Dynamic Behaviour of Simply Supported Non-Uniform Rayleigh Beam under Variable-Magnitude Accelerating Masses and Resting on Non-Uniform Bi-parametric Foundation

The dynamic behaviours of simply supported non-uniform Rayleigh beam under variable-magnitude accelerating masses and resting on non-uniform bi-parametric foundation is investigated in this paper. A partial differential equation of fourth order governs the situation. The governing equation is converted into a series of coupled second order ordinary differential equations with variable coefficients using the Galerkin approach, which is based on the series representation of the Heaviside function.Two instances are examined;(i) the moving force problem when the inertia term is neglected and (ii) the moving mass case when the inertia term is considered. Variation of parameters are employed to get the transverse displacement response in order to solve the moving force problem,the moving mass problem cannot be solved using the widely used Struble’s asymptotic method due to the variability of the load magnitude. Therefore, a numerical technique, specifically the Runge-Kutta of fourth order is used to obtain an approximate solution. The numerical solution of the moving force problem is compared with the analytical solution in order to verify the accuracy of the Runge-Kutta scheme, and it compares favorably. From the analytical and numerical result, it is observed that the amplitude of the deflection profile of simply supported non-uniform Rayleigh beam decreases with an increase in the value of some vital structural parameters such as rotatory inertia correction factor, axial force, shear modulus and foundation modulus.