外文翻譯--旋轉(zhuǎn)型行波超聲電機(jī)

1、<p><b>　　旋轉(zhuǎn)型行波超聲電機(jī)</b></p><p>　　帕薩迪納，CA91109，加利福尼亞理工學(xué)院噴氣推進(jìn)實(shí)驗(yàn)室；</p><p>　　科斯塔梅薩，CA92627，材料質(zhì)量檢測(cè)中心，威廉梅蘭迪亞。</p><p>　　摘要：旋轉(zhuǎn)型超聲波電機(jī)逐漸發(fā)展為太空飛船的微型驅(qū)動(dòng)器及其子系統(tǒng)。此技術(shù)應(yīng)用于有著嚴(yán)格要求的商業(yè)產(chǎn)品中，為

2、了更加有效地設(shè)計(jì)此類(lèi)電機(jī)而采用分析工具。分析模型用于檢測(cè)在旋轉(zhuǎn)超聲電機(jī)中激勵(lì)產(chǎn)生的彎曲行波。這個(gè)有限元分析模型為環(huán)形，被用于預(yù)測(cè)環(huán)形定子的振動(dòng)頻率和模態(tài)響應(yīng)。此模型給設(shè)計(jì)高效率的超聲波電機(jī)提供依據(jù)，定子的設(shè)計(jì)包括齒槽、壓電體、定子的幾何外形等方面，定子是由他們有機(jī)地組合而成。理論計(jì)算值與實(shí)驗(yàn)結(jié)果的比較表明這將是一個(gè)值得世人所關(guān)注的課題。與此同時(shí)，超聲波電機(jī)還被用于機(jī)械臂，他們是否能夠在火星的環(huán)境下正常運(yùn)行的研究還在進(jìn)行中。</p&

3、gt;<p>　　關(guān)鍵詞：驅(qū)動(dòng)器，彈性體，壓電電機(jī)，超聲波電機(jī)，定子與轉(zhuǎn)子，模態(tài)分析。</p><p><b>　　2. 緒論</b></p><p>　　當(dāng)前，美國(guó)國(guó)家航空和宇宙航行局一直致力于縮小未來(lái)太空飛船的體積和減少其質(zhì)量的研究。為了與這變化想適應(yīng)，超聲波電機(jī)逐漸成為機(jī)械裝置簡(jiǎn)化的一個(gè)重要的手段。傳統(tǒng)的微型電磁式電機(jī)由于受制造工藝的限制，一般這類(lèi)

4、電機(jī)為了達(dá)到速度與扭矩相適應(yīng)需要使用齒輪減速機(jī)構(gòu)，采用這個(gè)將會(huì)增加設(shè)備的質(zhì)量、體積和機(jī)構(gòu)的復(fù)雜性，同時(shí)增加系統(tǒng)的部件也會(huì)降低系統(tǒng)的可靠度?，F(xiàn)在所介紹的旋轉(zhuǎn)壓電電機(jī)將是微型設(shè)備中的未來(lái)潛在驅(qū)動(dòng)裝置，這種馬達(dá)具有低速大轉(zhuǎn)矩，堵轉(zhuǎn)力矩高、結(jié)構(gòu)簡(jiǎn)單、響應(yīng)快等特點(diǎn)，可以將外形制成環(huán)形（應(yīng)用于光學(xué)，配線通過(guò)中心的電子儀表組件）。目前，一個(gè)關(guān)于超聲波電機(jī)在宇宙環(huán)境中工作情況的課題正在研究中，換句話說(shuō)，它能夠在低溫和真空的環(huán)境下有效可靠地運(yùn)行。<

