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.