Slow and fast invariant manifolds, and normal modes in a two degree-of-freedom structural dynamical system with multiple equilibrium states

Abstract

Many problems in structural dynamics involve coupling between a stiff (high frequency) linear structure and a soft (low frequency) non-linear structure with multiple static equilibrium states. In this work, we analyze the slow and fast motions of a conservative structural system consisting of a non-linear oscillator with three equilibrium states coupled to a stiff linear oscillator. We combine analysis (singular perturbations) with geometry (manifolds) and computation to show that the system possesses invariant manifolds supporting either slow or fast motions. In particular, under appropriate conditions, a global slow invariant manifold passes through the three static equilibrium states of the system. The slow manifold is non-linear, orbitally stable, and it carries a continuum of in-phase periodic motions, including a homoclinic motion. We generalize the classical notion of vibrations-in-union to include systems with multiple equilibria, and thus identify the slow invariant manifold with a slow, non-linear normal mode of vibration.