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两齿差摆线行星齿轮传动的设计【中文3031字】

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译自:Design of cycloid planetary gear drives with tooth number difference of two两齿差摆线行星齿轮传动的设计Shyi-Jeng TsaiLing-Chiao ChangChing-Hao Huang摘要:摆线行星齿轮减速器在自动化机械中有着广泛的应用。即使该减速器具有高传动比、多齿对啮合、减震能力强等优点,如何提高传动装置的功率密度仍然是当今摆线行星齿轮减速器发展的重要课题。为此,提出了两齿数差的概念。本文的目的是系统地分析这种摆线行星齿轮传动的载荷接触特性,以评价其可行性。本文首先导出了摆线轮廓、齿面接触和摆线阶段比滑动的基本方程。从建立的基于影响系数法的模型出发,提出了一种载荷齿接触分析方法。通过算例,系统地分析了设计参数对接触特性的影响。所得出的这些结果也与传统的齿数差为 1 的传动装置进行了比较。分析结果表明,该概念具有较大的偏心率和较小的销钉半径,既能有效地扩大接触比,又能有效地减小比滑、载荷和接触应力。虽然轴承载荷的径向部分也可以相应地减小,但是利用两齿数差的概念不能有效地降低总周期时变轴承载荷。1. 引言摆线行星齿轮减速器是动力和精密运动传动的重要传动装置。目前,设计为两级偏心差速器的齿轮机构,即所谓的“RV-传动”,在自动化机械中得到了广泛的应用。如图 1 所示的结构图所示,这种齿轮传动类型由一个渐开线行星级传动和一个两个圆盘的摆线行星级传动组成。每一个渐开线行星都安装在曲柄轴上,以产生摆线盘的旋转运动。这种设计结构不仅具有齿轮比高的优点,而且由于多齿对的接触,在分担载荷和减震能力方面具有良好的性能。然而,目前齿轮减速器设计的趋势是除了对精密运动的要求外,还要增大功率密度。为此,必须减少接触齿副和曲柄上的载荷。图 1 摆线行星齿轮传动结构在各种的措施中,通过采用较大的齿数差的这种设计理念可以为改善负载接触特性提供一种可能性。换句话说,摆线齿轮副的齿数差的可能选择为两种,而不是常规应用中经常使用的一种。在实际应用中,齿数差为 2(z=2)的齿轮传动常用于小减速比的传动,但是齿数差为 2 的齿轮传动在传动比比较高的情况下则很少使用。因此,评价这种替代驱动概念的可行性是很有趣的。本文探讨了设计参数对加载接触特性的影响,并与常规传动进行了比较。载荷分析的基本工作是导出几何和运动学之间的关系。对摆线轮廓线数学模型的研究在许多文献中都有发现。根据齿轮传动理论或运动学方法,也可以对齿轮啮合进行分析,比如即时中心方法。齿轮传动的另一个评价标准是啮合齿之间的滑动特性。例如齿侧的损伤,如点蚀、磨损或打分,这些可以用比滑动率来预测。然而,在有关摆线齿轮泵的相关研究文献中,较少涉及摆线齿轮减速器的研究。此外,载荷分析也是评价 z=2 齿轮传动可行性的一个重要问题。摆线齿轮接触应力分析的常用方法 是驱动器有限单元法。基于该分析方法而开发了另一种负荷分析方法。比如轧制作用力的计算方法,该方法是为了支持在滚动作用力下的的摆线盘的轴承。也有着重研究了制造误差对传输负载和传动误差的影响。还有一些人基于齿面啮合刚度假设,分析了加工误差对动载荷行为的影响。也有人进行了接触应力分析实验,验证了分析方法的理论分析结果。本文作者基于影响系数法开发了一种数值加载的齿面接触分析方法,该方法还成功地应用于摆线齿对以及考虑轴承刚度和摩擦力的全齿接触分析。因此,本文的目的是系统地研究设计参数对这种替代驱动概念的接触和负载特性的影响。本文首先导出了 z=2 的摆线齿廓、齿面接触和摆线阶段比滑动的基本方程。扩展了一种新的载荷分析方法。 从发展的数值加载的齿面接触分析模型,进一步系统地分析了设计参数对接触特性的影响,即:接触比、比滑动、载荷分担、接触应力。 本文讨论了周期时变轴承载荷并且将这些结果与 z=1 的常规驱动器进行了比较。2 摆线行星齿轮传动分析方法的基本原理2.1两齿差摆线盘的构造基本齿廓的定义:z=2 行星摆线级的齿廓可以看作是两个具有相同基座摆线廓线的两个圆盘的组合,其角度为 ,其值等于基摆线轮廓的 ,见图 2。coC摆线轮廓的基本设计参数如下:螺距圆半径为 的销轮;cR针的半径 ;pr曲柄的偏心度;传动的齿轮比 e,这里等于 。zZp图 2 z=2 的摆线盘的构造2.2 齿轮啮合分析摆线齿轮啮合的分析可以将圆片视为固定的,当销轮的中心相对于曲柄轴角 围绕盘的中心移动c时,销轮本身也以角 旋转。相对运动可以用图 3 所示的几何关系来表示。)( ucp图 3 z=2 的摆线盘齿面接触的基本几何关系2.3 两种齿数差的典型齿廓由于基摆线轮廓由凹型和凸型组成,在 z1 的情况下,可以找到两种轮廓类型,即凹型或凹型一凸型。根据拐点位置与针尖指向之间的关系,对这两种摆线轮廓进行了划分。凹型轮廓类型, tinfP凹型凸型轮廓类型, 。tif一般情况下,凹面轮廓有利于齿面接触,但接触比也相应降低。3 数值算例综述为了分析设计参数的影响,在表 1 中列出了必要的齿轮数据。本文所考虑的设计参数是销半径 和偏心度 e,相应的分析值列于表 2。这里不考虑更大的销半径,因为在销轮上安装销的可Pr用空间是有限的。表 1 数值分析中的基本齿轮传动数据名称 数值 备注销轮的节圆半径 162.5mm摆线盘的齿数 78销轮齿数 80减少比率(卡勒固定) 40 Zp/z摆线盘厚度 31.5mm轴承孔中心半径 90mm输出扭矩 4000Nm表 2 影响分析的设计参数名称 数值针形半径 3,4,5,(6)偏心 2.5,3,3.54设计参数的影响分析4.1 齿廓与接触比设计参数 e 和 对尖端位置和接触比率的影响表示在图 4 中。由于齿面接触比与齿形变量Pr呈线性相关,因此图中的曲线同时说明了齿面接触和齿形变量这两个因素。Pt一般情况下,z=2 的摆线传动具有较小的引脚和较大的偏心比,且具有较大的接触比。在这种情况下,摆线轮廓具有凸凹部分。另一方面,较大的销钉半径,例如, ,其降低了接触比,7rP增加了偏心率。偏心 e(mm)图 4 电子和电子技术对针尖定位及触点比的影响4.2 比滑动图 5 和图 6 分别说明了针半径 和偏心 e 对比滑动的影响。摆线盘厚度 的比滑动远大于针轮厚Pr c度 。 的比滑动单调地从开始 A 增加到 E。相反, 的比滑动先单调下降,然后渐近于奇点附P c近的一点,其中传输角 等于 15。当齿对进一步啮合时, 从无穷大减小到一个局部极限值。图 5 销径 对比滑动的影响Pr图 6 偏心率 e 对比滑动的影响4.3 接触齿对之间的共同载荷因为活齿对的传动角 是不同的,正常载荷在接触齿对之间的分配也不均匀。销半径对载荷分配的影响较小,相反,一个较小的偏心会导致不均匀地分担负载。5 与齿数差为 1 的摆线行星齿轮传动比较为了比较其传动特性,使 z=1 和 2 的减速器的偏心度 e 为相同的值,而使其销半径的值不同。6 总结和展望为了提高高传动比摆线行星齿轮传动的传动性能,提出了齿数差为 2(z=2)的概念。利用建立的基于影响系数法的加载齿接触分析模型,探讨了该传动方式的载荷接触特性。分析了设计参数对偏心率和销径的影响,并与 z=1 的摆线行星齿轮传动进行了比较。因此得到了一些结果,分析结果使我们得出以下结论:(1)采用合适的设计参数可以提高 z=2 的摆线行星齿轮传动的理论接触比,并期望采用较小的针半径和较大的偏心度。(2)偏心率对比滑动有显著影响。当偏心较大时,具有无限大比滑动 的点将向齿根方向移动,c即摆线轮廓,且 值减小,c(3)z=2 载荷和接触应力的共同作用侧翼,且在 z=1 的情况下可以显着地降低接触应力,小数值的销半径和较大偏心度适合于这种传动装置的设计。(4)Z2 的侧向载荷和接触应力的变化与循环渐开线齿轮传动一样,类似于渐开线直齿圆柱齿轮传动的现象。(5)用 z=2 的摆线齿轮传动对于减小曲柄上的轴承载荷的效果不是很显著,因为环向力的影响要大得多。然而,由于间隙存在的必要性,这种改进后的摆线廓形常被应用于实际应用中,而不是应用于理论中。因此为了进一步设计 z=2 的摆线传动的侧面改进,本文得出了一些好的结果。同时,本文提出的分析方法也是一种有效的工具。