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Thickness-shear vibration analysis of circular quartz crystal plates by a differential quadrature hierarchical finite element method

机译：圆石英晶体板厚度剪切振动的微分正交分层有限元分析

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摘要

Piezoelectric crystals are widely used to make acoustic wave resonators. A large portion of quartz crystal resonators are operated with the so-called thickness-shear (TS) modes of a plate, which is characterized by high frequency vibration that needs both huge computational cost and high accuracy. The situation is even more difficult for circular resonators because the scalar differential equation is not separable in polar coordinates due to material anisotropy. In this paper, TS vibration analysis of circular quartz plates was carried out by a differential quadrature hierarchical finite element method (DQHFEM). The DQHFEM adopts the formulations of the DQFEM that are compact and highly accurate. The numerical stability problem of the high-order or very high-order hierarchical basis is overcome by using the recursion formula of Legendre polynomials and more than 2000th order can be reached. The high frequency thickness-shear vibration of a circular quartz plate can be modeled by using only one or several DQHFEM elements, which greatly simplifies pre- and post-processing. Numerical comparison of the DQHFEM results with the results in literatures validated the high accuracy of the DQHFEM. (C) 2015 Elsevier Ltd. All rights reserved.
机译：压电晶体广泛用于制造声波谐振器。石英晶体谐振器的大部分以板的所谓的厚度剪切（TS）模式工作，其特征在于高频振动需要大量的计算成本和高精度。对于圆形谐振器，情况更加困难，因为由于材料的各向异性，标量微分方程在极坐标中不可分离。本文采用差分正交分层有限元方法（DQHFEM）对圆形石英板的TS振动进行了分析。 DQHFEM采用紧凑且高度精确的DQFEM公式。通过使用勒让德多项式的递推公式，可以解决高阶或超高阶层次结构的数值稳定性问题，并且可以达到2000阶以上。圆形石英板的高频厚度剪切振动可以仅使用一个或几个DQHFEM元素进行建模，从而大大简化了预处理和后期处理。 DQHFEM结果与文献结果的数值比较证实了DQHFEM的高精度。 （C）2015 Elsevier Ltd.保留所有权利。

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