Nonlinear Vibrations of a Beam-Spring-Mass System

Abstract

The nonlinear response of a simply supported beam with an attached spring-mass system to a primary resonance is investigated, taking into account the effects of beam midplane stretching and damping. The spring-mass system has also a cubic nonlinearity. The response is found by using two different perturbation approaches. In the first approach, the method of multiple scales is applied directly to the nonlinear partial differential equations and boundary conditions. In the second approach, the Lagrangian is averaged over the fast time scale, and then the equations governing the modulation of the amplitude and phase are obtained as the Euler-Lagrange equations of the averaged Lagrangian. It is shown that the frequency-response and force-response curves depend on the midplane stretching and the parameters of the spring-mass system. The relative importance of these effects depends on the parameters and location of the spring-mass system.