the dynamic analysis of nonuniformly pretwisted timoshenko

International Journal of Mechanical Sciences 43 (2001) 2385–2405

The dynamic analysis of nonuniformly pretwisted Timoshenkobeams with elastic boundary conditions

Shueei-Muh Lina ; ∗, Wen-Rong Wanga, Sen-Yung LeebaMechanical 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 di.erential equations and the general elastic boundary conditions for the cou-pled bending–bending forced vibration of a nonuniform pretwisted Timoshenko beam are derived byHamilton’s principle. The closed-form static solution for the general system is obtained. The relationbetween the static solution and the 3eld transfer matrix is derived. Further, a simple and accurate mod-i3ed 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 Bernoulli–Euler beams can be easily examined by taking the correspondinglimiting procedures. The results are compared with those in the literature. Finally, the e.ects 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 reviewof the subject can be found in the literature [1]. For many purposes, it is satis3ed to neglectshear deformation and rotary inertia of beams and to use the proposed method given by Lin[2], based on Bernoulli–Euler beam theory. However, signi3cant errors are introduced if noaccount is taken of them for thicker beams. It is well known [2,3] that a pretwisted Bernoulli–Euler beam system is composed of two coupled governing di.erential equations and eightboundary conditions. A modi3ed transfer matrix method is a very useful tool to investigatethe vibrations of pretwisted blades where exact solutions are di;cult to obtain even for the

∗ Corresponding author.

0020-7403/01/$ - see front matter ? 2001 Elsevier Science Ltd. All rights reserved.PII: S 0020-7403(01)00018-2

2386 S.-M. Lin et al. / International Journal of Mechanical Sciences 43 (2001) 2385–2405

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) Young’s modulus of beam materialf(x; t) external transverse load in the z directionF(�; �) dimensionless external transverse load in the z direction,

f(x; t)L3=[E(0)Iyy(0)]I(x) area moment inertia of the beamKyTL, Ky�L, KyTR, Ky�R and translational and rotational spring constants at the leftKzTL, Kz�L, KzTR, Kz�R and the right end of the beam in the y and z directions,

respectivelyDK kinetic energyL length of the beamM (�) dimensionless mass, �(x)A(x)=[�(0)A(0)]p(x; t) external transverse load in the z directionP(�; �) 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 3eld and station transfer matrix[ DT ] overall transfer matrixu(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 3xed frame coordinates$1; $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,Kz�LL=[E(0)Iyy(0)], KzTLL3=[E(0)Iyy(0)], Ky�LL=[E(0)Iyy(0)],KyTLL3=[E(0)Iyy(0)]; Kz�RL=[E(0)Iyy(0)]; KzTRL3=[E(0)Iyy(0)],Ky�RL=[E(0)Iyy(0)], KyTRL3=[E(0)Iyy(0)]

% strain& dimensionless rotary inertia, Iyy(0)=[A(0)L2]� angle between principal and 3xed 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

S.-M. Lin et al. / International Journal of Mechanical Sciences 43 (2001) 2385–2405 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 di.erential equations and eight boundary conditions. A Timoshenko beam system ismore complicated than a Bernoulli–Euler beam system. Hence, it is necessary to develop anaccurate and simple method to solve the complicated problem and to 3nd the performance.

For Bernoulli–Euler beams, Dawson [4], Dawson and Carnegie [5] used the Rayleigh–Ritzmethod and transformation techniques to study the e.ects of uniform pretwist on the frequenciesof cantilever blades. Carnegie and Thomas [6] and Rao [7,8] used the Rayleigh–Ritz methodand Ritz–Galerkin method to study the e.ects of uniform pretwist and the taper ratio on thefrequencies of cantilever blades, respectively. Sabuncu [9] found by using the 3nite elementmethod that the e.ect of trigonometric pretwist angle on the frequencies increased as the pretwistangle increased. Rosard and Lestar [10] and Rao and Carnegie [11] used the transfer matrixmethod 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 di;culties 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 inIuence of the pretwist angle on thenatural frequencies of the beam with nonuniform pretwist is greater than those of the beam withuniform pretwist. The inIuence of the pretwist angle on the natural frequencies of higher modesis greater than on those of lower modes. The sti.er the boundary supports are, the greater theinIuence 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 inIuence of the shear deformation and the rotary inertia havebeen considered. Carnegie [12] determined the fundamental frequency of a cantilever beamby using Rayleigh’s principle. Dawson et al. [13] used the transformation method to studythe e.ects of shear deformation and rotary inertia on the natural frequencies. Gupta and Rao[14] and Abbas [15] used the 3nite element method to determine the natural frequencies ofuniformly pretwisted tapered cantilever blading. Subrahmanyam et al. [16] and Subrahmanyamand Rao [17] used the 3nite element method and the Reissner method to determine the natural


