In this paper, nonlinear vibration control of a plate is investigated using a nonlinear modified positive position feedback method that is applied through a piezoelectric layer on the plate. Based on the classical theory of displacement and strain relations with von Karman, intended equations of motion for the smart plate have been obtained. In this model, transverse vibrations are studied and stimulations are performed for the primary resonance. Boundary conditions of the smart plate are simply supported. The plate is thickness symmetrical. Using the Galerkin method the temporal nonlinear equations governing the system have been found. Then, the free and forced vibrations of the structure with the nonlinear modified positive position feedback controller have been solved using the Method of Multiple Scales to obtain an analytical solution. Results show that this controller reduces the amplitude of the vibration by inducing an increase in the damping coefficient. In addition, this provides a higher level of suppression in the overall frequency domain response by increasing the compensator gain. Finally, the results of the analytical solution for the closed-loop nonlinear modified positive position feedback controller are presented and compared with the result of the conventional positive position feedback controller and nonlinear integral resonant controller. The results show that the performance of the nonlinear modified positive position feedback controller is better than other controllers and significantly reduces the vibration amplitude.
