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翻譯部分
英文原文
Parametric study of dynamics of worm and worm-gear set under
suddenly applied rotating angle
M.Y. Chung, D. Shaw
Department of Power Mechanical Engineering, National Tsing-Hua University, No. 101 Kung Fu Road Section II, Hsin Chu, Taiwan, ROC
Received 13 September 2006; received in revised form 23 January 2007; accepted 25 February 2007
Available online 17 April 2007
Abstract
The 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. Introduction
When 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 elastic
deformation 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 Archard’s 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 stick–slip 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 worm’s 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 the
comparison 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 set
2.1. The equilibrium equations of the worm and worm-gear set
The 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 worm’s body is y2 (due to the elastic deformation of the worm shaft; y16?y2). The rotation angle of worm-gear body is y3. The moment of inertia of the worm is Jw. The pitch radius of the worm’s helical tooth is rw. The worm’s 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 shaft
The 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 teeth
The 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 worm’s helical tooth.
Fig.2 The relation of axial direction displacements of bearing and axial loads
3. Experimental setup
Experimental 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 Compumotor’s 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 1
Speci.cations of worm-gear set (modulus M2)
3.1. Experimental results
The 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. Conclusions
In 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 of
the 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.
中文譯文
應(yīng)用參數(shù)動力學(xué)研究蝸桿與蝸輪的旋轉(zhuǎn)角度
