Research ArticleDynamic Response of a Thick Piezoelectric Circular CylindricalPanel An Exact Solution

Atta Oveisi Mohammad Gudarzi and Seyyed Mohammad Hasheminejad

Department of Mechanical Engineering Iran University of Science and Technology Narmak Tehran 1684613114 Iran

Correspondence should be addressed to Mohammad Gudarzi gudarziiustacir

Received 26 September 2012 Accepted 19 November 2012 Published 27 May 2014

Academic Editor Hamid Ahmadian

Copyright copy 2014 Atta Oveisi 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

One of the interesting fields that attracted many researchers in recent years is the smart structures The piezomaterials because oftheir ability in converting both mechanical stress and electricity to each other are very applicable in this field However most of theworks available used various inexact two-dimensional theories with certain types of simplification which are inaccurate in someapplications such as thick shells while in some applications due to request of large displacementstress thick piezoelectric panelis needed and two-dimensional theories have not enough accuracy This study investigates the dynamic steady state response andnatural frequency of a piezoelectric circular cylindrical panel using exact three-dimensional solutions based on this decompositiontechnique In addition the formulation is written for both simply supported and clamped boundary conditions Then the naturalfrequencies mode shapes and dynamic steady state response of the piezoelectric circular cylindrical panel in frequency domainare validated with commercial finite element software (ABAQUS) to show the validity of the mathematical formulation and theresults will be compared finally

1 Introduction

Piezoelectric materials have been extensively used as trans-ducers and sensors due to their intrinsic direct and conversepiezoelectric effects that take place between electric fieldand mechanical deformation An important geometry inapplied engineering problems is circular cylindrical panelbecause of its widespread application in actual structuressuch as aircraft wings submarines missiles vessels and highpressure cylindrical containers The application of piezo-material structures in this field is mainly concentrated onvibration suppression and acoustic noise reduction Becauseof practical applications piezoelectric circular cylindricalshells have attracted a considerable amount of researchinterests Haskins and Walsh analyzed the free vibrationof piezoelectric cylindrical shells with radially polarizedtransverse isotropy [1] Martin investigated the vibration oflongitudinally polarized piezoelectric cylindrical tubes andpointed out the limitations of the assumption [2] Drumhellerand Kalnins presented a coupled theory for the vibrationof piezoceramic shells of revolution and analyzed the freeaxisymmetrical vibration of a circular cylindrical shell [3]

Burt simplified the circular cylinder to a two-dimensionalmodel and then investigated the voltage response of radiallypolarized ceramic [4] Tzou and Zhong gave a linear theoryof piezoelectric shell vibration which can be simplified toaccount for spheres [5] Ebenezer and Abraham presented anEigen function approach to determine the response of radi-ally polarized piezoelectric cylindrical shells of finite lengthsubjected to electrical excitation [6] Many other researchesby the methods of three-dimensional theory concentratedon the axisymmetrical and radial vibrations of cylinderssuch as Stephenson [7 8] and Adelman et al [9 10] Paulderived the frequency equation of a piezoelectric cylindricalshell without presenting numerical results [11] Paul andVenkatesan employed the same method to obtain the naturalfrequencies of infinite piezoelectric cylindrical shells [12]However some frequencies were missed in their calculationRecently Ding et al exactly investigated the free vibrationof hollow piezoelectric cylindrical shells on the basis of adecomposition formula for displacements exactly [13] Yanget al considered the theory of the basic vibration characteris-tics of a circular cylindrical shell piezoelectric transducer [14]They solved the vibration problemnumerically for electrically

Hindawi Publishing CorporationShock and VibrationVolume 2014 Article ID 592165 8 pageshttpdxdoiorg1011552014592165

2 Shock and Vibration

L

z

r

r0

r1

1205790

120579

Figure 1 Cylindrical panel and its geometry

forced case Li et al considered the spillover and harmoniceffect in real active vibration control and they presented anovel composite controller based on disturbance observer(DOB) for the all-clamped panel [15] Kumar and Singhaimed to examine through experiments vibration controlof curved panel treated with optimally placed active orpassive constrained layer damping patches and they foundthe optimum location for the application of ACLDPCLDpatches [16]

The main subject of this study is to investigate the freeand forced vibration of transversely isotropic piezoelectriccylindrical panels Based on the general solution for coupledequations for piezoelectricmedia presented inDing et al [17]three-dimensional exact solutions are obtained through thevariable separation method A numerical example is finallypresented

2 Problem Formulation

For dynamic modelling of piezoelectric layers two displace-ment functions Ψ and 119865 are considered [17] Figure 1 showsthe panel geometry

21 Basic Equations In circular cylindrical coordinates(119903 120579 119911) if the media is axially polarized the general solutioncan be written as

119906119875119894

119903=1

119903

120597120595

120597120579minus

120597

120597119903A1119865

119906119875119894

120579= minus

120597Ψ

120597119903minus1

119903

120597

120597120579A1119865

119908119875119894= A2119865 120601 = A

3119865

(1)

where 119906119875119894119903 119906119875119894120579 and 119908

119875119894 are three displacement components120601 is the electric potential and the differential operators A

1

A2 andA

3are

A1= [(119888

119875119894

13+ 119888119875119894

44) 12057633+ (11989015+ 11989031) 11989033]1205973

1205971199113

+ [(119888119875119894

13+ 119888119875119894

44) 12057611+ (11989015+ 11989031) 11989015] Λ

120597

120597119911

A2= 119888119875119894

4412057633

1205974

1205971199114

+ [119888119875119894

1112057633+ 119888119875119894

4412057611+ (11989015+ 11989031)2Λ minus 120588120576

33

1205972

1205971199052]

1205972

1205971199112

+ 119888119875119894

1112057611ΛΛ minus 120588120576

11Λ1205972

1205971199052

A3= 119888119875119894

4411989033

1205974

1205971199114

+ [119888119875119894

1111989033+ 119888119875119894

4411989015minus (119888119875119894

13+ 119888119875119894

44) (11989015+ 11989031)] Λ

minus12058811989033

1205972

1205971199052

1205972

1205971199112

+ 119888119875119894

1111989015ΛΛ minus 120588119890

15Λ1205972

1205971199052

(2)

where Λ = 12059721205971199032+ (1119903)120597120597119903 + (1119903

2)12059721205971205792 is the two-

dimensional Laplacian The displacement functions Ψ and 119865must satisfy the following two equations

(119888120588

66Λ + 119888119875119894

44

1205972

1205971199112minus 120588119875119894 1205972

1205971199052)120595 = 0 119871

0119865 = 0 (3)

1198710= 1198864ΛΛΛ + (119886

3

1205972

1205971199112+ 1198866

1205972

1205971199052)ΛΛ

+ (1198862

1205974

1205971199114+ 1198865

1205974

1205971199054+ 1198867

1205974

12059711991121205971199052)Λ + 119886

1

1205976

1205971199116

+ 1198868

1205976

12059711991141205971199052+ 1198869

1205976

12059711991121205971199054

(4)

Shock and Vibration 3

Here 119886119899(119899 = 1 2 9) can be expressed in terms of

elastic constants 119888119875119894119894119895 dielectric constants 120576

119894119895 and piezoelectric

coefficients 119890119894119895as follows

1198861= 119888119875119894

44(1198902

33+ 119888119875119894

3312057633)

1198864= 119888119875119894

11(1198902

15+ 119888119875119894

4412057611) 119886

5= 120588212057611

1198862= 119888119875119894

33[119888119875119894

4412057611+ (11989015+ 11989031)2]

+ 12057633[119888119875119894

11119888119875119894

33+ 1198882

44minus (119888119875119894

11+ 119888119875119894

44)2

]

+ 11989033[2119888119875119894

4411989015+ 119888119875119894

1111989033minus 2 (119888

119875119894

13+ 119888119875119894

44) (11989015+ 11989031)]

1198863= 119888119875119894

44[119888119875119894

1112057633+ (11989015+ 11989031)2]

+ 12057611[119888119875119894

1111988833+ 1198882

44minus (119888119875119894

13+ 119888119875119894

44)2

]

+ 11989015[2119888119875119894

1111989033+ 119888119875119894

4411989015minus 2 (119888

119875119894

13+ 119888119875119894

44) (11989015+ 11989031)]

1198866= minus120588 [119890

2

15+ (119888119875119894

11+ 119888119875119894

44) 12057611]

1198868= minus120588 [119890

2

33+ (119888119875119894

44+ 119888119875119894

33) 12057633] 119886

9= 120588212057633

1198867= minus 120588 [2119890

1511989033+ (119888119875119894

44+ 119888119875119894

33) 12057611+ (119888119875119868

11+ 119888119875119894

44) 12057633

+(11989015+ 11989031)2]

(5)

The circular cylindrical coordinates as well as a circularcylindrical panel with outer radius 119887

119875119894 inner radius 119886119875119894

circular center angle 120572119875119894 and length 119871119875119894 are shown in Figure 1If the panel is vibrating with a resonant frequency 120596 thedisplacement functions can be assumed as

119865 =1199035

0

1198881112057633

119875 (120585119875)Θ (120583120579)119885 (120573120577

119875) 119890119894120596119905

120595 = 1199032

01198754(120585119875)Θ1015840(120583120579)119885

1015840(120573120577119875) 119890119894120596119905

(6)

where 120585119875119894

= 119903119875119894119877119875119894 120577119875119894 = 119911

119875119894119871119875119894 are the dimensionless

coordinates in 119903 and 119911 directions and Θ1015840(120583120579119875119894) and 119885

1015840(120573120577119875119894)

denote the derivation of Θ(120583120579119875119894) with respect to 120583120579119875119894 and

the derivation of 119885(120573120577119875119894) with respect to 120573120577119875119894 respectively

In addition

Θ(120583120579) = 1198621cos (120583120579) + 119862

2sin (120583120579)

119885 (120573120577) = 1198623sin (120573120577) + 119888

4cos (120573120577)

(7)

where 119862119898(119898 = 1 2 3 4) are constants Substitution of (6)

into (3) yields

(Δ + 1198962

4) 1198754 (120585) = 0 (8)

(Δ + 1198962

1) (Δ + 119896

2

2) (Δ + 119896

2

3) 1198754 (120585) = 0 (9)

where Δ = 1205972120597(120585119875119894)2

+ (1120585119875119894)120597120597(120585

119875119894) minus 1205832(120585119875119894)2 and

1198962

4=Ω2119888119901119894

11

119888119901119894

66

minus1205742119888119901119894

44

119888119901119894

66

(Ω119875119894)2

=1205881198751198941205962

1199032

0

11988811

120574 = 1205731199051 1199051=

1199030

ℎ0

(10)

and (119896119875119894

119898)2 (119898 = 1 2 3) (assuming Re[119896119875119894

119898] ge 0) are the

eigenvalues of the following equation

11988641198966+ (1198866(Ω119875119894)2

+ 11988631205742) 1198964

+ (11988621205744+ 11988671205742(Ω119875119894)2

+ 1198865(Ω119875119894)4

) 1198962

+ (11988611205746+ 11988681205744(Ω119875119894)2

+ 1205742(Ω119875119894)4

) = 0

(11)

in which

119886119899=

119886119899

((119888119875119894

11)212057633)

(119899 = 1 2 3 4)

119886119899=

119886119899

(120588119888119875119894

1112057633) (119899 = 6 7 8)

1198865=

1198865

((120588119875119894)212057633)

(12)

The solution of (9) can be assumed as

119875 (120585) = 1198751 (120585) + 119875

2 (120585) + 1198753 (120585) (13)

where 119875119898(120585119875119894) is obtained as [18]

Substituting (6) into (1) gives the mechanical displace-ments and electric potential as follows

119906119901119894

119903= minus1199030[120583

1205851198754 (120585) +

3

sum

119898=1

12057211198981198751015840

119898(120585)]Θ (120583120579)119885

1015840(120573120577) 119890

119894120596119905

(14)

119906119901119894

120579= minus1199030[1198751015840

4(120585) +

120583

120585

3

sum

119898=1

1205721119898119875119898 (120585)]Θ

1015840(120583120579)119885

1015840(120573120577) 119890

119894120596119905

(15)

119908119901119894= 1199030[

3

sum

119898=1

1205722119898119875119898 (120585)]Θ (120583120579)119885 (120573120577) 119890

119894120596119905 (16)

Φ = 1199030radic119888119901119894

11

12057633

[

3

sum

119898=1

1205723119898119875119898 (120585)]Θ (120583120579)119885 (120573120577) 119890

119894120596119905 (17)

4 Shock and Vibration

where1205721119898

= minus ([(119888119875119894

13+ 11988844) (120576111198962

119898+ 120576331205742)

+ (11989015+ 11989031) (119890151198962

119898+ 119890331205742) ] 120574)

times (119888119875119894

1112057633)minus1

1205722119898

= ([(119888119875119894

111198962

119898+ 119888119875119894

441205742minus 119888119875119894

11Ω2

1) (120576111198962

119898+ 120576331205742)

+(11989015+ 11989031)21198962

1198981205742])

times (119888119875119894

1112057633)minus1

1205723119898

= ([ (119888119875119894

111198962

119898+ 119888119875119894

441205742minus 119888119875119894

11Ω2

1) (119890151198962

119898+ 119890331205742)

minus(119888119875119894

13+ 119888119875119894

44) (11989015+ 11989031)2

1198962

1198981205742])

times (119888119875119894

11radic119888119875119894

1112057633)

minus1

(119898 = 1 2 3)

(18)

Utilizing the constitutive relations of piezoelectricity and(14)ndash(17) the stress components and electric displacementcomponents can be derived as

120590119901119894

119903=

(119888119901119894

12minus 119888119901119894

11) [

120583

1205851198751015840

4(120585) minus

120583

12058521198754 (120585)]

+ (119888119901119894

12minus 119888119901119894

11)

3

sum

119898=1

120572111989811987510158401015840

119898(120585)

+

3

sum

119898=1

(1198881212057211198981198962

119898+ 119888119901119894

131205741205722119898

+ 11989031radic119888119901119894

11

12057633

1205741205723119898)

times119875119898 (120585)

Θ(120583120579)1198851015840(120573120577) 119890

119894120596119905

(19)

120590119901119894

120579=

(119888119901119894

11minus 119888119901119894

12) [

120583

1205851198751015840

4(120585) minus

120583

12058521198754 (120585)]

+ (119888119901119894

11minus 119888119901119894

12)

3

sum

119898=1

120572111989811987510158401015840

119898(120585)

+

3

sum

119898=1

(119888119901119894

1112057211198981198962

119898+ 119888119901119894

131205741205722119898

+ 11989031radic119888119901119894

11

12057633

1205741205723119898)

times119875119898 (120585)

Θ(120583120579)1198851015840(120573120577) 119890

119894120596119905

(20)

120590119901119894

119911=

3

sum

119898=1

[[

[

(119888119901119894

1312057211198981198962

119898+ 119888119901119894

331205741205722119898

+ 11989033radic119888119901119894

11

12057633

1205741205723119898)

times 119875119898 (120585)

]]

]

Θ (120583120579)1198851015840(120573120577) 119890

119894120596119905

(21)

120591119901119894

120579119911=

119888119901119894

441205741198751015840

4(120585)

+120583

120585

3

sum

119898=1

[[

[

(119888119901119894

441205741205721119898

+ 119888119901119894

441205722119898

+ 11989015radic119888119901119894

11

12057633

1205723119898)

times119875119898 (120585)

]]

]

Θ1015840(120583120579)119885 (120573120577) 119890

119894120596119905

(22)

120591119901119894

119903119911=

119888119901119894

44120574120583

120585119876 (120585)

+[[

[

3

sum

119898=1

(119888119901119894

441205741205721119898

+ 119888119901119894

441205722119898

+ 11989015radic119888119901119894

11

12057633

1205723119898)

times1199011015840

119872(120585)

]]

]

Θ(120583120579)119885 (120573120577) 119890119894120596119905

(23)

120591119901119894

119903120579= 11988866[ minus 1198962

41198754 (120585) minus 2119875

10158401015840

4(120585)

+2120583

1205852

3

sum

119898=1

1205721119898119875119898 (120585) minus

2120583

120585

3

sum

119898=1

12057211198981199011015840

119872(120585)]

times Θ1015840(120583120579)119885

1015840(120573120577) 119890

119894120596119905

(24)

119863119901119894

119903=

11989015120574120583

1205851198754 (120585)

