dynamic analysis of non uniformly pretwisted timeshenko beams

7/28/2019 Dynamic Analysis of Non Uniformly Pretwisted Timeshenko Beams

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International Journal of Mechanical Sciences 43 (2001) 23852405

The dynamic analysis of nonuniformly pretwisted Timoshenkobeams with elastic boundary conditions

Shueei-Muh Lina ; , Wen-Rong Wanga, Sen-Yung Leeb

aMechanical Engineering Department, Kun Shan University of Technology, Tainan, Taiwan 710-03, ROC

bMechanical Engineering Department, National Cheng Kung University, Tainan, Taiwan 701, ROC

Received 2 February 2000; received in revised form 11 January 2001

Abstract

The coupled governing dierential equations and the general elastic boundary conditions for the cou-pled bendingbending forced vibration of a nonuniform pretwisted Timoshenko beam are derived byHamiltons principle. The closed-form static solution for the general system is obtained. The relationbetween the static solution and the eld transfer matrix is derived. Further, a simple and accurate mod-ied transfer matrix method for studying the dynamic behavior of a Timoshenko beam with arbitrarypretwist is presented. The relation between the steady solution and the frequency equation is revealed.

The systems of Rayleigh and BernoulliEuler beams can be easily examined by taking the correspondinglimiting procedures. The results are compared with those in the literature. Finally, the eects of theshear deformation, the rotary inertia, the ratio of bending rigidities, and the pretwist angle on the naturalfrequencies are investigated. ? 2001 Elsevier Science Ltd. All rights reserved.

1. Introduction

The analysis of the pretwisted beams is important in a number of designs of engineeringcomponents, e.g. turbine blades, helicopter rotor blades and gear teeth. An interesting review

of the subject can be found in the literature [1]. For many purposes, it is satised to neglectshear deformation and rotary inertia of beams and to use the proposed method given by Lin[2], based on BernoulliEuler beam theory. However, signicant errors are introduced if noaccount is taken of them for thicker beams. It is well known [2,3] that a pretwisted BernoulliEuler beam system is composed of two coupled governing dierential equations and eight

boundary conditions. A modied transfer matrix method is a very useful tool to investigatethe vibrations of pretwisted blades where exact solutions are dicult to obtain even for the

Corresponding author.

0020-7403/01/$ - see front matter? 2001 Elsevier Science Ltd. All rights reserved.PII: S 0 0 2 0 -7 4 0 3 (0 1 )0 0 0 1 8 - 2


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2386 S.-M. Lin et al. / International Journal of Mechanical Sciences 43 (2001) 23852405

Nomenclature

A(x) cross-sectional area of the beamBij() dimensionless bending rigidity, E(x)Iij(x)=[E(0)Iyy(0)];

i; j =x; yE(x) Youngs modulus of beam materialf(x; t) external transverse load in the z direction

F(; ) dimensionless external transverse load in the z direction,f(x; t)L3=[E(0)Iyy(0)]

I(x) area moment inertia of the beamKyTL,KyL,KyTR,KyR and translational and rotational spring constants at the leftKzTL, KzL, KzTR, KzR and the right end of the beam in the y and z directions,

respectivelyK kinetic energy

L length of the beamM() dimensionless mass, (x)A(x)=[(0)A(0)]p(x; t) external transverse load in the z direction

P(; ) dimensionless external transverse load in the y direction,p(x; t)L3=[E(0)Iyy(0)]

S() dimensionless shear rigidity, (x)G(x)A(x)=[(0)G(0)A(0)]t time variable[Tf]j; [Ts]j the jth eld and station transfer matrix

[ T] overall transfer matrix

u(x; t); v(x; t), and w(x; t) displacements in the x; y and z directions, respectivelyV(); W() dimensionless lateral displacement in the y and z directions,

respectively, v=L; w=LX; Y; Z principal frame coordinatesx; y; z xed frame coordinates1; 2; 3; 4; 5; 6; 7; 8 dimensionless rotational and translational spring constants

at the left and right of the beam in the y and z directions,respectively,

KzLL=[E(0)Iyy(0)], KzTLL3=[E(0)Iyy(0)], KyLL=[E(0)Iyy(0)],

KyTLL3=[E(0)Iyy(0)]; KzRL=[E(0)Iyy(0)]; KzTRL

3=[E(0)Iyy(0)],

KyRL=[E(0)Iyy(0)], KyTRL3

=[E(0)Iyy(0)] strain dimensionless rotary inertia, Iyy(0)=[A(0)L

2] angle between principal and xed frames dimensionless natural frequency,

L2

(0)A(0)=[E(0)Iyy(0)] dimensionless ratio between bending and shear rigidities,

E(0)Iyy(0)=[(0)G(0)A(0)L2]

dimensionless distance to the left end of the beam, x=L


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S.-M. Lin et al. / International Journal of Mechanical Sciences 43 (2001) 23852405 2387

total potential energy(x) the mass density per unit volume

stress dimensionless time, (t=L2)

E(0)Iyy(0)=[(0)A(0)]

tip pretwist angle of the beam, (L)y; z angle of rotation due to bending about the z and y directions,

respectively. natural frequency! dimensionless excitation frequency

simplest cases. Moreover, a pretwisted Timoshenko beam system is composed of four coupledgoverning dierential equations and eight boundary conditions. A Timoshenko beam system ismore complicated than a BernoulliEuler beam system. Hence, it is necessary to develop anaccurate and simple method to solve the complicated problem and to nd the performance.

