A nonlinear analysis of electrically forced vibrations of piezoelectric plates with viscous damping near the thickness-shear mode

Abstract

Based on the theory of nonlinear piezoelectricity, an approximate solution for electrically forced nonlinear thickness-shear vibrations of piezoelectric plates is introduced. The model considers an infinite piezoelectric plate subjected to an alternative voltage, incorporating 3rd-order and 4th-order elastic constants and viscous damping under finite deformation. The system's differential equations for steady-state forced vibrations are derived from the nonlinear pure thickness-shear vibration problem, and transformed so that the new electrical boundary conditions are free. After the application of Galerkin's method, the transformed differential equations lead to a cubic equation, capturing the nonlinear current-frequency curves of AT-cut quartz plates without the need of the quality factor Q, due to the introduced viscous damping. When the damping effect is ignored, the frequency response curves of the AT-cut quartz plate under a specific voltage are confirmed with the existing literature. The study emphasizes the presence of the critical voltage and the importance of the fourth-order elastic constant (c6666) in the stability, revealing lower c6666 values correspond to improved stability. Additionally, a proposed algorithm efficiently determines critical voltages for resonator stability.