Ultrasonic Liear Motor using Traveling Surface Acoustic Wave

Transcription

1 .9/ULTSYM.9.5 Ultrasonic Liear Motor using Traveling Surface Acoustic Wave Minoru Kuribayashi Kurosawa Dept. of Information Processing Tokyo Institute of Technology Yokohama, Japan Abstract A traveling wave type ultrasonic linear motor has been miniaturized successfully by using a surface acoustic wave (SAW) device. Experimental results of the SAW motor using traveling Rayleigh wave showed high performance in of.5 m/s, thrust of 3 N and mechanical power of W with a xx mm 3 lithium niobate stator and a x mm silicon slider. Theoretical approach for SAW motor from physical modeling and behavior modeling were introduced. From the simulation, it was found that the frictional drive efficiency would reach up to %. An equivalent block diagram of the SAW motor was derived. Keywords; linear motor; actuator; SAW; friction drive;mems I. INTRODUCTION (HEADING ) For miniaturization of ultrasonic transducers to fabricate piezoelectric motor, a surface acoustic wave device has an advantage in rigid mounting and high-power-density operation. A surface acoustic wave (SAW) motor has been investigated []-[], and its superior performances have been demonstrated: a of.5 m/s [5], a thrust of N [], and a stepping motion of.5 nm [], [7] using a xx mm 3 plate piezoelectric SAW transducer. In addition, to increase the application range, a higher operation frequency of MHz [] and a two-dimensional design [9] were investigated. In addition, to reduce power consumption, energy circulation driving has been proposed and investigated []. From investigations based on experiments, it was found that slider surface texture affects motor performances such as and thrust [3]-[5]. Physical modeling of the SAW motor had been attempted []. One slider projection was modeled including the compliance of the slider and stator materials []- [7], and also the stick and slip at the boundary due to friction force. Using the slider projection modeling, operations of the SAW motor were simulated, and then, the results were compared with the experimental results []. Another modeling based on behavior of the SAW motor has being carried out from the measurements of the motor response [9]. This kind modeling is necessary for design of a servo system of the SAW motor. In this paper, two types of traveling wave ultrasonic linear motors are summarized at first. Then, recent study on the SAW motor is described. II. TRAVELING WAVE LINEAR MOTOR Driving principle of a traveling wave ultrasonic linear motor shall be explained using a drawing such as Fig.. Surface particles motion of the elastic material for which guides a wave is required to be elliptical. For the elliptical motion flexural wave was used, and then, Rayleigh wave was introduced. A. Flexural Wave Motor For the comparison an SAW motor with a former traveling wave ultrasonic linear motor, a flexural wave linear motor shall be reminded. For an example as shown in Fig., an ultrasonic linear motor was investigated using a -mm square aluminum bar and two bolt-clamped Langevin transducers with horns []. Fig. 3 shows a photo with a slider. The vibration system was bulky, so that high power was consumed. For m/s slider, driving power of, W was required as shown in Fig.. In spite of high power drive, mechanical output force was small as indicated in Fig. 5. It was hard to expect to be efficient actuator system. B. Surface Acoustic Wave Motor Using a Rayleigh wave for an SAW motor, a basic device is constructed as shown in Fig.. The motor has only a thin plate transducer and a thin friction material. The plate transducer is an SAW device to generate a Rayleigh wave using two interdigital transducers (IDTs) to excite traveling wave in both nm Slider wave pre-load friction drive particle motion Elastic material μm Figure. Principle of traveling wave motor. This work was supported by a grant-in-aid for the Creation of Innovations through Business-Academic-Public Sector Cooperation, and a grant-in-aid for the Development of Innovative Technology of the Ministry of Education /9/$5. 9 IEEE 9 9 IEEE International Ultrasonics Symposium Proceedings

