Journal of Sound and Vibration (1995) 183(3), 403–415

VIBRATIONS OF ELASTICALLY RESTRAINEDNON-UNIFORM TIMOSHENKO BEAMS

S. Y. L

Department of Mechanical Engineering, National Cheng Kung University, Tainan,Taiwan 701, Republic of China

S. M. L

Department of Mechanical Engineering, Kung Shan Institute of Technology, Tainan,Taiwan 710, Republic of China

(Received 20 December 1993, and in final form 6 June 1994)

The free vibration of an elastically restrained symmetric non-uniform Timoshenko beamresting on a non-uniform elastic foundation and subjected to an axial load is studied. Thetwo coupled governing characteristic differential equations are reduced into two separatefourth order ordinary differential equations with variable coefficients in the angle ofrotation due to bending and the flexural displacement. The frequency equation is expressedin terms of the four normalized fundamental solutions of the associated differentialequation. A simple and efficient algorithm is developed to find the approximate fundamen-tal solutions of the governing characteristic differential equation. The relation betweenproblems with elastically restrained boundary conditions and those with tip-mass boundaryconditions is explored. Finally, several limiting cases are examined and examples are givento illustrate the validity and accuracy of the analysis. It is noted that the proposed analysiscan also be applied to stepped beam problems.

7 1995 Academic Press Limited

1. INTRODUCTION

In order to optimize the distribution of weight and strength, and sometimes to satisfyspecial architectural and functional requirements, non-uniform beams are widely used inmany structural applications. Therefore, the analysis of non-uniform beams is of interestto many mechanical, aeronautical, nuclear and civil engineers.

Based on the Bernoulli–Euler beam theory, the governing characteristic differentialequation of a non-uniform beam is a fourth order ordinary differential equation in theflexural displacement with variable coefficients. The problems of the static, dynamic andstability analysis of beams have been studied by many authors via many different methods.A brief review of these works can be found in the recent works by Lee et al. [1, 2].

For Timoshenko beams, the governing characteristic differential equations are twocoupled differential equations expressed in terms of two dependent variables: the flexuraldisplacement and the angle of rotation due to bending. It is known that if the coefficientsof the beam system are constants, then the two coupled differential equations can beuncoupled into two fourth order ordinary differential equations in the flexural displace-ment and the angle of rotation due to bending [3–9]. For non-uniform beams, the problemshave been mainly studied by approximate methods such as the finite element method[10, 11], the transfer matrix method [12] and the optimized Rayleigh–Ritz method [7].

403

0022–460X/95/280403+13 $12.00/0 7 1995 Academic Press Limited

. . . . 404

Recently, Lee and Lin [13] studied the free vibrations of non-uniform beams withattachments. They uncoupled the two coupled differential equations to provide one fourthorder ordinary differential equation in the angle of rotation due to bending. It was shownthat if the coefficients of the reduced differential equation are in polynomial form, thenthe exact solution for the free vibration can be obtained. However, in their study, axialloads and the elastic foundation were not considered.

In this paper, we extend the Lee and Lin study [13] and consider the free vibrations ofa general elastically restrained symmetric non-uniform Timoshenko beam with variablegeometrical and material properties, resting on a non-uniform Winkler elastic foundationand subjected to axial loads. The explicit relation between the flexural displacement andthe angle of rotation due to bending is established. The two coupled governing character-istic differential equations are reduced to one complete fourth order ordinary differentialequation with variable coefficients in the angle of rotation due to bending and another inthe flexural displacement. The frequency equation is expressed in terms of the fournormalized fundamental solutions of the associated differential equation. In order to beable to handle the problems, for which the closed form fundamental solutions of the systemare not available, a simple and efficient algorithm, which is an extension of the one givenby Lee and Kuo [14], is developed to find the approximate fundamental solutions. Inaddition, the relation between the problems with elastically restrained boundary conditionsand those with tip-mass boundary conditions is explored. Finally, several limiting casesare examined and examples are given to illustrate the validity and accuracy of the analysis.It should be noted that the proposed analysis can also be applied to stepped beamproblems.

2. ANALYSIS

2.1. -

Consider the free vibration of a general elastically restrained symmetric non-uniformTimoshenko beam, resting on a non-uniform Winkler elastic foundation K(x) andsubjected to axial loads, as shown in Figure 1. For time harmonic vibration with angularfrequency v, the two coupled dimensionless governing characteristic differential equationsof motion are

[−n(j)y'+ {q(j)/d}(y'−C)]'+ [s(j)V2 − k(j)]y=0, (1)

[r(j)C']'+ {q(j)/d}(y'−C)+ v(j)hV2C=0, j $ (0, 1), (2)

Figure 1. The geometry and co-ordinate of a general elastically end restrained non-uniform beam.