frequencies of uniformly pretwisted tapered cantilever blading, respectively. Celep and Turhan[18] used the Galerkin method to investigate the inIuence of nonuniform pretwisting on thenatural frequencies of uniform cross-sectional cantilever or simply supported beams. From theexisting 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 di.erential equations and the eight elastic bound-ary conditions for the coupled bending–bending forced vibration of a nonuniform pretwistedTimoshenko beam are derived by Hamilton’s principle. The closed-form static solution for thegeneral system is derived. A simple and accurate modi3ed transfer matrix method for studyingthe dynamic behavior of a Timoshenko beam with arbitrary pretwist is presented. The relationbetween the steady solution and the frequency equation is derived. The results are comparedwith those in the literature. Finally, the e.ects of the shear deformation, the rotary inertia, theratio 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 3elds of the beam are

u(x; t) =−(z-z(x; t) + y-y(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 DK of beam are

*=12

∫ L

0

∫A(+xx%xx + 2+xy%xy + 2+xz%xz) dA dx +

12Kz�L-2

z (0; t)

+12KzTLw2(0; t) +

12Ky�L-2

y(0; t) +12KyTLv2(0; t)

+12Kz�R-2

z (L; t) +12KzTRw2(L; t) +

12Ky�R-2

y(L; t) +12KyTRv2(L; t)

−∫ L

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

DK =12

∫ L

0

∫A

[(@w@t

)2+(@v@t

)2+(@u@t

)2]� dA dx: (3)

Application of Hamilton’s principle [2,3] yields the coupled governing di.erential equations andthe associated general elastic boundary conditions.


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

The four coupled dimensionless governing characteristic di.erential 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(�)@2-z@�2

− &Ryz(�)@2-y@�2

= 0; (6)

@@�

[Byz@-z@�

]+@@�

[Bzz@-y@�

]+S(�))

(@V@�

−-y)

−&Ryz(�)@2-z@�2

− &Rzz(�)@2-y@�2

= 0; � ∈ (0; 1); (7)

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

212

(Byy@-z@�

+ Byy@-y@�

)− 211-z = 0; (8)

222)

(@W@�

−-z)− 221W = 0; (9)

232

(Byz@-z@�

+ Bzz@-y@�

)− 231-y = 0; (10)

242)

(@V@�

−-y)− 241V = 0; (11)


at �= 1:

252

(Byy@-z@�

+ Byy@-y@�

)+ 251-z = 0; (12)

262S)

(@W@�

−-z)+ 261W = 0; (13)

272

(Byz@-z@�

+ Bzz@-y@�

)+ 271-z = 0; (14)

282S)

(@V@�

−-y)+ 281V = 0; (15)

where 2i1 = $i=(1 + $i) and 2i2 = 1=(1 + $i).Neglecting the shear deformation and the rotary inertia, the four coupled governing di.erential

equations are reduced to be the two coupled governing di.erential equations for forced vibrationof Bernoulli–Euler beam studied by Lin [3]. Moreover, neglecting the forcing terms, the systembecomes 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(�; �) andP(�; �) are given as follows:

F(�; �) = DF(�) cos!�; P(�; �) = DP(�) cos!�; (16)

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

V (�; �) = DV (�) cos!�; W (�; �) = DW (�) cos!�;

-y(�; �) = D-y(�) cos!�; -z(�; �) = D-z(�) cos!�:(17)

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

dd�

[S(�))

(d DWd�

− D-z

)]+!2M (�) DW =− DF(�); (18)


dd�

[S(�))

(d DVd�

− D-y

)]+!2M (�) DV =− DP(�); (19)

dd�

[Byy

d D-zd�

]+

dd�

[Byz

d D-yd�

]+S(�))

