外文文献翻译-两齿差摆线行星齿轮传动的设计【中文3031字】 【PDF+中文WORD】

外文文献翻译-两齿差摆线行星齿轮传动的设计【中文3031字】 【PDF+中文WORD】,中文3031字,PDF+中文WORD,外文文献翻译-两齿差摆线行星齿轮传动的设计【中文3031字】,【PDF+中文WORD】,外文,文献,翻译,两齿差,摆线,行星,齿轮,传动,设计,中文,3031,PDF 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 Tsai·Ling-Chiao Chang·Ching-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(Δz=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Δz=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Δz=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Δz=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：Δz=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 t 组合，其角度为tc ，其值等于基摆线轮廓的 摆线轮廓的基本设计参数如下： 螺距圆半径为 Rc 的销轮； 针的半径 rp ； 曲柄的偏心度； 传动的齿轮比 e，这里等于 Zp 。 Dz Co ，见图 2。 2 图 2 Δz=2 的摆线盘的构造 jc u 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jc 围绕盘的中心移动 时，销轮本身也以角j（p = ）旋转。相对运动可以用图 3 所示的几何关系来表示。 图 3 Δz=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Δz>1 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，qinf ³ qPt 凹型—凸型轮廓类型，qinf áqPt 。 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 rP 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 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/Δz 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 2.5,3,3.5 4 设计参数的影响分析 4.1 齿廓与接触比 设计参数 e 和 rP 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 qPt 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。 一般情况下，Δz=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， rP = 7 ，其降低了接触比， 增加了偏心率。 4.2 比滑动 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 图 5 和图 6 分别说明了针半径 rP 和偏心 e 对比滑动的影响。摆线盘厚度xc 的比滑动远大于针轮厚 度xP 。xP 的比滑动单调地从开始 A 增加到 E。相反，xc 的比滑动先单调下降，然后渐近于奇点附 近的一点，其中传输角g等于 15。当齿对进一步啮合时，xc 从·无穷大减小到一个局部极限值。 图 5 销径 rP 对比滑动的影响 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g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Δz=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(Δz=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Δz=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Δz=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xc 的点将向齿根方向移动， 即摆线轮廓，且xc 值减小， （3）Δz=2 载荷和接触应力的共同作用侧翼，且在Δz=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）ΔZ＝2 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Δz=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Δz=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。 Forsch Ingenieurwes(2017)81:325336 DOI 10.1007/s10010-017-0244-yORIGINALARBEITEN/ORIGINALSDesign of cycloid planetary gear driveswith tooth number differenceof twoA comparative study on contact characteristics and loadanalysisShyi-JengTsai1Ling-ChiaoChang1 Ching-HaoHuang1Received:30 March2017 Springer-VerlagGmbH Deutschland 2017Abstract The cycloid planetary gear reducers are widelyapplied in automation machinery.Even having the advan-tages of high gear ratio,multiple contact tooth pairs andshock absorbing ability,how to enlarge the power densityof the drives is still the essential development work today.Tothis end,the concept of tooth number difference of twois proposed.The aim of the paper is to analyze systemati-cally the loaded contact characteristic of such the cycloidplanetary gear drives so as to evaluate the feasibility.A setof essential equations for the cycloid profile,the tooth con-tact and the specific sliding of the cycloid stage are at firstderived in the paper.A loaded tooth contact analysis ap-proach is extended from a developed model based on theinfluence coefficient method.The influences of the designparameters on the contact characteristics are systematicallyanalyzed with an example.These results are also comparedwith the conventional drive having tooth numberdifferenceof one.The analysis results show that the proposed con-cept with a larger eccentricity and a smaller pin radius cannot only effectively enlarge the contact ratio,but also re-duce the specific sliding,the shared loads and the contactstress.Although the radial portion of the bearing load canbe also reduced accordingly,the total periodical time-vari-ant bearing load can not be reduced effectively by usingtheconcept of tooth number difference of two.Shyi-Jeng Tsaisjtsaicc.ncu.edu.tw1Department of Mechanical Engineering,NationalCentralUniversity,No.300,Jhong-Da Road,Jhong-Li District,TaoyuanCity 320,Taiwan1 IntroductionThe cycloid planetary gear reducers are importantdri-ves for power and/or precision motion