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and the associated boundary conditions are, at j=0,

bTLy=−n(0)y'+ (1/d)(y'−C), buLC=C', (3, 4)

and at j=1,

−buRC= r(1)C', −bTRy=−n(1)y'+ {q(1)/d}(y'−C). (5, 6)

Here primes indicate differentiation with respect to the dimensionless spatial variable j and

k(j)=K(j)L4/[E(0)I(0)], n(j)=N(j)L2/[E(0)I(0)],

q(j)= k(j)G(j)A(j)/[k(0)G(0)A(0)], r(j)=E(j)I(j)/[E(0)I(0)],

s(j)= r(j)A(j)/[r(0)A(0)], v(j)= J(j)/J(0),

y(j)=Y(j)/L, bTL =KTLL3/[E(0)I(0)],

bTR =KTRL3/[E(0)I(0)], buL =KuLL/[E(0)I(0)],

buR =KuRL/[E(0)I(0)], d=E(0)I(0)/[k(0)G(0)A(0)L2],

h= J(0)/[r(0)A(0)L2], j= x/L, V2 =m(0)v2L4/[E(0)I(0)], (7)

and where Y is the flexural displacement, C is the angle of rotation due to bending, G isthe shear modulus, E is Young’s modulus and k is the shear correction factor. A(x), I(x),J(x), L and r(j) represent the cross-sectional area, the area moment of inertia, the massmoment of inertia per unit length about the neutral axis, the length and the mass per unitvolume of the beam, respectively. KuL , KTL and KuR , KTR are the rotational and thetranslational spring constants at the left and right ends of the beam, respectively. N(x)is an axial compressive load. If the beam is subjected to an end axial load N0, then N(x)is a constant function N(x)=N0. Instead, if the beam is subjected to distributed axial loadsp(x), then N(x)= fL

x p(s) ds. For a constant cross-section Timoshenko beam, asp(x)= rAg, where g is the gravitational acceleration, the system becomes a hanging beamunder gravity [15].

It should be noted that if the two dimensionless elastic restraint constants buR and bTR

are replaced by two constants −aV2 and −gV2, respectively, where

a= Jm /[r(0)A(0)L3], g=M/[r(0)A(0)L], (8a, b)

in which Jm and M are the rotatory inertia of the attached mass and the concentrated massattached at the right end of the beam, respectively, the beam system with elastic restraintsat both ends becomes a beam system with elastic restraints at one end and tip-mass at theother end.

2.1.1. Governing differential equation in terms of the dimensionless flexural displacementEquation (1) can be written as

C'= (1/q){[(q− dn)y']'+ d(sV2 − k)y− q'C}. (9)

Upon substituting equation (9) into equation (2), one can explicitly express the angle ofrotation due to bending in terms of the dimensionless flexural displacement,

C=1z 6drq'

q2 [[(q− dn)y']'+ d(sV2 − k)y]− d$rq [[(q− dn)y']'+ d(sV2 − k)y]%'− qy'7,(10)

where z= d[r(q'/q)2 − (rq'/q)']+ dvhV2 − q. Substituting equation (10) back intoequation (1), one obtains a governing characteristic differential equation of motion which

. . . . 406

is a fourth order ordinary differential equation with variable coefficients, in terms of thedimensionless flexural displacement:

{(q/z){(rq'/q2)[[(q− dn)y']'+ d(sV2 − k)y]− [(r/q)[[(q− dn)y']'+ d(sV2 − k)y]]'

− [r(q'/q)2 − (rq'/q)'+ vhV2]y'}+ ny'}'− (sV2 − k)y=0. (11)

2.1.2. Governing differential equation in terms of the angle of rotation due to bendingFollowing the procedures used in the paper by Lee and Lin [13], one can explicitly

express the dimensionless flexural displacement in terms of the angle of rotation due tobending,

y=[1/(sV2 − k)]{(rC')0+(vhV2C)'− [(n/q)[d(rC')'+ (dvhV2 − q)C]}'}, (12)

and thus obtain a governing characteristic differential equation in terms of the angle ofrotation due to bending:

q{[1/(sV2 − k)][(rC')0+(vhV2C)'− [(n/q)[d(rC')'+ (dvhV2 − q)C]]']}'

+ d(rC')'+ (dvhV2 − q)C=0, j $ (0, 1). (13)

Substituting the relations between the dimensionless flexural displacement and the angleof rotation due to bending, equations (10) and (12), into the boundary conditions,equations (3)–(6), one can express the boundary conditions either for the dimensionlessflexural displacement or for the angle of rotation due to bending.

