Analytical solutions for steady state responses of an infinite Euler-Bernoulli beam on a nonlinear viscoelastic foundation subjected to a harmonic moving load

Abstract

The paper deals with the steady state responses of an infinite Euler-Bernoulli beam resting on a nonlinear foundation under a harmonic moving load. Greatly different from previous works, the nonlinear partial differential governing equation of the beam motion is converted to two nonlinear Volterra integral equations by using the Fourier transform, residues Theorem and the convolution theorem. The Volterra integral equations have four different expressions depending on the damping coefficient of the foundation, the linear part of the foundation stiffness and the frequency of the moving load. The modified Adomian decomposition method in conjunction with a simple iterative formula derived from the integral equation theorems are applied to obtain analytical solutions for the steady state responses of the beam. The closed form solutions presented in this paper do not contain complicated infinite integrals, which are better to show the influence of the nonlinear part of foundation stiffness. The parametric study shows that the nonlinear part of foundation stiffness affects not only qualitative but also quantitative analysis results of the infinite Euler-Bernoulli beam under a harmonic moving load.