Nonlinear Faraday resonance

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Abstract
A cylinder containing liquid with a free surface is subjected to a vertical oscillation of amplitude εg/ω2and frequency 2ω, where ω is within O(εω) of the natural frequency of a particular(primary)mode in the surface-wave spectrum and 0 < ε 1. A Lagrangian formulation, which includes terms of second and fourth order in the primary mode and second order in the secondary modes (which are excited by the primary mode), together with the method of averaging, leads to a Hamiltonian system for the slowly varying amplitudes of the primary mode. The fixed points (which correspond to harmonic motions) and phase-plane trajectories and their perturbations due to linear damping are determined. It is shown that ε > δ, where δ is the damping ratio (actual/critical) of the primary mode, is a necessary condition for subharmonic response of that mode. Explicit results are given for the dominant axisymmetric and antisymmetric modes in a circular cylinder. Internal resonance, in which a pair of modes have frequencies that approximate ω and 2ω, is discussed separately, and the fixed points and their stability for the special case ω2= 2ω1are determined. Internal resonance for ω2= ω1is discussed in an appendix.

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