+[[

[

3

sum

119898=1

(119890151205741205721119898

+ 119890151205722119898

+ 12057611radic119888119901119894

11

12057633

1205723119898)

times1199011015840

119872(120585)

]]

]

Θ(120583120579)119885 (120573120577) 119890119894120596119905

(25)

Shock and Vibration 5

119863119901119894

120579=

119890151205741198751015840

4(120585)

+120583

120585

[[

[

3

sum

119898=1

(119890151205741205721119898

+ 119890151205722119898

+ 12057611radic119888119901119894

11

12057633

1205723119898)

times119875119898 (120585)

]]

]

Θ1015840(120583120579)119885 (120573120577) 119890

119894120596119905

(26)

119863119901119894

119911= [

3

sum

119898=1

(1198903112057211198981198962

119898+ 119890331205741205722119898

minus radic119888119901119894

11120576331205741205723119898)119875119898 (120585)]

times Θ (120583120579)1198851015840(120573120577) 119890

119894120596119905

(27)

22 Boundary Conditions The piezoelectric panel has 8boundary conditions consist of 6 mechanical and 2 electricalones

By considering generalized simply support boundaryconditions at 120579119894 = 0 and 120579

119894= 120572 and (119894 = 119875119894) we will have

119908119894= 119906119894

119903= 0 120590

119894

120579= 0 (119894 = 119875119894) (28)

Note that for piezoelectric layers the following conditionis added

120601 = 0 (29)

One can take

119862119894

1= 0 119862

119894

2= 1 120583 =

(2119898 + 1) 120587

2120572 119898 = 0 1 2

(30)

And by considering generalized simply support boundaryconditions at 120577119894 = 0 and 120577

119894= 1 (119894 = 119875119894) we will have

119906119894

119903= 119906119894

120579= 0 120590

119894

119911= 0 (119894 = 119875119894) (31)

And for piezoelectric layers the following condition isadded

119863119911= 0 (32)

One can take

119862119894

3= 0 119862

119894

4= 1 120573 = 119899120587 119899 = 0 1 2 (33)

Without loss of generality we suppose that external forceacts on the outer surface of the actuator and inner surface ofsensor has free boundary condition So we have

120590119875119894

119903= 119875 120591

119875119894

119903120579= 120591119875119894

119903119911= 0 120601 = 119868 at 119903 = 119903

4

120590119875119894

119903= 120591119875119894

119903120579= 120591119875119894

119903119911= 0 120601 = 0 at 119903 = 119903

1

(34)

For obtaining steady state frequency response of thecylindrical panel under a harmonic external excitation wemust solve the following matrix equation

[119879]119898times119899119883119899times1 = 119865119898times1 (35)

where [119879]119898times119899

is the coefficient matrix Consider

119883119899times1 = [11986011198611119860211986121198603119861311986041198614] (36)

and 119860119894 119861119894 119894 = 1 2 3 4 are the unknown constants that are

in (19)ndash(27)The vector 119865

119898times1denotes the force vector that acts on

the structure This force consists of the surface force that isconsidered as disturbance and has the breed of mechanicalforce such as wind effect The effect of controller unit inthe dynamic response of the piezo-panel is considered as anexternal electrical potential applied on the upper surface ofthe panel These two external forces acted on the structureindependently however summation of their effects on thewhole structure is the same as the case that both of them acton the structure simultaneously So

119865119899times2 = [1198651 1198652]

1198651119899times1

= [0 0 0 0 0 0 119868 (1199030 120579 119911 120596) 0]

119879

1198652119899times1

= [119875 (1199030 120579 119911 120596) 0 0 0 0 0 0 0]

119879

(37)

where 119875(1199030 120579 119911 120596) acting over the area (119871

119902le 119909 le 119871

119902+119886119902) on

its top surface while it is traction-free at the bottom surfaceThus

119875 (119903 120579 119911 120596) =

infin

sum

119899=minusinfin

infin

sum

119898=0

119901119899119898 (119903 120596) Sin(

119898120587119911

119897) 119890119894(119899120579+120596119905)

119868 (119903 120579 119911 120596) =

infin

sum

119899=minusinfin

infin

sum

119898=0

120580119899119898 (119903 120596) Sin(

119898120587119911

119897) 119890119894(119899120579+120596119905)

(38)

where

119901119899119898 (119903 120596) =

119860119899119898 (120596) 119869119899 (119870119903) 119870

2gt 0

119860119899119898 (120596) 119903

119899 119870

2= 0

119860119899119898 (120596) 119868119899 (119870119903) 119870

2= minus1198702

lt 0

120580119899119898 (119903 120596) =

119861119899119898 (120596) 119869119899 (119870119903) 119870

2gt 0

119861119899119898 (120596) 119903

119899 119870

2= 0

119861119899119898 (120596) 119868119899 (119870119903) 119870

2= minus1198702

lt 0

(39)

in which 119870 = radic1198962 minus (120587119898119871)2 and 119869

119899and 119868

119899denote the

standard and modified cylindrical Bessel functions of firstkind respectively and 119860

119899119898(120596) and 119861

119899119898(120596) are the amplitude

of the applied forces Substituting (21) (25) and (26) into themechanical condition (39) and substituting (27) or (19) intothe electric condition (40) yields homogeneous equationswith respect to coefficients 119860

119898and 119861

119898 (119898 = 1 2 3 4)

After finding these unknown constants that are functions of

6 Shock and Vibration

EPOT+4658e + 07

+3881e + 07

+3105e + 07

+2329e + 07

+1553e + 07

+7763e + 06

minus3930e + 02

minus7763e + 06

minus1553e + 07

minus2329e + 07

minus3105e + 07

(a) First mode shape

EPOT+1260e + 07

+1050e + 07

+8402e + 06

+6301e + 06

+4201e + 06

+2100e + 06

+0000e + 00

minus2100e + 06

minus4201e + 06

minus6301e + 06

minus8402e + 06

(b) Second mode shape

EPOT+4049e + 06

+3374e + 06

+2699e + 06

+2025e + 06

+1350e + 06

+6754e + 05

+7571e + 02

minus6739e + 05

minus1349e + 06

minus2023e + 06

minus2698e + 06

(c) Third mode shape

EPOT+7004e + 07

+5837e + 07

+4670e + 07

+3502e + 07

+2335e + 07

+1167e + 07

minus6000e + 00

minus1167e + 07

minus2335e + 07

minus3502e + 07

minus4670e + 07

(d) Forth mode shape

EPOT+4064e + 08

+3725e + 08

+3387e + 08

+3048e + 08

+2709e + 08

+2371e + 08

+2032e + 08

+1693e + 08

+1355e + 08

+1016e + 08

+6773e + 07

(e) Fifth mode shape

Figure 2 Mode shapes of the five first natural frequencies

119898 119899 by replacing them in the displacement and stress andelectric displacement of corresponding equations (19)ndash(27)all of the system variables will be determined easily Howeverfor control purposes the voltage obtained from the piezolayeras a sensor is the measured output and it is calculated as

119902 = intArea

119863 sdot 119889119860Area (40)

where 119863 = 119863119903119903 + 119863

120579120579 + 119863

119911 is the electric displacement

vector in the principle cylindrical coordinates Area in theintegration stands for the place that the sensor layer is activeand voltage (control output) is measured and 119889119860Area = (119889119911 times

119889120579)119903 which simplifies the above equation as

119902119903= int

120579119904

2

120579119904

1

int

119911119904

2

119911119904

1

119863119903119889119911 119889120579 (41)

Moreover by considering the piezoelectric sensor layer asan electric capacity 119881 = 119902119888

119875119878 one can obtain

119881 =1

119888119875119878

int

120579119904

2

120579119904

1

int

119911119904

2

119911119904

1

119863119903119889119911 119889120579 (42)

where 119888119875119878

is the capacitance of the piezoelectric sensor

Table 1 First three nondimensional natural frequencies

119878 120583 = 18 120583 = 09

01 09366 18562 23634 05178 14580 1816002 08266 17995 23514 05109 14144 1821403 07214 17650 23616 05043 13565 1843404 06271 17193 23889 04975 12960 1866605 05408 16549 24273 04906 12385 18741

3 Results and Discussion

Table 1 shows the first three nondimensional natural frequen-cies of some panels by different geometries Mode shapes ofthe five first natural frequencies are shown in Figure 2 Thepanel dynamic responses under the aforementioned inputs(dynamic excitation and electric excitation) are shown inFigure 3 and are compared by FEM results

It is obvious that a good accommodation exist betweenanalytical solution and FEM (ABAQUS)method In additionthe dynamic response of the panel in 450Hz is shown in

Shock and Vibration 7

0 50 100 150 200 250 300 350 400 450

Frequency (Hz)

Am

plitu

de (d

B)

10minus8

10minus7

10minus6

10minus5

10minus4

AnalyticalFEM

(a)

102

104

106

105

108

107

103

Am

plitu

de (d

B)

0 50 100 150 200 250 300 350 400 450 500

Frequency (Hz)

FEMAnalytical

101

(b)

Figure 3 Panel dynamic response (a) Mechanical excitation (b) electrical excitation

+1413e minus 07

+1296e minus 07

+1178e minus 07

+1060e minus 07

+9422e minus 08

+8245e minus 08

+7067e minus 08

+5889e minus 08

+4711e minus 08

+3533e minus 08

+2356e minus 08

+1178e minus 08

U magnitude

(a)

+5119e + 00

+4266e + 00

+3413e + 00

+2560e + 00

+1706e + 00

+8532e minus 01

+3576e minus 07

minus8532e minus 01

minus1706e + 00

minus2560e + 00

minus3413e + 00

minus4266e + 00

EPOT

(b)

Figure 4 The dynamic response of the panel at 450Hz due to (a) mechanical excitation (b) electrical excitation

Figure 4 It can be seen that the dominant mode shape in thisfrequency is the third mode shape

4 Conclusion

Based on the general solution of the coupled equationsfor a piezoelectric media the displacement functions areexpanded in terms of trigonometric functions in 119911 and 120579

directions Three-dimensional exact solutions for the freevibration of a piezoelectric circular cylindrical panel are thenobtained under several boundary conditions Also the forcedvibration is solved The natural frequencies are comparedwith previous works The dynamic responses with mechan-ical and electrical excitation are validated with FEM and themode shapes are shown

Conflict of Interests

The authors declare that there is no conflict of interestsregarding the publication of this paper

References

[1] J F Haskins and J L Walsh ldquoVibrations of ferroelectriccylindrical shells with transverse isotropy I Radially polarizedcaserdquo The Journal of the Acoustical Society of America vol 29no 6 pp 729ndash734 1975

[2] G E Martin ldquoVibrations of longitudinally polarized ferroelec-tric cylindrical tubesrdquo The Journal of the Acoustical Society ofAmerica vol 35 no 4 pp 510ndash520 1963

[3] D S Drumheller and A Kalnins ldquoDynamic shell theory forferroelectric ceramics rdquo The Journal of the Acoustical Society ofAmerica vol 47 no 5 pp 1343ndash1353 1970

[4] J A Burt ldquoThe electroacoustic sensitivity of radially polarizedceramic cylinders as a function of frequencyrdquoThe Journal of theAcoustical Society of America vol 64 no 6 pp 1640ndash1644 1978

[5] H S Tzou and J P Zhong ldquoA linear theory of piezoelastic shellvibrationsrdquo Journal of Sound and Vibration vol 175 no 1 pp77ndash88 1994

[6] DD Ebenezer andPAbraham ldquoEigenfunction analysis of radi-ally polarized piezoelectric cylindrical shells of finite lengthrdquoThe Journal of the Acoustical Society of America vol 102 no 3pp 1549ndash1558 1997

8 Shock and Vibration

[7] C V Stephenson ldquoRadial vibrations in short hollow cylindersof barium titanaterdquo The Journal of the Acoustical Society ofAmerica vol 28 no 1 pp 51ndash56 1956

[8] C V Stephenson ldquoHigher modes of radial vibrations in shorthollow cylinders of barium titanaterdquoThe Journal of the Acousti-cal Society of America vol 28 no 5 pp 928ndash929 1956

[9] N T Adelman Y Stavsky and E Segal ldquoAxisymmetric vibra-tions of radially polarized piezoelectric ceramic cylindersrdquoJournal of Sound and Vibration vol 38 no 2 pp 245ndash254 1975

[10] N T Adelman Y Stavsky and E Segal ldquoRadial vibrations ofaxially polarized piezoelectric ceramic cylindersrdquo The Journalof the Acoustical Society of America vol 57 no 2 pp 356ndash3601975

[11] H S Paul ldquoVibrations of circular cylindrical shells of piezoelec-tric silver iodide crystalsrdquo The Journal of the Acoustical Societyof America vol 40 no 5 pp 1077ndash1080 1966

[12] H S Paul and M Venkatesan ldquoVibrations of a hollow circularcylinder of piezoelectric ceramicsrdquoThe Journal of the AcousticalSociety of America vol 82 no 3 pp 952ndash956 1987

[13] H-JDingW-QChen Y-MGuo andQ-DYang ldquoFree vibra-tions of piezoelectric cylindrical shells filled with compressiblefluidrdquo International Journal of Solids and Structures vol 34 no16 pp 2025ndash2034 1997

[14] Z Yang J Yang Y Hu and Q-M Wang ldquoVibration charac-teristics of a circular cylindrical panel piezoelectric transducerrdquoIEEE Transactions on Ultrasonics Ferroelectrics and FrequencyControl vol 55 no 10 pp 2327ndash2335 2008

[15] S Li J Qiu H Ji K Zhu and J Li ldquoPiezoelectric vibration con-trol for all-clamped panel using DOB-based optimal controlrdquoMechatronics vol 21 no 7 pp 1213ndash1221 2011

[16] N Kumar and S P Singh ldquoVibration control of curved panelusing smart dampingrdquo Mechanical Systems and Signal Process-ing vol 30 pp 232ndash247 2012

[17] H J Ding B Chen and J Liang ldquoGeneral solutions for coupledequations for piezoelectric mediardquo International Journal ofSolids and Structures vol 33 no 16 pp 2283ndash2298 1996

[18] H J Ding R Q Xu and W Q Chen ldquoFree vibration oftransversely isotropic piezoelectric circular cylindrical panelsrdquoInternational Journal of Mechanical Sciences vol 44 no 1 pp191ndash206 2002

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

2 Shock and Vibration

L

z

r

r0

r1

1205790

120579

Figure 1 Cylindrical panel and its geometry

forced case Li et al considered the spillover and harmoniceffect in real active vibration control and they presented anovel composite controller based on disturbance observer(DOB) for the all-clamped panel [15] Kumar and Singhaimed to examine through experiments vibration controlof curved panel treated with optimally placed active orpassive constrained layer damping patches and they foundthe optimum location for the application of ACLDPCLDpatches [16]

The main subject of this study is to investigate the freeand forced vibration of transversely isotropic piezoelectriccylindrical panels Based on the general solution for coupledequations for piezoelectricmedia presented inDing et al [17]three-dimensional exact solutions are obtained through thevariable separation method A numerical example is finallypresented

2 Problem Formulation

For dynamic modelling of piezoelectric layers two displace-ment functions Ψ and 119865 are considered [17] Figure 1 showsthe panel geometry

21 Basic Equations In circular cylindrical coordinates(119903 120579 119911) if the media is axially polarized the general solutioncan be written as

119906119875119894

119903=1

119903

120597120595

120597120579minus

120597

120597119903A1119865

119906119875119894

120579= minus

120597Ψ

120597119903minus1

119903

120597

120597120579A1119865

119908119875119894= A2119865 120601 = A

3119865

(1)

where 119906119875119894119903 119906119875119894120579 and 119908

119875119894 are three displacement components120601 is the electric potential and the differential operators A

1

A2 andA

3are

A1= [(119888

119875119894

13+ 119888119875119894

44) 12057633+ (11989015+ 11989031) 11989033]1205973

1205971199113

+ [(119888119875119894

13+ 119888119875119894

44) 12057611+ (11989015+ 11989031) 11989015] Λ

120597

120597119911

A2= 119888119875119894

4412057633

1205974

1205971199114

+ [119888119875119894

1112057633+ 119888119875119894

4412057611+ (11989015+ 11989031)2Λ minus 120588120576

33

1205972

1205971199052]