5、/p><p>　　超聲波電機(jī)按工作模式劃分，可以分為靜態(tài)和動(dòng)態(tài)兩種；按運(yùn)動(dòng)方式可以分為旋轉(zhuǎn)式和直線式兩種；按執(zhí)行機(jī)構(gòu)的形狀可以分為梁式、桿式和板式等等。盡管它們之間有區(qū)別，但是他們的工作原理都是一樣，即利用壓電效應(yīng)產(chǎn)生的激勵(lì)：彈性體（通常與壓電陶瓷結(jié)合）的細(xì)小變形通過(guò)精確靜態(tài)機(jī)構(gòu)或者動(dòng)態(tài)諧振的方法擴(kuò)大。一些超聲波馬達(dá)已經(jīng)在一些要求結(jié)構(gòu)緊湊和做間歇運(yùn)動(dòng)的領(lǐng)域進(jìn)行產(chǎn)業(yè)化應(yīng)用。這些應(yīng)用包括：照相機(jī)的鏡頭自動(dòng)調(diào)焦、手表馬達(dá)以

6、及結(jié)構(gòu)緊湊的打字機(jī)。傳統(tǒng)電磁電機(jī)為了得到和超聲波電機(jī)一樣轉(zhuǎn)矩—速度特性，需要添加齒輪減速機(jī)構(gòu)，因此增加電機(jī)的尺寸、質(zhì)量和傳動(dòng)裝置的復(fù)雜性。超聲波電機(jī)有高的自鎖力，它能提供精確的零位移。此外，由于這些電機(jī)是依靠摩擦力矩驅(qū)動(dòng)的，所以在無(wú)外力的作用下產(chǎn)生反驅(qū)動(dòng)，因此讓人關(guān)注的與其他電機(jī)相比更高的堵轉(zhuǎn)扭矩。電機(jī)的組成部件的數(shù)量少代表了潛在故障點(diǎn)的數(shù)目會(huì)相應(yīng)減少。超聲波電機(jī)的優(yōu)良特性被人們所看好，將其應(yīng)用于有著體積小，間歇運(yùn)動(dòng)要求的機(jī)器人上。&l

7、t;/p><p>　　圖1為超聲波電機(jī)（環(huán)形行波超聲波電機(jī)）的工作原理。行波形成于由環(huán)形彈性體構(gòu)成的定子的表面上，并在轉(zhuǎn)子的表面產(chǎn)生橢圓運(yùn)動(dòng)。 定子表面質(zhì)點(diǎn)的橢圓運(yùn)動(dòng)驅(qū)動(dòng)轉(zhuǎn)子和與之相聯(lián)的軸旋轉(zhuǎn)。在定子表面添加齒槽結(jié)構(gòu)是用于增大振動(dòng)幅度，以此提高電機(jī)的轉(zhuǎn)速。超聲波電機(jī)的運(yùn)轉(zhuǎn)依靠運(yùn)動(dòng)的定子和轉(zhuǎn)子之間的接觸面產(chǎn)生的摩擦。這也是設(shè)計(jì)如何延長(zhǎng)接觸面的使用壽命的關(guān)鍵問(wèn)題。</p><p>　　圖1 旋

8、轉(zhuǎn)型行波超聲波電機(jī)工作原理示意圖</p><p><b>　　3. 工作原理</b></p><p>　　超聲波電機(jī)一般的工作原理是通過(guò)擴(kuò)大和重復(fù)振子的細(xì)小應(yīng)變來(lái)產(chǎn)生總的機(jī)械運(yùn)動(dòng)。振子引起與轉(zhuǎn)子相接觸的定子接觸面上的質(zhì)點(diǎn)產(chǎn)生一個(gè)軌跡運(yùn)動(dòng)，和在轉(zhuǎn)子與定子之間的分界面產(chǎn)生的摩擦，以此擴(kuò)大微小運(yùn)動(dòng)來(lái)產(chǎn)生定子的大運(yùn)動(dòng)。這一結(jié)構(gòu)如圖1所示。振子是壓電陶瓷受到激勵(lì)在定子內(nèi)部產(chǎn)生行

