Bifurcation of nonlinear internal resonant normal modes of a class of multi-degree-of-freedom systems

Abstract

The method of multiple scales is applied for constructing nonlinear normal modes (NNMs) of a three-degree-of-freedom system which is discretized from a two-link flexible arm connected by a nonlinear torsional spring. The discrete system is with cubic nonlinearity and 1:3 internal resonance between the second and the third modes. The approximate solution for the NNM associated with internal resonance are presented. The NNMs determined here tend to the linear modes as the nonlinearity vanishes, which is significant for one to construct NNM. Greatly different from results of those nonlinear systems without internal resonance, it is found that the NNM involved in internal resonance include coupled and uncoupled two kinds. The bifurcation analysis of the coupled NNM of the system considered is given by means of the singularity theory. The pitchfork and hysteresis bifurcation are simultaneously found. Therefore, the number of NNM arising from the internal resonance may exceed the number of linear modes, in contrast with the case of no internal resonance, where they are equal. Curves displaying variation of the coupling extent of the coupled NNM with the internal-resonance-deturing parameter are proposed for six cases.