(d DWd�

− D-z

)+&!2Ryy(�) D-z+&!2Ryz(�) D-y=0;

(20)

dd�

[Byz

d D-zd�

]+

dd�

[Bzz

d D-yd�

]+S(�))

(d DVd�

− D-y

)

+ &!2Ryz(�) D-z + &!2Rzz(�) D-y = 0; � ∈ (0; 1); (21)

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

212

(Byy

d D-zd�

+ Byyd D-yd�

)− 211 D-z = 0; (22)

222)

(d DWd�

− D-z

)− 221 DW = 0; (23)

232

(Byz

d D-zd�

+ Bzzd D-yd�

)− 231 D-y = 0; (24)

242)

(d DVd�

− D-y

)− 241 DV = 0; (25)

at �= 1:

252

(Byy

d D-zd�

+ Byyd D-yd�

)+ 251 D-z = 0; (26)

262S)

(d DWd�

− D-z

)+ 261 DW = 0; (27)

272

(Byz

d D-zd�

+ Bzzd D-yd�

)+ 271 D-z = 0; (28)

282S)

(d DVd�

− D-y

)+ 281 DV = 0: (29)


3.1. Static analysis

When the frequency of excitation is zero, the inertia terms disappear and the system is reducedto be a static one. The corresponding shear forces are obtained by integrating the governingequations (18) and (19) once, respectively,

DQz(�) =S(�))

(d DWd�

− D-z

)=−

∫ �

�j

DF(4) d4+ c1; (30)

DQy(�) =S(�))

(d DVd�

− D-y

)=−

∫ �

�j

DP(4) d4+ c2: (31)

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

− DMz(�) = Byyd D-zd�

+ Byzd D-yd�

=∫ �

�j

∫ 4

�j

DF(6) d6 d4− c1(�− �j) + c3; (32)

− DMy(�) = Byzd D-zd�

+ Bzzd D-yd�

=∫ �

�j

∫ 4

�j

DP(6) d6 d4− c2(�− �j) + c4: (33)

Obviously, the coe;cients 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 D-zd�

=Bzz∫ �

�j

∫ 4

�j

DF(6) d6 d4− Byz∫ �

�j

∫ 4

�j

DP(6) d6 d4

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

(B2yz − BzzByy)d D-yd�

=Byz∫ �

�j

∫ 4

�j

DF(6) d6 d4− Byy∫ �

�j

∫ 4

�j

DP(6) d6 d4

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

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

D-z = 7p(�) + c171(�) + c272(�) + c373(�) + c474(�) + c5; (36)

D-y = ’p(�) + c1’1(�) + c2’2(�) + c3’3(�) + c4’4(�) + c6; (37)


where

7p(�) =∫ �

�j

Bzz(4)∫ 4

�j

∫ 6

�j

DF(9) d9 d6− Byz(4)∫ 4

�j

∫ 6

�j

DP(9) d9 d6

Bzz(4)Byy(4)− B2yz(4)d4;

’p(�) =∫ �

�j

Byz(4)∫ 4

�j

∫ 6

�j

DF(9) d9 d6− Byy(4)∫ 4

�j

∫ 6

�j

DP(9) d9 d6

B2yz(4)− Bzz(4)Byy(4)d4;

71(�) =∫ �

�j

−(4− �j)Bzz(4)Bzz(4)Byy(4)− B2yz(4)

d4;

72(�) =∫ �

�j

(4− �j)Byz(4)Bzz(4)Byy(4)− B2yz(4)

d4;

73(�) =∫ �

�j

Bzz(4)Bzz(4)Byy(4)− B2yz(4)

d4;

74(�) =∫ �

�j

−Byz(4)Bzz(4)Byy(4)− B2yz(4)

d4;

’1(�) = 72(�); ’3(�) = 74(�);

’2(�) =∫ �

�j

−(4− �j)Byy(4)Bzz(4)Byy(4)− B2yz(4)

d4;

’4(�) =∫ �

�j

Byy(4)Bzz(4)Byy(4)− B2yz(4)

d4:

(38)

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

DW = wp(�) + c1w1(�) + c2w2(�) + c3w3(�) + c4w4(�) + c5(�− �j) + c7; (39)

