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.

The time-delayed displacement feedback control is provided to restrain the superharmonic and subharmonic response of the elastic support beams. The nonlinear equations of the controlled elastic beam are obtained with the help of the Euler–Bernoulli beam principle and time-delayed feedback control strategy. Based on Galerkin method, the discrete nonlinear time-delayed equations are derived. Using the multiscale method, the first-order approximate solutions and stability conditions of three superharmonic and 1/3 subharmonic resonance response on controlled beams are derived. The influence of time-delayed parameters and control gain are obtained. The results show that the time-delayed displacement feedback control can effectively suppress the superharmonic and subharmonic resonance response. Selecting reasonably the time-delayed quantity and control gain can avoid the resonance region and unstable multi-solutions and improve the efficiency of the vibration control. Furthermore, with the purpose of suppressing the amplitude peak and governing the resonance stability, appropriate feedback gain and time delay are derived.