译文原文Forsch Ingenieurwes (2017) 81:325336 DOI 10.1007/s10010-017-0244-yDesign of cycloid planetary gear drives with tooth number difference of twoA comparative study on contact characteristics and load analysisShyi-Jeng Tsai1 Ling-Chiao Chang1 Ching-Hao Huang1Received: 30 March 2017 Springer-Verlag GmbH Deutschland 2017Abstract The cycloid planetary gear reducers are widely applied in automation machinery. Even having the advan- tages of high gear ratio, multiple contact tooth pairs and shock absorbing ability, how to enlarge the power density of the drives is still the essential development work today. To this end, the concept of tooth number difference of two is proposed. The aim of the paper is to analyze systemati- cally the loaded contact characteristic of such the cycloid planetary gear drives so as to evaluate the feasibility. A set of essential equations for the cycloid profile, the tooth con- tact and the specific sliding of the cycloid stage are at first derived in the paper. A loaded tooth contact analysis ap- proach is extended from a developed model based on the influence coefficient method. The influences of the design parameters on the contact characteristics are systematically analyzed with an example. These results are also compared with the conventional drive having tooth number difference of one. The analysis results show that the proposed con- cept with a larger eccentricity and a smaller pin radius can not only effectively enlarge the contact ratio, but also re- duce the specific sliding, the shared loads and the contact stress. Although the radial portion of the bearing load can be also reduced accordingly, the total periodical time-vari- ant bearing load can not be reduced effectively by using the concept of tooth number difference of two. Shyi-Jeng Tsai sjtsai.tw1 Department of Mechanical Engineering, National Central University, No. 300, Jhong-Da Road, Jhong-Li District, Taoyuan City 320, Taiwan1 IntroductionThe cycloid planetary gear reducers are important dri- ves for power and/or precision motion transmission. The gear mechanism designed in the type of two-stage eccentric differential, i. e. the so-called “RV-drive”, is today widely applied in automation machinery. As the structural dia- grams in Fig. 1 show, this gear drive type consists of an involute planetary stage and a cycloid planetary stage with two disks. Each involute planet is mounted on a crank shaft to generate the revolution motion of the cycloid disk. Such the design configuration has not only the advantages of high gear ratio, but also good performances in load sharing and shock absorbing ability because of multiple tooth pairs in contact. Nevertheless, the trend in designing the gear reducers today is to enlarge the power density, besides the requirements on precise motion. To this purpose, the loads acting on the contact tooth pairs and the cranks must be reduced.Among various measures, the design concept by using a larger tooth number difference (abbr. TND) can give a possibility to improve the loaded contact characteristics.Fig. 1 Structure of cycloid planetary gear drive 1 326 Forsch Ingenieurwes (2017) 81:325336In other words, the TND of the cycloid gear pair can be selected as two, not as one that is often used in the conven- tional application. The gear drives with TND of two (6z =2) are often applied in the transmission of small reduction ratios in the practice, but they are rarely used in the cases with a higher ratio. Therefore it is interesting to evaluate the feasibility of such the alternative drive concept. The influences of the design parameters on the loaded contact characteristics