Nonlinear dynamics of systems involving widely spaced frequencies

This document focuses on the dynamics of nonlinear oscillatory systems involvingwidely spaced frequencies. First, experiments on the forced oscillations of a cantileverbeam with a circular cross-section are presented. The beam is excited near its fifthnatural frequency but sustained large-amplitude oscillations of the first mode accom-panied by modulated oscillations of the fifth mode occur. Thus, energy is transferredfrom the fifth to the first mode of the beam, whose natural frequencies are roughlyin the ratio of 60: 1.Some simple two-degree-of-freedom systems are studied in an effort to explain thestrong nonlinear coupling between modes with widely spaced frequencies. In thesesystems, the coupling is found to give rise to some novel phenomena including staticdisplacements or sustained oscillations of the low-frequency mode in response to ahigh-frequency excitation.The possibility that resonances may occur between a high-frequency excitationand a single low-frequency mode is then investigated. A single-degree-of-freedomoscillator driven at a frequency much higher than its natural frequency is analyzed.It is found that, if the excitation is modulated, resonant excitation of the oscillatormay occur.