1205972

1205971199112

+ 119888119875119894

1112057611ΛΛ minus 120588120576

11Λ1205972

1205971199052

A3= 119888119875119894

4411989033

1205974

1205971199114

+ [119888119875119894

1111989033+ 119888119875119894

4411989015minus (119888119875119894

13+ 119888119875119894

44) (11989015+ 11989031)] Λ

minus12058811989033

1205972

1205971199052

1205972

1205971199112

+ 119888119875119894

1111989015ΛΛ minus 120588119890

15Λ1205972

1205971199052

(2)

where Λ = 12059721205971199032+ (1119903)120597120597119903 + (1119903

2)12059721205971205792 is the two-

dimensional Laplacian The displacement functions Ψ and 119865must satisfy the following two equations

(119888120588

66Λ + 119888119875119894

44

1205972

1205971199112minus 120588119875119894 1205972

1205971199052)120595 = 0 119871

0119865 = 0 (3)

1198710= 1198864ΛΛΛ + (119886

3

1205972

1205971199112+ 1198866

1205972

1205971199052)ΛΛ

+ (1198862

1205974

1205971199114+ 1198865

1205974

1205971199054+ 1198867

1205974

12059711991121205971199052)Λ + 119886

1

1205976

1205971199116

+ 1198868

1205976

12059711991141205971199052+ 1198869

1205976

12059711991121205971199054

(4)

Shock and Vibration 3

Here 119886119899(119899 = 1 2 9) can be expressed in terms of

elastic constants 119888119875119894119894119895 dielectric constants 120576

119894119895 and piezoelectric

coefficients 119890119894119895as follows

1198861= 119888119875119894

44(1198902

33+ 119888119875119894

3312057633)

1198864= 119888119875119894

11(1198902

15+ 119888119875119894

4412057611) 119886

5= 120588212057611

1198862= 119888119875119894

33[119888119875119894

4412057611+ (11989015+ 11989031)2]

+ 12057633[119888119875119894

11119888119875119894

33+ 1198882

44minus (119888119875119894

11+ 119888119875119894

44)2

]

+ 11989033[2119888119875119894

4411989015+ 119888119875119894

1111989033minus 2 (119888

119875119894

13+ 119888119875119894

44) (11989015+ 11989031)]

1198863= 119888119875119894

44[119888119875119894

1112057633+ (11989015+ 11989031)2]

+ 12057611[119888119875119894

1111988833+ 1198882

44minus (119888119875119894

13+ 119888119875119894

44)2

]

+ 11989015[2119888119875119894

1111989033+ 119888119875119894

4411989015minus 2 (119888

119875119894

13+ 119888119875119894

44) (11989015+ 11989031)]

1198866= minus120588 [119890

2

15+ (119888119875119894

11+ 119888119875119894

44) 12057611]

1198868= minus120588 [119890

2

33+ (119888119875119894

44+ 119888119875119894

33) 12057633] 119886

9= 120588212057633

1198867= minus 120588 [2119890

1511989033+ (119888119875119894

44+ 119888119875119894

33) 12057611+ (119888119875119868

11+ 119888119875119894

44) 12057633

+(11989015+ 11989031)2]

(5)

The circular cylindrical coordinates as well as a circularcylindrical panel with outer radius 119887

119875119894 inner radius 119886119875119894

circular center angle 120572119875119894 and length 119871119875119894 are shown in Figure 1If the panel is vibrating with a resonant frequency 120596 thedisplacement functions can be assumed as

119865 =1199035

0

1198881112057633

119875 (120585119875)Θ (120583120579)119885 (120573120577

119875) 119890119894120596119905

120595 = 1199032

01198754(120585119875)Θ1015840(120583120579)119885

1015840(120573120577119875) 119890119894120596119905

(6)

where 120585119875119894

= 119903119875119894119877119875119894 120577119875119894 = 119911

119875119894119871119875119894 are the dimensionless

coordinates in 119903 and 119911 directions and Θ1015840(120583120579119875119894) and 119885

1015840(120573120577119875119894)

denote the derivation of Θ(120583120579119875119894) with respect to 120583120579119875119894 and

the derivation of 119885(120573120577119875119894) with respect to 120573120577119875119894 respectively

In addition

Θ(120583120579) = 1198621cos (120583120579) + 119862

2sin (120583120579)

119885 (120573120577) = 1198623sin (120573120577) + 119888

4cos (120573120577)

(7)

where 119862119898(119898 = 1 2 3 4) are constants Substitution of (6)

into (3) yields

(Δ + 1198962

4) 1198754 (120585) = 0 (8)

(Δ + 1198962

1) (Δ + 119896

2

2) (Δ + 119896

2

3) 1198754 (120585) = 0 (9)

where Δ = 1205972120597(120585119875119894)2

+ (1120585119875119894)120597120597(120585

119875119894) minus 1205832(120585119875119894)2 and

1198962

4=Ω2119888119901119894

11

119888119901119894

66

minus1205742119888119901119894

44

119888119901119894

66

(Ω119875119894)2

=1205881198751198941205962

1199032

0

11988811

120574 = 1205731199051 1199051=

1199030

ℎ0

(10)

and (119896119875119894

119898)2 (119898 = 1 2 3) (assuming Re[119896119875119894

119898] ge 0) are the

eigenvalues of the following equation

11988641198966+ (1198866(Ω119875119894)2

+ 11988631205742) 1198964

+ (11988621205744+ 11988671205742(Ω119875119894)2

+ 1198865(Ω119875119894)4

) 1198962

+ (11988611205746+ 11988681205744(Ω119875119894)2

+ 1205742(Ω119875119894)4

) = 0

(11)

in which

119886119899=

119886119899

((119888119875119894

11)212057633)

(119899 = 1 2 3 4)

119886119899=

119886119899

(120588119888119875119894

1112057633) (119899 = 6 7 8)

1198865=

1198865

((120588119875119894)212057633)

(12)

The solution of (9) can be assumed as

119875 (120585) = 1198751 (120585) + 119875

2 (120585) + 1198753 (120585) (13)

where 119875119898(120585119875119894) is obtained as [18]

Substituting (6) into (1) gives the mechanical displace-ments and electric potential as follows

119906119901119894

119903= minus1199030[120583

1205851198754 (120585) +

3

sum

119898=1

12057211198981198751015840

119898(120585)]Θ (120583120579)119885

1015840(120573120577) 119890

119894120596119905

(14)

119906119901119894

120579= minus1199030[1198751015840

4(120585) +

120583

120585

3

sum

119898=1

1205721119898119875119898 (120585)]Θ

1015840(120583120579)119885

1015840(120573120577) 119890

119894120596119905

(15)

119908119901119894= 1199030[

3

sum

119898=1

1205722119898119875119898 (120585)]Θ (120583120579)119885 (120573120577) 119890

119894120596119905 (16)

Φ = 1199030radic119888119901119894

11

12057633

[

3

sum

119898=1

1205723119898119875119898 (120585)]Θ (120583120579)119885 (120573120577) 119890

119894120596119905 (17)

4 Shock and Vibration

where1205721119898

= minus ([(119888119875119894

13+ 11988844) (120576111198962

119898+ 120576331205742)

+ (11989015+ 11989031) (119890151198962

119898+ 119890331205742) ] 120574)

times (119888119875119894

1112057633)minus1

1205722119898

= ([(119888119875119894

111198962

119898+ 119888119875119894

441205742minus 119888119875119894

11Ω2

1) (120576111198962

119898+ 120576331205742)

+(11989015+ 11989031)21198962

1198981205742])

times (119888119875119894

1112057633)minus1

1205723119898

= ([ (119888119875119894

111198962

119898+ 119888119875119894

441205742minus 119888119875119894

11Ω2

1) (119890151198962

119898+ 119890331205742)

minus(119888119875119894

13+ 119888119875119894

44) (11989015+ 11989031)2

1198962

1198981205742])

times (119888119875119894

11radic119888119875119894

1112057633)

minus1

(119898 = 1 2 3)

(18)

Utilizing the constitutive relations of piezoelectricity and(14)ndash(17) the stress components and electric displacementcomponents can be derived as

120590119901119894

119903=

(119888119901119894

12minus 119888119901119894

11) [

120583

1205851198751015840

4(120585) minus

120583

12058521198754 (120585)]

+ (119888119901119894

12minus 119888119901119894

11)

3

sum

119898=1

120572111989811987510158401015840

119898(120585)

+

3

sum

119898=1

(1198881212057211198981198962

119898+ 119888119901119894

131205741205722119898

+ 11989031radic119888119901119894

11

12057633

1205741205723119898)

times119875119898 (120585)

Θ(120583120579)1198851015840(120573120577) 119890

119894120596119905

(19)

120590119901119894

120579=

(119888119901119894

11minus 119888119901119894

12) [

120583

1205851198751015840

4(120585) minus

120583

12058521198754 (120585)]

+ (119888119901119894

11minus 119888119901119894

12)

3

sum

119898=1

120572111989811987510158401015840

119898(120585)

+

3

sum

119898=1

(119888119901119894

1112057211198981198962

119898+ 119888119901119894

131205741205722119898

+ 11989031radic119888119901119894

11

12057633

1205741205723119898)

times119875119898 (120585)

Θ(120583120579)1198851015840(120573120577) 119890

119894120596119905

(20)

120590119901119894

119911=

3

sum

119898=1

[[

[

(119888119901119894

1312057211198981198962

119898+ 119888119901119894

331205741205722119898

+ 11989033radic119888119901119894

11

12057633

1205741205723119898)

times 119875119898 (120585)

]]

]

Θ (120583120579)1198851015840(120573120577) 119890

119894120596119905

(21)

120591119901119894

120579119911=

119888119901119894

441205741198751015840

4(120585)

+120583

120585

3

sum

119898=1

[[

[

(119888119901119894

441205741205721119898

+ 119888119901119894

441205722119898

+ 11989015radic119888119901119894

11

12057633

1205723119898)

times119875119898 (120585)

]]

]

Θ1015840(120583120579)119885 (120573120577) 119890

119894120596119905

(22)

120591119901119894

119903119911=

119888119901119894

44120574120583

120585119876 (120585)

+[[

[

3

sum

119898=1

(119888119901119894

441205741205721119898

+ 119888119901119894

441205722119898

+ 11989015radic119888119901119894

11

12057633

1205723119898)

times1199011015840

119872(120585)

]]

]

Θ(120583120579)119885 (120573120577) 119890119894120596119905

(23)

120591119901119894

119903120579= 11988866[ minus 1198962

41198754 (120585) minus 2119875

10158401015840

4(120585)

+2120583

1205852

3

sum

119898=1

1205721119898119875119898 (120585) minus

2120583

120585

3

sum

119898=1

12057211198981199011015840

119872(120585)]

times Θ1015840(120583120579)119885

1015840(120573120577) 119890

119894120596119905

(24)

119863119901119894

119903=

11989015120574120583

1205851198754 (120585)

+[[

[

3

sum

119898=1

(119890151205741205721119898

+ 119890151205722119898

+ 12057611radic119888119901119894

11

12057633

1205723119898)

times1199011015840

119872(120585)

]]

]

Θ(120583120579)119885 (120573120577) 119890119894120596119905

(25)

Shock and Vibration 5

119863119901119894

120579=

119890151205741198751015840

4(120585)

+120583

120585

[[

[

3

sum

119898=1

(119890151205741205721119898

+ 119890151205722119898

+ 12057611radic119888119901119894

11

12057633

1205723119898)

times119875119898 (120585)

]]

]

Θ1015840(120583120579)119885 (120573120577) 119890

119894120596119905

(26)

119863119901119894

119911= [

3

sum

119898=1

(1198903112057211198981198962

119898+ 119890331205741205722119898

minus radic119888119901119894

11120576331205741205723119898)119875119898 (120585)]

times Θ (120583120579)1198851015840(120573120577) 119890

119894120596119905

(27)

22 Boundary Conditions The piezoelectric panel has 8boundary conditions consist of 6 mechanical and 2 electricalones

By considering generalized simply support boundaryconditions at 120579119894 = 0 and 120579

119894= 120572 and (119894 = 119875119894) we will have

119908119894= 119906119894

119903= 0 120590

119894

120579= 0 (119894 = 119875119894) (28)

Note that for piezoelectric layers the following conditionis added

120601 = 0 (29)

One can take

119862119894

1= 0 119862

119894

2= 1 120583 =

(2119898 + 1) 120587

2120572 119898 = 0 1 2

(30)

And by considering generalized simply support boundaryconditions at 120577119894 = 0 and 120577

119894= 1 (119894 = 119875119894) we will have

119906119894

119903= 119906119894

120579= 0 120590

119894

119911= 0 (119894 = 119875119894) (31)

And for piezoelectric layers the following condition isadded

119863119911= 0 (32)

One can take

119862119894

3= 0 119862

119894

4= 1 120573 = 119899120587 119899 = 0 1 2 (33)

Without loss of generality we suppose that external forceacts on the outer surface of the actuator and inner surface ofsensor has free boundary condition So we have

120590119875119894

119903= 119875 120591

119875119894

119903120579= 120591119875119894

119903119911= 0 120601 = 119868 at 119903 = 119903

4

120590119875119894

119903= 120591119875119894

119903120579= 120591119875119894

119903119911= 0 120601 = 0 at 119903 = 119903

1

(34)

For obtaining steady state frequency response of thecylindrical panel under a harmonic external excitation wemust solve the following matrix equation

[119879]119898times119899119883119899times1 = 119865119898times1 (35)

where [119879]119898times119899

is the coefficient matrix Consider

119883119899times1 = [11986011198611119860211986121198603119861311986041198614] (36)

and 119860119894 119861119894 119894 = 1 2 3 4 are the unknown constants that are

in (19)ndash(27)The vector 119865

119898times1denotes the force vector that acts on

the structure This force consists of the surface force that isconsidered as disturbance and has the breed of mechanicalforce such as wind effect The effect of controller unit inthe dynamic response of the piezo-panel is considered as anexternal electrical potential applied on the upper surface ofthe panel These two external forces acted on the structureindependently however summation of their effects on thewhole structure is the same as the case that both of them acton the structure simultaneously So

119865119899times2 = [1198651 1198652]

1198651119899times1

= [0 0 0 0 0 0 119868 (1199030 120579 119911 120596) 0]

119879

1198652119899times1

= [119875 (1199030 120579 119911 120596) 0 0 0 0 0 0 0]

119879

(37)

where 119875(1199030 120579 119911 120596) acting over the area (119871

119902le 119909 le 119871

119902+119886119902) on

its top surface while it is traction-free at the bottom surfaceThus

119875 (119903 120579 119911 120596) =

infin

sum

119899=minusinfin

infin

sum

119898=0

119901119899119898 (119903 120596) Sin(

119898120587119911

119897) 119890119894(119899120579+120596119905)

119868 (119903 120579 119911 120596) =

infin

sum

119899=minusinfin

infin

sum

119898=0

120580119899119898 (119903 120596) Sin(

119898120587119911

119897) 119890119894(119899120579+120596119905)

(38)

where

119901119899119898 (119903 120596) =

119860119899119898 (120596) 119869119899 (119870119903) 119870

2gt 0

119860119899119898 (120596) 119903

119899 119870

2= 0

119860119899119898 (120596) 119868119899 (119870119903) 119870

2= minus1198702

lt 0

120580119899119898 (119903 120596) =

119861119899119898 (120596) 119869119899 (119870119903) 119870

2gt 0

119861119899119898 (120596) 119903

119899 119870

2= 0

119861119899119898 (120596) 119868119899 (119870119903) 119870

2= minus1198702

lt 0

(39)

in which 119870 = radic1198962 minus (120587119898119871)2 and 119869

119899and 119868

119899denote the

standard and modified cylindrical Bessel functions of firstkind respectively and 119860

119899119898(120596) and 119861

119899119898(120596) are the amplitude

of the applied forces Substituting (21) (25) and (26) into themechanical condition (39) and substituting (27) or (19) intothe electric condition (40) yields homogeneous equationswith respect to coefficients 119860

119898and 119861

119898 (119898 = 1 2 3 4)

