Second-order analysis of free vibrations of piezoelectric ceramic actuators

In an earlier paper [1998], Lee and Yu derived a system of two-dimensional (2-D) equations for electroded piezoelectric plates with thickness-graded material properties. A special series expansion for the electric potential was employed to accommodate the short circuit face condition at the electroded area. Closed form solutions for the flexural and thickness-shear vibrations of circular bimorph disks were obtained later based on the first-order 2-D theory. In this work, we reduce Lee and Yu's 2-D equations for the case of homogeneous piezoelectric ceramic plates with electrodes. A set of second-order 2-D equations is derived after disregarding the higher order terms and after the back substitution. Closed form solutions are then obtained for the extensional, edge, symmetric thickness-shear, and thickness-stretch vibrations of circular disks. Frequency spectra and modes are calculated, identified, and studied for a range of diameter-to-thickness ratios. Resonance frequencies of PZT-5 disks calculated from our solutions are compared with the predictions of finite element analysis and experimental data by Guo et al. [1991]. The agreement is close.