Self-excited oscillations of a string on an elastic foundation subject to a nonlinear feed-forward force

Abstract

We investigate the self-excited oscillations of a string on an elastic foundation that is subject to a nonlinear feed-forward force. The feed-forward follows that of a model first proposed by Steele and Baker [1] for an active cochlear, and is due to the gain factor profile which depends on the string displacement. In order to determine the bifurcation structure induced by the nonlinear feed-forward mechanism, we formulate a taut string initial-boundary-value problem with periodic boundary conditions which is reduced to a finite order modal dynamical system. We employ an asymptotic multiple-scales method to obtain slowly varying evolution equations that enable an analytical derivation of the periodic system response and analysis of its orbital stability. The resulting bifurcation structure includes multiple regions of both stable and unstable coexisting periodic solutions defined by primary and secondary Hopf stability thresholds. Numerical verification of the bifurcation structure determines the accuracy of the analytically predicted periodic self-excited response and reveals the existence of quasiperiodic combination-tone solutions and complex nonstationary solutions that emerge in a range of the asymptotically predicted unstable solutions. This analysis enables construction of a comprehensive analytical bifurcation structure and may shed light on mechanisms governing complex multi-component spectra that have been documented for spontaneous otoacoustic emissions in the mammalian inner ear.