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 () in the stability, revealing lower values correspond to improved stability. Additionally, a proposed algorithm efficiently determines critical voltages for resonator stability.

中文翻译：

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厚度剪切模式下粘性阻尼压电板电力强迫振动的非线性分析


基于非线性压电理论，提出了压电板电驱动非线性厚度剪切振动的近似解。该模型考虑了承受交变电压的无限压电板，结合了有限变形下的三阶和四阶弹性常数和粘性阻尼。系统的稳态受迫振动微分方程是从非线性纯厚度剪切振动问题导出的，并进行转换以使新的电边界条件是自由的。应用伽辽金方法后，将微分方程变换为三次方程，由于引入了粘性阻尼，无需品质因数Q即可捕获AT切割石英板的非线性电流-频率曲线。当忽略阻尼效应时，AT切割石英板在特定电压下的频率响应曲线与现有文献得到证实。该研究强调了临界电压的存在以及四阶弹性常数 () 在稳定性中的重要性，表明较低的值对应于稳定性的提高。此外，所提出的算法有效地确定了谐振器稳定性的临界电压。