Let Vj (j), j=1, 2, 3, 4, be four linearly independent fundamental solutions of thecorresponding governing characteristic differential equation (11) or (13). The homo-geneous solution of the differential equation can thus be expressed as

V(j)=C1V1(j)+C2V2(j)+C3V3(j)+C4V4(j), (14)

where C1, C2, C3 and C4 are four constants to be determined. If the four fundamentalsolutions are such that they satisfy the normalization condition

V1(0) V2(0) V3(0) V4(0) 1 0 0 0

V'1 (0) V'2 (0) V'3 (0) V'4 (0) 0 1 0 0GG

G

K

kV01 (0) V02 (0) V03 (0) V04 (0)

GG

G

L

l

=GG

G

K

k0 0 1 0

GG

G

L

l

, (15)

V11 (0) V12 (0) V13 (0) V14 (0) 0 0 0 1

then, after substituting the solution (14) into the associated boundary conditions, theassociated frequency equation is obtained. The frequency equations in the dimensionlessflexural displacement and in the angle of rotation due to bending for a Timoshenko beamwith general elastic restraints at both ends and four limiting cases of the general systemare given in Appendices I and II, respectively.

2.2.

For a uniform Timoshenko beam resting on a uniform elastic foundation and subjectedto a uniform axial load, q= r= s= v=1. The governing characteristic differentialequations (11) and (13) are thus reduced to

(1− dn)y2+[V2(h+ d− dhn)+ n− dk]y0+(V2 − k)(dhV2 −1)y=0 (16)

and

(1− dn)C2+[V2(h+ d− dhn)+ n− dk]C0+(V2 − k)(dhV2 −1)C=0. (17)

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If there is no Winkler elastic spring and no axial load, equations (16) and (17) are reducedto

y2+V2(h+ d)y0+(dhV4 −V2)y=0, (18)

and

C2+V2(h+ d)C0+(dhV4 −V2)C=0, (19)

respectively. Equations (18) and (19) are exactly the same as those given byHuang [3].

2.3. - –

For Bernoulli–Euler beams, neither the shear deformation nor the rotatory inertia effectare considered. By letting d= vh=0 in equations (11) and (13), the governing character-istic differential equations become

(ry0)0+(ny')'− (sV2 − k)y=0, (20)

and

{[1/(sV2 − k)][(rC')0+(nC)']}'−C=0. (21)

Equation (20) is consistent with that given by Lee and Kuo [2], as are the associatedboundary conditions and the frequency equation.

3. APPROXIMATE NORMALIZED FUNDAMENTAL SOLUTIONS

In the previous sections, the frequency equations have been expressed in terms of thefour normalized fundamental solutions of the associated differential equation. It has beenshown by Lee and Lin [13] that if the coefficients of the reduced differential equation arein polynomial forms, then the exact solution for free vibration can be obtained. However,in many cases, closed form fundamental solutions of the differential equation are notavailable. Hence, approximate fundamental solutions are required. In this paper, a simpleand efficient algorithm, which is an extension of the one given by Lee and Kuo [14], isdeveloped to find the approximate fundamental solutions.

Upon expanding the governing characteristic differential equations (11), (13), (20) or(21). One finds that they all can be expressed in the form

p4(j)V2(j)+ p3(j)V1(j)+ p2(j)V0(j)+ p1(j)V'(j)+ p0(j)V(j)=0, j $ (0, 1), (22)

where V(j) may represent the angle of rotation due to bending or the dimensionlessflexural displacement.

By following the same procedures given by Lee and Kuo [14], i.e., by approximatingeach coefficient of the differential equation (22) pz (j), z=0, 1, 2, 3, 4 by n piecewiseconstants pi,z (si ), i=1, 2, . . . , n, one obtains a fourth order ordinary differential equationwith piecewise constant coefficients. Here si can be any value between [ai−1, ai ] and ai

denotes the co-ordinate position at the end of the ith subdomain. Consequently, one cangenerate a set of linearly independent normalized fundamental solutions of each subdo-main, V�i,j (j), j=1, 2, 3, 4. After applying the continuity conditions which require V, V',V0 and V1 to be continuous at the common interface of two neighboring subdomains,

. . . . 408

one can generate the normalized fundamental solutions of the whole domain Vj (j),j=1, 2, 3, 4. They can thus be expressed as

V1,j (j), j $ (0, a1)...

