Application of Center Manifold Theory to Regulation of a Flexible Beam

[+] Author and Article Information
Amir Khajepour, M. Farid Golnaraghi

Department of Mechanical Engineering, University of Waterloo, Waterloo, Ontario N2L 3B9 Canada

Kirsten A. Morris

Applied Math Department, University of Waterloo, Waterloo, Ontario N2L 3B9 Canada

J. Vib. Acoust 119(2), 158-165 (Apr 01, 1997) (8 pages) doi:10.1115/1.2889697 History: Received August 01, 1994; Revised May 01, 1995; Online February 26, 2008


In this paper we consider the problem of regulation of a flexible lumped parameter beam. The controller is an active/passive mass-spring-dashpot mechanism which is free to slide along the beam. In this problem the plant/controller equations are coupled and nonlinear, and the linearized equations of the system have two uncontrollable modes associated with a pair of pure imaginary eigenvalues. As a result, linear control techniques as well as most conventional nonlinear control techniques can not be applied. In earlier studies Golnaraghi (1991) and Golnaraghi et al. (1994) a control strategy based on Internal resonance was developed to transfer the oscillatory energy from the beam to the slider, where it was dissipated through controller damping. Although these studies provided very good understanding of the control strategy, the analytical method was based on perturbation techniques and had many limitations. Most of the work was based on numerical techniques and trial and error. In this paper we use center manifold theory to address the shortcomings of the previous studies, and extend the work to a more general control law. The technique is based on reducing the dimension of system and simplifying the nonlinearities using center manifold and normal forms techniques, respectively. The simplified equations are used to investigate the stability and to develop a relation for the optimal controller/plant natural frequencies at which the maximum transfer of energy occurs. One of the main contributions of this work is the elimination of the trial and error and inclusion of damping in the optimal frequency relationship.

Copyright © 1997 by The American Society of Mechanical Engineers
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