For BernoulliEuler beams, Dawson [4], Dawson and Carnegie [5] used the RayleighRitzmethod and transformation techniques to study the eects of uniform pretwist on the frequenciesof cantilever blades. Carnegie and Thomas [6] and Rao [7,8] used the RayleighRitz methodand RitzGalerkin method to study the eects of uniform pretwist and the taper ratio on thefrequencies of cantilever blades, respectively. Sabuncu [9] found by using the nite elementmethod that the eect of trigonometric pretwist angle on the frequencies increased as the pretwistangle increased. Rosard and Lestar [10] and Rao and Carnegie [11] used the transfer matrix

method to determine the frequencies of vibration of the cantilever beam with uniform pretwist.Rosard and Lestar [10] assumed that the displacements at each element are linear. Rao andCarnegie [11] used an iteration procedure to determine the displacements at each element whilethe initial displacements were assumed to be linear. The diculties of the methods given byRosard and Lestar [10] and Rao and Carnegie [11] are overcome by Lin [2]. Lin [2] presenteda simple and accurate transfer matrix method for an elastically restrained nonuniform beamwith arbitrary pretwist. Moreover, it was found that the inuence of the pretwist angle on thenatural frequencies of the beam with nonuniform pretwist is greater than those of the beam withuniform pretwist. The inuence of the pretwist angle on the natural frequencies of higher modesis greater than on those of lower modes. The stier the boundary supports are, the greater theinuence of the pretwist angle on the natural frequencies. Lin [3] studied the force vibrationof an elastically restrained nonuniform beam with time-dependent boundary conditions. Thevibration control of a pretwisted beam with boundary inputs is investigated.

For Timoshenko beams the inuence of the shear deformation and the rotary inertia havebeen considered. Carnegie [12] determined the fundamental frequency of a cantilever beamby using Rayleighs principle. Dawson et al. [13] used the transformation method to studythe eects of shear deformation and rotary inertia on the natural frequencies. Gupta and Rao[14] and Abbas [15] used the nite element method to determine the natural frequencies ofuniformly pretwisted tapered cantilever blading. Subrahmanyam et al. [16] and Subrahmanyamand Rao [17] used the nite element method and the Reissner method to determine the natural


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frequencies of uniformly pretwisted tapered cantilever blading, respectively. Celep and Turhan[18] used the Galerkin method to investigate the inuence of nonuniform pretwisting on thenatural frequencies of uniform cross-sectional cantilever or simply supported beams. From the

existing literature, it can be found that all the previous investigations are restricted to cantileveror simply supported and tapered or uniform cross-sectional beams. There still is no study onthe analysis of an elastically restrained nonuniform Timoshenko beam with arbitrary pretwist.

In this paper, the four coupled governing dierential equations and the eight elastic bound-ary conditions for the coupled bendingbending forced vibration of a nonuniform pretwistedTimoshenko beam are derived by Hamiltons principle. The closed-form static solution for thegeneral system is derived. A simple and accurate modied transfer matrix method for studyingthe dynamic behavior of a Timoshenko beam with arbitrary pretwist is presented. The relation

between the steady solution and the frequency equation is derived. The results are comparedwith those in the literature. Finally, the eects of the shear deformation, the rotary inertia, the

ratio of bending rigidities, and the pretwist angle on the natural frequencies are investigated.

2. Governing equations and boundary conditions

Consider the forced vibration problem of a generally elastically restrained pretwisted nonuni-form Timoshenko beam as shown in Fig. 1. Both shear deformation and rotary inertia areconsidered. The displacement elds of the beam are

u(x; t) = (zz(x; t) + yy(x; t)); v(x; t) = v(x; t); w(x; t) = w(x; t); (1)

where u; v, and w are the displacements in the x; y, and z directions, respectively. y and zare the angle of rotation due to bending about the z and y directions, respectively. t is timevariable. The total potential energy and the kinetic energy K of beam are

=1

2

L0

A

(xxxx + 2xyxy + 2xzxz) dA dx +1

2KzL

2z (0; t)

+1

2KzTLw

2(0; t) +1

2KyL

2y (0; t) +

1

2KyTLv

2(0; t)

+1

2KzR

2z (L; t) +

1

2KzTRw

2(L; t) +1

2KyR

2y (L; t) +

1

2KyTRv

2(L; t)

L0

[f(x; t)w(x; t) + p(x; t)v(x; t)] dx; (2)

K =1

2

L0

A

@w

@t

2+

@v

@t

2+

@u

@t

2 dA dx: (3)

Application of Hamiltons principle [2,3] yields the coupled governing dierential equations andthe associated general elastic boundary conditions.


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Fig. 1. Geometry and coordinate system of a generally elastically restrained pretwisted beam.

