Asynchronous Parametric Excitation in Dynamical Systems

The overall objective of this thesis is to obtain a comprehensive understanding of the stability behavior of general parametrically excited systems. Even though the manifold resonance effects caused by time-periodic variation of system parameters have been intensively studied since the mid-19th century, several aspects still remain unexplored. While, historically, parametric excitation had been prominent predominantly for its destabilizing impact (resonance), in recent decades, also its stabilizing impact (anti-resonance) had attained significant attention. Owing to this historical development, the coexistence of resonance and anti-resonance at certain frequencies in the case of asynchronous excitation was not identified. Further, most of the existing studies dealing with the appearance of different resonance effects are limited to simple systems featuring neither circulatory nor gyroscopic terms, making the response of more complex systems to parametric excitation unpredictable. In the present contribution, the stability behavior of systems featuring circulatory and gyroscopic terms subject to asynchronous parametric excitation is investigated employing the semi-analytical method of normal forms. First, novel stability patterns are identified revealing global stabilizing and destabilizing effects. More importantly, it is shown that, contrary to the previous knowledge, resonance and anti-resonance may both simultaneously appear in the vicinity of certain resonance frequencies with a particularly steep transition between them. Even for complex systems featuring circulatory terms, these effects can be easily assessed qualitatively and quantitatively using the symbolic expressions derived for the most representative stability features. The results are validated on an electronic system following the simulation-based approach. Finally, with the enhanced understanding of the parametric stability phenomena, two exemplary mechanical systems, including a minimal model of a disk brake, are analyzed. The analysis emphasizes the practical significance of the coexistence of resonance and anti-resonance and advocates more accurate consideration of possible asymmetries, i.e., parametric excitation, in the brake squeal analysis