2 Figure. Schematic of traveling flexural wave motor. Figure. Relation between tangential vibration velocity, no-load motor and transmission power of flexural wave guide shown in Fig.. Figure 3. Traveling wave motor using flexural wave in rectangular cross section aluminum bar. Figure 5. Load characteristics of flexural wave ultrasonic linear motor shown in Fig.. direction. One IDT at a time is driven to excite the wave, and then, the slider moves to the IDT. The active IDT is changed for the alternative linear motion of the slider. For friction drive, preload on the slider is given as shown in Fig. 7. In the case of SAW, the back of the device does not vibrate, so that the SAW device is fixed tightly on a base. Material of the SAW device was degrees y-rotated xpropagation LiNbO3. The dimension of the device was mm long, mm wide, and mm thick. The chromium/aluminum electrodes were twenty pairs of IDT that were - m-wide line with a m gap, and were 9 mm in aperture. The resonance frequency, namely, the operation frequency, was 9. MHz. When the driving voltage was 5 V-p, vibration amplitude and velocity were nm in the direction normal to the surface and.3 m/s in tangential direction, respectively; the input power was 7 W. For stable friction driving conditions, many projections are fabricated on the surface of the slider []. The diameter of the projections was m or smaller to obtain a large thrust and a high close to vibration velocity. It has been found from experiments that the larger the projection diameters, such as 3 or 5 m, the lower the performance in terms of and thrust []. For the experiments, several types of silicon sliders were fabricated with the dry etching process. The slider dimensions were 5x5 mm; m high projections were fabricated in a x mm area. RF power For m/s motor with no-load, driving power of 7 W was required as shown in Fig.. Actual propagating power from IDT to the slider side was 35 W, because simple IDT radiate the wave in both directions. The required power was preload Slider Rayleigh wave Electrode (IDT) SAW device (Stator transducer) Figure. Schematic of surface acoustic wave motor. Figure 7. Photograph of surface acoustic wave motor IEEE International Ultrasonics Symposium Proceedings

3 Velocity [m/s] 3.3 vibration velocity motor velocity No-load..... Si AlO3 HADLC Driving power [W] Figure. Relation between tangential vibration velocity, no-load motor and driving power of SAW device; 9. MHz Load [N] Figure 9. Load characteristics of SAW motor. about one-thirtieth of the former flexural wave motor. Mechanical loaded characteristics were estimated from the transient curve of the motor as shown in Fig. 9. Thrust of the motor was same as the huge flexural wave motor. III. PERFORMANCE OF SAW MOTOR The output mechanical power of SAW motor was estimated from transient response. The no-load, zero thrust and mechanical output power are shown in Figs., and as a function of the preload [5], []. For these measurements, same SAW element as shown in Fig. 7 was used at the driving voltage of 5 V -p. The slider friction driving area was mm by mm of a 5 mm by 5mm square Si thin plate. The area of x mm was micro machined to fabricate m diameter.5 m high projections in interval of 5 m. One slider was Si material surface and two sliders were deposited AlO3 or DLC (diamond like carbon) on each surface. From the driving voltage of 5 V -p, the vibration velocity at the surface of the stator became.3 m/s in tangential direction. It means that the slider moved at almost same velocity of the Rayleigh wave vibration in the cases of and N preload conditions as can be seen in Fig.. By being increased of the preload, the motor decreased due to being enlarged of the contact area between the stator wave and the slider, not only around the crest of the wave but also the Preload [N] Figure. No-load vs. preload of 9. MHz SAW motor at driving voltage of 5 V o-p. Output Force [N] Si AlO3 HADLC Preload [N] Figure. Thrust vs. preload of 9. MHz SAW motor at driving voltage of 5 V o-p. valley as being given the image in Fig. 3. The increase of the preload resulted in the large thrust as shown in Fig.. At around the N preload, the maximum thrust were obtained from to 3 N. It depended on the surface material and other, not certain, conditions. Even in these high preload and large thrust conditions, the no-load was still high around.7 m/s as indicated in Fig.. Much larger preload conditions, the motor and thrust also decreased due to too much contact area. The output force against the weight of actuator is one of the important factors to evaluate actuators. From the device weight of the 3.9 g SAW transducer and the g Si slider. The ratio (output force)/((device weight) x 9. m/s ) becomes 7. This value is much larger than that of magnetic motors. From the characteristics of the motor that the linearly decreases against the thrust, the maximum mechanical output power becomes at the halves of the maximum and the maximum thrust. The estimated results are indicated in Fig.. The maximum output power was. to.5 W at the preload range from 3 to N. The power weight ratio, the output power divided by the device weight, was about 5 W/kg. 9 9 IEEE International Ultrasonics Symposium Proceedings