DV = vp(�) + c1v1(�) + c2v2(�) + c3v3(�) + c4v4(�) + c6(�− �j) + c8; (40)

where c5; c6; c7 and c8 are D-z(�j); D-y(�j); DW (�j) and DV (�j), respectively,

wp(�) =∫ �

�j

[7p(4)− )

S(4)

∫ 4

�j

DF(6) d6

]d4;

vp(�) =∫ �

�j

[’p(4)− )

S(4)

∫ 4

�j

DP(6) d6

]d4;

w1(�) =∫ �

�j

()S(4)

+ 71(4))d4; wi(�) =

∫ �

�j7i(4) d4; i = 2; 3; 4;

v2(�) =∫ �

�j

()S(4)

+ ’2(4))d4; vi(�) =

∫ �

�j’i(4) d4; i = 1; 3; 4:

(41)


Letting �j = 0 and � = 1, and substituting Eqs. (30)–(33) and (36)–(41) into the bound-ary conditions (22)–(29), the coe;cients 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 modi3ed transfer matrix method. The pre-sented method is the generalization of the method [2] which was derived to study only thefree vibration problem of pretwisted Bernoulli–Euler beams. It is assumed that the mass andthe rotary inertia of beam are concentrated in n + 1 lumps at n + 1 stations and the 3eldsbetween the concentrated masses are massless and without the e.ect of rotary inertia but posingvarying bending and shear sti.nesses. The beam is divided into n number of elements of equallength 1=n.

3.2.1. Extended 3eld transfer matrixLetting the domain of the jth 3eld is (�j; �j+1) and �= �j+1 in Eqs. (31)–(42), the jth 3eld

transfer matrix relation with arbitrarily varying coe;cients is obtained easily viaEqs. (30)–(41)

[ DWLj+1

D-Lz; j+1

DMLz; j+1

DQLz; j+1

DVLj+1

D-Ly;j+1

DMLy;j+1

DQLy;j+11]

T

= [Tf ]j[ DWRjD-Rz; j

DMRz; j

DQRz; j

DVRjD-Ry;j

DMRy;j

DQRy;j1]

T; (42)

where the superscript T is the transpose of matrix and the extended 3eld transfer matrixpertaining to the jth 3eld is

[Tf ]j=

1 M� −w3(�j+1) w1(�j+1) 0 0 −w4(�j+1) w2(�j+1) f1(�j+1)

0 1 −73(�j+1) 71(�j+1) 0 0 −74(�j+1) 72(�j+1) f2(�j+1)

0 0 1 M� 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 M� −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 M� f7(�j+1)

0 0 0 0 0 0 0 1 f8(�j+1)

0 0 0 0 0 0 0 0 1

;

(43)


in which M�= �j+1 − �j. The forcing terms aref1(�j+1) = wp(�j+1); f2(�j+1) = 7p(�j+1);

f3(�j+1) =−∫ �j+1

�j

∫ 4

�j

DF(6) d6 d4; f4(�j+1) =−∫ �j+1

�j

DF(4) d4;

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

f7(�j+1) =∫ �j+1

�j

∫ 4

�j

DP(6) d6 d4; f8(�j+1) =−∫ �j+1

�j

DP(4) d4:

(44)

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

d DQzd�

+!2m(�) DW (�) =− DF(�); (45)

d DQyd�

+!2m(�) DV (�) =− DP(�); (46)

−d DMzd�

+ DQz(�) + &!2Ryy(�) D-z(�) + &!2Ryz(�) D-y(�) = 0; (47)

−d DMyd�

+ DQy(�) + &!2Rzz(�) D-y(�) + &!2Ryz(�) D-z(�) = 0: (48)

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

DQRz; j = DQ

Lz; j −!2mj DWj − DFj; (49)

DQRy;j = DQ

Ly;j −!2mj DVj − DPj; (50)

DMRz; j = DM

Lz; j + &!

2Ryy;j D-z;j + &!2Ryz; j D-y;j; (51)

DMRy;j = DM

Ly;j + &!

