A Semi-Analytical Solution for the Thickness-Vibration of Centrally Partially-Electroded Circular AT-Cut Quartz Resonators.

Bin Wang,Xintao Zhao,Xiaoyun Dai,Zhenghua Qian

doi:10.3390/s17081820

Bin Wang, Xintao Zhao + Show 2 more

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https://doi.org/10.3390/s17081820

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Abstract

Vibration frequencies and modes for the thickness-shear vibrations of infinite partially-electroded circular AT-cut quartz plates are obtained by solving the two-dimensional (2D) scalar differential equation derived by Tiersten and Smythe. The Mathieu and modified Mathieu equations are derived from the governing equation using the coordinate transform and the collocation method is used to deal with the boundary conditions. Solutions of the resonant frequencies and trapped modes are validated by those results obtained from COMSOL software. The current study provides a theoretical way for figuring out the vibration analysis of circular quartz resonators.

Highlights

Nowadays, acoustic wave resonators are widely used for frequency generation and control in telecommunications, as well as mass and acceleration sensors
The results of the fundamental and third overtone TS modes obtained through Finite Element Method (FEM) simulation
The exact frequencies of thickness-shear and thickness-twist modes along with mode shapes for infinite, centrally partially-electroded circular AT-cut quartz plates were obtained through theoretical analysis, by the application of coordinate transform and using Mathieu and modified Mathieu functions

Summary

Introduction

Acoustic wave resonators are widely used for frequency generation and control in telecommunications, as well as mass and acceleration sensors. Quartz is the most widely used crystal for resonators, due to its piezoelectric effects. The desire for better resonators has increased with the rapid development of electric devices, which calls for thorough studies on the vibration of the quartz plate. Quartz crystal plate works with thickness-shear (TS). Because of the complicated properties of quartz crystal, such as anisotropy and electro-mechanical coupling, it is difficult to obtain an exact three-dimensional analytical solution. In order to solve the problem, Mindlin and his co-authors firstly developed a two-dimensional plate theory for solving the vibrations of elastic and piezoelectric plates [2,3,4,5], which is especially suitable for analyzing low-order modes of thickness-shear vibrations.

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