9、波，致使定子上的質(zhì)點(diǎn)做橢圓運(yùn)動(dòng)。在置于定子之上的轉(zhuǎn)子上施加預(yù)緊力和旋轉(zhuǎn)的定子和轉(zhuǎn)子之間產(chǎn)生摩擦力，依靠這些擴(kuò)大接觸面上的細(xì)微應(yīng)變。此運(yùn)動(dòng)的轉(zhuǎn)換過(guò)程與齒輪機(jī)構(gòu)類(lèi)似，產(chǎn)生與行波頻率相比更低的旋轉(zhuǎn)速度。</p><p>　　定子的下層的厚度設(shè)為，在定子粘有一定厚度的一組壓電體，這些壓電體按照一定的順序和位置與定子的后表面結(jié)合，壓電陶瓷的厚度設(shè)為?？偤穸葹椋@是壓電陶瓷的厚度與定子的厚度之和（其中粘結(jié)層厚度忽略不計(jì)）。整

10、體高度可以隨著徑向位置變化而變化。定子的外半徑為，內(nèi)孔半徑為。為了產(chǎn)生行波，由兩個(gè)相差四分之一的波長(zhǎng)信號(hào)構(gòu)成壓電陶瓷的極化方向，這樣的極化方式也能被用來(lái)消除定子的范圍和最大撓曲。定子上的齒槽在徑向位置上成環(huán)形分布。</p><p>　　為了在定子內(nèi)部產(chǎn)生行波，需要同時(shí)激勵(lì)出兩個(gè)相同的正交振型。在同一模式中，兩個(gè)極化節(jié)粘于定子上，以此構(gòu)成由壓電驅(qū)動(dòng)器，這就是模型。從幾何學(xué)上分析這個(gè)模型，結(jié)果表明激勵(lì)出兩個(gè)狀態(tài)分別為

11、和信號(hào)，將會(huì)產(chǎn)生頻率為的行波。同時(shí)，通過(guò)改變驅(qū)動(dòng)信號(hào)的工作狀態(tài)，行波的方向也會(huì)相應(yīng)地發(fā)現(xiàn)變化。</p><p><b>　　4. 理論模型</b></p><p>　　超聲波電機(jī)的運(yùn)動(dòng)方程源于漢密爾頓原理，這個(gè)分析模型被許多學(xué)者所推導(dǎo)過(guò)（比如Hagood、A. McFarland和Kagawa等）。定子的通用運(yùn)動(dòng)方程歸納如下：</p><p>

12、　　式中，[M]、[C]、[K]、[P]、[G]分別為質(zhì)量矩陣、阻尼矩陣、剛度矩陣、機(jī)電耦合矩陣和電容矩陣，矢量{x}、{j}、{}、{}和{Q}分別是模型的振幅、電勢(shì)正常外力向量、切向力矢量和電荷矢量。振幅矢量{x}和其他廣義矢量能夠通過(guò)能量平衡原理定義，如Rayleigh Ritz 原理。但是，這個(gè)方法忽略了定子上的齒槽的作用。環(huán)形定子也會(huì)隨著內(nèi)支撐板徑向位置的變化而變化，這可能會(huì)導(dǎo)致不合要求的結(jié)果出現(xiàn)。即使三維有限元分析方法（FE

13、M）可以精確預(yù)測(cè)模型的固有頻率和定子的瞬態(tài)響應(yīng)特性，但這是一個(gè)復(fù)雜的計(jì)算過(guò)程。此外，決定設(shè)計(jì)模型往往需要通過(guò)三維有限元分析軟件核實(shí)計(jì)算響應(yīng)模型和共振頻率。由于此方法的所提及的缺點(diǎn)，需要改進(jìn)過(guò)去所描述的周期性有限元，這也是基于超聲波馬達(dá)的對(duì)稱(chēng)特性。環(huán)形有限元如圖2所示，其中都是自由度。橫向移動(dòng)量穿過(guò)每個(gè)部分，其表現(xiàn)方程如下：</p><p>　　式中，表示徑向振動(dòng)頻率，指標(biāo)m、n分別是沿著q和r方向的模型。當(dāng)假設(shè)橫

