外文文獻(xiàn)翻譯-兩齒差擺線行星齒輪傳動(dòng)的設(shè)計(jì)【中文3031字】 【PDF+中文WORD】,中文3031字,PDF+中文WORD,外文文獻(xiàn)翻譯-兩齒差擺線行星齒輪傳動(dòng)的設(shè)計(jì)【中文3031字】,【PDF+中文WORD】,外文,文獻(xiàn),翻譯,兩齒差,擺線,行星,齒輪,傳動(dòng),設(shè)計(jì),中文,3031,PDF 譯自：Design of cycloid planetary gear drives with tooth number difference of two 兩齒差擺線行星齒輪傳動(dòng)的設(shè)計(jì) Shyi-Jeng Tsai·Ling-Chiao Chang·Ching-Hao Huang 摘要：擺線行星齒輪減速器在自動(dòng)化機(jī)械中有著廣泛的應(yīng)用。即使該減速器具有高傳動(dòng)比、多齒 對(duì)嚙合、減震能力強(qiáng)等優(yōu)點(diǎn)，如何提高傳動(dòng)裝置的功率密度仍然是當(dāng)今擺線行星齒輪減速器發(fā)展 的重要課題。為此，提出了兩齒數(shù)差的概念。本文的目的是系統(tǒng)地分析這種擺線行星齒輪傳動(dòng)的 載荷接觸特性，以評(píng)價(jià)其可行性。本文首先導(dǎo)出了擺線輪廓、齒面接觸和擺線階段比滑動(dòng)的基本 方程。從建立的基于影響系數(shù)法的模型出發(fā)，提出了一種載荷齒接觸分析方法。通過(guò)算例，系統(tǒng) 地分析了設(shè)計(jì)參數(shù)對(duì)接觸特性的影響。所得出的這些結(jié)果也與傳統(tǒng)的齒數(shù)差為 1 的傳動(dòng)裝置進(jìn)行 了比較。分析結(jié)果表明，該概念具有較大的偏心率和較小的銷釘半徑，既能有效地?cái)U(kuò)大接觸比， 又能有效地減小比滑、載荷和接觸應(yīng)力。雖然軸承載荷的徑向部分也可以相應(yīng)地減小，但是利用 兩齒數(shù)差的概念不能有效地降低總周期時(shí)變軸承載荷。 1. 引言 擺線行星齒輪減速器是動(dòng)力和精密運(yùn)動(dòng)傳動(dòng)的重要傳動(dòng)裝置。目前，設(shè)計(jì)為兩級(jí)偏心差速器 的齒輪機(jī)構(gòu)，即所謂的“RV-傳動(dòng)”，在自動(dòng)化機(jī)械中得到了廣泛的應(yīng)用。如圖 1 所示的結(jié)構(gòu)圖 所示，這種齒輪傳動(dòng)類型由一個(gè)漸開(kāi)線行星級(jí)傳動(dòng)和一個(gè)兩個(gè)圓盤的擺線行星級(jí)傳動(dòng)組成。每一 個(gè)漸開(kāi)線行星都安裝在曲柄軸上，以產(chǎn)生擺線盤的旋轉(zhuǎn)運(yùn)動(dòng)。這種設(shè)計(jì)結(jié)構(gòu)不僅具有齒輪比高的 優(yōu)點(diǎn)，而且由于多齒對(duì)的接觸，在分擔(dān)載荷和減震能力方面具有良好的性能。然而，目前齒輪減 速器設(shè)計(jì)的趨勢(shì)是除了對(duì)精密運(yùn)動(dòng)的要求外，還要增大功率密度。為此，必須減少接觸齒副和曲 柄上的載荷。 圖 1 擺線行星齒輪傳動(dòng)結(jié)構(gòu) 在各種的措施中，通過(guò)采用較大的齒數(shù)差的這種設(shè)計(jì)理念可以為改善負(fù)載接觸特性提供一種 可能性。換句話說(shuō)，擺線齒輪副的齒數(shù)差的可能選擇為兩種，而不是常規(guī)應(yīng)用中經(jīng)常使用的一 種。在實(shí)際應(yīng)用中，齒數(shù)差為 2(Δz=2)的齒輪傳動(dòng)常用于小減速比的傳動(dòng)，但是齒數(shù)差為 2 的 齒輪傳動(dòng)在傳動(dòng)比比較高的情況下則很少使用。因此，評(píng)價(jià)這種替代驅(qū)動(dòng)概念的可行性是很有趣 的。本文探討了設(shè)計(jì)參數(shù)對(duì)加載接觸特性的影響，并與常規(guī)傳動(dòng)進(jìn)行了比較。 載荷分析的基本工作是導(dǎo)出幾何和運(yùn)動(dòng)學(xué)之間的關(guān)系。對(duì)擺線輪廓線數(shù)學(xué)模型的研究在許多 文獻(xiàn)中都有發(fā)現(xiàn)。