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https://hdl.handle.net/2142/21390
Description
Title
Coupled oscillators near resonance
Author(s)
Arsenault, Lance Eric
Issue Date
1996
Doctoral Committee Chair(s)
Jackson, E.A.
Department of Study
Physics
Discipline
Physics
Degree Granting Institution
University of Illinois at Urbana-Champaign
Degree Name
Ph.D.
Degree Level
Dissertation
Keyword(s)
Engineering, Mechanical
Language
eng
Abstract
We study the dynamics of two conservative librating oscillators with perturbations from a linear displacement coupling and non-Hamiltonian forces such as damping. We examine the dynamics of these systems when they are near a resonance using secular perturbation theory. We show that near resonance a large class of driven oscillators and two coupled oscillators can be transformed to the same ordinary differential equations (ODEs). We consider two types of resonances: accidental and intrinsic. For an accidental resonance, we find that the dynamics near a resonance is a generalization of the standard Hamiltonian dynamics of two coupled conservative oscillators, which we call the standard equation. For an intrinsic resonance, we show that a primary resonance island can fill all of the available phase space. We derive expressions for the parameters in these ODEs. From a fixed-point analysis of these ODEs, we show that hard oscillators lock in-phase and soft oscillators lock out-of-phase. We develop a novel method for calculating accurate response curves for driven strongly nonlinear oscillators, where no existing method can give accurate results. We present a method for finding the steady state frequency of two coupled oscillators. We compare our theoretical predictions with computer simulations of many examples including: a sinusoidally driven highly nonlinear Duffing oscillator, and two coupled van der Pol oscillators with a highly nonlinear Duffing force.
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