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Vibration control for the nonlinear resonant response of a piezoelectric elastic beam via time-delayed feedback

The paper presents time-delayed feedback control to reduce the nonlinear resonant vibration of a piezoelectric elastic beam. Specifically, we examine three single-input linear time-delayed feedback control methodologies: displacement, velocity and acceleration time-delayed feedback. Moreover, the multi-input time-delayed feedback control methodologies are discussed. Utilizing the method of multiple scales, the modulation equation and the first order approximations of the primary resonances are derived and the effect of time delay on the resonances is analyzed. Then the effect of time delays and control gains on the stability, amplitude, frequency-response behavior, peak amplitude and critical excitation amplitude are investigated. Optimal values of the controllers gains and delay are obtained, simulated, and compared. The time-delayed feedback control acts as a vibration absorber at specific values of time delay. On the other hand, using mixed delay feedback controllers demonstrates an excellent improvement in mitigating the first-mode vibration.