C. Mei Beams and Wave Analysis of Timoshenko Beam Structures

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C. Meie-mail: [email protected]

Department of Mechanical Engineering,The University of Michigan-Dearborn,

4901 Evergreen Road,Dearborn, MI 48128

B. R. Macee-mail: [email protected]

Institute of Sound and Vibration Research,University of Southampton, Highfield,

Southampton SO17 1BJ, UK

Wave Reflection andTransmission in TimoshenkoBeams and Wave Analysis ofTimoshenko Beam StructuresThis paper concerns wave reflection, transmission, and propagation in Timoshenkobeams together with wave analysis of vibrations in Timoshenko beam structures. Thetransmission and reflection matrices for various discontinuities on a Timoshenko beamare derived. Such discontinuities include general point supports, boundaries, andchanges in section. The matrix relations between the injected waves and externally ap-plied forces and moments are also derived. These matrices can be combined to provide aconcise and systematic approach to vibration analysis of Timoshenko beams or complexstructures consisting of Timoshenko beam components. The approach is illustrated withseveral numerical examples. �DOI: 10.1115/1.1924647�

1 IntroductionThe vibrations of elastic structures, such as strings, beams, and

plates, can be described in terms of waves propagating and decay-ing in waveguides. Such waves are reflected and transmitted whenincident upon discontinuities �1,2�. As a general approach, thetraveling wave method has proved to be powerful for analyzingvibrations in complex structural networks �3,4�. The practical ap-plication of such an approach, however, relies on knowledge ofthe detailed propagation, reflection, and transmission characteris-tics of waves. The reflection and transmission matrices of wavesin Euler-Bernoulli beams corresponding to various discontinuitieswere derived by Mace �5�. Harland et al. �6� considered the re-flection and transmission characteristics in tunable fluid-filledEuler-Bernoulli beams. Wave reflection and transmission in anaxially strained, rotating Timoshenko shaft was studied by Tanand Kang �7,8�. Doyle considered the propagation of elasticwaves in various one-dimensional structures �9�.

The purpose of this paper is to derive expressions for the propa-gation, reflection, and transmission matrices for waves in a Ti-moshenko beam. In particular, the coupling effect of translationaland rotational motion at a general discontinuity is taken into ac-count. The vectors of wave amplitudes for waves excited by ex-ternal forces and moments acting on a Timoshenko beam are alsoderived. These are required for the analysis of forced vibrationsusing the wave approach. Together, these form a concise and sys-tematic approach for the vibration analysis of Timoshenko beamsor built-up structures consisting of Timoshenko beam compo-nents. Some of the results derived here, those concerning propa-gation for example, are relatively well known and are presentedhere for completeness. The main contribution of this paper is thederivation of general reflection and transmission matrices thatfully include the effects of both wave types—propagating anddecaying.

The study of the dynamics and vibrations of Timoshenko beamsis of practical importance because engineering structures are oftenfabricated from a number of components held together by struc-tural elements that can be modeled as one-dimensional beamstructures. The Euler-Bernoulli beam model, as is well known,considers only the lateral inertia and the elastic forces caused by

Contributed by the Technical Committee on Vibration and Sound for publicationin THE JOURNAL OF VIBRATION AND ACOUSTICS. Manuscript received June 8, 2004. Final
manuscript received December 17, 2004. Associate Editor: Chin An Tan.
382 / Vol. 127, AUGUST 2005 Copyright © 2

: https://vibrationacoustics.asmedigitalcollection.asme.org on 06/15/2019 Terms o

bending deflections; the effects of rotary inertia and shear distor-tion are neglected. As a result, the theory is not valid for higherfrequencies, typically when the transverse dimensions are not neg-ligible with respect to the wavelength. Rayleigh �10� introducedthe effect of rotary inertia and Timoshenko �11,12� extended it toinclude the effect of transverse shear deformation. Following Ti-moshenko, several authors obtained the frequency equations andthe mode shapes for various boundary conditions �13,14� using amodal approach. The modal vibration analysis becomes morecomplex when Timoshenko rather than Euler-Bernoulli beamtheory has to be used; that is, when the effects of rotary inertia ofthe mass and the shear distortion cannot be neglected. The diffi-culty increases considerably with the increasing complexity of thestructure. Vibrations of Timoshenko beams have also been ana-lyzed using Green’s function representations �15�, the dynamicstiffness approach �16�, and finite element models �17�.

If the propagation, reflection, transmission, and excitation ma-trices are known, the structural vibration analysis �for time-harmonic behavior� involves a number of matrix operations in-volving the relevant matrices for each part of the structure. Thesecan be built up systematically regardless of the complexity of thestructure. Reference �4� considers the most general situations. Thebenefit of the wave approach arises mainly from the fact that, bydefinition, the reflection and transmission matrices satisfy theboundary conditions at a discontinuity, and the wave componentsand propagation matrices themselves necessarily satisfying thedifferential equation of motion between discontinuities. Theanalysis involves �in principle� the same computational cost what-ever the frequency; which is in contrast to finite element analysis,for example, where the size of the mesh must be refined as fre-quency increases.

The wave approach provides a concise and systematic approachto the analysis of built-up structures. Furthermore, the solutioncan be systematically developed in a well-conditioned manner bytaking appropriate account of the direction in which decayingwave components attenuate.

This paper is organized as follows. In Sec. 2, the equation ofmotion for Timoshenko beams is presented and expressions forthe propagation of waves derived. In Sec. 3, the reflection andtransmission matrices at discontinuities caused by general pointsupports, boundaries, and change in sections are derived. The re-flection and transmission characteristics of some example discon-tinuities are presented at the end of this section, including a
pinned point on a uniform beam, a change in section, and an angle
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joint. Reflection characteristics of common boundaries are alsostudied. In Sec. 4, the vectors of wave amplitudes for waves gen-erated by externally applied point forces and moments in a Ti-moshenko beam are derived. In Sec. 5, these matrices are com-bined to provide a concise and systematic approach for vibrationanalysis of Timoshenko beams or structures consisting of Timosh-enko beam components. The approach is illustrated through sev-eral numerical examples including free and forced vibrationanalysis of a uniform and a stepped Timoshenko beam. Conclud-ing remarks are given in Sec. 6.

