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DOI10.1007/s00170-003-1741-8ORIGINALARTICLEIntJAdvManufTechnol(2004)24:789793FengXianyingWangAiqunLindaLeeStudyonthedesignprincipleoftheLogiXgeartoothprofileandtheselectionofitsinherentbasicparametersReceived:2January2003/Accepted:3March2003/Publishedonline:3November2004Springer-VerlagLondonLimited2004AbstractThedevelopmentofscientifictechnologyandpro-ductivityhascalledforincreasinglyhigherrequirementsofgeartransmissionperformance.Thekeyfactorinfluencingdynamicgearperformanceistheformofthemeshedgeartoothprofile.Toimproveagearstransmissionperformance,anewtypeofgearcalledtheLogiXgearwasdevelopedintheearly1990s.How-ever,forthisspecialkindofgearthereremainmanyunknowntheoreticalandpracticalproblemstobesolved.Inthispaper,thedesignprincipleofthisnewtypeofgearisfurtherstudiedandthemathematicalmoduleofitstoothprofilededuced.Thein-fluenceontheformofthistypeoftoothprofileanditsmeshperformancebyitsinherentbasicparametersisdiscussed,andreasonableselectionsforLogiXgearparametersareprovided.ThusthetheoreticalsysteminformationabouttheLogiXgeararedevelopedandenriched.Thisstudyimpactsmostsignificantlytheimprovementofloadcapacity,miniaturisationanddurabilityofmodernkinetictransmissionproducts.KeywordsBasicparameterDesignprincipleLogiXgearMinuteinvoluteToothprofile1IntroductionInordertoimprovegeartransmissionperformanceandsatisfysomespecialrequirements,anewtypeofgear1wasputfor-ward；itwasnamed“LogiX”inordertoimprovesomedemeritsofW-N(Wildhaver-Novikov)andinvolutegears.Besideshavingtheadvantagesofbothkindsofgearsmen-tionedabove,thenewtypeofgearhassomeotherexcellentF.Xianying(a117)W.AiqunSchoolofMechanicalEngineering,ShandongUniversity,P.R.ChinaE-mail:FXYTel.:+86-531-8395852(0)L.LeeSchoolofMechanical&ManufacturingEngineering,SingaporePolytechnic,Singaporecharacteristics.Onthisnewtoothprofile,thecontinuouscon-cave/convexcontactiscarriedoutfromitsdedendumtoitsad-dendum,wheretheengagementswitharelativecurvatureofzeroareassuredatmanypoints.Here,thiskindofpointiscalledthenull-point(N-P).ThepresenceofmanyN-PsduringthemeshprocessofLogiXgearscanresultinasmallerslidingcoeffi-cient,andthemeshtransmissionperformancebecomesalmostrollingfrictionaccordingly.Thusthisnewtypeofgearhasmanyadvantagessuchashighercontactintensity,longerlifeandalargertransmission-ratiopowertransferthanthestandardin-volutegear.Experimentalresultsshowedthat,givenacertainnumberofN-PsbetweentwomeshedLogiXgears,thecontactfatiguestrengthis3timesandthebendfatiguestrength2.5timeslargerthanthoseofthestandardinvolutegear.Moreover,theminimumtoothnumbercanalsobedecreasedto3,muchsmallerthanthatofthestandardinvolutegear.TheLogiXgear,regardedasanewtypeofgear,stillpresentssomeunsolvedproblems.Thedevelopmentofcomputernumer-icalcontrolling(CNC)technologymustalsobetakenintocon-siderationnewhigh-efficiencymethodstocutthisnewtypeofgear.Therefore,furtherstudyofthisnewtypeofgearmostsignificantlyimpactstheaccelerationofitsbroadandpracticalapplication.Thispaperhasthepotentialtousherinanewerainthehistoryofgearmeshtheoryandapplication.2DesignprincipleofLogiXtoothprofileAccordingtogearmeshandmanufacturingtheories,inordertosimplifyproblemanalysis,generallyagearsbasicrackisbegunwithsomestudies2.SohereletusdiscussthebasicrackoftheLogiXgearfirst.Figure1showsthedesignprincipleofdi-videdinvolutecurvesoftheLogiXrack.InFig.1,P.LrepresentsapitchlineoftheLogiXrack.OnepointO1isselectedtoformtheanglen0O1N1=0,P.LO1N1.ThepointsofintersectionbytworadialsO1n0andO1N1andthepitchlineP.LareN1andn0.LetO1n0=G1,extendO1n0toOprime1,andmaketwotan-gentbasiccircleswhosecentresareO1,Oprime1andradiiareequaltoG1.ThepointofintersectionbetweencircleO1andpitchline790Fig.1.DesignprincipleofLogiXracktoothprofileP.Lisn0.ThepointofintersectionbetweencircleO2andpitchlineP.Lisn1.Makethecommontangentg1s1ofbasiccircleO1andOprime1,thengeneratetwominuteinvolutecurvesm0s1ands1m1whosebasiccirclecentresareO1andOprime1.Theradiiofcurvatureatpointsm0andm1onthetoothprofileshouldbe:m0=m0n0,m1=m1n1,andthecentresaremetonthepitchline.MultipledifferentminuteinvolutesconsistingofaLogiXprofileshouldbearrangedforapropersequence.Thepressureangleofthenextminuteinvolutecurvem1m2shouldhaveanincrementcomparabletoitslastsegmentm0m1.Thecentresofcurvatureatextremepointsm1,m2,etc.shouldbeonthepitchline,andtheradiusofthebasiccircleisafunctionofpressure1itvariesfromG1toG2.Theconditionforjoiningfrontandrearcurvesisthattheradiusofcurvatureatpointm1mustbeequaltotheradiusofcurvaturejustafterpointm1,andtheradiusofcurvatureatpointm2mustbeequaltotheradiusofcurvaturejustafterpointm2.Figure2showstheconnectionandprocessofgeneratingminuteinvolutecurves.Accordingtotheabovedis-cussion,thewholetoothprofilecanbeformed.Fig.2.Connectionofminuteinvolutecurves3MathematicmoduleofLogiXtoothprofile3.1MathematicmoduleofthebasicLogiXrackAccordingtotheabove-mentioneddesignprinciple,thecurva-turecentreofeveryfinelydividedprofilecurveshouldbelocatedattherackpitchline,andthevalueoftherelativecurvatureateverypointconnectingdifferentminuteinvolutecurvesshouldbezero.Thedesignofthetoothprofileissymmetricalwithre-specttothepitchline,andtheaddendumisconvexwhilethededendumisconcave.ThusforthewholeLogiXtoothprofile,itcanbedealtwithbydividingitintofourparts,asshowninFig.3.SetupthecoordinatesasshowninFig.4,wheretheoriginofthecoordinatesOcoincideswiththepointofintersectionm0be-tweenrackpitchlineP.Landtheinitialdividedminuteinvolutecurve.AccordingtothecoordinatessetupinFig.4,theformationofinitialminuteinvolutecurvem0m1isshowninFig.5.Fig.3.LogiXracktoothprofileFig.4.Set-upofcoordinatesFig.5.Formationprocessofinitialminuteinvolutecurvem0m1791Here:n0nprime0O1Oprime1,n1nprime1O1Oprime1,n1n1n0nprime0,andthepa-rameters0,G1andm0aregivenasinitialconditions.Thecurvatureradiusoftheinvolutecurveatpoints1iss1=G1,ors1=m1+G11.Thusthecurvatureradiusandpressureangleoftheminuteinvolutecurveatpointm1areasfollows:m1=s1G11=G1(1)(1)1=0+1.(2)Accordingtothegeometricalrelationship,wecandeduce:tg(0+)=2G1G1cosG1cos1G1sinG1sin1=2(cos+cos1)sinsin1.(3)BasedonEqs.1,2and3andtheformingprocessoftheLogiXrackprofile,thecurvatureradiusformulaofanarbitrarypointontheprofileisdeduced:mi=mi1+Gi(i).Wheni=kandm0=0,itisexpressedasfollows:mk=G1(1)+G2(2)+Gk(k)=ksummationdisplayi=1Gi(i).(4)Similarly,thepressureangleonanarbitrarykpointofthetoothprofilecanbededucedasfollows:k=0+(+1)+(+2)+(+k)=0+ksummationdisplayi=1(+i)=0+k+ksummationdisplayi=1i.(5)Byni1ni=Gi(sinsini)/cos(i1+),Eq.5canbeobtained:n0nk=ksummationdisplayi=1ni1ni=ksummationdisplayi=1Gi(sinsini)cos(i1+).(6)ThusthemathematicalmodeloftheNo.2portionfortheLogiXrackprofileisasfollows:braceleftbiggx1=n0nkmkcosky1=mksink(No.2).(7)Similarly,themathematicalmodelsoftheotherthreesegmentscanalsobeobtainedasfollows:braceleftbiggx1=(n0nkmkcosk)y1=mksink(No.1)(8)braceleftbiggx1=s(n0nkmkcosk)y1=mksink(No.3)(9)braceleftbiggx1=s+n0nkmkcosky1=mksink(No.4).