Auto and Cross-Bispectral Analysis of a System of Two Coupled Oscillators With Quadratic Nonlinearities Possessing Chaotic Motion

[+] Author and Article Information
Charles Pezeshki, R. Krishna, T. D. Burton

Department of Mechanical and Materials Engineering, Washington State University, Pullman, WA 99164-2920

Steve Elgar

Department of Electrical and Computer Engineering, Washington State University, Pullman, WA 99164-2920

J. Appl. Mech 59(3), 657-663 (Sep 01, 1992) (7 pages) doi:10.1115/1.2893774 History: Received September 05, 1989; Revised June 04, 1990; Online March 31, 2008


Auto and cross-bispectral analyses of a two-degree-of-freedom system with quadratic nonlinearities having two-to-one internal (autoparametric) resonance are presented. Following the work of Nayfeh (1987), the method of multiple scales is used to obtain a first-order uniform expansion yielding four first-order nonlinear ordinary differential equations governing the modulation of the amplitudes and phases of the two modes. The particular case of parametric resonance of the first mode considered in this paper admits Hopf bifurcations and a pure period doubling route to chaos. Auto bicoherence spectra isolate the phase coupling between increasing numbers of triads of Fourier components for a pure period doubling route to chaos for the individual degrees-of freedom. Cross-bicoherence spectra, on the other hand, yield information about the phase coupling between the two degrees-of-freedom. The results presented here confirm the capacity of bispectral techniques to identify a quadratically nonlinear mechanical system that possesses chaotic motions. For the chaotic case, cross-bicoherence spectra indicate that most of the nonlinear energy transfer between the modes is owing to cross-coupling between phase modulations rather than between amplitude modulations.

Copyright Ā© 1992 by The American Society of Mechanical Engineers
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