Correction factors for the Mindlin plate equations for thickness vibrations of crystal plates with thicker electrodes

The Mindlin plate theory has been widely used for the study of high frequency vibrations of quartz crystal plates for resonator design and analysis. To improve the accuracy, the plate equations have to be corrected for the exact thickness-shear frequencies which in turn ensure the accurate results for other coupled modes. The correction procedure was suggested by Mindlin and the correction factors for the firstorder equations have been available. Lately, a systematic procedure for the correction factors up to the fifth-order has been established and two correction schemes with different correction factors have been adopted for the higher-order vibrations. These results have expanded applications of the Mindlin plate equations to the overtone vibration analysis which is required for the overtone resonators. One critical issue concerning the applications of the Mindlin plate with correction factors are the effect of electrodes, which are essential part of resonators and complications are resulted. With this objective in mind, we start from the Mindlin plate equations with a full set of vibration modes for the consideration of electrode mass and stiffness effects as a continuation of our earlier studies which only considered the mass effect in terms of mass ratios. The analysis regarding to the AT-cut of quartz crystal plates show that the consideration of electrode stiffness and mass will change the correction factors, thus making the equations more accurate in the calculation of frequency spectra and mode shapes. For very thin electrodes, the correction factors should be the same as suggested by earlier studies. The procedure is implemented to the SC-cut quartz crystal plates which have more enhanced couplings of vibration modes due to the material anisotropy. Correction factors for both AT- and SC-cut quartz crystal plates with consideration of electrode stiffness are given in polynomials of mass ratios for different metals.

I. Introduction

The Mindlin plate theory was developed for analysis thickness-shear vibrations of quartz crystal resonators without considering complication factors in the beginning [1], [2]. Since electrodes are important parts of quartz crystal resonators with great influence on the frequency and other properties, they should not be neglected when analyzing vibrations for practical applications. Mass effect of electrodes as the dominant feature has always been taken into consideration through mass ratio [3], [4], [5], [6], [7], [8], because thicknesses of electrodes are much thinner but heavier compared with crystal plates. While crystal blanks in resonators are shrinking in size and the electrodes remaining constant, the relative increase of influence of electrodes on the performance of plate resonators should be considered. In this case, it is natural that both stiffness and mass effects of electrodes are taken into consideration simultaneously. The stiffness and mass effects of electrodes can be considered through the ratio of thickness, density, and elastic constants of electrodes to those of quartz crystal plates, as demonstrated in an earlier study [9].