Analysis of trapped-energy resonators operating in overtones of coupled thickness-shear and thickness-twist

The equations of three-dimensional linear piezoelectricity are applied in the analysis of trapped-energy resonators with rectangular electrodes vibrating in coupled thickness shear and thickness twist in the vicinity of the fundmantal and odd overtone thickness-shear frequencies. Closed form asymptotic expressions for the frequency wave-number dispersion relations for the fundamental and odd overtone coupled thickness-shear and thickness-twist waves near cutoff are obtained for both the electroded and unelectroded regions of the trapped-energy resonator. The influence of piezoelectric stiffening, electrode mass loading, and electrical shorting is included in the analysis. Simple approximate boundary conditions at a junction between an electroded and unelectroded region of the plate are obtained in a manner exhibiting the natural limitations inherent in the approximation. In order that these boundary conditions can be satisfied at each such junction, in the adjacent regions the wave numbers in the direction of the junction line are assumed to be the same. The boundary conditions to be satisfied at the junctions between the unelectroded corner region and the unelectroded regions adjacent to the electroded region are obtained from an extended version of the variational principle of linear piezoelectricity. These latter conditions result in the form of the solution in the corner region. One result of the foregoing analysis is the determination of a two-dimensional condition which is a generalization of Bechmann’s number in one dimension. The above-mentioned dispersion relations and edge conditions are applied in the analysis of the steady-state vibrations of a trapped-energy resonator and a lumped parameter representation of the admittance, which is valid in the vicinity of a resonance, is obtained. Subject Classification: [43]40.24; [43]85.52, [43]85.32.