Effect of Mass per Unit Length on freely vibrating Simply Supported

Rayleigh Beam

OLASUNMBO O. AGBOOLA1, TALIB EH. ELAIKH2, JIMEVWO G. OGHONYON1, OLAJIDE

IBIKUNLE3

1Department of Mathematics, Covenant University, Ota, NIGERIA

2Department of Mechanical Engineering, College of Engineering, University of Thi-Qar, Thi-Qar,

IRAQ

3Department of Mathematics and Statistics, Gateway (ICT) Polytechnic, Saapade, Ogun State,

NIGERIA

Abstract: - In this paper, free vibration characteristics of a uniform Rayleigh beam are studied using the

differential transform method. The procedure entails transforming the partial differential equation governing

the motion of the beam under consideration and the associated boundary conditions. The transformation yields

a set of difference equations. Some simple algebraic operations are performed on the resulting difference

equations to determine any ith natural frequency and the closed-form series function for any ith mode shape.

Finally, one problem is presented to illustrate the implementation of the present method and analyse the effect

of mass per length on the natural frequencies of the beam.

Key-Words: - Differential transform, free vibration, Harmonic motion, Natural frequency, Mode shape,

Rayleigh beam, Mass per unit length

Received: June 25, 2021. Revised: August 6, 2022. Accepted: September 20, 2022. Published: October 13, 2022.

1 Introduction

Vibration of elastic bodies, because of its real-life

applications, has been studied by quite a number of

scholars. Many authors have considered the forced

vibrations of elastic bodies such as beams and

plates. A study on the influence of a moving load

with variable velocity on the dynamic response of a

simply supported Euler-Bernoulli beam was

undertaken by Awodola [1]. The beam was assumed

to be resting on a uniform foundation and excited by

a load moving with variable velocity. Oni and

Omolofe [2] investigated the transverse vibration of

a prismatic Rayleigh beam using generalized finite

integral transform and modified Struble’s

asymptotic method. The effects of boundary

conditions, slenderness ratio and elastic foundation

on which the beam rests were analysed. Auciello

and Lippiello [3] investigated the dynamic response

of a column partially immersed in water, using the

Rayleigh beam theory to model the column. Golas

[4] worked on the influence of the rotary inertia on

the eigenvalues of composite beams. It was found

that the influence of the rotary inertia is over ten

times smaller than the influence of shear

deformations. It was therefore suggested that rotary

inertia might be neglected.

Rajesh and Kumar [5] carried out a study on free

vibration behaviour of some viscoelastic sandwich

beams based on the Euler-Bernoulli beam model at

different end classical conditions. The viscoelastic

sandwich beams considered had aluminium and

mild steel as face material and the core material was

modelled using neoprene rubber. The study reveals

that higher natural frequencies are associated with

mild steel used as face material compared with

when aluminium is used. It was further shown that

the natural frequencies reduce when neoprene

rubber was used as the core material. Usman,

Ogunsan, Okusaga and Solanke [6] studied the

influence of damping coefficient on an Euler-

Bernoulli beam excited by distributed load using the

finite Fourier sine transform and finite difference

method. It was reported that an increase in the speed

of the load causes a decrease in the amplitude of the

beam’s deflection in the presence of damping

coefficient. Contrarily, it was found that the

amplitude of the beam’s deflection decreases as the

speed of the load increases when damping is

neglected.

Usman, et al. [7] used the series solution method

to obtain the response Euler-Bernoulli beam under

the excitation of a concentrated moving load. Jimoh

and Ajoge [8] employed Galerkin’s method and the

integral transform techniques to study the influence

of rotatory inertia and axial force on the vibration

characteristics of non-uniform beam. It was

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assumed that the beam resting on Pasternak

foundation was harmonically excited by moving

loads with varying magnitude. Jimoh and Ajoge [9]

considered the influence of rotatory inertial and

damping coefficient on the dynamic response of a

uniform Rayleigh beam traversed moving loads of

constant magnitude. The authors used the Fourier

Sine and Laplace Integral Transformations. It was

found that the beam’s amplitude of displacement

decreases due to increase in the values of rotatory

inertia and damping coefficient of the beam. The

effects of shear modulus, foundation modulus and

axial force on the beam’s amplitude of deflection

were also investigated.