4 Output Power [W] Si AlO3 HADLC Preload [N] Figure. Mechanical output power of 9. MHz SAW motor at driving voltage of 5 V o-p. The input power of the IDT was 7 W. Hence, a simply calculated efficiency becomes 3.5 %; it s very low. However, half of the power was wasted, because the wave propagates in both directions in case of simple IDT that was used in the stator. Therefore the effective power to drive the motor was 35 W. By using unidirectional IDT, the input power of the IDT will be reduced half of the simple IDT. In this case, the conversion efficiency from the electrical input power to the mechanical output power will be 7 %. There was one more apparent reason for low efficiency; the wave propagating width was about two times of the slider width. Therefore the effective wave power flowing beneath the slider was half of the previous discussed effective power of the 35 W. The flowing wave power beneath the slider was about W, so that the conversion efficiency from the traveling wave to the mechanical output is estimated to be %; it is not so low efficiency. The apparent efficiency estimated from the input and output seemed to be very low, however, actual conversion efficiency seems to be acceptable. IV. ANALYSIS OF SAW MOTOR To explain the principle of the actuator driving mechanism, a drawing in which the wave motion is enhanced is convenient to show the image. Actual wave motion is too small to present in the drawing; therefore, the displacement of wave motion is enlarged about,-fold in Fig. 3. Regarding the actual dimensions of wavelength and vibration displacements, the stator surface is almost flat; in the case of 9. MHz driving frequency, for example, the wavelength and vibration displacements are about m and nm, respectively. Physically, therefore, the contact between the stator and the slider contains elastic deformation, stick, and slip. Under actual driving conditions, the driving frequency and the preload are maintained to be constant, and then, driving voltage is changed to control slider by changing wave amplitude, namely, the vibration velocity at the crests. A. Modeling A physical modeling of frictional drive of a SAW motor has been carried out [3]-[7] on the basis of contact mechanics []. For the first step of the modeling, a slider elastic body, a rigid projection, and a stator elastic body were expressed using four springs, one rigid body connected to the elastic slider part, and frictional boundary surfaces, as shown in Fig. In the modeling drawing shown in Fig., a v, a h, P, and Q are the vertical wave amplitude, the horizontal wave amplitude, the vertical direction force acting on the projection, and the thrust force acting on the projection surface, respectively. The equivalent spring constant of the slider and the stator are indicated by k pn, k pt, k sn, and k st in the normal and tangential directions, respectively. The two springs in the sliders k pn and k pt can be written in the forms k = G Si a[ln(3 Si )/( Si )] () k pt = G Si a/[/+(- Si )/( )]. () In these equations, G Si, Si, and a are the slider material shear modulus, Poisson s ratio, and projection radius. In eq. (), = ln(3- Si )/. The other two springs in the stators k sn and k st can be written in the forms k sn = G LN a/(- LN ) (3) k st = G LN a/(- LN ). () In these equations, G LN and LN are the stator material shear modulus and the Poisson s ratio. Using the physical model of one projection, we carried out a numerical simulation in time domain including preload, friction coefficient, vibration amplitude, and so on [3]. From the simulation, the thrust between the stator and the slider were obtained, then, the mean and thrust of the slider were estimated. It is understood from Fig. that the friction drive slider motion slider k pn kpt Slider Projection traveling wave stator Stator P k st k sn Q a v a h Stator Motion Figure 3. Schematic drawings of contact between traveling wave in stator and projections on slider surface. Figure. Physical contact modelings of projection on slider and stator surface IEEE International Ultrasonics Symposium Proceedings