14、向切力和旋轉(zhuǎn)慣性效應(yīng)忽略不計(jì)，質(zhì)量和剛度矩陣能按照標(biāo)準(zhǔn)變化理論推導(dǎo)。因此，解決特征值問(wèn)題可以得到正常頻率和模型的外形。</p><p>　　用標(biāo)準(zhǔn)的公式表示，其中包括了定子齒槽的作用。其他廣義坐標(biāo)的制定細(xì)節(jié)也和這些類(lèi)似確定。這些將會(huì)在作者以后的出版物中提及。</p><p>　　5. 對(duì)壓電電機(jī)的分析</p><p>　　對(duì)非線性、定子—轉(zhuǎn)子之間的動(dòng)態(tài)聯(lián)接模型分析時(shí)

15、，主要討論的內(nèi)容包括預(yù)測(cè)電機(jī)的潛在穩(wěn)定狀態(tài)和在臨界設(shè)計(jì)參數(shù)的情況下電機(jī)的運(yùn)行瞬態(tài)性能，比如接觸面上的法向力、齒高、定子的徑向切面。有限元的運(yùn)算法則被融入分析軟件中，MATLAB的代碼被用于確定定子模型的特征。模型反應(yīng)出定子的形狀、壓電陶瓷的極化模式和定子齒的相關(guān)參數(shù)。一旦選定定子的每個(gè)細(xì)節(jié)，那么模型的響應(yīng)也確定了。這也可以在電腦中進(jìn)行實(shí)時(shí)監(jiān)測(cè)，如圖2所示，此時(shí)的模型中的參數(shù)已經(jīng)給定，（m，n）=（4，0）。利用電子點(diǎn)模式的干涉測(cè)量?jī)x驗(yàn)證

16、預(yù)測(cè)的模型響應(yīng)特性，結(jié)果非常直觀，如圖3（左）。</p><p>　　MATLAB成為觀察超聲波電機(jī)工作狀態(tài)一種新的工具，能夠在電腦上模擬仿真。該軟件能夠模擬旋轉(zhuǎn)電機(jī)中彎曲行波在定子中工作狀態(tài)（圖4）。</p><p>　　圖2 環(huán)形有限元分析模型</p><p>　　圖3 模態(tài)響應(yīng)和共振頻率（左圖）和實(shí)驗(yàn)檢測(cè)（右圖）</p><p>　　采

17、用有限元的分析模型，以此構(gòu)建馬達(dá)。表1為直徑為1.71英寸鋼結(jié)構(gòu)定子所預(yù)測(cè)的振型和精確的共振頻率。在此表中的結(jié)果顯示理論值和實(shí)際值相對(duì)吻合，為了</p><p>　　圖4 利用動(dòng)畫(huà)展示超聲波馬達(dá)的工作原理。</p><p>　　定子以行波的形式運(yùn)動(dòng)，轉(zhuǎn)子在定子的上面旋轉(zhuǎn)。</p><p>　　表1 一個(gè)超聲波馬達(dá)的共振頻率的理論值和實(shí)驗(yàn)值</p>&

18、lt;p>　　圖7 在溫度為與真空度為的環(huán)境下，</p><p>　　直徑為1.1英寸的超聲波電機(jī)的實(shí)驗(yàn)檢測(cè)到的轉(zhuǎn)矩—速度曲線</p><p>　　檢測(cè)真空和低溫對(duì)馬達(dá)的影響。一個(gè)直徑為1.1英寸的超聲波馬達(dá)在一個(gè)低溫實(shí)驗(yàn)室進(jìn)行測(cè)試，此實(shí)驗(yàn)利用SATEC系統(tǒng)，實(shí)驗(yàn)測(cè)試轉(zhuǎn)矩與速度的曲線如圖7所示。結(jié)果表明在進(jìn)行伺服控制的馬達(dá)能夠在溫度低于和真空度為的環(huán)境下非常穩(wěn)定的運(yùn)行。這一結(jié)果是一個(gè)

