Thickness shear, thickness twist, and flexural vibrations of rectangular AT-cut quartz plates with patch electrodes

Abstract

The influence of the size and the mass of patch (rectangular electrodes on the vibrations and energy trapping in a plate rectangular crystal plate is studied. Mindlin’s two-dimensional approximate equations of vibrations of crystal plates are employed for which the coupled modes of thickness shear, thickness twist, and flexure are retained. Numerical solutions that satisfy the free-edge conditions of the plate and the continuity conditions of stresses and displacements at the interface of the plated and unplated regions of plate are obtained by the finite element method using displacement formulation. Numerical computations are made for rectangular AT-cut quartz plates with symmetrically placd, patch electrodes. Resonance frequencies and their corresponding two-dimensional modes of vibrations are obtained for various dimensions of plate and electrodes, and for different masses of electrodes. Calculated results are compared with existing analytical results by Mindlin and Lee [Int. J. Solids Structures, 2, 125 (1966)], Lee and Spencer [J. Acoust. Soc. Am. 45, 637 (1969)], and Tiersten [J. Acoust. Soc. Am. 59, 879 (1976)], and experimental data by Curran and Koneval [J. Acoust. Soc. Am. 34, 981 (1967)]. A two-dimensional Bechmann’s number is obtrained which depends essentially on three parameters; the electrode length-to-plate thickness ratio, the electrode wide-to-plate thickness ratio, and the ratio of the mass of electrodes to the mass of the plate per unit area.