2Ryz; j D-z;j + &!2Rzz; j D-y;j; (52)

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

61 = n∫ �2

0(�2 − 4)6(4) d4;

6j = n∫ �j

�j−1

(4− �j−1)6(4) d4+ n∫ �j+1

�j(�j+1 − 4)6(4) d4; j = 2; : : : ; n;

6n+1 = n∫ 1

�n(4− �n)6(4) d4;

(53)

in which 6 represents among m; Ryy; Ryz and Rzz. The continuity conditions are

DVRj = DV

Lj = DVj; DW

Rj = DW

Lj = DWj;

D-Ry;j = D-

Ly;j = D-y;j; D-

Rz; j = D-

Lz; j = D-z;j:

(54)


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

DWRj

D-Rz; j

DMRz; j

DQRz; j

DVRj

D-Ry;j

DMRy;j

DQRy;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 − DFj0 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 − DPj0 0 0 0 0 0 0 0 1

DWLj

D-Lz; j

DMLz; j

DQLz; j

DVLj

D-Ly;j

DMLy;j

DQLy;j

1

; (55)

where

<1 =−!2mj; <2 = &!2Ryy;j; <3 = &!2Ryz; j;

<4 =−!2mj; <5 = &!2Rzz; j:(56)

The 3rst 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 3eld 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

[ DW (1) D-z(1) DMz(1) DQz(1) DV (1) D-y(1) DMy(1) DQy(1) 1 ]T

= [ DT ][ DW (0) D-z(0) DMz(0) DQz(0) DV (0) D-y(0) DMy(0) DQy(0) 1 ]T; (57)

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

[ DT ] = [Ts]n+1

1∏i=n

[T ]i =

DT 11 DT 12 DT 13 DT 14 DT 15 DT 16 DT 17 DT 18 Df1








0 0 0 0 0 0 0 0 1

: (58)


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

0 211 212 0 0 0 0 0

221 0 0 −222 0 0 0 0

0 0 0 0 0 231 232 0

0 0 0 0 241 0 0 −242g11 g12 g13 g14 g15 g16 g17 g18g21 g22 g23 g24 g25 g26 g27 g28g31 g32 g33 g34 g35 g36 g37 g38g41 g42 g43 g44 g45 g46 g47 g48

DW (0)D-z(0)DMz(0)DQz(0)DV (0)D-y(0)DMy(0)DQy(0)

=

0

0

0

0

252 Df 3 − 251 Df 2

−262 Df 4 − 261 Df 1

272 Df 7 − 271 Df 6

−282 Df 8 − 281 Df 5

; (59)

where

g1j = 251 DT 2j − 252 DT 3j; g2j = 261 DT 1j + 262 DT 4j;

g3j = 271 DT 6j − 272 DT 7j; g4j = 281 DT 5j + 282 DT 8j; 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 e.ect 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 Bernoulli–Euler beams the e.ects 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 arepresented.

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 DWd�4

+(&+ ))!2 d2 DWd�2

+(&)!4−!2) DW = (1−&)!2)p∗>(�−�0)−)p∗d2>(�−�0)d�2

; (61)


at �= 0:

DW = 0; (62)

)d3 DWd�3

+ (1 + &))d DWd�

= 0; (63)

at �= 1:

d2 DWd�2

+ )!2 DW = 0; (64)

d3 DWd�3

+ (&+ ))!2 d DWd�

= 0: (65)

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

w1 =1

921 + 922(922 cosh 91�+ 9

21 cos 92�);

w2 =1

921 + 922

(92291

sinh 91�+92192

sin 92�);

w3 =1

921 + 922(cosh 91�− cos 92�);

w4 =1

921 + 922

(191

sinh 91�− 192

sin 92�);

(66)

where

91 =

√√√√−R1 +√R21 − 4R22

; 92 =

√√√√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 function

obtained by Lin [19]

DW = c2w2(�)+c3w3(�)+c4w4(�)+p∗[−)w2(�−�0)+()2!2+1)w4(�−�0)]H (�−�0);(68)

where

c2 =f1<4 − f2<2<1<4 − <2<3 ; c3 =

f2<1 − f1<3<1<4 − <2<3 ; c4 =

−(1 + )2!2))

c2; (69)


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)

! DW (0:2) DW (0:4) DW (0:6) DW (0:8) DW (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) [w′′4 (1) + )!

2w4(1)];

<2 = )[w′′3 (1) + )!

