoAn exact series solutionfor the transverse vibration of rectangular plates with generalelastic boundary supportsrdquo Journal of Sound amp Vibration vol321 no 1-2 pp 254ndash269 2009
[40] X Shi D Shi W L Li and Q Wang ldquoA unified method forfree vibration analysis of circular annular and sector plateswith arbitrary boundary conditionsrdquo Journal of Vibration andControl 2014
[41] D Y Shi X J Shi W L Li and Q S Wang ldquoFree transversevibrations of orthotropic thin rectangular plates with arbitraryelastic edge supportsrdquo Journal of Vibroengineering vol 16 no 1pp 389ndash398 2014
[42] Y Chen G Jin and Z Liu ldquoFree vibration analysis of circu-lar cylindrical shell with non-uniform elastic boundary con-straintsrdquo International Journal of Mechanical Sciences vol 74pp 120ndash132 2013
[43] L Dai T Yang J Du W L Li and M J Brennan ldquoAn exactseries solution for the vibration analysis of cylindrical shellswith arbitrary boundary conditionsrdquo Applied Acoustics vol 74no 3 pp 440ndash449 2013
International Journal of
AerospaceEngineeringHindawi Publishing Corporationhttpwwwhindawicom Volume 2014
RoboticsJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Active and Passive Electronic Components
Control Scienceand Engineering
Journal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
International Journal of
RotatingMachinery
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Hindawi Publishing Corporation httpwwwhindawicom
Journal ofEngineeringVolume 2014
Submit your manuscripts athttpwwwhindawicom
VLSI Design
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Shock and Vibration
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Civil EngineeringAdvances in
Acoustics and VibrationAdvances in
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Electrical and Computer Engineering
Journal of
Advances inOptoElectronics
Hindawi Publishing Corporation httpwwwhindawicom
Volume 2014
The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014
SensorsJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Modelling amp Simulation in EngineeringHindawi Publishing Corporation httpwwwhindawicom Volume 2014
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Chemical EngineeringInternational Journal of Antennas and
Propagation
International Journal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Navigation and Observation
International Journal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
DistributedSensor Networks
International Journal of
22 Shock and Vibration
K1-5119904119901
=
1
2
int
119886
0
int
119887
0
119863(119909 119910) minus 21205821198861198981015840 sin 120582
1198861198981015840119909 cos 120582
1198871198991015840119910
times
2
sum
119897=1
1205771198971015840
119887(119910) cos 120582
119886119898119909
+ 2 (1 minus 120583) 1205821198861198981015840 sin 120582
1198861198981015840119909 cos 120582
1198871198991015840119910
times
2
sum
119897=1
1205771198971015840
119887(119910) cos 120582
119886119898119909
+ (1 minus 120583) 1205821198861198981205821198871198991015840 cos 120582
1198861198981015840119909 sin 120582
1198871198991015840119910
times
2
sum
119897=1
120577119897
119887(119910) sin 120582
119886119898119909119889119909119889119910
K1-6119904119902
=
1
2
int
119886
0
int
119887
0
119863(119909 119910)1205821198871198991205821198861198981015840 sin 120582
1198861198981015840119909 cos 120582
1198861198991015840119910
times
2
sum
119897=1
120577119897
119886(119909) sin 120582
1198871198991015840119910
minus 2 (1 minus 120583) 1205821198871198991205821198861198981015840 sin 120582
1198861198981015840119909 cos 120582
1198861198991015840119910
times
2
sum
119897=1
120577119897
119886(119909) sin 120582
1198871198991015840119910
+ 1205821198871198991015840 cos 120582
1198861198981015840119909 sin 120582
1198871198991015840119910
times
2
sum
119897=1
1205771198971015840
119886(119909) cos 120582
119887119899119910119889119909119889119910
K1-7119904119905
= minus
1
2
int
119886
0
int
119887
0
120581119866ℎ (119909 119910) 120582119886119898
cos 1205821198861198981015840119909 cos 120582
1198871198991015840119910
times sin 120582119886119898119909 cos 120582
119887119899119910119889119909119889119910
K1-8119904119901
= minus
1
2
int
119886
0
int
119887
0
120581119866ℎ (119909 119910) 1205821198861198981015840 cos 120582
1198861198981015840119909 cos 120582
1198871198991015840119910
times
2
sum
119897=1
120577119897
119887(119910) sin 120582
1198861198981015840119909 119889119909 119889119910
K1-9119904119902
=
1
2
int
119886
0
int
119887
0
120581119866ℎ (119909 119910) cos 1205821198861198981015840119909 cos 120582
1198871198991015840119910
times
2
sum
119897=1
1205771198971015840
119886(119909) cos 120582
119887119899119910119889119909119889119910
M1-1119904119905
=
1
24
120588int
119886
0
int
119887
0
ℎ3
(119909 119910) cos 120582119886119898119909 cos 120582
119887119899119910
times cos 1205821198861198981015840119909 cos 120582
1198871198991015840119910119889119909119889119910
M1-2119904119901
=
1
24
120588int
119886
0
int
119887
0
ℎ3
(119909 119910)
2
sum
119897=1
120577119897
119887(119910) cos 120582
119886119898119909
times cos 1205821198861198981015840119909 cos 120582
1198871198991015840119910119889119909119889119910
M1-3119904119902
=
1
24
120588int
119886
0
int
119887
0
ℎ3
(119909 119910)
2
sum
119897=1
120577119897
119886(119909) cos 120582
119887119899119910
times cos 1205821198861198981015840119909 cos 120582
1198871198991015840119910119889119909119889119910
M1-4119904119905 = M
1-5119904119901 = M1-6119904119902
= M1-7119904119905 = M
1-8119904119901 = M1-9119904119902 = 0
(A1)
B Useful Integrals for the Moderately ThickRectangular Plates with Variable Thicknessand Arbitrary Boundary Conditions
Consider
int
119871
0
cos 120582119898119909 cos 120582
119899119909 119889119909 =
0 119898 = 119899
119871
2
119898 = 119899
119871 119898 = 119899 = 0
int
119871
0
sin 120582119898119909 cos 120582
119899119909119889119909
=
