The correction factors of mindlin plate theory with and without electrodes for SC-cut quartz crystal plates

Mindlin plate theory was first developed to provide accurate solutions for vibrations of thickness-shear mode, which has a much higher frequency than usual flexure vibrations. It has been widely used in the analysis of high frequency vibrations of quartz crystal plates, which are the core of resonators. The vibration frequency solutions obtained with Mindlin plate theory are proven much closer to the exact solutions. However, due to the truncation and approximation, the plate equations need to be corrected, as compared with the three-dimensional elasticity solutions. This has been done for the high-order Mindlin plate theory with and without electrodes for the AT-cut quartz crystal plates, and correction factors have been obtained though both natural and symmetric procedures. The correction factors could be used in the dispersion relationship and frequency spectrum in the analytical solutions, while the symmetric correction factors can be used in the finite element method implementation. Both correction schemes can provide improved and accurate results in the analysis of quartz crystal resonators. The electrodes are considered though its inertia effect as mass ratio known in resonator analysis.