2w3(1)];

<3 = )[w′′′2 (1) + (&+ ))!2w′

2(1)]− (1 + )2!2) [w′′′4 (1) + (&+ ))!2w′

4(1)];

<4 = )[w′′′3 (1) + (&+ ))!2w′

3(1)];

f1 = )p∗{[)w′′2 (1− �0)− (1 + )2!2)w′′

4 (1− �0)]+)!2[)w2(1− �0)− (1 + )2!2)w4(1− �0)]}

f2 = )p∗{[)w′′′2 (1− �0)− (1 + )2!2)w′′′

4 (1− �0)]+ (&+ ))!2[)w′

2(1− �0)− (1 + )2!2)w′4(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 e;ciency and convergence of the proposed numerical method, the3rst 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


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 1110.0

a[A(0) = 0:4387 cm2, E = 206:85 GPa, G = 82:74 GPa,Iyy(0)= 0:0010906 cm4, IZZ(0)= 0:23586 cm4, 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 di.erence 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 + 61�); h(�) = h0(1 + 62�); (71)

where 61 and 62 are the taper ratios of the breadth and depth, respectively. The correspondingcoe;cients are

Ryy(�) = Byy(�) =(b0h0

)2(1 + 61�)3(1 + 62�) sin2 �+ (1 + 61�) (1 + 62�)3 cos2 �;

Rzz(�) = Bzz(�) =(b0h0

)2(1 + 61�)3(1 + 62�) cos2 �+ (1 + 61�) (1 + 62�)3 sin2 �;

Ryz(�) = Byz(�) =12sin 2�

[(b0h0

)2(1 + 61�)3(1 + 62�)− (1 + 61�) (1 + 62�)3

]:

(72)


Fig. 2. The inIuence of shear deformation, rotary inertia, ratio of bending rigidities and pretwist angle on thefrequencies of a cantilever beam [61 = 62 = 0, (——) )= 0:0031325, &= 0:001; (– – –) )= &= 0].

From Eq. (72), if the cross-section of the beam is square, i.e., b0 = h0, 61 = 62, 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 veri3cated in Fig. 2. When IYY =IZZ = 0:9, the inIuenceof the total pretwist angle on the frequencies is very small, the 3rst 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 e.ectof the rotary inertia and the shear deformation on the 3rst two natural frequencies. However,the e.ect 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 inIuence on the natural frequencies ofhigher modes is greater than on those of lower modes.

Fig. 3a shows the inIuence 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 Iuctuated with the variation of the total pretwist angle ,. The inIuence 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 di.erence of the 3rst 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 di.erent from that of the 3rst 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 3rst. Moreover, because the variation of the fourth mode shape


Fig. 3. (a) The inIuence of two kinds of nonuniform pretwist on the frequencies of a cantilever nonuniform beam[IYY =IZZ =0:05, 61 = 62 =−0:1, )=0:000031325, &=0:00001, (——) �= �(2− �),; (– – –) �= �,]; (b) The 3rstfour mode shapes of a cantilever nonuniform beam [IYY =IZZ = 0:05, 61 = 62 =−0:1, )= 0:000031325, &= 0:00001,�= B=2�, (——) W ; (– – –) V ]; (c) The 3rst four mode shapes of a cantilever nonuniform beam [IYY =IZZ = 0:05,61 = 62 =−0:1, )= 0:000031325, &= 0:00001, �= 35B=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 theinIuence of the total pretwist angle on the frequencies of beam with the spring constant 261of 0.9. It is observed that when the total pretwist angle is large, the 3rst frequency does notapproach the second. It is revealed in Fig. 4b that when ,= 480◦, the node points of the 3rstand second mode shapes are at �= 0 and � �= 0, respectively.


Fig. 4. (a) The inIuence of two kinds of nonuniform pretwist on the frequencies of a beam[261 = 0:9; 211 = 221 = 231 = 241 = 252 = 272 = 282 = 1, IYY =IZZ = 0:05, 61 = 62 = −0:1, ) = 0:000031325,& = 0:00001, (——) � = �(2 − �),; (– – –) � = �,]; (b) The 3rst two mode shapes of a nonuniform beam[261 = 0:9; 211 = 221 = 231 = 241 = 252 = 272 = 282 = 1, IYY =IZZ =0:05, 61 = 62 =−0:1, )=0:000031325, &=0:00001,�= 5B=3�, (——) W ; (– – –) V ].

