Dynamic response of beams under moving loads with finite deformation

Abstract

A novel material parameter-dependent model is proposed in this work to investigate the nonlinear vibration of a beam under a moving load within a finite deformation framework. For the planar vibration problem, the Lagrange strain is adopted and the resulting model equations for the beam are established by the Hamilton principle. Under appropriate assumptions, the coupled model equations are simplified into a nonlinear integro-partial differential equation which incorporates a material parameter and a geometrical parameter. The dynamic response of the beam is determined with the help of the Galerkin method. The solutions show that both the material parameter and geometrical parameter have the effect of reducing the amplitude of the forced vibration, increasing the speed of the fluctuation and the critical velocity for the moving load. In comparison with small deformation formulations, the effect of finite deformation herein is to reduce the vibration amplitude, especially for slender beams. If the vibration amplitude is relatively small, the nonlinear model may be replaced by the corresponding higher-order linear model while preserving the main features of the vibration problem.