Second-order theories for extensional vibrations of piezoelectric crystal plates and strips

An infinite system of two-dimensional (2-D) equations for piezoelectric plates with general symmetry and faces in contact with vacuum is derived from the 3-D equations of linear piezoelectricity in a manner similar to that of previous work, in which an infinite system of 2-D equations for plates with electroded faces was derived. By using a new truncation procedure, second-order equations for piezoelectric plates with faces in contact with either vacuums or electrodes are extracted from the aforementioned infinite systems of equations, respectively. The second-order equations for plates with or without electrodes are shown to predict accurate dispersion curves by comparing to the corresponding curves from the 3-D equations in a range up to the cut-off frequencies of the first symmetric thickness-stretch and the second symmetric thickness-shear modes without introducing any correction factors. Furthermore, a system of 1-D second-order equations for strips with rectangular cross section is deduced from the 2-D second-order equations by averaging variables across the narrow width of the plate. The present 1-D equations are used to study the extensional vibrations of barium titanate strips of finite length and narrow rectangular cross section. Predicted frequency spectra are compared with previously calculated results and experimental data.