Dynamics of nonlinear structures with multiple equilibria: A singular perturbation-invariant manifold approach

Abstract.

This work analyzes the motions of a stiff linear oscillator coupled to a soft nonlinear oscillator and subject to a forcing term. This system is a representative of a large class of structural dynamical systems with stiff and soft substructures and with multiple equilibrium states. Using the methodology of singular perturbations and the theory of invariant manifolds, we describe globally (in time) the motions in a finite neighborhood of the origin in phase space. It is shown that, every motion of the system depends on a slowly varying component and a component which rapidly decays with time. The long term behavior of the system is, thus, described by a lower order slow system, which is the restriction of the system to an invariant manifold that contains all the slow motions of the system. For slow periodic forcing, the slow manifold is periodic. The dynamics of the slow system on the invariant manifold, as well as the effects of the singular perturbation parameter on the validity of the slow manifold approximation are also explored.