Finite Element Vibration Analysis of Rotating Timoshenko Beams

Journal of Sound and <ibration (2001) 242(1), 103}124doi:10.1006/jsvi.2000.3362, available online at http://www.idealibrary.com on

FINITE ELEMENT VIBRATION ANALYSIS OF ROTATINGTIMOSHENKO BEAMS

S. S. RAO

Department of Mechanical Engineering, ;niversity of Miami, Coral Gables, F¸ 33124-0624, ;.S.A.

AND

R. S. GUPTA

Department of Mechanical Engineering, Punjab Engineering College, Chandigarh 11, India

(Received 12 January 2000, and in ,nal form 13 September 2000)

The sti!ness and mass matrices of a rotating twisted and tapered beam element arederived. The angle of twist, breadth and depth are assumed to vary linearly along the lengthof beam. The e!ects of shear deformation and rotary inertia are also considered in derivingthe elemental matrices. The "rst four natural frequencies and mode shapes inbending}bending mode are calculated for cantilever beams. The e!ects of twist, o!set, speedof rotation and variation of depth and breadth taper ratios are studied.

( 2001 Academic Press

1. INTRODUCTION

Generally, turbomachine blades are idealized as tapered cantilever beams. In order to re"nethe analysis the e!ects of pre-twist and rotation are also included. As the blades are short insome of the designs and may vibrate in higher-frequency ranges, the e!ects of sheardeformation and rotary inertia may be of considerable magnitude. Various investigators inthe "eld of turbine-blade vibrations have solved the di!erential equations of motion, bytaking into account one or more of the above-mentioned e!ects.

The analysis of tapered beams has been made by many investigators using di!erenttechniques. Rao [1] used the Galerkin method, Housner and Keightley [2] applied theMyklestad procedure, Rao and Carnegie [3] used the "nite di!erences approach, Martin[4] adopted a perturbation technique and Mabie and Rogers [5] solved the di!erentialequations using Bessel functions to "nd the natural frequencies of vibration of taperedcantilever beams.

In analysing the pre-twisted beams, various approaches have been used by variousinvestigators. Mendelson and Gendler [6] used the station functions, Rosard [7] appliedthe Myklestad method, Di Prima and Handleman [8] adopted the Rayleigh}Ritzprocedure, Carnegie and Thomas [9] used the "nite di!erence techniques and Rao [10]considered the Galerkin method for the analysis of pre-twisted beams. The bendingvibration of a twisted rotating beam was considered by Targo! [11].

The rotation e!ect in beam analysis has also been considered using di!erent formulationand solution procedures. Targo! applied the matrix method, Yntema [12] applied theRayleigh}Ritz procedure, Isakson and Eisley [13] used the Rayleigh}Southwell procedure,Banerjee and Rao [14] applied the Galerkin procedure, Carnegie et al. [15] applied the"nite di!erence scheme, Krupka and Baumanis [16] used the Myklestad method and Rao

0022-460X/01/160103#22 $35.00/0 ( 2001 Academic Press

104 S. S. RAO AND R. S. GUPTA

and Carnegie [17] used the extended Holzer method to "nd the vibration characteristics ofrotating cantilever beams.

The "nite element technique has also been applied by many investigators, mostly for thevibration analysis of beams of uniform cross-section. All these investigations di!er from oneanother in the nodal degrees of freedom taken for deriving the elemental sti!ness and massmatrices. McCalley [18] derived the consistent mass and sti!ness matrices by selecting thetotal de#ection and total slope as nodal co-ordinates. Kapur [19] took bending de#ection,shear de#ection, bending slope and shear slope as nodal degrees of freedom and derived theelemental matrices of beams with linearly varying inertia. Carnegie et al. [20] analyzeduniform beams by taking few internal nodes in it. Nickel and Sector [21] used totalde#ection, total slope and bending slope of the two nodes and the bending slope at themid-point of the beam as the degrees of freedom to derive the elemental sti!ness and themass matrices of order seven. Thomas and Abbas [22] analyzed uniform Timoshenkobeams by taking total de#ection, total slope, bending slope and the derivative of thebending slope as nodal degrees of freedom.

The "nite element analysis of vibrations of twisted blades based on beam theory wasconsidered by Sisto and Chang [23]. Sabuncu and Thomas [24] studied the vibrationcharacteristics of pre-twisted aerofoil cross-section blade packets under rotating conditions.An improved two-node Timoshenko beam "nite element was derived by Friedman andKosmatka [25]. The vibration of Timoshenko beams with discontinuities in cross-sectionwas investigated by Farghaly and Gadelab [26, 27]. Corn et al. [28] derived "nite elementmodels through Guyan condensation method for the transverse vibration of short beams.Gupta and Rao [29] considered the "nite element analysis of tapered and twistedTimoshenko beams.

In this work, the "nite element technique is applied to "nd the natural frequenciesand mode shapes of beams in the bending}bending mode of vibration by takinginto account the taper, the pre-twist and the rotation simultaneously. The coupling thatexists between the #exural and torsional vibration is not considered. The taper and theangle of twist are assumed to vary linearly along the length of the beam. The elementsti!ness and mass matrices are derived and the e!ects of o!set, rotation, pre-twist, depthand breadth taper ratios and rotary inertia and shear deformation on the naturalfrequencies are studied. Various special cases of beam vibration can be obtained from thegeneral equations derived.