After finding these unknown constants that are functions of

6 Shock and Vibration

EPOT+4658e + 07

+3881e + 07

+3105e + 07

+2329e + 07

+1553e + 07

+7763e + 06

minus3930e + 02

minus7763e + 06

minus1553e + 07

minus2329e + 07

minus3105e + 07

(a) First mode shape

EPOT+1260e + 07

+1050e + 07

+8402e + 06

+6301e + 06

+4201e + 06

+2100e + 06

+0000e + 00

minus2100e + 06

minus4201e + 06

minus6301e + 06

minus8402e + 06

(b) Second mode shape

EPOT+4049e + 06

+3374e + 06

+2699e + 06

+2025e + 06

+1350e + 06

+6754e + 05

+7571e + 02

minus6739e + 05

minus1349e + 06

minus2023e + 06

minus2698e + 06

(c) Third mode shape

EPOT+7004e + 07

+5837e + 07

+4670e + 07

+3502e + 07

+2335e + 07

+1167e + 07

minus6000e + 00

minus1167e + 07

minus2335e + 07

minus3502e + 07

minus4670e + 07

(d) Forth mode shape

EPOT+4064e + 08

+3725e + 08

+3387e + 08

+3048e + 08

+2709e + 08

+2371e + 08

+2032e + 08

+1693e + 08

+1355e + 08

+1016e + 08

+6773e + 07

(e) Fifth mode shape

Figure 2 Mode shapes of the five first natural frequencies

119898 119899 by replacing them in the displacement and stress andelectric displacement of corresponding equations (19)ndash(27)all of the system variables will be determined easily Howeverfor control purposes the voltage obtained from the piezolayeras a sensor is the measured output and it is calculated as

119902 = intArea

119863 sdot 119889119860Area (40)

where 119863 = 119863119903119903 + 119863

120579120579 + 119863

119911 is the electric displacement

vector in the principle cylindrical coordinates Area in theintegration stands for the place that the sensor layer is activeand voltage (control output) is measured and 119889119860Area = (119889119911 times

119889120579)119903 which simplifies the above equation as

119902119903= int

120579119904

2

120579119904

1

int

119911119904

2

119911119904

1

119863119903119889119911 119889120579 (41)

Moreover by considering the piezoelectric sensor layer asan electric capacity 119881 = 119902119888

119875119878 one can obtain

119881 =1

119888119875119878

int

120579119904

2

120579119904

1

int

119911119904

2

119911119904

1

119863119903119889119911 119889120579 (42)

where 119888119875119878

is the capacitance of the piezoelectric sensor

Table 1 First three nondimensional natural frequencies

119878 120583 = 18 120583 = 09

01 09366 18562 23634 05178 14580 1816002 08266 17995 23514 05109 14144 1821403 07214 17650 23616 05043 13565 1843404 06271 17193 23889 04975 12960 1866605 05408 16549 24273 04906 12385 18741

3 Results and Discussion

Table 1 shows the first three nondimensional natural frequen-cies of some panels by different geometries Mode shapes ofthe five first natural frequencies are shown in Figure 2 Thepanel dynamic responses under the aforementioned inputs(dynamic excitation and electric excitation) are shown inFigure 3 and are compared by FEM results

It is obvious that a good accommodation exist betweenanalytical solution and FEM (ABAQUS)method In additionthe dynamic response of the panel in 450Hz is shown in

Shock and Vibration 7

0 50 100 150 200 250 300 350 400 450

Frequency (Hz)

Am

plitu

de (d

B)

10minus8

10minus7

10minus6

10minus5

10minus4

AnalyticalFEM

(a)

102

104

106

105

108

107

103

Am

plitu

de (d

B)

0 50 100 150 200 250 300 350 400 450 500

Frequency (Hz)

FEMAnalytical

101

(b)

Figure 3 Panel dynamic response (a) Mechanical excitation (b) electrical excitation

+1413e minus 07

+1296e minus 07

+1178e minus 07

+1060e minus 07

+9422e minus 08

+8245e minus 08

+7067e minus 08

+5889e minus 08

+4711e minus 08

+3533e minus 08

+2356e minus 08

+1178e minus 08

U magnitude

(a)

+5119e + 00

+4266e + 00

+3413e + 00

+2560e + 00

+1706e + 00

+8532e minus 01

+3576e minus 07

minus8532e minus 01

minus1706e + 00

minus2560e + 00

minus3413e + 00

minus4266e + 00

EPOT

(b)

Figure 4 The dynamic response of the panel at 450Hz due to (a) mechanical excitation (b) electrical excitation

Figure 4 It can be seen that the dominant mode shape in thisfrequency is the third mode shape

4 Conclusion

Based on the general solution of the coupled equationsfor a piezoelectric media the displacement functions areexpanded in terms of trigonometric functions in 119911 and 120579

directions Three-dimensional exact solutions for the freevibration of a piezoelectric circular cylindrical panel are thenobtained under several boundary conditions Also the forcedvibration is solved The natural frequencies are comparedwith previous works The dynamic responses with mechan-ical and electrical excitation are validated with FEM and themode shapes are shown

Conflict of Interests

The authors declare that there is no conflict of interestsregarding the publication of this paper

References

[1] J F Haskins and J L Walsh ldquoVibrations of ferroelectriccylindrical shells with transverse isotropy I Radially polarizedcaserdquo The Journal of the Acoustical Society of America vol 29no 6 pp 729ndash734 1975

[2] G E Martin ldquoVibrations of longitudinally polarized ferroelec-tric cylindrical tubesrdquo The Journal of the Acoustical Society ofAmerica vol 35 no 4 pp 510ndash520 1963

[3] D S Drumheller and A Kalnins ldquoDynamic shell theory forferroelectric ceramics rdquo The Journal of the Acoustical Society ofAmerica vol 47 no 5 pp 1343ndash1353 1970

[4] J A Burt ldquoThe electroacoustic sensitivity of radially polarizedceramic cylinders as a function of frequencyrdquoThe Journal of theAcoustical Society of America vol 64 no 6 pp 1640ndash1644 1978

[5] H S Tzou and J P Zhong ldquoA linear theory of piezoelastic shellvibrationsrdquo Journal of Sound and Vibration vol 175 no 1 pp77ndash88 1994

[6] DD Ebenezer andPAbraham ldquoEigenfunction analysis of radi-ally polarized piezoelectric cylindrical shells of finite lengthrdquoThe Journal of the Acoustical Society of America vol 102 no 3pp 1549ndash1558 1997

8 Shock and Vibration

[7] C V Stephenson ldquoRadial vibrations in short hollow cylindersof barium titanaterdquo The Journal of the Acoustical Society ofAmerica vol 28 no 1 pp 51ndash56 1956

[8] C V Stephenson ldquoHigher modes of radial vibrations in shorthollow cylinders of barium titanaterdquoThe Journal of the Acousti-cal Society of America vol 28 no 5 pp 928ndash929 1956

[9] N T Adelman Y Stavsky and E Segal ldquoAxisymmetric vibra-tions of radially polarized piezoelectric ceramic cylindersrdquoJournal of Sound and Vibration vol 38 no 2 pp 245ndash254 1975

[10] N T Adelman Y Stavsky and E Segal ldquoRadial vibrations ofaxially polarized piezoelectric ceramic cylindersrdquo The Journalof the Acoustical Society of America vol 57 no 2 pp 356ndash3601975

[11] H S Paul ldquoVibrations of circular cylindrical shells of piezoelec-tric silver iodide crystalsrdquo The Journal of the Acoustical Societyof America vol 40 no 5 pp 1077ndash1080 1966

[12] H S Paul and M Venkatesan ldquoVibrations of a hollow circularcylinder of piezoelectric ceramicsrdquoThe Journal of the AcousticalSociety of America vol 82 no 3 pp 952ndash956 1987

[13] H-JDingW-QChen Y-MGuo andQ-DYang ldquoFree vibra-tions of piezoelectric cylindrical shells filled with compressiblefluidrdquo International Journal of Solids and Structures vol 34 no16 pp 2025ndash2034 1997

[14] Z Yang J Yang Y Hu and Q-M Wang ldquoVibration charac-teristics of a circular cylindrical panel piezoelectric transducerrdquoIEEE Transactions on Ultrasonics Ferroelectrics and FrequencyControl vol 55 no 10 pp 2327ndash2335 2008

[15] S Li J Qiu H Ji K Zhu and J Li ldquoPiezoelectric vibration con-trol for all-clamped panel using DOB-based optimal controlrdquoMechatronics vol 21 no 7 pp 1213ndash1221 2011

[16] N Kumar and S P Singh ldquoVibration control of curved panelusing smart dampingrdquo Mechanical Systems and Signal Process-ing vol 30 pp 232ndash247 2012

[17] H J Ding B Chen and J Liang ldquoGeneral solutions for coupledequations for piezoelectric mediardquo International Journal ofSolids and Structures vol 33 no 16 pp 2283ndash2298 1996

[18] H J Ding R Q Xu and W Q Chen ldquoFree vibration oftransversely isotropic piezoelectric circular cylindrical panelsrdquoInternational Journal of Mechanical Sciences vol 44 no 1 pp191ndash206 2002

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 3

Here 119886119899(119899 = 1 2 9) can be expressed in terms of

elastic constants 119888119875119894119894119895 dielectric constants 120576

119894119895 and piezoelectric

coefficients 119890119894119895as follows

1198861= 119888119875119894

44(1198902

33+ 119888119875119894

3312057633)

1198864= 119888119875119894

11(1198902

15+ 119888119875119894

4412057611) 119886

5= 120588212057611

1198862= 119888119875119894

33[119888119875119894

4412057611+ (11989015+ 11989031)2]

+ 12057633[119888119875119894

11119888119875119894

33+ 1198882

44minus (119888119875119894

11+ 119888119875119894

44)2

]

+ 11989033[2119888119875119894

4411989015+ 119888119875119894

1111989033minus 2 (119888

119875119894

13+ 119888119875119894

44) (11989015+ 11989031)]

1198863= 119888119875119894

44[119888119875119894

1112057633+ (11989015+ 11989031)2]

+ 12057611[119888119875119894

1111988833+ 1198882

44minus (119888119875119894

13+ 119888119875119894

44)2

]

+ 11989015[2119888119875119894

1111989033+ 119888119875119894

4411989015minus 2 (119888

119875119894

13+ 119888119875119894

44) (11989015+ 11989031)]

1198866= minus120588 [119890

2

15+ (119888119875119894

11+ 119888119875119894

44) 12057611]

1198868= minus120588 [119890

2

33+ (119888119875119894

44+ 119888119875119894

33) 12057633] 119886

9= 120588212057633

1198867= minus 120588 [2119890

1511989033+ (119888119875119894

44+ 119888119875119894

33) 12057611+ (119888119875119868

11+ 119888119875119894

44) 12057633

+(11989015+ 11989031)2]

(5)

The circular cylindrical coordinates as well as a circularcylindrical panel with outer radius 119887

119875119894 inner radius 119886119875119894

circular center angle 120572119875119894 and length 119871119875119894 are shown in Figure 1If the panel is vibrating with a resonant frequency 120596 thedisplacement functions can be assumed as

119865 =1199035

0

1198881112057633

119875 (120585119875)Θ (120583120579)119885 (120573120577

119875) 119890119894120596119905

120595 = 1199032

01198754(120585119875)Θ1015840(120583120579)119885

1015840(120573120577119875) 119890119894120596119905

(6)

where 120585119875119894

= 119903119875119894119877119875119894 120577119875119894 = 119911

119875119894119871119875119894 are the dimensionless

coordinates in 119903 and 119911 directions and Θ1015840(120583120579119875119894) and 119885

1015840(120573120577119875119894)

denote the derivation of Θ(120583120579119875119894) with respect to 120583120579119875119894 and

the derivation of 119885(120573120577119875119894) with respect to 120573120577119875119894 respectively

In addition

Θ(120583120579) = 1198621cos (120583120579) + 119862

2sin (120583120579)

119885 (120573120577) = 1198623sin (120573120577) + 119888

4cos (120573120577)

(7)

where 119862119898(119898 = 1 2 3 4) are constants Substitution of (6)

into (3) yields

(Δ + 1198962

4) 1198754 (120585) = 0 (8)

(Δ + 1198962

1) (Δ + 119896

2

2) (Δ + 119896

2

3) 1198754 (120585) = 0 (9)

where Δ = 1205972120597(120585119875119894)2

+ (1120585119875119894)120597120597(120585

119875119894) minus 1205832(120585119875119894)2 and

1198962

4=Ω2119888119901119894

11

119888119901119894

66

minus1205742119888119901119894

44

119888119901119894

66

(Ω119875119894)2

=1205881198751198941205962

1199032

0

11988811

120574 = 1205731199051 1199051=

1199030

ℎ0

(10)

and (119896119875119894

119898)2 (119898 = 1 2 3) (assuming Re[119896119875119894

119898] ge 0) are the

eigenvalues of the following equation

11988641198966+ (1198866(Ω119875119894)2

+ 11988631205742) 1198964

+ (11988621205744+ 11988671205742(Ω119875119894)2

+ 1198865(Ω119875119894)4

) 1198962

+ (11988611205746+ 11988681205744(Ω119875119894)2

+ 1205742(Ω119875119894)4

) = 0

(11)

in which

119886119899=

119886119899

((119888119875119894

11)212057633)

(119899 = 1 2 3 4)

119886119899=

119886119899

(120588119888119875119894

1112057633) (119899 = 6 7 8)

1198865=

1198865

((120588119875119894)212057633)

(12)

The solution of (9) can be assumed as

119875 (120585) = 1198751 (120585) + 119875

2 (120585) + 1198753 (120585) (13)

where 119875119898(120585119875119894) is obtained as [18]

Substituting (6) into (1) gives the mechanical displace-ments and electric potential as follows

119906119901119894

119903= minus1199030[120583

1205851198754 (120585) +

3

sum

119898=1

12057211198981198751015840

119898(120585)]Θ (120583120579)119885

1015840(120573120577) 119890

119894120596119905

(14)

119906119901119894

120579= minus1199030[1198751015840

4(120585) +

120583

120585

3

sum

119898=1

1205721119898119875119898 (120585)]Θ

1015840(120583120579)119885

1015840(120573120577) 119890

119894120596119905

(15)

119908119901119894= 1199030[

3

sum

119898=1

1205722119898119875119898 (120585)]Θ (120583120579)119885 (120573120577) 119890

119894120596119905 (16)

Φ = 1199030radic119888119901119894

11

12057633

[

3

sum

119898=1

1205723119898119875119898 (120585)]Θ (120583120579)119885 (120573120577) 119890

119894120596119905 (17)

4 Shock and Vibration

where1205721119898

= minus ([(119888119875119894

13+ 11988844) (120576111198962

119898+ 120576331205742)

+ (11989015+ 11989031) (119890151198962

119898+ 119890331205742) ] 120574)

times (119888119875119894

1112057633)minus1

1205722119898

= ([(119888119875119894

111198962

119898+ 119888119875119894

441205742minus 119888119875119894

11Ω2

1) (120576111198962

119898+ 120576331205742)

+(11989015+ 11989031)21198962

1198981205742])

times (119888119875119894

1112057633)minus1

1205723119898

= ([ (119888119875119894

111198962

119898+ 119888119875119894

441205742minus 119888119875119894

11Ω2

1) (119890151198962

119898+ 119890331205742)

minus(119888119875119894

13+ 119888119875119894

44) (11989015+ 11989031)2

1198962

1198981205742])

times (119888119875119894

11radic119888119875119894

1112057633)

minus1

(119898 = 1 2 3)

(18)

Utilizing the constitutive relations of piezoelectricity and(14)ndash(17) the stress components and electric displacementcomponents can be derived as

120590119901119894

119903=

(119888119901119894

12minus 119888119901119894

11) [

120583

1205851198751015840

4(120585) minus

120583

12058521198754 (120585)]

+ (119888119901119894

12minus 119888119901119894

11)

3

sum

119898=1

120572111989811987510158401015840

119898(120585)

+

3

sum

119898=1

(1198881212057211198981198962

119898+ 119888119901119894

131205741205722119898

+ 11989031radic119888119901119894

11

12057633

1205741205723119898)

times119875119898 (120585)

Θ(120583120579)1198851015840(120573120577) 119890

119894120596119905

(19)

120590119901119894

120579=

(119888119901119894

11minus 119888119901119894

12) [

120583

1205851198751015840

4(120585) minus

120583

12058521198754 (120585)]

+ (119888119901119894

11minus 119888119901119894

12)

3

sum

119898=1

120572111989811987510158401015840

119898(120585)

+

3

sum

119898=1

(119888119901119894

1112057211198981198962

119898+ 119888119901119894

131205741205722119898

+ 11989031radic119888119901119894

11

12057633

1205741205723119898)

times119875119898 (120585)

Θ(120583120579)1198851015840(120573120577) 119890

119894120596119905

(20)

