Wu Rongxing, Wang Wenjun, Chen Guijia, Du Jianke, Ma Tingfeng, Wang Ji
Piezoelectric Device Laboratory, School of Mechanical Engineering and Mechanics, Ningbo University, 818 Fenghua Road, Ningbo, Zhejiang 315211, China; Department of Architectural Engineering, Ningbo Polytechnic Institute, Beilun District, Ningbo 315800, Zhejiang, China.
Piezoelectric Device Laboratory, School of Mechanical Engineering and Mechanics, Ningbo University, 818 Fenghua Road, Ningbo, Zhejiang 315211, China.
Ultrasonics. 2016 Feb;65:338-44. doi: 10.1016/j.ultras.2015.09.008. Epub 2015 Sep 25.
Lee plate equations for high frequency vibrations of piezoelectric plates have been established and perfected over decades with the sole objective of obtaining accurate predictions of frequency and mode shapes to aid the analysis and design of quartz crystal resonators. The latest improvement includes extra terms related to derivatives of the flexural displacement to provide much accurate solutions for vibrations of the thickness-shear mode, which is the functioning mode of resonators and has much higher frequency than the flexural mode. The improved Lee plate equations have been used in the analysis of high frequency vibrations of quartz crystal plates as an essential step for analysis of AT- and SC-cut quartz crystal resonators after validations with fully electrode quartz crystal piezoelectric plates. In this study, closed-form solutions of free and forced vibrations of SC-cut quartz plates with partial electrodes are obtained. A procedure has been established for the calculation of dispersion relations, frequency spectra, selected vibration modes, and capacitance ratios of forced vibrations. The vibration solutions obtained with the first-order Lee plate equations are proven to be close to solutions from the Mindlin plate equations. It is now clear that both the Mindlin and Lee plate equations can be used in the analysis and design of quartz crystal resonators.
几十年来，用于压电板高频振动的李板方程已经建立并完善，其唯一目的是获得频率和振型的准确预测，以辅助石英晶体谐振器的分析和设计。最新的改进包括与弯曲位移导数相关的额外项，以便为厚度剪切模式的振动提供更精确的解，厚度剪切模式是谐振器的工作模式，其频率比弯曲模式高得多。经过全电极石英晶体压电板验证后，改进的李板方程已被用于石英晶体板高频振动分析，作为分析AT切和SC切石英晶体谐振器的重要步骤。在本研究中，获得了带有部分电极的SC切石英板自由振动和强迫振动的封闭形式解。已经建立了一种计算色散关系、频谱、选定振动模式和强迫振动电容比的程序。结果表明，用一阶李板方程得到的振动解与明德林板方程的解相近。现在很清楚，明德林板方程和李板方程都可用于石英晶体谐振器的分析和设计。