2. ELEMENT STIFFNESS AND MASS MATRICES

2.1. DISPLACEMENT MODEL

Figure 1(a) shows a doubly tapered, twisted beam element of length l with the nodes as1 and 2. The breadth, depth and the twist of the element are assumed to be linearly varyingalong its length. The breadth and depth at the two nodal points are shown as b

1, h

1and

b2, h

2respectively. The pre-twist at the two nodes is denoted by h

1and h

2. Figure 1(b)

shows the nodal degrees of freedom of the element where bending de#ection, bending slope,shear de#ections and shear slope in the two planes are taken as the nodal degrees offreedom. Figure 1(c) shows the angle of twist h at any section z. The beam is assumed torotate about the x}x-axis at a speed of X rad/s.

The total de#ections of the element in the y and x directions at a distance z from node 1,w(z) and v(z), are taken as

w (z)"wb(z)#w

s(z), v(z)"v

b(z)#v

s(z), (1)

Figure 1. (a) An element of tapered and twisted beam, (b) degrees of freedom of an element, (c) angle of twist h,(d) rotation of tapered beam.

ROTATING TIMOSHENKO BEAMS 105

where wb(z) and v

b(z) are the de#ections due to bending in the yz and xz planes respectively,

and ws(z) and v

s(z) are the de#ections due to shear in the corresponding planes.

The displacement models for wb(z), w

s(z), v

b(z) and v

s(z) are assumed to be polynomials of

third degree. They are similar in nature except for the nodal constants. These expressionsare given by

wb(z)"

u1

l3(2z3!3lz2#l3)#

u2

l3(3lz2!2z3)!

u3

l2(z3!2lz2#l2z)!

u4

l2(z3!lz2),

ws(z)"

u5

l3(2z3!3lz2!l3)#

u6

l3(3lz2!2z3)!

u7

l2(z3!2lz2#l2z)!

u8

l2(z3!lz2),


vb(z)"

u9

l3(2z3!3lz2!l3)#

u10l3

(3lz2!2z3)!u11l2

(z3!2lz2#l2z)

(2)

!

u12l2

(z3!lz2),

vs(z)"

u13l3

(2z3!3lz2!l3)#u14l3

(3lz2!2z3)!u15l2

(z3!2lz2#l2z)

!

u16l2

(z3!lz2),

where u1, u

2, u

3and u

4represent the bending degrees of freedom and u

5, u

6, u

7and u

8are

the shear degrees of freedom in the yz plane; u9, u

10, u

11and u

12represent the bending

degrees of freedom and u13

, u14

, u15

and u16

shear degrees of freedom in the xz plane.

2.2. ELEMENT STIFFNESS MATRIX

The total strain energy ; of a beam of length l, due to bending and shear deformationincluding rotary inertia and rotation e!ects is given by

;"Pl

0CG

EIxx

2 AL2w

bLz2 B

2#EI

xy

L2wb

Lz2

L2vb

Lz2#

EIyy

2 AL2v

bLz2 B

2

H#

kAG

2 GALw

sLz B

2#A

Lvs

LzB2

HD dz#1

2 Pl

0

P(z) ALw

bLz

#

Lws

Lz B2dz (3)

#

1

2 Pl

0

P (z) ALv

bLz

#

Lvs

LzB2dz!P

l

0

pw(z) (w

b#w

s) dz!P

l

0

pv(z) (v

b#v

s) dz.

where

P (z)"PL`e

e`ze`z

mX2m dm+oAX2

2g[(¸#e)2!(e#z

e#z)2]

(4)

"

oAX2

g CAe¸#

1

2¸2!ez

e!

1

2z2eB!(e#z

e)z!

1

2z2D ,

pw(z)"

oAX2

g(w

b#w

s), p

v(z)"

oAX2

g(v

b#v

s), (5, 6)

where e is the o!set and zeis the distance of the "rst node of the element from the root of the

beam as shown in Figure 1(d), and P (z) is the axial force acting at section z.


As the cross-section of the element changes with z and as the element is twisted, thecross-sectional area A, and the moments of inertia I

xx, I

yyand I

xywill be functions of z:

A(z)"b (z)h(z)"Gb1#(b2!b

1)z

lH Gh1#(h2!h

1)z

lH(7)

"

1

l2(c

1z2#c

2lz#c

3l2),

where

c1"(b

2!b

1) (h

2!h

1), c

2"b

1(h

2!h

1)#h

1(b

2!b

1), c

3"b

1h1, (8)

Ixx

(z)"Ix{x{

cos2 h#Iy{y{

sin2 h, Iyy

(z)"Iy{y{

cos2 h#Ix{x{

sin2 h,

Ixy

(z)"(Ix{x{

!Iy{y{

)sin 2h

2, (9)

where x@x@ and y@y@ are the axes inclined at an angle h, the angle of twist, at any point in theelement, to the original axes xx and yy as shown in Figure 1(c). The value of I

x{y{"0 and the

values of Ix{x{

and Iy{y{

can be computed as

Ix{x{

(z)"b(z)h3 (z)

12"

