Vibration amplitude control for a van der Pol–Duffing oscillator with time delay

Periodic solutions for the fundamental resonance response of a van der Pol–Duffing system under time-delayed position and velocity feedbacks are investigated. Using the asymptotic perturbation method, two slow-flow equations for the amplitude and phase of the fundamental resonance response are derived. Their fixed points correspond to limit cycles (phase-locked periodic solutions) for the original system. In the uncontrolled system, periodic solutions exist only for fixed values of amplitude and phase and depend on the system parameters and excitation amplitude. In many cases, the amplitudes of periodic solutions do not correspond to the technical requirements. It is demonstrated that, if the vibration control terms are added, stable periodic solutions with arbitrarily chosen amplitude can be accomplished. Therefore, the results obtained show that an effective vibration amplitude control is possible if appropriate time delay and feedback gains are chosen.