Edelweiss Applied Science and Technology 
ISSN: 2576-8484 
Vol. 2, No. 1, 1-9 
2018 

DOI: 10.33805/2576.8484.106 
© 2018 by the authors 

© 2018 by the authors 
History: Received: 11 November  2017; Accepted: 18 December 2017; Published: 1 January 2018. 
* Correspondence: samueladebola84@gmail.com  

 
 
 
 
 

Dynamic Response to Moving Distributed Masses of Pre-stressed Uniform 
Rayleigh Beam Resting on Variable Elastic Pasternak Foundation 

 
Adeoye AS1*, Awodola TO1  
1Department of Mathematical Sciences, Federal University of Technology, Akure, Nigeria; samueladebola84@gmail.com 
(A.A.). 

 
 

Abstract: The dynamic response to moving distributed masses of pre-stressed uniform Rayleigh beam resting on variable 

elastic Pasternak foundation is examined. The equation governing this problem is a fourth order partial differential equation 
with variable and singular co-efficients. To solve this cumbersome equation, the method of Galerkin approach is adopted to 
reduce the governing differential equation to a sequence of coupled second order ordinary differential equation which is then 
simplified further with modified asymptotic method of Struble. The more simplified equation is solved using the Laplace 
transformation technique. The closed form solutions obtained are analyzed in order to show the conditions of resonance, and to 
show that resonance is attained earlier in moving mass system than in the moving force system. The results in plotted graphs 
show that as the axial force, the rotatory inertia, foundation modulus and shear modulus increase, the deflection of the 
elastically supported non-uniform Rayleigh beam decreases in each case. The transverse deflections of the beam on variable 
Pasternak elastic foundation are higher under the action of moving masses than those when only the force effects of the moving 
load are considered. This implies that resonance is reached faster in moving mass problem than in moving force problem.  

Keywords: Pasternak foundation; Shear deformation; Resonance; Critical speed; Natural frequency; Axial force; Modified 
frequency. 

 
1. Introduction  

Transport structures such as railway or bridges are subjected to moving vehicles (loads) which vary in both space and 
time. The branch of transport has experienced great advances, characterized by increasing high speed and weights of vehicles. 

These structures on which the vehicles move have been subjected to vibration and dynamic stress more than ever before. 
Therefore, the moving load problem has been a fundamental problem in several fields of applied mathematics, mechanical 
engineering, applied physics and railway engineering. Rails and bridges are examples of structured elements to be designed to  
support moving masses. Most importantly, problems of this type are mathematically cumbersome when the inertial effect of the 

load is taken into consideration. The challenges of these designs have attracted the interest of many researchers in the fields of 
applied mathematics, mechanical engineering, applied physics and railway engineering.  Some of these researchers include 
Fryba [1] who studied the vibration of solids and structures under moving loads. Gbadeyan and Dada [2] examined the 
influence of elastic foundation on plate under a moving load without considering the influence of rotatory inertia and shear 
deformation on the plate. The work of Stanistic et al was taken up much later by Gbadeyan and Oni [3] who investigated the 
dynamic analysis of an elastic plate continuously supported by an elastic foundation and traversed by an arbitrary number of 
concentrated moving masses. Yavari [4] studied the generalized solution of beams with jump discontinuities on elastic 
foundation. Yin [5] also investigated the closed form solution for reinforced Timoshenko beam on elastic foundation. In the 
same vein, Teodoru [6] in his work, analyzed beam on elastic foundation by using finite difference approach.  Oni and Awodola 
[7] investigated the vibrations under a moving load of a non-uniform Rayleigh beam on variable elastic foundation. Oni and 
Awodola [8] also analyzed the dynamic response under a moving load of an elastically supported non-prismatic Bernoulli-
Euler beam on variable. elastic foundation. In the work of Oni and Omolofe [9] the dynamic analysis of a pre-stressed elastic 
beam with general boundary conditions under moving loads at varying velocities was investigated. The study on exact series 
solution for the transverse vibrations of rectangular plates with elastic boundary supports was carried out by Li [10]. Hsu 
[11] studied the vibration analysis of non-uniform beams resting on elastic foundation. The work of Ismail [12] on dynamic 
response of a beam due to an accelerating moving mass using moving finite element approximation cannot be ignored. 
Kargarmovin and Younesian [13] took further study on dynamic of Timoshenko beams on Pasternak foundation under 
moving load. Recently, Adeoye and Awodola [14] took a close studied on influence of rotatory inertial correction factor on the 
vibration of elastically supported non-uniform Rayleigh beam using Galerkin method and Struble technique.  Adeoye and 
Akintomide [15] investigated dynamic behavior of Bernoulli-Euler beam with elastically supported boundary conditions under 