2 Equation of Motion and Wave PropagationConsider the forces and moments acting on an element of a

beam lying along the x-axis as shown in Fig. 1. The equations ofmotion are �1�

GA�� x,t��x

−�2y�x,t�

�x2 � + �A�2y�x,t�

�t2 = q�x,t� �1a�

EI�2��x,t�

�x2 + GA�� y�x,t��x

− ��x,t�� − �I�2y�x,t�

�t2 = 0 �1b�

A list of notations is given in the Nomenclature. �y�x , t� /�x is theslope of the centerline of the beam and �y�x , t� /�x−��x , t� theshear angle. It can be seen that Eqs. �1a� and �1b� are coupledthrough the slope and the transverse deflection of the structure.

The shear force V�x , t� and bending moment M�x , t� at anysection of the beam are related to the transverse deflection y�x , t�and the slope ��x , t� by

V�x,t� = GA�� y�x,t��x

− ��x,t�� 2�

M�x,t� = − EI��x,t�

�x�3�

The coefficients

Cb =�EI

�A, Cs =�GA�

�A, Cr =� �I

�A�4�

which are related to the bending stiffness, shear stiffness and ro-tational effects, respectively, are now introduced. The shear beam,the Rayleigh beam, and the simple Euler-Bernoulli beam modelscan be obtained from the Timoshenko beam model by setting Crto zero �that is, ignoring the rotational effect�, Cs to infinity �ig-noring the shear effect�, and setting both Cr to zero and Cs to

Fig. 1 Free-body diagram and geometry of a beam element †1‡

infinity, respectively.

Journal of Vibration and Acoustics


2.1 Free Wave Propagation. Consider the free vibrationproblem when no external force is applied to the beam. Eliminat-ing ��x , t� from Eqs. �1�, one obtains the differential equation ofmotion

EI�4y�x,t�

�x4 + �A�2y�x,t�

�t2 − �I�1 +E

G� �4y�x,t�

�x2�t2 +�2I

G�

�4y�x,t��t4 = 0

�5�

The same equation holds for ��x , t�.Assuming time-harmonic motion and using separation of vari-

ables, the solution to Eq. �5� can be written in the form y�x , t�=y0e−ikxei�t, where � is the frequency and k the wavenumber.Substituting this into Eq. �5� gives a polynomial in k2, the disper-sion equation. The solution to the dispersion equation gives a setof wavenumbers that are functions of the frequency � as well asthe properties of the structure, namely,

k = ± 1

2�� 1

Cs2

+ �Cr

Cb2��2 ±��2

Cb2 +

1

4�� 1

Cs2

− �Cr

Cb2�2

�4�1/2

�6�

Waves in the beam travel in both the positive and negativedirections, as “�” outside the brackets indicates. It can be seenfrom Eq. �6� that the waves are dispersive, in that the phase ve-locity � /k is frequency dependent. From Eq. �6�, it is also evidentthat, in the absence of damping, one pair of wavenumbers is al-ways real, which corresponds to positive- and negative-goingpropagating waves. However, the other pair can be imaginary orreal, depending on the frequency range under consideration. Thismeans that the pair of waves is either decaying, evanescent, ornearfield waves �at low frequencies� or propagating waves �athigh frequencies�. There exists a wave-mode transition at a cutofffrequency �c, which is given by

�c =Cs

Cr�7�

Below the cutoff frequency �c, there exist a pair of propagatingwaves and a pair of decaying waves in the beam, while above thecutoff frequency �c, there are two pairs of propagating waves. Inaudio-frequency applications, the former case is overwhelminglythe most common.

With the time dependence ei�t suppressed, the solution to Eq.�5� can be written as

y�x� = a1+e−ik1x + a2

+e−k2x + a1−eik1x + a2

−ek2x �8a�

��x� = a1+e−ik1x + a2


−ek2x �8b�

where the wavenumbers are

k1 = 1

2�� 1

Cs2

+ �Cr

Cb2��2

+��2

Cb2 +

1

4�� 1

Cs2

− �Cr

Cb2�2

�4�1/2

�9a�

k2 = � 1

2�� 1

Cs2

+ �Cr

Cb2��2

−��2

Cb2 +

1

4�� 1

Cs2

− �Cr

Cb2�2

�4��1/2

�9b�

The signs of the roots are evaluated such that Im k1��0, Re k1��0 if Im k1�=0, and Re k2��0, Im k2��0 if Re k2�=0. Theinterpretation is that a1

+ and a1− correspond to positive- and

negative-going propagating waves, whereas a2+ and a2

− correspond
to positive and negative decaying waves for ��c �the most
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common situation�, and to positive- and negative-going propagat-ing waves for ��c.

The wave amplitudes a of y�x� and a of ��x� are related to eachother, and the relation can be found from Eqs. �1� �9�. For ex-ample, the amplitude of a positive-going propagating transversedeflection wave component a1

+ is related to that of a positive-goingpropagating bending slope wave component a1

+ as

�k12GA� − �A�2 − ik1GA�

− ik1GA� − k12EI − GA� + �I�2 ��a1

+

a1+� = 0 �10�

From which one has

a1+

a1+ = i

�A�2 − k12GA�

k1GA�= i

�2 − k12Cs

2

k1Cs2 �11�

The relations between the coefficients of wave components ofy�x� and those of ��x� are listed as follows:

a1+

a1+ = − iP,

a1−

a1− = iP,

a2+

a2+ = − N,

a2−

a2− = N �12�

where

P = k1�1 −�2

k12Cs

2, N = k2�1 +�2

k22Cs

2 �13�

2.2 Propagation Matrix. Consider two points A and B on aflexurally vibrating uniform beam a distance x apart, as shown inFig. 2. Waves propagate from one point to the other, with thepropagation being determined by the appropriate wavenumber.Denoting the positive- and negative-going wave vectors at pointsA and B as a+ and a− and b+ and b−, respectively, they are relatedby

a− = f�x�b−; b+ = f�x�a+ �14�where

f�x� = �e−ik1x 0

0 e−k2x � �15�

is the propagation matrix for a distance x.