Vj (j)=gG

G

G

G

F

f

Vi,j (j), j $ (ai−1, ai ), hG

G

G

G

J

j

, (23)...Vn,j (j), j $ (an−1, 1), j=1, 2, 3, 4

in which Vi,j (j), j=1, 2, 3, 4, are the fundamental solutions in the ith subdomain. Thefundamental solutions of a higher subdomain, (ai , ai+1), ie 1, can be generated from thoseof neighboring lower subdomain, (ai−1, ai ), by the recurrence relation

Vi+1,j (j)=Vi,j (ai )V�i+1,1(j− ai )+V'i,j (ai )V�i+1,2(j− ai )+V0i,j (ai )V�i+1,3(j− ai )

+V1i,j (ai )V�i+1,4(j− ai ), j=1, 2, 3, 4, i=1, 2, 3, . . . , n−1, (24)

where V1,j (a1)=V�1,j (a1), j=1, 2, 3, 4.It can be observed that the approximate fundamental solutions are linearly independent

and satisfy the normalization condition (14) at the origin of the co-ordinate system. Whenthe number of subdomains approaches infinity, the approximate fundamental solutionsbecome the exact solutions. They can be obtained at any desired level of accuracy by takinga suitable number of subdomains.

It should be noted here that, while numerically determining the natural frequencies, therate and tendency of the convergence of the solutions will be different at the co-ordinateposition si in the piecewise constants pi,z (si ), i=1, 2, . . . , n, is selected as different valuesbetween [ai−1, ai ] [14]. In this paper, for a better rate of convergence, si is taken to be(ai−1 + ai )/2.

4. VERIFICATION AND EXAMPLES

To illustrate the validity and accuracy of the analysis and to explore the physical relationbetween the vibration of a beam with elastically restrained boundary conditions and thatwith a tip-mass, several limiting cases and numerical results are presented.

For a uniform Timoshenko beam resting on a uniform Winkler elastic foundation andsubjected to an axial load, the governing characteristic differential equation is equation(16). The associated normalized fundamental solutions are exact and are given as

V1 =1

e2 + t2 (t2 cosh ej+ e2 cos tj), V2 =1

e2 + t2 0t2

esinh ej+

e2

tsin tj1,

V3 =1

e2 + t2 (cosh ej−cos tj), V4 =1

e2 + t2 01e sinh ej−1t

sin tj1, (25)

where

e=X−A+zA2 −4B2

, t=XA+zA2 −4B2

, (26)

in which A=[V2(h+ d− dhn)+ n− dk]/(1− dn) and B=(V2 − k)(dhV2 −1)/(1− dn).If there is no Winkler elastic foundation and no axial load, the reduced frequency equationof a uniform Timoshenko beam with general elastic restraints listed in Appendix I isexactly the same as that given by Maurizi et al. [6].

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Upon substituting the four fundamental solutions (25) into the associated frequencyequation of a simply supported beam listed in Appendix I, one obtains

sin t=0. (27)

Equation (27) can also be expressed as

dhV4 − [m2p2(h+ d− dhn)+ dhk+1]V2 − [m2p2(n− dk)−m4p4(1− dn)− k]=0,

(28)

where m=1, 2, . . . . For a Bernoulli–Euler beam, it becomes

V2 =m4p4 −m2p2n+ k. (29)

If there is no Winkler elastic foundation and no axial load, the reduced forms of equations(27)–(29) are exactly the same as those given by Tuma and Cheng [16]. By letting m=1and V=0 in equations (28) and (29), one can obtain the buckling loads for theTimoshenko beam, ncr1 and that for the Bernoulli–Euler beam, ncr2, respectively. Here,

ncr1 = [p4 + k(1+ dp2)]/(p2 + dp4) and ncr2 = (p4 + k)/p2. (30, 31)

If there is no Winkler elastic foundation, equation (31) becomes the well-known criticalbuckling load [17].

In Table 1, the first three dimensionless natural frequencies of a clamped–freeTimoshenko beam with constant width and linearly varying thickness and attachment areshown. The Winkler elastic foundation and the axial load are not considered and thematerial properties of the beam are assumed to be constants. The relationshipsq= s=(1−0·4j) and r= v=(1−0·4j)3 are taken. The results obtained by the presentanalysis are compared with those given by Rossi et al. [10], who evaluated the frequenciesby the finite element method, and the exact solutions given by Lee and Lin [13]. It canbe found that the natural frequencies of the beams as determined from the frequencyequation expressed in terms of the angle of rotation due to bending are the same as thosedetermined from the frequency equation expressed in terms of the flexural displacement.It can be observed that the natural frequencies determined by the proposed methodconverge very rapidly. Even when the number of subdomains is only five, the differences

