An analysis of the thickness vibration of an unelectroded doubly-rotated quartz circular plate

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.