Frequency Response Analysis of Parametric Resonance and Vibrational Stabilization

Periodically time-varying models are found across nature and engineered systems, from fluid dynamics, structures and MEMS devices to quantum mechanics and astrophysics. Such systems are known to exhibit parametric resonance, a kind of instability caused by fluctuating model parameters. Under conditions of instability, they can also be vibrationally stabilized with the right forcing. The question of interest here is variation in behavior within these two stable regimes, and whether certain parameter configurations are preferred from a design perspective. This motivation leads us to consider Mathieu’s equation with harmonic forcing as a canonical model. To address these questions, we use a lifting based approach to obtain a representation of the frequency response operator that is amenable to methods from LTI systems. We study the poles of the system as a function of its parameters, and obtain a description of the free response of Mathieu’s equation as the product of two simple functions. We also investigate the dependence of the ${\mathcal{H}_2}$ norm of Mathieu’s equation on its parameters. A considerable difference in ${\mathcal{H}_2}$ norm between the two regimes is found, as well as interesting behavior within each domain.