清華大學(xué) 電力機(jī)械工程系
摘要:
應(yīng)用于旋轉(zhuǎn)角度的動態(tài)蝸輪訂下瞬間是受多種因素的影響(包括摩擦力和輪齒表面之間的彈性變形) .結(jié)果發(fā)現(xiàn),這些因素造成了不可忽略的轉(zhuǎn)動定位誤差. 本研究的目的是: ( 1 )建立一個數(shù)學(xué)模型，( 2 )進(jìn)行數(shù)值模擬, ( 3 )進(jìn)行實驗并與數(shù)值計算的結(jié)果作比較. 實驗結(jié)果和實際數(shù)值是一致的. 在這項研究中,飛輪的慣性，摩擦力,剛性軸和輪齒的剛度也被研究. 結(jié)果,可作為誤差估計應(yīng)用于旋轉(zhuǎn)角度的相關(guān)的角度, 角速度和角加速度,并且用于建立誤差補(bǔ)償所需的位置控制.
1.概述
當(dāng)采用雷達(dá)跟蹤正以快速角度旋轉(zhuǎn)的飛機(jī)時，該雷達(dá)可能由于小振蕩的天線無法到達(dá)鎖定目標(biāo),有時,振蕩甚至導(dǎo)致失敗的雷達(dá)系統(tǒng).在跟蹤系統(tǒng)中，準(zhǔn)確定位是一個至關(guān)重要的問題. 然而,由于跟蹤傳導(dǎo)機(jī)制的設(shè)計受到空間和重量, 體積和重量限制.蝸桿齒輪集是一個很好的選擇,因為它具有小體積和高還原度的特點.由于高的精度定位雷達(dá)系統(tǒng)振動天線是一個重要的動力學(xué)表現(xiàn)，要理解這種表現(xiàn)應(yīng)用于大扭矩的動態(tài)蝸輪訂下瞬間是非常重要的.此表現(xiàn)是受若干因素影響,包括摩擦力和輪齒表面之間的彈性變形.為了了解這些因素的影響，大量的研究工作已進(jìn)行了.yuksel和Kahraman：[1]研究有關(guān)一個典型行星齒輪的動態(tài)特性表面磨損的影響。 磨損模型采用了準(zhǔn)靜態(tài)齒輪接觸模型計算接觸壓力,并結(jié)合Archard磨損模型來確定磨損深度的分布等. [2]分析了大范圍的運(yùn)行速度和力矩的動態(tài)響應(yīng)。并比較了其他研究者發(fā)表的揭示復(fù)雜的非線性現(xiàn)象實驗,通過調(diào)查采用鈮元/接觸力學(xué)模型的動態(tài)響應(yīng),提供了大量的科學(xué)的齒輪動態(tài)分析. [3]maliha創(chuàng)造了有關(guān)直齒圓柱齒輪的非線性動力學(xué)模型, 這是加上線性鈮元素模型的軸向載荷,與離散型號的軸承和軟盤.考慮在模型中的外部靜扭矩和內(nèi)部激勵引起的剛度變化，齒輪誤差和間隙占37.4%。[4]Britton制作一個超級的全齒(大約0.05毫米 ) ,在摩擦牽引實驗中理論上用薄鎊非牛頓微潤滑求解和鼓勵協(xié)定之間的摩擦和測量理論預(yù)測得到.[5]預(yù)策彈性接觸和蝸桿.用幾何之間的差距齒接觸三維彈性接觸仿真技術(shù)。已研制出相對輪齒表面計算的真實面積彈性接觸負(fù)荷.[6]研究諧波傳動,具有相當(dāng)?shù)姆蔷€性動力學(xué)體現(xiàn),在他設(shè)計的模型中，不僅動態(tài)模型包括精準(zhǔn)有意義的摩擦傳動,間隙和運(yùn)動學(xué)誤差的理解。
更重要的特點是,諧波傳動齒輪齒面幾何與互動.實驗觀察是用來指導(dǎo)發(fā)展的一個模型來描述諧波傳動運(yùn)行.更重要的是了解蝸輪在選擇潤滑劑應(yīng)用和保養(yǎng)方面.[7]討論了受最大靜摩擦力和動摩擦力兩者的動態(tài)伺服定位控制受.他們在報告中說發(fā)生了穩(wěn)定的定位誤差,以及停車狀態(tài),在跟蹤時的轉(zhuǎn)折點限制回路中的跟蹤 制度.當(dāng)系統(tǒng)以單方向的低速跟蹤時，這是可以做到的粘滑現(xiàn)象發(fā)生,事實上,由于摩擦力的速度曲線，似乎是一個負(fù)斜率的靜摩擦力. 同時,還發(fā)現(xiàn)了類似的現(xiàn)象發(fā)生在高速跟蹤中。[8]提出了有關(guān)摩擦力的負(fù)斜率。在參考中,相對運(yùn)動，從靜態(tài)到高速滑動被分為四個階段：在啟始階段中，外力小于最大靜摩擦力. 在第二階段,兩個接觸表面開始發(fā)生一個小的相對速度。在第三階段，開始增加兩物體的相對速度，潤滑劑進(jìn)入兩者的接觸面積,目的是降低摩擦力。在第四階段，潤滑油已電弧的接觸面積，粘性特性和摩擦力來增加速度。[9]包括幾個實證圖表和計算公式中的摩擦系數(shù)與滑動速度，允許誤差產(chǎn)生在兩個輪齒之間的差距中，同時也允許接觸壓力的產(chǎn)生。[10]討論了齒輪的節(jié)圓線速度以及傳輸摩擦系數(shù)。