19、鼓勵(lì)，同時(shí)也意味著在未來(lái)決定超聲波馬達(dá)能否在火星的模擬環(huán)境下運(yùn)轉(zhuǎn)的研究中還有需要的工作要做。</p><p><b>　　6. 結(jié)論</b></p><p>　　有限元模型被用來(lái)分析超聲波電機(jī)的光譜響應(yīng)，包括各式各樣的外形結(jié)構(gòu)和組成材料的超聲波電機(jī)。模態(tài)響應(yīng)和預(yù)測(cè)的共振情況可以利用實(shí)驗(yàn)的方法確定，其中有光譜測(cè)量法和干涉分析法。此外，還有像MATLAB這類(lèi)簡(jiǎn)單的分析平臺(tái)

20、的交互式用戶界面軟件分析超聲波電機(jī)的模態(tài)行為。同時(shí)還可以用于研究各種定子參數(shù)。</p><p><b>　　致謝</b></p><p>　　在此感謝MIT航空宇航研究中心的，Nesbitt .W .Hagood IV。感謝他在IRTWG項(xiàng)目的合作期間給我的幫助．本文中結(jié)果的原稿從行星靈巧的操作者的課題中獲得。這課題由Dr. Paul Schenker負(fù)責(zé)，由加利福尼

21、亞大學(xué)噴氣推進(jìn)實(shí)驗(yàn)中心出資，同時(shí)與美國(guó)航天宇航局簽訂協(xié)議。Mr. David和Dr. Chuck Weisbin是TRIGW項(xiàng)目的負(fù)責(zé)人。</p><p><b>　　參考文獻(xiàn)</b></p><p>　　M. Hollerbach, I. W. Hunter and J. Ballantyne, "A Comparative Analysis of Act

22、uator Technologies for Robotics." In Robotics Review 2, MIT Press, Edited by Khatib, Craig and Lozano-Perez (1991). </p><p>　　A. M. Flynn, et al "Piezoelectric Micromotors for Microrobots" J.

23、of MEMS, Vol. 1, No. 1, (1992), pp. 44-51. </p><p>　　E. Inaba, et al, "Piezoelectric Ultrasonic Motor," Proceedings of the IEEE Ultrasonics 1987 Symposium, pp. 747-756, (1987). </p><p>

24、;　　J. Wallashek, "Piezoelectric Motors," J. of Intelligent Materials Systems and Structures, Vol. 6, (Jan. 1995), pp. 71-83. </p><p>　　N. W. Hagood and A. McFarland, "Modeling of a Piezoelectr

25、ic Rotary Ultrasonic Motor," IEEE Transactions on Ultrasonics, Ferroelectrics and Frequency Control, Vol. 42, No. 2, 1995 pp. 210-224. </p><p>　　K. Kagawa, T. Tsuchiya and T. Kataoka, "Finite Eleme

26、nt Simulation of Dynamic Responses of Piezoelectric Actuators,", J. of Sound and Vibrations, Vol. 89 (4), 1996, pp. 519-538. </p><p>　　D. G. Gorman, "Natural Frequencies of Transverse Vibration of

27、Polar Orthotropic Variable Thickness Annular Plates, " J. of Sound and Vibrations</p><p>　　Rotary Ultrasonic Motors Actuated By Traveling Flexural Waves</p><p>　　Shyh-Shiuh Lih, Yoseph Bar-

28、Cohen,Jet Propulsion Laboratory, California Institute of Technology, Pasadena, CA 91109yosi@jpl.nasa.gov and Willem Grandia, Quality Material Inspection (QMI), Costa Mesa, CA 92627</p><p>　　1.ABSTRACT&

29、lt;/p><p>　　Ultrasonic rotary motors are being developed as actuators for miniature spacecraft instruments and subsystems. The technology that has emerged in commercial products requires rigorous analytical too

30、ls for effective design of such motors. An analytical model was developed to examine the excitation of flexural plate wave traveling in a rotary piezoelectrically actuated motor. The model uses annular finite elements th