120590119901119894

119911=

3

sum

119898=1

[[

[

(119888119901119894

1312057211198981198962

119898+ 119888119901119894

331205741205722119898

+ 11989033radic119888119901119894

11

12057633

1205741205723119898)

times 119875119898 (120585)

]]

]

Θ (120583120579)1198851015840(120573120577) 119890

119894120596119905

(21)

120591119901119894

120579119911=

119888119901119894

441205741198751015840

4(120585)

+120583

120585

3

sum

119898=1

[[

[

(119888119901119894

441205741205721119898

+ 119888119901119894

441205722119898

+ 11989015radic119888119901119894

11

12057633

1205723119898)

times119875119898 (120585)

]]

]

Θ1015840(120583120579)119885 (120573120577) 119890

119894120596119905

(22)

120591119901119894

119903119911=

119888119901119894

44120574120583

120585119876 (120585)

+[[

[

3

sum

119898=1

(119888119901119894

441205741205721119898

+ 119888119901119894

441205722119898

+ 11989015radic119888119901119894

11

12057633

1205723119898)

times1199011015840

119872(120585)

]]

]

Θ(120583120579)119885 (120573120577) 119890119894120596119905

(23)

120591119901119894

119903120579= 11988866[ minus 1198962

41198754 (120585) minus 2119875

10158401015840

4(120585)

+2120583

1205852

3

sum

119898=1

1205721119898119875119898 (120585) minus

2120583

120585

3

sum

119898=1

12057211198981199011015840

119872(120585)]

times Θ1015840(120583120579)119885

1015840(120573120577) 119890

119894120596119905

(24)

119863119901119894

119903=

11989015120574120583

1205851198754 (120585)

+[[

[

3

sum

119898=1

(119890151205741205721119898

+ 119890151205722119898

+ 12057611radic119888119901119894

11

12057633

1205723119898)

times1199011015840

119872(120585)

]]

]

Θ(120583120579)119885 (120573120577) 119890119894120596119905

(25)

Shock and Vibration 5

119863119901119894

120579=

119890151205741198751015840

4(120585)

+120583

120585

[[

[

3

sum

119898=1

(119890151205741205721119898

+ 119890151205722119898

+ 12057611radic119888119901119894

11

12057633

1205723119898)

times119875119898 (120585)

]]

]

Θ1015840(120583120579)119885 (120573120577) 119890

119894120596119905

(26)

119863119901119894

119911= [

3

sum

119898=1

(1198903112057211198981198962

119898+ 119890331205741205722119898

minus radic119888119901119894

11120576331205741205723119898)119875119898 (120585)]

times Θ (120583120579)1198851015840(120573120577) 119890

119894120596119905

(27)

22 Boundary Conditions The piezoelectric panel has 8boundary conditions consist of 6 mechanical and 2 electricalones

By considering generalized simply support boundaryconditions at 120579119894 = 0 and 120579

119894= 120572 and (119894 = 119875119894) we will have

119908119894= 119906119894

119903= 0 120590

119894

120579= 0 (119894 = 119875119894) (28)

Note that for piezoelectric layers the following conditionis added

120601 = 0 (29)

One can take

119862119894

1= 0 119862

119894

2= 1 120583 =

(2119898 + 1) 120587

2120572 119898 = 0 1 2

(30)

And by considering generalized simply support boundaryconditions at 120577119894 = 0 and 120577

119894= 1 (119894 = 119875119894) we will have

119906119894

119903= 119906119894

120579= 0 120590

119894

119911= 0 (119894 = 119875119894) (31)

And for piezoelectric layers the following condition isadded

119863119911= 0 (32)

One can take

119862119894

3= 0 119862

119894

4= 1 120573 = 119899120587 119899 = 0 1 2 (33)

Without loss of generality we suppose that external forceacts on the outer surface of the actuator and inner surface ofsensor has free boundary condition So we have

120590119875119894

119903= 119875 120591

119875119894

119903120579= 120591119875119894

119903119911= 0 120601 = 119868 at 119903 = 119903

4

120590119875119894

119903= 120591119875119894

119903120579= 120591119875119894

119903119911= 0 120601 = 0 at 119903 = 119903

1

(34)

For obtaining steady state frequency response of thecylindrical panel under a harmonic external excitation wemust solve the following matrix equation

[119879]119898times119899119883119899times1 = 119865119898times1 (35)

where [119879]119898times119899

is the coefficient matrix Consider

119883119899times1 = [11986011198611119860211986121198603119861311986041198614] (36)

and 119860119894 119861119894 119894 = 1 2 3 4 are the unknown constants that are

in (19)ndash(27)The vector 119865

119898times1denotes the force vector that acts on

the structure This force consists of the surface force that isconsidered as disturbance and has the breed of mechanicalforce such as wind effect The effect of controller unit inthe dynamic response of the piezo-panel is considered as anexternal electrical potential applied on the upper surface ofthe panel These two external forces acted on the structureindependently however summation of their effects on thewhole structure is the same as the case that both of them acton the structure simultaneously So

119865119899times2 = [1198651 1198652]

1198651119899times1

= [0 0 0 0 0 0 119868 (1199030 120579 119911 120596) 0]

119879

1198652119899times1

= [119875 (1199030 120579 119911 120596) 0 0 0 0 0 0 0]

119879

(37)

where 119875(1199030 120579 119911 120596) acting over the area (119871

119902le 119909 le 119871

119902+119886119902) on

its top surface while it is traction-free at the bottom surfaceThus

119875 (119903 120579 119911 120596) =

infin

sum

119899=minusinfin

infin

sum

119898=0

119901119899119898 (119903 120596) Sin(

119898120587119911

119897) 119890119894(119899120579+120596119905)

119868 (119903 120579 119911 120596) =

infin

sum

119899=minusinfin

infin

sum

119898=0

120580119899119898 (119903 120596) Sin(

119898120587119911

119897) 119890119894(119899120579+120596119905)

(38)

where

119901119899119898 (119903 120596) =

119860119899119898 (120596) 119869119899 (119870119903) 119870

2gt 0

119860119899119898 (120596) 119903

119899 119870

2= 0

119860119899119898 (120596) 119868119899 (119870119903) 119870

2= minus1198702

lt 0

120580119899119898 (119903 120596) =

119861119899119898 (120596) 119869119899 (119870119903) 119870

2gt 0

119861119899119898 (120596) 119903

119899 119870

2= 0

119861119899119898 (120596) 119868119899 (119870119903) 119870

2= minus1198702

lt 0

(39)

in which 119870 = radic1198962 minus (120587119898119871)2 and 119869

119899and 119868

119899denote the

standard and modified cylindrical Bessel functions of firstkind respectively and 119860

119899119898(120596) and 119861

119899119898(120596) are the amplitude

of the applied forces Substituting (21) (25) and (26) into themechanical condition (39) and substituting (27) or (19) intothe electric condition (40) yields homogeneous equationswith respect to coefficients 119860

119898and 119861

119898 (119898 = 1 2 3 4)

After finding these unknown constants that are functions of

6 Shock and Vibration

EPOT+4658e + 07

+3881e + 07

+3105e + 07

+2329e + 07

+1553e + 07

+7763e + 06

minus3930e + 02

minus7763e + 06

minus1553e + 07

minus2329e + 07

minus3105e + 07

(a) First mode shape

EPOT+1260e + 07

+1050e + 07

+8402e + 06

+6301e + 06

+4201e + 06

+2100e + 06

+0000e + 00

minus2100e + 06

minus4201e + 06

minus6301e + 06

minus8402e + 06

(b) Second mode shape

EPOT+4049e + 06

+3374e + 06

+2699e + 06

+2025e + 06

+1350e + 06

+6754e + 05

+7571e + 02

minus6739e + 05

minus1349e + 06

minus2023e + 06

minus2698e + 06

(c) Third mode shape

EPOT+7004e + 07

+5837e + 07

+4670e + 07

+3502e + 07

+2335e + 07

+1167e + 07

minus6000e + 00

minus1167e + 07

minus2335e + 07

minus3502e + 07

minus4670e + 07

(d) Forth mode shape

EPOT+4064e + 08

+3725e + 08

+3387e + 08

+3048e + 08

+2709e + 08

+2371e + 08

+2032e + 08

+1693e + 08

+1355e + 08

+1016e + 08

+6773e + 07

(e) Fifth mode shape

Figure 2 Mode shapes of the five first natural frequencies

119898 119899 by replacing them in the displacement and stress andelectric displacement of corresponding equations (19)ndash(27)all of the system variables will be determined easily Howeverfor control purposes the voltage obtained from the piezolayeras a sensor is the measured output and it is calculated as

119902 = intArea

119863 sdot 119889119860Area (40)

where 119863 = 119863119903119903 + 119863

120579120579 + 119863

119911 is the electric displacement

vector in the principle cylindrical coordinates Area in theintegration stands for the place that the sensor layer is activeand voltage (control output) is measured and 119889119860Area = (119889119911 times

119889120579)119903 which simplifies the above equation as

119902119903= int

120579119904

2

120579119904

1

int

119911119904

2

119911119904

1

119863119903119889119911 119889120579 (41)

Moreover by considering the piezoelectric sensor layer asan electric capacity 119881 = 119902119888

119875119878 one can obtain

119881 =1

119888119875119878

int

120579119904

2

120579119904

1

int

119911119904

2

119911119904

1

119863119903119889119911 119889120579 (42)

where 119888119875119878

is the capacitance of the piezoelectric sensor

Table 1 First three nondimensional natural frequencies

119878 120583 = 18 120583 = 09

01 09366 18562 23634 05178 14580 1816002 08266 17995 23514 05109 14144 1821403 07214 17650 23616 05043 13565 1843404 06271 17193 23889 04975 12960 1866605 05408 16549 24273 04906 12385 18741

3 Results and Discussion

Table 1 shows the first three nondimensional natural frequen-cies of some panels by different geometries Mode shapes ofthe five first natural frequencies are shown in Figure 2 Thepanel dynamic responses under the aforementioned inputs(dynamic excitation and electric excitation) are shown inFigure 3 and are compared by FEM results

It is obvious that a good accommodation exist betweenanalytical solution and FEM (ABAQUS)method In additionthe dynamic response of the panel in 450Hz is shown in

Shock and Vibration 7

0 50 100 150 200 250 300 350 400 450

Frequency (Hz)

Am

plitu

de (d

B)

10minus8

10minus7

10minus6

10minus5

10minus4

AnalyticalFEM

(a)

102

104

106

105

108

107

103

Am

plitu

de (d

B)

0 50 100 150 200 250 300 350 400 450 500

Frequency (Hz)

FEMAnalytical

101

(b)

Figure 3 Panel dynamic response (a) Mechanical excitation (b) electrical excitation

+1413e minus 07

+1296e minus 07

+1178e minus 07

+1060e minus 07

+9422e minus 08

+8245e minus 08

+7067e minus 08

+5889e minus 08

+4711e minus 08

+3533e minus 08

+2356e minus 08

+1178e minus 08

U magnitude

(a)

+5119e + 00

+4266e + 00

+3413e + 00

+2560e + 00

+1706e + 00

+8532e minus 01

+3576e minus 07

minus8532e minus 01

minus1706e + 00

minus2560e + 00

minus3413e + 00

minus4266e + 00

EPOT

(b)

Figure 4 The dynamic response of the panel at 450Hz due to (a) mechanical excitation (b) electrical excitation

Figure 4 It can be seen that the dominant mode shape in thisfrequency is the third mode shape

4 Conclusion

Based on the general solution of the coupled equationsfor a piezoelectric media the displacement functions areexpanded in terms of trigonometric functions in 119911 and 120579

directions Three-dimensional exact solutions for the freevibration of a piezoelectric circular cylindrical panel are thenobtained under several boundary conditions Also the forcedvibration is solved The natural frequencies are comparedwith previous works The dynamic responses with mechan-ical and electrical excitation are validated with FEM and themode shapes are shown

Conflict of Interests

The authors declare that there is no conflict of interestsregarding the publication of this paper

References

[1] J F Haskins and J L Walsh ldquoVibrations of ferroelectriccylindrical shells with transverse isotropy I Radially polarizedcaserdquo The Journal of the Acoustical Society of America vol 29no 6 pp 729ndash734 1975

[2] G E Martin ldquoVibrations of longitudinally polarized ferroelec-tric cylindrical tubesrdquo The Journal of the Acoustical Society ofAmerica vol 35 no 4 pp 510ndash520 1963

[3] D S Drumheller and A Kalnins ldquoDynamic shell theory forferroelectric ceramics rdquo The Journal of the Acoustical Society ofAmerica vol 47 no 5 pp 1343ndash1353 1970

[4] J A Burt ldquoThe electroacoustic sensitivity of radially polarizedceramic cylinders as a function of frequencyrdquoThe Journal of theAcoustical Society of America vol 64 no 6 pp 1640ndash1644 1978

[5] H S Tzou and J P Zhong ldquoA linear theory of piezoelastic shellvibrationsrdquo Journal of Sound and Vibration vol 175 no 1 pp77ndash88 1994

[6] DD Ebenezer andPAbraham ldquoEigenfunction analysis of radi-ally polarized piezoelectric cylindrical shells of finite lengthrdquoThe Journal of the Acoustical Society of America vol 102 no 3pp 1549ndash1558 1997

8 Shock and Vibration

[7] C V Stephenson ldquoRadial vibrations in short hollow cylindersof barium titanaterdquo The Journal of the Acoustical Society ofAmerica vol 28 no 1 pp 51ndash56 1956

[8] C V Stephenson ldquoHigher modes of radial vibrations in shorthollow cylinders of barium titanaterdquoThe Journal of the Acousti-cal Society of America vol 28 no 5 pp 928ndash929 1956

[9] N T Adelman Y Stavsky and E Segal ldquoAxisymmetric vibra-tions of radially polarized piezoelectric ceramic cylindersrdquoJournal of Sound and Vibration vol 38 no 2 pp 245ndash254 1975

[10] N T Adelman Y Stavsky and E Segal ldquoRadial vibrations ofaxially polarized piezoelectric ceramic cylindersrdquo The Journalof the Acoustical Society of America vol 57 no 2 pp 356ndash3601975

[11] H S Paul ldquoVibrations of circular cylindrical shells of piezoelec-tric silver iodide crystalsrdquo The Journal of the Acoustical Societyof America vol 40 no 5 pp 1077ndash1080 1966

[12] H S Paul and M Venkatesan ldquoVibrations of a hollow circularcylinder of piezoelectric ceramicsrdquoThe Journal of the AcousticalSociety of America vol 82 no 3 pp 952ndash956 1987

[13] H-JDingW-QChen Y-MGuo andQ-DYang ldquoFree vibra-tions of piezoelectric cylindrical shells filled with compressiblefluidrdquo International Journal of Solids and Structures vol 34 no16 pp 2025ndash2034 1997

[14] Z Yang J Yang Y Hu and Q-M Wang ldquoVibration charac-teristics of a circular cylindrical panel piezoelectric transducerrdquoIEEE Transactions on Ultrasonics Ferroelectrics and FrequencyControl vol 55 no 10 pp 2327ndash2335 2008

[15] S Li J Qiu H Ji K Zhu and J Li ldquoPiezoelectric vibration con-trol for all-clamped panel using DOB-based optimal controlrdquoMechatronics vol 21 no 7 pp 1213ndash1221 2011

[16] N Kumar and S P Singh ldquoVibration control of curved panelusing smart dampingrdquo Mechanical Systems and Signal Process-ing vol 30 pp 232ndash247 2012

[17] H J Ding B Chen and J Liang ldquoGeneral solutions for coupledequations for piezoelectric mediardquo International Journal ofSolids and Structures vol 33 no 16 pp 2283ndash2298 1996

[18] H J Ding R Q Xu and W Q Chen ldquoFree vibration oftransversely isotropic piezoelectric circular cylindrical panelsrdquoInternational Journal of Mechanical Sciences vol 44 no 1 pp191ndash206 2002

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

4 Shock and Vibration

where1205721119898

= minus ([(119888119875119894

13+ 11988844) (120576111198962

119898+ 120576331205742)

+ (11989015+ 11989031) (119890151198962

119898+ 119890331205742) ] 120574)

times (119888119875119894

1112057633)minus1

1205722119898

= ([(119888119875119894

111198962

119898+ 119888119875119894

441205742minus 119888119875119894

11Ω2

1) (120576111198962

119898+ 120576331205742)