The effect of variable prestress and foundation

constants on the natural frequencies of a simply

supported Rayleigh beam subjected to distributed

loads was analysed by Andi and Wilson [10]. The

generalized Galerkin’s and modified Struble’s

asymptotic methods were applied to solve the

vibration problem. The study reported that both the

natural and modified frequencies increase when the

values of prestress increase. Another finding of the

study has it that resonance is reached earlier for

lower values of prestress and lower values of the

foundation constant. The problem of dynamic

behaviour of two-steps nanobeam modelled using

the Rayleigh beam theory was studied by Hossain

and Lellep [11]. They analysed the influence of

rotatory inertia on the dynamic characteristics of the

system. The study shows that the effect of rotatory

inertia is highly significant in the nanobeam and its

influence rises with the increase of mode of

frequency. Omolofe and Adara [12] in a study

applied Galerkin’s residual method and Struble’s

asymptotic technique in conjunction with

Duhamel’s integral transform to analyse the

response of a beam under the action compressive

axial force and moving masses.

Differential transform method (DTM) used in

this paper has been proved to be highly effective in

solving both ordinary and partial differential

equations. Research works in which the method has

been successfully applied to solve problems in solid

mechanics and computational fluid mechanics.

These include the work of Opanuga, Adesanya,

Okagbue and Agboola [13] where DTM was used to

obtain semi-analytical solutions of the equations

governing the entropy generation of radiative

Magnetohydrodynamic mixed convection Casson

fluid. In another work by Opanuga et al. [14], DTM

was used to solve the velocity and energy equations

associated with the entropy generation of unsteady

hydromagnetic Couette flow through vertical

microchannel. Agboola et al. [15] used DTM to

study the entropy generation of a steady natural

convection flow between two vertical parallel

micro-channels with Hall effect.

The aim of this work is to analyse numerically

the vibration characteristics of a uniform Rayleigh

beam considering simply supported end conditions.

The effect of mass per unit length on the non-

dimensional frequencies of the freely vibrating

beam is also explored. It is pertinent to note that the

effects of rotary inertia and shear deformation are

neglected in the Euler-Bernoulli beam theory, which

make the theory applicable to an analysis of long

and slender beams only. On the other hand, the

Rayleigh beam theory takes cognizance of the effect

of rotary inertia, while the Timoshenko beam

theory, which is applicable to short and thick beams,

considers the effects of both rotary inertia and shear

deformation. In real life engineering application,

Rayleigh beams are used to model spinning beam.

[16].

2 Problem Formulation

The partial differential equation governing the free

vibration of a uniform Rayleigh beam is given by

4 2 4

4 2 2 2

( , ) ( , ) ( , ) 0



  

  

   

V x t V x t V x t

EI b

x t x t

(1)

Here,

is the Young’s modulus,

is the moment

of inertia of the cross section of the beam,



is the

mass per unit length of the beam,

is the rotatory

inertia of the beam,

( , )V x t

is the transverse

displacement of the beam at point

and time

The beam is assumed to be simply-supported at both

ends. Thus, the associated boundary conditions at

the two ends of the beam, that is at

0x

and

Lx 

(

being the length of the beam) for the

partial differential equation in (1) are as follows:

(0, ) 0Vt

, (2)

(0, ) 0





, (3)

( , ) 0V L t

, (4)

and

( , ) 0





V L t

. (5)

The initial conditions are given by

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( ,0)

( ,0) 0, 0







Vx t

. (6)

3 Problem Solution

3.1 Description of the Method of Solution

The differential transform method, which is based

on the Taylor series expansion, can be used to

obtain analytical solutions of differential equations.

