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Rotary Ultrasonic Motors Actuated By Traveling Flexural WavesShyh-Shiuh Lih, Yoseph Bar-Cohen,Jet Propulsion Laboratory, California Institute of Technology, Pasadena, CA 91109 andWillem Grandia, Quality Material Inspection (QMI), Costa Mesa, CA 926271.ABSTRACT 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 tools 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 that are applied to predict the excitation frequency and modal response of the annular stator. This model allows to design efficient ultrasonic motors (USMs) and it incorporates the details of the stator which include the teeth, piezoelectric crystals, stator geometry, etc. The theoretical predictions and the experimental corroboration showed a remarkable agreement. Parallel to this effort, USMs are made and incorporated into a robotic arm and their capability to operate at the environment of Mars is being studied. Key Words: Actuators, Active Materials, Piezoelectric Motors, Ultrasonic Motors (USMs), Stators and Rotors, Modal Analysis.2. INTRODUCTIONThe recent NASA efforts to reduce the size and mass of future spacecraft are straining the specifications of actuation and articulation 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 reducing 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 system components. The recent introduction of rotary piezoelectric motors is offering potential drive mechanisms for miniature instruments 1-5. These motors offer high torque density at low speed, high holding torque, simple construction, can be made in annular shape (for optical application, electronic packaging and wiring through the center), and have a quick response. A study is underway to develop such motors for operation at space environment, namely, operate effectively and reliably at temperatures down to cryogenic levels and vacuum.Ultrasonic motors 5 can be classified by their mode 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 together: 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 application in areas needing compact, efficient, and intermittent motion. Such applications include: camera auto focus lenses, watch motors and compact paper handling. To obtain the levels of torque-speed characteristics of USMs using conventional motors requires adding a gear system to reduce the speed, thus increasing the size, mass and complexity of the drive mechanism. USMs are fundamentally designed to have a high holding force, providing effectively zero backlash. Further, since these motors are driven by friction the torque that would cause them to be backdriven at zero power is significantly higher than the stall torque. The number of components needed to construct the motor is small minimizing the number of potential failure points. The general characteristic of USMs makes them attractive for robotic applications where small, intermittent motions are required. 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, which 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 attached to the stator, are intended to increase the moment arm to amplify the speed. The operation of USM depends on friction at the interface between the moving rotor and stator, which is a key issue in the design of this interface for extended lifetime. Figure 1. Principle of Operation of a Rotary Traveling Wave Motor. 3. PRINCIPLE OF OPERATIONThe 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 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 is illustrated in shown in Figure 1. The active material, which is a piezoelectric material excites a traveling flexural wave within the stator that leads to elliptical motion of the surface particles. Teeth are used to enhance the speed that is associated with the propelling effect of these particles. The rectification of the micro-motion an interface is provided by pressing the rotor on top of the stator and the frictional force between the two causes the rotor to spin. This motion transfer operates as a gear leads to a much lower rotation speed than the wave frequency. A stator substrate is assumed to have a thickness, tS, with a set of piezoelectric