Time-Delayed Feedback Control of Piezoelectric Elastic Beams under Superharmonic and Subharmonic Excitations

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.