To apply this method, certain transformation rules

are used to transform the governing equation of

motion and the associated boundary conditions of

the problem under consideration. This process will

yield a system of algebraic equations in terms of the

differential transforms of the original functions.

Solving this resulting system of algebraic equations

yields the desired solution of the transformed

problem. The differential transform method is

described as follows:

Let

be any point in a domain D. Also suppose

()Fx

is analytic in domain D. Then a power series

whose center is

can be used to represent the

function. The differential transform of the function

()fx

is given by

1 ( )

() !







d F x

Fk k dx

, (7)

where

()Fx

is the function to be transformed and

()Fk

called the transformed function is the new

function obtained after the transformation. The

inverse transformation is defined as

( ) ( ) ( )







k

F x x x F k

. (8)

To express

()Fx

by a finite series, Eqs. (7) and (8)

are combined to get the series

() ()

() !











kk

kxx

xx d F x

Fx k dx

. (9)

It shoulde be noted that the value of

depends

largely on the convergence of the natural

frequencies. Some of the theorems that are useful

when transforming the governing differential

equation are provided as follows.

Theorem 1:

( ) ( ) ( )F x G x H x

, then

( ) ( ) ( )F r G r H r

Theorem 2:

( ) ( )



F x G x

, then

( ) ( )



F r G r

Theorem 3:

If,

( ) ( ) ( )F x G x H x

, then

( ) ( ) ( )







F r G s H r s

Theorem 4:

()

()n

d G x

Fx dx

, then

 

( ) ( )





F r G r n

Theorem 5:

()n

F x x

, then

 

() 1







   



if r n

F r r n if r n

The basic DTM theorems that are used for

transforming boundary conditions, which are found

applicable in this paper are as follows:

Theorem 6:

(0) 0F

, then

(0) 0F

Theorem 7:

(0) 0

, then

(1) 0F

Theorem 8:

(1) 0F

, then

( ) 0







Theorem 9:

(1) 0

, then

( ) 0







kF k

Theorem 10:

2(1) 0

, then

( 1) ( ) 0









k k F k

3.2 Using the DTM to Analyze the Free

Vibration Problem of Rayleigh Beam

To obtain the solution of the differential equation

(1) subject to the given conditions, a sinusoidal

variation of

( , )V x t

is assumed and consequently

the function is approximated as

 

, ( ) 

it

V x t W x e

, (10)

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where

()Wx

is the modal deflection and



is the

circular natural frequency of the harmonic function

of time.

Using equation (10), then equations (1)-(5) can be

expressed as follows:

( ) ( )

( ) 0,



    



d W x d W x

EI W x b

dx dx

(11)

(0) 0W

, (12)

(0) 0

, (13)

( ) 0WL

, (14)

() 0

d W L

. (15)

Let us introduce the dimensionless quantities as

follows

()

, ( )





x W x

(16)

The governing equation (11) can then be written in

the following dimensionless form:

( ) ( ) ( ) 0







    

d w d w w

(17)

where the dimensionless coefficients are given by



 L





bL

. (18)

The boundary conditions in equations (12)–(15)

have the following dimensionless form:

(0) 0w

, (19)

(0) 0





, (20)

(1) 0w

, (21)

(1) 0





. (22)

Taking the differential transform of equation (11) in

accordance with the Theorems 1-4, one obtains

( 1)( 2)( 3)( 3) ( 4)

( 1)( 2) ( 2) ( ) 0



    

       

r r r r W r

r r W r W r

. (23)

The following recursive equation can be obtained

from equation (23):

( ) ( 1)( 2) ( 2)

( 4) ( 1)( 2)( 3)( 4)



    

  

   

W r r r W r

Wr r r r r

(24)

Now applying appropriate transformation theorems

6-8 and 10, the boundary conditions (19)-(22)

become

(0) 0W

, (25)

(2) 0W

, (26)

( ) 0





, (27)

and

( 1) ( ) 0







r r W r

. (28)

Let us define

(1) Wc

, (29)

(3)Wc

, (30)

as the unknown parameters.

