Nonlinear modal analysis of structural systems using multi-mode invariant manifolds

In this paper, an invariant manifold approach is introduced for the generation of reduced-order models for nonlinear vibrations of multi-degrees-of-freedom systems. In particular, the invariant manifold approach for defining and constructing nonlinear normal modes of vibration is extended to the case of multi-mode manifolds. The dynamic models obtained from this technique capture the essential coupling between modes of interest, while avoiding coupling from other modes. Such an approach is useful for modeling complex system responses, and is essential when internal resonances exist between modes. The basic theory and a general, constructive methodology for the method are presented. It is then applied to two example problems, one analytical and the other finite-element based. Numerical simulation results are obtained for the full model and various types of reduced-order models, including the usual projection onto a set of linear modes, and the invariant manifold approach developed herein. The results show that the method is capable of accurately representing the nonlinear system dynamics with relatively few degrees of freedom over a range of vibration amplitudes.