We study asymptotically the family of subharmonic responses of an essentially nonlinear oscillator forced by two closely spaced harmonics. By expressing the original oscillator in action-angle form, we reduce it to a dynamical system with three frequencies (two fast and one slow), which is amenable to a singular perturbation analysis. We then restrict the dynamics in neighborhoods of resonance manifolds and perform local bifurcation analysis of the forced subharmonic orbits. We find increased complexity in the dynamics as the frequency detuning between the forcing harmonics decreases or as the order of a secondary resonance condition increases. Moreover, we validate our asymptotic results by comparing them to direct numerical simulations of the original dynamical system. The method developed in this work can be applied to study the dynamics of strongly nonlinear (nonlinearizable) oscillators forced by multiple closely spaced harmonics; in addition, the formulation can be extended to the case of transient excitations.
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January 2011
Research Papers
Subharmonic Orbits of a Strongly Nonlinear Oscillator Forced by Closely Spaced Harmonics
Alexander F. Vakakis
Alexander F. Vakakis
Department of Mechanical Science and Engineering,
e-mail: avakakis@illinois.edu
University of Illinois at Urbana-Champaign
, 1206 West Green Street, Urbana, IL 61801
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Themistoklis P. Sapsis
Alexander F. Vakakis
Department of Mechanical Science and Engineering,
University of Illinois at Urbana-Champaign
, 1206 West Green Street, Urbana, IL 61801e-mail: avakakis@illinois.edu
J. Comput. Nonlinear Dynam. Jan 2011, 6(1): 011014 (10 pages)
Published Online: October 7, 2010
Article history
Received:
July 28, 2009
Revised:
November 16, 2009
Online:
October 7, 2010
Published:
October 7, 2010
Citation
Sapsis, T. P., and Vakakis, A. F. (October 7, 2010). "Subharmonic Orbits of a Strongly Nonlinear Oscillator Forced by Closely Spaced Harmonics." ASME. J. Comput. Nonlinear Dynam. January 2011; 6(1): 011014. https://doi.org/10.1115/1.4002337
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