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

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Absorption mechanism of porcus material for a sound pulse is analysed by means of the Fourier transformation of the incident and reflected pulses. Measurement is made by impinging a sound pulse to a specimen mounted into a pipe (Fig. 1). For a thicker specimen compared with the pulse length, one obtains discrete series of the reflected pulses spaced with a certain duration corresponded to the time propagating in the specimen (Fig. 4). When this pulse series of the reflected wave is analysed as a whole by the Fourier transformation (Eq. 2), the absorption coefficient agrees with the values by the ordinary standing wave method for the sinusoidal waves (Fig. 5). This means that the reflection coefficient of a specimen identifies with the transfer function of the specimen for a linear system (Eq. 8). On the other hand, if the power spectrums of the individual pulse in the reflected pulse series are analysed and the absorption coefficients are calculated from them (Eq. 3), then the frequency response is different from the values of the former (Fig. 6). However, if the total absorbed energies for both cases integrated over the whole frequency range are calculated (Eqs. 14-17), it results in good agreement between them. Therefore, even if the frequency characteristics of the absorption coefficient are different, its total absorbed energy is equal (Eq. 18), and then the total absorption of pulse energy is determined by the attenuation constant of the material, and the frequency characteristics of the absorption coefficient depend upon the acoustic situation in which a specimen is mounted.

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This paper attempts to describe the new sound sources for airborne ultrasonics with high sensitivity. They are improved sound sources of high-intensity ultrasonics which were developed previously by the authors. The structures of these sound sources consist of ultrasonic vibrators, a metalic horn, a vibration plate having flexural modes and four reflection plates. It is desired to vibrate the vibration plate of these sound sources in flexural modes as shown in Fig. 2, so called "stripes mode", in order to have efficient radiation for ultrasonics. Production of such vibration plate, however, require much experiences. But, they can be produced rather easily taking into account of the Eqs. (4) and (5), i. e. by combining the equation for bending vibration of the rectangular bars and some experimental means. The directivity of sound radiated from these plates have the beam form having four directions shown in Fig. 4 (a). These four beams can be synthesized into a single beam by the successive introduction of four reflection plates, which is shown in Fig. 7 (e). The new sound sources for airborne ultrasonics have been constructed by this way, which are capable of operating at the frequency of 19, 28, 50, 70 and 90kHz shown in Fig. 8. Fig. 9 shows the polar patterns of the sound pressure produced from the new sound sources. These patterns can be kept as far as the distance of about 10m. Fig. 11 shows the relation between the distance from the sound sources and the sound pressure level. The sound pressure level decreases monotonously as the distance increases. Table 3 shows the sound pressure produced by these sound sources compared with that of some old ones. As seen from the table, a remarkably high sensitivity was gained by the new sound sources. And, Fig. 12 shows the different waveforms of the ultrasonic pulses produced by prosound sources. It is noted that the new sound sources authors devised could be used these ducing ultrasonic pulses in general.

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This paper deals with the optical transmission characteristics and the dielectric effects of a nematic liquid crystal (MBBA) film subjected to an ultrasonic field, which is the counterpart of our previous work on the effects subjected to shear vibration. The experimental arrangement for the measurement is illustrated in Fig. 1. The changes of the dielectric constant and tan δ with the sound pressure are almost alike both at 25℃ and 32℃, which are however much smaller at 40℃ (Fig. 3). This is explained by the fact that the dielectric and conductive anisotropy is small near the transition temperature, that is, 43℃. The relative changes of the transmitted light intensity through the cell subjected to the sound pressure are shown in Figs. 5 (400kHz) and 8 (1MHz). A He-Ne laser is used as the light source together with a crossed Nicol system. Dielectric constant and tan δ as a function of the sound pressure are shown in Figs. 6(400kHz) and 9(1MHz). It is found that as in the case of application of shear vibration there exists a threshold sound pressure above which rapid changes not only for the transmitted light intensity but also for the dielectric constant and tan δ follow. They can all be explained by the same mechanism that the molecular alignment rotates under the radiation pressure of the incident sound. The threshold is almost inversely proportional to the film thickness as well as the frequency of the ultrasound, as seen from Fig. 10. Thus these tendency agrees well with that in Nagai's and Helfrich's theories. With increase of the sound pressure above the threshold, the transmitted optical intensity sharply increases and then fluctuation occurs alternately with maxima and minima. The dielectric constant increases while the tan δ decreases. In this range, the color of the liquid crystal film varies alternately from red to blue and then from blue to red with increasing sound pressure (Figs. 6 (a) and (b)). A parallel domain structure takes place with the vibratory displacement applied, however, does not appear in the present case. With further increase of the sound pressure the dielectric constant reaches saturation, and the transmitted optical intensity is decreased due to generation of flow in the film. This phenomenon is not so-called DSM but a flow along the layer (Fig. 6 (c)). On the other hand, the change of tan δ is not so simple as that. The tan δ is sharply increased with temperature, while the dielectric constant remains almost constant (Fig. 3).

