Instantaneous Center Manifolds and Nonlinear Modes of Vibration

Abstract

Nonlinear Normal Modes (NNM) have been defined in various ways; first by Rosenberg as a subset of periodic solutions of a nonlinear system and then by Shaw and Pierre as invariant manifolds tangent to the vector field of a nonlinear system at its equilibrium point. This work presents an alternative approach, namely Instantaneous Center Manifold (ICM), that extends the concept of modes of vibration to nonlinear systems, using both periodicity and invariance properties. Instantaneous Center Manifolds are invariant manifolds that contain all of the periodic invariant solutions of the nonlinear oscillatory system. The ICM approach is explained through three simple analytical examples, and is shown to be capable of finding solutions that have been remaining latent using the aforementioned approaches. New branches of nonlinear normal modes, separate from the main branches that are a continuation of linear modes, are illustrated. It is shown that these new branches connect the main branches of Rosenberg’s NNMs, and make it possible to travel from one main branch to another. Some natural extensions and applications of the ICM approach are briefly discussed in the conclusion.