Nonlinear beam behaviour with a moving mass

Abstract

Dynamics of a flexible cantilever beam carrying a moving mass-spring is investigated. The system is an idealisation of an important class of problems characterised by nonlinear interaction between a continuously distributed mass and stiffness subsystems and a lumped mass and stiffness sub-system. Two models are developed, one with the beam undergoing small oscillations under the presence of kinematic nonlinearities arising from the coupled mass-beam behaviour and a second model for large oscillations where geometric nonlinearities are also considered.

A closed form solution is obtained using the perturbation method of multiple scales and a parametric analysis is conducted. System response is investigated under internal resonance conditions between the moving mass and the beam. A new technique, based on the finite element method is presented for numerical solution of nonlinear partial differential equations. Using this method the dynamics of the mass-beam system is investigated numerically for both the small oscillation and the large oscillation models. Motion of a cantilever beam without the moving mass, undergoing large oscillations is also analysed. Time-Frequency analysis is used to investigate the spectral behaviour of the system.