This problem is identical to the one discussed in a report by Mercer et al. (1987). The structure is shown in Figure 6.1.1–1 and consists of a piezoelectric material PZT4 with brass end caps. The piezoelectric material is electroded on both the inner and outer surfaces.

The properties for PZT4 in a cylindrical system are:

Elasticity Matrix:

Piezoelectric Coupling Matrix (Stress Coefficients):

Dielectric Matrix:

The brass is elastic and isotropic with a Young's modulus of 104 GPa and a Poisson's ratio of 0.37.

The transducer is modeled with a variety of elements. It is modeled as an axisymmetric structure utilizing both the planar, axisymmetric elements and the three-dimensional elements. For the axisymmetric elements, five meshes employing 4-node, 6-node, and 8-node elements are used in the finite element discretization. The first two meshes use 4-node elements with two levels of refinement, the third mesh uses 6-node elements, and the last two meshes use the 8-node elements with two levels of refinement. Lumped mass matrices are used for the lower-order elements. Consistent mass matrices are used in the higher-order elements. The meshes used for the 4-node and 6-node axisymmetric elements are shown in Figure 6.1.1–2. The meshes used for the 8-node axisymmetric elements are shown in Figure 6.1.1–3.

The three-dimensional model uses a slice of the structure and applies axisymmetric boundary conditions; 8-node and 20-node brick elements are used. The discretization used for each model is shown in Figure 6.1.1–3. These models use the *ORIENTATION option to maintain the proper definitions of the material properties. Also, in order to prescribe the axisymmetric boundary conditions, the nodal degrees of freedom are transformed into a cylindrical coordinate system.

All the models are considered to be open-circuited. The potentials on the inside surface are restrained to zero. The frequencies correspond to those for anti-resonance.