crystals that are bonded 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 stators (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. To generate traveling wave, the piezoelectric crystals poling direction is structured such that quarter wavelength out-of-phase is formed. This poling pattern is also intended to eliminate extension in the stator and maximize bending. The teeth on the stator are arranged in a ring at the radial position.4. THEORETICAL MODELINGThe equation of motion of the ultrasonic motor can be derived from Hamiltons principle. The analytical model has been derived by many authors (e.g. Hagood and A. McFarland 5, Kagawa et al 6). The generalized equation of motion of the stator can be summarized aswhere M, C, K, P, G, are the mass, damping, stiffness, electromechanical coupling, and capacitance matrices, 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 amplitude x and other generalized coordinates can be defined through energy methods such as Rayleigh Ritz method 5. However, this method smears the contribution of the teeth and the variation of the stator ring as well as the support disk along the radial direction and may lead to undesirable results. Even though, 3-D finite element method (FEM) was reported 6 to be used to accurately predict the modal frequencies and transient response of the stator, it is computational intensive process. Further, the calculated response modes and associated frequencies that are determined by the 3-D FEM needs to be identified visually to find the designed mode. Due to the disadvantages for the methods mentioned above the modified annual finite element described in 7 is used and it is based on the symmetrical characteristics of the ultrasonic motors. The annular finite element is shown as in Fig. 2, where w1, w2 y 1, and y 2 are the degree of freedoms. The transverse displacement w across each element is assumed to be of the form given by the equation , for R1 R2 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 mass, stiffness can be derived using the standard variational methods. Thus, the natural frequency and modal shape can be found by solving the eigenvalue problem.Using consistent mass formulations, 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.5. ANALYSIS OF PIEZOELECTRIC MOTORSThe analysis of the nonlinear, coupled rotor-stator dynamic model discussed above has demonstrated the potential to predicting 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 the 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 piezoelectric poling pattern, and the teeth parameters. Once the details of the stators are selected the modal response is determined and is presented on the computer monitor, as shown for example in Figure 2, where the mode (m, n) = (4, 0) is presented. An electronic speckle pattern interferometry was used to corroborate the predicted modal response and the agreement seems to be very good as can be seen in Figure 3 on the left. Using MATLAB we developed an animation tool 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). Figure 2: An annular finite element.Figure 3: Modal response and resonance frequency (left) and experimental verification (right).Figure 4: Animation tool for viewing the operation of USM. The stator is shown with traveling wave and the rotor is rotating above the stator.Using this analytical model that employs finite element analysis, motors were constructed. The predicted resonance and measured 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 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 speed was measured as shown in Figure 7. The motor that was servo-controlled showed a remarkable stable performance down to about -48oC and vacuum at the level of 2x10-2 Torr. This result is very encouraging and more work will be done in the future to determine the requirements for operation of USMs at Mars simulated conditions.TABLE 1. The measured and calculated resonance frequencies of a USMs stator.Figure 7. Measured torque-speed curve for a 1.1-inch diameter USM at -48o C and 2x10-2 Torr.6. CONCLUSIONSA finite 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 corroborated 