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翻译部分英文原文Parametric study of dynamics of worm and worm-gear set undersuddenly applied rotating angleM.Y. Chung, D. ShawDepartment of Power Mechanical Engineering, National Tsing-Hua University, No. 101 Kung Fu Road Section II, Hsin Chu, Taiwan, ROCReceived 13 September 2006; received in revised form 23 January 2007; accepted 25 February 2007Available online 17 April 2007AbstractThe dynamics of a worm-gear set under instantly applied rotating angle are affected by several factors (including thefriction force and elastic deformation of the surface between gear teeth). It is found that those factors cause a nonnegligible rotational positioning error. The goals of this study are to (1) set up a mathematic model, (2) carry out anumerical simulation, and (3) carry out an experiment and compare it with the numerical results. The experiments and numerical results have very good agreement. In this research, themoment-inertia of fly wheel, friction, rigidity of shaft andrigidity of gear tooth are also studied. The results can be used as error estimations of relevant angles, angular speed andangular acceleration under a suddenly applied rotational angle, and are useful for establishing the error compensationrequired for position control.1. IntroductionWhen radar tracks an aircraft which turns at a sharp angle; the radar may not be able to lock onto the target due to the small oscillation of the antenna. Sometimes, the oscillation even causes a failure of the radar system . Accurate positioning is a crucial subject in a tracking system. However, due to the design of the trackingtransmission mechanism is restricted by both space and weight, the volume and the weight of the transmissionmechanism must be limited. The worm-gear set is a good choice due to its small volume and high reduction ratio. Owing to the oscillation of the antenna is an important dynamics behavior for high precision positioningof radar system, the dynamics of a worm-gear set under instantly applied large torque is very important tounderstand this behavior. This behavior is affected by several factors including the friction force and elasticdeformation of the surface between gear teeth. To understand the effect of those factors, a lot of research hasbeen conducted. Yuksel and Kahraman 1 studied the in.uence of surface wear on the dynamic behavior of a typical planetary gear set. The wear model employed a quasi-static gear contact model to compute contact pressures and Archards wear model to determine the wear depth distributions. Parker et al. 2 analyzed the dynamic response of a spur pair of a wide range of operating speeds and torques. Comparisons were made to other researchers published experiments that reveal complex nonlinear phenomena. The dynamic response of a spur pair was investigated using a .nite element/contact mechanics model that offers signi.cant advantages for dynamic gear analyses. Maliha 3 created a nonlinear dynamic model of a spur gear pair, which was coupled with linear .nite element models of the shafts carrying them, and with discrete models of bearings and disks. The excitations considered in the model were external static torque and internal excitation caused by mesh stiffness