5 has two parts: sticking with elastic deformation in the nanometer range and slipping at the boundaries. B. Peformance Depending on Projection Number Simulations of a SAW motor operation were carried out using the projection contact model [3]-[7] and results were compared [] with experimental results at 9. MHz motor operation. The driving voltage was 5 V -p, so that the vibration amplitude and velocity were nm in the direction normal to the surface and.3 m/s in tangential direction. The slider projection number was changed from 9 to in the case of a -μm-diameter slider. Then, performance differences depending on the slider projection diameters ranging from to 5 μm with same contact surface area of about 3 mm in by mm were compared with the experimental results [], [], [5]. The silicon material for the slider and lithium niobate for the stator are anisotropic materials. Hence, the material constants of the slider and the stator were simplified to be approximated isotropic elastic constants for use in equations () to (). The approximated isotropic elastic constants of rigidity modulus G Si and Poisson s ratio Si have been obtained using Voigt average [], namely, G Si =. GPa and Si =.. For the elastic constants of lithium niobate, we then approximated to the isotropy from the propagation velocity of the Rayleigh wave and the longitudinal wave; substituting those velocities to the equations of velocities of an isotropic material extracted the approximated isotropic elastic constants, which were G LN = 93.7 GPa and LN =.5. It is difficult to maintain a uniform contact condition for each of the projections distributed in a x mm area, owing to the small vibration amplitude and elastic deformation in the nanometer range. Hence, the effective factor of projection contact was investigated using m diameter, projections in the line by projections in the column slider. In the simulation, the number of projections was reduced by the effective factor in the line and column. For example, in the case of the effective factor of., the actual slider of x projections was reduced to x projections. It is found that the effective factor of. agreed with the experimental results. In addition, the projection distribution in the wave traveling direction was ignored, in the simulation. Namely, the attenuation of the wave by the friction drive was neglected to simplify the calculation. The friction coefficient between the slider and the stator was fixed to be. to make the incline of the thrust curves under low preloaded condition agree with the experimental results. Using five different sliders with projection numbers of,, 35, 5, and 9, thrust and that were estimated by the simulation were compared with experimental results; the projection diameter was μm. The thrust at null, namely, maximum thrust, depends on the preload and the number of the projections, as indicated in Fig. 5. There are five curves obtained for simulation and plots for experimental results, as indicated in Fig. 5 as follows: n=,, and so on. From the simulation, it is understood that the stiffness at the slider contact surface is important for a large thrust. This is because the stiffness is in proportion to the number of projections. The stiff surface is suitable for high operation, as shown in Fig.. In the case of the soft slider surface, the projections contact the bottom of the wave with a large preload. C. Performance Depending on Projection Diameter A difference in slider projection diameter exerting influence on the output force of the SAW motor indicated interesting result, as shown in Fig. 7. The simulation results are indicated with a dotted line, a dashed line, and a solid line for, 3, and 5 m projection diameter sliders, whereas the experimental results are indicated with circles, triangles, and squares, respectively. Three sliders had the same contact area of 3 mm but different projection diameters; the - m-diameter projection slider had projections. The small diameter projection slider has lower stiffness than the large projections. However, the total stiffness becomes higher if the total projection contact surface area is the same as the large. n= Thrust s [N] 35 Speed f [m/s]... n= Preload [N] Preload [N] Figure 5. Output force characteristics of -μm-diameter sliders with change in number of projections from by to 33 by 33; n= ( ), ( ), 35 ( ), 5 ( ), and 9 ( ) by experiment: dashed, dot-dashed, dotted, dashed, and solid curves by simulation. Figure. No-load characteristics of -μm-diameter sliders with change in number of projections from by to 33 by 33; n= ( ), ( ), 35 ( ), 5 ( ), and 9 ( ) by experiment: dashed, dot-dashed, dotted, dashed, and solid curves by simulation. 9 IEEE International Ultrasonics Symposium Proceedings

6 projection slider. It is clearly indicated in equations from () to () that the stiffness is proportional to the radius of projections, not to the square of projection radius. Thus, if the total projection areas are the same, the smaller projection slider has a higher surface stiffness. The high surface stiffness of slider induces a large thrust. The higher surface stiffness due to smalldiameter projections provides superior performance in terms of, as shown also in Fig., about three different-diameter sliders, similar to that shown in Fig. 7. It is clearly understood also from the experimental results that the -μm-diameter slider has superior performances than the 5-μm-diameter slider in terms of and thrust against the variation of the preload, as indicated also in Figs. 7 and. The three sliders had same contact area of 3 mm but different diameters: thus different contact stiffness values. The projection contact stiffness values are.5 and.5 times larger, in the cases of 3- and 5-μm-diameter sliders respectively, but the amounts of the projections are /.5 and /.5 for the - μm-diameter slider. As a result, the 3- and 5-μm-diameter projection sliders have.5 and.5 times lower stiffness. As a result, the smaller-diameter projection slider has a high contact stiffness. D. Output Power, Loss and Efficiency From experimental results, the slider projection diameter of m was small enough. This is because reduction of the slider projection diameter less than m did not bring the increase of thrust. However, from the simulation results, we have found that the smaller diameter of slider projection would bring the high conversion efficiency of the friction drive. From the estimation of mechanical loss due to the slip between the stator surface and the slider projection, and the mechanical output power of the motor, the conversion efficiency from the wave power to the motor output was evaluated. Three projection diameter sliders, namely, m shown in Figs 9 and, m in Figs and, and 5 m in Figs 3 and were estimated. These slider had the same contact area; m, m, and 5 m diameter sliders had by, by, and by projections, respectively. Thrust s [N] Pre-load [N] ø μm ø3 μm ø5 μm Figure 7. Differences in output force characteristics between projections of -μm- ( ), 35 projections of 3- m- ( ), and projections of 5-μm-diameter ( ) sliders by experiments: dotted, dotdashed, and solid curves by simulation. skef_9rmpl9f_p.... Mechanical output Mech out [W] Figure 9. Simulation results of, mechanical output power and slip loss vs. thrust for projections of - m-diameter slider. loss. skef_9rmpl9f_p 7 Speed f [m/s].... ø μm ø3 μm ø5 μm Pre-load [N] Figure. Differences in no-load characteristics between projections of -μm- ( ), 35 projections of 3- m- ( ), and projections of 5-μm-diameter ( ) sliders by experiments: dotted, dotdashed, and solid curves by simulation..... efficiency Efficiency [%] Figure. Simulation results of and efficiency vs. thrust for projections of - m-diameter slider. 9 IEEE International Ultrasonics Symposium Proceedings