T 1

First three dimensionless frequencies of a cantilever non-uniform Timoshenko beam withattachment [q= s=(1−0·4j), r= v=(1−0·4j)3, a=0, g=0·32]; N is the number ofsubdomains taken; (*) present method as determined from the frequency equation in the angleof rotation due to bending; the exact solution, given by Lee and Lin [13], *, determined fromthe frequency equation in the angle of rotation due to bending, and **, determined from the

frequency equation in the dimensionless flexural displacement; ( Rossi et al. [10]

Present method (*) ExactZXXXXXXXCXXXXXXXV ZXXXCXXXV

h V N=5 N=10 N=15 N=20 * ** (0·0016 V1 2·103 2·100 2·099 2·099 2·099 2·099 2·09

V2 13·59 13·59 13·56 13·55 13·55 13·55 13·54V3 36·89 36·79 36·78 36·77 36·76 36·76 36·76

0·01 V1 2·019 2·016 2·015 2·015 2·015 2·015 2·01V2 11·10 11·08 11·07 11·07 11·07 11·07 11·07V3 25·69 25·64 25·64 25·64 25·63 25·63 25·67

. . . . 410

between the present solutions and the converged solutions are less than 0·2%. Theconvergent solutions and those given by Rossi et al. [10] are very consistent.

In Table 2, the first five dimensionless natural frequencies of non-uniform cantileverbeams resting on a Winkler elastic foundation and subjected to an axial load are shown.Both the width and the depth of the beams vary linearly with the same taper ratio, l.Consequently, q= s=(1+ lj)2 and r= v=(1+ lj)4. The natural frequencies have beenevaluated via Timoshenko, Rayleigh and Bernoulli–Euler beam theories, respectively. Itcan be observed that the natural frequencies of the Rayleigh and the Bernoulli–Eulerbeams determined by employing the frequency equation expressed in terms of the angleof rotation due to bending are almost the same as those determined by employing thefrequency equation expressed in terms of the flexural displacement. Comparing the naturalfrequencies of the Bernoulli–Euler beams with those given by Lau [18] shows that theresults are very consistent.

In Figure 2, the first four natural frequencies of a clamped–translational springsupported and a clamped–attached mass non-uniform Timoshenko beam are shown. Thebeam considered is a doubly tapered beam with q=s=(1−0·2j) and r=v=(1−0·2j)3.The real and dashed lines denote the natural frequencies of the clamped–translationalspring supported beam and those of the clamped–attached mass beam without consideringthe rotatory inertia effect of the attached mass, i.e., a=0, respectively. It can be observedthat when the translation spring constant bTR is increased, the natural frequencies of theclamped–translational spring supported beam increase. The natural frequencies will allapproach constant values as bTR is increased to a certain value, and the system becomesa clamped–hinged system. For the same beam with attached mass and a=0, the naturalfrequencies decrease when the concentrated mass constant g is increased. When g is

T 2

First five dimensionless frequencies for a non-uniform cantilever beam [q= s=(1−0·1 j)2,r= v=(1−0·1 j)4]; *, determined from the frequency equation in the angle of rotation dueto bending; **, determined from the frequency equation in the dimensionless flexural

displacement; (, Lau [18]

Rayleigh Bernoulli–EulerTimoshenko h=0.008, h=0,h=0·0008, d=0 d=0d=0.0025 ZXXXCXXXV ZXXXXXXCXXXXXXV

k n * * ** * ** (0 0 V1/2

1 1·9095 1·9150 1·9151 1·9167 1·9167 1·9167V1/2

2 4·5401 4·6162 4·6163 4·6422 4·6423 4·6422V1/2

3 7·3151 7·5917 7·5918 7·6934 7·6935 7·6934V1/2

4 9·8614 10·486 10·486 10·742 10·742 10·742V1/2

5 12·193 13·288 13·288 13·797 13·797 13·797

5 0 V1/21 2·0931 2·0973 2·0973 2·0991 2·0991 —

V1/22 4·5507 4·6303 4·6303 4·6563 4·6564 —

V1/23 7·3127 7·5947 7·5948 7·6965 7·6966 —

V1/24 9·8566 10·487 10·487 10·744 10·744 —

V1/25 12·188 13·288 13·288 13·798 13·798 —

5 1 V1/21 1·9123 1·9161 1·9161 1·9179 1·9179 —

V1/22 4·4501 4·5318 4·5318 4·5574 4·5575 —

V1/23 7·2592 7·5450 7·5451 7·6461 7·6462 —

V1/24 9·8185 10·454 10·454 10·710 10·710 —

V1/25 12·157 13·264 13·264 13·772 13·773 —

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Figure 2. The first four natural frequencies of a clamped–translational spring supported and aclamped–attached mass beam [q= s=(1−0·2j), r= v=(1−0·2j)3, a= buR =0, bTL and buL:a, h=0·0004,d=0·001 25]. ——, bTR ; - - - - , g.