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The Hall effect magnetic head (thereafter called as a Hall head) has long been investigated because it has the following potential advantages: (1) higher sensitivity in the low frequency range and (2) compactness in size. But the Hall heads so far reported seem to have two serious drawbacks: firstly much smaller signal to noise ratio, and secondly poor frequency responce. The authors therefore investigated a thin film Hall element with high signal to noise ratio, and more efficient magnetic core in order to offset the above-mentioned drawbacks to adopt a Hall head for audio use (an 8-track 4-channel type, Fig. 4). As a semiconducting material for the Hall element, we selected InSb thin film because it has a high signal to noise ratio even when it is prepared as a thin film. We found, moreover, that it is necessary for the InSb thin film to be microzone-melted on a ferrite substrate. After this treatment the current noise is reduced sufficiently. The thin film 2μm thick thus treated is formed into a K-shaped Hall element (Fig. 2). On this Hall element a ferrite pole-piece is fastened and the front end is ground and lapped to form a front gap type Hall head with the gap length of 3μm (Fig. 1). The shape of pole-piece is so designed as shown in the plane view (Fig. 2) and in the side view (Fig. 1) to be triangular in order to enhance flux density on the Hall element and to reduce contour effect. Between the Hall elements a shield plate is placed (Fig. 3) to avoid cross-talk. The output level and noise level between 40Hz and 10kHz of the Hall head thus manufactured are measured by a B & K level meter using a standard tape of Japan Industrial Standard (JIS) specification (20mMaxwell/mm at 400Hz, 9. 5cm/s). Moreover, the crosstalk, contour effect and transient characteristics are measured. During this measurement DC control current is passed through the Hall element at 12mA. The results obtained are as follows: (1) Output level: As shown in Fig. 5, the frequency spectrum of the Hall head is quite similar to that of the recorded level in the tape (shown by dotted line). This aspect is characteristic to the Hall head. The output level at 400Hz is 1. 5mV and larger than that of the coil head, 0. 9mV. (2) Frequency response and SN ratio: In case of the Hall head, equalization has to be carried out in order to enhance the output level in the high frequency range. The frequency spectra of both the output level and the noise level (output level from the signalless tape) of the Hall head are measured and compared with those of a typical coil head (Fig. 6). It is found that there is no difference in these heads and the signal to noise ratio between 40Hz and 10kHz is 54dB. (3) Contour effect: As the top end of the pole-piece (point A in Fig. 1) is designed never to touch the magnetic tape, an echo signal is reduced as shown in Fig. 7. Therefore, the contour effect is also reduced to ±2dB. (4) Cross-talk: A permalloy shield plate 0. 6mm thick is enough to reduce the crosstalk below -50dB at 80Hz. (5) Transient characteristics: The Hall head shows lower distortion than coil heads when triangular waveform is played back (Fig. 8) due to the fact that a Hall element has no reactance. (6) Temperature coefficient: When the Hall head is driven under constant voltage condition, the temperature coefficient of the output around room temperature is ±0. 5%/deg, or equal to that of electron mobility of InSb (Table 2).

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We have devised bolted Langevin-type torsional vibrators for ultrasonic power applications, and also reported conformal mapping analyses on the distribution of equipotential lines and e. m. f. of a ceramic ring under polarization. These analyses are, however, based on a linearized assumption of D-E relation in the ceramic and the effect of adjacent electrodes is neglected. The present paper reports about a finite element approach for nonlinear problems. We have at first derived the expressions necessary for FEM in the nonlinear electrostatic field problems, secondly devised a method to reduce the instability that might occur in Newton-Raphson procedure, and finally obtained the following results after iterative numerical calculations: 1) Maximum electric field (electric field concentration) does not increase monotonously with increase of the external field (cf. Figs. 8 and 10). 2) Almost all parts of the ceramic can be effectively utilized by the "two-round polarization method" that we have developed. As is clearly seen from Fig. 1 or Fig. 3, this allocation of electrodes will provide nonuniform electric field distribution in the ceramic. If the dielectric constant of the ceramic was determined only spatially, which was independent of the electric field (the D-E curve was assumed linear within the triangular elements), the electric potential distribution would be obtained simply by solving simultaneous equations (4); the D-E relation is, however, usually nonlinear as shown in Fig. 4 so that consideration of the nonlinearity is necessary. Silvester et. al. employed Newton-Raphson iteration in the analysis on saturable magnetic field problems. So far as we know, such a method has not yet been applied to electrostatic problems. At first, we have derived the expressions necessary for the finite element method in nonlinear electrostatic problems as their counterpart. The D-E curve is assumed isotropic but nonlinear as shown in Fig. 4. In the nonlinear case, the most likely solution is the one which minimizes the potential energy of the system, too. In order to obtain such a solution, the Newton-Raphson iteration method is used. Results are shown in eq. (8) and eqs. from (13) to (16). In many cases of the electrostatic problems, the stability of Newton-Raphson iteration is more severe than for the magnetic counterpart because of its nature of the D-E relation as seen from Fig. 2. Then we devised a method to eliminate the instability that might occur in the process using relaxation technique to find an optimum acceleration factor ω in eq. (6) for which the potential energy is minimized. Such ω that satisfies the condition is given in eqs. (23) and (24). Figs. 5 and 6 show the calculated equipotential lines of the nonlinear electrostatic field shown in Fig. 3. Figures from 7 to 10 show the residual polarization and the maximum electric field against the applied electric field. Their special features are that the maximum electric field in the ceramic does not increase monotonously with increase of the external field as mentioned above. The regions face to the electrodes do not contribute to the torsional torque because the residual polarization there is not circumferentially directed. This dead-space sometimes amounts up to 30[%] of the whole ceramic. So we devised "Two-Round Polarization Method" making use of the nonlinear nature of the ceramic, which enables to highly utilize almost all parts of the ceramic (Fig. 12). Fig. 15 shows a small amplitude free admittance locus of a tested vibrator with the ceramic rings processed by this method (Fig. 14), for which the torque factor A=11. 7×10^<-3> [Newton meter/Volt]. If the ceramic rings are ideally circumferentially polarized, the torque factor will be 12. 5×10^<-3>, which means that the measured value amounts to about 94[%] of the ideal one, while a conventional "One-Round Polarization

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