A two-dimensional theory for high-frequency vibrations of piezoelectric crystal plates with or without electrodes

Abstract

Two-dimensional equations of motion of successively higher-order approximations for piezoelectric crystal plates with triclinic symmetry are deduced from the three-dimensional equations of linear piezoelectricity by expansion in series of trigonometric functions of the thickness coordinate of the plate. These equations, complemented by two additional relations: one, the usual relation of face tractions to the mass of electrodes, and the other relating face charges to face potentials and face displacements, can accommodate either the traction and charge boundary conditions at the faces of the plate without electrodes or the traction and potential boundary conditions at the faces of the plate with electrodes. Dispersion curves are obtained from the first- to fourth-order approximate plate equations for a rotated 45° Y-cut lithium tantalate plate without electrodes, and these curves are compared with those from the frequency equation of the three-dimensional equations with close agreement. Solutions of forced vibrations of an AT-cut quartz plate with electrodes are obtained from the first-order plate equations complemented by the two additional relations. It is shown that cutoff frequencies of the fundamental thickness-shear vibrations from the approximate equations are identical to the ‘‘exact’’ ones obtained from the three-dimensional equations.

Higher-Order Piezoelectric Plate Theory Derived from a Three-Dimensional Variational Principle

“…The boundary conditions at the edges are applied in an average sense as in other plate theories. Higher-order plate theories have also been formulated by Mindlin and Medick, 10 Vlasov, 11 Lo et al, 12 Kant, 13 Reddy, 14 Hanna and Leissa, 15 Lee and Yu, 16 and Lee et al 17 ; this list is by no means complete. Exceptions to the usual expansions of the mechanical displacements and the electric potential as a power series in the thickness coordinate are the works of Soldatos and Watson, 9 Mindlin and Medick, 10 and Lee et al 17 Soldatos and Watson 9 use exponential functions, Mindlin and Medick 10 use Legendre polynomials, and…”

Analysis of a monolithic crystal plate acoustic wave filter

“…In most cases approximate two-dimensional (2D) plate equations [5][6][7] which are much simpler than the three-dimensional equations of elasticity or piezoelectricity are used. There exist systematic theoretical results from the 2D plate equations for TSh modes with in-plane variation in one direction only, i.e., in the direction parallel to the TSh particle velocity.…”

Thickness-shear vibration of a rectangular quartz plate with partial electrodes

“…Equation (19) satisfies Eqs. (3) and (6). To apply the boundary conditions at the plate top and bottom in Eq.…”

“…Two approaches are often used. One is to develop approximate, twodimensional (2-D) plate equations [4][5][6] to simplify the problems so that theoretical analyses are possible. The other is to use numerical techniques like the finite element method.…”