transmission.Thegear mechanism designed in the type of two-stageeccentricdifferential,i.e.the so-called“RV-drive”,is today widelyapplied in automation machinery.As the structural dia-grams in Fig.1 show,this gear drive type consists of aninvolute planetary stage and a cycloid planetary stage withtwo disks.Each involute planet is mounted on a crankshaftto generate the revolution motion of the cycloid disk.Suchthe design configuration has not only the advantages ofhigh gear ratio,but also good performances in load sharingand shock absorbing ability because of multiple tooth pairsin contact.Nevertheless,the trend in designing the gearreducers today is to enlarge the power density,besides therequirements on precise motion.Tothis purpose,the loadsacting on the contact tooth pairs and the cranks must bereduced.Among various measures,the design concept byusinga larger tooth number difference(abbr.TND)can givea possibility to improve the loaded contact characteristics.Fig.1 Structure of cycloid planetary gear drive1326Forsch Ingenieurwes(2017)81:325336In other words,the TND of the cycloid gear pair can beselected 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 reductionratios in the practice,but they are rarely used in the caseswith a higher ratio.Therefore it is interesting to evaluatethe feasibility of such the alternative drive concept.Theinfluences of the design parameters on the loaded contactcharacteristics and the comparison with the conventionaldrives should be explored.The essential work for the load analysis is to derive thegeometrical and kinematic relations.The study on the ma-thematic model of the cycloid profile can be found in manyliteratures.The gear mesh can be also analyzed based onthe theory of gearing or the kinematic methods,e.g.thein-stant center method 2,3.Another evaluation criterion ofthe drives is the sliding characteristics between the engagedteeth.The damages on the tooth flanks,e.g.,pitting,wearor 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 inthe field of the cycloid gear reducer.Additionally,the load analysis is also an important is-sue for evaluating the feasibility of the gear drives with6z=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 onanalytical methods.For example,Dong et al.10proposeda calculation approach for the acting forces on the rollingbearings for supporting the cycloid disks.Blanche and Yang11,12 focused on the influences of the manufacturing er-rors on the transmitted load and transmission errors.Hidakaet al.13 analyzed the influences of manufacturing errorson the dynamic load behaviors based on theassumptionof contact mesh stiffness of tooth action.Gorla et al.14conducted an experiment to analyze the contact stress soas to validate the theoretical analysis results from the ana-lytical approach.The authors have developed an numericalloaded tooth contact analysis(LTCA)approach based onthe influence coefficient method 15.This approach isalsosuccessfully applied for analysis of the cycloid tooth pairs2,3 and also the complete tooth contact considering thebearing stiffness and the friction 16.The aim of the paper is therefore to study systematicallythe influences of the design parameters on the contact andloading characteristics of such the alternative drive concept.A set of essential equations for the cycloid profile,toothcontact and the specific sliding of the cycloid stage with6z=2 are at first derived in the paper.A new load analy-sis approach is extended from the developed LTCA model12.The influences of the design parameters on the contactcharacteristics are further systematically analyzed.Namely,the contact ratio,the specific sliding,the load sharing,thecontact stress and the periodical time-variant bearing loadsare discussed in the paper.These results are also comparedwith the conventional drive having 6z=1.2Fundamentals of the analysis methods forcycloid planetary geardrives2.1Construction of the cycloid disk for toothnumberdifference(TND)oftwo(1)Definition of the base tooth profiles.The toothprofileof the planetary cycloid stage with 6z=2 can be regardedas combination of two disks with the same base cycloidprofile rotated against each other with an angle C,whichis equal to C0/2 of the base cycloid profile.,see Fig.2.Theessential design parameters for the cycloid profile are asfollows,the pitch circle radius RCof the pin-wheel,the radius rPof thepins,the eccentricity e of thecrank,the gear ratio u of the drive,here equal to zP/6z.Thebase cycloidprofile isin accordancewiththecycloidprofile with the same design parameters but with 6z=1,i.e.,owns the coordinates 2,see Fig.3:xC=RCcos&e cos.u&/rPcos.