Vibrations of Doubly-Rotated Piezoelectric Crystal Strip with a Pair of Electrode-Plated, Traction-Free Edges

Abstract

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 ...

An analysis of thickness-shear vibrations of doubly-rotated quartz crystal plates with the corrected first-order mindlin plate equations

“…In addition, the corrected equations can also be implemented with the finite element method based on the mindlin plate theory for more general analysis of sc-cut quartz crystal resonators. The results from this approach have been verified by known results reported by lee and collaborators with the lee plate theory as the needed validation [9]- [12]. This study expands applications of the mindlin plate theory to the analysis of sc-cut quartz crystal plates through necessary correction and modification with the method and procedure we are familiar with.…”

“…all of these efforts have established a relatively adequate procedure for the analysis and design of aT-cut quartz crystal resonators using the mindlin plate theory. of course, initiatives such as the lee plate theory, based on the trigonometric expansion of mechanical displacements and electrical potential, should also be considered as improvements because, as we shall explain later, the correction factors accompanying the mindlin plate equations are eliminated with a trade-off of relative elegance and simplicity [9]- [12]. In summary, the analysis of aT-cut quartz crystal resonators can be performed with the mindlin plate equations based on decades of study for their improvement and perfection, and it is an important part of the design procedure.…”

“…research efforts have been concentrated on the analysis of aT-cut quartz crystal resonators partly because of the many demands in oscillator and sensor applications, but also because of the relatively simple equations and couplings. as an attempt to improve the complications of correction factors related to the exact cut-off frequencies of the mindlin plate equations, the lee plate theory was introduced, based on the trigonometric expansions, with improved results without correction factors [9]- [12]. The latest work by lee and collaborators has proven that the high-frequency vibrations of sc-cut plates can be adequately analyzed with the first-order equations, demonstrating that the standard procedure for the analysis of aT-cut quartz crystal resonators can be also adopted for sc-cut types.…”

“…later, threedimensional, approximate, and numerical methods were used in the analysis of novel applications such as sensors [18]- [21]. as an alternative, studies on sc-cut crystal plate vibrations close to this objective were done with the lee plate equations by lee and colleagues to avoid the complicated correction scheme required [9]- [12]. From our studies on the higher-order mindlin plate equations for vibrations of overtone modes of aT-cut quartz crystal plates, a systematic procedure for the correction and truncation of the mindlin plate equations has been established and correction factors up to the fifth-order have been obtained [22]- [24].…”

Free and forced vibrations of SC-cut quartz crystal rectangular plates with the first-order Mindlin plate equations

“…Of course, the Lee plate theory based on the trigonometric expansion is similar in nature and also offers equally accurate results of analysis as an alternative approach [24][25][26][27][28][29][30][31][32]. Consequently, many efforts have been made on the correction, truncation, and simplification of the Mindlin plate theory with extensive work on AT-cut quartz crystal plates with and without electrodes for resonator applications by Wang and collaborators [15][16][17][18][19].…”