Trapped thickness-shear modes in a contoured, partially electroded AT-cut quartz resonator

Abstract

We perform a theoretical analysis of a contoured crystal plate resonator with nonuniform thickness. The resonator is made from AT-cut quartz and is partially electroded in the central region. Based on the variational formulation established in a previous paper and the Ritz method with trigonometric functions as basis functions, free vibration resonant frequencies and thickness-shear modes trapped in the central electroded region are obtained. The effect of the curvature of the contour is examined. It is also found that the classical frequency prediction given by Tiersten et al. in 1996 from an approximate analysis has an inaccuracy of the order of 40 parts per million for the fundamental mode, significant in resonator design and application.