3 Reflection and Transmission of Waves in Timosh-enko Beams

Waves incident upon discontinuities are reflected and transmit-ted. In this section the reflection and transmission of waves in aTimoshenko beam are considered. The cases considered includegeneral point supports, boundaries, and changes in section.

Fig. 2 Wave propagation

Fig. 3 Wave scattering at a point support

384 / Vol. 127, AUGUST 2005


3.1 Point Supports. Consider the point support at x=0 shownin Fig. 3. The support exerts both translational and rotational con-straints on the beam. These are described by translational and

rotational dynamic stiffnesses KT and KR. Furthermore, there maybe coupling between translation and rotation, in that the supportmight induce moments in response to a translation and a force inresponse to a rotation. These are described by the transfer dy-

namic stiffnesses KTR and KRT, which are equal. These dynamicstiffnesses are, in general, complex and frequency dependent, andcan include the effects of mass, stiffness, damping, etc.

A set of positive-going waves a+ is incident upon the supportand gives rise to transmitted and reflected waves b+ and a−, whichare related to the incident waves through the transmission andreflection matrices t and r by

b+ = ta+, a− = ra+ �16�

where

a+ = a1+

a2+�, a− = a1

−

a2−�, b+ = b1

+

b2+� �17�

Denoting the transverse displacements and the bending slope ofthe beam on the left- and right-hand sides of the point support asy−, y+, �−, and �+, respectively, one has

y− = a1+e−ik1x + a2


−ek2x �18a�

y+ = b1+e−ik1x + b2

+e−k2x �18b�

�− = − iPa1+e−ik1x − Na2

+e−k2x + iPa1−eik1x + Na2

−ek2x �19a�

�+ = − iPb1+e−ik1x − Nb2

+e−k2x �19b�The continuity of the beam requires �i� the displacement of the

neutral axes on either side of the attachment to be the same and�ii� the cross sections of the beam immediately on either side ofthe attachment to remain in contact and, hence, have the samebending angle. Accordingly one has

y− = y+, �− = �+ �20�

Therefore,

� 1 1

− iP − N�b+ + � − 1 − 1

− iP − N�a− = � 1 1

− iP − N�a+ �21�

Furthermore, by considering the equilibrium of the support, onehas

KTy± + KTR�± = V+ − V− = GA�� y+

�x− �+ − GA�� y−

�x− �−

�22�

KR�± + KRTy± = M− − M+ = − EI� ��−

�x−

��+

�x �23�

Thus,

� − iPKTR + KT + ik1 − NKTR + KT + k2

− KRT + iKRP − k1P − KRT + KRN + k2N�b+ + � ik1 k2

k1P − k2N�a−

= � ik1 k2

− k1P k2N�a+ �24�

where

KT = KT/GA�, KR = KR/EI, KTR = KTR/GA�, KRT = KRT/EI

�25�Equations �16�, �21�, and �24�, can be solved to obtain the re-

flection and transmission matrices, which, in the general case, are

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too lengthy to be shown. However, if the coupling effects are

negligible �i.e., KTR and KRT are negligible�, the transmission andreflection matrices become

t = I + c + �d, r = c − �d �26�

where I is the identity matrix and

c = �iN iN

P P�, d = �− iP − N

iP N� �27�

=KT

2�− Pk2 + Nk1� + �− P − iN�KT,

� =KR

2�− Pk1 − Nk2� + �− N + iP�KR�28�

The parameters and � represent the effects of the transla-tional and rotational constraints, respectively, and are frequencydependent. At low frequency �that is, below �c�, the first columnsof t and r are the transmission and reflection coefficients for anincident propagating wave. The second columns yield the coeffi-cients for an incident decaying wave. At high frequency �that is,above �c�, the two columns of both t and r are the transmissionand reflection coefficients corresponding to the two types of inci-dent propagating waves.

For the special case where KT→�, KR=0, the support becomespinned and the transmission and reflection matrices are given by

t =1

P + iN� P − iN

− P iN�

r =1

P + iN�− iN − iN

− P − P� �29�

Note that in this case the transmission matrix is singular. Aspointed out by Mace �5�, the transmission matrix will always besingular when the beam displacement, rotation, bending moment,or shear force is zero at a particular discontinuity. This has con-sequences for alternate analysis approaches, such as those based
on transfer matrices.
waves at the junction. However, the displacement, slope, bending



3.2 Reflections at Boundaries. A general boundary conditionis shown in Fig. 4. The incident waves a+ give rise to reflectedwaves a−, which are related by

a− = ra+ �30�

The reflection matrix r can be determined by considering equilib-rium at the boundary; that is,

− V = − GA�� y

�x− � = KTy + KTR� �31�

M = − EI��

�x= KR� + KRTy �32�

where

y = a1+e−ik1x + a2


−ek2x �33a�

� = − iPa1+e−ik1x − Na2


−ek2x �33b�

If the boundary is at x=0, then the equilibrium conditions be-come

�− iP�1 − KTR� + ik1 + KT − N�1 − KTR� + k2 + KT

iKRP − k1P + KRT KRN + k2N + KRT�a−

+ �iP�1 − KTR� − ik1 + KT N�1 − KTR� − k2 + KT

− iKRP − k1P + KRT − KRN + k2N + KRT�a+ = 0

�34�From Eqs. �30� and �34�, it follows that

r = − �iP�1 − KTR� + ik1 + KT − N�1 − KTR� + k2 + KT

iKRP − k1P + KRT KRN + k2N + KRT�−1

�iP�1 − KTR� − ik1 + KT N�1 − KTR� − k2 + KT

− iKRP − k1P + KRT − KRN + k2N + KRT� �35�

Three common boundary conditions of interest are simply sup-ported, clamped, and free boundaries. Corresponding to theseboundary conditions, both KTR and KRT are 0, while KT and KR are
either zero or infinite. The reflection matrices are found to be
rs = �− 1 0