Substituting equations (29) and (30) into equation

(24), we have

For

0r

(4) 0W

(31)

For

1r

(5) 5!







(32)

By following the same recursive procedure,

(6)W

up to

()Wm

can be evaluated; where

is to be

determined by the convergence of natural

frequencies.

Substituting

()Wj

, for

0, 1, ,jm

into equations

(27) and (28), we obtain a system of two algebraic

equations which can be put in the matrix form as

follows:

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[ ] [ ]

11 12

[ ] [ ]

21 22

( ) ( )













 



, (33)

where

[]

11 ()

[]

12 ()

[]

21 ()

and

[]

22 ()

are

polynomials of



For the nontrivial solutions of equation (33), the

determinant of the coefficient matrix is set to zero.

Thus, we have

[ ] [ ]

11 12

[ ] [ ]

21 22

( ) ( ) 0

( ) ( )







(34)

The dimensionless natural frequencies are then

calcuated by solving equation (34).

[]

m

is the

estimated

jth

dimensionless natural frequency that

corresponds to

. The value of

is decided by the

following convergence criterion:

[ ] [ 1]





   

, (35)

where

[ 1]

m

is the jth estimated dimensionless

natural frequency corresponding to

1m

and



is a

predefined small value.

3.3 Verification and Case Study

Setting



M

and the first dimensionless

natural frequency and mode shape for

demonstration, the computations and results

corresponding to

16m

are described as follows:

Substituting Equations (25), (26), (29) and (30) and

0r

into Eq. (24), we have

(4) 0W

(36)

Substituting Eqs. (25), (26), (29)-(31) and

1r

into

Eq. (24), we have

 

(5) 6

  W c c

. (37)

Substituting Equations (25), (26), (29)-(31) and

2r

into Eq. (24), we have

(6) 0W

. (38)

Substituting Equations (25), (26), (29)-(32) and

3r

into Eq. (24), we have

4 2 2

(7) (1 ) 6(2 )

7! 

      



W c c

(39)

Following the same recursive procedure, we

calculate up to the 20th term

(16)W

. Substituting

( ), 0,1,2, , 16W j j

into Eqs. (27) and (28) and

using Eq. (34), we have the frequency equation as

follows

9 12 7 10

1.126964276 10 1.700514259 10

0.00001858512347 0.001325165135

0.05481481481 1.066666667 0.6



   









  

   

(40)

Solving Eq. (34), we have the first two roots

 

12.9936

(41)

 

25.9019

(42)

When

15j

, by the same method we obtain

 

12.9936

(43)

From Eqs. (35) and (37), we have

[20] [19]

| | 0



    

, (44)

which implies convergence.

So,

12.9936

is taken as the first dimensionless

natural frequency.

Substituting

12.9936

into

( ), 0,1, , 16V j j

and using

( ) ( )





j

V V j

, we obtain the closed

form series solution of the first mode shape.

11 13

( ) 1.644936963 0.8117470647

0.1907536301 0.02614819400

0.002346120939 0.001484318771

0.000006976044909

   





  





(45)

Following the same routine demonstrated above,

one can determine the other natural frequencies and

their associated mode shapes. As the number of

terms, denoted by m increases, the first five non-

dimensional natural frequencies



up to



of the

Rayleigh beam converge to 2.993593837,

6.205088447, 9.372169410, 12.52676961,

15.72132054. The predefined value of



used to

monitor the convergence of the natural frequencies

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0.0001





. In the example considered, the first

five natural frequencies converge quickly one by

one without missing any frequency. The natural

frequencies are then used to determine their

corresponding mode shapes. The first three mode

shapes of the freely vibrating beam with the given

configuration are shown in Figs. 2, 3 and 4. The

combination of all the first three mode shapes is

given in Fig. 5.