5. Conclusion

The coupled governing di.erential equations and the general elastic boundary conditions forthe coupled bending–bending forced vibration of a nonuniform pretwisted Timoshenko beamare derived by Hamilton’s principle. The closed-form static solution for the general system isobtained. A simple and accurate modi3ed 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 inIuence of the taper ratio, thespring constants, the rotary inertia and the shear e.ect on the natural frequencies of highermodes is greater than on those of lower modes. The inIuence of the nonuniform pretwiston the frequencies is greater than that of the uniform pretwist. The systems of Rayleigh andBernoulli–Euler 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 3nite 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).


Appendix A. Static solution

The coe;cients of the general solutions (36)–(37) and (39)–(40) are

c1c2c3c4c5c6c7c8

=

0 0 212 0 211 0 0 0

−222 0 0 0 0 0 221 0

0 0 0 232 0 231 0 0

0 −242 0 0 0 0 0 241g1 g2 g3 g4 251 0 0 0

g5 g6 g7 g8 261 0 261 0

g9 g10 g11 g12 0 271 0 0

g13 g14 g15 g16 0 281 0 281

−1

0

0

0

0

f1

f2

f3

f4

; (A.1)

where

g1 = 25171(1)− 252; g2 = 25172(1); g3 = 25173(1)− 252;g4 = 25174(1); g5 = 261w1(1) + 262; g6 = 261w2(1);

g7 = 261w3(1); g8 = 261w4(1); g9 = 271’1(1);

g10 = 271’2(1)− 272; g11 = 271’3(1); g12 = 271’4(1)− 272;g13 = 281v1(1); g14 = 281v2(1) + 282; g15 = 281v3(1);

g16 = 281v4(1); f1 =−252e2 − 2517p(1); f2 = 262e1 − 261wp(1);

f3 =−272e4 − 271’p(1); f4 = 282e3 − 281vp(1);

(A.2)

in which

e1 =∫ 1

0

DF(4) d4; e2 =∫ 1

0

∫ 4

0

DF(6) d6 d4; e3 =∫ 1

0

DP(4) d4; e4 =∫ 1

0

∫ 4

0

DP(6) d6 d4:

(A.3)

References

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[3] Lin SM. Pretwisted nonuniform beams with time dependent elastic boundary conditions. AIAA Journal1998;36(8):1516–23.

[4] Dawson B. Coupled bending vibrations of pretwisted cantilever blading treated by Rayleigh–Ritz method.Journal of Mechanical Engineering Science 1968;10(5):381–6.

[5] Dawson B, Carnegie W. Modal curves of pre-twisted beams of rectangular cross-section. Journal of MechanicalEngineering Science 1969;11(1):1–13.

[6] Carnegie W, Thomas J. The coupled bending–bending vibration of pre-twisted tapered blading. Journal ofEngineering Industry 1972;94:255–66.


[7] Rao JS. Flexural vibration of pretwisted tapered cantilever blades. Journal of Engineering Industry1972;94:343–6.

[8] Rao JS. Advanced theory of vibration. New York: Wiley, 1992. p. 330–8.[9] Sabuncu M. Coupled vibration analysis of blades with angular pretwist of cubic distribution. AIAA Journal

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lateral vibration of blades allowing for the e.ect of pre-twist and rotation. International Journal of MechanicalEngineering Education 1973;1(1):37–47.

[12] Carnegie W. Vibrations of pretwisted cantilever blading. Proceeding of the Institution of Mechanical Engineers,vol. 173. London, 1959. p. 343–74.

[13] Dawson B, Ghosh NG, Carnegie W. E.ect of slenderness ratio on the natural frequencies of pre-twistedcantilever beams of uniform rectangular cross-section. Journal of Mechanical Engineering Science1971;13(1):51–9.

[14] Gupta RS, Rao JS. Finite element eigenvalue analysis of tapered and twisted Timoshenko beams. Journal ofSound and Vibration 1978;56(2):187–200.

[15] Abbas B. Simple 3nite elements for dynamic analysis of thick pre-twisted blades. The Aeronautical Journal1979;83:450–3.

[16] Subrahmanyam KB, Kulkarni SV, Rao JS. Coupled bending–bending vibrations of pretwisted cantilever bladingallowing for shear deIection and rotary inertia by the Reissner method. International Journal of MechanicalSciences 1981;23(9):517–30.

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the dynamic analysis of nonuniformly pretwisted timoshenko

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