moving distributed masses on constant bi-parametric foundation using Galerkin method and Struble technique.  Akintomide 
and Awodola [16] analyzed the dynamic response to variable-magnitude moving distributed masses of Bernoulli-Euler beam 
on bi-parametric foundation and they obtained the closed form solution using Runge-Kutta technique.  

In our recent research work, Adeoye and Awodola [14] effort was made to investigate the influence of rotatory inertial 
correction factor on the vibration of elastically supported non-uniform Rayleigh beam on variable foundation. The objective of 
this paper is to extend this research work to elastically supported uniform Rayleigh beam on variable elastic bi-parametric 
foundation. This paper therefore investigates dynamic response to moving distributed masses of pre-stressed uniform Rayleigh 
beam resting on variable elastic Pasternak foundation. 
 
1.1. Governing Equation 

Considering the dynamic response to moving distributed masses of pre-stressed uniform Rayleigh beam on variable elastic 
Pasternak foundation; the governing equation of motion is given by the fourth order partial differential equation Fryba [1]. 

𝒜𝐼
𝜕4

𝜕𝑥4 𝒷(𝑥, 𝑡)
𝜕

𝜕𝑥
[𝒩𝑜

𝜕

𝜕𝑥
𝒷(𝑥, 𝑡)] 𝔅(𝑥)

𝜕2

𝜕𝑡 2 𝒷(𝑥, 𝑡) 𝔅(𝑥)ℛ 𝑜
𝜕4

𝜕𝑥2𝜕𝑡 2 𝒷(𝑥, 𝑡) + 𝛿(𝑥, 𝑡) =     𝒫(𝑥, 𝑡) (1) 

where 𝑥 is the spatial co-ordinate, t  is the time co-ordinate, 𝒷(𝑥, 𝑡) is the transverse displacement, 𝒜𝐼 is the flexural rigidity of 

the structure, 𝔅(𝑥) is the variable mass per unit length of the non-uniform beam, 𝒩𝑜is the constant axial force, ℛ 𝑜  is the 

rotatory inertial correction factor, 𝛿(𝑥, 𝑡) is the variable foundation reaction  𝒫(𝑥, 𝑡) is the moving distributed load. 

The relationship between the foundation reaction and lateral deflection 𝒷(𝑥, 𝑡) is 

𝛿(𝑥, 𝑡) = 𝒮(𝑥)𝒷(𝑥, 𝑡) −
𝜕

𝜕𝑥
[ℋ(𝑥) 𝜕

𝜕𝑥
𝒷(𝑥, 𝑡)]  (2) 

where, 𝒮(𝑥)  and ℋ(𝑥) are two variable parameters of the elastic foundation. That is,  𝒮(𝑥)  is the variable foundation stiffness 
(foundation modulus) and  ℋ(𝑥) is the variable shear modulus. 
where  

𝒮(𝑥) = 𝒮𝑂(4𝑥 − 3𝑥2 + 𝑥3)                               (3) 



2 

 

 
Edelweiss Applied Science and Technology 

ISSN: 2576-8484 

Vol. 2, No. 1: 1-9, 2018 
DOI: 10.33805/2576.8484.106 
© 2018 by the authors 
 

ℋ(𝑥) = ℋ𝑂(12 − 13𝑥 + 6𝑥2 + 𝑥3)              (4) 

𝔅(𝑥) = 𝔅𝑜 (1 + sin
𝜋𝑥

𝐿
)                                    (5) 

Substituting equations (3) , (4)  and (5) into equation (1), one obtains 

 

( ) ( ) ( )

( ) ( ) ( )