7 To maintain the maximum thrust of to 7 N for each condition, contact area of the slider projections were changed. As a result, in case of smaller projection slider, the contact area reduces due to lower compliance of the slider surface. It means that stiff surface slider contacted narrow area only around the wave crest. As a result, the mechanical loss decreased less than W in the medium thrust in case of smaller diameter projection slider. In addition, at the large thrust range, smaller projection diameter sliders had lower decrease, so that they had larger mechanical output power. For example, 5 m projection diameter slider had high conversion efficiency % and mechanical output of W; twice of the m projection diameter slider. V. BEHAIVIOR MODELING Modeling from the response of an energy circulation SAW motor was carried out [9]. For a servo drive system of SAW motor, precise response modeling is required. A. Energy Circulation Drive A schematic view of the energy circulation SAW device is illustrated in Fig. 5. Two RF electrical power sources are used to excite Rayleigh wave. Namely, one way traveling Rayleigh wave is excited with two IDTs. The Rayleigh wave arrive at a unidirectional IDT is converted to electrical power with less reflection. Then, the electrical power is converted into elastic wave at the other unidirectional IDT again. If we change the phase of one power source, the wave propagation direction is alternated. With this mechanism, the Rayleigh wave is excited efficiently. A fabricated SAW device and a motor setup is shown in Fig.. The SAW device is the stator of the motor; the dimensions were x3x mm 3 and the material was LiNbO 3. The electrodes were designed to obtain less standing wave at.33 MHz Mechanical output skef_9r5mpl9f_p5 loss Mech out [W]..... Mechanical output skef_9rd5mplf_p5 loss Mech out [W] Figure. Simulation results of, mechanical output power and slip loss vs. thrust for projections of - m-diameter slider. Figure 3. Simulation results of, mechanical output power and slip loss vs. thrust for projections of 5- m-diameter slider.. skef_9r5mpl9f_p5. skef_9rd5mplf_p5... efficiency Efficiency [%]... efficiency Efficiency [%] Figure. Simulation results of and efficiency vs. thrust for projections of - m-diameter slider. Figure. Simulation results of and efficiency vs. thrust and efficiency for projections of 5- m-diameter slider. 9 IEEE International Ultrasonics Symposium Proceedings

