Response of a three-degree-of-freedom system with cubic non-linearities to harmonic excitation

Abstract

An investigation of the response of a three-degree-of-freedom-system with cubic non-linearities and the autoparametric resonances ω 3≅3 ω 2 and ω 2≅3 ω 1 to a harmonic excitation of the third mode, where the ω n are the linear natural frequencies of the system, is presented. The method of multiple scales is used to determine six first-order non-linear ordinary differential equations that govern the time variation of the amplitudes and phases of the interacting modes. The fixed points of these equations are obtained and their stability is determined. Numerical solutions are presented. Discussion of the figures is given.