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

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Abstract
We have combined the theories of geometric singular perturbation and proper orthog- onal 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 ∞exible 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 suciently sti 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 ∞exi- bility 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. Key words. mechanics, linear/nonlinear dynamics, singular perturbation, slow invariant man-

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