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 , who studied the vibration of solids and structures under moving loads. Gbadeyan and Dada  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 , 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 , studied the generalized solution of beams with jump discontinuities on elastic foundation. Yin  also investigated the closed form solution for reinforced Timoshenko beam on elastic foundation. In the same vein, Teodoru  in his work, analyzed beam on elastic foundation by using finite difference approach.
Oni and Awodola  investigated the vibrations under a moving load of a non-uniform Rayleigh beam on variable elastic foundation. Oni and Awodola  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 , 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 . Hsu , studied the vibration analysis of non-uniform beams resting on elastic foundation. The work of Ismail  on dynamic response of a beam due to an accelerating moving mass using moving finite element approximation cannot be ignored. Kargarmovin and Younesian  took further study on dynamic of Timoshenko beams on Pasternak foundation under moving load.
Recently, Adeoye and Awodola  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  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  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  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. Substituting equations (3) , (4)and (5) into equation (1), one obtains
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
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
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;
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.
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 (33) represents the transverse deflection of uniform Rayleigh beam under moving distributed force and resting on variable Pasternak elastic foundation.
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).
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.
Using equations (75), (76) and (77), the frequency equation for the dynamical problem is obtained as
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.
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 , the moment of inertia
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.
For complete eqautions go through the pdf
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