The four coupled dimensionless governing characteristic dierential equations of motion areobtained

@

@

S()

@W

@ z

M()

@2W

@2= F(; ); (4)

@

@

S()

@V

@ y

M()

@2V

@2= P(; ); (5)

@

@

Byy

@z@

+

@

@

Byz

@y@

+

S()

@W

@ z

Ryy()@2z

@2 Ryz()

@2y

@2= 0; (6)

@

@

Byz

@z@

+

@

@

Bzz

@y@

+

S()

@V

@ y

Ryz()@2z@2

Rzz()@2y@2

= 0; (0; 1); (7)

and the associated dimensionless elastic boundary conditions areat = 0:

12

Byy

@z

@ +Byy

@y

@ 11z = 0; (8)

22

@W

@ z

21W = 0; (9)

32

Byz

@z@

+Bzz@y@

31y = 0; (10)

42

@V

@ y

41V = 0; (11)


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at = 1:

52

Byy

@z

@ +Byy

@y

@

+ 51z = 0; (12)

62S

@W

@ z

+ 61W = 0; (13)

72

Byz

@z@

+Bzz@y@

+ 71z = 0; (14)

82S

@V

@

y+ 81V = 0; (15)where i1 = i=(1 + i) and i2 = 1=(1 + i).

Neglecting the shear deformation and the rotary inertia, the four coupled governing dierentialequations are reduced to be the two coupled governing dierential equations for forced vibrationof BernoulliEuler beam studied by Lin [3]. Moreover, neglecting the forcing terms, the system

becomes to be an eigenvalue system studied by Lin [3].

3. Solution method

The steady solution of the generally elastically restrained pretwisted Timoshenko beamsubjected to a harmonic excitation is derived. The harmonic transverse excitations F(; ) and

P(; ) are given as follows:

F(; ) = F()cos !; P (; ) = P()cos !; (16)

where ! is the dimensionless frequency of excitation. The steady solutions can be assumed totake the form

V(; ) =

V()cos !; W (; ) =

W()cos !;y(; ) = y()cos !; z(; ) = z()cos !:

(17)

Substituting Eqs. (16)(17) into the governing equations (4)(7) and the boundary conditions(8)(15), the following governing ordinary dierential equations and boundary conditions areobtained, respectively,

d

d

S()

d W

d z

+ !2M() W = F(); (18)


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d

d

S()

d V

d y

+ !2M() V = P(); (19)

d

d

Byy

d zd

+

d

d

Byz

d yd

+

S()

d W

d z

+!2Ryy() z+!

2Ryz() y=0;

(20)

d

d

Byz

d zd

+

d

d

Bzz

d yd

+

S()

d V

d y

+ !2Ryz() z + !2Rzz() y = 0; (0; 1); (21)

and the associated dimensionless general elastic boundary conditions areat = 0:

12

Byy

d zd

+Byyd yd

11 z = 0; (22)

22

d W

d z

21 W = 0; (23)

32

Byzd zd +B

zz

d yd

31 y = 0; (24)

42

d V

d y

41 V = 0; (25)

at = 1:

52

Byy

d zd

+Byyd yd

+ 51 z = 0; (26)

62S

d W

d z

+ 61 W = 0; (27)

72

Byz

d zd

+Bzzd yd

+ 71 z = 0; (28)

82S

d V

d y

+ 81 V = 0: (29)


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3.1. Static analysis

When the frequency of excitation is zero, the inertia terms disappear and the system is reduced

to be a static one. The corresponding shear forces are obtained by integrating the governingequations (18) and (19) once, respectively,

Qz() =S()

d W

d z

=

j

F() d + c1; (30)

Qy() =S()

d V

d y

=

j

P() d + c2: (31)

Substituting Eqs. (30) and (31) into Eqs. (20) and (21), respectively, and integrating theseonce,

Mz() =Byyd zd

+Byzd yd

=

j

j

F() d d c1( j) + c3; (32)

My() =Byzd zd

+Bzzd yd

=

j

j

P() d d c2( j) + c4: (33)

Obviously, the coecients c1; c2; c3 and c4 are the corresponding shear forces and moments at

= j in the z and y directions, respectively. The following equations can be obtained easilyvia Eqs. (32) and (33)

(BzzByy B2yz)

d zd

=Bzz

j

j

F() d d Byz

j

j

P() d d

c1( j)Bzz + c2( j)Byz + c3Bzz c4Byz; (34)

(B2yzBzzByy)d yd

=Byz

j

j

F() d d Byy

j

j

P() d d

c1( j)Byz + c2( j)Byy + c3Byz c4Byy: (35)

Integrating Eqs. (34) and (35) once, respectively, one obtains

z = p() + c11() + c22() + c33() + c44() + c5; (36)

y = p() + c11() + c22() + c33() + c44() + c6; (37)


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where

p() =

j

Bzz()

j

j

F() d d Byz()

j

j

P() d d

Bzz()Byy() B2yz()d;

p() =

j

Byz()

j

j

F() d d Byy()

j

j

P() d d

B2yz() Bzz()Byy()d;

1() =

j

( j)Bzz()

Bzz()Byy() B2yz()d;

