Atypical parametric instability in linear and nonlinear systems

Abstract

The linear parts of the equations of motion of parametrically excited mechanical systems are characterized by the M, D, G, K, N matrices which may all be time-periodic (mass, damping, gyroscopic, stiffness and circulatory matrices, respectively). The stability of these systems can be studied via Floquet theory. A typical property of parametric instability behavior is the existence of combination resonances. It has been known for a long time that the type of parametric resonance depends very much on whether the excitation is in the in the K or in the N matrices, or simultaneously in both of them, the other matrices being constant. In general, problems of parametric excitation are studied for the case in which all the excitation terms are in phase. If this is not the case, an atypical behavior may occur: The linear system may then be unstable for all frequencies of the parametric excitation, and not only in the neighborhood of certain discrete frequencies. Examples of differential equations of this type were first given about 70 years ago by Lamberto Cesari, but seem largely to have fallen into oblivion since then. It was recently observed that the linearized equations of motion for a minimal model of a squealing disk brake have such out of phase parametric excitation. Additional nonlinearities are introduced in the equations and the corresponding limit cycles are calculated using normal form theory.