On Internal Resonance of Two Damped Oscillators

The damped motion of two oscillators with natural frequencies ω 1 and ω 2 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 ω 2 −2ω 1 = O ( εω 1 ), where ε is a dimensionless measure of the displacements; the logarithmic decrements of the two modes, λ 1 and λ 1 , are O ( ε ). The motion may be described by modulated sine waves with carrier frequencies ω 1,2 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 ω 2 =2ω 1 and λ 1,2 <0; they admit two invariants and can be completely integrated in terms of elliptic functions if ω 2 =2ω 1 and 2λ 2 =λ 1 . Numerical results are presented for typical parametric combinations.