MULTIPLE RESONANT OR NON-RESONANT PARAMETRIC EXCITATIONS FOR NONLINEAR OSCILLATORS

The transient and steady state response of a very general non-linear oscillator subject to a finite number of parametric excitations is considered by the asymptotic perturbation method. Three main cases are examined: (1) the parametric excitations frequencies are not close to each other or close to the principal parametric resonance of the oscillator; (2) the parametric excitations frequencies are close to each other but not close to the principal resonance; and (3) all the parametric excitations frequencies are close to the principal resonance. Both the conditions for the quenching of the oscillation and the conditions for its persistence are determined. The main conclusion is that the oscillation in systems with one degree of freedom cannot be fully quenched due to the action of parametric excitation, because the only change is a shift in the oscillator frequencies. Analytical approximate results are checked by numerical integration.