Optimal time delayed control of the combination resonances of viscoelastic graphene sheets under dual-frequency excitation

Abstract

In this study, the nonlinear combination vibration of a single-layer graphene sheet (SLGS) resting on a visco-Pasternak foundation and exposed to dual-frequency excitation was investigated. The time delayed feedback proportional-derivative (PD) controller was applied to an SLGS for the first time, and three combination resonances with two different time delays were examined. The approximate analytical solutions of nonlinear vibration equations were obtained using the multiple scales perturbation method, and their stability under the condition of combination resonance was investigated. The optimal time delay for the controller of the system was analyzed and calculated. Finally, the results of this study were compared with those of other studies. The results showed that the dual-frequency excitation intensifies the nonlinear behavior of the system. Further, the influence of the feedback gain coefficients and time delays on the combination resonance of the system was examined, and the results showed that the efficiency of vibration control can be improved by adjusting the feedback gain and time delay parameters. This study aims to provide theoretical guidance for the application of viscoelastic graphene sheets.