An analysis of thickness-shear vibrations of doubly-rotated quartz crystal plates with the corrected first-order mindlin plate equations

Abstract

The Mindlin plate equations have been widely used in the analysis of high-frequency vibrations of quartz crystal resonators with accurate solutions, as demonstrated by the design procedure based on analytical results in terms of frequency, mode shapes, and optimal parameters for the AT-cut quartz crystal plate, which is the core element in a resonator structure. Earlier studies have been focused on the AT-cut (which is one type of rotated Y-cut) quartz crystal plates because it is widely produced and has relatively simple couplings of vibration modes at thickness-shear frequencies of the fundamental and overtone modes. The simplified equations through the truncation, correction, and modification of the Mindlin plate equations have been widely accepted for practical applications, and further efforts to expand their applications to similar problems of other material types, such as doubly-rotated quartz crystals, with the SC-cut being a typical and popular one, are also naturally expected. We have found out that the Mindlin plate theory can be truncated and corrected for the SC-cut quartz crystal plates in a manner similar to the AT-cut plates. The analytical results show that the corrected Mindlin plate equations are equally accurate and convenient for obtaining essential design parameters of resonators for the thickness-shear vibrations of SC-cut quartz crystal plates.

Free and forced vibrations of SC-cut quartz crystal rectangular plates with the first-order Mindlin plate equations

“…The correction factors for the first-order Mindlin plate equations with the consideration of material constants of SC-cut quartz crystal are [12][13][14]16] j ð0Þ…”

“…From the beginning, it has been widely utilized in the vibration analysis of different kinds of piezoelectric plates with the AT-cut of quartz crystal as a typical example [8][9][10][11][12][13][14]. For couplings of flexural and thickness-shear modes in finite crystal plates, a correction of Mindlin plate theory through inserting correction factors for Y-and AT-cut quartz crystal plates were done by Mindlin and others [15][16][17][18][19], effectively extending the corrected plate theory to include the thickness-twist, face-extension, and face-shear modes.…”

“…For couplings of flexural and thickness-shear modes in finite crystal plates, a correction of Mindlin plate theory through inserting correction factors for Y-and AT-cut quartz crystal plates were done by Mindlin and others [15][16][17][18][19], effectively extending the corrected plate theory to include the thickness-twist, face-extension, and face-shear modes. The improved Mindlin plate theory is used for the analysis of crystal plates in the fundamental and overtone TSh vibrations including the consideration of effects of electrodes, temperature variation, and even the viscosity [9][10][11][12][13][14][20][21][22][23].…”

“…In this paper, the corrected Mindlin plate theory is used with the SC-cut quartz crystal plates of rectangular shapes for the analysis of thickness-shear vibrations under alternating driving voltage on electrodes. Because the SC-cut quartz crystal has enhanced material anisotropy, or more couplings of vibration modes, the analysis is generally more challenging in meeting the requirements of resonator design [12].…”

“…In the analytical process, we have also found that the higher-order equations with correction procedure and factors are applicable to the analysis of thickness-shear vibrations of SC-cut quartz crystal plate [12][13][14]. With the selected vibration modes and truncated equations, we can take the advantage of the known procedure as in the analysis of AT-cut quartz crystal plates.…”