在這項研究中,只有兩齒表面之間的摩擦力和彈力的變化是重要的，一般的ZK型蝸輪集是用來消除反彈力的。在以往的研究工程中，齒輪和齒輪套均集中極限承載，在壓力分布和壓力允許中裝載齒輪,并且影響到兩者的摩擦系數(shù)和滑動速度。在大多數(shù)調(diào)查中：假定輪齒之間的相對滑動速度是不變的。這項研究中著重考察了彈性變形的齒面和系統(tǒng)動力學(xué)非線性摩擦。在發(fā)達(dá)國家中研究動態(tài)的蝸輪集和動態(tài)模型的蝸輪集現(xiàn)在越來越廣泛。在實際值和測量值之間進(jìn)行了大量的實驗。最后,探討了矩慣性輪齒摩擦，剛性軸和剛度齒的非線性特性的傳輸機(jī)制進(jìn)行了研究。
2 . 蝸輪的動態(tài)模型
2.1蝸桿與蝸輪集的平衡方程
幾何蝸輪載列于圖. 1 . 有蝸桿,蝸輪,兩軸,四軸承和一個輪齒，軸蝸桿驅(qū)動的角度的Y1。旋轉(zhuǎn)角度的旋轉(zhuǎn)角度的本體y2 (由于彈性變形的蝸桿軸; Y16組?)。轉(zhuǎn)角蝸輪傳動機(jī)構(gòu)是扭矩和慣性矩。圓形半徑的斜齒是DVD+RW的. 他的轉(zhuǎn)動慣量為剛體JG。圓形半徑的蝸輪是體操。目前具有慣性的輪齒是J4。在這項研究中?？梢钥闯龊鲆暱茖W(xué)的影響，論在理論分析和實驗中都應(yīng)用ZK型蝸輪與蝸桿。
簡化的理論模型,其基本假設(shè)如下：
( 1 )輪齒的平衡行；
( 2 )蝸桿與蝸輪齒的彈性；
( 3 )肌體中的蝸輪；
( 4 )蝸桿與蝸輪相互垂直；
( 5 )輪齒的間隙被忽略；
( 6 )兩對齒之間沒有誤差產(chǎn)生；
( 7 )單方向的輪齒；
( 8 )在蝸桿與蝸輪軸的轉(zhuǎn)動中動態(tài)表現(xiàn)可以忽略；
( 9 )平行自由度的所有元素不予考慮；
( 10 )考慮摩擦沒有其他的阻尼效應(yīng)的影響。
2.2 . 軸承、蝸桿與蝸輪軸的軸向彈簧常數(shù)
實驗結(jié)果對軸承列負(fù)荷和軸向位移圖2。蝸桿軸承與蝸輪軸承的軸向彈簧常數(shù)軸承得到了良好的效果，如圖2。此外, klws和kbtws可視為兩個串聯(lián)的輪齒以及klgs 和kbtgs也可視為兩個串聯(lián)的輪齒。方程列如下：
2.3.兩摩擦輪齒之間的滑動
摩擦系數(shù)的非接觸齒之間的相對運(yùn)動齒蝸桿與蝸輪牙齒被視為純滑動, 所以摩擦起著重要作用,對經(jīng)營業(yè)績的蝸輪定。外嚙合齒輪組直接承受壓力。
3.實驗裝置：
實驗裝置如列圖.3和陽離子的蝸輪載列于表1。一般的ZK型蝸輪套是用來消除反彈力的，減少比率齒輪集( 1:56 )和模塊齒輪2。在步進(jìn)電機(jī)中,沒有型號 (轉(zhuǎn)速[ 0.361/step ] )，是用于蝸桿的驅(qū)動的。加速度( ks77 )優(yōu)越的位置上的蝸桿齒輪軸被用來測量角度，加速度使用y2，角位移的使用y2，獲得了綜合測量角加速度兩次. 鎳USB和6009數(shù)據(jù)采集系統(tǒng)配備實驗室瀏覽軟件是用來挑取電壓數(shù)據(jù)和加速度。
在最初的實驗中,該系統(tǒng)是一套在休息狀態(tài)，采樣率的數(shù)據(jù)采集量達(dá)到標(biāo)準(zhǔn)值。當(dāng)實驗開始后，蝸輪加速到其最大速度沿蝸桿軸以下所需的加速度曲線運(yùn)動，再停止后投入角度到達(dá)的Y1量，數(shù)據(jù)采集系統(tǒng)記錄了加速度的加速度而1.6mm的蝸桿和齒輪。用幾何關(guān)系可以計算出加速度角加速度。
3.1實驗測量結(jié)果：
實驗測量結(jié)果的圓形角加速度和輪齒的蝸輪軸(曲線)列于圖.4。加速度之間的零線. 角加速度對Y 2和Y 4所綜合兩次獲得角位移，列圖.5。
4.結(jié)論
在這項研究中,運(yùn)用動力平衡方程考慮蝸輪平衡輪齒的發(fā)展. 分析的結(jié)果與實測數(shù)據(jù)表明，此數(shù)學(xué)模型是合理正確的。隨著對轉(zhuǎn)動慣量,摩擦力,剛度軸的固定。研究參數(shù)結(jié)論如下:
1 . 直徑的輪齒逐漸減少,振幅振蕩也逐漸減少；
2 . 較大的模塊齒輪,較高的最大振幅角度使用y2；
3 . 對于蝸桿的振蕩,蝸桿軸不是最重要的；
4 . 齒輪的效果；
5 . 與y2相比，摩擦的影響較大。
上述結(jié)論表明,當(dāng)設(shè)計高精度定位的蝸輪集時，蝸輪的平衡性是非常重要的，該蝸輪的材料應(yīng)采用較好的剛性材料。