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A New Theory for Electroded Piezoelectric Plates and Its Finite Element Application for the Forced Vibrations Analysis of Quartz Crystal Resonators
, 1998
"... For crystal resonators, it is always desirable to calculate the electric properties accurately for application purposes. As an extension of the Mindlin plate theory based finite element analysis of crystal resonators, a new theory for the electroded plates is derived and the piezoelectrically forced ..."
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Cited by 7 (7 self)
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For crystal resonators, it is always desirable to calculate the electric properties accurately for application purposes. As an extension of the Mindlin plate theory based finite element analysis of crystal resonators, a new theory for the electroded plates is derived and the piezoelectrically forced vibrations are formulated and implemented in this paper in a manner similar to our previous work. The effect of the electrodes and the electric boundary conditions are taken into considerations through the modification of the higherorder plate equations by changing the expansion function of the electric potential for this particular problem. Through the conventional discretization of the new plate theory, the linear equations for the piezoelectric plate under thickness excitation are constructed and solved with efficient numerical computation techniques such as the sparse matrix handling. Numerical examples showing good predictions of the resonance frequency and capacitance ratio of electr...
A fast analysis of vibrations of crystal plates for resonator design applications
 in Proc. IEEE Int. Freq. Contr. Symp. Exposition
"... ABSTRACT Mindlin plate equations are the foundation of quartz crystal resonator analysis, and tremendous efforts have been made to improve the equations and obtain efficient and accurate solutions in past decades. It is clear now that the simplified resonator model can usually be studied with Mindl ..."
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Cited by 1 (0 self)
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ABSTRACT Mindlin plate equations are the foundation of quartz crystal resonator analysis, and tremendous efforts have been made to improve the equations and obtain efficient and accurate solutions in past decades. It is clear now that the simplified resonator model can usually be studied with Mindlin equations of chosen order with only one spatial variable, and this has been used for both theoretical studies and practical design work. As an improvement, many work have been done with the finite element implementation of the equations, and better solutions that can consider the actual configuration and electrodes have been obtained with intensive numerical computation. Of course, these solutions are not widely utilized due to the stringent requirement on the computing resources and less familiarity of Mindlin plate equations in general. On the other hand, we have found that the advances of the finite element method and computer software have made the precise solutions of the equations more affordable technically and financially. In this paper, we implement the wellknown Mindlin firstorder equations in Femlab environment, and find that the useful solutions can be quickly obtained to examine the mode shapes that are important in the design of quartz crystal resonators. Such solutions are hard to obtain with traditional finite element method, thus offering a rare opportunity to use the complete solutions for product design, improvement, and optimization. We showed the applications of these equations to a simple ATcut quartz crystal strip resonator model. I.
Higherorder Plate Theory Based Finite Element Analysis of the Frequencytemperature Relations of Quartz Crystal Resonators
, 1998
"... The frequencytemperature characteristics of quartz crystal resonators, particularly the frequency stability in a specific temperature range in which the vibration modes could be strongly coupled, has been an important requirement in most applications. The analytical work on the frequencytemperatur ..."
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Cited by 1 (0 self)
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The frequencytemperature characteristics of quartz crystal resonators, particularly the frequency stability in a specific temperature range in which the vibration modes could be strongly coupled, has been an important requirement in most applications. The analytical work on the frequencytemperature relations has been done over last decades in many aspects, ranging from fundamental theory of the thermal effect to the simplified plate equations of a few strongly coupled vibration modes. However, it has been clearly observed that due to the complication of resonator structures, such as the presence of mounting structure and asymmetric electrodes, simple and analytical solutions will not be able to consider all the factors which will have inevitable and noticeable effects on the resonators. In this paper, we incorporate the frequencytemperature theory for crystal plates based on incremental thermal field theory by Lee and Yong into our finite element analysis program, which can analyze ...
Correspondence Consideration of Stiffness and Mass Effects of Relatively Thicker Electrodes with Mindlin Plate
"... Abstract—Mindlin plate theory has been widely used in the highfrequency vibrations of piezoelectric crystal plates with emphasis on its applications in crystal resonator analysis and design. The plate equations were derived without considering the effect of electrodes from the beginning. But contin ..."
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Abstract—Mindlin plate theory has been widely used in the highfrequency vibrations of piezoelectric crystal plates with emphasis on its applications in crystal resonator analysis and design. The plate equations were derived without considering the effect of electrodes from the beginning. But continuing efforts have been made to include the mechanical effect, or the mass loading, through the consideration of the mass ratio of the electrodes and crystal blank. Such a consideration has been effective for relatively thin electrodes before, but the everincreasing mass ratio has been pressing further improvement to take into account relatively thicker electrodes. To extend Mindlin plate equations for these applications, we derive the plate equations systematically with the approximation of displacements in electrodes with those in the crystal blank. As a result, both mass and stiffness effects of electrodes are considered through ratios of the thickness, density, and elastic constants of the electrodes to those of the crystal blank, respectively, and the plate equations are modified accordingly. A practical design of the electrodes and crystal blank are analyzed to demonstrate the necessity of such modifications to Mindlin plate equations. I.
ATwodimensional analysis of surface acoustic waves in finite plates with eigensolutions
 J. of Applied Electromagnetics and Mechanics
"... It is generally known that surface acoustic waves, or Rayleigh waves, have different mode shapes in infinite plates. To be precise, there are both exponentially decaying and growing components in plates appearing in pairs, representing symmetric and antisymmetric modes in a plate. As the plate thic ..."