1

12l4[a

1z4#a

2lz3#a

3l2z2#a

4l3z#a

5l4], (10)

where

a1"(b

2!b

1) (h

2!h

1)3, a

2"b

1(h

2!h

1)3#3(b

2!b

1) (h

2!h

1)2h

1,

a3"3Mb

1h1(h

2!h

1)3#(b

2!b

1) (h

2!h

1)h2

1, (11)

a4"3b

1h21(h

2!h

1)#(b

2!b

1)h3

1, a

5"b

1h31,

Iy{y{

(z)"h(z)b3 (z)

12"

1

12l4[d

1z4#d

2lz3#d

3l2z2#d

4l3z#d

5l4], (12)

where

d1"(h

2!h

1) (b

2!b

1)3, d

2"h

1(b

2!b

1)3#3(h

2!h

1) (b

2!b

1)2b

1,

d3"3Mh

1b1(b

2!b

1)3#(h

2!h

1) (b

2!b

1)b2

1, (13)

d4"3h

1b21(b

2!b

1)#(h

2!h

1)b3

1, d

5"h

1b31.

By substituting the expressions of wb, w

s, v

b, v

s, A, I

xx, I

xyand I

yyfrom equations (2), (7)

and (9) into equation (3), the strain energy ; can be expressed as

;"12uT[K]u, (14)


where u is the vector of nodal displacements u1, u

2,2, u

16, and [K] is the elemental

sti!ness matrix of order 16. Denoting the integrals

Pl

0

EIxxA

L2wb

Lz2 B2

dz"[u1

u2

u3

u4]T[AK][u

1u2

u3

u4], (15)

Pl

0

EIyyA

L2vb

Lz2B2

dz"[u9

u10

u11

u12

]T[BK][u9

u10

u11

u12

], (16)

Pl

0

kAGALw

sLz B

2dz"[u

5u6

u7

u8]T[CK][u

5u6

u7

u8], (17)

Pl

0

EIxyA

L2wb

Lz2 B AL2v

bLz2B dz"[u

1u2

u3

u4]T[DK][u

9u10

u11

u12

], (18)

Pl

0

P (z)ALw

bLz B

2dz"[u

1u2

u3

u4]T[EK][u

1u2

u3

u4] (19)

and

Pl

0

2oAX2

g(w2

b) dz"[u

1u2

u3

u4]T[FK][u

1u2

u3

u4], (20)

the element sti!ness matrix can be expressed as

[K]"

[AK]#[EK]![FK] [EK]![FK] [DK] [0]

[EK]![FK] [CK]#[EK]![FK] [0] [0]

[DK] [0] [BK]#[EK]![FK] [EK]![FK]

[0] [0] [EK]![FK] [CK]#[EK]![FK]

,

(21)

where [AK], [BK], [CK], [DK], MEK] and [FK] are symmetric matrices of order 4 andtheir elements are formulated in Appendix B. [0] is a null matrix of order 4.

2.3. ELEMENT MASS MATRIX

The kinetic energy of the element ¹ including the e!ects of shear deformation and rotaryinertia is given by

¹"Pl

0CoA

2g ALw

bLt

#

Lws

Lt B2#

oA

2g ALv

bLt

#

Lvs

Lt B2#

oIyy

2g AL2v

bLz LtB

(22)

#

oIxy

g AL2w

bLz LtB A

L2vb

Lz LtB#oI

xx2g A

L2wb

Lz LtB2

D dz.


By de"ning

Pl

0

oA

g ALw

bLt B

2dz"[uR

1uR2

uR3

uR4]T[AM][uR

1uR2

uR3

uR4], (23)

Pl

0

oIxx

g AL2w

bLz LtB

2dz"[uR

1uR2

uR3

uR4]T[BM][uR

1uR2

uR3

uR4], (24)

Pl

0

oIyy

g AL2v

bLz LtB

2dz"[uR

9uR10

uR11

uR12

]T[CM][uR9

uR10

uR11

uR12

], (25)

and

Pl

0

oIxy

g AL2w

bLz LtB A

L2vb

Lz LtB dz"[uR1

uR2

uR3

uR4]T[DM][uR

9uR10

uR11

uR12

], (26)

where uRl

denotes the time derivative of the nodal displacement ui, i"1, 2,2 , 16, the

kinetic energy of the element can be expressed as

¹"12u5 [M]u5 , (27)

where [M] is the mass matrix given by

[M]"

16]16

[AM]#[BM] [AM] [DM] [0]

[AM] [AM] [AM] [0]

[DM] [AM] [AM]#[CM] [AM]

[0] [0] [AM] [AM]

(28)

and [AM], [BM] [CM] and [DM] are symmetric matrices of order 4 whose elements arede"ned in Appendix A.

2.4. BOUNDARY CONDITIONS

The following boundary conditions are to be applied depending on the type of endconditions.

Free endLw

sLz

"0,Lv

sLz

"0,L2v

bLz2

"0,L2w

bLz2

"0. (29)

Clamped end ws"0, w

b"0, v

s"0, v

b"0,

Lwb

Lz"0,

Lvb

Lz"0. (30)

Hinged end ws"0, w

b"0, v

s"0, v

b"0.