increased to a certain value, the first natural frequency will vanish and the naturalfrequencies of the nth modes, nq 1, will be the same as those of the (n−1)th modes ofthe clamped–hinged system. This reveals that a clamped–hinged system is equivalent toa clamped–attached mass system with a=0 and g approaching infinity.

In Figure 3, for the same beam considered in Figure 2, the first four natural frequenciesof a clamped–elastic spring supported and a clamped–attached mass non-uniform

Figure 3. The first four natural frequencies of a clamped–elastic spring supported and a clamped–attachedmass beam [q= s=(1−0·2j), r= v=(1−0·2j)3, g, bTR , bTL and buL:a, h=0·0004, d=0·001 24]. ——, buR ;- - - , a.

. . . . 412

Timoshenko beam are shown. The real lines denote the natural frequencies of theclamped–elastic spring supported beam with the translational spring constant bTR beinginfinite and the rotational spring constant buR having various values. The dashed linesdenote the natural frequencies of the clamped–attached mass beam with g being infiniteand a having various values. When both buR and bTR approach infinity, a clamped–elasticspring supported system becomes a clamped–clamped system. This shows that aclamped–clamped system is equivalent to a clamped–attached mass system with g beinginfinite and a approaching infinity.

5. CONCLUSIONS

The two coupled governing characteristics differential equations for the free vibrationof a general elastically restrained symmetric non-uniform Timoshenko beam resting onnon-uniform Winkler elastic foundation and subjected to an axial load have been reducedinto two separate fourth order ordinary differential equations in the angle of rotation dueto bending and the flexural displacement, respectively. The relation between the flexuraldisplacement and the angle of rotation due to bending is derived. The frequency equationis expressed in terms of the four normalized fundamental solutions of the associateddifferential equation. These four approximate fundamental solutions can be obtainedthrough a simple and efficient algorithm.

ACKNOWLEDGMENT

This research work was supported by the National Science Council of Taiwan, R.O.C.,under grant NSC81-0401-E006-577 and is gratefully acknowledged.

REFERENCES

1. S. Y. L, H. Y. K and Y. H. K 1990 Journal of Sound and Vibration 142, 15–29. Analysisof non-uniform beam vibration.

2. S. Y. L and Y. H. K 1992 Transactions of the American Society of Mechanical EngineersJournal of Applied Mechanics 59, 205–212. Exact solutions for the analysis of general elasticallyrestrained non-uniform beams.

3. T. C. H 1961 Transactions of the American Society of Mechanical Engineers Journal ofApplied Mechanics 28, 579–584. The effect of rotatory inertia and of shear deformation on thefrequency and normal mode equations of uniform beams with simple end conditions.

4. T. J. R and F. S. W 1984 American Society of Civil Engineers Journal of EngineeringMechanics 111, 416–430. Timoshenko beam with rotational end constraints.

5. J. C. B and T. P. M 1987 Journal of Sound and Vibration 114, 341–345. Vibrationsof a mass-loaded clamped–free Timoshenko beam.

6. M. J. M, R. E. R and P. M. B 1990 Journal of Sound and Vibration 141, 359–362.Free vibrations of uniform Timoshenko beams with ends elastically restrained against rotationand translation.

7. R. H. G, P. A. A. L and R. E. R 1991 Journal of Sound and Vibration 145,341–344. Fundamental frequency of vibration of a Timoshenko beam of non-uniform thickness.

8. R. H. G 1991 Applied Acoustics 33, 141–152. Free vibrations of a Timoshenko beamof non-uniform cross-section elastically restrained against translation and rotation at both ends.

9. H. A and O. H 1992 Journal of Sound and Vibration 154, 67–80. Vibrationof a uniform cantilever Timoshenko beam with translational and rotational springs and witha tip mass.

10. R. E. R, P. A. A. L and R. H. G 1990 Journal of Sound and Vibration 143,491–502. A note on transverse vibrations of a Timoshenko beam of non-uniform thicknessclamped at one end and carrying a concentrated mass at the other.

11. W. L. C and B. T 1992 Journal of Sound and Vibration 152, 461–470. Finiteelement formulation of a tapered Timoshenko beam for free lateral vibration analysis.