and the comparison with the conventional drives should be explored.The essential work for the load analysis is to derive the geometrical and kinematic relations. The study on the ma- thematic model of the cycloid profile can be found in many literatures. The gear mesh can be also analyzed based on the theory of gearing or the kinematic methods, e. g. the in- stant center method 2, 3. Another evaluation criterion of the drives is the sliding characteristics between the engaged teeth. The damages on the tooth flanks, e. g., pitting, wear or scoring, can be predicted by the ratio of specific sliding. However, the related research is often found in some artic- les on trochoidal gear pumps 4, and is less mentioned in the field of the cycloid gear reducer.Additionally, the load analysis is also an important is- sue for evaluating the feasibility of the gear drives with 6z = 2. The often applied method for analysis of the con- tact stress of the cycloid gear drives is FEM, e. g. 59. Another approach for load analysis is developed based on analytical methods. For example, Dong et al. 10 proposed a calculation approach for the acting forces on the rolling bearings for supporting the cycloid disks. Blanche and Yang 11, 12 focused on the influences of the manufacturing er- rors on the transmitted load and transmission errors. Hidaka et al. 13 analyzed the influences of manufacturing errors on the dynamic load behaviors based on the assumption of contact mesh stiffness of tooth action. Gorla et al. 14 conducted an experiment to analyze the contact stress so as to validate the theoretical analysis results from the ana- lytical approach. The authors have developed an numerical loaded tooth contact analysis (LTCA) approach based on the influence coefficient method 15. This approach is also successfully applied for analysis of the cycloid tooth pairs 2, 3 and also the complete tooth contact considering the bearing stiffness and the friction 16.The aim of the paper is therefore to study systematically the influences of the design parameters on the contact and loading characteristics of such the alternative drive concept. A set of essential equations for the cycloid profile, tooth contact and the specific sliding of the cycloid stage with 6z = 2 are at first derived in the paper. A new load analy- sis approach is extended from the developed LTCA model 12. The influences of the design parameters on the contact characteristics are further systematically analyzed. Namely, the contact ratio, the specific sliding, the load sharing, thecontact stress and the periodical time-variant bearing loads are discussed in the paper. These results are also compared with the conventional drive having 6z = 1.2 Fundamentals of the analysis methods for cycloid planetary gear drives2.1 Construction of the cycloid disk for tooth number difference (TND) of two(1) Definition of the base tooth profiles. The tooth profile of the planetary cycloid stage with 6z = 2 can be regarded as combination of two disks with the same base cycloid profile rotated against each other with an angle C, which is equal to C0/2 of the base cycloid profile., see Fig. 2. The essential design parameters for the cycloid profile are as follows, the pitch circle radius RC of the pin-wheel, the radius rP of the pins, the eccentricity e of the crank, the gear ratio u of the drive, here equal to zP/6z.The base cycloid profile is in accordance with the cycloid profile with the same design parameters but with 6z = 1,i. e., owns the coordinates 2, see Fig. 3:xC = RC cos otherwise, if the cranks rotate in the clockwise direction, the tooth pairs with negative eqi are in contact.