High-frequency vibration of beveled crystal plates by using subregional geometric fitting method

The beveling process is an important process in the manufacturing of resonators, which has a significant impact on the frequency stability of resonators. Without understanding the frequency characteristics of the resonator after beveling, it is impossible to accurately design the beveled resonators. Thus, in order to investigate the vibration characteristics of AT-cut beveled resonators, we investigated the high-frequency vibration in this work by using the subregional geometric fitting method (SGFM) based on Mindlin's plate theory. Quartz crystal plates with nonuniform thicknesses are partitioned into three regions and each region is fitted by using the polynomial functions based on the measured geometric morphology data. The governing equations are obtained based on Mindlin's two-dimensional theory and the coupled vibrations are further solved using the partial differential equation module of COMSOL. In the numerical calculation, we compare the results obtained by the SGFM with those obtained by the global fitting method and the measured data. The accuracy and effectiveness of the SGFM are also verified. It is found that the frequencies obtained by the SGFM are slightly higher than the frequencies obtained by the global fitting method, and the results of SGFM are closer to the measured results. Meanwhile, as the beveling time increases, the frequency increases and the energy trapping effect becomes more significant. The proposed method can significantly improve the computational efficiency of thickness-shear vibration while ensuring accuracy. It is expected to provide a new geometric fitting method for the analysis of beveled crystal resonators.