Extensional, Flexural, and Width-Shear Vibrations of Thin Rectangular Crystal Plates

Abstract

An approximate one-dimensional theory is derived for thin crystal plates or bars with narrow rectangular cross section, and for cases in which the stiffness matrix of the crystal cij referring to the plate axes exhibits monoclinic symmetry. A set of five equations of motion is separated into two groups: the extensional, width-length flexural, and width-shear motion; and the transverse shear and width-twist motion. Closed form solutions of coupled extensional, flexural, and width-shear vibration are obtained for thin rectangular plates with free edges. Calculated resonant frequencies as a function of length-to-width ratio of the plate are compared with detailed measurements by Spears on +5° X-cut quartz plates. For the use in design, explicit algebraic formulas for predicting extensional and flexural frequencies are obtained in terms of elastic stiffnesses, density, and width-to-length ratio of the plates.