+(11989015+ 11989031)21198962

1198981205742])

times (119888119875119894

1112057633)minus1

1205723119898

= ([ (119888119875119894

111198962

119898+ 119888119875119894

441205742minus 119888119875119894

11Ω2

1) (119890151198962

119898+ 119890331205742)

minus(119888119875119894

13+ 119888119875119894

44) (11989015+ 11989031)2

1198962

1198981205742])

times (119888119875119894

11radic119888119875119894

1112057633)

minus1

(119898 = 1 2 3)

(18)

Utilizing the constitutive relations of piezoelectricity and(14)ndash(17) the stress components and electric displacementcomponents can be derived as

120590119901119894

119903=

(119888119901119894

12minus 119888119901119894

11) [

120583

1205851198751015840

4(120585) minus

120583

12058521198754 (120585)]

+ (119888119901119894

12minus 119888119901119894

11)

3

sum

119898=1

120572111989811987510158401015840

119898(120585)

+

3

sum

119898=1

(1198881212057211198981198962

119898+ 119888119901119894

131205741205722119898

+ 11989031radic119888119901119894

11

12057633

1205741205723119898)

times119875119898 (120585)

Θ(120583120579)1198851015840(120573120577) 119890

119894120596119905

(19)

120590119901119894

120579=

(119888119901119894

11minus 119888119901119894

12) [

120583

1205851198751015840

4(120585) minus

120583

12058521198754 (120585)]

+ (119888119901119894

11minus 119888119901119894

12)

3

sum

119898=1

120572111989811987510158401015840

119898(120585)

+

3

sum

119898=1

(119888119901119894

1112057211198981198962

119898+ 119888119901119894

131205741205722119898

+ 11989031radic119888119901119894

11

12057633

1205741205723119898)

times119875119898 (120585)

Θ(120583120579)1198851015840(120573120577) 119890

119894120596119905

(20)

120590119901119894

119911=

3

sum

119898=1

[[

[

(119888119901119894

1312057211198981198962

119898+ 119888119901119894

331205741205722119898

+ 11989033radic119888119901119894

11

12057633

1205741205723119898)

times 119875119898 (120585)

]]

]

Θ (120583120579)1198851015840(120573120577) 119890

119894120596119905

(21)

120591119901119894

120579119911=

119888119901119894

441205741198751015840

4(120585)

+120583

120585

3

sum

119898=1

[[

[

(119888119901119894

441205741205721119898

+ 119888119901119894

441205722119898

+ 11989015radic119888119901119894

11

12057633

1205723119898)

times119875119898 (120585)

]]

]

Θ1015840(120583120579)119885 (120573120577) 119890

119894120596119905

(22)

120591119901119894

119903119911=

119888119901119894

44120574120583

120585119876 (120585)

+[[

[

3

sum

119898=1

(119888119901119894

441205741205721119898

+ 119888119901119894

441205722119898

+ 11989015radic119888119901119894

11

12057633

1205723119898)

times1199011015840

119872(120585)

]]

]

Θ(120583120579)119885 (120573120577) 119890119894120596119905

(23)

120591119901119894

119903120579= 11988866[ minus 1198962

41198754 (120585) minus 2119875

10158401015840

4(120585)

+2120583

1205852

3

sum

119898=1

1205721119898119875119898 (120585) minus

2120583

120585

3

sum

119898=1

12057211198981199011015840

119872(120585)]

times Θ1015840(120583120579)119885

1015840(120573120577) 119890

119894120596119905

(24)

119863119901119894

119903=

11989015120574120583

1205851198754 (120585)

+[[

[

3

sum

119898=1

(119890151205741205721119898

+ 119890151205722119898

+ 12057611radic119888119901119894

11

12057633

1205723119898)

times1199011015840

119872(120585)

]]

]

Θ(120583120579)119885 (120573120577) 119890119894120596119905

(25)

Shock and Vibration 5

119863119901119894

120579=

119890151205741198751015840

4(120585)

+120583

120585

[[

[

3

sum

119898=1

(119890151205741205721119898

+ 119890151205722119898

+ 12057611radic119888119901119894

11

12057633

1205723119898)

times119875119898 (120585)

]]

]

Θ1015840(120583120579)119885 (120573120577) 119890

119894120596119905

(26)

119863119901119894

119911= [

3

sum

119898=1

(1198903112057211198981198962

119898+ 119890331205741205722119898

minus radic119888119901119894

11120576331205741205723119898)119875119898 (120585)]

times Θ (120583120579)1198851015840(120573120577) 119890

119894120596119905

(27)

22 Boundary Conditions The piezoelectric panel has 8boundary conditions consist of 6 mechanical and 2 electricalones

By considering generalized simply support boundaryconditions at 120579119894 = 0 and 120579

119894= 120572 and (119894 = 119875119894) we will have

119908119894= 119906119894

119903= 0 120590

119894

120579= 0 (119894 = 119875119894) (28)

Note that for piezoelectric layers the following conditionis added

120601 = 0 (29)

One can take

119862119894

1= 0 119862

119894

2= 1 120583 =

(2119898 + 1) 120587

2120572 119898 = 0 1 2

(30)

And by considering generalized simply support boundaryconditions at 120577119894 = 0 and 120577

119894= 1 (119894 = 119875119894) we will have

119906119894

119903= 119906119894

120579= 0 120590

119894

119911= 0 (119894 = 119875119894) (31)

And for piezoelectric layers the following condition isadded

119863119911= 0 (32)

One can take

119862119894

3= 0 119862

119894

4= 1 120573 = 119899120587 119899 = 0 1 2 (33)

Without loss of generality we suppose that external forceacts on the outer surface of the actuator and inner surface ofsensor has free boundary condition So we have

120590119875119894

119903= 119875 120591

119875119894

119903120579= 120591119875119894

119903119911= 0 120601 = 119868 at 119903 = 119903

4

120590119875119894

119903= 120591119875119894

119903120579= 120591119875119894

119903119911= 0 120601 = 0 at 119903 = 119903

1

(34)

For obtaining steady state frequency response of thecylindrical panel under a harmonic external excitation wemust solve the following matrix equation

[119879]119898times119899119883119899times1 = 119865119898times1 (35)

where [119879]119898times119899

is the coefficient matrix Consider

119883119899times1 = [11986011198611119860211986121198603119861311986041198614] (36)

and 119860119894 119861119894 119894 = 1 2 3 4 are the unknown constants that are

in (19)ndash(27)The vector 119865

119898times1denotes the force vector that acts on

the structure This force consists of the surface force that isconsidered as disturbance and has the breed of mechanicalforce such as wind effect The effect of controller unit inthe dynamic response of the piezo-panel is considered as anexternal electrical potential applied on the upper surface ofthe panel These two external forces acted on the structureindependently however summation of their effects on thewhole structure is the same as the case that both of them acton the structure simultaneously So

119865119899times2 = [1198651 1198652]

1198651119899times1

= [0 0 0 0 0 0 119868 (1199030 120579 119911 120596) 0]

119879

1198652119899times1

= [119875 (1199030 120579 119911 120596) 0 0 0 0 0 0 0]

119879

(37)

where 119875(1199030 120579 119911 120596) acting over the area (119871

119902le 119909 le 119871

119902+119886119902) on

its top surface while it is traction-free at the bottom surfaceThus

119875 (119903 120579 119911 120596) =

infin

sum

119899=minusinfin

infin

sum

119898=0

119901119899119898 (119903 120596) Sin(

119898120587119911

119897) 119890119894(119899120579+120596119905)

119868 (119903 120579 119911 120596) =

infin

sum

119899=minusinfin

infin

sum

119898=0

120580119899119898 (119903 120596) Sin(

119898120587119911

119897) 119890119894(119899120579+120596119905)

(38)

where

119901119899119898 (119903 120596) =

119860119899119898 (120596) 119869119899 (119870119903) 119870

2gt 0

119860119899119898 (120596) 119903

119899 119870

2= 0

119860119899119898 (120596) 119868119899 (119870119903) 119870

2= minus1198702

lt 0

120580119899119898 (119903 120596) =

119861119899119898 (120596) 119869119899 (119870119903) 119870

2gt 0

119861119899119898 (120596) 119903

119899 119870

2= 0

119861119899119898 (120596) 119868119899 (119870119903) 119870

2= minus1198702

lt 0

(39)

in which 119870 = radic1198962 minus (120587119898119871)2 and 119869

119899and 119868

119899denote the

standard and modified cylindrical Bessel functions of firstkind respectively and 119860

119899119898(120596) and 119861

119899119898(120596) are the amplitude

of the applied forces Substituting (21) (25) and (26) into themechanical condition (39) and substituting (27) or (19) intothe electric condition (40) yields homogeneous equationswith respect to coefficients 119860

119898and 119861

119898 (119898 = 1 2 3 4)

After finding these unknown constants that are functions of

6 Shock and Vibration

EPOT+4658e + 07

+3881e + 07

+3105e + 07

+2329e + 07

+1553e + 07

+7763e + 06

minus3930e + 02

minus7763e + 06

minus1553e + 07

minus2329e + 07

minus3105e + 07

(a) First mode shape

EPOT+1260e + 07

+1050e + 07

+8402e + 06

+6301e + 06

+4201e + 06

+2100e + 06

+0000e + 00

minus2100e + 06

minus4201e + 06

minus6301e + 06

minus8402e + 06

(b) Second mode shape

EPOT+4049e + 06

+3374e + 06

+2699e + 06

+2025e + 06

+1350e + 06

+6754e + 05

+7571e + 02

minus6739e + 05

minus1349e + 06

minus2023e + 06

minus2698e + 06

(c) Third mode shape

EPOT+7004e + 07

+5837e + 07

+4670e + 07

+3502e + 07

+2335e + 07

+1167e + 07

minus6000e + 00

minus1167e + 07

minus2335e + 07

minus3502e + 07

minus4670e + 07

(d) Forth mode shape

EPOT+4064e + 08

+3725e + 08

+3387e + 08

+3048e + 08

+2709e + 08

+2371e + 08

+2032e + 08

+1693e + 08

+1355e + 08

+1016e + 08

+6773e + 07

(e) Fifth mode shape

Figure 2 Mode shapes of the five first natural frequencies

119898 119899 by replacing them in the displacement and stress andelectric displacement of corresponding equations (19)ndash(27)all of the system variables will be determined easily Howeverfor control purposes the voltage obtained from the piezolayeras a sensor is the measured output and it is calculated as

119902 = intArea

119863 sdot 119889119860Area (40)

where 119863 = 119863119903119903 + 119863

120579120579 + 119863

119911 is the electric displacement

vector in the principle cylindrical coordinates Area in theintegration stands for the place that the sensor layer is activeand voltage (control output) is measured and 119889119860Area = (119889119911 times

119889120579)119903 which simplifies the above equation as

119902119903= int

120579119904

2

120579119904

1

int

119911119904

2

119911119904

1

119863119903119889119911 119889120579 (41)

Moreover by considering the piezoelectric sensor layer asan electric capacity 119881 = 119902119888

119875119878 one can obtain

119881 =1

119888119875119878

int

120579119904

2

120579119904

1

int

119911119904

2

119911119904

1

119863119903119889119911 119889120579 (42)

where 119888119875119878

is the capacitance of the piezoelectric sensor

Table 1 First three nondimensional natural frequencies

119878 120583 = 18 120583 = 09

01 09366 18562 23634 05178 14580 1816002 08266 17995 23514 05109 14144 1821403 07214 17650 23616 05043 13565 1843404 06271 17193 23889 04975 12960 1866605 05408 16549 24273 04906 12385 18741

3 Results and Discussion

Table 1 shows the first three nondimensional natural frequen-cies of some panels by different geometries Mode shapes ofthe five first natural frequencies are shown in Figure 2 Thepanel dynamic responses under the aforementioned inputs(dynamic excitation and electric excitation) are shown inFigure 3 and are compared by FEM results

It is obvious that a good accommodation exist betweenanalytical solution and FEM (ABAQUS)method In additionthe dynamic response of the panel in 450Hz is shown in

Shock and Vibration 7

0 50 100 150 200 250 300 350 400 450

Frequency (Hz)

Am

plitu

de (d

B)

10minus8

10minus7

10minus6

10minus5

10minus4

AnalyticalFEM

(a)

102

104

106

105

108

107

103

Am

plitu

de (d

B)

0 50 100 150 200 250 300 350 400 450 500

Frequency (Hz)

FEMAnalytical

101

(b)

Figure 3 Panel dynamic response (a) Mechanical excitation (b) electrical excitation

+1413e minus 07

+1296e minus 07

+1178e minus 07

+1060e minus 07

+9422e minus 08

+8245e minus 08

+7067e minus 08

+5889e minus 08

+4711e minus 08

+3533e minus 08

+2356e minus 08

+1178e minus 08

U magnitude

(a)

+5119e + 00

+4266e + 00

+3413e + 00

+2560e + 00

+1706e + 00

+8532e minus 01

+3576e minus 07

minus8532e minus 01

minus1706e + 00

minus2560e + 00

minus3413e + 00

minus4266e + 00

EPOT

(b)

Figure 4 The dynamic response of the panel at 450Hz due to (a) mechanical excitation (b) electrical excitation

Figure 4 It can be seen that the dominant mode shape in thisfrequency is the third mode shape

4 Conclusion

Based on the general solution of the coupled equationsfor a piezoelectric media the displacement functions areexpanded in terms of trigonometric functions in 119911 and 120579

directions Three-dimensional exact solutions for the freevibration of a piezoelectric circular cylindrical panel are thenobtained under several boundary conditions Also the forcedvibration is solved The natural frequencies are comparedwith previous works The dynamic responses with mechan-ical and electrical excitation are validated with FEM and themode shapes are shown

Conflict of Interests

The authors declare that there is no conflict of interestsregarding the publication of this paper

References

[1] J F Haskins and J L Walsh ldquoVibrations of ferroelectriccylindrical shells with transverse isotropy I Radially polarizedcaserdquo The Journal of the Acoustical Society of America vol 29no 6 pp 729ndash734 1975

[2] G E Martin ldquoVibrations of longitudinally polarized ferroelec-tric cylindrical tubesrdquo The Journal of the Acoustical Society ofAmerica vol 35 no 4 pp 510ndash520 1963

[3] D S Drumheller and A Kalnins ldquoDynamic shell theory forferroelectric ceramics rdquo The Journal of the Acoustical Society ofAmerica vol 47 no 5 pp 1343ndash1353 1970

[4] J A Burt ldquoThe electroacoustic sensitivity of radially polarizedceramic cylinders as a function of frequencyrdquoThe Journal of theAcoustical Society of America vol 64 no 6 pp 1640ndash1644 1978

[5] H S Tzou and J P Zhong ldquoA linear theory of piezoelastic shellvibrationsrdquo Journal of Sound and Vibration vol 175 no 1 pp77ndash88 1994

[6] DD Ebenezer andPAbraham ldquoEigenfunction analysis of radi-ally polarized piezoelectric cylindrical shells of finite lengthrdquoThe Journal of the Acoustical Society of America vol 102 no 3pp 1549ndash1558 1997

8 Shock and Vibration

[7] C V Stephenson ldquoRadial vibrations in short hollow cylindersof barium titanaterdquo The Journal of the Acoustical Society ofAmerica vol 28 no 1 pp 51ndash56 1956

[8] C V Stephenson ldquoHigher modes of radial vibrations in shorthollow cylinders of barium titanaterdquoThe Journal of the Acousti-cal Society of America vol 28 no 5 pp 928ndash929 1956

[9] N T Adelman Y Stavsky and E Segal ldquoAxisymmetric vibra-tions of radially polarized piezoelectric ceramic cylindersrdquoJournal of Sound and Vibration vol 38 no 2 pp 245ndash254 1975

[10] N T Adelman Y Stavsky and E Segal ldquoRadial vibrations ofaxially polarized piezoelectric ceramic cylindersrdquo The Journalof the Acoustical Society of America vol 57 no 2 pp 356ndash3601975

[11] H S Paul ldquoVibrations of circular cylindrical shells of piezoelec-tric silver iodide crystalsrdquo The Journal of the Acoustical Societyof America vol 40 no 5 pp 1077ndash1080 1966

[12] H S Paul and M Venkatesan ldquoVibrations of a hollow circularcylinder of piezoelectric ceramicsrdquoThe Journal of the AcousticalSociety of America vol 82 no 3 pp 952ndash956 1987

