Non-linear oscillations of circular plates near a critical speed resonance

Abstract

The non-linear response of an axisymmetric, thin elastic circular plate subject to a constant, space-fixed transverse force and rotating near a critical speed of an asymmetric mode, is analyzed. A small-stretch, moderate-rotation plate theory of Nowinski [J. Appl. Mech. (1964) 72–78], leading to von Kármán-type field equations is used. This leads to non-linear modal interactions of a pair of 1–1 internally resonant, asymmetric modes which are studied through first-order averaging. The resulting amplitude equations represent a system whose O(2) symmetry is broken by a resonant rotating force. The non-linear coupling of the modes induces steady-state solutions that have no apparent evolution from any previous linear analyses of this problem. For undamped disks, the analysis of the averaged Hamiltonian predicts two codimension-two bifurcations that give rise to three sets of doubly degenerate, one-dimensional manifolds of steady mixed wave motions. On the addition of the smallest damping, the branches of the backward travelling waves with equal modal content become isolated, and it is proved that these are the only steady motions possible. A simple experiment is used to confirm the analytical predictions.