Seminar Series - 1999-2000

Order Reduction and Nonlinear Energy Pumping in Coupled Oscillators

Professor Alexander F. Vakakis
Department of Mechanical and Industrial Engineering
University of Illinois
Urbana-Champaign, IL
 
Sponsored by the UNL Research Council and the Department of Engineering Mechanics

Date:  Tuesday, March 28, 2000
Time:  3:30 p.m.
Place:  W128 Nebraska Hall


We consider the method of proper orthogonal decomposition (POD) to compute spatial coherent structures in the dynamics of a forced multibay truss. We then use the POD modes to develop reduced order modes for the truss. We show that the truss dynamics can be reconstructed by low-order POD-based models. This approach enables system identification of the dynamics of structures with high modal densities where traditional modal analysis methods have limited applicability. We then present numerical evidence of nonlinear energy pumping in coupled mechanical oscillators, i.e., of one-way (irreversible) ‘channeling’ of externally imparted energy from the linear to the nonlinear part of the system. This occurs provided that the energy is above a critical level. To understand the phenomenon we transform the equations of motion into action-angle form, and apply two-frequency averaging. We show that energy pumping is due to resonance capture on a 1:1 resonance manifold of the system, and perform a perturbation analysis in an O() neighborhood of this manifold to study the attracting region responsible for this phenomenon. A second analytical approach,  based on the assumption of 1:1 internal resonance in the fast dynamics of the system, complexifies and averages the fast dynamics. This leads to satisfactory analytical approximations to the nonlinear transient responses of the system in the energy pumping regime. The practical applications of the nonlinear energy pumping phenomenon to the active/passive control of systems are discussed.


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