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.

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A b s t r a c t Two-dimensional equations o f m o t i o n o f p i e z o e l e ct r i c c r y s t a l p l a t e s , d e r i v e d p r e v i o u s l y b y S y n q e l l a k i s a n d L e e , f r o m t h e t h r e e -d i m e n s i o n a l e q u a t i o n s o f t h e l i n e a r p i e z o e l e c t r i c i t y b y e x p a n s i o n i n t r i g o n o m e t r i c s e r i e s o f t h e t h i c k n e s s c o o r d i n a t e o f t h e p l a t e , a r e employed t o s t u d y t h e v i b r a t i o n s d r i v e n b y t h e l a t e r a l e l e c t r i c f i e l d a p p l i e d t o a p a i r o f e l e c t r o d e -p l a t e d and t r a c t i o n -f r e e edges o f d o u b l y -r o t a t e d q u a r t z s t r i p . S t r a i g h t -c r e s t e d v i b r a t i o n a l s o l u t i o n s i n c l o s e d f o r m a r e o b t a i n e d f o r t h e f i r s t e i g h t c o u p l e d e q u a t i o n s w h i c h c o n s t i t u t e t h e t w o -d i m e n s i o n a l , f i r s t -o r d e r a pp r o x i m a t e e q u a t i o n s o f m o t i o n o f p i e z o e l e c t r i c c r y s t a l plates. D i s p e r s i o n c u r v e s , f r e q u e n c y s p e c t r u m f o r e l e c t r ic a l l y s h o r t e d and t r a c t i o n -f r e e edge conditions, and mode shapes a t d i f f e r e n t resonances and f o r v a r i o u s w i d t h -t o -t h i c k n e s s r a t i o s o f t h e p l a t e a r e c o m p u t e d and p l o t t e d f o r SC-ctlt q u a r t z s t r i p s . I n t r o d u c t i o n a t o e t a1 [3]. All these studies were based on the s o l u t i o n o f t h e t h r e e -d i m e n s i o n a l e q u a t i o n s o f p i e z oe l e c t r i c i t y . Some o f t h e r e a s o n s f o r t h e r e c e n t s t u d y and i n t e r e s t i n t h e l a t e r a l e x c i t a t i o n a n d a comprehens i v e l i s t o f r e f e r e n c e s o n LE a r e g i v e n i n [S].F o r c e d v i b r a t i o n s o f p i e z o e l e c t r i c c r y s t a l p l a t e s with f i n i t e w i d t h ( o r l e n t h ) were studied by Tiersten and M i n d l i n [4], M i n d l i n !S], [6]. I n t h e s e s t u d i e s , two-dimensional plate equations based on power series expansion were employed t o i n v e s t i g a t e t h i c k n e s s -f i e l d e x c i t a t i o n . I n t h e p r e s e n t p a p e r , t h e t w o -d i m e n s i o n a l p l a t e equations based on trigonometric series expansion by S y n q e l l a k i s and Lee 17J are employed t o s t u d y t h e l a t e r a l -f i e l d e x c i t a t i o n o f d o u b l y -r o t a t e d q u a r t z s t r i p . G e n e r a l s o l u t i o n i n c l o s e d f o r m f o r t h e e i g h t c o u p l e d e q u a t i o n s o f m o t i o n i s o b t a i n e d a s t h e sum o f t h e s o l u t i o n s o f homogeneous equations and the solution o f t h e inhomogeneous equations. The e l e c t r i c a l l y f o r c e d and t r a c t i o n -f r e e edge c o n d i t i o n s a r e s a t i s f i e d . Two-Dimensional Plate Equations Consider a d o u b l y -r o t a t e d c r y s t a l p l a t e , r e f e r r e d t o a r e ...