Controlling parametric resonance: Induction and stabilization of unstable motions

Abstract

Parametric resonance is a resonant phenomenon which takes place in systems characterized by periodic variations of some parameters. While seen as a threatening condition, whose onset can drive a system into instability, this chapter advocates that parametric resonance may become an advantage if the system undergoing it could transform the large amplitude motion into, for example, energy. Therefore the development of control strategies to induce parametric resonance into a system can be as valuable as those which aim at stabilizing the resonant oscillations. By means of a mechanical equivalent the authors review the conditions for the onset of parametric resonance, and propose a nonlinear control strategy in order to both induce the resonant oscillations and to stabilize the unstable motion. Lagrange's theory is used to derive the dynamics of the system and input-output feedback linearization is applied to demonstrate the feasibility of the control method.