minus
(minus1 + (minus1)119898+119899
) 119871119898
(1198982minus 1198992) 120587
119898 = 119899
0 119898 = 119899
int
119871
0
sin 120582119898119909 sin 120582
119899119909 119889119909 =
0 119898 = 119899
119871
2
119898 = 119899
0 119898 = 119899 = 0
int
119871
0
cos 120582119898119909120577119897
119886(119909) 119889119909
=
41198712
31205872 119897 = 1 119898 = 0
minus
41198712
(41198982
minus 3)
(9 minus 401198982+ 16119898
4) 1205872
119897 = 1 119898 = 0
minus
41198712
31205872 119897 = 2 119898 = 0
41198712
(41198982
minus 3) (minus1)119898
(9 minus 401198982+ 16119898
4) 1205872
119897 = 2 119898 = 0
Shock and Vibration 23
int
119871
0
sin 120582119898119909120577119897
119886(119909) 119889119909
=
0 119897 = 1 119898 = 0
161198712
119898(minus1)119898
(9 minus 401198982+ 16119898
4) 1205872
119897 = 1 119898 = 0
0 119897 = 2 119898 = 0
161198981198712
(9 minus 401198982+ 16119898
4) 1205872
119897 = 2 119898 = 0
int
119871
0
cos 1205821198981199091205771198971015840
119886(119909) 119889119909
=
0 119897 = 1 119898 = 0
161198711198982
(minus1)119898
(9 minus 401198982+ 16119898
4) 120587
119897 = 1 119898 = 0
0 119897 = 2 119898 = 0
161198711198982
(9 minus 401198982+ 16119898
4) 120587
119897 = 2 119898 = 0
int
119871
0
sin 1205821198981199091205771198971015840
119886(119909) 119889119909
=
0 119897 = 1 119898 = 0
4119871119898 (minus3 + 41198982
)
(9 minus 401198982+ 16119898
4) 120587
119897 = 1 119898 = 0
0 119897 = 2 119898 = 0
119871119898(
1
120587 minus 41198982120587
+
3
9120587 minus 41198982120587
) (minus1)119898
119897 = 2 119898 = 0
int
119871
0
1205771198971
119886(119909) 1205771198972
119886(119909) 119889119909 =
1198713
41205872 1198971= 1 1198972= 1
minus
21198713
31205873 1198971= 1 1198972= 2
minus
21198713
31205873 1198971= 2 1198972= 1
1198713
41205872 1198971= 2 1198972= 2
int
119871
0
1205771198971
119886(119909) 1205771198971015840
2
119886(119909) 119889119909 =
0 1198971= 1 1198972= 1
minus
1198712
8120587
1198971= 1 1198972= 2
1198712
8120587
1198971= 2 1198972= 1
0 1198971= 2 1198972= 2
int
119871
0
1205771198971015840
1
119886(119909) 1205771198971015840
2
119886(119909) 119889119909 =
5119871
16
1198971= 1 1198972= 1
minus
119871
2120587
1198971= 1 1198972= 2
minus
119871
2120587
1198971= 2 1198972= 1
5119871
16
1198971= 2 1198972= 2
int
119871
0
119909119899 cos 120582
119898119909 119889119909
=
1198711+119899
1 + 119899
119898 = 0
(1198861+119899HypergeometricPFQtimes [ 12 + 1198992 12 32 + 1198992
minus (14)1198982
1205872
]) times (1 + 119899)minus1
119898 gt 0
(B1)
Conflict of Interests
The authors declare that there is no conflict of interestsregarding the publication of this paper
Acknowledgments
The authors would like to thank the anonymous reviewersfor their very valuable comments The authors also gratefullyacknowledge the financial support from the National NaturalScience Foundation of China (no 51209052) HeilongjiangProvince Youth Science Fund Project (no QC2011C013) andHarbin Science and Technology Development InnovationFoundation of Youth (no 2011RFQXG021)
References
[1] K Liew Y Xiang and S Kitipornchai ldquoResearch on thick platevibration a literature surveyrdquo Journal of Sound and Vibrationvol 180 no 1 pp 163ndash176 1995
[2] J H Chung T Y Chung and K C Kim ldquoVibration analysisof orthotropic mindlin plates with edges elastically restrainedagainst rotationrdquo Journal of Sound and Vibration vol 163 no 1pp 151ndash163 1993
[3] Y K Cheung and D Zhou ldquoVibrations of moderately thickrectangular plates in terms of a set of static Timoshenko beamfunctionsrdquoComputers and Structures vol 78 no 6 pp 757ndash7682000
[4] CMWang ldquoNatural frequencies formula for simply supportedMindlin platesrdquo Journal of Vibration and Acoustics vol 116 no4 pp 536ndash540 1994
[5] K N Saha R C Kar and P K Datta ldquoFree vibration analysisof rectangular Mindlin plates with elastic restraints uniformlydistributed along the edgesrdquo Journal of Sound andVibration vol192 no 4 pp 885ndash902 1996
24 Shock and Vibration
[6] D J Gorman ldquoAccurate free vibration analysis of shear-deformable plates with torsional elastic edge supportrdquo Journalof Sound and Vibration vol 203 no 2 pp 209ndash218 1997
[7] D J Gorman ldquoFree vibration analysis of Mindlin plates withuniform elastic edge support by the superposition methodrdquoJournal of Sound andVibration vol 207 no 3 pp 335ndash350 1997
[8] D J Gorman ldquoAccurate free vibration analysis of point sup-ported mindlin plates by the superposition methodrdquo Journal ofSound and Vibration vol 219 no 2 pp 265ndash277 1999
[9] Y Xiang ldquoVibration of rectangular Mindlin plates resting onnon-homogenous elastic foundationsrdquo International Journal ofMechanical Sciences vol 45 no 6-7 pp 1229ndash1244 2003
[10] Y Xiang and G W Wei ldquoExact solutions for buckling andvibration of stepped rectangular Mindlin platesrdquo InternationalJournal of Solids and Structures vol 41 no 1 pp 279ndash294 2004
[11] Y L Yeh M J Jang and C C Wang ldquoAnalyzing the freevibrations of a plate using finite difference and differential trans-formationmethodrdquoAppliedMathematics andComputation vol178 no 2 pp 493ndash501 2006
[12] Y Xiang S K Lai L Zhou and C W Lim ldquoDSC-Ritz elementmethod for vibration analysis of rectangular Mindlin plateswith mixed edge supportsrdquo European Journal of MechanicsmdashASolids vol 29 no 4 pp 619ndash628 2010
[13] H Nguyen-Xuan C H Thai and T Nguyen-Thoi ldquoIsogeo-metric finite element analysis of composite sandwich platesusing a higher order shear deformation theoryrdquo Composites BEngineering vol 55 pp 558ndash574 2013
[14] C H Thai A J M Ferreira E Carrera and H Nguyen-XuanldquoIsogeometric analysis of laminated composite and sandwichplates using a layerwise deformation theoryrdquo Composite Struc-tures vol 104 pp 196ndash214 2013