2() =

j

( j)Byz()

Bzz()Byy() B2yz()d;

3() =

j

Bzz()

Bzz()Byy() B2yz()d;

4() =

j

Byz()

Bzz()Byy() B2yz()d;

1() = 2(); 3() = 4();

2() =

j

( j)Byy()

Bzz()Byy() B2yz()d;

4() =

j

Byy()Bzz()Byy() B2yz()

d:

(38)

Substituting Eqs. (36) and (37) back into Eqs. (30) and (31), respectively, the displacementsare obtained

W = wp() + c1w1() + c2w2() + c3w3() + c4w4() + c5( j) + c7; (39)

V = vp() + c1v1() + c2v2() + c3v3() + c4v4() + c6( j) + c8; (40)

where c5; c6; c7 and c8 are z(j); y(j); W(j) and V(j), respectively,

wp() =

jp()

S()

j

F() dd;vp() =

j

p()

S()

j

P() d

d;

w1() =

j

S()+ 1()

d; wi() =

j

i() d; i = 2; 3; 4;

v2() =

j

S()+ 2()

d; vi() =

j

i() d; i = 1; 3; 4:

(41)


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Letting j = 0 and = 1, and substituting Eqs. (30)(33) and (36)(41) into the bound-ary conditions (22) (29), the coecients ci; i = 1; 2; : : : ; 8, of the general static solutions(36)(37) and (39)(40) are obtained and tabulated in Appendix A. It is shown that if the in-

tegrals (38) and (41) are done analytically, then the exact static solution is obtained. Otherwise,a semi-exact solution can be easily obtained by using the numerical integration method.

3.2. Dynamic analysis

When the frequency of excitation is not zero, the system composed of Eqs. (18)(29) presentsthe steady motion of a pretwisted Timoshenko beam subjected to harmonic transverse loadswith an excitation frequency !. The steady solution, the eigenvalue and the eigenfunctionsof the system are derived by using the following modied transfer matrix method. The pre-sented method is the generalization of the method [2] which was derived to study only the

free vibration problem of pretwisted BernoulliEuler beams. It is assumed that the mass andthe rotary inertia of beam are concentrated in n + 1 lumps at n + 1 stations and the eldsbetween the concentrated masses are massless and without the eect of rotary inertia but posingvarying bending and shear stinesses. The beam is divided into n number of elements of equallength 1=n.

3.2.1. Extended eld transfer matrixLetting the domain of the jth eld is (j; j+1) and = j+1 in Eqs. (31)(42), the jth eld

transfer matrix relation with arbitrarily varying coecients is obtained easily viaEqs. (30)(41)

[ WL

j+1 L

z; j+1 ML

z; j+1 QL

z; j+1 VL

j+1 L

y; j+1 ML

y; j+1 QL

y; j+11]T

= [Tf]j[ WR

j

R

z; jM

R

z; jQ

R

z; jV

R

j

R

y; jM

R

y; jQ

R

y; j1]T; (42)

where the superscript T is the transpose of matrix and the extended eld transfer matrixpertaining to the jth eld is

[Tf]j=

1 w3(j+1) w1(j+1) 0 0 w4(j+1) w2(j+1) f1(j+1)

0 1 3(j+1) 1(j+1) 0 0 4(j+1) 2(j+1) f2(j+1)

0 0 1 0 0 0 0 f3(j+1)

0 0 0 1 0 0 0 0 f4(j+1)

0 0 v3(j+1) v1(j+1) 1 v4(j+1) v2(j+1) f5(j+1)

0 0 3(j+1) 1(j+1) 0 1 4(j+1) 2(j+1) f6(j+1)

0 0 0 0 0 0 1 f7(j+1)

0 0 0 0 0 0 0 1 f8(j+1)

0 0 0 0 0 0 0 0 1

;

(43)


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in which = j+1 j. The forcing terms are

f1(j+1) = wp(j+1); f2(j+1) = p(j+1);

f3(j+1) = j+1

j

j

F() d d; f4(j+1) = j+1

j

F() d;

f5(j+1) = vp(j+1); f6(j+1) = p(j+1);

f7(j+1) =

j+1j

j

P() d d; f8(j+1) =

j+1j

P() d:

(44)

3.2.2. Extended station transfer matrixThe dimensionless governing dierential equations (19)(22) can be written as

d Qz

d

+ !2m() W() = F(); (45)

d Qyd

+ !2m() V() = P(); (46)

d Mzd

+ Qz() + !2Ryy() z() + !

2Ryz() y() = 0; (47)

d My

d+ Qy() + !

2Rzz() y() + !2Ryz() z() = 0: (48)

The distributions of the mass and the rotary inertia are determined by statics [2]. Eqs.(45)(48) in dierence form, applied at the jth station, yields

Q

R

z; j =

Q

L

z; j !2

mj

Wj

Fj; (49)Q

R

y; j =Q

L

y; j !2mj Vj Pj; (50)

MR

z; j = ML

z; j + !2Ryy;j z; j + !

2Ryz;j y; j; (51)

MR

y; j = ML

y; j + !2Ryz;j z; j + !

