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This study investigated the force–frequency characteristics of quartz wafers inside a cantilever beam frame. Firstly, the force–frequency coefficient formula of quartz wafers with fixed ends under axial force was analyzed. Firstly, the formula for the force–frequency coefficient of quartz wafers with fixed ends under axial force was analyzed. A force–frequency coefficient formula suitable for cantilever beam structures was derived by considering the changes in surface stress and stiffness of quartz wafers with fixed ends and one end under force on the other. Subsequently, the formula's accuracy was verified by experiments, and the accuracy was more than 92%. In addition, strain simulation analysis was performed on three different shapes of quartz wafers, and experimental verification was carried out on two of them. The results revealed that trapezoidal quartz wafers and cantilever beam structures exhibited superior stress distribution to rectangular chips. Furthermore, by positioning electrodes at various locations on the surface of the quartz chip, it was observed that, as the electrodes moved closer to the fixed end, the force–frequency coefficient of the rectangular quartz chip increased, along with an increase in chip strain under the cantilever structure. In summary, this study provides a new approach for designing cantilever quartz resonator sensors in the future. [ABSTRACT FROM AUTHOR]

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 (c 6666) in the stability, revealing lower c 6666 values correspond to improved stability. Additionally, a proposed algorithm efficiently determines critical voltages for resonator stability. • A cubic equation predicts the nonlinear behavior of piezoelectric resonators. • The decrease of c 6666 improves the stability of piezoelectric resonators. • The amplitude-frequency curve has a hysteresis region when voltage is that high. • An efficient algorithm for the critical voltage is suggested. [ABSTRACT FROM AUTHOR]