( ) ( ) ( ) ( )

4 2 2

4 2 2

4
2 3

2 2

2
2 3

2
1

  , , 1 sin ,

   1 sin   , 4 3 ,

1
[ 12 13 6 , ] [1 , ]     (6)

o

o o O

N

O

i

x
I b x t b x t b x t

x x L t

x
b x t x x x b x t

L x t

d
x x x b x t MgH x vt b x t

x x g dt





=

   
− + + 

   

 
− + + − + 

  

 
− − + + = − −
 



B

B

A N

R S

H
 

The boundary condition of the structure under consideration is first taken to be arbitrary and the initial condition without any 
loss of generality is taken as 

𝒷(𝑥, 0) = 0 =
𝜕

𝜕𝑡
𝒷(𝑥, 𝑜)                                                                     (7) 

 
2. Analytical Approximate Solution 

Due to complex nature of equation (1), no conventional method can be used to solve the partial differential equation and 
till this moment, there is no exact closed form solution to equation(1). Therefore, an approximate solution is sought. The 
method of Galerkin is used to reduce equation(1) to second order coupled ordinary differential equations, and this takes the 

form   

𝒷𝑖(𝑥, 𝑡) = ∑ 𝓌𝑖 (𝑡)𝒰𝑖(𝑥)

𝑁

𝑖=1

                                                                      (8) 

where  

𝒰𝑖(𝑥) =  sin
𝔇𝑖𝑥

𝐿
+ 𝐴𝑖 cos

𝔇𝑖𝑥

𝐿
+ 𝐵𝑖 sinh

𝔇𝑖𝑥

𝐿
+ 𝐶𝑖 cosh

𝔇𝑖𝑥

𝐿
         (9) 

( ) ( ) ( ) ( ) ( )

( ) ( ) ( ) ( ) ( )( )

( ) ( ) ( )

( ) ( ) ( ) ( )

¨
'' ''

1

'' 2 3

0

' ' 2 '

'' '' 2 '' 3 ''

sin   sin

 
4 3

4

13 12 3

12 13 6

( )

N

ii i o i i

m

iv o O
i i i i i

o o

i i iO

o i i i i

i

x x
x x x x t

L L

I
x x x x x x

W

x x

x x x x x

x x x x x x x

w t

 



=

  
+ − + +   

  

 
− + − + 

 
 

 − + − −     + − + −  

+



B B

B

U U R U U

N SA
U U U U U

U U UH

U U U U

( ) ( ) ( ) ( ) ( ) ( )

( )

¨
' 2 ''( )  2

0

ii i i i i

o

o

M
H Wx vt x t v x w t v x w t

Mg
H x vt

 
− + + 

 


− − =



B

B

U U U

  

 
Equation (11) can be re-written as 

 
Where 

𝐹𝐴 =
𝒜𝐼

4𝔅𝑜
, 𝐸1(𝑖 , 𝑗) = ∫ 𝒰𝑖 (𝑥)𝒰𝑗(𝑥)𝑑𝑥

𝐿

0
, 𝐸2(𝑖 , 𝑗) = ∫ sin

𝜋𝑥

𝐿

𝐿

0
𝒰𝑖(𝑥)𝒰𝑗(𝑥)𝑑𝑥      (13) 

 



3 

 

 
Edelweiss Applied Science and Technology 

ISSN: 2576-8484 

Vol. 2, No. 1: 1-9, 2018 
DOI: 10.33805/2576.8484.106 
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In order to evaluate the integrals in E_17 (i,j),E_18 (i,j)  and E_19 (i,j), one makes use of the Fourier  series 
representation for the Heaviside function in the form; 

𝐻(𝑥 − 𝑣𝑡) =
1

4
+

1

𝜋
∑

sin(2𝑛 + 1)𝜋(𝑥 − 𝑣𝑡)

2𝑛 + 1

∞

𝑛=1

,   0 < 𝑥 < 1   (23) 

To solve E20(i,j), one makes use of  the definition of Heaviside function and substitute the result  into (12), one obtains, 

 
Therefore, equation (24) becomes 

 
Where  

 
Equation (25) is re-written as 

 
Equation (29) is the transformed equation governing the problem of supported beam on variable bi-parametric elastic 

foundation. This coupled non-homogeneous second order ordinary differential equation is assumed to have arbitrary boundary 
conditions. 
 
