Internal Resonance Modes Research Of Nonlinear Vibration Of Beams

The extensive applications of linear modal analysis and the existing nonlinear factors in vibratory systems make it necessary that nonlinear modal analysis is developed. Nonlinear modal analysis has a great significance for structural designs and is also a main approach for one to obtain approximate solutions for nonlinear systems with multi-degree-of-freedom. In this paper, the method of multiple scales is successively used to construct NNMs of systems with internal resonance which has been a puzzle for a long time.At first, the advance in the study on NNMs is proposed completely, which involves the character of NNMs and the effect of internal resonance on constructing NNMs.Secondly, the theory for multi-flexible dynamical systems is introduced and applied, and the simplified system obtained is cubic nonlinear, with 1:3 internal resonance for certain parameter condition. Next, the NNMs of the system studied are computed by using the method of multiple scales. The NNMs associated with internal resonance can be classified into two kinds: coupled an uncoupled. The coupled normal modes only exist in nonlinear systems with internal resonance.At last, the idea used for constructing NNMs of discrete systems is also valid for continuous systems, and this mothed is substantiated by two examples.