(6) Sliding velocity. on the contact point plays an import- ant role for evaluation of tooth scuffing. The sliding velocity of the ith contact tooth pair can be determined based on theThe specific sliding Ci will become infinitely great, andPi = 1, if the transmission angle i is equal to /2.2.3 Typical tooth profile for TND of twoBecause the base cycloid profile consists of both concave and convex profile, two profile types, i. e., either concave or concave-convex profile, can be found in the case of 6z 1. These two type of the cycloid profile are divided according to the relation between the location of the inflection point and tip pointing, i. e., concave profile, inf pt; concave-convex profile, inf pt.In general, the concave profile is good for tooth contact, but the contact ratio is reduced accordingly.i:4 5Forsch Ingenieurwes (2017) 81:325336 3292.4 Loaded tooth contact analysisn 0m1 TX X (1) Basic LTCA model for cycloid stage. The contact pro- blem of multiple tooth pairs under loading is statically in-i =1s qi j =1pi j A =(18)edeterminate. The shared loads on the tooth pairs can be solved by using two types of equations, namely the equati- ons of load equilibrium as well as the equations of loaded deformation and displacement 15. To this end, a numerical approach for loaded tooth contact analysis of cycloid plane- tary gear drives is developed by the authors. This approachis based on the influence coefficient method to express thewhere the factor qi is equal to qi = sini .The set of the deformation-displacement equations and the load equilibrium equation can be summarized in a form of matrix equation 2, as the expression:relation of the deformation of any specific point i on the6 :engaged flanks due to the influence of all the distributed77 6qnI Ppressures pj, i. e., 4 q s I q s I 5 4q s I 0nwi = X .fi j p j . (16)j =1where fij is the influence coefficient for the condition, that1 1 2 2 n n e2 H 1 36 H 2 76 76 76 76 H n 7(19)the deformation on point i is caused by a load acting on thepoint j. .A Q P . . H=T= .u e/.(20)The relations of displacement-deformation is thus validfor the specific point YiS 0 e T= .u e/wi + hi = eqi (17)where hi is the separation distance between the engaged flanks at the discrete point, more detail see 2.Another relation for the loaded tooth contact is the load equilibrium equation, i. e., the sum of all the acting forces in the tangential direction (see Fig. 6) must be equal to the equivalent force T/(u e), i. e.,The sub-matrices in Eqs. 19 and 20 are defined as fol- lows: Ai contains all the influence coefficients fPi for the tooth pair i; all these sub-matrices are summarized in the ma- trix A in Eq. 20. I is either the column or the row unit vector. Pi as a column vector contains all the contact stresses on the discrete units of the tooth pair i; all these sub-vectors are summarized in a column vector P in Eq. 20. Hi as a column vector contains all the separation distan- ces between the engaged tooth flanks of the tooth pair i according to the discrete points; all these sub-vectors are summarized in a column vector H in Eq. 20. S in Eq. 20 combines all the row vectors with a value ofqisi.Because the contact region of two engaged flanks are divided into small discretized areas for load analysis, the actual contact pattern and distributed contact stresses can be simulated. More details can be found in 2, 3, 15, 16.