2.1. Case I: Moving Force Problem 

In moving force problem, only the load is being transferred to the structure. In this case, the inertia effect is negligible. 

Setting ϖ=0 in the transformed equation (27), one obtains 

 
Equation (28) can be rewritten as, 

 
 
Where 

 
Equation (29) is an approximate model, which assumes the inertia effect of the moving mass as negligible. 
Further rearrangement of (29) yields 

 

Where  𝛺𝑗 =
𝔇𝑗𝑣

𝐿
     (32) 

Solving equation (31) using Laplace transformation techniques and taking into account equation (7) one obtains  

 
Equation (33) represents the transverse deflection of uniform Rayleigh beam under moving distributed force and resting 

on variable Pasternak elastic foundation. 
 
 

 



4 

 

 
Edelweiss Applied Science and Technology 

ISSN: 2576-8484 

Vol. 2, No. 1: 1-9, 2018 
DOI: 10.33805/2576.8484.106 
© 2018 by the authors 
 

2.2. Case II:  Moving Mass Problem 
In moving mass problem, the moving load is assumed to be rigid, and the weight and as well as inertia forces are 

transferred to the moving load. That is the inertia effect is not negligible. Thus  ϖ≠0 and so it is required to solve the entire 
equation (27). Thus, equation (27) takes the form 

 
On further rearrangements, one obtains 

 
Obviously, unlike the moving force problem, an exact analytical solution to equation (35) is not possible. In order to obtain 

approximate analytical solution, one makes use of a modification of the asymptotic method of Struble. By this method, one seeks 
the modified frequency corresponding to the frequency of the free system due to the presence of the effect of the moving mass . 
An equivalent system operator defined by the modified frequency then replaces equation (35). 
We shall consider a parameter ϖ0 < 1 for any arbitrary mass ratio defined by 

𝜛0 =
𝜛

1 + 𝜛
     (36) 

By using binomial theorem and truncating after second terms, one obtains  𝜛0 = 𝜛 − o(𝜛2)   (37) 

Equation (45) becomes 𝜛 = 𝜛0       (38) 
to 𝑜(𝜛) only and from equation (38) 

1

1 + 𝜛o (
1

4

𝒢1(𝑖,𝑗)

Ҕo(𝑖,𝑗)
+

1

𝜋
∑ cos

(2𝑛+1)𝜋𝑣𝑡

2𝑛+1

𝒢2(𝑖,𝑗)

Ҕo(𝑖,j)
−

1

𝜋
∑ sin

(2𝑛+1)𝜋𝑣𝑡

2𝑛+1

𝒢3(𝑖,𝑗)

Ҕo(𝑖,𝑗)
∞
𝑛=0

∞
𝑛=0 )

      (39) 

[1 − 𝜛o (1

4

𝒢1(𝑖,𝑗)

Ҕo(𝑖,𝑗)
+

1

𝜋
∑ cos

(2𝑛+1)𝜋𝑣𝑡

2𝑛+1

𝒢2(𝑖,𝑗)

Ҕo(𝑖,j)
−

1

𝜋
   

1

𝜋
∑ sin

(2𝑛+1)𝜋𝑣𝑡

2𝑛+1

𝒢3(𝑖,𝑗)

Ҕo(𝑖,𝑗)
∞
𝑛=0

∞
𝑛=0 ) + ⋯ ]       (40)  

Where 

|𝜛 (
1

4

𝒢1(𝑖, 𝑗)

Ҕo(𝑖, 𝑗)
+

1

𝜋
∑ cos

(2𝑛 + 1)𝜋𝑣𝑡

2𝑛 + 1

𝒢2(𝑖, 𝑗)

Ҕo(𝑖, j)
−

1

𝜋
∑ sin

(2𝑛 + 1)𝜋𝑣𝑡

2𝑛 + 1

𝒢3(𝑖, 𝑗)

Ҕo(𝑖, 𝑗)

∞

𝑛=0

∞

𝑛=0

)| < 1      (41) 