2Rzz; j y; j; (52)

where Fj and Pj are the concentrated forces at the jth station. The parameters mj ; Ryy;j ; Ryz;j,and Rzz; j are

1 = n 2

0

(2 )() d;

j = n

jj1

( j1)() d + n

j+1j

(j+1 )() d; j = 2; : : : ; n ;

n+1 = n

1n

( n)() d;

(53)

in which represents among m; Ryy; Ryz and Rzz. The continuity conditions are

VR

j = VL

j = Vj; WR

j = WL

j = Wj;

R

y; j =

L

y; j =y; j;

R

z; j =

L

z; j =z; j:

(54)


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Using Eqs. (50)(53), the extended station transfer matrix relation is obtained

WR

j

Rz; j

MR

z; j

QR

z; j

VR

j

R

y; j

MR

y; j

QR

y; j

1

=

1 0 0 0 0 0 0 0 0

0 1 0 0 0 0 0 0 0

0 2 1 0 0 3 0 0 0

1 0 0 1 0 0 0 0 Fj

0 0 0 0 1 0 0 0 0

0 0 0 0 0 1 0 0 0

0 3 0 0 0 5 1 0 0

0 0 0 0 4 0 0 1 Pj

0 0 0 0 0 0 0 0 1

WL

j

Lz; j

ML

z; j

QL

z; j

VL

j

L

y; j

ML

y; j

QL

y; j

1

; (55)

where

1 = !2mj; 2 = !

2Ryy;j ; 3 = !2Ryz;j ;

4 = !2mj; 5 = !

2Rzz; j:(56)

The rst square matrix at the right-hand side of Eq. (55) is the station transfer matrixpertaining to the jth station, denoted as [Ts]j. The extended transfer matrix [T]j for the jthstation and eld is the product of [Tf]j and [Ts]j. Letting = = 0 and neglecting the forcingterms, the extended transfer matrices are reduced to be those given by Lin [2].

3.2.3. Harmonic solution and frequency equationAccording to the above results, the overall transfer matrix relation is obtained

[ W(1) z(1) Mz(1) Qz(1) V(1)y(1) My(1) Qy(1) 1 ]

T

= [ T][ W(0) z(0) Mz(0) Qz(0) V(0)y(0) My(0) Qy(0) 1 ]

T; (57)

where the extended overall transfer matrix [ T] can be expressed as

[ T] = [Ts]n+1

1i=n

[T]i =

T11 T12 T13 T14 T15 T16 T17 T18 f1T21 T22 T23 T24 T25 T26 T27 T28 f2

T31 T32 T33 T34 T35 T36 T37 T38 f3T41 T42 T43 T44 T45 T46 T47 T48 f4T51 T52 T53 T54 T55 T56 T57 T58 f5T61 T62 T63 T64 T65 T66 T67 T68 f6T71 T72 T73 T74 T75 T76 T77 T78 f7T81 T82 T83 T84 T85 T86 T87 T88 f8

0 0 0 0 0 0 0 0 1

: (58)


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Substituting Eqs. (57) and (58) into the boundary conditions (22)(29), the following relationis obtained as follows:

0 11 12 0 0 0 0 0

21 0 0 22 0 0 0 0

0 0 0 0 0 31 32 0

0 0 0 0 41 0 0 42

g11 g12 g13 g14 g15 g16 g17 g18

g21 g22 g23 g24 g25 g26 g27 g28

g31 g32 g33 g34 g35 g36 g37 g38

g41 g42 g43 g44 g45 g46 g47 g48

W(0)z(0)

Mz(0)

Qz(0)

V(0)

y(0)

My(0)

Qy(0)

=

0

0

0

0

52 f 3 51f 2

62 f 4 61f 1

72 f 7 71f 6

82 f 8 81f 5

; (59)

where

g1j = 51 T2j 52 T3j; g2j = 61 T1j + 62 T4j;

g3j = 71 T6j 72 T7j; g4j = 81 T5j + 82 T8j; j = 1; 2; : : : ; 7; 8:(60)

The square matrix in Eq. (59) is denoted as G. Multiplying Eq. (59) by the inverse of thematrix, the state vector at = 0 can be obtained. For free vibration problems, the frequencyequation can be obtained by letting the determinant of the matrix G equal to zero.

For Rayleigh beams the eect of rotary inertia is considered and that of shear deformation

is neglected. By letting = 0, the corresponding steady response and frequency equation areobtained easily. For BernoulliEuler beams the eects of rotary inertia and shear deformationare neglected. Similarly, by letting = = 0, the corresponding steady response and frequencyequation are obtained easily. The reduced frequency equation is the same as that given byLin [2].

4. Numerical results and discussion

To illustrate the application of the method, compare the results with those in the exist-ing literature and explore the physical phenomena of the system, the following examples are

presented.

