Frequency-Temperature Behavior of Thickness Vibrations of Doubly-Rotated Quartz Plates Affected by Plate Dimensions and Orientations

Abstract

Three‐dimensional linear equations of motion for small vibrations superposed on thermal deformations induced by steady, uniform temperature change in quartz are obtained. The material properties of quartz, such as the elastic stiffnesses and thermal expansion coefficients, are assumed temperature dependent and expressible by third‐degree polynomials in temperature change. From the solutions of third‐order perturbations of these equations for the thickness resonances of infinite quartz plates, six values of the effective third temperature derivatives of elastic stiffnesses C(3)pq are calculated by the use of the measured temperature coefficients of frequency by Bechmann, Ballato, and Lukaszek [Proc. IRE 50, 1812 (1962)] for various doubly rotated cuts and the values of the first temperature derivatives C(1)pq and the effective second temperature derivatives C(2)pq obtained in a previous study. An infinite system of two‐dimensional equations of motion is derived by Mindlin’s method of power‐series expansi...