Positive Position Feedback Control for High-Amplitude Vibration of a Flexible Beam to a Principal Resonance Excitation

The application of active linear absorber based on positive position feedback control strategy to suppress the high-amplitude response of a flexible beam subjected to a primary external excitation is developed and investigated. A mathematical nonlinear model that describes the single-mode dynamic behavior of the beam is considered. The perturbation method of multiple scales is employed to find the general nonlinear response of the system and four first-order differential equations governing the amplitudes and phases of the responses are derived. Then a stability analysis is conducted for the open- and closed-loop responses of the system and the performance of the control strategy is analyzed. A parametric investigation is carried out to investigate the effects of changing the damping ratio of the absorber and the value of the feedback gain as well as the effect of detuning the frequency of the absorber on the responses of the system. It is demonstrated that the positive position feedback control technique is effective in reducing the high-amplitude vibration of the model and the control scheme possesses a wide suppression bandwidth if the absorber's frequency is properly tuned. Finally, the numerical simulations are performed to validate the perturbation solutions.