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 and allowing the study of their response to various stator parameters. ACKNOWLEDGMENT The authors would like to thank Nesbitt. W. Hagood IV, Aeronautics and Astronautics, MIT, for his assistance 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 by a JPL, Caltech, contract with NASA Headquarters, Code S, Mr. David Lavery and Dr. Chuck Weisbin are the Managers of TRIWG.REFERENCES1. M. Hollerbach, I. W. Hunter and J. Ballantyne, A Comparative Analysis of Actuator Technologies for Robotics. In Robotics Review 2, MIT Press, Edited by Khatib, Craig and Lozano-Perez (1991). 2. A. M. Flynn, et al Piezoelectric Micromotors for Microrobots J. of MEMS, Vol. 1, No. 1, (1992), pp. 44-51. 3. E. Inaba, et al, Piezoelectric Ultrasonic Motor, Proceedings of the IEEE Ultrasonics 1987 Symposium, pp. 747-756, (1987). 4. J. Wallashek, Piezoelectric Motors, J. of Intelligent Materials Systems and Structures, Vol. 6, (Jan. 1995), pp. 71-83. 5. N. W. Hagood 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. 6. K. Kagawa, 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. 7. D. G. Gorman, Natural Frequencies of Transverse Vibration of Polar Orthotropic Variable Thickness Annular Plates, J. of Sound and Vibrations旋转型行波超声电机帕萨迪纳，CA91109，加利福尼亚理工学院喷气推进实验室；科斯塔梅萨，CA92627，材料质量检测中心，威廉梅兰迪亚。摘要：旋转型超声波电机逐渐发展为太空飞船的微型驱动器及其子系统。此技术应用于有着严格要求的商业产品中，为了更加有效地设计此类电机而采用分析工具。分析模型用于检测在旋转超声电机中激励产生的弯曲行波。这个有限元分析模型为环形，被用于预测环形定子的振动频率和模态响应。此模型给设计高效率的超声波电机提供依据，定子的设计包括齿槽、压电体、定子的几何外形等方面，定子是由他们有机地组合而成。理论计算值与实验结果的比较表明这将是一个值得世人所关注的课题。与此同时，超声波电机还被用于机械臂，他们是否能够在火星的环境下正常运行的研究还在进行中。关键词：驱动器，弹性体，压电电机，超声波电机，定子与转子，模态分析。2. 绪论当前，美国国家航空和宇宙航行局一直致力于缩小未来太空飞船的体积和减少其质量的研究。为了与这变化想适应，超声波电机逐渐成为机械装置简化的一个重要的手段。传统的微型电磁式电机由于受制造工艺的限制，一般这类电机为了达到速度与扭矩相适应需要使用齿轮减速机构，采用这个将会增加设备的质量、体积和机构的复杂性，同时增加系统的部件也会降低系统的可靠度。现在所介绍的旋转压电电机将是微型设备中的未来潜在驱动装置，这种马达具有低速大转矩，堵转力矩高、结构简单、响应快等特点，可以将外形制成环形（应用于光学，配线通过中心的电子仪表组件）。目前，一个关于超声波电机在宇宙环境中工作情况的课题正在研究中，换句话说，它能够在低温和真空的环境下有效可靠地运行。超声波电机按工作模式划分，可以分为静态和动态两种；按运动方式可以分为旋转式和直线式两种；按执行机构的形状可以分为梁式、杆式和板式等等。尽管它们之间有区别，但是他们的工作原理都是一样，即利用压电效应产生的激励：弹性体（通常与压电陶瓷结合）的细小变形通过精确静态机构或者动态谐振的方法扩大。一些超声波马达已经在一些要求结构紧凑和做间歇运动的领域进行产业化应用。这些应用包括：照相机的镜头自动调焦、手表马达以及结构紧凑的打字机。传统电磁电机为了得到和超声波电机一样转矩速度特性，需要添加齿轮减速机构，因此增加电机的尺寸、质量和传动装置的复杂性。超声波电机有高的自锁力，它能提供精确的零位移。此外，由于这些电机是依靠摩擦力矩驱动的，所以在无外力的作用下产生反驱动，因此让人关注的与其他电机相比更高的堵转扭矩。电机的组成部件的数量少代表了潜在故障点的数目会相应减少。超声波电机的优良特性被人们所看好，将其应用于有着体积小，间歇运动要求的机器人上。图1为超声波电机（环形行波超声波电机）的工作原理。行波形成于由环形弹性体构成的定子的表面上，并在转子的表面产生椭圆运动。 定子表面质点的椭圆运动驱动转子和与之相联的轴旋转。在定子表面添加齿槽结构是用于增大振动幅度，以此提高电机的转速。超声波电机的运转依靠运动的定子和转子之间的接触面产生的摩擦。这也是设计如何延长接触面的使用寿命的关键问题。图1 旋转型行波超声波电机工作原理示意图3. 工作原理超声波电机一般的工作原理是通过扩大和重复振子的细小应变来产生总的机械运动。振子引起与转子相接触的定子接触面上的质点产生一个轨迹运动，和在转子与定子之间的分界面产生的摩擦，以此扩大微小运动来产生定子的大运动。这一结构如图1所示。振子是压电陶瓷受到激励在定子内部产生行波，致使定子上的质点做椭圆运动。在置于定子之上的转子上施加预紧力和旋转的定子和转子之间产生摩擦力，依靠这些扩大接触面上的细微应变。此运动的转换过程与齿轮机构类似，产生与行波频率相比更低的旋转速度。定子的下层的厚度设为，在定子粘有一定厚度的一组压电体，这些压电体按照一定的顺序和位置与定子的后表面结合，压电陶瓷的厚度设为。总厚度为，这是压电陶瓷的厚度与定子的厚度之和（其中粘结层厚度忽略不计）。整体高度可以随着径向位置变化而变化。定子的外半径为，内孔半径为。为了产生行波，由两个相差四分之一的波长信号构成压电陶瓷的极化方向，这样的极化方式也能被用来消除定子的范围和最大挠曲。定子上的齿槽在径向位置上成环形分布。为了在定子内部产生行波，需要同时激励出两个相同的正交振型。在同一模式中，两个极化节粘于定子上，以此构成由压电驱动器，这就是模型。从几何学上分析这个模型，结果表明激励出两个状态分别为和信号，将会产生频率为的行波。同时，通过改变驱动信号的工作状态，行波的方向也会相应地发现变化。4. 理论模型超声波电机的运动方程源于汉密尔顿原理，这个分析模型被许多学者所推导过（比如Hagood、A. McFarland和Kagawa等）。定子的通用运动方程归纳如下：式中，M、C、K、P、G分别为质量矩阵、阻尼矩阵、刚度矩阵、机电耦合矩阵和电容矩阵，矢量x、j、和Q分别是模型的振幅、电势正常外力向量、切向力矢量和电荷矢量。振幅矢量x和其他广义矢量能够通过能量平衡原理定义，如Rayleigh Ritz 原理。但是，这个方法忽略了定子上的齿槽的作用。环形定子也会随着内支撑板径向位置的变化而变化，这可能会导致不合要求的结果出现。即使三维有限元分析方法（FEM）可以精确预测模型的固有频率和定子的瞬态响应特性，但这是一个复杂的计算过程。此外，决定设计模型往往需要通过三维有限元分析软件核实计算响应模型和共振频率。由于此方法的所提及的缺点，需要改进过去所描述的周期性有限元，这也是基于超声波马达的对称特性。环形有限元如图2所示，其中都是自由度。横向移动量穿过每个部分，其表现方程如下：式中，表示径向振动频率，指标m、n分别是沿着q和r方向的模型。当假设横向切力和旋转惯性效应忽略不计，质量和刚度矩阵能按照标准变化理论推导。因此，解决特征值问题可以得到正常频率和模型的外形。用标准的公式表示，其中包括了定子齿槽的作用。其他广义坐标的制定细节也和这些类似确定。这些将会在作者以后的出版物中提及。5. 对压电电机的分析对非线性、定子转子之间的动态联接模型分析时，主要讨论的内容包括预测电机的潜在稳定状态和在临界设计参数的情况下电机的运行瞬态性能，比如接触面上的法向力、齿高、定子的径向切面。有限元的运算法则被融入分析软件中，MATLAB的代码被用于确定定子模型的特征。模型反应出定子的形状、压电陶瓷的极化模式和定子齿的相关参数。一旦选定定子的每个细节，那么模型的响应也确定了。这也可以在电脑中进行实时监测，如图2所示，此时的模型中的参数已经给定，（m，n）=（4，0）。利用电子点模式的干涉测量仪验证预测的模型响应特性，结果非常直观，如图3（左）。MATLAB成为观察超声波电机工作状态一种新的工具，能够在电脑上模拟仿真。该软件能够模拟旋转电机中弯曲行波在定子中工作状态（图4）。图2 环形有限元分析模型图3 模态响应和共振频率（左图）和实验检测（右图）采用有限元的分析模型，以此构建马达。表1为直径为1.71英寸钢结构定子所预测的振型和精确的共