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Dynamic Green Function for Response of TimoshenkoBeam with Arbitrary Boundary ConditionsAmin Ghannadiasl a & Massood Mofid aa Department of Civil Engineering , Sharif University of Technology , Tehran , IranAccepted author version posted online: 09 Sep 2013.

To cite this article: Mechanics Based Design of Structures and Machines (2013): Dynamic Green Function for Response ofTimoshenko Beam with Arbitrary Boundary Conditions, Mechanics Based Design of Structures and Machines: An InternationalJournal, DOI: 10.1080/15397734.2013.836063

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Dynamic Green Function for Response of Timoshenko Beam with Arbitrary BoundaryConditions

Amin Ghannadiasl and Massood Mofid

Department of Civil Engineering, Sharif University of Technology, Tehran, Iran

Abstract

This paper presents the dynamic response of uniform Timoshenko beams with arbitrary

boundary conditions using Dynamic Green Function. An exact and direct modeling

technique is stated to model beam structures with arbitrary boundary conditions subjected

to the external load which is an arbitrary function of time t and coordinate x and the

concentrated moving load. This technique is based on the Dynamic Green Function. The

effect of different boundary condition, load and other parameters is assessed. Finally,

some numerical examples are shown to illustrate the efficiency and simplicity of the new

formulation based on the Dynamic Green Function.

KEYWORDS: Arbitrary boundary conditions; Dynamic green function; Timoshenko

beam.

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1. INTRODUCTION

The transverse vibration of Timoshenko beams due to steady and moving loads has been a

highly important research subject in recent years. The Timoshenko beam model is suitable

for describing the behavior of short beams, sandwich composite beams or beams under

high-frequency excitation load when the wavelength approaches the thickness of the beam

because it considers the shear deformation and the rotational inertia effects. The resulting

differential equation of the motion is of 4th order; however unlike ordinary Bernoulli-

Euler beam theory, there is also a second order spatial derivative present. The literature

concerning the forced vibration analysis of Timoshenko beams is sparse. The most used

method for determining the forced vibrations is the expansion of the applied loads and

the dynamic responses in terms of the eigenfunctions of the undamped beams. Abbas

and Thomas (1984) presents the problem of free vibration of Timoshenko beams with

elastically supported ends. The problem is solved using the unique finite element model.

The presented method can satisfy all the geometric and natural boundary conditions of

an elastically restrained Timoshenko beam. The effects of the translational and rotational

support flexibilities are investigated on the natural frequencies of free vibration of

Timoshenko beam. Based on the double Laplace transformation, a method is presented

for dynamic analysis of a damped Bernoulli-Euler uniform beam subjected to the action

of a moving concentrated force for simply supported boundary condition by Hamada

(1981). In this study, an exact solution for the dynamic deflection of the considered beam

is obtained in closed form. It is shown that the forced vibration can be expanded in a

double power series. The second-order variational formalism is demonstrated to derive the

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Timoshenko beam equation with consistent boundary conditions by Nesterenko (1993).

Properties of the second high-frequency mode are investigated for vibrations predicted in

the Timoshenko theory. In addition, a simple method is suggested to take the effect of the

deformation of the beam cross section into account during the vibrations on its natural

frequencies.

Mohamed (1994) presents the exact method to determine the dynamic response of Euler-

Bernoulli beams along with the attached masses and given springs, using Green’s functions.

For beams of several common boundary conditions, these functions are tabulated.

Furthermore, the method is applicable to multi-span beams, and to the important class of

periodic structures such as coupled bladed disk assemblies of a turbine shaft. The closed

form expressions for Green functions of the uniform Timoshenko beam are provided for

the special boundary conditions by Lueschen et al. (1996). It is shown that For Euler-

Bernoulli beams with and without a constant axial preload; these Green’s functions are

accurate.

Esmailzadeh and Ghorashi (1997) studied the analysis of the Timoshenko beam under

uniform partially distributed moving masses. The finite difference method is applied to

solve Equations of Motion. In this study, the effects of rotary inertia and shear deformation

are neglected. For simply supported beam with a short length of load distribution, the

computations and results are found to be in a reasonable agreement. Foda and Abduljabbar

(1998) present a simple and direct technique to treat the problem of an undamped simply

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supported Euler–Bernoulli beam under a moving mass. The technique is based on the

Dynamic Green Function. A simple matrix expression is produced for the deflection of

the beam by the proposed method. In this study, the inertia of the mass is neglected.