Substituting equations (40) and (41) into equation (35), one obtains 

 
to 0(𝜛0)  only 
Applying method of Struble technique to eqation (42) one obtains 

 
Where 

𝜎𝑖𝑖 = 𝜎𝑖 −
𝜛0

2
[
𝒢1(𝑖, 𝑗)𝜎𝑖

2 − 𝑣2𝒢7(𝑖, 𝑗)

4𝜎𝑖Ҕo(𝑖, 𝑗)
]      (44) 

Solving equation (43) using Laplace transformation techniques and taking into account equation (7) one obtains 

 
Equation (45) represents the transverse deflection of uniform Rayleigh beam under moving distributed mass and resting 

on variable Pasternak elastic foundation. 
 
3. Discussion of the Analytical Solutions 

For this undamped system, it is desirable to examine the phenomenon of resonance. From equation(33), it is clearly shown 
that the beam resting on variable bi-parametric elastic foundation and traversed by a moving distributed force reaches a state of 
resonance whenever   

𝜎𝑖 = 𝛺𝑗                                             (46) 

Where  𝛺𝑗 =
𝔇𝑗𝑣

𝐿
            (47) 

that is  𝜎𝑖 = 
𝔇𝑗𝑣

𝐿
             (48) 

Equation (45) shows that the same beam under the action of moving distributed mass experiences resonance effect 
whenever 

𝜎𝑖𝑖 =
𝔇𝑗𝑣

𝐿
                                       (49) 

From equation (44) 



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Edelweiss Applied Science and Technology 

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Vol. 2, No. 1: 1-9, 2018 
DOI: 10.33805/2576.8484.106 
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𝜎𝑖𝑖 = 𝜎𝑖 −
𝜛

2
[
𝒢1(𝑖, 𝑗)𝜎𝑖

2 − 𝑣2𝒢7(𝑖, 𝑗)

4𝜎𝑖Ҕo(𝑖, 𝑗)
] =   

𝔇𝑗𝑣

𝐿
     (50)   

It is therefore clear that for the same natural frequency, the critical speed for the system consisting of elastically supported 
uniform Rayleigh beam resting on variable elastic foundation and traverse by moving distributed force with uniform speed is 
greater than that of moving distributed mass problem. Thus for the same natural frequency, resonance is reached faster in the 
moving distributed mass system than in the moving distributed force system. 

 

3.1. Illustrative Examples 
3.1.1. Clamped-Elastic Boundary Conditions 

At a clamped end, both deflection and slope vanish. Thus, when the Rayleigh beam is clamped at 𝑥 = 0 and elastically 

supported at 𝑥 = 𝐿, the conditions are expressed as 
𝒷(0, 𝑡) = 0 = 𝒷 ′(0, 𝑡)                     (59) 

at the end 𝑥 = 0 and   

𝒷 ′′ − 𝑘1𝒷 ′(𝐿, 𝑡) = 0 = 𝒷 ′′′(𝐿, 𝑡) + 𝑘2𝒷(𝐿, 𝑡)           (60) 
at the end 𝑥 = 𝐿 and for the normal modes  

𝒰𝑖(0) = 0 = 𝒰𝑖
′(0)           (61) 

at the end 𝑥 = 0 and  

𝒰𝑖
′′(𝐿) − 𝑘1𝒰𝑖

′(𝐿) = 0 = 𝒰𝑖
′′′(𝐿) + 𝑘2𝒰𝑖(𝐿)      (62) 

at end 𝑥 = 𝐿 which implies that 
𝒰𝑗(0) = 0 = 𝒰𝑗

′(0)               (63) 

and  
𝒰𝑗

′′(𝐿) − 𝑘1𝒰𝑗
′(𝐿) = 0 = 𝒰𝑗

′′′(𝐿) + 𝑘2𝒰𝑗(𝐿)       (64) 

Using equations (59) and (60), it can be shown that at 𝑥 = 0, 

𝐴𝑖 = −𝐶𝑖 and  𝐵𝑖 = −1               (65) 
 and at 𝑥 = 𝐿, using (64) 