Fig. 1: First mode shape

Fig. 2: Second mode shape

Fig. 3: Third mode shape

Fig. 4: The first, second and third mode shapes

Table 1 shows the effect of mass per unit length of

the beam on the first five natural frequencies of the

simply supported Rayleigh beam. The results reveal

that the non-dimensional frequencies calculated

become smaller with the increase in the mass per

unit length of the beam. The implication of this is

that there will be a decrease in natural frequencies

of excitation of the beam if there is an increase in

the beam’s mass per unit length.

Table 1. Effect of Mass per Unit length of the beam

on Non-dimensional frequencies of Simply-

Supported Rayleigh Beam (





)



2.9940

2.5595

2.2141

1.9791



6.2051

5.9195

5.6124

5.3487



9.3722

9.1702

8.9352

8.7174



12.5268

12.3703

12.1864

12.0089



15.7213

15.5955

15.4428

15.2943

4 Conclusion

The differential transform method has been used to

the closed form series solution of the freely

vibrating uniform Rayleigh beam. As earlier noted,

solving the problem using DTM involves three main

steps. The first step is to transform the equation

governing the motion as well as the boundary

conditions into a system of algebraic difference

equations. The second step entails solving the set of

algebraic difference equations in step one. Finally,

the solution of the transformed problem is inverted

using the inverse differential transform to determine

the natural frequencies and their associated closed

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form series solution of the mode shape. The

influence of mass per unit length on the natural

frequencies of a simply supported Rayleigh beam

freely vibrating has also been investigated. The

governing differential equation is solved and the

dimensionless natural frequencies for various values

of the mass per unit length of the beam were

obtained and presented in table. It was found that

the natural frequency of the beam decreases with

increase in the value of the mass per unit length. It is

recommended that the further studies on the

influence of rotary inertia on the vibration

characteristics of freely vibrating Rayleigh beam

should be carried out.

Nomenclature

Young’s modulus,

Moment of inertia of the cross section of the

beam



Mass per unit length of the beam,

Rotatory inertia of the beam

x spatial location along the beam

T time

( , )V x t

Transverse displacement of the beam at

point

and time

L length/span of the beam



Circular natural frequency of harmonic

function of time

()Wx

Modal deflection of the beam



Non-dimensional parameter of the spatial

location along the beam

()



Non-dimensional parameter of the

modal deflection of the beam



Non-dimensional parameter of the mass per

unit lenght of the beam



Non-dimensional rotary inertia

Acknowledgments:

The authors appreciate Covenant University for

supporting this research financially.

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Olasunmbo O. Agboola, Talib Eh. Elaikh,

Jimevwo G. Oghonyon, Olajide Ibikunle

E-ISSN: 2224-347X

179

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[13] A.A. Opanuga, S.O. Adesanya, H.I. Okagbue,

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Contribution of Individual Authors to the

Creation of a Scientific Article (Ghostwriting

Policy)

-Olasunmbo O. Agboola carried out the formulation

of the problem and performed the numerical

caclculations.

-Talib Eh. Elaikh contributed to the interpretation

and discussion of the results.

-Jimevwo G. Oghonyon and Olajide Ibikunle

contributed to the analysis of the results and to the

writing of the manuscript.

Sources of Funding for Research Presented in a

Scientific Article or Scientific Article Itself

Covenant University, Ota, Nigeria provided funding

for the publication of this article.

Creative Commons Attribution License 4.0

(Attribution 4.0 International, CC BY 4.0)

This article is published under the terms of the

Creative Commons Attribution License 4.0

https://creativecommons.org/licenses/by/4.0/deed.en

_US

WSEAS TRANSACTIONS on FLUID MECHANICS

DOI: 10.37394/232013.2022.17.17

Olasunmbo O. Agboola, Talib Eh. Elaikh,

Jimevwo G. Oghonyon, Olajide Ibikunle

E-ISSN: 2224-347X

180

Volume 17, 2022