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On Internal Resonance of Two Damped Oscillators

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The damped motion of two oscillators with natural frequencies ω1 and ω2 is studied on the assumptions that: the oscillators are uncoupled for infinitesimal displacements and quadratically coupled for finite displacements; the frequencies are approximately in the ratio 2:1, such that ω2−2ω1=O(εω 1), where ε is a dimensionless measure of the displacements; the logarithmic decrements of the two modes, λ1 and λ1, are O(ε). The motion may be described by modulated sine waves with carrier frequencies ω1,2 and slowly varying energies and phases that satisfy four first‐order, nonlinear differential equations. These equations admit one invariant and may be reduced to two first‐order equations if ω2=2ω1 and λ1,2<0; they admit two invariants and can be completely integrated in terms of elliptic functions if ω2=2ω1 and 2λ21. Numerical results are presented for typical parametric combinations.
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Document Type: Research Article

Affiliations: University of California, San Diego

Publication date: December 1, 1976

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