Green Function of the beam on an elastic foundation is obtained using Fourier transform

method by Sun (2001). In this study, the theory of linear partial differential equation is

used to represent the beam displacement in terms of convolution of the Green Function.

The theory of complex function is employed to evaluate this convolution analytically by

seeking the poles of the integrand of the generalized integral. The displacement response of

beam-type structures is provided as closed-form and numerical computation is performed.

Sun (2001) formulated an analytical expression of the dynamic amplification factor and

characteristic response spectrum for the Euler-Bernoulli beams with various boundary

conditions under the action of successive moving loads.

Ekwaro-Osire et al. (2001) presented the series expansion solution to the Timoshenko

beam. In this paper, a particular natural frequency and its corresponding mode shape

illustrate one particular solution to the boundary value problem of the Timoshenko beam.

A method for determining the dynamic response of Euler–Bernoulli beams subjected

to distributed and concentrated loads is presented by Abu-Hilal (2003). The method is

used to solve single and multi-loaded beams, single and multi-span beams, and statically

determinate and indeterminate beams. In addition, the Green Functions are given for

beams with different homogenous and elastic boundary conditions.

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Dyniewicz and Bajer (2010) exhibited the semi-analytical solution for on the simply

supported Timoshenko beam under a moving mass using the Lagrange equation of the

second kind. However, the presented solution of the problem cannot simply be applied

to complex problems, for example strings, beams, subjected to a system of masses or

composed of segments with variable rigidity. Majkut (2009) presented the approach to

free and forced vibrations of the Timoshenko beam by a single equation. It was shown

that the form of solution approach to the differential equation depends on the vibration

frequency. In addition, production of the Dynamic Green Functions was proposed to

solve the problem of vibration amplitudes excited by an arbitrary function of time t.

The free vibration problem of beams on an elastic foundation of the Winkler type was

studied by Motaghian et al. (2011). The elastic foundation is distributed over a particular

length of the Euler–Bernoulli beam. The governing differential equations of beam were

solved as closed form solutions. Moreover, to find the precise analytical solution of the

free vibration of beams with mixed boundary conditions, an innovative mathematical

approach was proposed. An exact solution of Timoshenko beams resting on two-parameter

elastic foundations is obtained using Green’s functions by Wang et al. (1998). In this

study, a unified formulation is presented for bending, buckling, and vibration problems of

Timoshenko and Euler-Bemoulli beams. In addition, an electro-rheological (ER) layer is

adhered to the beam on Pasternak elastic foundation to control the vibration of the beam.

Kargarnovin et al. (2012) presented the dynamic response of a delaminated composite

beam under the action of moving oscillatory mass. In this paper, the Poisson’s effect, shear

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deformation and rotary inertia is considered. The method is used to solve simply supported

beam with a constant oscillator travelling with various prescribed constant speeds. The

response of an inclined beam subjected to sprung mass is studied by Mohebpoura et al.

(2013). The effects of inertial forces like Coriolis and centrifugal forces are considered

in the governing differential equations. The moving system is modelled as one degree of

freedom system by two masses located at the ends of the moving load system, one spring

and viscous damping.

In previous studies, on dynamic analysis of the Timoshenko beam under the external load

that is an arbitrary function of time t and coordinate x using the Dynamic Green Function,

only the beam free vibration has been analyzed. On the other hand, previous researchers’

solution cannot be generalized to different boundary conditions. Therefore, the objectives

of this paper are:

- To present a highly simple and practical analytical–numerical technique to determine

the dynamic response of Timoshenko beams, with various boundary conditions, under

the external load which is an arbitrary function of time t and coordinate x and the

concentrated moving load.

- To state exact solutions in closed forms using the Dynamic Green Function

2. GREEN FUNCTION SOLUTION

In this paper, a uniform Timoshenko beam of length L is considered which is partially

restrained against translation and rotation at its ends. The translational restraint is

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characterized by the spring constant KTL, at one end and KTR at the other end. At the same

time, the rotational restraint is characterized by the spring constants KRL at one end and

KTR at the other end, as shown in Fig. 1. By using Hamilton principle, the coupled system

of differential equations can be given by (Eftekhar Azam et al., 2013):

�Ga(��x − w�xx

)+ �Aw�tt = q �x� t�+ f�x� t� (1)

EI��xx + �GA(w�x − �

)− �I��tt = 0 (2)

where w �x� t� represents the transverse displacement of the mid-surface in the z-direction;

��x� t� is the anticlockwise angle of rotation of the normal to the mid-surface of the beam;

q�x� t� presents the external load which is an arbitrary function of time t and coordinate x

and F�x� t� is the concentrated moving load.