根據(jù)齒輪傳動(dòng)理論或運(yùn)動(dòng)學(xué)方法，也可以對(duì)齒輪嚙合進(jìn)行分析，比如即時(shí)中心 方法。齒輪傳動(dòng)的另一個(gè)評(píng)價(jià)標(biāo)準(zhǔn)是嚙合齒之間的滑動(dòng)特性。例如齒側(cè)的損傷，如點(diǎn)蝕、磨損或 打分，這些可以用比滑動(dòng)率來(lái)預(yù)測(cè)。然而，在有關(guān)擺線齒輪泵的相關(guān)研究文獻(xiàn)中，較少涉及擺線 齒輪減速器的研究。 此外，載荷分析也是評(píng)價(jià)Δz=2 齒輪傳動(dòng)可行性的一個(gè)重要問(wèn)題。擺線齒輪接觸應(yīng)力分析的 常用方法 是驅(qū)動(dòng)器有限單元法?；谠摲治龇椒ǘ_(kāi)發(fā)了另一種負(fù)荷分析方法。比如軋制作用 力的計(jì)算方法，該方法是為了支持在滾動(dòng)作用力下的的擺線盤的軸承。也有著重研究了制造誤差 對(duì)傳輸負(fù)載和傳動(dòng)誤差的影響。還有一些人基于齒面嚙合剛度假設(shè)，分析了加工誤差對(duì)動(dòng)載荷行 為的影響。也有人進(jìn)行了接觸應(yīng)力分析實(shí)驗(yàn)，驗(yàn)證了分析方法的理論分析結(jié)果。本文作者基于影 響系數(shù)法開(kāi)發(fā)了一種數(shù)值加載的齒面接觸分析方法，該方法還成功地應(yīng)用于擺線齒對(duì)以及考慮軸 承剛度和摩擦力的全齒接觸分析。 因此，本文的目的是系統(tǒng)地研究設(shè)計(jì)參數(shù)對(duì)這種替代驅(qū)動(dòng)概念的接觸和負(fù)載特性的影響。本 文首先導(dǎo)出了Δz=2 的擺線齒廓、齒面接觸和擺線階段比滑動(dòng)的基本方程。擴(kuò)展了一種新的載荷 分析方法。 從發(fā)展的數(shù)值加載的齒面接觸分析模型，進(jìn)一步系統(tǒng)地分析了設(shè)計(jì)參數(shù)對(duì)接觸特性 的影響，即：接觸比、比滑動(dòng)、載荷分擔(dān)、接觸應(yīng)力。 本文討論了周期時(shí)變軸承載荷并且將這 些結(jié)果與Δz=1 的常規(guī)驅(qū)動(dòng)器進(jìn)行了比較。 2 擺線行星齒輪傳動(dòng)分析方法的基本原理 2.1 兩齒差擺線盤的構(gòu)造 基本齒廓的定義：Δz=2 行星擺線級(jí)的齒廓可以看作是兩個(gè)具有相同基座擺線廓線的兩個(gè)圓盤的 t 組合，其角度為tc ，其值等于基擺線輪廓的 擺線輪廓的基本設(shè)計(jì)參數(shù)如下： 螺距圓半徑為 Rc 的銷輪； 針的半徑 rp ； 曲柄的偏心度； 傳動(dòng)的齒輪比 e，這里等于 Zp 。 Dz Co ，見(jiàn)圖 2。 2 圖 2 Δz=2 的擺線盤的構(gòu)造 jc u 2.2 齒輪嚙合分析 擺線齒輪嚙合的分析可以將圓片視為固定的,當(dāng)銷輪的中心相對(duì)于曲柄軸角jc 圍繞盤的中心移動(dòng) 時(shí)，銷輪本身也以角j（p = ）旋轉(zhuǎn)。相對(duì)運(yùn)動(dòng)可以用圖 3 所示的幾何關(guān)系來(lái)表示。 圖 3 Δz=2 的擺線盤齒面接觸的基本幾何關(guān)系 2.3 兩種齒數(shù)差的典型齒廓 由于基擺線輪廓由凹型和凸型組成，在Δz>1 的情況下，可以找到兩種輪廓類型，即凹型或 凹型一凸型。根據(jù)拐點(diǎn)位置與針尖指向之間的關(guān)系，對(duì)這兩種擺線輪廓進(jìn)行了劃分。 凹型輪廓類型，qinf 3 qPt 凹型—凸型輪廓類型，qinf áqPt 。 一般情況下，凹面輪廓有利于齒面接觸，但接觸比也相應(yīng)降低。 3 數(shù)值算例綜述 為了分析設(shè)計(jì)參數(shù)的影響，在表 1 中列出了必要的齒輪數(shù)據(jù)。本文所考慮的設(shè)計(jì)參數(shù)是銷半 徑 rP 和偏心度 e，相應(yīng)的分析值列于表 2。這里不考慮更大的銷半徑，因?yàn)樵阡N輪上安裝銷的可 用空間是有限的。 