&/(1)yC=RCsin&e sin.u&/rPsin.&/(2)with the pressure angle.uesin.u 1/&.=arctan(3)RCuecos.u 1/&or with the factor k=u e/RCFig.2 Construction ofthe cycloid discwith6z=2Forsch Ingenieurwes(2017)81:3253363272.2Gear meshinganalysisThe cycloid gear mesh can be analyzed considering the diskas stationary,while the center of the pin wheel OPmovesaround the center of the disk OCrelatively with the crank-shaft angle C,and the pin wheel itself rotates also withan angle P(=C/u).The relative motion can be illustratedwith the geometric relation shown in Fig.4.Some relatedissues are discussed as follows.Fig.3 Definition of the base cycloidprofile=arctan.ksin.u 1/&.1 kcos.u 1/&(4)(1)Determination of contact points.The contact pointsof the cycloid-pin tooth pairs Pican be determined with aidof the instant center of velocity.In general the analysis canbebasedon eachof the two cycloid profiles respectively,he-re the terms“odd-numbered”and“even-numbered flanks”are used for distinction.As the relation in Fig.4 shows,theequations of the profile variables for the contact pointson both the cycloid flanks are listed in Table 1.The profile variable for the cycloid curve is definedfrom the rotation of the generating circle,as the relationshown in Fig.3.The gear ratio u of the base profile isequal to zP0.In the case of 6z=2,the gear ratio u is equalto zP/2,also equal to zP0.The curvature radius of the toothprofile can be obtained 2 as3=2(2)Theoretic contact ratio.Because the variable of thecycloid profile is linearly associated with the rotation angleCof the crank,the meshing period is thus equal to ptforthe case with 6z 1.The theoretic contact ratio,which isdefined as the average number of contact tooth pairsduringgear meshing,can be expressed asRC.1+k22kcos.u 1/&/.1+uk2.1+iC/kcos.u 1/&rP(5)=&ptA(8)The inflection point on the cycloid profile owns the pro-perty of the infinite curvature radius,i.e.,the correspondingvariable infmust be equal towhere the pitch angle6is equal to the relation4 vA=zPzC(9)&inf=1arccos.1+uk2.(6)u1.1+u/k(2)Intersecting point of the two base cycloid profiles.Ingeneral,the pointing tip of the tooth profile in the case 6z 1 is usually rounded with a circular arc.In order to simplifythe analysis,the case of rounding is not considered in thestudy.Based on the symmetrical relation,the separationangle of the intersection point Yptof the two base cycloidprofiles to the x-axis is equal to a half-pitch angle C/2,as the relation shown in Fig.2.The corresponding profilevariable ptof the point Yptcan be determined with theequation,yC.&pt.#CvarctanxC.&pt.=2zC(7)Fig.4 Basic geometric relation for tooth contact of the cycloid discwith6z=2p=h328Forsch Ingenieurwes(2017)81:325336Table1 Essential equations for determination of tooth contactRelationToothpairEquationOdd-numberedflanksVariablefor contactpoint1st81I=P=C=uith8iI=81I+.i 1/P0Contactconditionith.j8iIjmodC0/8ptEven-numberedflanksVariablefor contactpoint1st81II=81I+P=81I+P0=2ith8iII=81II+.i 1/P0Contactconditionith.j8iIIjmodC0/8pt(3)Transmission angle.The conversion of the loaded dis-placements from the angular displacement of the cranks aswell as the decomposition of the acting forces are based onthe transmission angle ibetween the normal of the con-tact tooth pair and the line OPOC.As the relation in Fig.4shows,the transmission angle ifor odd-numbered flankscan be determined asiI=.&iIiI/Cv(10)Fig.5 Velocityrelation for the cycloid discinstant center IC.According to the geometric relationinFig.5,the sliding velocity at the instant contact point Miisequal to the multiple of the rotation speed of the pin-wheelwith the distance ICMi,while for even-numbered flanks isvPiC=!PRCp1+k2 2kcos.&iC/rP(13)iII=.&iIIiII/.CC/v(11)The specific sliding Ciof the driving cycloid disc iscalculated by the expression:vPiCp1+k2 2kcos.