0 − 1� �36a�

rc = �P − iN

P + iN

− 2iN

P + iN

− 2P

P + iN−

P − iN

P + iN� �36b�

r f = �− Pk1�− N + k2� + ik2N�k1 − P�Pk1�− N + k2� + ik2N�k1 − P�

2Nk2�− N + k2�Pk1�− N + k2� + ik2N�k1 − P�

2iPk1�− P + k1�Pk1�− N + k2� + ik2N�k1 − P�

Pk1�− N + k2� − ik2N�k1 − P�Pk1�− N + k2� + ik2N�k1 − P�

� �36c�

for simply supported, clamped, or free boundary conditions, re-spectively.

3.3 Change in Section. Let two beams of different propertiesbe joined at x=0, as shown in Fig. 5. It is assumed that theirneutral axes coincide. Because of impedance mismatching, inci-dent waves from one beam give rise to reflected and transmitted

moment, and shear force are all continuous at the junction. Thereflection and transmission matrices can then be obtained from thecontinuity and equilibrium conditions.

Denoting the parameters related to the incident and transmittedsides of the junction with subscripts L and R, respectively, choos-ing the origin at the point where the section changes, then at x=0, one has

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yL = yR, �L = �R, ML = MR, VL = VR �37�Considering Eq. �16�, Eqs. �37� can be put into matrix form in

terms of the reflection and the transmission matrices rLL and tLRas follows:

� 1 1

iPL NL�rLL + �− 1 − 1

iPR NR�tLR = �− 1 − 1

iPL NL�

� PLkL1 − NLkL2

i�− PL + kL1� − NL + kL2�rLL

+ � − �RLPRkR1 �RLNRkR2

i�RL�− PR + kR1� �RL�− NR + kR2� �tLR

= � − PLkL1 NLkL2

i�− PL + kL1� − NL + kL2� �38�

where �RL= �EI�R / �EI�L and �RL= �GA��R / �GA��L.The equations can be solved to find the reflection and transmis-

sion matrices rLL and tLR, which, in the general case, are toolengthy to be shown.

3.4 An Angle Joint. Wave transmission and reflection at anangle joint is of practical interest and, in general, introduces wavemode conversion. An incident wave of one type can induce re-flected and transmitted waves of other types. For example, anincident bending wave might induce reflected and transmittedbending, axial and torsional waves in all the members attached tothe joint. Furthermore, the joint might be modeled in a variety ofways. The simplest of models is a point connecting the neutralaxes of the beams, with the dimensions of the joint and the thick-ness of the beams being assumed to be negligible. However, forfrequencies where a Timoshenko model might be required, thefinite dimensions of the joint can be important in determiningwave reflection and transmission, and such a model is consideredbelow. Finally, flexibility in the joint might be included, typicallyby matching a finite element model of the joint to wave motion inthe beams. Given that there may be many members attached at

Fig. 4 Wave reflection at a general boundary

386 / Vol. 127, AUGUST 2005


arbitrary angles at a joint in a three-dimensional structure, therange of possible situations is vast and a comprehensive analysisbeyond the scope of this paper. Here a simple case is considered—that where two Timoshenko beams are joined at a right angle, withthe dimensions and mass of the joint being taken into consider-ation. The analysis of an arbitrary multimember joint, thoughmore cumbersome, follows similar procedures, in principle.

Consider the beams joined as shown in Fig. 6. The structure isseparated into the two beams connected by a rectangular piece ofmaterial �i.e., the joint�. This has dimensions h1 by h2 �which offset the surfaces of the beams�, a mass m and a mass moment ofinertia J. In-plane bending waves and axial waves are assumed toexist. Out-of-plane bending waves would couple with torsionalwaves in the members—the analysis follows the same lines. �Insome circumstances the axial waves might be ignored and thejoint then be allowed to only rotate.�

The equations of motion of the joint in Fig. 6 are

F2 − V1 = myJ

− V2 − F1 = muJ

M2 − M1 + V1h1

2+ V2

h2

2= J�J �39�

where F is the axial force in the beam and h the beam thickness.Subscripts 1 and 2 and refer to beam 1 and beam 2, respectively,while uJ, yJ, and �J are the displacements and rotation of the jointas indicated in Fig. 6. The first two of these equations include themass of the joint, whereas the third includes the moment of inertiaof the joint and the moments induced by the offset shear forces.

The continuity equations at the joint are

u1 = uJ, u2 = yJ

y1 = yJ −h1

2�J, y2 = − uJ +

h2

2�J

�1 = �J, �2 = �J �40�

Fig. 5 Wave reflection and transmission at change of section

Fig. 6 Beams jointed at right angle



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The axial force, shear force, and bending moment in Fig. 6,respectively, are related to the displacements by

F1 = �EA�1�u1

�x1, F2 = �EA�2

�u2

�x2

V1 = �GA��1� �y1

�x1− �1, V2 = �GA��2� �y2

�x2− �2

M1 = − �EI�1��1

�x1, M2 = − �EI�2

��2

�x2�41�

where x is the distance along each beam axis.A set of positive going waves A+ is incident upon the joint from

beam 1 and gives rise to transmitted and reflected waves B+ andA−, which are related to the incident waves through the transmis-sion and reflection matrices t and r by

B+ = tA+, A− = rA+ �42�

where

A+ = �a+

aN+

c+ �, A− = �a−

aN−

c− �, B+ = �b+

bN+

d+ � �43�

Here a+, a−, and b+ denote the incident, reflected, and transmittedpropagating bending waves, aN

+, aN−, and bN

+ the transition bendingwaves, and c+, c−, and d+ the incident, reflected, and transmittedaxial propagating waves, respectively. For axial waves, the el-ementary, one-dimensional theory of �18� is adopted here as it is

that is, a decaying mode at frequencies below the cutoff frequency



typically valid for frequencies up to twice the cutoff frequency ofTimoshenko bending waves.