L2vb

Lz2"0,

L2wb

Lz2"0. (31)

TABLE 1

Natural frequencies (Hz)

No ofelements

First mode Second mode Third mode Fourth mode

1 304.3 1191)66 2327)07 4552)232 296)22 1161)70 1779)50 4106)833 295)03 1155)97 1746)55 3747)044 294)85 1154)94 1741)39 3697)055 294)78 1154)67 1739)99 3689)826 294)78 1154)58 1739)40 3683)987 294)78 1154)54 1739)25 3683)918 294)78 1154)50 1739)10 3683)85

Data: length of beam"0)1524 m, breadth at root"0)0254 m, depth at root"0)0046 m, depth taperratio"2)29, breadth taper ratio"2)56, twist angle"453, shear coe$cient"0)833, E"2)07]1011 N/m2,G"E/2)6, o!set"0, mass density"800 kg/m3.

Figure 2. E!ect of depth taper ratio and shear deformation on frequency ratio of rotating beam:**, withoutshear deformation; } } } }, with shear deformation.



3. SPECIAL CASES

The various special cases of the beam vibration problem can be solved by applying one ormore of the following four conditions:

(a) For non-rotating beams: X"0 which results in [EK]"[FK]"[0]. (32)

(b) For uniform beams: by setting b2"b

1and h

2"h

1, one obtains

c1"c

2"0 and c

3"b

1h1, a

1"a

2"a

3"a

4"0 and a

5"b

1h31, (33)

d1"d

2"d

3"d

4"0 and d

5"h

1b31.

Figure 3. E!ect of breadth taper ratio and shear deformation on frequency ratio of rotating beam: **, withshear deformation; } } } }, without shear deformation.


(c) For neglecting the e!ect of shear deformation: ws"v

s"0 so that equations (1) and (2)

become

w (z)"wb(z) and v (z)"v

b(z). (34)

Due to this, the order of [K] and [M] matrices reduces from 16 to 8.(d) For beams without pre-twist: in this case, there will be no coupling between the moment

of inertia terms and one obtains

Ixx"I

x{x{, I

yy"I

y{y{, I

xy"0. (35)

For vibration in the yz plane, vb"v

s"0; for vibration in the xz plane, w

b"w

s"0.

This condition further reduces the size of matrices [K] and [M] by half. Thus, for thegeneral case (with shear deformation) the matrices will be of order 8 and, if coupled withcondition (c) these will be of order 4.

If the conditions (a), (b) and (d) are applied one gets the following expressions for [K] and[M] for non-rotating uniform beams without pre-twist but with a consideration of rotaryinertia and shear deformation e!ects (for vibration in the y}z plane):

[K]"EI

xxl3

12 !12 !6l !6l 0 0 0 0

12 6l 6l 0 0 0 0

4l2 2l2 0 0 0 0

4l2 0 0 0 0

36J !36J !3lJ !3lJ

Symmetric 36J 3lJ 3lJ

4l2J !l2J

4l2J

, (36)

[M]"oAI

420g

156#36PM 54!36PM 22l!3lPM 13l!3lPM 156 54 22l 13l

156#36PM !13l#3lPM 22l#3lPM 54 156 !13l 22l

4l2#4l2PM !3l2!l2PM 22l !13l 4l2 !3l2

4l2!4l2PM 13l 22l !3l2 4l2

156 54 22l 13l

156 !13l 22l

4l2 !3l2

4l2

, (37)

where J"kGb1h1l2/(30EI

xx) , and PM "14I

xx/(b

1h1l2),


Equations (36) and (37) further reduce to the following well-known equations if the e!ectsof shear deformation and rotary inertia are neglected:

[K]"EI

xxl3

12 !12 !6l !6l

12 6l 6l

Symmetric 4l2 2l2

4l2

, (38)

[M]"ob

1h1

420g

156 54 !22l 13l

156 !13l 22l

Symmetric 4l2 !3l

4l2

. (39)

4. NUMERICAL RESULTS

The element sti!ness and mass matrices developed are used for the dynamic analysis ofcantilever beams. By using the standard procedures of structural analysis, the eigenvalue

Figure 4. E!ect of rotation and twist on "rst and second natural frequencies.


problem can be stated as

([K3]!u2[M

3])U

3"0, (40)

where [K3] and [M

3] denote the sti!ness and mass matrices of the structure, respectively,

U3

indicates nodal displacement vector of the structure, and u is the natural frequency ofvibration.

A study of the convergence properties of the general element developed has been madeand the results are given in Table 1. It is seen that the results converge well even with onlyfour elements. The e!ects of shear deformation and depth taper ratio on the naturalfrequencies of a rotating twisted beam are shown in Figure 2 for a beam of length 0)254 m,o!set zero, depth at root 0)00865 m, breadth at root 0)0173 m, twist 453, rotation 100 r.p.s.

Figure 5. E!ect of rotation and twist on third and fourth natural frequencies: **, third mode; } } }, fourthmode.


and breadth taper ratio 3. The material properties of the beam are taken the same as thosegiven in Table 1. The e!ect of shear deformation is found to reduce the frequencies at highermodes while at lower modes the results are nearly una!ected. There is an increase in thefrequencies of vibration with an increase in the depth taper ratio in the "rst, second andfourth modes while a decrease has been observed in the case of third mode (vibration ina perpendicular plane). Figure 3 shows the variation of natural frequencies with breadthtaper ratio. In this case the data are the same as in the case of Figure 2.