Example 1. Consider a cantilever unpretwisted uniform beam subjected to a unit concentratedforce in the z direction at = 0. The following governing equation and boundary conditionscan be obtained from Eqs. (18)(29)

d4 W

d4+( + )!2

d2 W

d2+(!4!2) W = (1!2)p(0)p

d2(0)

d2; (61)


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at = 0:

W = 0; (62)

d3 W

d3+ (1 + )

d W

d= 0; (63)

at = 1:

d2 W

d2+ !2 W = 0; (64)

d3 W

d3+ ( + )!2

d W

d= 0: (65)

The four fundamental solutions of Eq. (61) are obtained

w1 =1

21 + 22

(22 cosh 1 + 21 cos 2);

w2 =1

21 + 22

221

sinh 1 +212

sin 2

;

w3 =1

2

1

+ 2

2

(cosh 1 cos 2);

w4 =1

21 + 22

1

1sinh 1

1

2sin 2

;

(66)

where

1 =

R1 +R21 4R22

; 2 =

R1 +R21 4R22

; (67)

in which R1 = ( + )!2 and R2 = !

4 !2.

The exact solution of the system can be obtained by using the generalized Green functionobtained by Lin [19]

W = c2w2()+c3w3()+c4w4()+p[w2(0)+(

2!2+1)w4(0)]H(0);

(68)

where

c2 =f14 f2214 23

; c3 =f21 f1314 23

; c4 =(1 + 2!2)

c2; (69)


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Table 1The response of the cantilever unpretwisted uniform Timoshenko beam subjected to a unitconcentrated force in the z direction at 0 = 0:5. (p

= 0:1)

! W(0:2) W(0:4) W(0:6) W(0:8) W(1:0)

= 0:0001 1 a 0.00094 0.00317 0.00588 0.00864 0.01140

= 0:000312 b 0.00094 0.00318 0.00589 0.00864 0.01140

2 a 0.00121 0.00416 0.00787 0.01176 0.01571b 0.00122 0.00417 0.00787 0.01177 0.01571

3 a 0.00278 0.00978 0.01913 0.02946 0.04009b 0.00279 0.00980 0.01915 0.02949 0.04009

= 0:001 1 a 0.00100 0.00329 0.00604 0.00880 0.01158

= 0:00312 b 0.00100 0.00330 0.00605 0.00881 0.01158

2 a 0.00129 0.00434 0.00813 0.01208 0.01607b 0.00130 0.00435 0.00814 0.01209 0.01607

3 a 0.00306 0.01056 0.02051 0.03146 0.04270b 0.00307 0.01058 0.02053 0.03149 0.04270

aExact solutions.bResults obtained by the proposed method.

in which

1 = [w

2 (1) + !2w2(1)] (1 +

2!2) [w4 (1) + !2w4(1)];

2 = [w

3 (1) + !2w3(1)];

3 = [w

2 (1) + ( + )!2w2(1)] (1 +

2!2) [w4 (1) + ( + )!2w4(1)];

4 = [w

3 (1) + ( + )!2w3(1)];

f1 = p{[w2 (1 0) (1 +

2!2)w4 (1 0)]

+ !2[w2(1 0) (1 + 2!2)w4(1 0)]}

f2 = p{[w2 (1 0) (1 +

2!2)w4 (1 0)]

+ ( + )!2[w2(1 0) (1 + 2!2)w4(1 0)]}:

(70)

Its numerical results are listed in the rows with the mark a of Table 1. Those in therows b are determined by using the proposed method. As can be seen, these results are veryconsistent.

Example 2. To demonstrate eciency and convergence of the proposed numerical method, therst four frequencies are determined for a cantilever uniform pretwisted beam. For compari-son, the natural frequencies obtained by the proposed method as well as those given by other


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Table 2Convergence pattern and comparison of the frequencies of acantilever pretwisted uniform Timoshenko beam (Hz)a

n 1 2 3 4

4 60.3 282.4 820.4 1117.910 61.7 300.9 917.0 1175.120 61.9 303.8 933.1 1183.730 62.0 304.3 936.2 1185.340 62.0 304.5 937.3 1185.750 62.0 304.6 937.7 1186.060 62.0 304.6 938.0 1186.2

b 62.0 305.1 955.1 1214.7c 61.9 304.7 937.0 1205.1d

59.0 290.0 920.0 1110.0a[A(0) = 0:4387 cm2, E = 206:85 GPa, G = 82:74 GPa,

Iyy(0)=0:0010906 cm4, IZZ(0)=0:23586 cm

4, L = 15:24 cm, = 45

, = 0:847458, = 7857:6 kg=m3].bPotential energy method by Subrahmanyam et al. [16].cReissner method by Subrahmanyam et al. [16].dExperiment by Carnegie [12].

investigators are tabulated in Table 2. It is observed that fair agreement is obtained betweenthe numerical and experimental results and those by the proposed method. The natural fre-

quencies determined by the proposed method converge very rapidly. Even when the number ofsubsections is only 10, the dierence between the present fourth frequency and the convergedfrequency is less than 0.94%.

