7.5. A Ceramic Ring

In this section we analyze vibrations of a rotating ceramic ring poled in the z direction as shown in Fig. 7.5.1 [82].

Fig. 7.5.1: A ceramic ring as a gyroscope.

One possible arrangement of eight pairs of identical electrodes at z= c are shown in the areas bounded by the thick lines which can be used to excite and detect a particular pair of modes in Fig. 7.5.2. Equivalently, radially polarized ceramics can also be used which will require electrodes to be placed at ?= a.

Fig. 7.5.2: Radial displacements of the primary and secondary modes of a non-rotating ring.

7.5.1. Governing equations

In Fig. 7.5.1, V ₁ is the input voltage across four pairs of driving electrodes and V ₂ is the output voltage across the other four pairs of electrodes. We use one-dimensional equations for a piezoelectric ring [20]. The one-dimensional electric potentials are

where

and are complex constants. We have assumed harmonic time dependence and introduced the complex notation. Since the electrodes are equally spaced and that the electric potential is a constant on an electrode, the voltage distribution due to the electrode configuration in Fig. 7.5.1 is essentially piecewise constant. In Eq. (7.5.2), this piecewise constant voltage distribution is approximated by a trigonometric function. generates