31、at are applied to predict the excitation frequency and modal response of the annul</p><p>　　Key Words: Actuators, Active Materials, Piezoelectric Motors, Ultrasonic Motors (USMs), Stators and Rotors, Modal A

32、nalysis.  </p><p>　　2. INTRODUCTION </p><p>　　The recent NASA efforts to reduce the size and mass of future spacecraft are straining the specifications of actuation and articulati

33、on mechanisms that drive planetary instruments. The miniaturization of conventional electromagnetic motors is limited by manufacturing constrains. Generally, these type of motors compromise speed for torque using speed r

34、educing gears. The use of gear adds mass, volume and complexity as well as reduces the system reliability due the increase in the number of the sys</p><p>　　Ultrasonic motors [5] can be classified by their m

35、ode of operation (static or resonant), type of motion (rotary or linear) and shape of implementation (beam, rod, disk, etc.). Despite the distinctions, the fundamental principles of solid-state actuation tie them togethe

36、r: microscopic material deformations (usually associated with piezoelectric materials) are amplified through either quasi-static mechanical or dynamic/resonant means. Several of the motor classes have seen commercial app

37、lication i</p><p>　　In Figure 1 the principle of operation of an ultrasonic motor (flexural traveling wave ring-type motor) is shown as an example. A traveling wave is established over the stator surface, wh

38、ich behaves as an elastic ring, and produces elliptical motion at the interface with the rotor. This elliptical motion of the contact surface propels the rotor and the drive-shaft connected to it. The teeth, which are at

39、tached to the stator, are intended to increase the moment arm to amplify the speed. The opera</p><p>　　Figure 1. Principle of Operation of a Rotary Traveling Wave Motor. </p><p>　　3. PRINCIPLE O

40、F OPERATION</p><p>　　The general principle of the operation of ultrasonic motors is to generate gross mechanical motion through the amplification and repetition of micro-deformations of active material. The

41、active material induces an orbital motion of the stator at the rotor contact points and frictional interface between the rotor and stator rectifies the micro-motion to produce macro-motion of the stator. This mechanism i

42、s illustrated in shown in Figure 1. The active material, which is a piezoelectric material exc</p><p>　　A stator substrate is assumed to have a thickness, tS, with a set of piezoelectric crystals that are bo

43、nded to the back surface of the stator in a given pattern of poling sequence and location. The thickness of the piezoelectric crystals is tp. The total height, h, is the sum of the thickness of the crystals and the stato

44、rs (bonding layer is neglected). The overall height of the stator is also allowed to vary with radial position. The outer radius of the disk is b and the inner hole radius is a.</p><p>　　4. THEORETICAL MODEL

45、ING</p><p>　　The equation of motion of the ultrasonic motor can be derived from Hamilton’s principle. The analytical model has been derived by many authors (e.g. Hagood and A. McFarland [5], Kagawa et al [6]

46、). The generalized equation of motion of the stator can be summarized as </p><p>　　where [M], [C], [K], [P], [G], are the mass, damping, stiffness, electromechanical coupling, and capacitance matrices,

47、respectively. The vectors {x }, {j }, {FN} , {FT}, and {Q} are the model amplitude, the electric potential vectors the normal external force, the tangential external force and the charge vectors, respectively. The modal

48、amplitude {x } and other generalized coordinates can be defined through energy methods such as Rayleigh Ritz method [5]. However, this method smears the contribu</p><p>　　, for R1 < R2 </p><p&g

49、t;　　where w nm is the radial resonance frequency and the index m, n are mode along the q and r direction, respectively. If we assume that the transverse shear and rotary inertial effects are negligible, the elemental mas

50、s, stiffness can be derived using the standard variational methods. Thus, the natural frequency and modal shape can be found by solving the eigenvalue problem. </p><p>　　Using consistent mass formulatio

51、ns, the effect of the stator teeth can also be included. Details of the formulation of other generalized coordinates are treated similar to those in [7] and will be presented by the authors’ in a future publication. 