𝐴𝑖 =

𝜙𝑖

𝐿
[ sin 𝜙𝑖 +   sinh 𝜙𝑖] + 𝑘1[cos 𝜙𝑖 − cosh 𝜙𝑖]

𝜙𝑖

𝐿
[cos 𝜙𝑖 + cosh 𝜙𝑖] −  𝑘1[sin 𝜙𝑖 +  sinh 𝜙i]

=

𝜙𝑖
3

𝐿3
[cos 𝜙𝑖 +  cosh 𝜙𝑖] +  𝑘2[sinh 𝜙𝑖 −  sin 𝜙𝑖]

−𝜙𝑖
3

𝐿3
[sin 𝜙𝑖 − sinh 𝜙𝑖] +  𝑘2[cos 𝜙𝑖 −  cosh 𝜙𝑖]

 

= −𝐶𝑖             (66) 

From (66)  one obtains tan 𝜙𝑖 = tanh 𝜙𝑖          (67) 
Hence, we have 𝜙1 = 3.927,𝜙2 = 7.069,𝜙3 = 10.21 …               (68) 

Putting equations (65), (66) and (68)  into equations (41)and (53), one obtains the displacement response respectively 
to a moving force and a moving mass of clamped-elastic ends Rayleigh beam on a variable foundation. 
 
3.2. Elastically Supported Conditions at Both Ends 

For the case when the beam is elastically supported both at 𝑥 = 0 and 𝑥 = 𝐿, the conditions are expressed as  
 𝒷 ′′(0, 𝑡) − 𝑘1 𝒷 ′(0, 𝑡) = 0 =  𝒷 ′′′(0, 𝑡) + 𝑘2 𝒷(0, 𝑡)    (69)   

at  𝑥 = 0 and  

 𝒷 ′′(𝐿, 𝑡) − 𝑘1 𝒷 ′(𝐿, 𝑡) = 0 =  𝒷 ′′′(𝐿, 𝑡) + 𝑘2 𝒷(𝐿, 𝑡)     (70) 

 at 𝑥 = 𝐿          
Similarly, for normal modes 

𝒰𝑖
′′(0) − 𝑘1𝒰𝑖

′(0) = 0 = 𝒰𝑖
′′′(0) + 𝑘2𝒰𝑖 (0)                    (71) 

at 𝑥 = 0  and  
𝒰𝑖

′′(𝐿) − 𝑘1𝒰𝑖
′(𝐿) = 0 = 𝒰𝑖

′′′(𝐿) + 𝑘2𝒰𝑖(𝐿)                   (72) 

at 𝑥 = 𝐿 which implies that 
 

𝒰𝑗
′′(0) − 𝑘1𝒰𝑗

′(0) = 0 = 𝒰𝑗
′′′(0) + 𝑘2𝒰𝑗(0)                     (73) 

at 𝑥 = 0 and 

𝒰𝑗
′′(𝐿) − 𝑘1𝒰𝑗

′(𝐿) = 0 = 𝒰𝑗
′′′(𝐿) + 𝑘2𝒰𝑗(𝐿)                    (74) 

at 𝑥 = 𝐿 using (71)  and (72),  it can be shown that 

 
𝐴𝑖 = 𝑟1𝐶𝑖 + 𝑟2 and 𝐵𝑖 = 𝑟3𝐶𝑖 + 𝑟1          (76) 
Where 

𝑟1 =

𝜙𝑖
4

𝐿4+𝑘1𝑘2

𝜙
𝑖
4

𝐿4−𝑘1𝑘2

; 𝑟2 =
− 

2𝑘1𝜙𝑖
3

𝐿3

𝜙
𝑖
4

𝐿4−𝑘1𝑘2

  and 𝑟3 =
− 

2𝑘1𝜙𝑖

𝐿
𝜙

𝑖
4

𝐿4−𝑘1𝑘2

         (77) 

Using equations (75), (76) and (77), the frequency equation for the dynamical problem is obtained as 

tan 𝜙𝑖 = tanh 𝜙𝑖                                                             (78) 

Hence. We have        𝜙1 = 3.927, 𝜙2 = 7.069, 𝜙3= 10.21 …                       (79) 
Substituting equations (75), (76), (77) and (78) into equations (41) and (53)  one obtains the displacement response respectively 
to a moving force and a moving mass of Rayleigh beam elastically supported at both ends and resting on a variable foundation. 
 
