Nonlinear dynamics of systems involving widely spaced frequencies

This document focuses on the dynamics of nonlinear oscillatory systems involving widely spaced frequencies. First, experiments on the forced oscillations of a cantilever beam with a circular cross-section are presented. The beam is excited near its fifth natural frequency but sustained large-amplitude oscillations of the first mode accompanied by modulated oscillations of the fifth mode occur. Thus, energy is transferred from the fifth to the first mode of the beam, whose natural frequencies are roughly in the ratio of 60:1.

Some simple two-degree-of-freedom systems are studied in an effort to explain the strong nonlinear coupling between modes with widely spaced frequencies. In these systems, the coupling is found to give rise to some novel phenomena including static displacements or sustained oscillations of the low-frequency mode in response to a high-frequency excitation.

The possibility that resonances may occur between a high-frequency excitation and a single low-frequency mode is then investigated. A single-degree-of-freedom oscillator 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 oscillator may occur.