Considering beams 1 and 2 to have the same dimensions andmaterial properties, one has

y1 = a+e−ik1x1 + aN+e−k2x1 + a−eik1x1 + aN

−ek2x1 �44a�

u1 = c+e−ik3x1 + c−eik3x1 �44b�

�1 = − iPa+e−ik1x1 − NaN+e−k2x1 + iPa−eik1x1 + NaN

−ek2x1 �44c�

y2 = b+e−ik1x2 + bN+e−k2x2 �45a�

u2 = d+e−ik3x2 �45b�

�2 = − iPb+e−ik1x2 − NbN+e−k2x2 �45c�

where k3=�� /E is the axial wavenumber.From the continuity conditions, one has

�− 1 − iPh

2− 1 − N

h

20

iPh

2N

h

21

− iP − N 0�B+ − � 0 0 1

1 1 0

iP N 0�A−

= � 0 0 1

1 1 0

− iP − N 0�A+ �46�

The equilibrium conditions give

� i�AG�k1 − P� �AG�k2 − N� 0

0 0 − ik3EA + m�2

EIPk1 − i�AG�k1 − P�h/2 − iJ�2P − EINk2 − �AG�k2 − N�h/2 − J�2N 0�B+

− � 0 0 ik3EA − m�2

i�AG�k1 − P� �AG�k2 − N� 0

EIPk1 − i�AG�k1 − P�h/2 − EINk2 − �AG�k2 − N�h/2 0�A−

= � 0 0 − ik3EA − m�2

− i�AG�k1 − P� − �AG�k2 − N� 0

EIPk1 + i�AG�k1 − P�h/2 − EINk2 + �AG�k2 − N�h/2 0�A+ �47�

The transmission and reflection matrices can be obtained fromsolving Eqs. �42�, �46�, and �47�. A numerical example is consid-ered below.

3.5 Numerical Examples. In this section, various numericalexamples are presented. These include wave reflection and trans-mission at a pinned point support on a uniform beam, a change insection, and an angle joint. Reflection characteristics of the threecommon boundaries are also considered.

The physical properties of the beam are chosen as follows:Young’s modulus E is 190 GN/m2; shear modulus G77.5 GN/m2; Poisson ratio � 0.29; mass density � 7680 mg/m3;and the width b, thickness h, and length L0 of the beam being 0.6,0.4, and 2.0 m, respectively. Shear coefficient � is related to thePoisson ratio � by �=10�1+�� / �12+11�� 19�. In the discussionsthat follow, “propagation mode” is used to denote the wave com-ponent that is always propagating, whereas “transition mode” re-fers to the wave component that involves wave mode transition;

�c and a propagating mode at higher frequencies.

3.5.1 Pin Joint. Figure 7 shows the reflection and transmissioncoefficients at a pin support. It can be seen that reflection coeffi-cients for the propagation mode are, by far, the largest, especiallyat higher frequencies, whereas the transmission coefficients arelargest for an incident transition mode. As a result, the decayingcomponents of the transition mode arising from a second discon-tinuity attached in close proximity to a pinned support in a struc-ture can produce substantial transmitted propagating waves. Wavemode transition does not seem to affect the transmission and re-flection much under this circumstance.

3.5.2 Change in Section. Because of a change of section, thereexist two cutoff frequencies associated with the two different sec-tions, respectively. The wave mode transition within a bandformed by the two cutoff frequencies is complicated by the factthat it depends on the direction of the incident wave. Figure 8shows the variation of the reflection and transmission coefficients
at a step change with thickness ratio of 0.7 �right/left�. Unlike the
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pin support, the propagation mode contributes little to the re-flected waves, whereas the transition mode dominates the re-flected waves, especially at frequencies in the band between thetwo cutoff frequencies. The propagation mode is mostly transmit-ted as a propagation mode and does not seem to be affected by thecutoff frequencies much. On the contrary, the transmission of anincident transition mode is strongly related to the cutoff frequen-cies, maxima and minima being observed at these frequencies.

3.5.3 An Angle Joint. Figure 9 shows the transmission andreflection coefficients for incident longitudinal, bending propagat-ing, and bending transition waves. It can be seen that wave modeconversion exists regardless of the incident wave mode. Theangled joint is seen to strongly reflect an incident axial propagat-ing wave, especially at high frequencies. An incident bendingpropagation wave generates strong axial propagating wave in thetransmitted waves. An incident bending transition wave is mostlyreflected at frequencies higher than the cutoff frequency. In themean time, it generates a very strong propagating bending wavecomponent in the transmitted waves. Hence from vibration reduc-tion standpoint, care should be taken in avoiding discontinuitiesbeing placed in the adjacency of an angle change. At the cutofffrequency, unless the incident wave is the transition wave itself,there is no transition wave generated in both the reflected andtransmitted waves.

3.5.4 Reflection at Boundaries. As can be seen from the re-flection matrix given by Eq. �36a�, the reflection characteristics ata simple support are the same as those predicted by elementaryEuler-Bernoulli beam theory. That is, both incident propagatingand transition modes are fully reflected independently in the origi-nal wave types with 180 deg phase change. The reflection is fre-quency independent. For a clamped support and a free end, how-ever, the situation is different. At lower frequencies, both incidentpropagating and transition modes are reflected in the incidentwave types, however, with a frequency-dependent phase angle.Furthermore, they introduce waves of the other type. An incidenttransition mode gives rise to a propagation mode of significantamplitude; thus, a significant amount of energy in the propagatingcomponents may arise from the incidence of a transition modeupon boundaries.

