Vibration Control for Two Nonlinearly Coupled and Parametrically Excited van der Pol Oscillators

We investigate the parametric resonance of two nonlinearly coupled van der Pol oscillators under state feedback control with a time delay. Using the asymptotic perturbation method, we obtain four slow flow equations on the amplitude and phase of the oscillators. Their fixed points correspond to a two-period quasi-periodic motion for the starting system and we show parametric excitation-response and frequency-response curves. We analyze the effect of time delay and feedback gains from the viewpoint of vibration control and use energy considerations to study existence and characteristics of limit cycles of the slow flow equations. A limit cycle corresponds to a three-period quasi-periodic modulated motion for the starting system and in order to reduce the amplitude peak of the parametric resonance and to exclude the existence of three-period quasi-periodic motion, we find the appropriate choices for the feedback gains and the time delay. Analytical results are verified with numerical simulations.