52、; </p><p>　　5. ANALYSIS OF PIEZOELECTRIC MOTORS </p><p>　　The analysis of the nonlinear, coupled rotor-stator dynamic model discussed above has demonstrated the potential to predicting

53、 motor steady state and transient performance as a function of critical design parameters such as interface normal force, tooth height, and stator radial cross section. A finite element algorithm was incorporated into th

54、e analysis and a MATLAB code was developed to determine the modal characteristics of the stator. The model accounts for the shape of the stator, the piezo</p><p>　　Using MATLAB we developed an animation tool

55、 to view the operation of USMs on the computer display. The tool allows to show the rotation of the rotor while a flexural wave is traveling on the stator (Figure 4). </p><p>　　Figure 2: An annular fini

56、te element.</p><p>　　Figure 3: Modal response and resonance frequency (left) and experimental verification (right).</p><p>　　Figure 4: Animation tool for viewing the operation of USM. The stator

57、 is shown with traveling wave and the rotor is rotating above the stator.</p><p>　　Using this analytical model that employs finite element analysis, motors were constructed. The predicted resonance and measu

58、red resonance frequency for a 1.71-in diameter steel stator are represented in Table 1. The results that are presented in this table are showing an excellent agreement between the calculated and measured data. To examine

59、 the effect of vacuum and low temperatures, a 1.1 inch USM was also tested in a cryo-vac chamber that was constructed using a SATEC system and the torque sp</p><p>　　TABLE 1. The measured and calculated reso

60、nance frequencies of a USM’s stator. </p><p>　　Figure 7. Measured torque-speed curve for a 1.1-inch diameter USM at -48o C and 2x10-2 Torr.</p><p>　　6. CONCLUSIONS</p><p>　　A f

61、inite element model was developed to analyze the spectral response of ultrasonic motors with various geometrical configurations and construction materials. The modal response and the predicted resonance conditions were c

62、orroborated experimentally using spectral measurements and interferometric analysis. Further, user interface interactive tools were developed for a MATLAB platform simplifying the analysis of the modal behavior of USMs a

63、nd allowing the study of their response to various stator</p><p>　　ACKNOWLEDGMENT</p><p>　　The authors would like to thank Nesbitt. W. Hagood IV, Aeronautics and Astronautics, MIT, for his assis

64、tance in this study under a TRIWG contract. The results reported in this manuscript were obtained under the Planetary Dexterous Manipulator Task, that is managed by Dr. Paul Schenker and it is a TRIWG task that is funded

65、 by a JPL, Caltech, contract with NASA Headquarters, Code S, Mr. David Lavery and Dr. Chuck Weisbin are the Managers of TRIWG.</p><p>　　REFERENCES </p><p>　　M. Hollerbach, I. W. Hunter and

66、J. Ballantyne, "A Comparative Analysis of Actuator Technologies for Robotics." In Robotics Review 2, MIT Press, Edited by Khatib, Craig and Lozano-Perez (1991). </p><p>　　A. M. Flynn, et al "P

67、iezoelectric Micromotors for Microrobots" J. of MEMS, Vol. 1, No. 1, (1992), pp. 44-51. </p><p>　　E. Inaba, et al, "Piezoelectric Ultrasonic Motor," Proceedings of the IEEE Ultrasonics 1987 Sy

68、mposium, pp. 747-756, (1987). </p><p>　　J. Wallashek, "Piezoelectric Motors," J. of Intelligent Materials Systems and Structures, Vol. 6, (Jan. 1995), pp. 71-83. </p><p>　　N. W. Hagood

69、 and A. McFarland, "Modeling of a Piezoelectric Rotary Ultrasonic Motor," IEEE Transactions on Ultrasonics, Ferroelectrics and Frequency Control, Vol. 42, No. 2, 1995 pp. 210-224. </p><p>　　K. Kaga

70、wa, T. Tsuchiya and T. Kataoka, "Finite Element Simulation of Dynamic Responses of Piezoelectric Actuators,", J. of Sound and Vibrations, Vol. 89 (4), 1996, pp. 519-538. </p><p>　　D. G. Gorman, &qu