Asynchronous parametric excitation, total instability and its occurrence in engineering structures

Abstract In mechanical engineering systems self-excited and parametrically excited vibrations are in general unwanted and sometimes dangerous. There are many systems exhibiting such vibrations which up to this day cannot be completely avoided, such as brake squeal, the galloping vibrations of overhead transmission lines, the ground resonance in helicopters and others. In general, problems of parametric excitation are studied for the case in which all the time-periodic terms are synchronous. In this case the stability behavior is well understood. However, if the time-periodic terms are asynchronous, 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 (total instability). Until recently it was believed that such atypical behavior would not appear in mechanical systems. The present paper discusses some recent insights and results obtained for linear and nonlinear systems with asynchronous parametric excitation. The method of normal forms is used to prove total instability and to calculate limit cycles of a generalized nonlinear system. Further, a mechanical example of a minimal disk brake model featuring such out of phase parametric excitation is presented. The example outlines the importance of the observed effects from the engineering point of view, since similar terms are also expected in the equations of motion of disk brakes with disks with ventilation channels and most likely also in other physical systems