The analysis of the third-order thickness-shear overtone vibrations of quartz crystal plates with Mindlin plate theory

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

“…In practical applications, these equations have to be truncated to a set of lower-order equations for solutions. Details on the truncation and simplification of these equations can be found in earlier studies [9][10][11].…”

“…Finally, we can study the effects of driving voltage on electrodes of a resonator so complete solutions of vibrations can be obtained for the calculation of electrical properties. This has been a standard procedure practiced in the analysis of AT-cut quartz crystal resonators in the development of theoretical foundation of the analytical technique and design method [9][10][11][37][38][39][40][41][42]. Equations utilized in this procedure have been accepted and integrated to product development process with different methods and various implementations.…”

“…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].…”

“…From our studies on the higher-order Mindlin plate equations for vibrations of overtone modes of AT-cut quartz crystal plates [9][10][11], a complete procedure on the correction and truncation of the Mindlin plate equations has been established and correction factors up to the fifth-order have been obtained for both analytical and FEM procedures [15][16][17]. 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].…”