[15] C H Thai A Ferreira S Bordas T Rabczuk and HNguyen-Xuan ldquoIsogeometric analysis of laminated compositeand sandwich plates using a new inverse trigonometric sheardeformation theoryrdquo European Journal of Mechanics A Solidsvol 43 pp 89ndash108 2014
[16] H Luong-Van T Nguyen-Thoi G R Liu and P Phung-VanldquoA cell-based smoothed finite elementmethod using three-nodeshear-locking free Mindlin plate element (CS-FEM-MIN3) fordynamic response of laminated composite plates on viscoelasticfoundationrdquo Engineering Analysis with Boundary Elements vol42 pp 8ndash19 2014
[17] T Nguyen-Thoi T Rabczuk T Lam-Phat V Ho-Huu andP Phung-Van ldquoFree vibration analysis of cracked Mindlinplate using an extended cell-based smoothed discrete sheargap method (XCS-DSG3)rdquo Theoretical and Applied FractureMechanics 2014
[18] H Nguyen-Xuan G R Liu and C a Thai-Hoang ldquoAnedge-based smoothed finite element method (ES-FEM) withstabilized discrete shear gap technique for analysis of Reissner-Mindlin platesrdquo Computer Methods in Applied Mechanics andEngineering vol 199 no 9ndash12 pp 471ndash489 2010
[19] P Phung-Van T Nguyen-Thoi T Bui-Xuan and Q Lieu-XuanldquoA cell-based smoothed three-node Mindlin plate element(CS-FEM-MIN3) based on the 119862
0 -type higher-order sheardeformation for geometrically nonlinear analysis of laminatedcomposite platesrdquo Computational Materials Science 2014
[20] NNguyen-Thanh T RabczukHNguyen-Xuan and S BordasldquoAn alternative alpha finite element method with discrete sheargap technique for analysis of isotropicMindlin-Reissner platesrdquoFinite Elements inAnalysis andDesign vol 47 no 5 pp 519ndash5352011
[21] KM Liew Y Xiang and S Kitipornchai ldquoTransverse vibrationof thick rectangular plates-I Comprehensive sets of boundaryconditionsrdquo Computers and Structures vol 49 no 1 pp 1ndash291993
[22] D Zhou ldquoVibrations of Mindlin rectangular plates with elasti-cally restrained edges using static Timoshenko beam functionswith the Rayleigh-Ritz methodrdquo International Journal of Solidsand Structures vol 38 no 32-33 pp 5565ndash5580 2001
[23] D Zhou SH Lo F T KAu andYK Cheung ldquoVibration anal-ysis of rectangular Mindlin plates with internal line supportsusing static Timoshenko beam functionsrdquo International Journalof Mechanical Sciences vol 44 no 12 pp 2503ndash2522 2002
[24] H S Shen J Yang and L Zhang ldquoFree and forced vibrationof Reissner-Mindlin plates with free edges resting on elasticfoundationsrdquo Journal of Sound and Vibration vol 244 no 2 pp299ndash320 2001
[25] Y Xing and B Liu ldquoClosed form solutions for free vibrations ofrectangular Mindlin platesrdquo Acta Mechanica Sinica vol 25 no5 pp 689ndash698 2009
[26] P Gagnon C Gosselin and L Cloutier ldquoA finite strip elementfor the analysis of variable thickness rectangular thick platesrdquoComputers amp Structures vol 63 no 2 pp 349ndash362 1997
[27] T Mlzusawa ldquoVibration of rectangular mindlin plates withtapered thickness by the spline strip methodrdquo Computers andStructures vol 46 no 3 pp 451ndash463 1993
[28] H Nguyen-Xuan and T Nguyen-Thoi ldquoA stabilized smoothedfinite element method for free vibration analysis of Mindlin-Reissner platesrdquo Communications in Numerical Methods inEngineering with Biomedical Applications vol 25 no 8 pp 882ndash906 2009
[29] K M Liew L X Peng and S Kitipornchai ldquoVibration anal-ysis of corrugated Reissner-Mindlin plates using a mesh-freeGalerkin methodrdquo International Journal of Mechanical Sciencesvol 51 no 9-10 pp 642ndash652 2009
[30] Y Hou GWWei and Y Xiang ldquoDSC-Ritz method for the freevibration analysis of Mindlin platesrdquo International Journal forNumerical Methods in Engineering vol 62 no 2 pp 262ndash2882005
[31] F- Liu and K M Liew ldquoVibration analysis of discontinuousMindlin plates by differential quadrature element methodrdquoJournal of Vibration and Acoustics Transactions of the ASMEvol 121 no 2 pp 204ndash208 1999
[32] P Malekzadeh G Karami and M Farid ldquoA semi-analyticalDQEM for free vibration analysis of thick plates with twoopposite edges simply supportedrdquoComputerMethods inAppliedMechanics and Engineering vol 193 no 45ndash47 pp 4781ndash47962004
[33] P Malekzadeh and S A Shahpari ldquoFree vibration analysisof variable thickness thin and moderately thick plates withelastically restrained edges by DQMrdquo Thin-Walled Structuresvol 43 no 7 pp 1037ndash1050 2005
[34] R E Diaz-Contreras and S Nomura ldquoGreenrsquos function appliedto solution ofMindlin platesrdquoComputers and Structures vol 60no 1 pp 41ndash48 1996
[35] T Sakiyama and M Huang ldquoFree vibration analysis of rect-angular plates with variable thicknessrdquo Journal of Sound andVibration vol 216 no 3 pp 379ndash397 1998
[36] M Huang X Q Ma T Sakiyama H Matuda and C MoritaldquoFree vibration analysis of orthotropic rectangular plates withvariable thickness and general boundary conditionsrdquo Journal ofSound and Vibration vol 288 no 4-5 pp 931ndash955 2005
Shock and Vibration 25
[37] I Shufrin and M Eisenberger ldquoVibration of shear deformableplates with variable thicknessmdashfirst-order and higher-orderanalysesrdquo Journal of Sound and Vibration vol 290 no 1-2 pp465ndash489 2006
[38] S A Eftekhari and A A Jafari ldquoAccurate variational approachfor free vibration of variable thickness thin and thick plates withedges elastically restrained against translation and rotationrdquoInternational Journal of Mechanical Sciences vol 68 pp 35ndash462013
[39] W L Li X Zhang J Du and Z Liu ldquoAn exact series solutionfor the transverse vibration of rectangular plates with generalelastic boundary supportsrdquo Journal of Sound amp Vibration vol321 no 1-2 pp 254ndash269 2009
[40] X Shi D Shi W L Li and Q Wang ldquoA unified method forfree vibration analysis of circular annular and sector plateswith arbitrary boundary conditionsrdquo Journal of Vibration andControl 2014
[41] D Y Shi X J Shi W L Li and Q S Wang ldquoFree transversevibrations of orthotropic thin rectangular plates with arbitraryelastic edge supportsrdquo Journal of Vibroengineering vol 16 no 1pp 389ndash398 2014