(2) Conversion of the basic LTCA model for the case with 6z= 2. Considering two disks with a separation angle of C, the LTCA model for analysis of the case of 6z = 2 can be expanded by the expression:2AI 0 QI 3 2 PI 3 2 HI 34 0 AII QII5 4PII5 = 4HII5 (21)Fig. 6 Relation of acting forces on the cycloid diskS I SII 0 e T =eu2 A10 0 q1I3 2P1 36 0 A2: 0: q2I7 6P2 766 : : : : : : : 7 6 : 7 6 :776 0 0 An7 6 7n 5=330 Forsch Ingenieurwes (2017) 81:325336The contact areas with distributed stresses of the contact tooth pairs based on Eqs. 19 or 21 are solved iteratively until the convergent condition is fulfilled, i. e., all the contact stresses are positive.(3) Load sharing of the drive is distinguished between the load sharing among the tooth pairs at a specific angular position as well as the shared loads distributed on an indi- vidual tooth flank within a meshing cycle. The normal load FNiI,II acting on the ith odd/even-numbered contact tooth pair is determined as the sum of all the distributed contact stresses which are solved from the LTCA approach basedFt =Fc =Tout3 u eTout .u 1/3 u rRFig. 7 Relation of acting forces on the crank(30)(31)on Eq. 21, namely,m nFNi I;II = X si pi j (22)j =1(4) Bearing loads on the cranks can be divided into three types of forces based on the load equilibrium conditions, see Fig. 6 and 2, 10: Force equilibrium,mFr = X qri FNi =3 (23)ionly the radial force Fr can be changed by using suitable design parameters so as to reduce the bearing loads.3 Overview of the numerical exampleIn order to analyzed the influences of the design parameter, the essential gear data are listed in Table 2. The design parameters considered in the paper are the pin radius rP and the eccentricity e, the corresponding values used for the analysis are listed Table 3. A larger pin radius is not considered here, because the available space for installation of the pins in the pin wheel is limited.mFt = X qti FNi =3 (24)i4 Influence analysis of the design parameters4.1 Tooth profile and contact ratio Moment equilibrium, the circumferential force: The influences of the design parameters e and rP on themFc = X qti FNi .u 1/ e= .3 rR / (25)iwhere the factors qri and qti in the above equation are deter- mined as follows,qri = sini (26)qti = cosi (27)The result radial forces FCRr and tangential forces FCRt acting on the crank j are equal to the following relations respectively, see Fig. 7,FCRrj = Fr + Fc sin C + 2v .j 1/=3 (28)FCRtj = Ft Fc cos C + 2v .j 1/=3 (29)Because the tangential Ft and circumferential force Fcare directly associated with the output torque Tout, i. e.,location of tip pointing and the contact ratio are represented in Fig. 8. Because the contact ratio is linearly associated with the profile variable pt for tooth pointing, the curves in the diagram illustrate both the two factors at the same time.Table 2 Essential gearing data for numerical analysisItems/symbols Value RemarksPitch circle radius of pin wheel RC 162.5 mm Tooth number of the cycloid disk zC 78 Tooth number of the pin wheel zP 80 Reduction ratio u (Carrier fixed) 40 zP/6zThickness of the cycloid disk t 31.5 mm Radius of the bearing hole center rR 90 mm Output torque T 4000 Nm Table 3 Design parameters for influence analysis Items/symbols Value mmRadius of the pin rP 3, 4, 5, (6)Eccentricity e 2.5, 3, 3.5ProfileGenerationAngle Forsch Ingenieurwes (2017) 81:325336 331262422201816141210864200.5 1 1.5 2 2.5 3 3.5Eccentricity e mm32.521.510.50Fig. 8 Influence of e and rP on the location of tip pointing and the contact ratio Fig. 9 Influence of e a
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本文标题:两齿差摆线行星齿轮传动的设计【中文3031字】
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