variation, gear errors and gear tooth pro.le modi.cation. Britton 4 produced a super .nished gear teeth (with a approximately 0.05 mm Ra Film) and the friction traction in the experiments were simulated theoretically using a thin .lm non-Newtonian micro-elastohydrodynamic lubrication solver and encouraging agreement between friction measurements and theoretical predictions was obtained. Kong 5 predicted elastic contact and elastohydrodynamic .lm thickness in worm gears. Using the undeformed geometry of the gap between gear teeth in contact a three-dimensional elastic contact simulation technique had been developed for calculation of the true area of elastic contact under load relative the wheel and worm surfaces. Tuttle 6 studied the harmonic drives, which exhibited very nonlinear dynamic behavior, in his model not only dynamic models include accurate representation of transmission friction, compliance and kinematical error were understand, but also important features of harmonic-drive gear-tooth geometry and interaction. Experimental observations were used to guide the development of a model to describe harmonic-drive operation.It is important to be aware that worm-gear performance is in.uenced by the lubricant applied and maintained. Helouvry 7 discussed the issue of the servo-orientation control being affected by a surge of both maximum static friction force and dynamic friction force. They reported the occurrence of a stable positioning error, as well as the stopping (or stoking) during tracking at the turning point of a limiting loop of the tracking system. When the system was in a one directional low-speed tracking, it was possible for the stickslip phenomenon to occur due to the fact that the frictional force to velocity curve appears to be a negative slope and in.uences the static friction force. It was also found that a similar phenomenon occurs at high-speed tracking. Doupont 8 proposed a frictional force model with negative slope. In the reference, relative motion from static to high-speed sliding was divided into four stages. In the .rst stage, the external force was less than the maximum static friction force. In the second stage, two contacting surfaces started to develop a small relative velocity. In the third stage, the relative velocity started to increase between two objects, and lubricant got into the contact area and decreased the frictional force. In the fourth stage, the lubricant had fully .lled the contact area, and the viscous characteristic and the frictional force increased with the velocity. Oguri design handbook 9 included several empirical charts and calculation formulae of the friction coef.cient vs. sliding velocity, the allowable error of manufacturing, the recommended value for the gaps between two teeth, and the allowable contact pressure. Shigley 10 discussed the worms pitch circle line velocity as well as the transmission frictional coef.cient.In this study, only the friction force and elastic deformation of the surface between gear teeth are interested. The ZK-type worm-gear set is used to eliminate the backlash. Previous research works on gears and gear sets were focused on the limiting loading, pressure distribution and allowable pressure loading of gears, and on the