Wang Wenjun, Wu Rongxing, Wang Ji, Du Jianke, Yang Jiashi
IEEE Trans Ultrason Ferroelectr Freq Control. 2013 Jun;60(6):1192-8. doi: 10.1109/TUFFC.2013.2681.
We study free vibrations of an elliptical crystal resonator of AT-cut quartz with an optimal ratio between the semi-major and semi-minor axes as defined by Mindlin. The resonator is contoured with a quadratic thickness variation. The scalar equation for thickness-shear modes in an AT-cut quartz plate by Tiersten and Smythe is used. Analytical solutions for the frequencies and modes to the scalar equation are obtained using a power series expansion that converges rapidly. The frequencies and modes are exact in the sense that they can satisfy the scalar differential equation and the free edge condition to any desired accuracy. They are simple and can be used conveniently for further studies on other effects on frequencies and modes of contoured resonators.
我们研究了由明德林定义的半长轴与半短轴具有最佳比例的AT切石英椭圆晶体谐振器的自由振动。该谐振器的轮廓具有二次厚度变化。使用了蒂尔斯特恩和斯迈思给出的AT切石英板中厚度剪切模式的标量方程。通过快速收敛的幂级数展开获得了该标量方程频率和模式的解析解。这些频率和模式在能够以任何所需精度满足标量微分方程和自由边缘条件的意义上是精确的。它们很简单，可方便地用于进一步研究对轮廓谐振器频率和模式的其他影响。
