The Fifth-order overtone vibrations of crystal plates with corrected higher-order Mindlin plate equations

An analysis of overtone quartz crystal resonators is required as part of the design process which has been turning out products of higher-order overtone for many years. It has been known that current analysis is generally oversimplified and the trial-and-error approach has been the only choice by engineers as the practical analytical method and tools are not available, but we found that the Mindlin plate equations in the design and analysis of the fundamental thickness-shear type of resonators can be improved and utilized. Through extensive improvements of the Mindlin plate equations, we can now analyze vibrations for mode couplings, electrode effect, optimal sizes, and thermal behavior, among others. Since it has been proven that the Mindlin plate equations can be used for the vibration analysis of plates at the higher-order overtone modes with accurate prediction of frequency and dispersion relations in the vicinity of cut-off frequencies, we have extended the equations to the third-order for the modal behavior and frequency spectra. The results show that earlier knowledge on the proper selection of the sizes of electrode can be theoretically proven from our analysis. In addition, the spatial variation and end effects of displacements, particularly of the working mode, can be used in the optimal selection of a quartz crystal blank. The design changes can be used as a way to improve the resonator performance, which has been increasingly degenerating for higher-order overtone types, to meet more stringent requirements. We now extend the Mindlin plate equations with latest corrections to the fifth-order so the design principle and guidelines can be summarized from analytical results of vibration analysis. As a research objective for years, we are getting improved frequency solutions as expected after extensive improvements and corrections of the Mindlin plate equations, and the frequency spectra and vibration modes will be compared with known fundamental and third-order solutions to extract needed design guidelines.