relationship between friction coef.cient and sliding velocity. Most investigations are base on the assumption that the tooth is rigid. This study is focused on the effects of the elastic deformation of the gear tooth surface and the nonlinear friction coef.cient on the system dynamics. To study the dynamics of the worm-gear set, a dynamic model of the worm-gear set is developed and the comparison between the experimental results and the numerical results is also studied. The parameters of the dynamic model are adjusted by the .ndings of thecomparison of the analysis and the experiment. Finally, the effects of moment-inertia of .y wheel, friction, rigidity of shaft and rigidity of gear tooth on the nonlinear behavior of the transmission mechanism are also studied.2. The dynamic models of the worm-gear set2.1. The equilibrium equations of the worm and worm-gear setThe geometric of worm-gear set is shown in Fig. 1. There are a worm, a worm gear, two shafts, four bearings and a .ywheel. The shaft of worm is driven by an angle y1. The rotational angle of the .ywheel is y4. The rotational angle of the worms body is y2 (due to the elastic deformation of the worm shaft; y16y2). The rotation angle of worm-gear body is y3. The moment of inertia of the worm is Jw. The pitch radius of the worms helical tooth is rw. The worms rotational inertia as a rigid body is Jg. The pitch radius of the worm gear is rg. The moment-inertia of the .ywheel is J4. In this study, to neglect the effect of backlash, ZK-type worm and worm gear are used both in analysis and experiment.To simplify the theoretical model, the basic assumptions are listed as follows:(1) The .ywheel is rigid.(2) The worms and worm-gear teeth are elastic.(3) The body of the worm gear is rigid.(4) Worm and worm gear are perpendicular to each other.(5) The clearance between teeth is ignored.(6) No error on the gear tooth pro.le.(7) The worm is single thread.(8) The worm shaft and worm-gear shaft are two rotational spring. The dynamic behavior is neglected.(9) Translational degree-of-freedom for all the elements is not considered.(10) Only friction is considered, no other damping effect is considered.2.2. The axial spring constant of bearings, worm shaft and worm-gear shaftThe results of experiment to .nd the relation between load and the axial displacement of bearing are shown in Fig. 2. The axial spring constant of worm bearing (KbTws) and worm-gear bearing (KbTgs) are obtained by using Fig. 2. Furthermore, KLws and KbTws can be treat as two series connected spring. As well as KLgs and KbTgs can also be treat as two series connected spring. Therefore, kws can be expressed in terms of KLws with KbTws and kgs can also be expressed in terms of KLgs with KbTgs. The equations are listed as following:Fig. 1. Geometric of worm and worm gear.2.3. The friction coefficient of contact-gear teethThe relative motion between worm teeth and worm-gear teeth is considered pure sliding, so the friction plays an important role on the performance of worm-gear set. The ef.ciency of gear set is directly affected by the coef.cient of friction. The coef.cient of friction is in.uenced by the surface sliding speed of the both teeth;the friction is reduced as the relative sliding velocity