[13] H-JDingW-QChen Y-MGuo andQ-DYang ldquoFree vibra-tions of piezoelectric cylindrical shells filled with compressiblefluidrdquo International Journal of Solids and Structures vol 34 no16 pp 2025ndash2034 1997

[14] Z Yang J Yang Y Hu and Q-M Wang ldquoVibration charac-teristics of a circular cylindrical panel piezoelectric transducerrdquoIEEE Transactions on Ultrasonics Ferroelectrics and FrequencyControl vol 55 no 10 pp 2327ndash2335 2008

[15] S Li J Qiu H Ji K Zhu and J Li ldquoPiezoelectric vibration con-trol for all-clamped panel using DOB-based optimal controlrdquoMechatronics vol 21 no 7 pp 1213ndash1221 2011

[16] N Kumar and S P Singh ldquoVibration control of curved panelusing smart dampingrdquo Mechanical Systems and Signal Process-ing vol 30 pp 232ndash247 2012

[17] H J Ding B Chen and J Liang ldquoGeneral solutions for coupledequations for piezoelectric mediardquo International Journal ofSolids and Structures vol 33 no 16 pp 2283ndash2298 1996

[18] H J Ding R Q Xu and W Q Chen ldquoFree vibration oftransversely isotropic piezoelectric circular cylindrical panelsrdquoInternational Journal of Mechanical Sciences vol 44 no 1 pp191ndash206 2002

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 5

119863119901119894

120579=

119890151205741198751015840

4(120585)

+120583

120585

[[

[

3

sum

119898=1

(119890151205741205721119898

+ 119890151205722119898

+ 12057611radic119888119901119894

11

12057633

1205723119898)

times119875119898 (120585)

]]

]

Θ1015840(120583120579)119885 (120573120577) 119890

119894120596119905

(26)

119863119901119894

119911= [

3

sum

119898=1

(1198903112057211198981198962

119898+ 119890331205741205722119898

minus radic119888119901119894

11120576331205741205723119898)119875119898 (120585)]

times Θ (120583120579)1198851015840(120573120577) 119890

119894120596119905

(27)

22 Boundary Conditions The piezoelectric panel has 8boundary conditions consist of 6 mechanical and 2 electricalones

By considering generalized simply support boundaryconditions at 120579119894 = 0 and 120579

119894= 120572 and (119894 = 119875119894) we will have

119908119894= 119906119894

119903= 0 120590

119894

120579= 0 (119894 = 119875119894) (28)

Note that for piezoelectric layers the following conditionis added

120601 = 0 (29)

One can take

119862119894

1= 0 119862

119894

2= 1 120583 =

(2119898 + 1) 120587

2120572 119898 = 0 1 2

(30)

And by considering generalized simply support boundaryconditions at 120577119894 = 0 and 120577

119894= 1 (119894 = 119875119894) we will have

119906119894

119903= 119906119894

120579= 0 120590

119894

119911= 0 (119894 = 119875119894) (31)

And for piezoelectric layers the following condition isadded

119863119911= 0 (32)

One can take

119862119894

3= 0 119862

119894

4= 1 120573 = 119899120587 119899 = 0 1 2 (33)

Without loss of generality we suppose that external forceacts on the outer surface of the actuator and inner surface ofsensor has free boundary condition So we have

120590119875119894

119903= 119875 120591

119875119894

119903120579= 120591119875119894

119903119911= 0 120601 = 119868 at 119903 = 119903

4

120590119875119894

119903= 120591119875119894

119903120579= 120591119875119894

119903119911= 0 120601 = 0 at 119903 = 119903

1

(34)

For obtaining steady state frequency response of thecylindrical panel under a harmonic external excitation wemust solve the following matrix equation

[119879]119898times119899119883119899times1 = 119865119898times1 (35)

where [119879]119898times119899

is the coefficient matrix Consider

119883119899times1 = [11986011198611119860211986121198603119861311986041198614] (36)

and 119860119894 119861119894 119894 = 1 2 3 4 are the unknown constants that are

in (19)ndash(27)The vector 119865

119898times1denotes the force vector that acts on

the structure This force consists of the surface force that isconsidered as disturbance and has the breed of mechanicalforce such as wind effect The effect of controller unit inthe dynamic response of the piezo-panel is considered as anexternal electrical potential applied on the upper surface ofthe panel These two external forces acted on the structureindependently however summation of their effects on thewhole structure is the same as the case that both of them acton the structure simultaneously So

119865119899times2 = [1198651 1198652]

1198651119899times1

= [0 0 0 0 0 0 119868 (1199030 120579 119911 120596) 0]

119879

1198652119899times1

= [119875 (1199030 120579 119911 120596) 0 0 0 0 0 0 0]

119879

(37)

where 119875(1199030 120579 119911 120596) acting over the area (119871

119902le 119909 le 119871

119902+119886119902) on

its top surface while it is traction-free at the bottom surfaceThus

119875 (119903 120579 119911 120596) =

infin

sum

119899=minusinfin

infin

sum

119898=0

119901119899119898 (119903 120596) Sin(

119898120587119911

119897) 119890119894(119899120579+120596119905)

119868 (119903 120579 119911 120596) =

infin

sum

119899=minusinfin

infin

sum

119898=0

120580119899119898 (119903 120596) Sin(

119898120587119911

119897) 119890119894(119899120579+120596119905)

(38)

where

119901119899119898 (119903 120596) =

119860119899119898 (120596) 119869119899 (119870119903) 119870

2gt 0

119860119899119898 (120596) 119903

119899 119870

2= 0

119860119899119898 (120596) 119868119899 (119870119903) 119870

2= minus1198702

lt 0

120580119899119898 (119903 120596) =

119861119899119898 (120596) 119869119899 (119870119903) 119870

2gt 0

119861119899119898 (120596) 119903

119899 119870

2= 0

119861119899119898 (120596) 119868119899 (119870119903) 119870

2= minus1198702

lt 0

(39)

in which 119870 = radic1198962 minus (120587119898119871)2 and 119869

119899and 119868

119899denote the

standard and modified cylindrical Bessel functions of firstkind respectively and 119860

119899119898(120596) and 119861

119899119898(120596) are the amplitude

of the applied forces Substituting (21) (25) and (26) into themechanical condition (39) and substituting (27) or (19) intothe electric condition (40) yields homogeneous equationswith respect to coefficients 119860

119898and 119861

119898 (119898 = 1 2 3 4)

After finding these unknown constants that are functions of

6 Shock and Vibration

EPOT+4658e + 07

+3881e + 07

+3105e + 07

+2329e + 07

+1553e + 07

+7763e + 06

minus3930e + 02

minus7763e + 06

minus1553e + 07

minus2329e + 07

minus3105e + 07

(a) First mode shape

EPOT+1260e + 07

+1050e + 07

+8402e + 06

+6301e + 06

+4201e + 06

+2100e + 06

+0000e + 00

minus2100e + 06

minus4201e + 06

minus6301e + 06

minus8402e + 06

(b) Second mode shape

EPOT+4049e + 06

+3374e + 06

+2699e + 06

+2025e + 06

+1350e + 06

+6754e + 05

+7571e + 02

minus6739e + 05

minus1349e + 06

minus2023e + 06

minus2698e + 06

(c) Third mode shape

EPOT+7004e + 07

+5837e + 07

+4670e + 07

+3502e + 07

+2335e + 07

+1167e + 07

minus6000e + 00

minus1167e + 07

minus2335e + 07

minus3502e + 07

minus4670e + 07

(d) Forth mode shape

EPOT+4064e + 08

+3725e + 08

+3387e + 08

+3048e + 08

+2709e + 08

+2371e + 08

+2032e + 08

+1693e + 08

+1355e + 08

+1016e + 08

+6773e + 07

(e) Fifth mode shape

Figure 2 Mode shapes of the five first natural frequencies

119898 119899 by replacing them in the displacement and stress andelectric displacement of corresponding equations (19)ndash(27)all of the system variables will be determined easily Howeverfor control purposes the voltage obtained from the piezolayeras a sensor is the measured output and it is calculated as

119902 = intArea

119863 sdot 119889119860Area (40)

where 119863 = 119863119903119903 + 119863

120579120579 + 119863

119911 is the electric displacement

vector in the principle cylindrical coordinates Area in theintegration stands for the place that the sensor layer is activeand voltage (control output) is measured and 119889119860Area = (119889119911 times

119889120579)119903 which simplifies the above equation as

119902119903= int

120579119904

2

120579119904

1

int

119911119904

2

119911119904

1

119863119903119889119911 119889120579 (41)

Moreover by considering the piezoelectric sensor layer asan electric capacity 119881 = 119902119888

119875119878 one can obtain

119881 =1

119888119875119878

int

120579119904

2

120579119904

1

int

119911119904

2

119911119904

1

119863119903119889119911 119889120579 (42)

where 119888119875119878

is the capacitance of the piezoelectric sensor

Table 1 First three nondimensional natural frequencies

119878 120583 = 18 120583 = 09

01 09366 18562 23634 05178 14580 1816002 08266 17995 23514 05109 14144 1821403 07214 17650 23616 05043 13565 1843404 06271 17193 23889 04975 12960 1866605 05408 16549 24273 04906 12385 18741

3 Results and Discussion

Table 1 shows the first three nondimensional natural frequen-cies of some panels by different geometries Mode shapes ofthe five first natural frequencies are shown in Figure 2 Thepanel dynamic responses under the aforementioned inputs(dynamic excitation and electric excitation) are shown inFigure 3 and are compared by FEM results

It is obvious that a good accommodation exist betweenanalytical solution and FEM (ABAQUS)method In additionthe dynamic response of the panel in 450Hz is shown in

Shock and Vibration 7

0 50 100 150 200 250 300 350 400 450

Frequency (Hz)

Am

plitu

de (d

B)

10minus8

10minus7

10minus6

10minus5

10minus4

AnalyticalFEM

(a)

102

104

106

105

108

107

103

Am

plitu

de (d

B)

0 50 100 150 200 250 300 350 400 450 500

Frequency (Hz)

FEMAnalytical

101

(b)

Figure 3 Panel dynamic response (a) Mechanical excitation (b) electrical excitation

+1413e minus 07

+1296e minus 07

+1178e minus 07

+1060e minus 07

+9422e minus 08

+8245e minus 08

+7067e minus 08

+5889e minus 08

+4711e minus 08

+3533e minus 08

+2356e minus 08

+1178e minus 08

U magnitude

(a)

+5119e + 00

+4266e + 00

+3413e + 00

+2560e + 00

+1706e + 00

+8532e minus 01

+3576e minus 07

minus8532e minus 01

minus1706e + 00

minus2560e + 00

minus3413e + 00

minus4266e + 00

EPOT

(b)

Figure 4 The dynamic response of the panel at 450Hz due to (a) mechanical excitation (b) electrical excitation

Figure 4 It can be seen that the dominant mode shape in thisfrequency is the third mode shape

4 Conclusion

Based on the general solution of the coupled equationsfor a piezoelectric media the displacement functions areexpanded in terms of trigonometric functions in 119911 and 120579

directions Three-dimensional exact solutions for the freevibration of a piezoelectric circular cylindrical panel are thenobtained under several boundary conditions Also the forcedvibration is solved The natural frequencies are comparedwith previous works The dynamic responses with mechan-ical and electrical excitation are validated with FEM and themode shapes are shown

Conflict of Interests

The authors declare that there is no conflict of interestsregarding the publication of this paper

References

[1] J F Haskins and J L Walsh ldquoVibrations of ferroelectriccylindrical shells with transverse isotropy I Radially polarizedcaserdquo The Journal of the Acoustical Society of America vol 29no 6 pp 729ndash734 1975

[2] G E Martin ldquoVibrations of longitudinally polarized ferroelec-tric cylindrical tubesrdquo The Journal of the Acoustical Society ofAmerica vol 35 no 4 pp 510ndash520 1963

[3] D S Drumheller and A Kalnins ldquoDynamic shell theory forferroelectric ceramics rdquo The Journal of the Acoustical Society ofAmerica vol 47 no 5 pp 1343ndash1353 1970

[4] J A Burt ldquoThe electroacoustic sensitivity of radially polarizedceramic cylinders as a function of frequencyrdquoThe Journal of theAcoustical Society of America vol 64 no 6 pp 1640ndash1644 1978

[5] H S Tzou and J P Zhong ldquoA linear theory of piezoelastic shellvibrationsrdquo Journal of Sound and Vibration vol 175 no 1 pp77ndash88 1994

[6] DD Ebenezer andPAbraham ldquoEigenfunction analysis of radi-ally polarized piezoelectric cylindrical shells of finite lengthrdquoThe Journal of the Acoustical Society of America vol 102 no 3pp 1549ndash1558 1997

8 Shock and Vibration

[7] C V Stephenson ldquoRadial vibrations in short hollow cylindersof barium titanaterdquo The Journal of the Acoustical Society ofAmerica vol 28 no 1 pp 51ndash56 1956

[8] C V Stephenson ldquoHigher modes of radial vibrations in shorthollow cylinders of barium titanaterdquoThe Journal of the Acousti-cal Society of America vol 28 no 5 pp 928ndash929 1956

[9] N T Adelman Y Stavsky and E Segal ldquoAxisymmetric vibra-tions of radially polarized piezoelectric ceramic cylindersrdquoJournal of Sound and Vibration vol 38 no 2 pp 245ndash254 1975

[10] N T Adelman Y Stavsky and E Segal ldquoRadial vibrations ofaxially polarized piezoelectric ceramic cylindersrdquo The Journalof the Acoustical Society of America vol 57 no 2 pp 356ndash3601975

[11] H S Paul ldquoVibrations of circular cylindrical shells of piezoelec-tric silver iodide crystalsrdquo The Journal of the Acoustical Societyof America vol 40 no 5 pp 1077ndash1080 1966

[12] H S Paul and M Venkatesan ldquoVibrations of a hollow circularcylinder of piezoelectric ceramicsrdquoThe Journal of the AcousticalSociety of America vol 82 no 3 pp 952ndash956 1987

[13] H-JDingW-QChen Y-MGuo andQ-DYang ldquoFree vibra-tions of piezoelectric cylindrical shells filled with compressiblefluidrdquo International Journal of Solids and Structures vol 34 no16 pp 2025ndash2034 1997

[14] Z Yang J Yang Y Hu and Q-M Wang ldquoVibration charac-teristics of a circular cylindrical panel piezoelectric transducerrdquoIEEE Transactions on Ultrasonics Ferroelectrics and FrequencyControl vol 55 no 10 pp 2327ndash2335 2008

[15] S Li J Qiu H Ji K Zhu and J Li ldquoPiezoelectric vibration con-trol for all-clamped panel using DOB-based optimal controlrdquoMechatronics vol 21 no 7 pp 1213ndash1221 2011

[16] N Kumar and S P Singh ldquoVibration control of curved panelusing smart dampingrdquo Mechanical Systems and Signal Process-ing vol 30 pp 232ndash247 2012

[17] H J Ding B Chen and J Liang ldquoGeneral solutions for coupledequations for piezoelectric mediardquo International Journal ofSolids and Structures vol 33 no 16 pp 2283ndash2298 1996

[18] H J Ding R Q Xu and W Q Chen ldquoFree vibration oftransversely isotropic piezoelectric circular cylindrical panelsrdquoInternational Journal of Mechanical Sciences vol 44 no 1 pp191ndash206 2002

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

6 Shock and Vibration

EPOT+4658e + 07

+3881e + 07

+3105e + 07

+2329e + 07

+1553e + 07

+7763e + 06

minus3930e + 02

minus7763e + 06

minus1553e + 07

minus2329e + 07

minus3105e + 07

(a) First mode shape

EPOT+1260e + 07

+1050e + 07

+8402e + 06

+6301e + 06

+4201e + 06

+2100e + 06

+0000e + 00

minus2100e + 06

minus4201e + 06

minus6301e + 06

minus8402e + 06

(b) Second mode shape

EPOT+4049e + 06

+3374e + 06

+2699e + 06

+2025e + 06

+1350e + 06

+6754e + 05

+7571e + 02

minus6739e + 05

minus1349e + 06

minus2023e + 06

minus2698e + 06

(c) Third mode shape

EPOT+7004e + 07

+5837e + 07

+4670e + 07

+3502e + 07

+2335e + 07

+1167e + 07

minus6000e + 00

minus1167e + 07

minus2335e + 07

minus3502e + 07

minus4670e + 07

(d) Forth mode shape

EPOT+4064e + 08

+3725e + 08

+3387e + 08

+3048e + 08

+2709e + 08

+2371e + 08

+2032e + 08

+1693e + 08

+1355e + 08

+1016e + 08

+6773e + 07

(e) Fifth mode shape

Figure 2 Mode shapes of the five first natural frequencies

119898 119899 by replacing them in the displacement and stress andelectric displacement of corresponding equations (19)ndash(27)all of the system variables will be determined easily Howeverfor control purposes the voltage obtained from the piezolayeras a sensor is the measured output and it is calculated as

