Nonlinear vibrations of four-degrees of freedom for piezoelectric functionally graded graphene-reinforced laminated composite cantilever rectangular plate with PPF control strategy

The nonlinear vibration suppression of the piezoelectric functionally graded graphene-reinforced laminated composite cantilever (PFG-GRLCC) rectangular plate with positive position feedback (PPF) control strategy is investigated firstly. The material properties of the graphene-reinforced structure are calculated through the Halpin–Tsai micromechanical model. Considering the transverse external excitation and the converse effect of piezoelectricity, the governing equations of motion are formulated through von Karman large deformation theory, the classical laminated plate theory, and Hamilton principle. After adding the PPF controllers, a four-degrees of freedom model for the close-loop vibration control system is achieved via Galerkin truncation technique. The average equations in the case of the primary resonance and 1:1:3:3 internal resonance can be obtained using the multiple scale perturbation (MSP) method. The amplitude–frequency response curves are studied by the numerical continuation algorithm. The detailed parametric analyses show that the PPF controller can effectively reduce the nonlinear vibration response amplitudes of the PFG-GRLCC rectangular plate. In addition, the results reveal the energy transform between the host system and the PPF controller. This work is expected to provide theoretical guidance for nonlinear large amplitude vibration reduction of graphene-reinforced structure.