[42] Y Chen G Jin and Z Liu ldquoFree vibration analysis of circu-lar cylindrical shell with non-uniform elastic boundary con-straintsrdquo International Journal of Mechanical Sciences vol 74pp 120ndash132 2013
[43] L Dai T Yang J Du W L Li and M J Brennan ldquoAn exactseries solution for the vibration analysis of cylindrical shellswith arbitrary boundary conditionsrdquo Applied Acoustics vol 74no 3 pp 440ndash449 2013
International Journal of
AerospaceEngineeringHindawi Publishing Corporationhttpwwwhindawicom Volume 2014
RoboticsJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Active and Passive Electronic Components
Control Scienceand Engineering
Journal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
International Journal of
RotatingMachinery
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Hindawi Publishing Corporation httpwwwhindawicom
Journal ofEngineeringVolume 2014
Submit your manuscripts athttpwwwhindawicom
VLSI Design
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Shock and Vibration
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Civil EngineeringAdvances in
Acoustics and VibrationAdvances in
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Electrical and Computer Engineering
Journal of
Advances inOptoElectronics
Hindawi Publishing Corporation httpwwwhindawicom
Volume 2014
The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014
SensorsJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Modelling amp Simulation in EngineeringHindawi Publishing Corporation httpwwwhindawicom Volume 2014
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Chemical EngineeringInternational Journal of Antennas and
Propagation
International Journal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Navigation and Observation
International Journal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
DistributedSensor Networks
International Journal of
Shock and Vibration 23
int
119871
0
sin 120582119898119909120577119897
119886(119909) 119889119909
=
0 119897 = 1 119898 = 0
161198712
119898(minus1)119898
(9 minus 401198982+ 16119898
4) 1205872
119897 = 1 119898 = 0
0 119897 = 2 119898 = 0
161198981198712
(9 minus 401198982+ 16119898
4) 1205872
119897 = 2 119898 = 0
int
119871
0
cos 1205821198981199091205771198971015840
119886(119909) 119889119909
=
0 119897 = 1 119898 = 0
161198711198982
(minus1)119898
(9 minus 401198982+ 16119898
4) 120587
119897 = 1 119898 = 0
0 119897 = 2 119898 = 0
161198711198982
(9 minus 401198982+ 16119898
4) 120587
119897 = 2 119898 = 0
int
119871
0
sin 1205821198981199091205771198971015840
119886(119909) 119889119909
=
0 119897 = 1 119898 = 0
4119871119898 (minus3 + 41198982
)
(9 minus 401198982+ 16119898
4) 120587
119897 = 1 119898 = 0
0 119897 = 2 119898 = 0
119871119898(
1
120587 minus 41198982120587
+
3
9120587 minus 41198982120587
) (minus1)119898
119897 = 2 119898 = 0
int
119871
0
1205771198971
119886(119909) 1205771198972
119886(119909) 119889119909 =
1198713
41205872 1198971= 1 1198972= 1
minus
21198713
31205873 1198971= 1 1198972= 2
minus
21198713
31205873 1198971= 2 1198972= 1
1198713
41205872 1198971= 2 1198972= 2
int
119871
0
1205771198971
119886(119909) 1205771198971015840
2
119886(119909) 119889119909 =
0 1198971= 1 1198972= 1
minus
1198712
8120587
1198971= 1 1198972= 2
1198712
8120587
1198971= 2 1198972= 1
0 1198971= 2 1198972= 2
int
119871
0
1205771198971015840
1
119886(119909) 1205771198971015840
2
119886(119909) 119889119909 =
5119871
16
1198971= 1 1198972= 1
minus
119871
2120587
1198971= 1 1198972= 2
minus
119871
2120587
1198971= 2 1198972= 1
5119871
16
1198971= 2 1198972= 2
int
119871
0
119909119899 cos 120582
119898119909 119889119909
=
1198711+119899
1 + 119899
119898 = 0
(1198861+119899HypergeometricPFQtimes [ 12 + 1198992 12 32 + 1198992
minus (14)1198982
1205872
]) times (1 + 119899)minus1
119898 gt 0
(B1)
Conflict of Interests
The authors declare that there is no conflict of interestsregarding the publication of this paper
Acknowledgments
The authors would like to thank the anonymous reviewersfor their very valuable comments The authors also gratefullyacknowledge the financial support from the National NaturalScience Foundation of China (no 51209052) HeilongjiangProvince Youth Science Fund Project (no QC2011C013) andHarbin Science and Technology Development InnovationFoundation of Youth (no 2011RFQXG021)
References
[1] K Liew Y Xiang and S Kitipornchai ldquoResearch on thick platevibration a literature surveyrdquo Journal of Sound and Vibrationvol 180 no 1 pp 163ndash176 1995
[2] J H Chung T Y Chung and K C Kim ldquoVibration analysisof orthotropic mindlin plates with edges elastically restrainedagainst rotationrdquo Journal of Sound and Vibration vol 163 no 1pp 151ndash163 1993
[3] Y K Cheung and D Zhou ldquoVibrations of moderately thickrectangular plates in terms of a set of static Timoshenko beamfunctionsrdquoComputers and Structures vol 78 no 6 pp 757ndash7682000
[4] CMWang ldquoNatural frequencies formula for simply supportedMindlin platesrdquo Journal of Vibration and Acoustics vol 116 no4 pp 536ndash540 1994
[5] K N Saha R C Kar and P K Datta ldquoFree vibration analysisof rectangular Mindlin plates with elastic restraints uniformlydistributed along the edgesrdquo Journal of Sound andVibration vol192 no 4 pp 885ndash902 1996
24 Shock and Vibration
[6] D J Gorman ldquoAccurate free vibration analysis of shear-deformable plates with torsional elastic edge supportrdquo Journalof Sound and Vibration vol 203 no 2 pp 209ndash218 1997
[7] D J Gorman ldquoFree vibration analysis of Mindlin plates withuniform elastic edge support by the superposition methodrdquoJournal of Sound andVibration vol 207 no 3 pp 335ndash350 1997
[8] D J Gorman ldquoAccurate free vibration analysis of point sup-ported mindlin plates by the superposition methodrdquo Journal ofSound and Vibration vol 219 no 2 pp 265ndash277 1999
