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Resonant Frequency Calculations Using a Hybrid Perturbation-Galerkin Technique

[+] Author and Article Information
James F. Geer

Department of Systems Science, Watson School of Engineering, Applied Science and Technology, State University of New York, Binghamton NY 13902

Carl M. Andersen

Department of Mathematics, College of William and Mary in Virginia, Williamsburg VA 23185

Appl. Mech. Rev 44(11S), S76-S88 (Nov 01, 1991) doi:10.1115/1.3121376 History: Online April 30, 2009

Abstract

A two step hybrid perturbation-Galerkin technique is applied to the problem of determining the resonant frequencies of one- or several-degree(s)-of-freedom nonlinear systems involving a parameter. In step one, the Lindstedt-Poincaré method is used to determine perturbation solutions which are formally valid about one or more special values of the parameter (e.g. for small or large values of the parameter). In step two, a subset of the perturbation coordinate functions determined in step one is used in a Galerkin type approximation. The technique is illustrated for several one-degree-of-freedom systems, including the Duffing and van der Pol oscillators, as well as for the compound pendulum. For all of the examples considered, it is shown that the frequencies obtained by the hybrid technique using only a few terms from the perturbation solutions are significantly more accurate than the perturbation results on which they are based, and they compare very well with frequencies obtained by purely numerical methods.

Copyright © 1991 by American Society of Mechanical Engineers
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