On Internal Resonance of Two Damped Oscillators

摘要

The damped motion of two oscillators with natural frequencies ω1and ω2is 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(εω1), where ε is a dimensionless measure of the displacements; the logarithmic decrements of the two modes, λ1and λ1, areO(ε). The motion may be described by modulated sine waves with carrier frequencies ω1,2and 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=2ω1and λ1,2<0; they admit two invariants and can be completely integrated in terms of elliptic functions if ω2=2ω1and 2λ2=λ1. Numerical results are presented for typical