The Analysis Of High Frequency Vibrations Of Doubly-rotated Quartz Crystal Plates

In the high frequency vibration analysis of quartz crystal plates, higher-order plate theories,include Mindlin plate theory based on the power series expansion and Lee plate theory based onthe trigonometric series expansion of the plate thickness coordinate, have been utilized in bothanalytical method with straight-crested waves and finite element method. Although theMindlin plate theory has been a standard choice in the analysis of high frequency vibrations ofquartz crystal resonators, the required modification through a set of correction factors has beenknown and efforts has been made to add more for improved accuracy in cut-off frequencies ofhigher-order modes. To have an accurate plate theory without correction factors, Lee hasimproved the earlier trigonometric expansion scheme by including a set of extra terms related tothe gradient of flexural mode. The modified Lee plate theory has been applied to the analysisof high frequency vibrations of crystal plates with good accuracy.Earlier work has been concentrated on the AT-cut of quartz crystal plates for many reasonssuch as the wide production of AT-cut type resonators. From studies on the higher-orderMindlin plate equations for vibrations of overtone modes of AT-cut quartz crystal plates, asystematic procedure on the correction and truncation of the Mindlin plate equations have beenestablished and correction factors up to the fifth-order have been obtained. We found that thehigher-order equations with correction are also applicable to the analysis of thickness-shearvibrations of SC-cut quartz crystal plates with accuracy.We start from the three-dimensional equations of linear piezoelectricity, the accuratedispersion relations of high frequency vibrations of doubly-rotated quartz crystal plates areobtained. Then, we start from the first-order equations of plate vibrations with all coupledvibration modes for a finite SC-cut quartz crystal rectangular plate, for the dispersion relations,frequency spectra relations and selected vibration modes. By considing the forced vibrationswith an alternating voltage applied to the electrode faces of the SC-cut quartz crystal rectangularplate, we obtained the capacitance ratio of forced vibrations of SC-cut plates. Finally, weanalyzed the frequency-temperature behavior of free vibrations of SC-cut quartz crystal plateswith the incremental thermal field theory which was proposed by Lee and Yong.