In addition, EI, A, E, G, I, � and � are, the rigidity of the beam, cross-sectional area of the

beam, Young’s modulus of elasticity, shear modulus, the second moment of area, sectional

shear coefficient, beam material density, respectively.

For a linear elastic, isotropic and homogeneous beam, these two equations can be combined

after several transformations. The moving load can be defined as:

F �x� t� = F0 n� �x − xF �t�� nei�t (3)

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where � is Dirac Delta function and xF �t� is the equation of the trajectory of the moving

load which is defined as:

xF �t� = x0 + vxt +12axt

2 (4)

where x0 is the point of application of the load, vx is the initial speed in the x direction,

and ax is the constant acceleration in the x direction. This function describes a uniform

decelerating or accelerating motion. The forced vibration equations for the Timoshenko

beam can be obtained in a form dependent only on the functions of the displacement w(x,

t) and the rotation ��x� t�:

[EI

4

x4− �I

(1+ E

�G

)4

t2x2+ �2I

�G

4

t4+ �A

2

t2

]w �x� t�

=[

�I

�AG

2

t2− EI

�AG

2

x2+ 1

]q �x� t�+ F�x� t�� (5)

[EI

4

x4− �I

(1+ E

�G

)4

t2x2+ �2I

�G

4

t4+ �A

2

t2

]� �x� t�

= q �x� t�

x+ F �x� t�

x(6)

For a beam with constant cross-section, the differential equations of the motion can be

rewritten as follows:

EIw�xxxx −mI

A

(1+ E

�G

)w�xxtt +

m2I

A2�Gw�tttt +mw�tt

= q + mI

�A2Gq�tt −

EI

�AGq�xx + F + mI

�A2GF�tt −

EI

�AGF�xx (7)

EI��xxxx −mI

A

(1+ E

�G

)��xxtt +

m2I

A2�G��tttt +m��tt = q�x + F�x (8)

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where m = �A is the mass per unit length of the beam. In this study, the initial conditions

and the general boundary conditions associated with the Timoshenko beam theory are

provided below:

∀ t@x = 0 � EI��x�0� t� = KRLw�x�0� t�

�GA(w�x�0� t�− ��0� t�

) = −KTLw�0� t�

∀ t@x = L � EI��x �L� t� = −KRRw�x�L� t�

�GA(w�x�L� t�− ��L� t�

) = KTRw�L� t�

∀ x � �I��t���tt0 = 0 and�Aw�t�w�tt0 = 0

The external load q�x� t� is given as:

q �x� t� = Q�x� ei�t (9)

where Q�x� and � are the amplitude of the applied load in point x and the external

load frequency, respectively. It is assumed that each function w �x� t� and � �x� t� can be

introduced in the form of a product of a function dependent on the coordinate x and a

function dependent on time t (with the same time function):

w �x� t� = W �x� ei�t (10)

� �x� t� = �x� ei�t (11)

where W �x�, �x� and � are the beam displacement amplitude at point x, the amplitude

angle of rotation of the normal to the mid-surface of the beam at point x and the circular

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frequency of the Timoshenko beam, respectively. Substituting Eqs. (9), (10) and (11) into

Eqs. (7) and (8) results in:

EIW�xxxx +mI�2

A

(1+ E

�G

)W�xx +

(m2I�4

A2�G−m�2

)W

=(1− �2mI

�A2G

)Q− EI

�AGQ�xx +

(1− �2mI

�A2G

)F0� �x − xF �t��

− EI

�AGF0��xx �x − xF �t�� (12)

EI �xxxx +mI�2

A

(1+ E

�G

) �xx +

(m2I�4

A2�G−m�2

)

= Q�x + F0��x �x − xF �t�� (13)

The spatial distribution of the forcing function can be taken into account by using an

integral equation, i.e. by using the superposition rule. The kernel of the integral equation is

the Green function.

