Sun Zhenbo, Wang Zhe, Li Zhen, Guo Yan, Huang Bin
Zhejiang-Italy Joint Lab for Smart Materials and Advanced Structures, Ningbo University, Ningbo, 315211, China.
TXC (Ningbo) Corp, Ningbo, China.
Sci Rep. 2024 Jul 25;14(1):17131. doi: 10.1038/s41598-024-67846-5.
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.
倒角工艺是谐振器制造中的一个重要工艺，对谐振器的频率稳定性有重大影响。如果不了解倒角后谐振器的频率特性，就无法精确设计倒角谐振器。因此，为了研究AT切型倒角谐振器的振动特性，我们在这项工作中采用基于Mindlin板理论的子区域几何拟合方法（SGFM）研究了高频振动。将厚度不均匀的石英晶体板划分为三个区域，并根据测量的几何形态数据，用多项式函数对每个区域进行拟合。基于Mindlin二维理论得到控制方程，并使用COMSOL的偏微分方程模块进一步求解耦合振动。在数值计算中，我们将SGFM得到的结果与整体拟合方法得到的结果以及测量数据进行了比较。SGFM的准确性和有效性也得到了验证。结果发现，SGFM得到的频率略高于整体拟合方法得到的频率，且SGFM的结果更接近测量结果。同时，随着倒角时间的增加，频率升高，能量俘获效应变得更加显著。所提出的方法在保证精度的同时，可以显著提高厚度剪切振动的计算效率。有望为倒角晶体谐振器的分析提供一种新的几何拟合方法。
