Forced vibrations of piezoelectric crystal plates

Abstract

1. Introduction. In a previous paper [1] (referred to in the sequel as I), Cauchy's two-dimensional equations of coupled flexural and extensional motion of crystal plates were extended to the next higher order of approximation so as to accommodate the two lowest thickness-shear modes. In the present paper a further extension is made to include the piezoelectric relations and the electric field equations. The equations obtained are also extensions of previous equations [2] in which the thickness-shear modes, the piezoelectric effect and the electric field equations were taken into account, but coupling with extensional modes was omitted. The new equations are deduced from the three-dimensional, linear, piezoelectric equations by a procedure based, as in 7, on the series expansion methods of Cauchy and Poisson and the variational method of Kirchhoff. The theorems of uniqueness and orthogonality established in I for the approximate equations are extended to include coupling with the electric field. The solution of problems of steady, forced vibration is considered and is reduced to the solution of the associated free vibration problem plus some quadratures. The case of a plate driven by an ac-voltage applied to electrodes on its faces is investigated and a formula is given for the total surface charge. An application is made to the forced vibrations of rectangular, rotated F-cut quartz plates. 2. Three-dimensional equations. The three-dimensional equations, from which the plate equations will be deduced, are: the variational forms of the equations of motion and electrostatics; the strain-displacement and electric field-potential relations; and the linear, piezoelectric constitutive relations. They may be written, respectively, as