Dynamics of Large Scale Coupled Structural/ Mechanical Systems: A Singular Perturbation/ Proper Orthogonal Decomposition Approach

Abstract

We have combined the theories of geometric singular perturbation and proper orthogonal decomposition to study systematically the dynamics of coupled systems in mechanics involving coupling between continuous structures and nonlinear oscillators. Here we analyze a prototypical structural/mechanical system consisting of a planar nonlinear pendulum coupled to a flexible rod made of linear viscoelastic material. We cast the equations of motion in a singularly perturbed set of oscillators and compute analytic approximations to an attractive global invariant manifold in phase space of the coupled system. The invariant manifold, two-dimensional for the unforced system and three-dimensional for the forced system, carries a continuum of slow motions. For a sufficiently stiff rod, a proper orthogonal decomposition of any long time motion extracts a single structure for the spatial coherence of the dynamics, which is a realization of the slow invariant manifold. As the flexibility of the rod increases, the energy of periodic and chaotic motions is shown to spread to multiple coherent structures, indicating a high degree-of-freedom attractor.