Nonlinear Effects on Parametric Resonance of a Beam Subjected to Periodic Mass Transition

In this paper, dynamic stability of a simply supported beam excited by a sequence of moving masses is investigated by preserving nonlinear terms in the analysis. This type of loading is important in problems such as motion of vehicles on bridges, high-speed transportation on rails, machining processes, conveying pipelines and barrel dynamics, so its investigation is important from practical viewpoint. The intermittent loading across the beam results in a periodic time-varying equation system. The effects of convective mass acceleration beside large deformation beam theory are both taken into account in the derivation of governing equations which is performed through adopting Hamilton's principle for mass-varying systems. In order to deal with the coupling between longitudinal and transversal deflections, the inextensibility assumption is implicitly introduced into the Hamiltonian formulation and an appropriate interpretation is presented to maintain this approximation reasonable. The method of multiple scales is implemented to find the domains of stability and instability of the problem in a parameter space. The results of applying the method forecast a qualitative change in beam behavior due to nonlinear terms. Results of different numerical simulations show the validity of the analytical approach obtained by the applied perturbation method.