3. DYNAMIC GREEN FUNCTION

The Dynamic Green Function is employed to find the solution for Eqs. (12) and (13).

It is a function of the beam vibration amplitudes excited by the unit harmonic load. By

applying the superposition rule, the spatial distribution of the loading function can be taken

into account by using an integral equation. Therefore, if G(x,u) was the Dynamic Green

Function for the presented problem, the solution of Eq. (12) can be taken the form as:

W �x� =∫ L

0F �u�G �x� u� du (14)

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Solving the above integration is complex in many cases and it has not been presented

analytically as an answer in a number of cases. There are other cases where integration

has a complex form and complicates the solution of the stated problem. In this case, it

might have to resort to a numerical procedure to solve the integration and/or it might be

able to extract the approximate behavior from the integral. The Dynamic Green Function,

G�x� u� � is the solution of the differential equation:

EIG�xxxx +mI�2

A

(1+ E

�G

)G�xx +

(m2I�4

A2�G−m�2

)G

=(1− �2mI

�A2G

)� �x − u� − EI

�AG��xx �x − u�+

(1− �2mI

�A2G

)� �x − xF �t��

− EI

�AG��xx �x − xF �t�� (15)

where ��x − u� is the Dirac delta function which is defined as:

��x − u� ={+� ifx = u

0 ifx �= u

In addition, ��xx�x − u� is the second order generalized derivative of the Dirac delta

function, which is described as (Falsone, 2002):

� �x − u� = d

dx�H �x − u�� or

∫ x

−�� �y − u� dy = H �x − u�

where H �x − u� is the Heaviside unit function which is defined as:

H �x − u� ={0 x < u

1 x ≥ u

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Therefore, ��xx is calculated as follows (Bracewell, 2000):

��xx �x − u� = 2

�x − u�2� �x − u�

The function G�x� u� will be determined as a sum of the particular solution for the non-

homogeneous equation and the general solution to the related homogeneous equation.

Therefore, if G0 �x� is the solution to the homogeneous equation and G1 �x� u� is the

solution to the inhomogeneous equation, the Dynamic Green Function can be given by:

G�x� u� = G0 �x�+G1 �x� u�+G1 �x� xF �t�� (16)

The free vibration equation of Timoshenko beam is obtained in the form:

G�xxxx + �2(r2 + �2

)G�xx − �2

(1− �2r2�2

)G = 0 (17)

where �� r and � are the parameter proportional to the natural frequency (�2 = �2mEI

�, the

radius of gyration of the beam cross section (r2 = IA� and the parameter proportional to the

rigidity of the beam (�2 = EI�AG

�, respectively. The general solution of free vibration Eq. (17)

can be stated as:

G0 �x� = C1 sin ���1x�+ C2 cos ���1x�+ C3 sinh ���2x�+ C4 cosh ���2x� (18)

where x ∈ 0� l� and �1 and �2 are calculated as:

�1 =

√√√√r2 + �2

2+

√(r2 − �2

2

)2

+ 1�2

(19)

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�2 =

√√√√−r2 + �2

2+

√(r2 − �2

2

)2

+ 1�2

(20)

It is evident from Eq. (19) that for all values of �, �21. is greater than zero. Therefore, �22.

is positive for �. satisfying r2�2�2 < 1. (or � < A√

�GmI�. and �22. is negative for �. satisfying

r2�2�2 > 1. (or � > A√

�GmI�. In this case, �22. can be defined as �22 = −�∗22 , where �∗2. is

real and it can also be concluded that, in this range of frequencies, Eq. (18) will contain

only trigonometric terms. In addition, C1� � � � � C4. are the integration constants which are

evaluated such that the Green Function G0 �x�. satisfies two boundary conditions at each

end of the beam depending on the type of end support:

M �0� = −KRL� �0� V �0� = KTLW�0�

M �l� = −KRR� �l� V �l� = KTRW�l�

where M and Q are the bending moment, M = EI��x, a the shear force are V =�AG

(w�x − �

), respectively. After separation of variable, it is expressed as:

�AG

�2r2 − 1�2

(�2

(r2 + �2

)G�x �0�+G�xxx �0�

)− KTLG�0� = 0 (21a)

G�xx �0�+ �2�2G�0�+ KRE

EI

(�2�4 + 1

)G�x �0�+ �2G�xxx �0�

�2r2�2 − 1= 0 (21b)

