## AB. Special Case of a Tangentially Polarized Piezoelectric Segmented Ring

The objective is to model a tangentially polarized piezoelectric segmented ring (Fig. 1) with a two-dimensional mesh such as wouldbe used if the structure were axisymmetric. This objective corresponds to theneed to reduce the problem size when dealing with radiation problems. Thesuggested method of solution is as follows:

- Build the two-dimensional mesh of an equivalent axisymmetric ringof the same geometric dimensions, with radial poling
- Modify the elastic, piezoelectric and dielectric tensors to takeinto account the polarization direction change. After transformation the initial tensor:
  becomes:
- Modify the displacement and pressure magnitudes computed in a harmonic analysis to take into account the electrical field amplitudedifference due to the difference between the thickness of a segment and the thickness of the ring. The correct values (subscript 3D) of the displacement uand of the pressure p are obtained from the value computed with theaxisymmetrical model (subscript 2D) by:
  where d is the stave thickness and e is the ring thickness.
- Modify the electrical impedance values in the harmonic analysisto take the same effect into account. The correct value (subscript 3D) of theimpedance Z is obtained from the value computed with the axisymmetrical model (subscript 2D) by:

Notes

Since the potential degree-of-freedom is actually a scalar,the difference between radial and tangential poling is simply a mechanicaltransformation of ratio d/e.

This way of dealing with a tangentially polarized ring canbe used only if the displacement field of the real structure is nearlyaxisymmetrical. It is, for example, impossible to obtain circumferential modesusing this method.

Fig. 1: Segmented free-flooded ring