In Figures 4 and 5, the variation of frequency ratio with rotation and pre-twist isstudied. It can be seen that the frequency ratio changes slightly with the rotationbut appreciably with the twist. At higher modes (in Figure 5) the e!ect of twist can be seento be more pronounced. It is also observed that the frequency ratio increases withan increase in the twist in the case of "rst and third modes while it decreases with an increaseof the twist in the case of second and fourth modes of vibration. In Figure 6, the e!ect ofo!set is studied for a twisted blade having 603 twist with the other data the same as that ofFigure 4. It is observed that an increase in o!set changes the frequency ratio more at highervalues of rotation. The frequency ratio has been found to increase with an increase in theo!set.

Figure 6. E!ect of o!set and rotation on frequency ratio. O!set 1: e"0 m; o!set 2: e"0.0254 m; o!set 3:e"0.0508 m; o!set 4: e"0.0762 m.


5. CONCLUSION

The mass and sti!ness matrices of a thick rotating beam element with taper and twist aredeveloped for the eigenvalue analysis of rotating, doubly tapered and twisted Timoshenkobeams. The element has been found to give reasonably accurate results even with four "niteelements. The e!ects of breadth and depth taper ratios, twist angle, shear deformation, o!setand rotation on the natural frequencies of vibration of cantilever beams are found. Theconsideration of shear deformation is found to reduce the values of the higher naturalfrequencies of vibration of the beam. An increase in the breadth and depth taper ratios isfound to increase the "rst two modes of vibration. The frequency ratio is found to changeonly slightly with rotation but appreciably with the twist of the beam.

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14. S. BANERJEE and J. S. RAO American Society of Mechanical Engineers Paper No. 76-GT.43.Coupled bending-torsion vibrations of rotating blades.

15. W. CARNEGIE, C. STIRLING and J. FLEMING 1966 Paper No. 32. Applied Mechanics Convention,Cambridge, MA. Vibration characteristics of turbine blading under rotation-results of an initialinvestigation and details of a high-speed test installation.

16. R. M. KRUPKA and A. M. BAUMANIS 1969 Journal of Engineering for Industry American Society ofMechanical Engineers 91, 1017. Bending bending mode of a rotating tapered twistedturbomachine blade including rotary inertia and shear de#ection.

17. J. S. RAO and W. CARNEGIE 1973 International Journal of Mechanical Engineering Education 1,37}47. Numerical procedure for the determination of the frequencies and mode shapes of lateralvibration of blades allowing for the e!ects of pretwist and rotation.

18. R. MCCALLEY 1963 Report No. DIG SA, 63}73, General Electric Company, Schenectady, New>ork. Rotary inertia correction for mass matrices.

19. K. K. KAPUR 1966 Journal of Acoustical Society of America 40, 1058}1063. Vibrations ofa Timoshenko beam, using "nite element approach.


20. W. CARNEGIE, J. THOMAS and E. DOCUMAKI 1969 Aeronautical Quarterly 20, 321}332. Animproved method of matrix displacement analysis in vibration problems.

21. R. NICKEL and G. SECOR 1972 International Journal of Numerical Methods in Engineering 5,243}253. Convergence of consistently derived Timoshenko beam "nite elements.

22. J. THOMAS and B. A. H. ABBAS 1975 Journal of Sound and <ibration 41, 291}299. Finite elementmodel for dynamic analysis of Timoshenko beam.

23. F. SISTO and A. T. CHANG 1984 American Institute of Aeronautics and Astronautics Journal 22,1646}1651. A "nite element for vibration analysis of twisted blades based on beam theory.

24. M. SABUNCU and J. THOMAS 1992 American Institute of Aeronautics and Astronautics Journal 30,241}250. Vibration characteristics of pretwisted aerofoil cross-section blade packets underrotating conditions.

25. Z. FRIEDMAN and J. B. KOSMATKA 1993 Computers and Structures 47, 473}481. An improvedtwo-node Timoshenko beam "nite element.

26. S. H. FARGHALY Journal of Sound and <ibration 174, 591}605. Vibration and stability analysis ofTimoshenko beams with discontinuities in cross-section.

27. S. H. FARGHALY and R. M. GADELRAB 1995 Journal of Sound and <ibration 187, 886}896. Freevibration of a stepped composite Timoshenko cantilever beam.

28. S. CORN, N. BUHADDI and J. PIRANDA 1997 Journal of Sound and <ibration 201, 353}363.Transverse vibrations of short beams: "nite element models obtained by a condensation method.

29. R. S. GUPTA and S. S. RAO 1978 Journal of Sound and <ibration 56, 187}200. Finite elementeigenvalue analysis of tapered and twisted Timoshenko beams.

APPENDIX A: EXPRESSIONS FOR STRAIN AND KINETIC ENERGIES

A.1. EXPRESSION FOR STRAIN ENERGY (;)

A.1.1. Strain energy due to bending

If the bending de#ections in yz and xz planes of a beam are wband v

b, respectively, the

axial strain and stress induced due to wband v

bare given by e

xxdue to w

b"!y (L2w

b/Lz2);

exx

due to vb"!xL2v

b/Lz2 and p

xxdue to w

b"!EyL2w

b/Lz2; p

xxdue to v

b"

!ExL2vb/Lz2 .

