Exponentially Varying Load on Rayleigh Beam Resting on Pasternak

This paper investigates the dynamic behaviour of uniform Rayleigh beam resting on Pasternak foundation and subjected to exponentially varying magnitude moving the load. The solution techniques are based on finite Fourier sine transformed Laplace transformation and convolution theorem. The results show that for a fixed value of axial force, damping coefficient and rotatory inertia, increases in shear modulus and foundation modulus reduces the response amplitude of the dynamical system. It was also found that increases in axial force, rotary inertia, and damping coefficient for fixed values of shear modulus and foundation modulus lead to decreases in the deflection profile of the Rayleigh beam resting on Pasternak foundation. Finally, it was found that the effect of shear modulus is more noticeable that that of the foundation modulus.


INTRODUCTION
The response of structural and flexible members to moving loads has been the subject of numerous researches owing to its relevance in many diverse areas. The structure has commonly been modelled either as a beam or a plate in most of the analytical studies in engineering, applied mathematics, and mathematical physics. The dynamic response analysis is very complex in such structural members under the passage of moving loads due to the interaction between the passing load and the structure. The response of railroads to moving trains, the response of bridges to moving vehicles, belt drives to the conveyor and computer tape drive to floppy disks are some of the application of such moving load problem Adams [1] and Shabnam et al. [2].
Moreover, modern means of transport are ever faster and heavier, while the structure over which they move is ever more slender and lighter Fryba [3]. The dynamic stresses produce are larger by far than the static ones. Theism has continued to motivate a lot of research activities in this area.
The dynamic response of a simply supported beam, traversed by a constant force moving at a uniform speed was first studied by Kyrylo [4]. He used the method of expansion of eigenfunction to obtain his results. Later Timoshenko [5] used energy methods to obtain solutions in series form for the simply supported finite beam on an elastic foundation subjected to the time dependent points loads moving with uniform velocities across the beam. Kenny [6] similarly investigated the dynamic response of infinite elastic beams on elastic foundation under the influence of load moving at constant speeds. He included the effects of viscous damping in the governing differential equation.
In a recent development, some of the researchers that made tremendous feat in the dynamic study of structures under moving loads includes Oni and Awodola [7], Liu and Chang [8], Oni and Omolefe [9], Misra [10], Oni and Omolefe [11], Achawakorn and Jearsiri Pongkul [12]. However, all the researchers aforementioned considered only one parameter model which has various shortcoming because it predicts discontinuities in the deflection of the surface of the foundation at the ends of a finite beam and this is in contradiction to an observation made in practice.
Several researchers in recent time that considered two parameters models in their studies so as to overcome the shortcoming of the one parameter model include Jimoh and Ajoge [13], Jimoh and Ajoge [14], Oni and Jimoh [15], Oni and Jimoh [16], Oni and Jimoh [17], Oni and Ogunbamike [18].
In all those researches, exponentially varying magnitude moving load were not considered. The present paper is concerned with the response of uniform Rayleigh beam resting on Pasternak foundation and traversed by exponentially varying magnitude moving load.

THE INITIAL BOUNDARY-VALUE PROBLEM
The governing partial differential equation for a uniform Rayleigh beam length L on Pasternak foundation and traversed by a moving load Q(x,t) of mass, m moving with velocity c and damping term included is given by Frybal [3]  The boundary condition of the structure under consideration is simply supported and is given as And the initial conditions are taken as The load moving on the elastic beam is assumed to be an exponentially varying magnitude moving load of the form

SOLUTION OF THE MATHEMATICAL PROBLEM
In this section, we proceed to solve the initial boundary value problem described by 2, 3, and 5. We remark that the integral transform techniques have proved suitable and effectively applicable to solving moving load problems such as the one under investigation [3]. Therefore, this method is adopted in the solution of the IBVP. Specifically, the Fourier transformation for the length coordinate and the Laplace transformation for the time coordinate with boundary and initial conditions are used in this work.

FINITE FOURIER TRANSFORMED GOVERNING EQUATION
Here, we proceed to take the Fourier transform of the governing partial differential equation (5). We find, however, that the boundary conditions in equation (2) may be accommodated only by using a finite Fourier sine transform, so we shall apply the finite Fourier sine integral transformation for the length coordinate, and this is defined as With the inverse transform defined as Thus, by invoking equation (6) Equation (9) represents the finite Fourier transform governing equations of the uniform Rayleigh beam resting on Pasternak foundation and subjecting to exponentially varying magnitude moving load moving with constant velocity

LAPLACE TRANSFROMED SOLUTIONS
We apply the method of the Laplace integral transformation for the time coordinate between 0 and ∞ to solve equation (9). The operator of the Laplace transform is here indicated by Equation (14) can be re-written as Equation (15) can be simplified to obtain

THE INVERSE INTEGRAL TRANSFORMATION
We proceed in this section to obtain the inverse transformation of equation (18). To obtain the inverse transformation of (18), we shall adopt the following representation. Using equation (27) in equation (7), we have

NUMERICAL ANALYSIS AND DISCUSSION OF RESULTS
To illustrate the theory described in this paper numerically, the values of the physical constants and parameters considered are The responses of the uniform Rayleigh beam to exponentially varying magnitude moving load for various values of foundation modulus (K), shear modulus (G), axial force (N), Rotatory inertia (Ro) and damping coefficient are shown in the following figures (Figures 1-5) The response of the beam to exponentially varying magnitude moving load for various values of the shear modulus (K) at fixed value of rotatory inertia (Ro), shear modulus (G), damping coefficient ( ) and axial force (N) is shown in Figure 1 while figure 2 shows the dynamic response of the system. Figures 1 and 2 shows that the response amplitude of the beam decreases with increases values of shear modulus and foundation modulus, respectively.

CONCLUSION
The dynamic response of uniform Rayleigh beam resting on Pasternak foundation and subjected to exponentially varying magnitude moving load has been studied in this paper. The governing partial differential equations of the problem have been solved by using the finite Fourier integral sine transformed, Laplace transformation and convolution theorem and the prescribed initial and boundary conditions. It is assumed that the beam is of uniform cross-section and of constant mass. The effects of shear modulus, foundation modulus, axial force, rotatory inertia, and damping coefficient on the deflections of the beams are highlighted. It is found that the amplitudes of the deflections profiles of the beam decrease with increasing values of each of the structural parameters. Furthermore, it is also found that the effect of shear modulus is more noticeable than that of the foundation modulus.