4 Waves Generated by Externally Applied PointForces and Moments

Applied forces and moments have the effect of injecting waves

Fig. 7 „a… Reflection coefficients at a pin joint „� r11a pin joint „Modulus: � t11 and t21, … t12, and t22; Pha

into the structure. Consider the flexural waves injected by a point

388 / Vol. 127, AUGUST 2005


force Q and moment M applied at x=0, as shown in Fig. 10. Atx=0 there are discontinuities in the shear force and bending mo-ment in the beam with resulting discontinuities in the waves a andb on either side of the excitation point. The relations between theapplied forces and the waves are described by the following con-tinuity and equilibrium conditions:

y− = y+, �− = �+ �48�

Q = GA�� y−

�x− �− − � �y+

�x− �+�, M = EI� ��−

�x−

��+

�x

�49�

where

y− = a1+e−ik1x + a2


−ek2x �50�

y+ = b1+e−ik1x + b2

+e−k2x + b1−eik1x + b2

−ek2x �51�

�− = − iPa1+e−ik1x − Na2


−ek2x �52�

�+ = − iPb1+e−ik1x − Nb2

+e−k2x + iPb1−eik1x + Nb2

−ek2x �53�

The continuity and the equilibrium conditions can be written inmatrix form as

� 1 1

iP N�b+ + � 1 1

− iP − N�b− + � − 1 − 1

− iP − N�a+ + �− 1 − 1

iP N�a−

= 0 �54�

� ik1 k2

− k1P k2N�b+ + � − ik1 − k2

− k1P k2N�b− + �− ik1 − k2

k1P − k2N�a+

+ � ik1 k2

k1P − k2N�a− = �Q/GA�

M/EI� �55�

from which it follows that

b+ − a+ = q + m, a− − b− = q − m �56�

where the vectors of the excited wave amplitudes are

q = iN

P� Q

2GA��k2P − k1N�, m = − 1

1� M

2EI�k1P + k2N�

r12, … r21, and r22…; „b… transmission coefficients at� t11, �.�.�.t21, … t12, and � � � t22…

and

�57�



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5 Vibration Analysis Using Wave ApproachThe transmission and reflection matrices for waves incident

upon various discontinuities were derived above. The waves in-jected by externally applied forces and/or moments were alsofound in matrix form. These matrices can be combined to providea concise and systematic approach for vibration analysis of Ti-moshenko beams or complex structures consisting of Timoshenkobeam components. The systematic approach is illustrated throughtwo numerical examples below, namely, free and forced vibrationanalysis of a uniform and a stepped Timoshenko beam.

5.1 Free and Forced Vibration Analysis of a Uniform Ti-moshenko Beam.

5.1.1 Free Response: The Natural Frequencies of the Beam.Figure 11 shows a general beam structure with boundaries A andB. Now consider the free response of this structure. Denoting theincident and reflected waves at the boundaries A and B as a−, a+,b+, and b−, respectively, the relationships between the different

Fig. 8 „a… Reflection coefficients at a step change wr12, and � � � r22…; „b… transmission coefficients at a ste�.�.�.t21, … t12, and � � � t22…; „c… Reflection coefficients„ � r11, �.�.�. r21, … r12, and � � � r22…; „d… transmission coleft side „ � t11, �.�.�. t21, … t12, and � � � t22…

waves are given as



a+ = rAa−, b− = rBb+, a− = f�L�b−, b+ = f�L�a+, �58�

where rA and rB are the reflection matrices at boundaries A and B,respectively, and f�L� is the propagation matrix between A and B,which are a distance L apart.

Solving Eqs. �58� gives

�rAf�L�rBf�L� − I�a+ = 0 �59�

where I denotes the identity matrix. For nontrivial solution, itfollows that free vibration occurs at frequencies for which thedeterminant

�rAf�L�rBf�L� − I� = 0 �60�

This is the characteristic equation from which the natural frequen-cies of the beam can be found.

5.1.2 Forced Response. Figure 12 shows a cantilever beamwith a point force and a moment applied at point C. The distur-bance force dividing the structure into two regions, namely, region

wave incident from the right side „ � r11, �.�.�. r21, …change with wave incident from the right side „ � t11,a step change with wave incident from the left sidecients at a step change with wave incident from the

ithpateffi

1 and 2. The incoming and outgoing waves at the disturbance

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Fig. 9 „a… Transmission coefficients at a right-angle joint with longitudinal incident wave „ � bending propa-gation, �.�.�. bending transition, and … longitudinal propagation components…; „b… reflection coefficients at aright-angle joint with longitudinal incident wave „ � bending propagation, �.�.�. bending transition, and …longitudinal propagation components…; „c… transmission coefficients at a right-angle joint with bendingpropagating incident wave „ � bending propagation, �.�.�. bending transition, and … longitudinal propagationcomponents…; „d… reflection coefficients at a right-angle joint with bending propagating incident wave „ �bending propagation, �.�.�. bending transition, and … longitudinal propagation components…; „e… transmis-sion coefficients at a right-angle joint with bending transition incident wave „ � bending propagation, �.�.�.bending transition, and … longitudinal propagation components…; „f… reflection coefficients at a right-anglejoint with bending transition incident wave „ � bending propagation, �.�.�. bending transition, and … longitu-
dinal propagation components…
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applied point C are denoted as c1+, c1

−, c2+, and c2

−.The waves on both sides of the disturbances, namely, c1

± and c2±,

are related to the point force Q, the applied moment M and theproperties of the structure, as described in Sec. 4. Also c1

+ is re-lated to c1

− by reflection at boundary A and propagation throughregion 1. Similarly c2

+ is related to c2− by reflection at boundary B

and propagation through region 2. The forced responses can beobtained from the propagation and reflection matrices.

