Non-linear oscillations of continuous systems with quadratic and cubic non-linerities using non-linear normal modes

Abstract

Nonlinear oscillations of continuous structures with curvature, such as arches and shells, are discussed in this chapter. The motion of these structures, at moderately large vibration amplitudes, is governed by PDEs that include quadratic and cubic nonlinearities. Using nonlinear normal modes (NNMs) defined as invariant manifolds in phase space that allow one to exhibit reduced-order models and capture the essential properties of the dynamics. NNMs, which are defined as invariant manifolds, are introduced through the Normal Form theory. In a conservative framework, it is shown that all NNMs, as well as the attendant dynamics onto the manifolds, are computed in a single operation. The general third-order approximation of the dynamics onto a single NNM is derived. It is underlined that single linear mode truncation can lead to erroneous results that are corrected when considering NNMs. These results are illustrated by studying the vibrations of a linear beam resting on a nonlinear elastic foundation.