Instability and Piezoelectric Resonance of a Damped S-FGP Plate Resting on a Foundation with Shear Layer Using Modified FSDT Theory

In this study, dynamic instability and nonlinear behavior of sigmoidal-functionally Graded piezoelectric (S-FGP) plates resting on linear elastic foundation under parametric harmonic piezoelectric excitation is investigated. Based on Modified FSDT, applying Hamilton’s principle, the governing nonlinear coupled partial differential equations are derived. By considering six vibration modes, the Galerkin’s procedure is used to reduce the equations of motion to nonlinear Mathieu equations, and dynamic instability of the problem in absence/presence of the Rayleigh's proportional damping model and foundation is analyzed. In the absence of foundation, the validity of the modeling for analyzing the maximum nonlinear deflection and the modified shear correction factors are accomplished by comparing the results with those of the literature. It is shown that as the parameters of the foundation increase, the piezoelectric instability occurs at higher excitation frequencies and increasing the damping makes the region of the instability to be shrunk. By applying the perturbation method, steady-state response equations are derived and the parametric resonance of the system under the piezoelectric excitation is analyzed. Then, the conditions of existence and stability of trivial/nontrivial solutions are discussed. Moreover, the effects of the system parameters, including excitation frequency, amplitude of harmonic excitation voltage, foundation parameters and damping, on the nonlinear dynamics of the S-FGP plate are studied and it is shown that the presence of the damping/foundation has a considerable influence on the resonance characteristic curves.