�AG

�2r2 − 1�2

(�2

(r2 + �2

)G�x �l�+G�xxx �l�

)− KTRG�l� = 0 (21c)

G�xx �l�+ �2�2G�l�− KRR

EI

(�2�4 + 1

)G�x �l�+ �2G�xxx �l�

�2r2�2 − 1= 0 (21d)

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By applying the boundary conditions at each end, the matrix equation is given as:A11 A12 A13 A14A21 A22 A23 A24A31 A32 A33 A34A41 A42 A43 A44

C1C2C3C4

=

0000

(22)

A11 =�2��1�r

2 + �2 − �21��3

−1+ r2�2�2A12 =

KTL

AG

A13 =�2��2�r

2 + �2 + �22��3

−1+ r2�2�2A14 =

KTL

AG

A21 =KRL�1�

EI

1+ �2��2 − �21��2

−1+ r2�2�2A22 = ��2 − �21��

2

A23 =KRL�2�

EI

1+ �2��2 + �22��2

−1+ r2�2�2A24 = ��2 + �22��

2

A31 = A11 cos ���1L�−KTR

AGsin ���1L� A32 = −KTR

AGcos ���1L�− A11 sin ���1L�

A33 = A13 cosh ���2L�−KTR

AGsinh ���2L� A34 = −KTR

AGcosh ���2L�+ A13 sin h ���2L�

A41 = −KRR�1�

EI

1+ �2��2 − �21��2

−1+ r2�2�2cos ���1L�+ A22 sin ���1L�

A42 = A22 cos ���1L�+KRR�1�

EI

1+ �2��2 − �21��2

−1+ r2�2�2sin ���1L�

A43 = −KRR�2�

EI

1+ �2(�2 + �22

)�2

−1+ r2�2�2cosh ���2L�+ A24 sinh ���2L�

A44 = A24 cosh ���2L�−KRR�2�

EI

1+ �2��2 + �22��2

−1+ r2�2�2sinh ���2L�

The nontrivial solution to Eq. (22) is obtained from the condition that the main matrix

determinant is equal to zero. The force vibration equation of Timoshenko beam is obtained

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in the form as:

G�xxxx + �2(r2 + �2

)G�xx − �2

(1− �2r2�2

)G

= 1− �2r2�2

EI� �x − u� − �2

EI��xx �x − u�+ 1− �2r2�2

EI� �x − xF �t��

− �2

EI��xx �x − xF �t�� (23)

In the forced vibration, the negligence of damping leads to large errors in the determined

vibration amplitudes, especially in the high frequency. Therefore, in this study, the forced

vibration solution is searched only for frequencies below the cut-off frequency, i.e. for � <

A√

�GmI. Hence, the internal and external damping of the beam is neglected in this study. The

general solution of force vibration Eq. (17) can be stated as for x > u and in the absence

of a moving load:

G1 �x� u� = D1 sin ���1 �x − u��+D2 cos ���1 �x − u��+D3 sin ���2 �x − u��

+D4 cosh ���2 �x − u���H �x − u� (24)

where D1� � � � � D4 are the integration constants which are evaluated such that the Green

Function G1 �x� u� satisfies the continuity conditions of displacement, slope and moment

along with the jump condition of shear force at x = u:

G1�u+� u� = G1�u

−� u� (25a)

��u+� = ��u−� (25b)

M�u+� = M�u−� (25c)

V(u+)− V �u−� = 1 (25d)

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By considering the relationships between the individual physical quantities and the Green

Function, the continuity conditions and the jump condition can be rewritten as follows:

(�2�4 + 1

)G1�x

(u+� u

)+ �2G1�xxx

(u+� u

)= (

�2�4 + 1)G1�x �u

−� u�+ �2G1�xxx �u−� u� (26a)

G1�xx�u+� u� = G1�xx�u

−� u� (26b)

�AG

�2r2 − 1�2

(�2

(r2 + �2

) (G1�x

(u+� u

)−G1�x �u−� u�

)+ (

G1�xxx

(u+� u

)−G1�xxx �u−� u�

)) = 1 (26c)

By applying the continuity conditions and the jump condition, the Green Function G1 �x� u�

for the Timoshenko beam is finally expressed as:

G1 �x� u� = D1 sin ���1 �x − u��+D2 sinh ���2 �x − u�� (27)

where:

D1 =1+ �4�2 + �2�2�22

�AG�2�3�1(�21 + �22

)D2 = − 1+ �4�2 − �2�2�21

�AG�2�3�2(�21 + �22

)The Green Function is obtained by following the above procedure, and has a general form.

