二次元電気・機械振動子の有限要素シミュレーション

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Piezoelectric plates are widely used as electromechanical resonators and filters and their multi-mode utilization is now in fashion. Lloyd and Redwood have applied the well-known finite difference method to the analysis of square and rectangular plates. Their treatment, however, becomes troublesome for plates of irregular shape or having complicated boundary conditions. The technique of applying the finite elememt method to coupled electromechanical systems has been developed, and has been used to successfully analyze electromechanical resonators and filters of lexure type. The present paper describes a simular technique which can be applied to the viblation simulation of two-dimensioal electromechanical thin plates which vibrate in plane(Fig. 1). Finite element formulation is made with the inclusion of electromechanical coupling. The division of a plate is approximated by the assemblage of traingular elements, so that the formulation is applicable to piezoelectric or electrostrictive thin plate resonators and filters of arbitrary shape. A linear approximation is employed to describe the displacements of a triangular element of the plate, as illustrated in Fig. 2. To demonstrate the validity and the availability of the procedure, the natural frequencies of planarly isotropic rectangular plates are calculated for the symmetrical modes (Fig. 5). The results are discussed in conjunction with those of other investigators [(a) in Fig. 6: the finite element method using a 2nd order polynomial to approximate the displacement functions, (b): the finite element method using a linear displacement approximation, (c): the finite difference method, (d): the series expansion method]. Good agreement is obtained for the L1 (longitudinal dominant) mode. For the other modes, discrepancies are found within the range of a few percent. Effect on the natural frequencies due to electrical termination is less than 1% for ordinary electrostrictive materials such as barium titanite. The vibrational modes including the asymmetrical ones, are then calculated for the plates having length-to-width ratios (r_<ab>)of 1 : 2 and 1 : 1. The model shapes are illustrated in Figs. 7 and 8. For plates of arbitrary shape, a calculation is also made for the plates with one of the corners rounded off. The modal shapes and the corresponding natural frequencies are illustrated in Figs. 9 and 10. The effect of rounding off is clearly seen which, as shown in Fig. 10, resolves the degenerate modes (F1/L2, F3/L2') as expected. In the present analysis, the input admittance at the electrical terminals can be directly obtained, because electromechanical coupling is included. In Fig. 11, the frequency characterristics of motional admittance are shown for a rectangular plate fully electroded on both surfaces (r_<ab>=2). For the plates with partial electrodes and those with one rounded corner, they are shown in Figs. 12 and 13, respectively. The motional admittance for the square plates are shown in Figs. 15 to 18. Full electrode arrangements are likely to excite only the symmetrical modes while partial arrangements excite all associated modes. From the curve, the excitation strength of each mode is predicted, and the equivalent stiffness and mass in its vicinity can also be evaluated from the slope. In the numerical demonstration, the convergence does not appear to be complete. This is due to the limited capacity of the available computer, and not an essential defect of the procedure. This can be overcome by making the divisions smaller or employing a higher order polynomial to approximate the displacements. The technique introduced proves to be a powerful means for simulating and analysing two-dimensional electromechanical resonater problems. Once the computer program is developed, wide applications are possible with little modification.
社団法人日本音響学会の論文
1974-07-01

二次元電気・機械振動子の有限要素シミュレーション

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