Forced vibrations of SC-cut quartz crystal rectangular plates with partial electrodes by the Lee plate equations

Lee plate equations for high frequency vibrations of piezoelectric plates have been established and perfected over decades with the sole objective of obtaining accurate predictions of frequency and mode shapes to aid the analysis and design of quartz crystal resonators. The latest improvement includes extra terms related to derivatives of the flexural displacement to provide much accurate solutions for vibrations of the thickness-shear mode, which is the functioning mode of resonators and has much higher frequency than the flexural mode. The improved Lee plate equations have been used in the analysis of high frequency vibrations of quartz crystal plates as an essential step for analysis of AT- and SC-cut quartz crystal resonators after validations with fully electrode quartz crystal piezoelectric plates. In this study, closed-form solutions of free and forced vibrations of SC-cut quartz plates with partial electrodes are obtained. A procedure has been established for the calculation of dispersion relations, frequency spectra, selected vibration modes, and capacitance ratios of forced vibrations. The vibration solutions obtained with the first-order Lee plate equations are proven to be close to solutions from the Mindlin plate equations. It is now clear that both the Mindlin and Lee plate equations can be used in the analysis and design of quartz crystal resonators.