用Mindlin板方程分析环板的厚度剪切振动
An Analysis of Thickness-shear Vibrations of an Annular Plate with the Mindlin Plate Equations
摘要：
本文用考虑厚度剪切变形为自变量的Mindlin板方程分析了矩形和圆形石英晶体谐振器的振动。Mindlin或Lee板理论将厚度剪切变形视为二维变量耦合系统中的独立高阶振动模式，是分析理论的选择。对于圆板，我们推导了Mindlin板方程，在一个系统的方式证明了Mindlin和其他人，并得到截断的二维方程的密切耦合模式在极坐标系中。对厚度-剪切基频附近的振型方程进行了简化，并验证了简化方程和方法的正确性。为了探索石英晶体谐振器的新结构，我们利用Mindlin板方程分析具有固定内边缘和自由外边缘的环形板的频谱。通过对圆板振动的归一化频率与尺寸参数关系的详细分析，为基于强厚度剪切模态和最小化其他模态的原理进行参数优化选择提供了参考，从而通过保持强的、纯的厚度剪切振动对温度和初应力等复杂因素的不敏感性来增强能量捕获。
英文摘要：
The Mindlin plate equations with the consideration of thickness-shear deformation as an independent variable have been used for the analysis of vibrations of quartz crystal resonators of both rectangular and circular types. The Mindlin or Lee plate theories that treat thickness-shear deformation as an independent higher-order vibration mode in a coupled system of two-dimensional variables are the choice of theory for analysis. For circular plates, we derived the Mindlin plate equations in a systematic manner as demonstrated by Mindlin and others and obtained the truncated two-dimensional equations of closely coupled modes in polar coordinates. We simplified the equations for vibration modes in the vicinity of fundamental thickness-shear frequency and validated the equations and method. To explore newer structures of quartz crystal resonators, we utilized the Mindlin plate equations for the analysis of annular plates with fixed inner and free outer edges for frequency spectra. The detailed analysis of vibrations of circular plates for the normalized frequency versus dimensional parameters provide references for optimal selection of parameters based on the principle of strong thickness-shear mode and minimal presence of other modes to enhance energy trapping through maintaining the strong and pure thickness-shear vibrations insensitive to some complication factors such as thermal and initial stresses.
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