The strain energy stored in the beam due to bending is given by: ; due to bending

"

1

2 PV

exx

pxx

d<

"

1

2 PV

(exx

due to wb#e

xxdue to v

b) (p

xxdue to w

b#p

xxdue to v

b) d<

"

E

2 Pl

0

dz PAAy

L2wb

Lz2#x

L2vb

Lz2B2dA

"

E

2 Pl

0

dzCAL2w

bLz2 B

2

PA

y2 dA#AL2v

bLz2B

2

PA

x2 dA#2L2w

bLz2

L2vb

Lz2 PA

xy dAD

"Pl

0CEI

xx2 A

L2wb

Lz2 B2#EI

xy

L2wb

Lz2

L2vb

Lz2#

EIyy

2 AL2v

bLz2B

2

D dz, (A.1)

where < is the volume, l is the length and A is the cross-sectional area of the beam.


A.1.2. Strain energy due to shearing

Let Fx

and Fy

be the shear forces that produce the shear de#ections dvsand dw

sin an

element of length dz respectively. Then the strain energy of the beam due to shearing isgiven by

;due to shearing

"

1

2 Pl

0AFx

dvs

dz#F

y

dws

dz B dz.

By substituting AGk dvs/dz and AGk dw

s/dz for F

xand F

y, respectively, one obtains

;due to shearing

"Pl

0

kAG

2 CAdw

sdz B

2#A

dvs

dzB2

Ddz. (A.2)

A.1.3. Strain energy due to rotation

The rotation of a beam induces an axial force P in the beam due to centrifugal action. Ifthe beam is bending in the yz plane (Figure A1), the change in the horizontal projection ofan element of length ds is given by

ds!dz"G(dz)2#ALw

LzdzB

2

H1@2

!dz+1

2 ALw

LzB2dz.

Since the axial force P acts against the changes in the horizontal projection, the work doneby P is given by

;due to P andw

"!

1

2 Pl

0

P(z) Adw

dzB2dz. (A.3)

The work done by the transverse distributed force pw(z) can be written as

;due to pw

"Pl

0

pw(z)w dz. (A.4)

Figure A.1. An element of the beam in equilibrium.


The expressions corresponding to the bending of the beam in the xz plane can be obtainedsimilarly as

;due to P and v

"

1

2 Pl

0

P(z) Adv

dzB2

dz, ;due to pv

"Pl

0

pv(z)v dz. (A.5, 6)

The total strain energy of the beam can be obtained as given in equation (3) by combiningequations (A.1)}(A.6).

A.2. EXPRESSION FOR KINETIC ENERGY (¹)

Consider a small element of area dA and length dz at a point in the cross-section havingco-ordinates (x, y) with respect to x- and y-axes. The kinetic energy of this element is givenby

o2g

[(wR 2#vR 2)#(y/Qx#x/Q

y)2] dz dA,

where /xand /

ydenote the bending slopes, Lv

b/Lz and Lw

b/Lz, respectively, and a dot over

a symbol represents derivative with respect to time. Integrating this equation over the beamcross-section, the kinetic energy of an element of length dz can be obtained as

d¹"

o2g

[A(wR 2#vR 2)#(Ixx

/Q 2y#2I

xy/Qx/Q

y#I

yy/Q 2x)] dz.

The kinetic energy of the entire beam (¹ ) can be expressed as

¹"Pl

0CoA

2g(wR 2#vR 2)#

o2g

(Ixx

/Q 2y#2I

xy/Q

x/Qy#I

yy/Q 2

x)] dz, (A.7)

which can be seen to be the same as in equation (22).

APPENDIX B: EXPRESSIONS FOR [AK], MBK],2 , [DM]

The following notation is used for convenience:

;i"P

l

0

zi~1 dz, ¸i"li~1, i"1, 2,2, n, (B.1, 2)

<i"P

l

0

zi~1 cos2C(h2!h1)z

l#h

1D dz, i"1, 2,2 , n, (B.3)

Si"P

l

0

zi~1 sin 2C(h2!h1)z

l#h

1D dz, i"1, 2,2 , n, (B.4)

where h1

and h2

denote the values of pre-twist at nodes 1 and 2, respectively, of the element.As the nature of w

b, w

s, v

band v

sis the same except for their positions in the sti!ness and

mass matrices, one can use wN to denote any one of the quantities wb, w

s, v

bor v

sand in

TABLE B1

<alues of Hi,j

, Ri, j,k

, Qi,j,k

Ri, j,k

Qi, j,k

i j Hi,j

k"1 k"2 k"3 k"1 k"2 k"3 k"4 k"5

1 1 0 144)0 !144)0 36)0 36)0 !72)0 36)0 0)0 0)01 2 0 !144)0 144)0 !36)0 !36)0 72)0 !36)0 0)0 0)01 3 1 !72)0 84)0 !24)0 !18)0 42)0 !30)0 6)0 0)01 4 1 !72)0 60)0 !12)0 !18)0 30)0 !12)0 0)0 0)02 2 0 144)0 !144)0 36)0 36)0 !72)0 36)0 0)0 0)02 3 1 72)0 !84)0 24)0 18)0 !42)0 30)0 !6)0 0)02 4 1 72)0 !60)0 12)0 18)0 !30)0 12)0 0)0 0)03 3 2 36)0 !48)0 16)0 9)0 !24)0 22)0 !8)0 1)03 4 2 36)0 !36)0 8)0 9)0 !18)0 11)0 !2)0 0)04 4 2 36)0 !24)0 4)0 9)0 !12)0 4)0 0)0 0)0