Example 3. Consider a doubly tapered beam subjected to a unit concentrated force in the zdirection. The breadth and depth of the beam are

b() = b0(1 + 1); h() = h0(1 + 2); (71)

where 1 and 2 are the taper ratios of the breadth and depth, respectively. The corresponding

coecients are

Ryy() =Byy() =

b0h0

2(1 + 1)

3(1 + 2)sin2 + (1 + 1) (1 + 2)

3 cos2 ;

Rzz() =Bzz() =

b0h0

2(1 + 1)

3(1 + 2)cos2 + (1 + 1) (1 + 2)

3 sin2 ;

Ryz() =Byz() =1

2sin 2

b0h0

2(1 + 1)

3(1 + 2) (1 + 1) (1 + 2)3

:

(72)


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Fig. 2. The inuence of shear deformation, rotary inertia, ratio of bending rigidities and pretwist angle on thefrequencies of a cantilever beam [1 = 2 = 0, () = 0:0031325, = 0:001; ( ) = = 0].

From Eq. (72), if the cross-section of the beam is square, i.e., b0 = h0, 1 = 2, IYY=IZZ = 1,and Byz =Ryz = 0, then the original system is uncoupled into two identical subsystems that areindependent of the pretwist angle. The fact for a cantilever beam twisted uniformly along itslength by an angle , i.e., = , is vericated in Fig. 2. When IYY=IZZ = 0:9, the inuence

of the total pretwist angle on the frequencies is very small, the rst frequency approaches thesecond, and the third approaches the fourth. The reason is that when the ratio IYY=IZZ approachesunity, the original system is uncoupled into two identical subsystems. There is almost no eectof the rotary inertia and the shear deformation on the rst two natural frequencies. However,the eect of those on the third and fourth natural frequencies is large especially for a beam witha small ratio of bending rigidities, IYY=IZZ. Moreover, its inuence on the natural frequencies ofhigher modes is greater than on those of lower modes.

Fig. 3a shows the inuence of nonuniform pretwist on the frequencies of a cantilever taperbeam. The real lines represent the frequencies of the beam with the nonuniform pretwistedangle of = (2 ). The dashed lines represent the frequencies of the beam with the uni-

form pretwisted angle of = . The variation of the pretwisted angle of the previous isgreater than that of the latter near = 0. It is observed that the frequencies of the higher modeare uctuated with the variation of the total pretwist angle . The inuence of the nonuni-form pretwist on the higher frequencies is greater than that of the uniform pretwist. Moreover,when the total pretwist angle is increased the dierence of the rst two frequencies and thefourth frequency are decreased. It is revealed in Figs. 3b and 3c that when = 90

, the nodepoints of the second mode shape are at = 0. When = 700

, the displacement v() ofthe second mode shape is dierent from that of the rst mode shape. But the node point of thesecond mode shape is at = 0. It is the reason that when the total pretwist angle is large, thesecond frequency approaches the rst. Moreover, because the variation of the fourth mode shape


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Fig. 3. (a) The inuence of two kinds of nonuniform pretwist on the frequencies of a cantilever nonuniform beam[IYY=IZZ = 0:05, 1 = 2 =0:1, = 0:000031325, = 0:00001, () = (2 ); ( ) = ]; (b) The rstfour mode shapes of a cantilever nonuniform beam [IYY=IZZ = 0:05, 1 = 2 =0:1, = 0:000031325, = 0:00001,

= =2, () W; ( ) V]; (c) The rst four mode shapes of a cantilever nonuniform beam [IYY=IZZ = 0:05,1 = 2 = 0:1, = 0:000031325, = 0:00001, = 35=9, () W; ( ) V]:

of the beam with = 90

is greater than that of the beam with = 700

, the fourth frequencyof the beam with = 90

is greater than that of the beam with = 700

. Fig. 4a shows theinuence of the total pretwist angle on the frequencies of beam with the spring constant 61of 0.9. It is observed that when the total pretwist angle is large, the rst frequency does notapproach the second. It is revealed in Fig. 4b that when = 480

, the node points of the rstand second mode shapes are at = 0 and = 0, respectively.


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Fig. 4. (a) The inuence of two kinds of nonuniform pretwist on the frequencies of a beam[61 = 0:9; 11 = 21 = 31 = 41 = 52 = 72 = 82 = 1, IYY=IZZ = 0:05, 1 = 2 = 0:1, = 0:000031325, = 0:00001, () = (2 ); ( ) = ]; (b) The rst two mode shapes of a nonuniform beam[61 = 0:9; 11 = 21 = 31 = 41 = 52 = 72 = 82 = 1, IYY=IZZ = 0:05, 1 = 2 =0:1, = 0:000031325, = 0:00001, = 5=3, () W; ( ) V].

5. Conclusion

The coupled governing dierential equations and the general elastic boundary conditions forthe coupled bendingbending forced vibration of a nonuniform pretwisted Timoshenko beamare derived by Hamiltons principle. The closed-form static solution for the general system isobtained. A simple and accurate modied transfer matrix method for studying the steady motionof the elastically restrained nonuniform Timoshenko beam with arbitrary pretwist is proposed.The relation between the steady solution and the frequency equation is revealed. When theratio of bending rigidities IYY=IZZ is unity, the general system is uncoupled into two identicalsubsystems that are independent to the pretwist angle. The inuence of the taper ratio, thespring constants, the rotary inertia and the shear eect on the natural frequencies of highermodes is greater than on those of lower modes. The inuence of the nonuniform pretwiston the frequencies is greater than that of the uniform pretwist. The systems of Rayleigh and

BernoulliEuler beams can be easily examined by taking the corresponding limiting procedures.It is well known that when the variations of the parameters including the pretwist angle arelarge, the required matrix dimension of the nite element method is increased greatly. However,the disadvantage does not exist in the proposed method.

