Nonlinear Oscillations of Circular Plates Near Critical Speed

Abstract

The nonlinear 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 (1964), leading to von Kármán type field equations is used. This leads to nonlinear 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 nonlinear 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 codimension-two bifurcations that give rise to sets of doubly-degenerate, one-dimensional manifolds of steady mixed wave motions. These manifolds of steady motions are bounded in phase space by either mode localized backward travelling wave branches or forward travelling wave branches. On the addition of the smallest damping, the branches of the backward travelling waves with equal modal content become isolated, and numerical investigations indicate the absence of any other types of steady motions.