外文文獻翻譯-兩齒差擺線行星齒輪傳動的設計【中文3031字】 【PDF+中文WORD】,中文3031字,PDF+中文WORD,外文文獻翻譯-兩齒差擺線行星齒輪傳動的設計【中文3031字】,【PDF+中文WORD】,外文,文獻,翻譯,兩齒差,擺線,行星,齒輪,傳動,設計,中文,3031,PDF 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.