Xie Longtao, Wang Shaoyun, Zhang Chuanzeng, Wang Ji
Piezoelectric Device Lab, School of Mechanical Engineering & Mechanics, Ningbo University, Ningbo, 315211, China.
Department of Civil Engineering, University of Siegen, Siegen, D-57068, Germany.
J Acoust Soc Am. 2018 Aug;144(2):814. doi: 10.1121/1.5050609.
The scalar differential equation in the thickness eigendisplacement for the doubly-rotated quartz plates is applied to analyze thickness vibrations of an unelectroded circular plate with free edges at the neighborhood of the pure thickness vibration mode. The scalar differential equation is transformed into an elliptical coordinate system. With the boundary conditions of free edges, the frequencies and the modes are solved in terms of the Mathieu function and the Modified Mathieu function. The results of frequencies of the fundamental harmonic and its third overtone of an AT-cut quartz circular plate by the present approach agree well with the existing theoretical results and the experiment results. The frequencies and modes of an SC-cut quartz circular plate are investigated by the present approach. The frequencies are close to each other when the order of the harmonics is the same. A rotation angle of the symmetric axes of the vibration modes are observed that is dependent on the anisotropic material constants and the order of the harmonics. This approach has potential applications in the design of the doubly-rotated quartz circular resonators.
应用双旋转石英板厚度本征位移的标量微分方程，在纯厚度振动模式附近分析自由边无电极圆形板的厚度振动。将该标量微分方程变换到椭圆坐标系中。利用自由边的边界条件，借助马修函数和修正马修函数求解频率和振型。用本方法得到的AT切石英圆形板基波及其三次谐波的频率结果与现有理论结果和实验结果吻合良好。用本方法研究了SC切石英圆形板的频率和振型。当谐波阶数相同时，频率彼此接近。观察到振动模式对称轴的旋转角度取决于各向异性材料常数和谐波阶数。该方法在双旋转石英圆形谐振器设计中具有潜在应用。