By coming close to the spring constants of the translational and rotational restraint to

extreme values (infinity and/or zero), thus the suitable Green Function for the desired

combinations of end boundary conditions can be obtained (i.e. simply supported, clamped

and free boundary conditions).

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4. NUMERICAL EXAMPLES

4.1. The Frequency Parameters of Timoshenko Beam

To verify this new formulation, a beam with different boundary ends has been studied and

the results are compared with the results obtained by Abbas and Thomas (1984). The beam

is assumed with the following characteristics:

� = 0�3 � = 0�85

r2/L2 = 0�0016 r2 = I

A�2 = 2 �1+ �� I

�A

where L stands for the beam length, � is designated as the poisson ratio, and � represents

the shear factor.

Table 1 compares the frequency parameters of free vibration of a clamped-clamped and

simply supported at both ends and a cantilever Timoshenko beam using the Dynamic

Green Function and a Finite Element Model (Abbas and Thomas 1984). It is seen that the

results are fairly close and the maximum difference is 0.3%.

4.2. The Influence of the Spring Supports on the Frequency of Timoshenko Beam

In this problem, the influence of the spring supports behavior is evaluated for free vibration

characteristics of Timoshenko beam. For this purpose, a Timoshenko beam with general

boundary conditions, KT and KR, are considered. The stiffness of translational restraint

and rotational restraint are taken as having the same values at all the supports. The beam

characteristics are as follows:

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� = 0�3 � = 0�85

r/L = 0�04 r2 = I

A�2 = 2 �1+ �� I

�A

For simplicity, the spring supports are assumed to be as:

KTL = KTR = KT KRL = KRR = KR

The first frequency parameter (�� of free vibration of Timoshenko beam is illustrated

in Table 2. In addition, it is evident from the obtained values of frequency parameter

that when the values of KT/GA and KR/EI are greater than 1000 then the beam can be

considered as fixed-fixed at the both ends.

4.3. A Cantilever Beam Under a Moving Load

The presented new formulation for dynamic analysis of the Timoshenko beam with moving

load is successfully supplied into a cantilever beam. Furthermore, comparison of the results

with the results obtained by Lueschen et al. (1996) is shown. The beam is considered with

the following characteristics:

� = 7�28× 10−4 lb�s2/in E = 3× 107 psi � = 0�3

� = 0�83 I = 1�33in4 A = 4�0in2

L = 20�0 in m = �× A

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where, “m” represents the total mass of the beam, “�” as the density of beam, “E” as the

Young’s modulus, “I” as the moment of inertia, “A” as the beam cross-sectional area.

Figure 2 compares the magnitude of the Green’s function of the cantilever beam under a

unit point load in mid-span calculated using present study of the Green function approach

and the results reported by Lueschen et al. (1996). It is observed that there are no

slight differences between the results and the maximum difference is less than 6% for the

maximum magnitude of the Green’s function. In this problem, the Green’s function of

beam has been computed at frequencies of 100, 600 and 10000Hz.

5. CONCLUSIONS

This paper presents the dynamic response of uniform Timoshenko beams with a different

boundary condition using Dynamic Green Function. An exact and direct modeling

technique is stated to model beam structures with various boundary conditions subjected to

the external load which is an arbitrary function of time t and coordinate x. This technique

is based on the Dynamic Green Function. The method of Green Functions is more efficient

and simple in comparison with the other methods (e.g. series method) because essentially

the Green Function yields exact solutions in closed forms. In addition, the boundary

conditions are embedded in the Green Functions by the Green Functions method. The

effect of different boundary condition, load and other parameters is determined. Finally,

a number of numerical examples are presented to illustrate the efficiency and simplicity of

the new formulation based on the Dynamic Green Function.

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REFERENCES

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Abu-Hilal, M. (2003). Forced vibration of Euler–Bernoulli beams by means of dynamic

Green functions. Journal of Sound and Vibration 267(2):191–207.

Bracewell, R. (2000). The Fourier Transform and Its Applications. New York: McGraw-Hill.

