Extensional vibrations of piezoelectric crystal plates and strips

A system of approximate second-order governing equations are extracted from an infinite system of 2-D equations for piezoelectric plates with general symmetry and charge-free faces which is deduced from the 3-D equations of linear piezoelectricity using a similar approach as that in Lee, Yu and Lin. These equations improve the accuracy of calculated dispersion curves as compared with those from the 3-D equations because a new truncation procedure has been employed, and they are applicable for frequencies up to and including the cutoff frequencies of the first symmetric thickness-stretch and the second symmetric thickness-shear modes. Furthermore, a system of 1-D second-order equations for the vibrations of piezoelectric strips with thin rectangular cross-section is deduced by averaging the 2-D equations over the narrow width of the plate. Closed form solutions of these 1-D equations are obtained for the second-order extensional vibrations of a barium titanate strip with finite length and narrow rectangular cross-section. Predicted frequency spectra are compared with the experimental data by Onoe and Pao and calculated results by Medick and Pao with good agreement. The present second-order theory for plates and strips is an improvement in that it contains no correction factors, includes the piezoelectric effect and is applicable for any crystal symmetry