increases. No formulae can be used to calculate the friction coef.cient precisely. In this study, the formula is based on the sliding velocity of the mean worm diameter 8,9,18:In above formula, n the relative sliding velocity of the mean worm diameter, the relative Iangular velocity of the mean worm diameter and rw the pitch radius of the worms helical tooth.Fig.2 The relation of axial direction displacements of bearing and axial loads3. Experimental setupExperimental setup is shown in Fig. 3.and the speci.cations of the worm-gear set are listed in Table 1. The ZK-type worm-gear set was used to eliminate the backlash. The reduction ratio of gear set is (1:56) and the module of gear is 2. In Fig. 3, a Compumotors stepping motor, model no. OEM57-83, (with resolution of 1000 steps/rev 0.361/step) was used to drive the worm shaft. An accelerometer (KS77) positioned on the upper location of the worm-gear shaft was used to measure the angle acceleration of y2 and y4. The angle displacement of y2 and y4 are obtained by integrated the measured angular acceleration twice. The NI USB-6009 DAQ system equipped with Lab-View software was employed to pick and fetch the voltage data from accelerometer.At the beginning of the experiment, the system was set at rest condition The sampling rate of data acquisition was 25_10_6 s. When the experiment started, the motor was accelerated to its maximum speed along the worm axis following the required acceleration curve, then stop after the input angle reaches y1 (y1.0.1130973 rad, 18 steps). The data-acquisition system recorded the acceleration of the accelerometer which attached on the bar end of worm and gear. The angular acceleration can be calculated from acceleration by using geometric relation.Fig. 3. The experiment device of worm-gear set.Table 1Speci.cations of worm-gear set (modulus M2)3.1. Experimental resultsThe measurement results of the worm angle acceleration (y2 curve) and the .ywheel of the worm-gear axis (y4 curve) are shown in Fig. 4. The acceleration oscillates between the zero line. The angular acceleration of y_2and y_4 are integrated twice to get angular displacement, which is shown in Fig. 5.Fig. 4. The measured results: (a) angular acceleration y2 and (b) angular acceleration y4.Fig. 5. The angular displacement obtained by integrating the acceleration twice: (a) angle y2 and (b) angle y4.4. ConclusionsIn this study, the dynamic equilibrium equations of worm-gear set considering the rigidity of the tooth were developed. The analyzed results were compared with the measured data. The comparison shows that the mathematics model is reasonable correct. The in.uence of moment-inertia, friction, rigidity shaft and rigiditygear tooth were also studied. The parameters study has following conclusions:1. The diameter of the .ywheel is reduced gradually, the amplitude of oscillating reduced gradually.2. The larger the module of gear, the higher the maximum amplitude of angles y2 and y4.3. Most oscillation of y2 comes from worm shaft, not from gear tooth. As to the oscillating of y4, the effect ofthe rigidity of the shaft is not important.4. The effect of the rigidity of the gear tooth of the worm is not as much as that of worm gear.5. The effect of friction is larger for y2 than that for y4.Above conclusions indicate that when design a worm-gear set for high precision positioning, the rigidity of the worm gear is very important, that the material of the worm gear should be used more rigid material than that of the worm.