Abstract
Finite element method has been proved to be a useful means for the analysis and design of electromechanical devices. Since the technique is essentially based on the energy principle, coupled electromechanical problems can be treated without imposing any assumption such as weak coupling. Piezoelectric or electrostrictive resonator and filter problems have been dealt with for devices of flexure-type, inplane vibration type, trapped-energy type, bi-material type, and Langevin type. In each case, however, uniform electric distribution was assumed between a pair of electrodes provided. The present paper is concerned with more general cases, in which the electric field distribution that couples to the mechanical system is not reasonably defined in advance and the electrode configuration and arrangement are arbitrary. The finite element formulation is given within the scope of two-dimensions and the second order polynomial is assumed as a trial function. (Fig. 1). The formulation in which a linear trial function is assumed can be obtained by a little modification of the above formula. As numerical examples, the natural frequencies, modal patterns and electric potential distributions corresponding to the modes are calculated for a thickness shear vibrator (Fig. 2) and an interdigital transducer (Fig. 11). In these calculations, barium titanite ceramic is taken as the material. The first example is a trapped-energy resonator with electrode loading. In Figs. 3 and 4, odd shear modes and the corresponding potential distributions are illustrated. The energy-trapping is observed for the first mode and the coupling to the bending is seen at the edges for higher modes. The effect of the some restriction on the modal patterns and the natural frequencies are then discussed. In Figs. 5 and 6, the modal displacements in the x-and y-directions and the corresponding natural frequencies are shown for partial and full (or non) electrode configurations respectively. Those shown in Figs. 7 and 8 are the cases when the displacements in the y-direction are suppressed by the assumption of the uniform electric field distribution between a pair of the electrodes. For the last case, sinusoidal distribution is assumed for the displacements in the x-direction with the thickness, which was considered in our previous paper. The uniform electric field assumption is more influential than the displacement suppression in the y-direction. The coupling to bending is rather small except in the edge region. By at large, the displacements in the x-direction are very much alike for all of them and the difference of the natural frequencies are within a few percent for this very thin plate. In Fig. 9, the damped electric field distribution is illustrated in which the second order polynomial is assumed as a trial function and the frequency characteristics of motional admittance is shown in Fig. 10. The second example is concerned with the wave propagation in an interdigital transducer of infinite length (Fig. 11). In order to apply the finite element method, a finite brock with proper boundary conditions are considered by virtue of periodicity and symmetricity of wave propagation. Frequency characteristic of the normalized motional admittance is illustrated in Fig. 12. The first ((1)) and 16th ((16)) modes are surface wave ones. In Figs. 13 and 14, the modal patterns and the corresponding electric potential distributions at each resonant frequency are illustrated respectively. The damped electric potential distribution is shown in Fig. 15. For the surface wave modes, electric potential waves propagate only in the vicinity of the surface, while they propagate as in an electromagnetic wave guide for the bulk wave modes, which would not be present when the medium is of a half space. Finally in Fig. 16, the electric potential distribution and the displacements of the surface wave mode ((1)) are shown for the depth from the surface of the transducer. The phenomena are clearly pr