8 E =E cos t E =E sin t Unidirectional IDT.3 Fitting Measurement.5 V 5 V 7 V Reflector Rayleigh Wave Stator Transducer Velocity [m/sec]. 55 V.5 5 V. Figure 5. Schemaic of energy circulation drive device for SAW motor Driving Force [N] Figure 7. Load characteristics of SAW motor estimated from transient response... Figure. Setup for energy circulation SAW motor. B. Estimation of Motor Characteristics For estimation of the motor characteristics, transient motion of the slider was measured by changing the driving voltage. The output force of the motor was obtained from the acceleration and the inertia of the moving part at each of the transient curves. The relations between the and the driving force are plotted on Fig. 7 from the transient responses. The curves were jagged due to the performance of a observer that estimated the slider from a linear position sensor. Hence, the -force curves indicated in Fig. 7 are winding a little bit. At the starting acceleration, namely, at null time point in transient response, the null driving force was characterized as showing in Fig.. At the lower driving voltage, the motor slider doesn t move. The dead zone is caused by the friction drive principle of the motor. The SAW motor does not have positive thrust in the dead zone. The generating traction of the motor was characterized as a function of the driving voltage indicated in Fig.. Characterization of the motor response in the dead zone was carried out by being measured the response as shown in Fig. 9. The driving voltage was reduced from the higher voltage down to the dead zone or the boundary driving voltage. From these responses, the force- relations in the dead zone were obtained. Driving Force [N] Input Voltage [V pkpk ] Figure. SAW motor output force dependence on driving voltage. The static, namely, non-acceleration condition, motor response was characterized for the next process. The force relations as indicated in Figs. 7 and 9 were approximated in linear relation using a damping factor D as indicated below, F d = F - D v. (5) Here, F d, F, and v are the output force, null force, and the of the motor. The damping factor is a nonlinear function of the driving voltage. However, the factor of /D has a linear relation with the driving voltage as indicated in Fig. 3. Therefore, the factor of /D has a linear function of the driving voltage as below, /D = av+b. () Here, V is the driving voltage. Constant factors of a and b are obtained from Fig. 3; an incline is a and the extrapolated value of V= is b. 3 9 IEEE International Ultrasonics Symposium Proceedings

9 To be simplified the notation, a factor D o was introduced; D = /(av+b) = D o /( V+). (7) By using the new factor D o, the relation between the motor and output force is indicated as F d = F - (D o /( V+)) v. () By taking account of the inertia of the motor M, the motor response is expressed as F = (/M) (dv/dt) + (D o /( V+)) v. (9) The notation of Eq. (9) describes the starting up of the transient response of the motor. From the transient responses as indicated by thin lines in Fig. 3, the stopping motion of the motor, namely, after. s in time in the graph, was simple friction condition. The friction force depends on the friction coefficient and the preload. Hence, a constant friction force acts on the slider. This effect is expressed as a limiter of the traction. C. Block Daiagram and Discussion As a result, a block diagram of an energy circulation drive SAW motor was obtained as shown in Fig. 3 [9]. From the driving voltage V, the motor v is obtained in time domain from the start to the stop. For example, the transient response of the motor in time domain, the model simulated the motor response as indicated in Fig. 3 by using thick lines. The load characteristics are also indicated in Fig. 33. From these results, it is found that the saturated was different from the measurements. It is certain that if the motor force- characteristics are linear relation, the simulation correspond with the measurement. In sometimes, however, the force- is not linear. For a versatile modeling, the block diagram indicated in Fig. 3 should be improved in that case. Velocity [m/sec] V- 3V V- 3V 35V- 3V V friction drive F αv+ + Fd v Ms Fb traction limit D damping factor Driving Force [N] Figure 3. A block daiagram of SAW motor modeling. Figure 9. Output force vs. motor loci obtained from transient responses..3 Measurement 7 V /D [(m/sec)/n] Velocity [m/sec] Simulation 5 V V 55 V 5 V Input Voltage [V pkpk ] Time [sec] Figure 3. Reciprocal value of damping factor D of SAW motor against the driving voltage. Figure 3. Transient responses of SAW motor; experimental results (thin lines) and simulated by modeling (thick lines). 9 IEEE International Ultrasonics Symposium Proceedings

10 Velocity [m/sec] Measurement Simulation Driving Force [N] VI. CONCLUSION Traveling wave type ultrasonic linear motor has been miniaturized successfully by using SAW device. The SAW motor experimental results showed high performance in and thrust. Theoretical approach for SAW motor from physical modeling and behavior modeling were introduced. ACKNOWLEDGMENT The author thanks to Dr. T. Shigematsu, Mr. M. Okano, Mr. Y. Nakamura and Mr. H. Itoh for usage of their research results. REFERENCES [] M. Kurosawa, M. Takahashi, and T. Higuchi, "Ultrasonic linear motor using surface acoustic wave," IEEE Trans. on UFFC, vol. 3(5), pp. 9-9, 99. [] M. K. Kurosawa, M. Takahashi and T. Higuchi, "Elastic Contact Conditions to Optimize Friction Drive of Surface Acoustic Wave Motor," IEEE Trans. on UFFC, vol. 5, no. 5, pp. 9-37, 99. [3] K. Asai, M. K. Kurosawa and T. 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