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12. T. I, G. Y and I. T 1980 Journal of Sound and Vibration 70, 503–512.Vibration and stability of a non-uniform Timoshenko beam subjected to a follower force.

13. S. Y. L and S. M. L 1992 American Institute of Aeronautics and Astronautics Journal 30(12),2930–2934. Exact vibration solutions for nonuniform Timoshenko beams with attachments.

14. S. Y. L and Y. H. K 1992 Journal of Sound and Vibration 154, 441–451. Bending vibrationsof a rotating beam with an elastically restrained root.

15. T. Y 1990 Journal of Sound and Vibration 141, 245–258. Vibrations of a hangingTimoshenko beam under gravity.

16. J. J. T and F. Y. C 1983 Dynamic Structural Analysis. New York: McGraw-Hill.17. S. T and D. H. Y 1968 Elements of Strength of Materials. New York: Van

Nostrand.18. J. H. L 1984 Journal of Sound and Vibration 97, 513–521. Vibration frequencies for a

non-uniform beam with end mass.

APPENDIX I: THE FREQUENCY EQUATION OF NON-UNIFORM TIMOSHENKOBEAMS IN THE DIMENSIONLESS FLEXURAL DISPLACEMENT

1:

For this case, KuL , KTL , KuR and KTR are all constants and the frequency equation is

F14 F13 F12 F11

F24 F23 F22 F21

p=GG

G

G

GG34 G33 G32 G31

GG

G

G

G

=0,

G44 G43 G42 G41

where

F11 =B0/d+ bTLz=j=0, F12 = (B1 − z)/d− nz=j=0, F13 =B2/d=j=0,

F14 =B3/d=j=0, F21 = (buL + q')B0 − d(V2 − k)z=j=0,

F22 = (buL + q')B1 − (q'− dn')z=j=0, F23 = (buL + q')B2 − (1− dn)z=j=0,

F24 = (buL + q')B3=j=0, F31 = (−qbuR /r+ q')B0 − d(sV2 − k)z=j=1,

F32 = (−qbuR /r+ q')B1 − (q'− dn')z=j=1, F33 = (−qbuR /r+ q')B2 − (q− dn)z=j=1,

F34 = (−qbuR /r+ q')B3=j=1, F41 = qB0/d− bTRz=j=1,

F42 = q(B1 − z)/d− nz=j=1, F43 = qB2/d=j=1, F44 = qB3/d=j=1,

B0 = d2[(sV2 − k)(2rq'− qr')/q2 − (s'V2 − k')r/q],

B1 = d{(q'− dn')(2rq'− qr')/q2 − [(q0− dn0)+ d(sV2 − k)]r/q}− q,

B2 = d[(q− dn)(2rq'− qr')/q2 −2(q'− dn')r/q], B3 =−d(q− dn)r/q,

G3j =F34V1j (1)+F33V0j (1)+F32V'j (1)+F31Vj (1),

G4j =F44V1j (1)+F43V0j (1)+F42V'j (1)+F41Vj (1), j=1, 2, 3, 4.

2: –

For this case, KuL =KuR =0, KTL and KTR:a, the frequency equation is

p=F22G33G44 +F23G34G42 +F24G32G43 −F22G34G43

−F23G32G44 −F24G33G42 =0,

. . . . 414

where

F21 = q'B0 − d(V2 − k)z=j=0, F22 = q'B1 − (q'− dn')z=j=0,

F23 = q'B2 − (1+ dn)z=j=0, F24 = q'B3=j=0, F31 = q'B0 − d(sV2 − k)z=j=1,

F32 = q'B1 − (q'+ dn')z=j=1, F33 = q'B2 − (q+ dn)z=j=1, F34 = q'B3=j=1,

G3j =F34V1j (1)+F33V0j (1)+F32V'j (1)+F31Vj (1), G4j =Vj (1), j'=1, 2, 3, 4.

3: –

For this case, KuR =0, KuL , KTL and KTR:a, the frequency equation is the same as thatof case 2 where F21 =B0(0), F22 =B1(0), F23 =B2(0), F24 =B3(0) and the other parametersare the same as those of case 2.

4: –

For this case, KuL and KTL:a, KuR =KTR =0 and the frequency equation is the sameas that of case 2, where

F21 =B0(0), F22 =B1(0), F23 =B2(0), F24 =B3(0),

F41 = qB0/d=j=1, F42 = q(B1 − z)/d− nz=j=1, F43 = qB2/d=j=1,

F44 = qB3/d=j=1, G3j =F34V1j (1)+F33V0j (1)+F32V'j (1)+F31Vj (1),

G4j =F44V1j (1)+F43V0j (1)+F42V'j (1)+F41Vj (1), j=1, 2, 3, 4,

and the other parameters are the same as those of case 2.