At boundaries A and B, one has

a+ = rAa−, b− = rBb+ �61�

where rA and rB are the reflection matrices at the boundaries.Furthermore, by the propagation relations it follows that

c1+ = f�L1�a+, a− = f�L1�c1

−, b+ = f�L2�c2+, c2

− = f�L2�b−

�62�

For convenience, the reflection matrices �1 and �2 can be de-fined to relate the amplitudes of incoming and outgoing waves atthe external force applied point, such that

c1+ = �1c1

−, c2− = �2c2

+ �63�

Equations �61�–�63� give

�1 = f�L1�rAf�L1�, �2 = f�L2�rBf�L2� �64�Considering the forced vibration of the beam due to an applied

force Q only �that is, assume the moment M =0 in Fig. 12�, fromSec. 4 it is known that the waves at the force applied point arerelated to the force by

c2+ − c1

+ = q, c1− − c2

− = q �65�

Equations �63� and �65� can be solved for the wave amplitudes at

point C in terms of Q. They are found to be

c1+ = �1�I − �2�1�−1��2 + I�q, c1

− = �I − �2�1�−1��2 + I�q

c2+ = �I − �2�1�−1��1 + I�q, c2

− = �2�I − �2�1�−1��1 + I�q�66�

The deflection of any point along the beam can then be found.For example, the deflection of a point in region 1 that is a distancex from the excitation point is given by

Fig. 10 Point excitations

Fig. 12 Forced response of a clampe



y− = �1 1�f�x�c1− + �1 1�f�− x�c1

+ �67�

Similarly, the deflection of a point in region 2 that is a distance xfrom the excitation point is

y+ = �1 1�f�x�c2− + �1 1�f�− x�c2

+ �68�

Figure 13 shows the forced response of the beam at 1.6 m witha disturbance force applied at 0.8 m from the clamped end. Thefigure also shows the response predicted by Euler-Bernoulli beamtheory. It is clear that Euler-Bernoulli beam theory is only accu-rate at low frequencies, as is well known.

5.2 Free and Forced Vibration Analysis of a Stepped Ti-moshenko Beam.

5.2.1 Free Responses: The Natural Frequencies of the Beam.As a final example, Fig. 14 shows a stepped cantilever Timosh-enko beam. The geometric discontinuity is at point D. The analy-sis follows the same lines as above, except that there is also wavereflection and transmission at the step. The incident and reflectedwaves at the clamped boundary, free boundary, and the left- andright-hand sides of D are denoted by a�, b±, d2

±, and d3�, respec-

tively. The relationships between the incident and reflected wavesat the boundaries are as described in Eq. �61�. At the geometricdiscontinuity D, the incident, reflected, and transmitted waves arerelated as follows:

d2− = r11d2

+ + t21d3−, d3

+ = r22d3− + t12d2

+ �69�

where r and t are the reflection and transmission matrices of thechange of section, as discussed in Sec. 3.3. The subscripts of r andt identify the incident and transmitted sides of the junction.

The propagation relations are

d2+ = f�L1�a+, a− = f�L1�d2

−, b+ = f�L2�d3+, d3

− = f�L2�b−

�70�

where f�L1� and f�L2� are the propagation matrices between ADand DB, respectively. Solving Eqs. �61�, �69�, and �70� gives

�I − r11f�L1�rAf�L1��t12f�L1�rAf�L1��−1�I − r22f�L2�rBf�L2��

− t21f�L2�rBf�L2��d3+ = 0 �71�

For a nontrivial solution, it follows that

Fig. 11 Free response of a uniform Timoshenko beam

d free uniform Timoshenko beam

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a s

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��I − r11f�L1�rAf�L1��t12f�L1�rAf�L1��−1�I − r22f�L2�rBf�L2��

− t21f�L2�rBf�L2�� = 0 �72�

Equation �72� is the characteristic equation from which the naturalfrequencies of the stepped Timoshenko beam can be found.

5.2.2 Forced Response. Figure 15 shows the stepped beamwith a point force and a moment applied at point C. The definitionof wave amplitudes and the analysis are as above, the only differ-ence being in the propagation relations

c1+ = f�L11�a+, a− = f�L11�c1

−, d2+ = f�L12�c2

+

c2− = f�L12�d2

−, b+ = f�L2�d3+, d3

− = f�L2�b− �73�

where f�L11� and f�L12� are the propagation matrices between ACand CD, respectively. Solving Eqs. �61�, �63�, �69�, and �73� gives

�1 = f�L11�rAf�L11�

Fig. 13 Frequency responses of a„�… and Euler Bernoulli „…… beam th

Fig. 14 Free response of

Fig. 15 Forced response of a

392 / Vol. 127, AUGUST 2005


�2 = f�L12�r11f�L12� + f�L12�t21f�L2�rBf�L2�

�I − r22f�L2�rBf�L2��−1t12f�L12� �74�Following the same procedure as described in Sec. 5.1.2, the

deflection of any point along the beam can be found. Figure 16shows the forced response of a beam with a step change of 0.7�right/left� in the thickness at its center, that is, L1=L2=1 m. Thephysical properties are the same as those described in Sec. 3.5.The response shown is the response at 1.6 m with a disturbanceforce applied at 0.8 m. The frequency is normalized to �cL, thecutoff frequency of the beam on the clamped side.

6 ConclusionsIn this paper Timoshenko beam theory was used to describe

wave propagation, transmission and reflection in beams undervarious circumstances. Cases considered included discontinuities

iform beam based on Timoshenkory

tepped Timoshenko beam

uneo

stepped Timoshenko beam



of

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caused by general point supports, boundaries, and change in sec-tions. The matrix relations between the injected waves and exter-nally applied point forces and/or moments were also derived. Ti-moshenko beam theory is required at higher frequencies when theeffects of rotary inertia and shear distortion become important.Under these circumstances Euler-Bernoulli beam theory is inac-curate.

When Timoshenko beam theory must be used, the vibrationanalysis becomes more complex and further difficulties arise fromincreasing complexity of structures. However, with the availabil-ity of propagation, reflection, and transmission matrices, the vi-bration analysis becomes systematic and concise: it involves onlya number of matrix operations related to the reflection and trans-mission matrices at the discontinuities. The procedures were illus-trated by various numerical examples.