4. Numerical Results and Discussions 

To illustrate the analysis presented in this work, the uniform Rayleigh beam is taken to be of length L = 12.192 m, the 

load velocity c = 8.128 m/s and modulus of elasticity 𝐸 = 2.109 × 109𝑘𝑔/𝑚, the moment of inertia 𝐼𝑜 = 2.87698 × 10−3𝑚4. 
 
4.1 Graphs for Free-Elastic Boundary Conditions 

Figures 6.1 and 6.2 display the effect of axial force N on the deflection profile of free elastic Rayleigh beam under the 
action of load moving at constant velocity in both cases of moving distributed forces and moving distributed masses 
respectively. The graphs show that the response amplitude decreases as the value of N increases.  
 



6 

 

 
Edelweiss Applied Science and Technology 

ISSN: 2576-8484 

Vol. 2, No. 1: 1-9, 2018 
DOI: 10.33805/2576.8484.106 
© 2018 by the authors 
 

 
Figure 6.1. 
Deflection profile of a free elastic uniform Rayleigh beam on 
variable foundation and traversed by moving distributed force for 

various values of 𝑵. 
 

 
Figure 6.2. 
Deflection profile of a free elastic uniform Rayleigh beam on 
variable foundation and traversed by moving distributed mass 

for various values of 𝑵. 
 

Figures 6.3 and 6.4 display the effect of rotatory inertia R on the deflection profile of free elastic Rayleigh beam under the 
action of load moving at constant velocity in both cases of moving distributed forces and moving distributed masses 
respectively. The graphs show that the response amplitude decreases as the value of  R  increases. Figures 6.5 and 6.6 display 
the effect of foundation modulus So on the deflection profile of free elastic Rayleigh beam under the action of load moving at 
constant velocity in both cases of moving distributed forces and moving distributed masses respectively. The graphs show that 
the response amplitude decreases as the value of So increases. 
 

 
Figure 6.3. 
Deflection profile of a free elastic uniform Rayleigh 
beam on variable foundation and traversed by moving 

distributed force for various values of 𝑹. 
 

 
Figure 6.4. 
Deflection profile of a free elastic uniform Rayleigh 
beam on variable foundation and traversed by 

moving distributed force for various values of 𝑹. 
 

 
Figure 6.5.  
Deflection profile of a free elastic uniform Rayleigh beam on 
variable foundation and traversed by moving distributed 

forces for various values of 𝑺𝒐. 
 

 
Figure 6.6. 
Deflection profile of a free elastic uniform Rayleigh beam on 
variable foundation and traversed by moving distributed mass 

for various values of 𝑺𝒐. 



7 

 

 
Edelweiss Applied Science and Technology 

ISSN: 2576-8484 

Vol. 2, No. 1: 1-9, 2018 
DOI: 10.33805/2576.8484.106 
© 2018 by the authors 
 

Figures 6.7 and 6.8 display the effect of shear modulus H on the deflection profile of free elastic Rayleigh beam under the 
action of load moving at constant velocity in both cases of moving distributed forces and moving distributed masses 
respectively. The graphs show that the response amplitude decreases as the value of H increases.  
 

 
Figure 6.7. 
Deflection profile of a free elastic uniform Rayleigh beam 
on variable foundation and traversed by moving 
distributed force for various values of H. 

 

 
Figure 6.8. 
Deflection profile of a free elastic uniform Rayleigh 
beam on variable foundation and traversed by 

moving distributed mass for various values of 𝓗. 
 
Figure 6.9. shows the comparison of the moving distributed forces and moving distributed masses for fixed values of Ho, N, So 
and R. 
 

 
Figure 6.9.  
Comparison of the deflection profile of moving force and 

moving mass for a free elastic uniform Rayleigh beam. 
 
4.2. Graphs for Elastic-Elastic Boundary Conditions 

Figures 6.10 and 6.11 display the effect of axial force N  on the deflection profile of elastic- elastic Rayleigh beam under 
the action of load moving at constant velocity in both cases of moving distributed forces and moving distributed masses 
respectively. The graphs show that the response amplitude decreases as the value of rotatory inertia increases. 
 

