A theory of longitudinally polarised piezocomposite rod based on Mindlin-Herrmann model

Abstract

The conventional theory of the piezoelectric rod is based on an assumption that its lateral vibrations are negligible. In this case the rod, vibration could be described in terms of one-dimensional wave equation and a set of mechanical and electric boundary conditions. However, the main assumption of the model is only valid in the case of long and relatively thin rods. As a rule, the linear dimensions of piezoelectric components used in real transducers are comparable with the characteristic dimensions of their cross-sections, and hence, their lateral displacements have to be taken into consideration. In the present paper, the theory of the piezoelectric rod is developed on the basis of the Mindlin-Herrmann model. In the frame of this theory, the longitudinal and lateral displacements are described by two independent functions and vibration of the rod is obtained in terms of a system of two partial differential equations. The Hamilton variational principle is used for derivation of the system of equations of motion and for obtaining the mechanical and electric boundary conditions. On the basis of the formulated Mindlin-Herrmann model, the electric impedance of the piezoelectric rod is calculated. Possible generalizations of the proposed approach are considered and conclusions are formulated. An example of the application of the piezoelectric rod based on the Mindlin-Herrmann theory is given.