This work describes a matrix model of the radial mode of a thin piezoceramic ring capable of predicting the dynamic behavior when the two main surfaces are stress free, while the lateral, inner, and outer are loaded by an external medium. The ring is modeled as a three-port system with two mechanical ports and one electrical port. With this approach it is easy to compute the resonance frequency spectrum, the radial displacement, and the electric impedance of a thin ring. Good agreement between the computed and the measured electric impedance is found. The resonance frequency spectrum is computed as a function of the inner-to-outer radius ratio G: when the inner radius vanishes, the resonances of the ring coincide with those of a disk, while, increasing G up to one, the first-mode frequencies decrease approaching the value obtained with a lumped mode model. The frequencies of the higher-order modes, on the other hand, increase to infinity, justifying the lumped mode approximation. The spatial distribution of the displacement in the radial direction is also computed; it has a Bessel function shape which, as expected, becomes linear by increasing the inner radius. Finally, the behavior of the effective coupling factor k/sub eff/ with G is examined. It is shown that, when G/spl rarr/1, k/sub eff/ approaches the material coupling factor k/sub 31/, while when G/spl rarr/0, k/sub eff/ is proportional to the planar coupling factor k/sub p/. Further it is shown that for G>0.6, the approximation of the ring to a lumped mode system is quite acceptable.

In this paper we describe an approximate two-dimensional model of a single piezoelectric array element. Substantially, the element is a two-dimensional structure whose vibrations can be described by two coupled differential wave equations with coupled boundary conditions. We take as a solution of this system two orthogonal wave functions which depend only on one axis, corresponding to the propagation direction, and which satisfy the boundary conditions only in an integral form. This solution makes it necessary to neglect the piezoelectric coupling in the transverse direction; nevertheless this approximation does not substantially affect the computed results because the transverse elastic coupling is much stronger than the piezoelectric one. With our model, the external behavior of the element in the frequency domain can be described by a 5/spl times/5 matrix from which all the transfer functions of the element can be easily computed. We compared our results with those of the classical Mason-Sittig one dimensional model, finding, as expected, a small difference in the principal thickness resonance frequency and the presence of a lateral mode. To verify the results obtained with our model, we computed the frequency spectrum of the element varying the width/thickness ratio. We found a good agreement with the low branch of the spectrum computed by the coupled mode theory. Finally, we computed, in the frequency domain, the dynamic electromechanical coupling coefficient k/sub U/, evaluating the internal energy distribution of the element and applying the definition reported by Berlincourt. We compared the values at resonance with the appropriate quasistatic coupling factors k/sub 33/' and k/sub 31/' and with the effective coupling factor k/sub eff/ computed by the same model.<>