Thickness-shear and flexural vibrations of contoured crystal strip resonators

A system of two-dimensional equations of motion of successively higher-order approximations for contoured crystal plates are deduced from the three-dimensional equations of elasticity by expansion of displacements in a series of trigonometrical functions of the thickness coordinate of the plates. By removing the first-order thickness-stretch mode and the first-order x(2)x(3) (or fast) thickness-shear mode and all the higher ones, a set of first-order equations of motion is obtained for contoured crystal plates and for frequencies up to and including those of the fundamental x(1)x(2) (or slow) thickness-shear mode. The coupled thickness-shear and flexural vibrations are studied for contoured quartz strip resonators having the shape of (a) a double wedge and (b) a plate with beveled edges. Exact solutions in terms of infinite power series are obtained by Frobenius method for the plates with linearly varying thickness in the x(1) direction. Frequency spectra and mode shapes are computed for both types of resonators. The effects of the contour on the frequencies and modes are examined. (C) 1996 American Institute of Physics.