 
Figure. 6.10. 
Deflection profile of an elastic- elastic uniform 
Rayleigh beam on variable foundation and traversed 

by moving distributed forces for various values of 𝑵. 
 

 
Figure 6.11.  
Deflection profile of an elastic- elastic uniform Rayleigh beam on variable 

foundation and traversed by moving distributed forces for various values of 𝑵. 
 

Figures 6.12 and 6.13 display the effect of rotatory inertia R  on the deflection profile of elastic- elastic Rayleigh beam 
under the action of load moving at constant velocity in both cases of moving distributed forces and moving distributed masses  
respectively. The graphs show that the response amplitude decreases as the value of rotatory inertia increases. 

 



8 

 

 
Edelweiss Applied Science and Technology 

ISSN: 2576-8484 

Vol. 2, No. 1: 1-9, 2018 
DOI: 10.33805/2576.8484.106 
© 2018 by the authors 
 

 
Figure 6.12. 
Deflection profile of an elastic- elastic uniform Rayleigh beam on variable 

foundation and traversed by moving distributed forces for various values of 𝑹. 
 

 
Figure 6.13. 
Deflection profile of an elastic-elastic uniform Rayleigh beam on variable foundation and 

traversed by moving distributed mass for various values of 𝑹. 
 

Figures 6.14 and 6.15 display the effect of shear modulus 𝓗 on the deflection profile of clamped elastic Rayleigh beam 
under the action of load moving at constant velocity in both cases of moving distributed forces and moving distributed masses 

respectively. The graphs show that the response amplitude decreases as the value of 𝓗 increases. 
 

 
Figure 6.14. 
Deflection profile of an elastic-elastic uniform Rayleigh beam on 
variable foundation and traversed by moving distributed force for 

various values of 𝓗.  
 

 
Figure 6.15. 
Deflection profile of an elastic-elastic uniform Rayleigh 
beam on variable foundation and traversed by moving 
distributed masses for various values of 𝓗. 

 

Figures 6.16 and 6.17 display the effect of foundation modulus 𝑺𝒐 on the deflection profile of clamped elastic Rayleigh 
beam under the action of load moving at constant velocity in both cases of moving distributed forces and moving distributed 

masses respectively. The graphs show that the response amplitude decreases as the value of 𝑺𝒐 increases. 
 

 
Figure 6.16. 
Deflection profile of an elastic-elastic uniform Rayleigh beam on 
variable foundation and traversed by moving distributed force for 

various values of 𝑺𝒐. 
 

 
Figure 6.17. 
Deflection profile of an elastic-elastic uniform Rayleigh beam on 
variable foundation and traversed by moving distributed mass for 

various values of 𝑺𝒐. 



9 

 

 
Edelweiss Applied Science and Technology 

ISSN: 2576-8484 

Vol. 2, No. 1: 1-9, 2018 
DOI: 10.33805/2576.8484.106 
© 2018 by the authors 
 

Figure 6.18. shows the comparison of the moving distributed forces and moving distributed masses for fixed values of 𝑺𝒐 , 𝑵 , 
𝓗  and 𝑹. 
 

 
Figure 6.18. 
Comparison of the deflection profile of moving force and 
moving mass for elastic-elastic    uniform Rayleigh beam for 

fixed values of   𝑺𝒐, 𝑵𝒐, 𝑹𝒐, and 𝓗. 
 
5. Conclusion 

In this research work, the problem associating with the dynamic response to moving distributed masses of pre-stressed 
uniform Rayleigh beam resting on variable elastic Pasternak foundation has been studied. The closed form solutions of the 
fourth order partial differential equations with variable and singular co-efficients are obtained for both cases of moving force 
and moving mass. From the closed form solutions obtained, the conditions of resonance are obtained for both moving force  

and moving mass. Also from the closed form solutions, the effects of beam parameters such as rotatory inertia, axial force, 
shear modulus and foundation modulus on the beam for both cases of moving distributed force and moving distributed mass 
were investigated.  

 
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