[9] Y Xiang ldquoVibration of rectangular Mindlin plates resting onnon-homogenous elastic foundationsrdquo International Journal ofMechanical Sciences vol 45 no 6-7 pp 1229ndash1244 2003
[10] Y Xiang and G W Wei ldquoExact solutions for buckling andvibration of stepped rectangular Mindlin platesrdquo InternationalJournal of Solids and Structures vol 41 no 1 pp 279ndash294 2004
[11] Y L Yeh M J Jang and C C Wang ldquoAnalyzing the freevibrations of a plate using finite difference and differential trans-formationmethodrdquoAppliedMathematics andComputation vol178 no 2 pp 493ndash501 2006
[12] Y Xiang S K Lai L Zhou and C W Lim ldquoDSC-Ritz elementmethod for vibration analysis of rectangular Mindlin plateswith mixed edge supportsrdquo European Journal of MechanicsmdashASolids vol 29 no 4 pp 619ndash628 2010
[13] H Nguyen-Xuan C H Thai and T Nguyen-Thoi ldquoIsogeo-metric finite element analysis of composite sandwich platesusing a higher order shear deformation theoryrdquo Composites BEngineering vol 55 pp 558ndash574 2013
[14] C H Thai A J M Ferreira E Carrera and H Nguyen-XuanldquoIsogeometric analysis of laminated composite and sandwichplates using a layerwise deformation theoryrdquo Composite Struc-tures vol 104 pp 196ndash214 2013
[15] C H Thai A Ferreira S Bordas T Rabczuk and HNguyen-Xuan ldquoIsogeometric analysis of laminated compositeand sandwich plates using a new inverse trigonometric sheardeformation theoryrdquo European Journal of Mechanics A Solidsvol 43 pp 89ndash108 2014
[16] H Luong-Van T Nguyen-Thoi G R Liu and P Phung-VanldquoA cell-based smoothed finite elementmethod using three-nodeshear-locking free Mindlin plate element (CS-FEM-MIN3) fordynamic response of laminated composite plates on viscoelasticfoundationrdquo Engineering Analysis with Boundary Elements vol42 pp 8ndash19 2014
[17] T Nguyen-Thoi T Rabczuk T Lam-Phat V Ho-Huu andP Phung-Van ldquoFree vibration analysis of cracked Mindlinplate using an extended cell-based smoothed discrete sheargap method (XCS-DSG3)rdquo Theoretical and Applied FractureMechanics 2014
[18] H Nguyen-Xuan G R Liu and C a Thai-Hoang ldquoAnedge-based smoothed finite element method (ES-FEM) withstabilized discrete shear gap technique for analysis of Reissner-Mindlin platesrdquo Computer Methods in Applied Mechanics andEngineering vol 199 no 9ndash12 pp 471ndash489 2010
[19] P Phung-Van T Nguyen-Thoi T Bui-Xuan and Q Lieu-XuanldquoA cell-based smoothed three-node Mindlin plate element(CS-FEM-MIN3) based on the 119862
0 -type higher-order sheardeformation for geometrically nonlinear analysis of laminatedcomposite platesrdquo Computational Materials Science 2014
[20] NNguyen-Thanh T RabczukHNguyen-Xuan and S BordasldquoAn alternative alpha finite element method with discrete sheargap technique for analysis of isotropicMindlin-Reissner platesrdquoFinite Elements inAnalysis andDesign vol 47 no 5 pp 519ndash5352011
[21] KM Liew Y Xiang and S Kitipornchai ldquoTransverse vibrationof thick rectangular plates-I Comprehensive sets of boundaryconditionsrdquo Computers and Structures vol 49 no 1 pp 1ndash291993
[22] D Zhou ldquoVibrations of Mindlin rectangular plates with elasti-cally restrained edges using static Timoshenko beam functionswith the Rayleigh-Ritz methodrdquo International Journal of Solidsand Structures vol 38 no 32-33 pp 5565ndash5580 2001
[23] D Zhou SH Lo F T KAu andYK Cheung ldquoVibration anal-ysis of rectangular Mindlin plates with internal line supportsusing static Timoshenko beam functionsrdquo International Journalof Mechanical Sciences vol 44 no 12 pp 2503ndash2522 2002
[24] H S Shen J Yang and L Zhang ldquoFree and forced vibrationof Reissner-Mindlin plates with free edges resting on elasticfoundationsrdquo Journal of Sound and Vibration vol 244 no 2 pp299ndash320 2001
[25] Y Xing and B Liu ldquoClosed form solutions for free vibrations ofrectangular Mindlin platesrdquo Acta Mechanica Sinica vol 25 no5 pp 689ndash698 2009
[26] P Gagnon C Gosselin and L Cloutier ldquoA finite strip elementfor the analysis of variable thickness rectangular thick platesrdquoComputers amp Structures vol 63 no 2 pp 349ndash362 1997
[27] T Mlzusawa ldquoVibration of rectangular mindlin plates withtapered thickness by the spline strip methodrdquo Computers andStructures vol 46 no 3 pp 451ndash463 1993
[28] H Nguyen-Xuan and T Nguyen-Thoi ldquoA stabilized smoothedfinite element method for free vibration analysis of Mindlin-Reissner platesrdquo Communications in Numerical Methods inEngineering with Biomedical Applications vol 25 no 8 pp 882ndash906 2009
[29] K M Liew L X Peng and S Kitipornchai ldquoVibration anal-ysis of corrugated Reissner-Mindlin plates using a mesh-freeGalerkin methodrdquo International Journal of Mechanical Sciencesvol 51 no 9-10 pp 642ndash652 2009
[30] Y Hou GWWei and Y Xiang ldquoDSC-Ritz method for the freevibration analysis of Mindlin platesrdquo International Journal forNumerical Methods in Engineering vol 62 no 2 pp 262ndash2882005
[31] F- Liu and K M Liew ldquoVibration analysis of discontinuousMindlin plates by differential quadrature element methodrdquoJournal of Vibration and Acoustics Transactions of the ASMEvol 121 no 2 pp 204ndash208 1999
[32] P Malekzadeh G Karami and M Farid ldquoA semi-analyticalDQEM for free vibration analysis of thick plates with twoopposite edges simply supportedrdquoComputerMethods inAppliedMechanics and Engineering vol 193 no 45ndash47 pp 4781ndash47962004
[33] P Malekzadeh and S A Shahpari ldquoFree vibration analysisof variable thickness thin and moderately thick plates withelastically restrained edges by DQMrdquo Thin-Walled Structuresvol 43 no 7 pp 1037ndash1050 2005
[34] R E Diaz-Contreras and S Nomura ldquoGreenrsquos function appliedto solution ofMindlin platesrdquoComputers and Structures vol 60no 1 pp 41ndash48 1996
[35] T Sakiyama and M Huang ldquoFree vibration analysis of rect-angular plates with variable thicknessrdquo Journal of Sound andVibration vol 216 no 3 pp 379ndash397 1998
