Nonlinear Normal Modes and Reduced Order Models

Nonlinear normal modes (NNMs) have been used in the last decades as an additional tool for the analysis of multi degree of freedom (mdof) dynamical systems and for the derivation of reduced order models. Examples in literature show that only a few NNMs are usually required to model, in many problems, their forced dynamics. In this chapter a brief literature review with the seminal contributions in this field and recent contributions are presented. Then the definitions of NNMs based on the works by Rosenberg and Shaw and Pierre are given. According to the latter definition, a nonlinear vibration mode is a two-dimensional invariant manifold in the phase space of the system. This is approximated by asymptotic expansions in terms of a pair of master coordinates or through a Galerkin expansion. This methodology and additional numerical tools are presented here together with didactical examples that illustrate some unique characteristics of NNMs as compared to linear normal modes. Also Poincaré sections are used for the identification of NNMs and their stability and continuation methods are used to obtain frequency-energy plots and frequency-amplitude relations. The concept of multi-mode is briefly presented.