中文译文应用参数动力学研究蜗杆与蜗轮的旋转角度清华大学 电力机械工程系摘要:应用于旋转角度的动态蜗轮订下瞬间是受多种因素的影响(包括摩擦力和轮齿表面之间的弹性变形) .结果发现,这些因素造成了不可忽略的转动定位误差. 本研究的目的是: ( 1 )建立一个数学模型，( 2 )进行数值模拟, ( 3 )进行实验并与数值计算的结果作比较. 实验结果和实际数值是一致的. 在这项研究中,飞轮的惯性，摩擦力,刚性轴和轮齿的刚度也被研究. 结果,可作为误差估计应用于旋转角度的相关的角度, 角速度和角加速度,并且用于建立误差补偿所需的位置控制.1.概述当采用雷达跟踪正以快速角度旋转的飞机时，该雷达可能由于小振荡的天线无法到达锁定目标,有时,振荡甚至导致失败的雷达系统.在跟踪系统中，准确定位是一个至关重要的问题. 然而,由于跟踪传导机制的设计受到空间和重量, 体积和重量限制.蜗杆齿轮集是一个很好的选择,因为它具有小体积和高还原度的特点.由于高的精度定位雷达系统振动天线是一个重要的动力学表现，要理解这种表现应用于大扭矩的动态蜗轮订下瞬间是非常重要的.此表现是受若干因素影响,包括摩擦力和轮齿表面之间的弹性变形.为了了解这些因素的影响，大量的研究工作已进行了.yuksel和Kahraman：1研究有关一个典型行星齿轮的动态特性表面磨损的影响。 磨损模型采用了准静态齿轮接触模型计算接触压力,并结合Archard磨损模型来确定磨损深度的分布等. 2分析了大范围的运行速度和力矩的动态响应。并比较了其他研究者发表的揭示复杂的非线性现象实验,通过调查采用铌元/接触力学模型的动态响应,提供了大量的科学的齿轮动态分析. 3maliha创造了有关直齿圆柱齿轮的非线性动力学模型, 这是加上线性铌元素模型的轴向载荷,与离散型号的轴承和软盘.考虑在模型中的外部静扭矩和内部激励引起的刚度变化，齿轮误差和间隙占37.4%。4Britton制作一个超级的全齿(大约0.05毫米 ) ,在摩擦牵引实验中理论上用薄镑非牛顿微润滑求解和鼓励协定之间的摩擦和测量理论预测得到.5预策弹性接触和蜗杆.用几何之间的差距齿接触三维弹性接触仿真技术。已研制出相对轮齿表面计算的真实面积弹性接触负荷.6研究谐波传动,具有相当的非线性动力学体现,在他设计的模型中，不仅动态模型包括精准有意义的摩擦传动,间隙和运动学误差的理解。更重要的特点是,谐波传动齿轮齿面几何与互动.实验观察是用来指导发展的一个模型来描述谐波传动运行.更重要的是了解蜗轮在选择润滑剂应用和保养方面.7讨论了受最大静摩擦力和动摩擦力两者的动态伺服定位控制受.他们在报告中说发生了稳定的定位误差,以及停车状态,在跟踪时的转折点限制回路中的跟踪 制度.当系统以单方向的低速跟踪时，这是可以做到的粘滑现象发生,事实上,由于摩擦力的速度曲线，似乎是一个负斜率的静摩擦力. 同时,还发现了类似的现象发生在高速跟踪中。8提出了有关摩擦力的负斜率。在参考中,相对运动，从静态到高速滑动被分为四个阶段：在启始阶段中，外力小于最大静摩擦力. 在第二阶段,两个接触表面开始发生一个小的相对速度。在第三阶段，开始增加两物体的相对速度，润滑剂进入两者的接触面积,目的是降低摩擦力。在第四阶段，润滑油已电弧的接触面积，粘性特性和摩擦力来增加速度。9包括几个实证图表和计算公式中的摩擦系数与滑动速度，允许误差产生在两个轮齿之间的差距中，同时也允许接触压力的产生。10讨论了齿轮的节圆线速度以及传输摩擦系数。在这项研究中,只有两齿表面之间的摩擦力和弹力的变化是重要的，一般的ZK型蜗轮集是用来消除反弹力的。在以往的研究工程中，齿轮和齿轮套均集中极限承载，在压力分布和压力允许中装载齿轮,并且影响到两者的摩擦系数和滑动速度。在大多数调查中：假定轮齿之间的相对滑动速度是不变的。这项研究中着重考察了弹性变形的齿面和系统动力学非线性摩擦。在发达国家中研究动态的蜗轮集和动态模型的蜗轮集现在越来越广泛。在实际值和测量值之间进行了大量的实验。最后,探讨了矩惯性轮齿摩擦，刚性轴和刚度齿的非线性特性的传输机制进行了研究。2 . 蜗轮的动态模型2.1蜗杆与蜗轮集的平衡方程几何蜗轮载列于图. 1 . 有蜗杆,蜗轮,两轴,四轴承和一个轮齿，轴蜗杆驱动的角度的Y1。旋转角度的旋转角度的本体y2 (由于弹性变形的蜗杆轴; Y16组)。转角蜗轮传动机构是扭矩和惯性矩。圆形半径的斜齿是DVD+RW的. 他的转动惯量为刚体JG。圆形半径的蜗轮是体操。目前具有惯性的轮齿是J4。在这项研究中。可以看出忽视科学的影响，论在理论分析和实验中都应用ZK型蜗轮与蜗杆。 简化的理论模型,其基本假设如下：( 1 )轮齿的平衡行；( 2 )蜗杆与蜗轮齿的弹性；( 3 )肌体中的蜗轮； ( 4 )蜗杆与蜗轮相互垂直；( 5 )轮齿的间隙被忽略；( 6 )两对齿之间没有误差产生；( 7 )单方向的轮齿；( 8 )在蜗杆与蜗轮轴的转动中动态表现可以忽略；( 9 )平行自由度的所有元素不予考虑；( 10 )考虑摩擦没有其他的阻尼效应的影响。2.2 . 轴承、蜗杆与蜗轮轴的轴向弹簧常数实验结果对轴承列负荷和轴向位移图2。蜗杆轴承与蜗轮轴承的轴向弹簧常数轴承得到了良好的效果，如图2。此外, klws和kbtws可视为两个串联的轮齿以及klgs 和kbtgs也可视为两个串联的轮齿。方程列如下：2.3.两摩擦轮齿之间的滑动 摩擦系数的非接触齿之间的相对运动齿蜗杆与蜗轮牙齿被视为纯滑动, 所以摩擦起着重要作用,对经营业绩的蜗轮定。外啮合齿轮组直接承受压力。 3.实验装置：实验装置如列图.3和阳离子的蜗轮载列于表1。一般的ZK型蜗轮套是用来消除反弹力的，减少比率齿轮集( 1:56 )和模块齿轮2。在步进电机中,没有型号 (转速 0.361/step )，是用于蜗杆的驱动的。加速度( ks77 )优越的位置上的蜗杆齿轮轴被用来测量角度，加速度使用y2，角位移的使用y2，获得了综合测量角加速度两次. 镍USB和6009数据采集系统配备实验室浏览软件是用来挑取电压数据和加速度。在最初的实验中,该系统是一套在休息状态，采样率的数据采集量达到标准值。当实验开始后，蜗轮加速到其最大速度沿蜗杆轴以下所需的加速度曲线运动，再停止后投入角度到达的Y1量，数据采集系统记录了加速度的加速度而1.6mm的蜗杆和齿轮。用几何关系可以计算出加速度角加速度。3.1实验测量结果：实验测量结果的圆形角加速度和轮齿的蜗轮轴(曲线)列于图.4。加速度之间的零线. 角加速度对Y 2和Y 4所综合两次获得角位移，列图.5。4.结论在这项研究中,运用动力平衡方程考虑蜗轮平衡轮齿的发展. 分析的结果与实测数据表明，此数学模型是合理正确的。随着对转动惯量,摩擦力,刚度轴的固定。研究参数结论如下: 1 . 直径的轮齿逐渐减少,振幅振荡也逐渐减少；2 . 较大的模块齿轮,较高的最大振幅角度使用y2；3 . 对于蜗杆的振荡,蜗杆轴不是最重要的；4 . 齿轮的效果；5 . 与y2相比，摩擦的影响较大。上述结论表明,当设计高精度定位的蜗轮集时，蜗轮的平衡性是非常重要的，该蜗轮的材料应采用较好的刚性材料。
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