[36] M Huang X Q Ma T Sakiyama H Matuda and C MoritaldquoFree vibration analysis of orthotropic rectangular plates withvariable thickness and general boundary conditionsrdquo Journal ofSound and Vibration vol 288 no 4-5 pp 931ndash955 2005
Shock and Vibration 25
[37] I Shufrin and M Eisenberger ldquoVibration of shear deformableplates with variable thicknessmdashfirst-order and higher-orderanalysesrdquo Journal of Sound and Vibration vol 290 no 1-2 pp465ndash489 2006
[38] S A Eftekhari and A A Jafari ldquoAccurate variational approachfor free vibration of variable thickness thin and thick plates withedges elastically restrained against translation and rotationrdquoInternational Journal of Mechanical Sciences vol 68 pp 35ndash462013
[39] W L Li X Zhang J Du and Z Liu ldquoAn exact series solutionfor the transverse vibration of rectangular plates with generalelastic boundary supportsrdquo Journal of Sound amp Vibration vol321 no 1-2 pp 254ndash269 2009
[40] X Shi D Shi W L Li and Q Wang ldquoA unified method forfree vibration analysis of circular annular and sector plateswith arbitrary boundary conditionsrdquo Journal of Vibration andControl 2014
[41] D Y Shi X J Shi W L Li and Q S Wang ldquoFree transversevibrations of orthotropic thin rectangular plates with arbitraryelastic edge supportsrdquo Journal of Vibroengineering vol 16 no 1pp 389ndash398 2014
[42] Y Chen G Jin and Z Liu ldquoFree vibration analysis of circu-lar cylindrical shell with non-uniform elastic boundary con-straintsrdquo International Journal of Mechanical Sciences vol 74pp 120ndash132 2013
[43] L Dai T Yang J Du W L Li and M J Brennan ldquoAn exactseries solution for the vibration analysis of cylindrical shellswith arbitrary boundary conditionsrdquo Applied Acoustics vol 74no 3 pp 440ndash449 2013
International Journal of
AerospaceEngineeringHindawi Publishing Corporationhttpwwwhindawicom Volume 2014
RoboticsJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Active and Passive Electronic Components
Control Scienceand Engineering
Journal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
International Journal of
RotatingMachinery
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Hindawi Publishing Corporation httpwwwhindawicom
Journal ofEngineeringVolume 2014
Submit your manuscripts athttpwwwhindawicom
VLSI Design
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Shock and Vibration
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Civil EngineeringAdvances in
Acoustics and VibrationAdvances in
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Electrical and Computer Engineering
Journal of
Advances inOptoElectronics
Hindawi Publishing Corporation httpwwwhindawicom
Volume 2014
The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014
SensorsJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Modelling amp Simulation in EngineeringHindawi Publishing Corporation httpwwwhindawicom Volume 2014
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Chemical EngineeringInternational Journal of Antennas and
Propagation
International Journal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Navigation and Observation
International Journal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
DistributedSensor Networks
International Journal of
24 Shock and Vibration
[6] D J Gorman ldquoAccurate free vibration analysis of shear-deformable plates with torsional elastic edge supportrdquo Journalof Sound and Vibration vol 203 no 2 pp 209ndash218 1997
[7] D J Gorman ldquoFree vibration analysis of Mindlin plates withuniform elastic edge support by the superposition methodrdquoJournal of Sound andVibration vol 207 no 3 pp 335ndash350 1997
[8] D J Gorman ldquoAccurate free vibration analysis of point sup-ported mindlin plates by the superposition methodrdquo Journal ofSound and Vibration vol 219 no 2 pp 265ndash277 1999
[9] Y Xiang ldquoVibration of rectangular Mindlin plates resting onnon-homogenous elastic foundationsrdquo International Journal ofMechanical Sciences vol 45 no 6-7 pp 1229ndash1244 2003
[10] Y Xiang and G W Wei ldquoExact solutions for buckling andvibration of stepped rectangular Mindlin platesrdquo InternationalJournal of Solids and Structures vol 41 no 1 pp 279ndash294 2004
[11] Y L Yeh M J Jang and C C Wang ldquoAnalyzing the freevibrations of a plate using finite difference and differential trans-formationmethodrdquoAppliedMathematics andComputation vol178 no 2 pp 493ndash501 2006
[12] Y Xiang S K Lai L Zhou and C W Lim ldquoDSC-Ritz elementmethod for vibration analysis of rectangular Mindlin plateswith mixed edge supportsrdquo European Journal of MechanicsmdashASolids vol 29 no 4 pp 619ndash628 2010
[13] H Nguyen-Xuan C H Thai and T Nguyen-Thoi ldquoIsogeo-metric finite element analysis of composite sandwich platesusing a higher order shear deformation theoryrdquo Composites BEngineering vol 55 pp 558ndash574 2013
[14] C H Thai A J M Ferreira E Carrera and H Nguyen-XuanldquoIsogeometric analysis of laminated composite and sandwichplates using a layerwise deformation theoryrdquo Composite Struc-tures vol 104 pp 196ndash214 2013
[15] C H Thai A Ferreira S Bordas T Rabczuk and HNguyen-Xuan ldquoIsogeometric analysis of laminated compositeand sandwich plates using a new inverse trigonometric sheardeformation theoryrdquo European Journal of Mechanics A Solidsvol 43 pp 89ndash108 2014
[16] H Luong-Van T Nguyen-Thoi G R Liu and P Phung-VanldquoA cell-based smoothed finite elementmethod using three-nodeshear-locking free Mindlin plate element (CS-FEM-MIN3) fordynamic response of laminated composite plates on viscoelasticfoundationrdquo Engineering Analysis with Boundary Elements vol42 pp 8ndash19 2014
[17] T Nguyen-Thoi T Rabczuk T Lam-Phat V Ho-Huu andP Phung-Van ldquoFree vibration analysis of cracked Mindlinplate using an extended cell-based smoothed discrete sheargap method (XCS-DSG3)rdquo Theoretical and Applied FractureMechanics 2014
