Mindlin plate equations forthe thickness-shear vibrations of circular elastic plates

The systematic derivation of Mindlin plate equations has been fully presented in the monograph of Mindlin and some of his papers. Most applications followed are focused on the straight-crested wave in rectangular quartz crystal plates due to the necessity in the design of rectangular type resonators and relative simplicity of analysis. Some applications concerning circulate plates vibrating at the thickness-shear mode or the shear effects are analyzed by transforming the relatively simple equations of the thickness-shear and flexural modes to the cylindrical coordinates with a simple procedure, as we can find from a few papers and books. The systematic derivation by following a rigorous procedure of Mindlin plate equations are not presented before, and subsequent applications to coupled vibrations of circular plates at thickness-shear modes have not be studied with the Mindlin plate equations for circular type quartz crystal resonators. Particularly, vibrations of overtone modes with more coupled displacements require a systematic derivation which is more clear with the cylindrical coordinates.