Acknowledgements

The support of the National Science Council of Taiwan, ROC, is gratefully acknowledged(Grant number: Nsc88-2212-E168-003).


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Appendix A. Static solution

The coecients of the general solutions (36)(37) and (39)(40) are

c1

c2

c3

c4

c5

c6

c7

c8

=

0 0 12 0 11 0 0 0

22 0 0 0 0 0 21 0

0 0 0 32 0 31 0 0

0 42 0 0 0 0 0 41

g1 g2 g3 g4 51 0 0 0

g5 g6 g7 g8 61 0 61 0

g9 g10 g11 g12 0 71 0 0

g13 g14 g15 g16 0 81 0 81

1

0

0

0

0

f1

f2

f3

f4

; (A.1)

where

g1 = 511(1) 52; g2 = 512(1); g3 = 513(1) 52;

g4 = 514(1); g5 = 61w1(1) + 62; g6 = 61w2(1);

g7 = 61w3(1); g8 = 61w4(1); g9 = 711(1);

g10 = 712(1) 72; g11 = 713(1); g12 = 714(1) 72;

g13 = 81v1(1); g14 = 81v2(1) + 82; g15 = 81v3(1);

g16 = 81v4(1); f1 = 52e2 51p(1); f2 = 62e1 61wp(1);

f3 = 72e4 71p(1); f4 = 82e3 81vp(1);

(A.2)

in which

e1 =

10

F() d; e2 =

10

0

F() d d; e3 =

10

P() d; e4 =

10

0

P() d d:

(A.3)

References

[1] Rosen A. Structural and dynamic behavior of pretwisted rods and beams. Applied Mechanics Review

1991;44(12):483515.[2] Lin SM. Vibrations of elastically restrained nonuniform beams with arbitrary pretwist. AIAA Journal

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[3] Lin SM. Pretwisted nonuniform beams with time dependent elastic boundary conditions. AIAA Journal1998;36(8):151623.

[4] Dawson B. Coupled bending vibrations of pretwisted cantilever blading treated by RayleighRitz method.Journal of Mechanical Engineering Science 1968;10(5):3816.

[5] Dawson B, Carnegie W. Modal curves of pre-twisted beams of rectangular cross-section. Journal of MechanicalEngineering Science 1969;11(1):113.

[6] Carnegie W, Thomas J. The coupled bendingbending vibration of pre-twisted tapered blading. Journal ofEngineering Industry 1972;94:25566.


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[7] Rao JS. Flexural vibration of pretwisted tapered cantilever blades. Journal of Engineering Industry1972;94:3436.

[8] Rao JS. Advanced theory of vibration. New York: Wiley, 1992. p. 3308.

[9] Sabuncu M. Coupled vibration analysis of blades with angular pretwist of cubic distribution. AIAA Journal1985;23(9):142430.[10] Rosard DD, Lester PA. Natural frequencies of twisted cantilever beams. ASME Journal of Applied Mechanics

1953;20:2414.[11] Rao JS, Carnegie W. A numerical procedure for the determination of the frequencies and mode shapes of

lateral vibration of blades allowing for the eect of pre-twist and rotation. International Journal of MechanicalEngineering Education 1973;1(1):3747.

[12] Carnegie W. Vibrations of pretwisted cantilever blading. Proceeding of the Institution of Mechanical Engineers,vol. 173. London, 1959. p. 34374.

[13] Dawson B, Ghosh NG, Carnegie W. Eect of slenderness ratio on the natural frequencies of pre-twistedcantilever beams of uniform rectangular cross-section. Journal of Mechanical Engineering Science1971;13(1):519.

[14] Gupta RS, Rao JS. Finite element eigenvalue analysis of tapered and twisted Timoshenko beams. Journal ofSound and Vibration 1978;56(2):187200.

[15] Abbas B. Simple nite elements for dynamic analysis of thick pre-twisted blades. The Aeronautical Journal1979;83:4503.

[16] Subrahmanyam KB, Kulkarni SV, Rao JS. Coupled bendingbending vibrations of pretwisted cantilever bladingallowing for shear deection and rotary inertia by the Reissner method. International Journal of MechanicalSciences 1981;23(9):51730.

[17] Subrahmanyam KB, Rao JS. Coupled bendingbending cantilever beams treated by the Reissner method.Journal of Sound and Vibration 1982;82(4):57792.

[18] Celep Z, Turhan D. On the inuence of pretwisting on the vibration of beams including the shear and rotatoryinertia eects. Journal of Sound and Vibrations 1986;110(3):5238.

[19] Lin SM. Exact solutions for extensible circular Timoshenko beams with nonhomogeneous elastic boundaryconditions. ACTA Mechanic 1998;130:6779.

dynamic analysis of non uniformly pretwisted timeshenko beams

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