Vibration control for the parametrically excited van der Pol oscillator by nonlocal feedback

Abstract

A nonlocal feedback is used for the control of nonlinear vibrations in a parametrically excited van der Pol oscillator. A nonlocal control force is introduced in order to obtain a third-order nonlinear differential equation (jerk dynamics). Using the asymptotic perturbation method, two slow flow equations on the amplitude and phase of the response are obtained, and subsequently the performance of the control strategy is investigated. Parametric excitation–response and frequency–response curves are shown. Uncontrolled and controlled systems are compared, and the appropriate choices of the feedback gains for reducing the amplitude peak of the response are found. Energy considerations are used in order to study the existence and characteristics of limit cycles of the slow flow equations. A limit cycle corresponds to a two-period modulated motion for the van der Pol oscillator. To exclude the possibility of quasi-periodic motion and to reduce the amplitude peak of the parametric resonance, appropriate choices of the feedback gains are found. Numerical simulation confirms the validity of the new method.