119902 = intArea

119863 sdot 119889119860Area (40)

where 119863 = 119863119903119903 + 119863

120579120579 + 119863

119911 is the electric displacement

vector in the principle cylindrical coordinates Area in theintegration stands for the place that the sensor layer is activeand voltage (control output) is measured and 119889119860Area = (119889119911 times

119889120579)119903 which simplifies the above equation as

119902119903= int

120579119904

2

120579119904

1

int

119911119904

2

119911119904

1

119863119903119889119911 119889120579 (41)

Moreover by considering the piezoelectric sensor layer asan electric capacity 119881 = 119902119888

119875119878 one can obtain

119881 =1

119888119875119878

int

120579119904

2

120579119904

1

int

119911119904

2

119911119904

1

119863119903119889119911 119889120579 (42)

where 119888119875119878

is the capacitance of the piezoelectric sensor

Table 1 First three nondimensional natural frequencies

119878 120583 = 18 120583 = 09

01 09366 18562 23634 05178 14580 1816002 08266 17995 23514 05109 14144 1821403 07214 17650 23616 05043 13565 1843404 06271 17193 23889 04975 12960 1866605 05408 16549 24273 04906 12385 18741

3 Results and Discussion

Table 1 shows the first three nondimensional natural frequen-cies of some panels by different geometries Mode shapes ofthe five first natural frequencies are shown in Figure 2 Thepanel dynamic responses under the aforementioned inputs(dynamic excitation and electric excitation) are shown inFigure 3 and are compared by FEM results

It is obvious that a good accommodation exist betweenanalytical solution and FEM (ABAQUS)method In additionthe dynamic response of the panel in 450Hz is shown in

Shock and Vibration 7

0 50 100 150 200 250 300 350 400 450

Frequency (Hz)

Am

plitu

de (d

B)

10minus8

10minus7

10minus6

10minus5

10minus4

AnalyticalFEM

(a)

102

104

106

105

108

107

103

Am

plitu

de (d

B)

0 50 100 150 200 250 300 350 400 450 500

Frequency (Hz)

FEMAnalytical

101

(b)

Figure 3 Panel dynamic response (a) Mechanical excitation (b) electrical excitation

+1413e minus 07

+1296e minus 07

+1178e minus 07

+1060e minus 07

+9422e minus 08

+8245e minus 08

+7067e minus 08

+5889e minus 08

+4711e minus 08

+3533e minus 08

+2356e minus 08

+1178e minus 08

U magnitude

(a)

+5119e + 00

+4266e + 00

+3413e + 00

+2560e + 00

+1706e + 00

+8532e minus 01

+3576e minus 07

minus8532e minus 01

minus1706e + 00

minus2560e + 00

minus3413e + 00

minus4266e + 00

EPOT

(b)

Figure 4 The dynamic response of the panel at 450Hz due to (a) mechanical excitation (b) electrical excitation

Figure 4 It can be seen that the dominant mode shape in thisfrequency is the third mode shape

4 Conclusion

Based on the general solution of the coupled equationsfor a piezoelectric media the displacement functions areexpanded in terms of trigonometric functions in 119911 and 120579

directions Three-dimensional exact solutions for the freevibration of a piezoelectric circular cylindrical panel are thenobtained under several boundary conditions Also the forcedvibration is solved The natural frequencies are comparedwith previous works The dynamic responses with mechan-ical and electrical excitation are validated with FEM and themode shapes are shown

Conflict of Interests

The authors declare that there is no conflict of interestsregarding the publication of this paper

References

[1] J F Haskins and J L Walsh ldquoVibrations of ferroelectriccylindrical shells with transverse isotropy I Radially polarizedcaserdquo The Journal of the Acoustical Society of America vol 29no 6 pp 729ndash734 1975

[2] G E Martin ldquoVibrations of longitudinally polarized ferroelec-tric cylindrical tubesrdquo The Journal of the Acoustical Society ofAmerica vol 35 no 4 pp 510ndash520 1963

[3] D S Drumheller and A Kalnins ldquoDynamic shell theory forferroelectric ceramics rdquo The Journal of the Acoustical Society ofAmerica vol 47 no 5 pp 1343ndash1353 1970

[4] J A Burt ldquoThe electroacoustic sensitivity of radially polarizedceramic cylinders as a function of frequencyrdquoThe Journal of theAcoustical Society of America vol 64 no 6 pp 1640ndash1644 1978

[5] H S Tzou and J P Zhong ldquoA linear theory of piezoelastic shellvibrationsrdquo Journal of Sound and Vibration vol 175 no 1 pp77ndash88 1994

[6] DD Ebenezer andPAbraham ldquoEigenfunction analysis of radi-ally polarized piezoelectric cylindrical shells of finite lengthrdquoThe Journal of the Acoustical Society of America vol 102 no 3pp 1549ndash1558 1997

8 Shock and Vibration

[7] C V Stephenson ldquoRadial vibrations in short hollow cylindersof barium titanaterdquo The Journal of the Acoustical Society ofAmerica vol 28 no 1 pp 51ndash56 1956

[8] C V Stephenson ldquoHigher modes of radial vibrations in shorthollow cylinders of barium titanaterdquoThe Journal of the Acousti-cal Society of America vol 28 no 5 pp 928ndash929 1956

[9] N T Adelman Y Stavsky and E Segal ldquoAxisymmetric vibra-tions of radially polarized piezoelectric ceramic cylindersrdquoJournal of Sound and Vibration vol 38 no 2 pp 245ndash254 1975

[10] N T Adelman Y Stavsky and E Segal ldquoRadial vibrations ofaxially polarized piezoelectric ceramic cylindersrdquo The Journalof the Acoustical Society of America vol 57 no 2 pp 356ndash3601975

[11] H S Paul ldquoVibrations of circular cylindrical shells of piezoelec-tric silver iodide crystalsrdquo The Journal of the Acoustical Societyof America vol 40 no 5 pp 1077ndash1080 1966

[12] H S Paul and M Venkatesan ldquoVibrations of a hollow circularcylinder of piezoelectric ceramicsrdquoThe Journal of the AcousticalSociety of America vol 82 no 3 pp 952ndash956 1987

[13] H-JDingW-QChen Y-MGuo andQ-DYang ldquoFree vibra-tions of piezoelectric cylindrical shells filled with compressiblefluidrdquo International Journal of Solids and Structures vol 34 no16 pp 2025ndash2034 1997

[14] Z Yang J Yang Y Hu and Q-M Wang ldquoVibration charac-teristics of a circular cylindrical panel piezoelectric transducerrdquoIEEE Transactions on Ultrasonics Ferroelectrics and FrequencyControl vol 55 no 10 pp 2327ndash2335 2008

[15] S Li J Qiu H Ji K Zhu and J Li ldquoPiezoelectric vibration con-trol for all-clamped panel using DOB-based optimal controlrdquoMechatronics vol 21 no 7 pp 1213ndash1221 2011

[16] N Kumar and S P Singh ldquoVibration control of curved panelusing smart dampingrdquo Mechanical Systems and Signal Process-ing vol 30 pp 232ndash247 2012

[17] H J Ding B Chen and J Liang ldquoGeneral solutions for coupledequations for piezoelectric mediardquo International Journal ofSolids and Structures vol 33 no 16 pp 2283ndash2298 1996

[18] H J Ding R Q Xu and W Q Chen ldquoFree vibration oftransversely isotropic piezoelectric circular cylindrical panelsrdquoInternational Journal of Mechanical Sciences vol 44 no 1 pp191ndash206 2002

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 7

0 50 100 150 200 250 300 350 400 450

Frequency (Hz)

Am

plitu

de (d

B)

10minus8

10minus7

10minus6

10minus5

10minus4

AnalyticalFEM

(a)

102

104

106

105

108

107

103

Am

plitu

de (d

B)

0 50 100 150 200 250 300 350 400 450 500

Frequency (Hz)

FEMAnalytical

101

(b)

Figure 3 Panel dynamic response (a) Mechanical excitation (b) electrical excitation

+1413e minus 07

+1296e minus 07

+1178e minus 07

+1060e minus 07

+9422e minus 08

+8245e minus 08

+7067e minus 08

+5889e minus 08

+4711e minus 08

+3533e minus 08

+2356e minus 08

+1178e minus 08

U magnitude

(a)

+5119e + 00

+4266e + 00

+3413e + 00

+2560e + 00

+1706e + 00

+8532e minus 01

+3576e minus 07

minus8532e minus 01

minus1706e + 00

minus2560e + 00

minus3413e + 00

minus4266e + 00

EPOT

(b)

Figure 4 The dynamic response of the panel at 450Hz due to (a) mechanical excitation (b) electrical excitation

Figure 4 It can be seen that the dominant mode shape in thisfrequency is the third mode shape

4 Conclusion

Based on the general solution of the coupled equationsfor a piezoelectric media the displacement functions areexpanded in terms of trigonometric functions in 119911 and 120579

directions Three-dimensional exact solutions for the freevibration of a piezoelectric circular cylindrical panel are thenobtained under several boundary conditions Also the forcedvibration is solved The natural frequencies are comparedwith previous works The dynamic responses with mechan-ical and electrical excitation are validated with FEM and themode shapes are shown

Conflict of Interests

The authors declare that there is no conflict of interestsregarding the publication of this paper

References

[1] J F Haskins and J L Walsh ldquoVibrations of ferroelectriccylindrical shells with transverse isotropy I Radially polarizedcaserdquo The Journal of the Acoustical Society of America vol 29no 6 pp 729ndash734 1975

[2] G E Martin ldquoVibrations of longitudinally polarized ferroelec-tric cylindrical tubesrdquo The Journal of the Acoustical Society ofAmerica vol 35 no 4 pp 510ndash520 1963

[3] D S Drumheller and A Kalnins ldquoDynamic shell theory forferroelectric ceramics rdquo The Journal of the Acoustical Society ofAmerica vol 47 no 5 pp 1343ndash1353 1970

[4] J A Burt ldquoThe electroacoustic sensitivity of radially polarizedceramic cylinders as a function of frequencyrdquoThe Journal of theAcoustical Society of America vol 64 no 6 pp 1640ndash1644 1978

[5] H S Tzou and J P Zhong ldquoA linear theory of piezoelastic shellvibrationsrdquo Journal of Sound and Vibration vol 175 no 1 pp77ndash88 1994

[6] DD Ebenezer andPAbraham ldquoEigenfunction analysis of radi-ally polarized piezoelectric cylindrical shells of finite lengthrdquoThe Journal of the Acoustical Society of America vol 102 no 3pp 1549ndash1558 1997

8 Shock and Vibration

[7] C V Stephenson ldquoRadial vibrations in short hollow cylindersof barium titanaterdquo The Journal of the Acoustical Society ofAmerica vol 28 no 1 pp 51ndash56 1956

[8] C V Stephenson ldquoHigher modes of radial vibrations in shorthollow cylinders of barium titanaterdquoThe Journal of the Acousti-cal Society of America vol 28 no 5 pp 928ndash929 1956

[9] N T Adelman Y Stavsky and E Segal ldquoAxisymmetric vibra-tions of radially polarized piezoelectric ceramic cylindersrdquoJournal of Sound and Vibration vol 38 no 2 pp 245ndash254 1975

[10] N T Adelman Y Stavsky and E Segal ldquoRadial vibrations ofaxially polarized piezoelectric ceramic cylindersrdquo The Journalof the Acoustical Society of America vol 57 no 2 pp 356ndash3601975

[11] H S Paul ldquoVibrations of circular cylindrical shells of piezoelec-tric silver iodide crystalsrdquo The Journal of the Acoustical Societyof America vol 40 no 5 pp 1077ndash1080 1966

[12] H S Paul and M Venkatesan ldquoVibrations of a hollow circularcylinder of piezoelectric ceramicsrdquoThe Journal of the AcousticalSociety of America vol 82 no 3 pp 952ndash956 1987

[13] H-JDingW-QChen Y-MGuo andQ-DYang ldquoFree vibra-tions of piezoelectric cylindrical shells filled with compressiblefluidrdquo International Journal of Solids and Structures vol 34 no16 pp 2025ndash2034 1997

[14] Z Yang J Yang Y Hu and Q-M Wang ldquoVibration charac-teristics of a circular cylindrical panel piezoelectric transducerrdquoIEEE Transactions on Ultrasonics Ferroelectrics and FrequencyControl vol 55 no 10 pp 2327ndash2335 2008

[15] S Li J Qiu H Ji K Zhu and J Li ldquoPiezoelectric vibration con-trol for all-clamped panel using DOB-based optimal controlrdquoMechatronics vol 21 no 7 pp 1213ndash1221 2011

[16] N Kumar and S P Singh ldquoVibration control of curved panelusing smart dampingrdquo Mechanical Systems and Signal Process-ing vol 30 pp 232ndash247 2012

[17] H J Ding B Chen and J Liang ldquoGeneral solutions for coupledequations for piezoelectric mediardquo International Journal ofSolids and Structures vol 33 no 16 pp 2283ndash2298 1996

[18] H J Ding R Q Xu and W Q Chen ldquoFree vibration oftransversely isotropic piezoelectric circular cylindrical panelsrdquoInternational Journal of Mechanical Sciences vol 44 no 1 pp191ndash206 2002

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

8 Shock and Vibration

[7] C V Stephenson ldquoRadial vibrations in short hollow cylindersof barium titanaterdquo The Journal of the Acoustical Society ofAmerica vol 28 no 1 pp 51ndash56 1956

[8] C V Stephenson ldquoHigher modes of radial vibrations in shorthollow cylinders of barium titanaterdquoThe Journal of the Acousti-cal Society of America vol 28 no 5 pp 928ndash929 1956

[9] N T Adelman Y Stavsky and E Segal ldquoAxisymmetric vibra-tions of radially polarized piezoelectric ceramic cylindersrdquoJournal of Sound and Vibration vol 38 no 2 pp 245ndash254 1975

[10] N T Adelman Y Stavsky and E Segal ldquoRadial vibrations ofaxially polarized piezoelectric ceramic cylindersrdquo The Journalof the Acoustical Society of America vol 57 no 2 pp 356ndash3601975

[11] H S Paul ldquoVibrations of circular cylindrical shells of piezoelec-tric silver iodide crystalsrdquo The Journal of the Acoustical Societyof America vol 40 no 5 pp 1077ndash1080 1966

[12] H S Paul and M Venkatesan ldquoVibrations of a hollow circularcylinder of piezoelectric ceramicsrdquoThe Journal of the AcousticalSociety of America vol 82 no 3 pp 952ndash956 1987

[13] H-JDingW-QChen Y-MGuo andQ-DYang ldquoFree vibra-tions of piezoelectric cylindrical shells filled with compressiblefluidrdquo International Journal of Solids and Structures vol 34 no16 pp 2025ndash2034 1997

[14] Z Yang J Yang Y Hu and Q-M Wang ldquoVibration charac-teristics of a circular cylindrical panel piezoelectric transducerrdquoIEEE Transactions on Ultrasonics Ferroelectrics and FrequencyControl vol 55 no 10 pp 2327ndash2335 2008

[15] S Li J Qiu H Ji K Zhu and J Li ldquoPiezoelectric vibration con-trol for all-clamped panel using DOB-based optimal controlrdquoMechatronics vol 21 no 7 pp 1213ndash1221 2011

[16] N Kumar and S P Singh ldquoVibration control of curved panelusing smart dampingrdquo Mechanical Systems and Signal Process-ing vol 30 pp 232ndash247 2012

[17] H J Ding B Chen and J Liang ldquoGeneral solutions for coupledequations for piezoelectric mediardquo International Journal ofSolids and Structures vol 33 no 16 pp 2283ndash2298 1996

[18] H J Ding R Q Xu and W Q Chen ldquoFree vibration oftransversely isotropic piezoelectric circular cylindrical panelsrdquoInternational Journal of Mechanical Sciences vol 44 no 1 pp191ndash206 2002

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