An Alternative Perturbation Procedure of Multiple Scales for Nonlinear Dynamics Systems

Abstract

An alternative perturbation procedure of multiple scales is presented in this paper which is capable of treating various periodic and almost periodic steady-state vibrations including combination resonance of nonlinear systems with multiple degrees-of-freedom. This procedure is a generalization of the Lindstedt-Poincaré method. To show its essential features a typical example of cubic nonlinear systems, the clamped-hinged beam, is analyzed. The numerical results for the almost periodic-free vibration are surprisingly close to that obtained by the incremental harmonic balance (IHB) method, and the analytical formulae for steady-state solution are, in fact, identical with that of conventional method of multiple time scales. Moreover, detail calculations of this example revealed some interesting behavior of nonlinear responses, which is of significance for general cubic systems.