Response of Three Oscillators Coupled by a Weak Nonlinearity

Abstract

As a simplified model of the exchange processes occurring among resonance modes in physical systems, such as a piezoelectric crystal plate or an acoustic interferometer, a study is made of the response of three oscillators that are coupled by a weak nonlinearity and whose frequencies satisfy the condition ω1 + ω2 ≅ ω3 . The transient behavior is obtained by a perturbation expansion. There exist three integral constraints on the amplitude and phase variation of the oscillations for a conservative system, and the solution of the response can be reduced to quadrature. The phase diagram describing the motion indicates that the high frequency oscillation is unstable; the energy associated with it, under certain conditions, can be diverted to lower frequency oscillations. For nonconservative systems, the effects of dissipation and detuning are examined for their role in limiting the energy exchange among the oscillations and in determining the steady-state response to forcing. Predictions from this analysis are compared with results of a reported experiment in which a piezoelectric crystal plate is forced to oscillate at amplitudes sufficient to generate coupled subharmonics.