表 1 數(shù)值分析中的基本齒輪傳動(dòng)數(shù)據(jù) 名稱 數(shù)值 備注 銷輪的節(jié)圓半徑 162.5mm 擺線盤的齒數(shù) 78 銷輪齒數(shù) 80 減少比率(卡勒固定) 40 Zp/Δz 擺線盤厚度 31.5mm 軸承孔中心半徑 90mm 輸出扭矩 4000Nm 表 2 影響分析的設(shè)計(jì)參數(shù) 名稱 數(shù)值 針形半徑 3,4,5，（6） 偏心 2.5,3,3.5 4 設(shè)計(jì)參數(shù)的影響分析 4.1 齒廓與接觸比 設(shè)計(jì)參數(shù) e 和 rP 對(duì)尖端位置和接觸比率的影響表示在圖 4 中。由于齒面接觸比與齒形變量 qPt 呈線性相關(guān)，因此圖中的曲線同時(shí)說(shuō)明了齒面接觸和齒形變量這兩個(gè)因素。 一般情況下，Δz=2 的擺線傳動(dòng)具有較小的引腳和較大的偏心比，且具有較大的接觸比。在這種 情況下，擺線輪廓具有凸凹部分。另一方面，較大的銷釘半徑，例如， rP = 7 ，其降低了接觸比， 增加了偏心率。 4.2 比滑動(dòng) 偏心 e（mm） 圖 4 電子和電子技術(shù)對(duì)針尖定位及觸點(diǎn)比的影響 圖 5 和圖 6 分別說(shuō)明了針半徑 rP 和偏心 e 對(duì)比滑動(dòng)的影響。擺線盤厚度xc 的比滑動(dòng)遠(yuǎn)大于針輪厚 度xP 。xP 的比滑動(dòng)單調(diào)地從開(kāi)始 A 增加到 E。相反，xc 的比滑動(dòng)先單調(diào)下降，然后漸近于奇點(diǎn)附 近的一點(diǎn)，其中傳輸角g等于 15。當(dāng)齒對(duì)進(jìn)一步嚙合時(shí)，xc 從·無(wú)窮大減小到一個(gè)局部極限值。 圖 5 銷徑 rP 對(duì)比滑動(dòng)的影響 圖 6 偏心率 e 對(duì)比滑動(dòng)的影響 4.3 接觸齒對(duì)之間的共同載荷 因?yàn)榛铨X對(duì)的傳動(dòng)角g是不同的，正常載荷在接觸齒對(duì)之間的分配也不均勻。銷半徑對(duì)載荷分配 的影響較小，相反，一個(gè)較小的偏心會(huì)導(dǎo)致不均勻地分擔(dān)負(fù)載。 5 與齒數(shù)差為 1 的擺線行星齒輪傳動(dòng)比較 為了比較其傳動(dòng)特性，使Δz=1 和 2 的減速器的偏心度 e 為相同的值，而使其銷半徑的值不 同。 6 總結(jié)和展望 為了提高高傳動(dòng)比擺線行星齒輪傳動(dòng)的傳動(dòng)性能，提出了齒數(shù)差為 2(Δz=2)的概念。利用 建立的基于影響系數(shù)法的加載齒接觸分析模型，探討了該傳動(dòng)方式的載荷接觸特性。分析了設(shè)計(jì) 參數(shù)對(duì)偏心率和銷徑的影響，并與Δz=1 的擺線行星齒輪傳動(dòng)進(jìn)行了比較。因此得到了一些結(jié)果， 分析結(jié)果使我們得出以下結(jié)論： （1）采用合適的設(shè)計(jì)參數(shù)可以提高Δz=2 的擺線行星齒輪傳動(dòng)的理論接觸比，并期望采用較小 的針半徑和較大的偏心度。 （2）偏心率對(duì)比滑動(dòng)有顯著影響。當(dāng)偏心較大時(shí)，具有無(wú)限大比滑動(dòng)xc 的點(diǎn)將向齒根方向移動(dòng)， 即擺線輪廓，且xc 值減小， （3）Δz=2 載荷和接觸應(yīng)力的共同作用側(cè)翼，且在Δz=1 的情況下可以顯著地降低接觸應(yīng)力，小 數(shù)值的銷半徑和較大偏心度適合于這種傳動(dòng)裝置的設(shè)計(jì)。 （4）ΔZ＝2 的側(cè)向載荷和接觸應(yīng)力的變化與循環(huán)漸開(kāi)線齒輪傳動(dòng)一樣，類似于漸開(kāi)線直齒圓柱 齒輪傳動(dòng)的現(xiàn)象。 （5）用Δz=2 的擺線齒輪傳動(dòng)對(duì)于減小曲柄上的軸承載荷的效果不是很顯著，因?yàn)榄h(huán)向力的影 響要大得多。 然而，由于間隙存在的必要性，這種改進(jìn)后的擺線廓形常被應(yīng)用于實(shí)際應(yīng)用中，而不是應(yīng)用 于理論中。因此為了進(jìn)一步設(shè)計(jì)Δz=2 的擺線傳動(dòng)的側(cè)面改進(jìn)，本文得出了一些好的結(jié)果。同時(shí)， 本文提出的分析方法也是一種有效的工具。 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.