Nonlinear Dynamics of a Beam Subjected to a Moving Mass and Resting on a Viscoelastic Foundation Using Optimal Homotopy Analysis Method

Abstract

Studying the dynamics of beams subjected to a moving mass is important due to their wide applications, including railways, machining processes, and microelectromechanical systems (MEMS). Various numerical and analytical approaches have been used for modeling such structures. In this analytical study, we have used a combination of the Optimal homotopy analysis method (Optimal HAM) and enriched multiple scales (MS) to analytically study the dynamics of a simply supported Euler–Bernoulli beam traversed by a moving mass and resting on a viscoelastic foundation. The viscoelastic foundation contributes to the modeling by adding a linear and nonlinear term to the formulation. Further, we have considered a fifth-order nonlinear term to account for the bending vibration of the flexible beam. Using the Galerkin method, we have formed the corresponding ordinary differential equation (ODE). Then, we used the enriched MS Optimal HAM to calculate the dynamic response of the beam. After validating our method by comparing our results with the dynamic results of the beam obtained from finite element analysis (FEA), we investigated the effects of the determining parameters on the beam dynamic response. The effects of the foundation nonlinear and linear terms, the moving load weight, and its velocity have been investigated by studying the variation of the normalized beam lateral deflection versus the normalized moving mass instantaneous position in each case. We showed that the difference between linear and nonlinear modeling results is pronounced, and it becomes more pronounced for faster and heavier moving loads.