a similar manner the set (uN1, uN

2, uN

3, uN

4) can be used to represent any one of the sets

(u1, u

2, u

3, u

4), (u

5, u

6, u

7, u

8), (u

9, u

10, u

11, u

12) or (u

13, u

14, u

15, u

16). Thus,

wN (z)"uN1

l3(2z3!3lz2#l2)!

uN3

l2(z3!2lz2#l2z)#

uN2

l3(3lz2!2z3)!

uN4

l2(z3!lz2), (B.5)

dwNdz

"

uN1

l3(6z2!6lz)!

uN3

l2(3z2!4lz#l2)#

uN2

l3(6lz!6z2)!

uN4

l2(3z2!2lz), (B.6)

d2wNdz2

"

uN1

l3(12z!6l)!

uN3

l2(6z!4l)#

uN2

l3(6l!12z)!

uN4

l2(6z!2l). (B.7)

By letting Pi,j,k

(i"1,2, 4; j"1,2, 4; k"1,2, 7) denote the coe$cient of zk~1 l7~k forthe uN

iuNj

term in the expression of wN 2, Qi, j,k

(i"1,2, 4; j"1,2, 4; k"1,2, 5), thecoe$cient of zk~1 l5~k for the uN

iuNjterm in the expression of (dwN /dz)2, R

i, j,k(i"1,2, 4;

j"1,2, 4; k"1,2, 3), the coe$cient of zk~1 l3~k for the uNiuNjterm in the expression of

(d2wN /dz2)2, Hi,j

(i"1,2, 4; j"1,2, 4), the index coe$cient of l to account for thedi!erence in the index of l due to multiplication of rotational degrees of freedom uN

1and

uN2

and the displacement degrees of freedom uN3

and uN4, the values of P

i, j,k, Q

i, j,k, R

i, j,kand

Hi,j

can be obtained as shown in Tables B1 and B2.

B.1. EVALUATION OF [BK]

As the procedure for the derivation of [AK], [BK],2, [DM] is the same, the expressionfor [BK] is derived here as an illustration:

[u9

u10

u11

u12

]T[BK][u9

u10

u11

u12

]"Pl

0

EIyyA

L2vb

Lz2B2dz

(B.8)

"Pl

0

EIyy A

L2wNLz2B

2dz,

TABLE B2

<alues of Pi, j,k

Pi, j,k

i j k"1 k"2 k"3 k"4 k"5 k"6 k"7

1 1 4)0 !12)0 9)0 4)0 !6)0 0)0 1)01 2 !4)0 12)0 !9)0 !2)0 3)0 0)0 0)01 3 !2)0 7)0 !8)0 2)0 2)0 !1)0 0)01 4 !2)0 5)0 !3)0 !1)0 1)0 0)0 0)02 2 4)0 !12)0 9)0 0)0 0)0 0)0 0)02 3 2)0 !7)0 8)0 !3)0 0)0 0)0 0)02 4 2)0 !5)0 3)0 0)0 0)0 0)0 0)03 3 1)0 !4)0 6)0 !4)0 1)0 0)0 0)03 4 1)0 !3)0 3)0 !1)0 0)0 0)0 0)04 4 1)0 !2)0 1)0 0)0 0)0 0)0 0)0


where

wN "vb

and GuN1

uN2

uN3

uN4H"G

u9

u10

u11

u12H .

Substituting the values of Iyy

and wN into equation (B.8),

Pl

0

EIyy A

L2wNLz2B

2dz"P

l

0

ECIx{x{#(Iy{y{

!Ix{x{

) cos2G(h2!h1)z

l#h

1HD(B.9)

CuN1

l3(12z!6l)!

uN3

l2(6z!4l)#

uN2

l3(6l!12z)!

uN4

l2(6z!2l)D

2dz

with

BK1,1

"coe$cient of uN1uN1"coe$cient of u

9u9

"

E

12l10lHi,j P

l

0CT(a1z4#a

2lz3#a

3l2z2#a

4l3z#a

5l4)

#M(d1z4#d

2lz3#d

3l2z2#d

4l3z#d

5l4)

!(a1z4#a

2lz3#a

3l2z2#a

4l3z#a

5l4)N cos2G(h2

!h1)z

l#h

1HU]MR

1,1,1z2#R

1,1,2lz#R

1,1,3l2ND dz


"

E

12l10¸

(H1,1`1)

5+i/1

ai[R

1,1,1;

8~i#¸

i`1R

1,1,2;

7~i#¸

i`2R

1,1,3<6~i

]

(di!a

i) [¸

iR

1,1,1<8~i

#¸i`1

R1,1,2<7~i

#¸i`2

R1,1,3<6~i

] (B.10)

"

E

12l10¸

(H1,1`1)

5+i/1

3+j/1

Mai¸i`j~1

;9~i~j

R1,1, j

N#(di!a

i)M¸

i`j~1<9~i~j

R1,1,j

N .

