An Analysis of Nonlinear Thickness-shear Vibrations of Quartz Crystal Plates by Two-Dimensional Finite Element Method

TL;DR

This paper develops a nonlinear finite element method to analyze high-frequency thickness-shear vibrations in quartz crystal plates, enabling better prediction of nonlinear behaviors in miniaturized resonators.

Contribution

It extends the kinematic and constitutive equations to include nonlinear effects and formulates a finite element approach for analyzing nonlinear vibrations in quartz plates.

Findings

Backbone curves and mode shapes are obtained and analyzed.

The developed program predicts nonlinear characteristics of quartz resonators.

The method improves understanding of nonlinear vibrational behavior in miniaturized devices.

Abstract

A nonlinear analysis of high-frequency thickness-shear vibrations of AT-cut quartz crystal plates is presented with the two-dimensional finite element method. We expanded both kinematic and constitutive nonlinear Mindlin plate equations and then truncated them to the first-order equations as an approximation, which is used later for the formulation of nonlinear finite element analysis with all zeroth- and first-order displacements and electric potentials. The matrix equation of motion is established with the first-order harmonic approximation and the generalized nonlinear eigensystem is solved by a direct iterative procedure. A backbone curve and corresponding mode shapes are obtained and analyzed. The nonlinear finite element program is developed based on earlier linear edition and can be utilized to predict nonlinear characteristics of miniaturized quartz crystal resonators in the design process.