Vibration under variable magnitude moving distributed masses of non-uniform Bernoulli-Euler beam resting on Pasternak elastic foundation

Abstract

The dynamic response to variable magnitude moving distributed masses of simply supported non-uniform Bernoulli-Euler beam resting on Pasternak elastic foundation is investigated in this paper. The problem is governed by fourth-order partial differential equation with variable and singular coefficients. The main objective of this work is to obtain closed form solution to this class of dynamical problem. In order to obtain the solution, a technique based on the method of Galerkin with the series representation of Heaviside function is first used to reduce the equation to second order ordinary differential equations with variable coefficients. Thereafter the transformed equations are simplified using (i) The Laplace transformation technique in conjunction with convolution theory to obtain the solution for moving force problem and (ii) finite element analysis in conjunction with Newmark method to solve the analytically unsolvable moving mass problem because of the harmonic nature of the moving load. The finite element method is first used to solve the moving force problem and the solution is compared with the analytical solution of the moving force problem in order to validate the accuracy of the finite element method in solving the analytically unsolvable moving mass problem. The numerical solution using the finite element method is shown to compare favorably with the analytical solution of the moving force problem. The displacement response for moving distributed force and moving distributed mass models for the dynamical problem are calculated for various time t and presented in plotted curves.