This relation can be generalized as

BKI,J

"

E

12l10¸

(H1,J`1)

5+i/1

3+j/1

Mai¸

(i`j~1);

(9~i~j)R

I,J,jN

#(di!a

i)M¸

(i`j~1)<(9~i~j )

RI,J,j

N, I"1,2, 4, J"I ,2, 4,

(B.11)

"

E

12l10

5+i/1

3+j/1

[Mai¸(i`j`HI,J)

;(9~i~j)

RI,J,j

N

#(di!a

i)M¸

(i`j`HI,J)<(9~i~j)

RI,J,j

N], I"1,2, 4, J"1,2, 4.

Similarly,

AKI,J

"

E

12l10

5+i/1

3+j/1

[Mdi¸

(i`j`HI,J);

(9~i~j)R

I,J,jN

(B.12)

#(ai!d

i)M¸

(i`j`HI,J)<(9~i~j)

RI,J,j

N], I"1,2, 4, J"1,2, 4,

CKI,J

"

kG

l8

5+i/1

3+j/1

[ci¸(i`j`HI,J)

;(9~i~j)

QI,J,j

], I"1,2, 4, J"1,2, 4, (B.13)

DKI,J

"

E

12l10

5+i/1

3+j/1

[(ai!d

i)¸

(i`j`HI,J)S(9~i~j)

RI,J,j

], I"1,2, 4, J"1,2, 4,

(B.14)

AMI,J

"

ogl8

3+i/1

7+j/1

[ci¸

(i`j`HI,J);

(11~i~j)PI,J,j

], I"1,2, 4, J"1,2, 4, (B.15)

BMI,J

"

o12gl10

5+i/1

5+j/1

[Mdi¸(i`j`HI,J)

;(11~i~j)

QI,J,j

N

(B.16)

#(ai!d

i)M¸

(i`j`HI,J)<(11~i~j)

QI,J,j

N], I"1,2, 4, J"1,2, 4,

CMI,J

"

o12gl10

5+i/1

5+j/1

[Mai¸(i`j`HI,J)

;(11~i~j)

QI,J,j

N

(B.17)

#(di!a

i)M¸

(i`j`HI,J)<(11~i~j)

QI,J,j

N], I"1,2, 4, J"1,2, 4,


DMI,J

"

o12gl10

5+i/1

5+j/1

[(ai!d

i)¸

(i`j`HI,J)S(11~i~j)

QI,J,j

],

(B.18)I"1,2, 4, J"1,2, 4.

B.2. EVALUATION OF [EK] AND [FK]

[u1

u2

u3

u4]T[EK] [u

1u2

u3

u4]"P

l

0

P(z) ALw

bLz B

2dz

"Pl

0

oAX2

g CAe¸#

1

2¸2!ez

e!z2

eB!(e#ze)z!

1

2z2D A

Lwb

Lz B2dz

(B.19)

"Pl

0

oAX2

g Ae¸#

1

2¸2!ez

e!z2

eB ALw

bLz B

2dz

!Pl

0

oAX2

g(e#z

e) z A

Lwb

Lz B2dz!P

l

0

1

2

oAX2

gz2 A

Lwb

Lz B2dz.

Thus,

EKI,J

"

oX2

gl8 Ae¸#

1

2¸2!ez

e!z2

eB3+i/1

5+j/1

[Ci¸(i`j`HI,J)

;(9~i~j)

QI,J, j

]

!

oX2

gl8(e#z

e)

3+i/1

5+j/1

[Ci¸

(i`j`HI,J);

(10~i~j)Q

I,J,j]

(B.20)

!

oX2

2gl8

3+i/1

5+j/1

[Ci¸(i`j`HI,J)

;(11~i~j)

QI,J,j

], I"1,2, 4, J"1,2, 4.

Similarly,

FKI,J

"

2oX2

gl8

3+i/1

7+j/1

[Ci¸(i`j`HI,J)

;(11~i~j)

PI,J,j

], I"1,2, 4, J"I ,2, 4. (B.21)

APPENDIX C: NOMENCLATURE

A area of cross-sectionb breadth of beame o!setE Young's modulusg acceleration due to gravityG shear modulush depth of beamIxx

, Iyy

, Ixy

moment of inertia of beam cross-section about xx-, yy- and xy-axis[K] element sti!ness matrixl length of an element¸ length of total beam[M] element mass matrixt time parameteru nodal degrees of freedom


; strain energyv displacement in xz planew displacement in yz planex, y co-ordinate axesz co-ordinate axis and length parameterze

distance of the "rst node of the element from the root of the beamfrequency ratio ratio of modal frequency to frequency of fundamental mode of uniform beam with

the same root cross-section and without shear deformation e!ectsa depth taper ratio"h

1/h

2b breadth taper ratio"b1/b

2h angle of twisto weight densityk shear coe$cientX rotational speed of the beam (rad/s)

Subscriptsb bendings shear

Finite Element Vibration Analysis of Rotating Timoshenko Beams

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