Equations for high-frequency vibrations of piezoelectric plates derived from a semi-mixed variational principle and applications in resonators

Abstract

We derive two-dimensional equations for high-frequency vibrations of piezoelectric plates from a semi-mixed vari- ational principle with the mechanical displacement vector, the electric potential, and the electric displacement vector as inde- pendent fields. This variational formulation allows independent expansions of the three fields into power series of the plate thickness coordinate. Similar to the shear correction factors used in Mindlin's equations for piezoelectric plates to adjust the thickness-shear frequencies predicted by the plate equations slightly so that they become the same as the exact frequencies from the three-dimensional equations, a charge correction factor is conveniently introduced in the plate equations in the present paper so that the inaccuracy of the plate equations in the literature in predicting the charge or current on the electrodes of an electroded plate is corrected. The final equations obtained are convenient and accurate to use for both electroded and unelectroded plates, and for calculating both the voltage and the charge on the electrodes at the plate surfaces in the case of an electroded plate.