A New Three-Dimensional Model for Circular Piezoelectric Transducers

Abstract

Piezoelectric ceramic disks are usually employed as active components in ultrasonic transducers for acoustical imaging and not destructive testing. Thickness mode is mainly need for all the applications in which high frequencies are required. For inspection of highly scattering and absorbing materials, however, lower frequencies must be used. To this end two solutions can be adopted: the Langevin composite transducer or a single disk working on its radial mode. In fact, with this vibration mode a significant strain also occurs in the transverse direction due to the elastic coupling. In this work, we develop a new three-dimensional model which takes both radial and thickness displacements into account; it can be used to optimize the geometry of the disk in order to maximize the thickness displacement of the radial mode. The vibration of a piezoceramic disk is described, in terms of cylindrical coordinates, by a system of two coupled differential wave equations with coupled boundary conditions. The solution of this problem is very difficult and approximation methods are usually required. An approximate solution was obtained by Brissaud [1], who proposed an approximated 3-D model based on the assumption that the displacements along the radial and thickness directions are dependent only on the related coordinate. In this model the stress free boundary conditions are satisfied only in an approximate way. Our approach also consists in considering two orthogonal wave functions as solutions of the differential equations, each depending only on the corresponding axis but, for the thickness mode, we choose a more general solution which permits to consider different transmission media for each external surface. As far as the boundaries are concerned, we apply integral conditions, obtaining an approximate model of the external behavior of the disk. Following this approximation we compute modified elastic and elastoelectric constants for the material. With this approach we model the plate in the frequency domain as a four ports system with one electric and three mechanical ports, one for each external surface.