Correction factors of the Mindlin plate equations for vibrations of SC-cut quartz crystal plates

The Mindlin plate equations have been the primary choice for the analysis of high frequency vibrations of AT- and SC-cut quartz crystal plates in the design of resonators for predictions of vibration frequency, mode couplings, and bias effects. However, the plate equations have to be modified to yield exact solutions in certain frequency range and correction factors are chosen in a systematic manner with validation from three-dimensional solutions. We have obtained correction factors up to the fifth-order plate equations in terms of electrode mass ratio as polynomials up to cubic terms through curve fitting for SC-cut quartz crystal. For partially electroded quartz plates, capacitance ratios have also been compared with that from the Lee plate equations in good agreement. We conclude that the procedure have been established to take the advantage of simplicity and elegance of the corrected Mindlin plate equations for analytical and numerical solutions.