Correction factors for mindlin plate equations with the consideration of stiffness and mass effects of electodes for SC-cut quartz crystal plates

Mindlin plate theory has been playing a vital role in the analysis of high frequency vibration of quartz crystal resonators, since the coupled equations with infinite number of vibration modes have been truncated to retain essential ones with strong couplings for simplified analysis, and correction factors have been introduced to ensure exact solutions at cut-off frequencies as part of the validation procedure. Corrections must be made at the cut-off frequency with stiffness and mass effects of electrodes considered in the plate equations, but correction factors are different between the AT- and SC-cut quartz crystal. The systematic study of correction factors is readily available to the analytical equations with selected modes of strong couplings, and equally convenient for the implementation of the finite element analysis with the Mindlin plate equations.