&/r=R(4)Equivalent displacement.is defined as thecomplianceof a contact tooth pair along its contact normal due to thetranslational displacement e of the cycloid discunder$Ci=vCcosi=iCPCkcosi(14)loading.This displacement e is caused by the motionofthe cranks with an angular displacement at each instant.while the specific sliding Piof the driven pin is equal to$CiThe equivalent displacement eqi-I(II)of odd-/even-numberedtooth pair i can be determined with the transmission angle$Pi=$Ci1(15)i(see Eqs.10 or 11)from Fig.4,i.e.,eqiI.II/=esiniI.II/(12)(5)Determination of active contact tooth pairs.In orderto calculate the acting force,it is essential to determinewhich tooth pairs are in contact.The following relationscan be applied:if the cranks rotate in the counter-clockwise direction,thetooth pairs with positive eqiare in contact,see Eq.12;otherwise,if the cranks rotate in the clockwise direction,the toothpairs with negative eqiare incontact.(6)Sliding velocity.on the contact point plays an import-ant rolefor evaluation of tooth scuffing.The sliding velocityof the ith contact tooth pair can be determined based on theThe specific sliding Ciwill become infinitely great,andPi=1,if the transmission angle iis equal to/2.2.3Typical tooth profile for TND oftwoBecause the base cycloid profile consists of both concaveand convex profile,two profile types,i.e.,either concave orconcave-convex profile,can be found in the case of 6z 1.These two type of the cycloid profile are divided accordingto the relation between the location of the inflection pointand tip pointing,i.e.,concave profile,inf pt;concave-convex profile,infpt.In general,the concave profile is good for tooth contact,but the contact ratio is reduced accordingly.i:45Forsch Ingenieurwes(2017)81:3253363292.4Loaded tooth contactanalysisn0m1TXX(1)Basic LTCAmodelfor cycloidstage.The contact pro-blem of multiple tooth pairs under loading is staticallyin-i=1sqij=1pijA=(18)edeterminate.The shared loads on the tooth pairs can besolved by using two types of equations,namely the equati-ons of load equilibrium as well as the equations of loadeddeformation and displacement 15.Tothis end,a numericalapproach 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 qiis equal to qi=sini.The set of the deformation-displacement equations andthe load equilibrium equation can be summarized in a formof 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 distributed776qnIPpressures pj,i.e.,4q s Iq s I54q s I0nwi=X.fijpj.(16)j=1where fijis the influence coefficient for the condition,that1 12 2 n ne2H136H276767676Hn7(19)the deformation on point i is caused by a load acting on thepoint j.A Q.P.H=T=.ue/.(20)The relations of displacement-deformation is thus validfor the specific point YiS0eT=.ue/wi+hi=eqi(17)where hiis the separation distance between the engagedflanks at the discrete point,more detail see 2.Another relation for the loaded tooth contact is the loadequilibrium equation,i.e.,the sum of all the acting forcesin the tangential direction(see Fig.6)must be equal to theequivalent force T/(u e),i.e.,The sub-matrices in Eqs.19 and 20 are defined as fol-lows:Aicontains all the influence coefficients fPifor the toothpair i;all these sub-matrices are summarized in the ma-trix A in Eq.20.I is either the column or the row unitvector.Pias a column vector contains all the contact stresses onthe discrete units of the tooth pair i;all thesesub-vectorsare summarized in a column vector P in Eq.20.Hias a column vector contains all the separation distan-ces between the engaged tooth flanks of the tooth pair iaccording to the discrete points;all these sub-vectorsaresummarized 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 aredivided into small discretized areas for load analysis,theactual contact pattern and distributed contact stresses canbe simulated.More details can be found in 2,3,15,16.(2)ConversionofthebasicLTCAmodelforthecasewith6z=2.Considering two disks with a separation angle ofC,the LTCA model for analysis of the case of 6z=2 canbeexpandedbytheexpression:2AI0QI32PI32HI340AIIQII5 4PII5=4HII5(21)Fig.6 Relation of acting forces on the cycloiddiskSISII0eT=e2A100q1I32P1360A2:0:q2I76P2766:7 6:7 6:77600An767n5=u330Forsch Ingenieurwes(2017)81:325336The contact areas with distributed stresses of the contacttooth pairs based on Eqs.19 or 21 are solved iteratively untilthe convergent condition is fulfilled,i.e.,all the contactstresses are positive.