Thickness Vibrations of Piezoelectric Plates

The thickness vibrations of an infinite anisotropic plate with electrodes coated on both surfaces are investigated using the linear piezoelectric equations. In general, the three fundamental solutions of the differential equations are found to couple at the traction-free surfaces of the plate. The solution reduces to the well-known uncoupled solution for the purely elastic case when the piezoelectric constants are zero. The transcendental equation which determines the resonant frequencies is derived. The theory is applied to a ferroelectric ceramic in two special cases: one with the plate polarized in the thickness direction, and the other with the plate polarized in its plane. In either of these special cases, the three solutions of the differential equations uncouple. Two of these solutions are identical with the purely elastic solutions and cannot be excited electrically, whereas the third differs from the purely elastic solution and can be excited electrically. The frequency obtained from this piezoelectric solution shows that the resonant frequencies are not integral multiples of the fundamental.