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It is generally known that surface acoustic waves, or Rayleigh waves, have different mode shapes in infinite plates. To be precise, there are both exponentially decaying and growing components in plates appearing in pairs, representing symmetric and antisymmetric modes in a plate. As the plate thickness increases, the combined modes will approach to the Rayleigh mode in a semiinfinite solid, exhibiting surface acoustic wave deformation and velocity. As a result, for plates with finite thickness, we need to consider the effect of two modes in the analysis. In this study, the twodimensional theory for surface acoustic waves in finite plates is extended to include exponentially growing modes in the expansion function, creating a twodimensional equation system for plates with finite thickness. Since additional expansion functions are also exponential, the twodimensional equations keep the same appearance, implying the same evaluation and solution procedure. These results are important in the improvement of twodimensional analysis of surface acoustic waves in finite solids, which are the essential problem in surface acoustic wave resonator analysis and design.
A TwoDimensional Analysis Of Surface Acoustic Waves In
 J. of Applied Electromagnetics and Mechanics
"... INTRODUCTION The analysis of surface waves in elastic solids has been clearly demonstrated with semiinfinite solids by assuming displacements decay exponentially along the thickness coordinate, and solutions including the velocity are obtained by applying the tractionfree boundary conditions [1]. ..."
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INTRODUCTION The analysis of surface waves in elastic solids has been clearly demonstrated with semiinfinite solids by assuming displacements decay exponentially along the thickness coordinate, and solutions including the velocity are obtained by applying the tractionfree boundary conditions [1]. While surface waves found in some occasions like in earthquakes can be treated like travelling in infinite media, in other applications such as electronic devices utilizing surface acoustic waves (SAW) for frequency control have to be considered as in bounded elastic or piezoelectric solids requiring precise evaluation of complication factors like the periodic electrodes known as interdigital transducers. It is known that the wave propagation in bounded elastic solids is difficult to analyze because of the reflections in the boundaries produce many modes and their overtones, making both deformation and vibration frequency have many strong couplings. One typical method for the analysis of b
Jointly with the 17th European Frequency and Time Forum Thickness Stretch Vibrations of Piezoelectric Ceramic Plates for Resonator Applications
"... AbstractThe thickness stretch vibrations of piezoelectric ceramic plates are analyzed by solving the firstorder Mindlin plate equations with finite element method in the twodimensional domain. The precise resonance frequency and distribution of displacements are obtained from the analysis in detai ..."
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AbstractThe thickness stretch vibrations of piezoelectric ceramic plates are analyzed by solving the firstorder Mindlin plate equations with finite element method in the twodimensional domain. The precise resonance frequency and distribution of displacements are obtained from the analysis in detail. The results based on the twodimensional solutions are more important particularly in the evaluation of the energytrapping feature of ceramic resonators because the accurate mapping of the displacements including the vital thickness stretch mode is of great practical interests. We start from the firstorder Mindlin plate theory for PZT type ceramic plates for resonator applications. The cutoff frequency of the thickness stretch vibrations is obtained from the coupled equations. Then these equations are reformatted with the known fundamental resonance frequency and related elastic constants for finite element solutions. With given geometry of a resonator model, the numerical solutions include the resonance frequencies and associated mode shapes are calculated. The thickness stretch vibrations and the associated frequency are of great importance because these results can be directly used for the determination of the frequency of a resonator and the optimal layout of the electrodes for best performance. All these analyses are intended for the direct applications in the design of ceramic resonators. Further considerations of the effects of electrodes, support structures, and other complications can be readily included in current finite element analysis. I.
Modified Lee Plate Equations for the Vibration Analysis of Piezoelectric Plates with Consideration of Stiffness and Mass of Electrodes����
"... ABSTRACT Lee plate equations for high frequency vibrations of piezoelectric plates have been established and improved over the decades with the sole objective to obtain the accurate prediction of frequency and mode shapes to aid crystal resonator design. The latest improvement includes extra terms ..."
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ABSTRACT Lee plate equations for high frequency vibrations of piezoelectric plates have been established and improved over the decades with the sole objective to obtain the accurate prediction of frequency and mode shapes to aid crystal resonator design. The latest improvement includes extra terms related to derivatives of the flexural displacement to adjust the accuracy and for the consideration of the electrode for practical applications. As part of the efforts to make the equations more practical for resonator design with the improved of frequency accuracy and consideration of electrodes, we derived Lee plate equations for electroded plates by changing the integration limits in the dimension reduction procedure to signify the dominant role of the crystal plate. As a result, the equations are modified for the inclusion of the electrode effects. To improve the accuracy in the vicinity of thicknessshear vibration frequency of electroded plates, we modified the density terms in plate equations to reflect the contribution of both electrode stiffness and density, which makes the frequency more accurate for commonly used electrode materials. I.
1999 Joint Meeting EFTF IEEE IFCS A Layerwise Plate Theory for the Vibrations of Electroded Crystal Plates
"... Abstract Electrodes on a crystal resonator has been traditionally considered as mass addition to the crystal plate, thus resulting the neglect of their stiffness. This assumption is considered reasonable if electrodes are thin in comparison with the crystal in terms of the mass ratio, the relative ..."
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Abstract Electrodes on a crystal resonator has been traditionally considered as mass addition to the crystal plate, thus resulting the neglect of their stiffness. This assumption is considered reasonable if electrodes are thin in comparison with the crystal in terms of the mass ratio, the relative mass of electrodes. For thicker eiectrodes oi high frequency resonators, this assumption has to be reexamined for better prediction on their effects on I.