Dyniewicz, B., Bajer, C. I. (2010). New feature of the solution of a Timoshenko beam

carrying the moving mass particle. Archives of Mechanics 62(5):327–341.

Eftekhar Azam, S., Mofid, M., Afghani Khorasghani, R. (2013). Dynamic response

of Timoshenko beam under moving mass. Scientia Iranica, Transactions A: Civil

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Ekwaro-Osire, S., Maithripala, D. H. S, Berg, J. M. (2001). A series expansion approach

to interpreting the spectra of the Timoshenko beam. Journal of Sound and Vibration

240(4):667–678.

Esmailzadeh, E., Ghorashi, M. (1997). Vibration analysis of Timoshenko beam subjected

to a travelling mass. Journal of Sound and Vibration 199(4):615–628.

Falsone, G. (2002). The use of generalised functions in the discontinuous beam bending

differential equations. International Journal of Engineering Education 18(3):337–343.

Foda, M. A., Abduljabbar, Z. (1998). A dynamic Green function formulation for the

response of a beam structure to a moving mass. Journal of Sound and Vibration

210(3):295–306.

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Hamada, R. (1981). Dynamic analysis of a beam under a moving force: a double Laplace

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Kargarnovin, M. H., Ahmadian, M. T., Jafari-Talookolaei, R. A. (2012). Dynamics of

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Lueschen, G. G. G., Bergman, L. A., McFarland, D. M. (1996). Green’s functions for

uniform Timoshenko beams. Journal of Sound and Vibration 194(1):93–102.

Majkut, L. (2009). Free and forced vibrations of Timoshenko beams described by single

difference equation. Journal of Theoretical and Applied Mechanics 47(1):193–210.

Mohamed, A. S. (1994). Tables of Green’s function for the theory of beam vibrations

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Mohebpoura, S. R., Daneshmand, F., Mehregan, H. A. (2013). Numerical analysis

of inclined flexible beam carrying one degree of freedom moving mass including

centrifugal and coriolis accelerations and rotary inertia effects. Mechanics Based Design

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Motaghian, S. E., Mofid, M., Alanjari, P. (2011). Exact solution to free vibration of beams

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Nesterenko, V. V. (1993). A theory for transverse vibrations of the Timoshenko beam.

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Savin, E. (2001). Dynamic amplification factor and response spectrum for the evaluation

of vibrations of beams under successive moving loads. Journal of Sound and Vibration

248(2):267–288.

Sun, L. (2001). Dynamic displacement response of beam-type structures to moving line

loads. International Journal of Solids and Structures 38(48–49):8869–8878.

Wang, C. M., Lam, K. Y., He, X. Q. (1998). Exact solutions for Timoshenko beams on

elastic foundations using Green’s functions. Mechanics Based Design of Structures and

Machines 26(1):101–113.

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Table 1: The frequency parameters of free vibration of Timoshenko beams different

boundary ends

mode clamped simply supported cantileverPresentstudy

(%Error)

Abbas &Thomas(1984)

Presentstudy

(%Error)

Abbas &Thomas(1984)

Presentstudy

(%Error)

Abbas &Thomas(1984)

1 19.9387 (0.0016) 19.939 9.57097 (0.0003) 9.571 3.46434 (0.0098) 3.4642 48.8750 (0.0183) 48.884 35.3589 (0.0088) 35.362 20.0408 (0.0010) 20.0413 85.1183 (0.0607) 85.170 71.6566 (0.0438) 71.688 50.6848 (0.0181) 50.6944 125.449 (0.1535) 125.642 113.845 (0.1325) 113.996 88.5021 (0.0688) 88.5635 168.339 (0.3193) 168.878 159.136 (0.3176) 159.643 130.541 (0.1797) 130.776

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Table 2: Variation of the first modal frequency parameter of free vibration of

Timoshenko beam

KR/EI

0 1 10 100 1000

KT/GA 0 0.000 4.334 8.135 9.393 9.5531 10.35 12.35 19.63 24.94 25.7810 9.644 11.22 16.35 19.88 20.46100 9.578 11.12 16.05 19.38 19.931000 9.572 11.11 16.03 19.34 19.88

simply supported Clamped

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Figure 1: Timoshenko beam with general boundary conditions.

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Figure 2: Magnitude of the Green’s function of the cantilever beam under a unit point

load in mid-span.

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