Vibrations of AT-cut quartz strips of narrow width and finite length

A system of one-dimensional equations of motion for AT-cut quartz strip resonators with narrow width and for frequencies up to and including the fundamental thickness-shear is deduced from the two dimensional, first order equations for piezoelectric crystal plates. This is accomplished by expanding the mechanical displacements and electric potentials in a series of trigonometric functions of the width coordinate. Four groups of coupled equations of motion are obtained. For the equations of each group, closed form solutions are obtained and the traction-free conditions at four edges are accommodated. Dispersion curves and the resonance frequency as a function of the length-to-thickness ratio or as a function of the width-to-thickness ratio of the quartz strips are computed. Predicted results are compared with experimental data