An approximated 3-D model of cylinder-shaped piezoceramic elements for transducer design

In this paper an approximated 3-D model of cylinder shaped piezoceramics is described. In the hypothesis of axial symmetry, the element vibration in the extensional and radial directions is described by two coupled differential wave equations. The model is obtained choosing, as solution of these equations, two orthogonal wave functions, each depending only on one axis, corresponding to the propagation direction. The mechanical boundary conditions are applied imposing continuity between the stresses and the external forces on the surfaces of the element in an integral way, while, as far as the electrical boundary condition is concerned, two possibilities are explored: to neglect the piezoelectric constant in the transverse direction and to impose an integral condition also for the electric field. Comparisons with experimental results show this last approach to give better results. The model predicts with sufficient accuracy only the first radial and the first thickness modes of the cylinder-shaped piezoceramic element of arbitrary aspect ratio; but, for these modes, it is able to compute all the relations between the input applied voltage and the output forces and velocities on every external surface. Because only these two modes are of relevance in the practical applications of piezoceramic elements as ultrasonic transducers, the model can be used as a simple and useful tool in transducer design and optimization. Experimental validations of the model are also shown in the work.