Thickness stretch vibrations of piezoelectric ceramic plates for resonator applications

The thickness stretch vibrations of piezoelectric ceramic plates are analyzed by solving the first-order Mindlin plate equations with finite element method in the two-dimensional domain. The precise resonance frequency and distribution of displacements are obtained from the analysis in detail. The results based on the two-dimensional solutions are more important particularly in the evaluation of the energy-trapping feature of ceramic resonators because the accurate mapping of the displacements including the vital thickness stretch mode is of great practical interests. We start from the first-order Mindlin plate theory for PZT type ceramic plates for resonator applications. The cut-off frequency of the thickness stretch vibrations is obtained from the coupled equations. Then these equations are reformatted with the known fundamental resonance frequency and related elastic constants for finite element solutions. With given geometry of a resonator model, the numerical solutions include the resonance frequencies and associated mode shapes are calculated. The thickness stretch vibrations and the associated frequency are of great importance because these results can be directly used for the determination of the frequency of a resonator and the optimal layout of the electrodes for best performance. All these analyses are intended for the direct applications in the design of ceramic resonators. Further considerations of the effects of electrodes, support structures, and other complications can be readily included in current finite element analysis.