5: –

For this case, KuR , KuL , KTL and KTR:a, the frequency equation is the same as thatof case 2, where

F21 =B0(0), F22 =B1(0), F23 =B2(0), F24 =B3(0),

G3j =B3(0)V1j (1)+B2(0)V0j (1)+B1(0)V'j (1)+B0(0)Vj (1),

G4j =Vj (1), j=1, 2, 3, 4

APPENDIX II: THE FREQUENCY EQUATION OF NON-UNIFORM TIMOSHENKOBEAMS IN THE ANGLE OF ROTATION DUE TO BENDING

1:

For this case, KuL , KTL , KuR and KTR are all constants and the frequency equation is

p=−(g1F3H1 + g4F4H3 + g2F1H4 − g4F3H4 − g2F4H1 − g1F1H3)+ g5(g1F3H2

+g3F4H3 + g2F2H4 − g3F3H4 − g2F4H2 − g1F2H3)=0,

where

g1 = bTL (1− dn)=j=0, g2 = {bTL [2r'(1− dn)+ d(n/q)']+ (V2 − k)(1− dn)}=j=0,

g3 = {bTL [r0(1− dn)+ hV2 − dr'(n/q)'− n(dhV2 −1)]+ r'(V2 − k)(1− dn)}=j=0,

g4 = {bTL [v'hV2(1− dn)+ nq'− (n/q)'(dhV2 −1)]+ [hV2(1− dn)+ n](V2 − k)}=j=0,

g5 =−buL , g6 =−buR /r(1), g7 = [bTRr(1− dn/q)]=j=1,

- 415

g8 = {bTR [2r'(1− dn/q)− dr(n/q)']− r(1− dn/q)(sV2 − k)}=j=1,

g9 = {bTR [(r0+ vhV2)(1− dn/q)+ n− dr(n/q)']− r'(1− dn/q)(sV2 − k)}=j=1,

g10 = {bTR [v'hV2(1− dn/q)+ nq'/q−(n/q)'(dvhV2 − q)]

−[vhV2(1− dn/q)+ n](sV2 − k)}=j=1,

Fj =V'j (1)+ g6Vj (1), Hj = g7V1j (1)+ g8V0j (1)+ g9V'j (1)+ g10Vj (1),

j=1, 2, 3, 4.

2: –

For this case, KuL =KuR =0, KTL and KTR:a, the frequency equation is

p= g1F3H1 + g4F4H3 + g2F1H4 − g4F3H4 − g2F4H1 − g1F1H3 =0,

where

g1 = (1− dn)=j=0, g2 = [2r'(1− dn)− d(n/q)']=j=0,

g3 = [r0(1− dn)+ hV2 − dr'(n/q)'− n(dhV2 −1)]=j=0,

g4 = [v'hV2(1− dn)+ nq'− (n/q)'(dhV2 −1)]=j=0,

g6 =0, g7 = r(1− dn/q)=j=1, g8 = [2r'(1− dn/q)− dr(n/q)']=j=1,

g9 = [(r0+ vhV2)(1− dn/q)+ n− dr(n/q)']=j=1,

g10 = [v'hV2(1− dn/q)+ nq'/q−(n/q)'(dvhV2 − q)]=j=1,

Fj =V'j (1)+ g6Vj (1), Hj = g7 V1j (1)+ g8V0j (1)+ g9V'j (1)+ g10Vj (1),

j=1, 2, 3, 4.

3: –

For this case, KuR =0, KuL , KTL and KTR:a, the frequency equation is

p= g1F3H2 + g3F4H3 + g2F2H4 − g3F3H4 − g2F4H2 − g1F2H3 =0,

and all the parameters are the same as those of case 2.

4: –

For this case, KuL and KTL:a, KuR =KTR =0, and the frequency equation is the sameas that of case 3, where

g7 =0, g8 =−r(1− dn/q)=j=1, g9 =−r'(1− dn/q)=j=1,

g10 =−[vhV2(1− dn/q)+ n]=j=1, Fj =V'j (1), j=1, 2, 3, 4,

and the other parameters are the same as those of case 2.

5: –

For this case, KuR , KuL , KTL and KTR:a, the frequency equation is the same as thatof case 3, where Fj =Vj (1), j=2, 3, 4, and the other parameters are the same as those ofcase 2.