NomenclatureA � cross-sectional area

A±, B± � wave vectorsai

±, bi±, ci

±, di± � positive and negative going wave components,

i=1,2 ,3a±, b±, c±, d± � wave vectors

b � widthCb � bending-stiffness-related coefficientCs � shear-stiffness-related coefficientCr � rotational-effects-related coefficientE � Young’s modulusF � axial force

f�x� � propagation matrix for a distance x, f�x�=� e−ik1x 0

0 e−k2x �G � shear modulush � beam thicknessI � area moment of inertia of cross sectionI � identity matrixJ � mass moment of inertiak � wavenumber �k1,k2, transverse wavenumbers;

k3, axial wavenumber�KR � rotational dynamic stiffness

KT � translational dynamic stiffness

KTR , KRT � transfer dynamic stiffness

Fig. 16 Frequency responses

L � distance



L0 � lengthM�x , t� � bending moment

M � externally applied momentm � massm � vector of moment excited wave amplitudes

P ,N � coefficients relating amplitudes of wave com-ponents of y�x� and ��x�, P=k1�1−�2 / �k1

2Cs2��,

N=k2�1+�2 / �k22Cs

2��Q � externally applied point force

q�x , t� � external forceq � vector of point force excited wave amplitudesr � reflection matrix

rc � reflection matrix at clamped boundaryr f � reflection matrix at free boundaryrs � reflection matrix at simply supported boundaryt � timet � transmission matrixu � axial deflection

V�x , t� � shear forcex � position along the beam axis

y�x , t� � transverse deflection of the centerline of thebeam

Greek Symbols�RL � �EI�R / �EI�L, bending stiffness ratio�RL � �GA��R / �GA��L, shear stiffness ratio

� � parameter representing the effect of rotationalconstraint

� � shear coefficient � parameter representing the effect of transla-

tional constraint� � Poisson ratio� � mass density

�1 ,�2 � reflection matrices relating the amplitudes ofincoming and outgoing waves at the externalforce�s� and/or moment�s� applied point

��x , t� � slope due to bending� � frequency

� /k � phase velocity � /k�c � cutoff frequency, which is given by �c=Cs /Cr

a stepped Timoshenko beam

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Subscripts1, 2 � beam 1 and beam 2 or region 1 and region 2

J � angle jointL ,R � the incident and transmitted side of a

discontinuityN � transition wave component

�, � � right- and left-hand sides of the chosen origin

Superscripts�, � � positive- and negative-going waves

References�1� Graff, K. F., 1975, Wave Motion in Elastic Solids, Ohio State University Press.�2� Cremer, L., Heckl, M., and Ungar, E. E., 1987, Structure-Borne Sound,

Springer-Verlag, Berlin.�3� Miller, D. W., and Von Flotow, A., 1989, “A Traveling Wave Approach to

Power Flow in Structural Networks,” J. Sound Vib., 128�1�, pp. 145–162.�4� Beale, L. S., and Accorsi, M. L., 1995, “Power Flow in Two- and Three-

Dimensional Frame Structures,” J. Sound Vib., 185�4�, pp. 685–702.�5� Mace, B. R., 1984, “Wave Reflection and Transmission in Beams,” J. Sound

Vib., 97, pp. 237–246.�6� Harland, N. R., Mace, B. R., and Jones, R. W., 2001, “Wave Propagation,

Reflection and Transmission in Tunable Fluid-Filled Beams,” J. Sound Vib.,241�5�, pp. 735–754.

�7� Tan, C. A., and Kang, B., 1998, “Wave Reflection and Transmission in anAxially Strained, Rotating Timoshenko Shaft,” J. Sound Vib., 213�3�, pp.
483–510.
394 / Vol. 127, AUGUST 2005


�8� Tan, C. A., and Kang, B., 1999, “Free Vibration of Axially Loaded, RotatingTimoshenko Shaft Systems by the Wave-Train Closure Principle,” Int. J. Sol-ids Struct., 36, pp. 4031–4049.

�9� Doyle, J. F., 1989, Wave Propagation in Structures, Spring-Verlag, New York.�10� Rayleigh, L., 1926, Theory of Sound, Macmillan, New York.�11� Timoshenko, S. P., 1921, “On the Correction for Shear of the Differential

Equation for Transverse Vibrations of Prismatic Bars,” Philos. Mag., 41, pp.744–746.

�12� Timoshenko, S. P., 1922, “On the Transverse Vibrations of Bars of UniformCross Sections,” Philos. Mag., 43, pp. 125–131.

�13� Huang, T. C., 1961, “The Effect of Rotatory Inertia and of Shear Deformationon the Frequency and Normal Mode Equations of Uniform Beams with SimpleEnd Conditions,” ASME J. Appl. Mech., 28, pp. 579–584.

�14� Anderson, R. A., 1953, “Flexural Vibrations in Uniform Beams according tothe Timoshenko Theory,” ASME J. Appl. Mech., 75, pp. 504–510.

�15� Lueschen, G. G. G., Bergman, L. A., and McFarland, D. M., 1996, “Green’sFunctions for Uniform Timoshenko Beams,” J. Sound Vib., 194�1�, pp. 93–102.

�16� Banerjee, J. R., 2004, “Development of an Exact Dynamic Stiffness Matrix forFree Vibration Analysis of a Twisted Timoshenko Beam,” J. Sound Vib., 270,pp. 379–401.

�17� Corn, S., Bouhaddi, N., and Piranda, J., 1997, “Transverse Vibrations of ShortBeams: Finite Element Models Obtained by a Condensation Method,” J.Sound Vib., 201�3�, pp. 353–363.

�18� Wang, C. H., and Rose, L. R. F., 2003, “Wave Reflection and Transmission inBeams Containing Delamination and Inhomogeneity,” J. Sound Vib., 264, pp.851–872.

�19� Cowper, G. R., 1966, “The Shear Coefficient in Timoshenko’s Beam Theory,”ASME J. Appl. Mech., 33, pp. 335–340, June.



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