(3)Load sharing of the drive is distinguishedbetweenthe load sharing among the tooth pairs at a specific angularposition as well as the shared loads distributed on an indi-vidual tooth flank within a meshing cycle.The normal loadFNiI,IIacting on the ith odd/even-numbered contact toothpair is determined as the sum of all the distributed contactstresses which are solved from the LTCAapproachbasedFt=Fc=Tout3 ueTout.u 1/3urRFig.7 Relation of acting forceson thecrank(30)(31)on Eq.21,namely,mnFNiI;II=Xsipij(22)j=1(4)Bearing loads on the cranks can be divided into threetypes of forces based on the load equilibrium conditions,see Fig.6 and 2,10:Forceequilibrium,mFr=XqriFNi=3(23)ionly the radial force Frcan be changed by using suitabledesign parameters so as to reduce the bearing loads.3Overview of the numericalexampleIn order to analyzed the influences of the design parameter,the essential gear data are listed in Table 2.The designparameters considered in the paper are the pin radius rPand the eccentricity e,the corresponding values used forthe analysis are listed Table 3.A larger pin radius is notconsidered here,because the available space forinstallationof the pins in the pin wheel is limited.mFt=XqtiFNi=3(24)i4Influenceanalysisof the designparameters4.1Toothprofile and contactratioMoment equilibrium,the circumferentialforce:The influences of the design parameters e and rPon themFc=XqtiFNi.u 1/e=.3rR/(25)iwhere the factors qriand qtiin the above equation aredeter-mined as follows,qri=sini(26)qti=cosi(27)The result radial forces FCRrand tangential forces FCRtacting on the crank j are equal to the following relationsrespectively,see Fig.7,FCRrj=Fr+FcsinC+2v.j1/=3(28)FCRtj=FtFccosC+2v.j1/=3(29)Because the tangential Ftand circumferential forceFcare directly associated with the output torque Tout,i.e.,location of tip pointing and the contact ratio are representedin Fig.8.Because the contact ratio is linearly associatedwith the profile variable ptfor tooth pointing,the curvesinthe diagram illustrate both the two factors at the sametime.Table2 Essential gearing data for numerical analysisItems/symbolsValueRemarksPitch circle radius of pin wheelRC162.5mmToothnumber of the cycloid diskzC78Toothnumber of the pin wheelzP80Reduction ratio u(Carrierfixed)40zP/6zThickness of the cycloid diskt31.5mmRadius of the bearing hole centerrR90mmOutput torqueT4000NmTable3 Design parameters for influence analysisItems/symbolsValue mmRadius of the pinrP3,4,5,(6)Eccentricitye2.5,3,3.5Profile GenerationAngle Forsch Ingenieurwes(2017)81:325336331262422201816141210864200.511.522.533.5Eccentricity e mm32.521.510.50Fig.8 Influence of e and rPon the location of tip pointing and thecontact ratio Fig.9 Influence of e and rPon theprofileIn general,the cycloid drive with 6z=2 having smallerpins and a larger eccentricity owns a larger contact ratio.Insuch the case,the cycloid profile has a convex and concaveportion.On the other hand,a larger pin radius,e.g,rP=7,lowers the contact ratio with an increased eccentricity.How these parameters affect the profile can be furtheridentified from Fig.9.A larger pin radius rPunder thesameeccentricity e causes a smaller tooth thickness tCof cycloidflank and a closer location of the inflection point Yinfandthe pointing tip Ypt.On the other hand,the eccentricity eaffects the shape of the tooth profile strongly.The toothdepth hCis enlarged with an increased eccentricity e.Fig.10 Influences of the pin radius rPon the specificslidingFig.11 Influences of the eccentricity e on the specific sliding4.2SpecificslidingThe influences of the pin radius rPand the eccentricity eon the specific sliding are illustrated in Figs.10 and 11,respectively.The specific sliding of the cycloid disk Cismuch larger than that of the pin P.The specific sliding Pincreases monotonously from the begin A to the end E ofcontact.The specific sliding C,by contrast,decreases atfirst monotonously,and then asymptotically to1nearbythe singular point,where the transmission angle iis equalto/2.As the tooth pair engages further,Cdecreases withrP=3456Inflection Point7Contact Ratio 332Forsch Ingenieurwes(2017)81:325336Fig.12 Influences of the pin radius rPon the load sharing among thecontact toothpairsFig.13 Influences of the eccentricity e on the load sharing among thecontact toothpairschange of the sign from+1to the a local extremum on thetip E 4.The pin radius has almost no influences as the curvesin Fig.10 show.