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 by Lee, Syngellakis, and Hou [J. Appl. Phys. 61, 1249 (1987)] by expanding the mechanical displacements and electric potentials in a series of trigonometric functions of the width coordinate. By neglecting the piezoelectric coupling and the weak mechanical coupling through c56, 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 frequency spectrum are computed for quartz strips. The predicted frequency as a function of the length-to-thickness ratio and as a function of the width-to-thickness ratio of the quartz strips is compared with experimental data with good agreement.