Lee Peter C Y, Huang Rui
Department of Civil and Environmental Engineering, Princeton University, NJ 08544, USA.
IEEE Trans Ultrason Ferroelectr Freq Control. 2002 May;49(5):604-11. doi: 10.1109/tuffc.2002.1002459.
Thickness-shear vibrations of a rectangular AT-cut quartz with one face in contact with a layer of Newtonian (linearly viscous and compressible) fluid are studied. The two-dimensional (2-D) governing equations for vibrations of piezoelectric crystal plates given previously are used in the present study. The solutions for 1-D shear wave and compressional wave in a liquid layer are obtained, and the stresses at the bottom of the liquid layer are used as approximations to the stresses exerting on the crystal surface in the plate equations. Closed form solutions are obtained for both free and piezoelectrically forced thickness-shear vibrations of a finite, rectangular AT-cut quartz in contact with a liquid layer of finite thickness. From the present solutions, a simple and explicit formula is deduced for the resonance frequency of the fundamental thickness-shear mode, which includes the effects of both shear and compressional waves in the liquid layer and the effect of the thickness-to-length ratio of the crystal plate. The formula reduces to the widely used frequency equation obtained by many previous investigators for infinite plates. The resonance frequency of a rectangular AT-cut quartz, computed as a function of the thickness of the adjacent liquid layer, agrees closely with the experimental data measured by Schneider and Martin.
研究了一个面与一层牛顿流体（线性粘性且可压缩）接触的矩形AT切石英的厚度剪切振动。本研究采用了先前给出的压电晶体板振动的二维控制方程。得到了液层中一维剪切波和压缩波的解，并将液层底部的应力用作板方程中作用在晶体表面应力的近似值。对于与有限厚度液层接触的有限矩形AT切石英的自由和压电强迫厚度剪切振动，都得到了封闭形式的解。从当前的解中，推导出了基波厚度剪切模式共振频率的一个简单明了的公式，该公式包括了液层中剪切波和压缩波的影响以及晶体板厚度与长度比的影响。该公式简化为许多先前研究者针对无限板得到的广泛使用的频率方程。作为相邻液层厚度的函数计算得到的矩形AT切石英的共振频率，与施耐德和马丁测量的实验数据非常吻合。
