電気系と機械系が結合している振動系への有限素子法の応用 : 電わい変換器を貼付した音片振動子の解析

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The most commonly used technique for the analysis and design of electromechanical vibrators and filters is that employing the equivalent electrical circuit analogy. The equivalent circuit for vibrators and filters making use of extensional or torsional vibrations is four terminal network, so that the well established electrical network theory can by used for analysis and design. Konno et al investigated the equivalent electrical circuits for bars in transverse vibration with electrostrictive transducers but did not include any electromechanical coupling. Transverse vibrations involve not only forces and displacements but also moments and slopes, so that equivalent electrical circuits are eight terminal networks. These circuits are so complicated that the advantages which circuits usually offer for electrical engineers are lost. Moreover, when the system has a complicated shape and higher modes have to be taken into account, the circuits become even more involved. With recent developments in digital computers, various means for the numerical analysis of mechanical structures have been developed. In this paper, a composite vibrator of flexure-type with electrical and mechanical effects coupled, is analysed by means of the finite element method. The finite element method is a variant of the Rayleigh-Ritz method in which the assumed modes are built up from simple modes relating to each of a number of elements into which the system is divided. When a mechanical system is coupled with an electrical system, the electrical terminals appear as boundary conditions. The example analysed is similar to the transversely vibrating bar with electrostrictive transducers considered by Konno et al (See Fig. 1 and Fig. 3(a)). The natural frequencies of the system are calculated for two extreme electrical boundary terminations, namely, short-circuited and open, and the results are compared with those of Konno et al. The vibrators is divided into 10 elements, so that the length of the transducer can easily be changed in the numerical calculation. The dimensional values of the vibrator are drawn from Konno's example; r_b=(b_t)/(b_m)=1. 0, r_h=(h_t)/(h_m)=1. 5, r_<hL>=(h_m)/L=0. 00527, r_p=(ρ_t)/(ρ_m)=0. 962, r_y=(Y_l)/(Y_m)=0. 381 where b, h, ρ and Y are width, thickness, density and Young's modulus. The suffices t and m indicate those of the transducers and the vibrator. h_m and L are the half-thickness and the total length of the vibrator. The effect of the electrical terminals on the natural frequencies is not great, at most 10% in the example. However, if the transducer has a larger electrical mechanical coupling coefficient, the effect cannot always be neglected. It is seen that Konno's results are quite close to the results for the terminals open, whereas in fact they should be compared with those for the terminals short-circuited. These last named results are a few percent lower than Konno's, because of the negative stiffness effect (See Table 1). The reason why his results are rather close to those for the terminals open is that the two additional terms introduced by the coupling, namely the negative stiffness and the term due to the electrical charges on the electrodes, partially cancel each other in our present example. Since, the vibrator is symmetrical, the same results can be obtained for the half length under the "free-sliding" boundary conditions. The computed results for the two element case are shown in the same table whose percent errors are less than 0. 9% for the first modes. The calculation of input admittance at the electrical terminals, which is an important measure of an electrical device, is also developed. The motional admittances are shown in Fig. 4-6 in the normalized form. The suppression of the third mode for the particular dimension is clearly demonstrated in Fig. 6.
社団法人日本音響学会の論文
1970-03-10

電気系と機械系が結合している振動系への有限素子法の応用 : 電わい変換器を貼付した音片振動子の解析

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