[18] H Nguyen-Xuan G R Liu and C a Thai-Hoang ldquoAnedge-based smoothed finite element method (ES-FEM) withstabilized discrete shear gap technique for analysis of Reissner-Mindlin platesrdquo Computer Methods in Applied Mechanics andEngineering vol 199 no 9ndash12 pp 471ndash489 2010
[19] P Phung-Van T Nguyen-Thoi T Bui-Xuan and Q Lieu-XuanldquoA cell-based smoothed three-node Mindlin plate element(CS-FEM-MIN3) based on the 119862
0 -type higher-order sheardeformation for geometrically nonlinear analysis of laminatedcomposite platesrdquo Computational Materials Science 2014
[20] NNguyen-Thanh T RabczukHNguyen-Xuan and S BordasldquoAn alternative alpha finite element method with discrete sheargap technique for analysis of isotropicMindlin-Reissner platesrdquoFinite Elements inAnalysis andDesign vol 47 no 5 pp 519ndash5352011
[21] KM Liew Y Xiang and S Kitipornchai ldquoTransverse vibrationof thick rectangular plates-I Comprehensive sets of boundaryconditionsrdquo Computers and Structures vol 49 no 1 pp 1ndash291993
[22] D Zhou ldquoVibrations of Mindlin rectangular plates with elasti-cally restrained edges using static Timoshenko beam functionswith the Rayleigh-Ritz methodrdquo International Journal of Solidsand Structures vol 38 no 32-33 pp 5565ndash5580 2001
[23] D Zhou SH Lo F T KAu andYK Cheung ldquoVibration anal-ysis of rectangular Mindlin plates with internal line supportsusing static Timoshenko beam functionsrdquo International Journalof Mechanical Sciences vol 44 no 12 pp 2503ndash2522 2002
[24] H S Shen J Yang and L Zhang ldquoFree and forced vibrationof Reissner-Mindlin plates with free edges resting on elasticfoundationsrdquo Journal of Sound and Vibration vol 244 no 2 pp299ndash320 2001
[25] Y Xing and B Liu ldquoClosed form solutions for free vibrations ofrectangular Mindlin platesrdquo Acta Mechanica Sinica vol 25 no5 pp 689ndash698 2009
[26] P Gagnon C Gosselin and L Cloutier ldquoA finite strip elementfor the analysis of variable thickness rectangular thick platesrdquoComputers amp Structures vol 63 no 2 pp 349ndash362 1997
[27] T Mlzusawa ldquoVibration of rectangular mindlin plates withtapered thickness by the spline strip methodrdquo Computers andStructures vol 46 no 3 pp 451ndash463 1993
[28] H Nguyen-Xuan and T Nguyen-Thoi ldquoA stabilized smoothedfinite element method for free vibration analysis of Mindlin-Reissner platesrdquo Communications in Numerical Methods inEngineering with Biomedical Applications vol 25 no 8 pp 882ndash906 2009
[29] K M Liew L X Peng and S Kitipornchai ldquoVibration anal-ysis of corrugated Reissner-Mindlin plates using a mesh-freeGalerkin methodrdquo International Journal of Mechanical Sciencesvol 51 no 9-10 pp 642ndash652 2009
[30] Y Hou GWWei and Y Xiang ldquoDSC-Ritz method for the freevibration analysis of Mindlin platesrdquo International Journal forNumerical Methods in Engineering vol 62 no 2 pp 262ndash2882005
[31] F- Liu and K M Liew ldquoVibration analysis of discontinuousMindlin plates by differential quadrature element methodrdquoJournal of Vibration and Acoustics Transactions of the ASMEvol 121 no 2 pp 204ndash208 1999
[32] P Malekzadeh G Karami and M Farid ldquoA semi-analyticalDQEM for free vibration analysis of thick plates with twoopposite edges simply supportedrdquoComputerMethods inAppliedMechanics and Engineering vol 193 no 45ndash47 pp 4781ndash47962004
[33] P Malekzadeh and S A Shahpari ldquoFree vibration analysisof variable thickness thin and moderately thick plates withelastically restrained edges by DQMrdquo Thin-Walled Structuresvol 43 no 7 pp 1037ndash1050 2005
[34] R E Diaz-Contreras and S Nomura ldquoGreenrsquos function appliedto solution ofMindlin platesrdquoComputers and Structures vol 60no 1 pp 41ndash48 1996
[35] T Sakiyama and M Huang ldquoFree vibration analysis of rect-angular plates with variable thicknessrdquo Journal of Sound andVibration vol 216 no 3 pp 379ndash397 1998
[36] M Huang X Q Ma T Sakiyama H Matuda and C MoritaldquoFree vibration analysis of orthotropic rectangular plates withvariable thickness and general boundary conditionsrdquo Journal ofSound and Vibration vol 288 no 4-5 pp 931ndash955 2005
Shock and Vibration 25
[37] I Shufrin and M Eisenberger ldquoVibration of shear deformableplates with variable thicknessmdashfirst-order and higher-orderanalysesrdquo Journal of Sound and Vibration vol 290 no 1-2 pp465ndash489 2006
[38] S A Eftekhari and A A Jafari ldquoAccurate variational approachfor free vibration of variable thickness thin and thick plates withedges elastically restrained against translation and rotationrdquoInternational Journal of Mechanical Sciences vol 68 pp 35ndash462013
[39] W L Li X Zhang J Du and Z Liu ldquoAn exact series solutionfor the transverse vibration of rectangular plates with generalelastic boundary supportsrdquo Journal of Sound amp Vibration vol321 no 1-2 pp 254ndash269 2009
[40] X Shi D Shi W L Li and Q Wang ldquoA unified method forfree vibration analysis of circular annular and sector plateswith arbitrary boundary conditionsrdquo Journal of Vibration andControl 2014
[41] D Y Shi X J Shi W L Li and Q S Wang ldquoFree transversevibrations of orthotropic thin rectangular plates with arbitraryelastic edge supportsrdquo Journal of Vibroengineering vol 16 no 1pp 389ndash398 2014
[42] Y Chen G Jin and Z Liu ldquoFree vibration analysis of circu-lar cylindrical shell with non-uniform elastic boundary con-straintsrdquo International Journal of Mechanical Sciences vol 74pp 120ndash132 2013
[43] L Dai T Yang J Du W L Li and M J Brennan ldquoAn exactseries solution for the vibration analysis of cylindrical shellswith arbitrary boundary conditionsrdquo Applied Acoustics vol 74no 3 pp 440ndash449 2013
International Journal of
AerospaceEngineeringHindawi Publishing Corporationhttpwwwhindawicom Volume 2014
RoboticsJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Active and Passive Electronic Components
Control Scienceand Engineering
Journal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
International Journal of
RotatingMachinery
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Hindawi Publishing Corporation httpwwwhindawicom
Journal ofEngineeringVolume 2014
Submit your manuscripts athttpwwwhindawicom
VLSI Design
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Shock and Vibration
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Civil EngineeringAdvances in
Acoustics and VibrationAdvances in
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Electrical and Computer Engineering
Journal of
Advances inOptoElectronics
Hindawi Publishing Corporation httpwwwhindawicom
Volume 2014
The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014
SensorsJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Modelling amp Simulation in EngineeringHindawi Publishing Corporation httpwwwhindawicom Volume 2014
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Chemical EngineeringInternational Journal of Antennas and
Propagation
International Journal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Navigation and Observation
International Journal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
DistributedSensor Networks
International Journal of
Shock and Vibration 25
[37] I Shufrin and M Eisenberger ldquoVibration of shear deformableplates with variable thicknessmdashfirst-order and higher-orderanalysesrdquo Journal of Sound and Vibration vol 290 no 1-2 pp465ndash489 2006
[38] S A Eftekhari and A A Jafari ldquoAccurate variational approachfor free vibration of variable thickness thin and thick plates withedges elastically restrained against translation and rotationrdquoInternational Journal of Mechanical Sciences vol 68 pp 35ndash462013
[39] W L Li X Zhang J Du and Z Liu ldquoAn exact series solutionfor the transverse vibration of rectangular plates with generalelastic boundary supportsrdquo Journal of Sound amp Vibration vol321 no 1-2 pp 254ndash269 2009
[40] X Shi D Shi W L Li and Q Wang ldquoA unified method forfree vibration analysis of circular annular and sector plateswith arbitrary boundary conditionsrdquo Journal of Vibration andControl 2014
[41] D Y Shi X J Shi W L Li and Q S Wang ldquoFree transversevibrations of orthotropic thin rectangular plates with arbitraryelastic edge supportsrdquo Journal of Vibroengineering vol 16 no 1pp 389ndash398 2014
[42] Y Chen G Jin and Z Liu ldquoFree vibration analysis of circu-lar cylindrical shell with non-uniform elastic boundary con-straintsrdquo International Journal of Mechanical Sciences vol 74pp 120ndash132 2013
[43] L Dai T Yang J Du W L Li and M J Brennan ldquoAn exactseries solution for the vibration analysis of cylindrical shellswith arbitrary boundary conditionsrdquo Applied Acoustics vol 74no 3 pp 440ndash449 2013
International Journal of
AerospaceEngineeringHindawi Publishing Corporationhttpwwwhindawicom Volume 2014
RoboticsJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Active and Passive Electronic Components
Control Scienceand Engineering
Journal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
International Journal of
RotatingMachinery
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Hindawi Publishing Corporation httpwwwhindawicom
Journal ofEngineeringVolume 2014
Submit your manuscripts athttpwwwhindawicom
VLSI Design
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Shock and Vibration
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Civil EngineeringAdvances in
Acoustics and VibrationAdvances in
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Electrical and Computer Engineering
Journal of
Advances inOptoElectronics
Hindawi Publishing Corporation httpwwwhindawicom
Volume 2014
The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014
SensorsJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Modelling amp Simulation in EngineeringHindawi Publishing Corporation httpwwwhindawicom Volume 2014
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Chemical EngineeringInternational Journal of Antennas and
Propagation
International Journal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Navigation and Observation
International Journal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
DistributedSensor Networks
International Journal of
International Journal of
AerospaceEngineeringHindawi Publishing Corporationhttpwwwhindawicom Volume 2014
RoboticsJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Active and Passive Electronic Components
Control Scienceand Engineering
Journal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
International Journal of
RotatingMachinery
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Hindawi Publishing Corporation httpwwwhindawicom
Journal ofEngineeringVolume 2014
Submit your manuscripts athttpwwwhindawicom
VLSI Design
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Shock and Vibration
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Civil EngineeringAdvances in
Acoustics and VibrationAdvances in
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Electrical and Computer Engineering
Journal of
Advances inOptoElectronics
Hindawi Publishing Corporation httpwwwhindawicom
Volume 2014
The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014
SensorsJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Modelling amp Simulation in EngineeringHindawi Publishing Corporation httpwwwhindawicom Volume 2014
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Chemical EngineeringInternational Journal of Antennas and
Propagation
International Journal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Navigation and Observation
International Journal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
DistributedSensor Networks
International Journal of
![Research Article Free Vibration Analysis of …downloads.hindawi.com/journals/sv/2014/572395.pdfproblems of moderately thick plates with four free edges. Xing and Liu [ ] presented](https://static.fdocuments.net/doc/165x107/5f7329d97c810e1e240870b5/research-article-free-vibration-analysis-of-problems-of-moderately-thick-plates.jpg)