(10)Fig.6.MeshcoordinatesofLogiXgearanditsba-sicrack3.2MathematicalmoduleoftheLogiXgearThecoordinatesO1X1Y1,O2X2Y2andPXYaresetupasshowninFig.6toexpressthemeshrelationshipbetweentheLogiXrackandtheLogiXgear.Here,O1X1Y1isfixedontherack,andO1isthepointofintersectionbetweentheracktoothprofileanditspitchline.O2X2Y2isfixedonthemeshedgear,andO2isthegearscentre.PXYisanabsolutecoordinate,andPisthepointofintersectionoftherackspitchlineandthegearspitchcircle.Inaccordancewithgearmeshingtheories3,iftheabovemodeloftheLogiXracktoothprofileischangedfromcoordinateO1X1Y1toOXY,andthenagaintoO2X2Y2,anewtypeofgearprofilemodelcanbededucedasfollows:braceleftbiggx2=mkcoskcos2(mksinkr2)sin2y2=mkcosksin2+(mksinkr2)cos2.(11)Herethepositivedirectionof2isclockwise,andonlythemodeloftheLogiXgeartoothprofileinthefirstquadrantofthecoordi-natesisgiven.4EffectontheperformanceoftheLogiXgearbyitsinherentparametersandtheirreasonableselectionBesidesthebasicparametersofthestandardinvoluterack,theLogiXtoothprofilehasinherentbasicparameterssuchasinitialpressureangle0,relativepressureangle,initialbasiccircleradiusG0,etc.Theselectionoftheseparametershasagreatin-fluenceontheformoftheLogiXtoothprofile,andtheformdirectlyinfluencesgeartransmissionperformance.Thustherea-sonableselectionofthesebasicparametersisveryimportant.4.1Influenceandselectionofinitialpressureangle0Consideringthehighertransmissionefficiencyinpracticalde-sign,theinitialpressureangle0shouldbeselectedas0.ButthefinalcalculationresultshowedthattheLogiXgeartoothpro-filecutbytheracktoolwhoseinitialpressureanglewasequaltozerowouldbeovercutonthepitchcirclegenerally.Thustheinitialpressureangle0cannotbezero.Comparingtherelativedoublecircle-arcgear3,wecanalsodeducethatthesmaller792theinitialpressureangle0,thelargerthegearnumberforpro-ducingtheovercut.Thustheinitialpressureangle0shouldnotonlynotbezero,butshouldnotbetoosmall,either.FromEqs.3,4and5,theinfluenceof0ontheLogiXtoothprofilecanbedirectlydescribedbyFig.7.Obviously,increasingtheini-tialpressureanglewillcausethecurvatureoftheLogiXracktoothprofiletobecomelarger.Iftherackselectsalargermod-uleandtoosmallaninitialpressureangle0,itsaddendumwillbecometoonarroworevenovercut.ThustheLogiXtoothpro-filethatselectsalargermoduleshouldselectasmaller0,andtheprofilethatselectsasmallermoduleshouldselectalarger0.Generally,bypracticalcalculationexperience,theselected0shouldbelocatedwithinarangeof212,andthelargertheLogiXgearmodule,thesmallershouldbeitsinitialpressureangle0.4.2InfluenceandselectionofinitialbasiccircleradiusG0AccordingtotheempiricalformulaGi=G01sin(0.6i)1,therearetwoparametersaffectingthebasiccircleradiusGioftheLogiXgearatdifferentpositionsoftoothprofile:oneistheG0andtheotheristheinitialpressureanglei.Figure8showstheinfluenceofG0ontheLogiXtoothprofilewhencertainvaluesofparameter0andareselected.Obviously,asG0in-creases,thecurvatureofthenewtypeofgeartoothprofilewillbecomesmallerandsmaller.Conversely,itwillbecomeincreas-inglylargerasG0decreases.ThusthenewtypeofrackwithalargemoduleparametershouldselectalargeG0value,andonehavingasmallmoduleparametershouldselectasmallG0value.4.3InfluenceandselectionofrelativepressureangleFigure9showsthevariableofthetoothprofileaffectedbytheparameter.AccordingtotheformingprocessoftheLogiXtoothFig.7.Influenceof0onLogiXtoothprofileFig.8.InfluenceofG0onLogiXtoothprofileFig.9.InfluenceofonLogiXtoothprofileprofile,thesmallertheselectedparameter,thelargerthenum-berofN-PsmeshingonthetoothprofileoftwoLogiXgears.FromSect.2.1theformuladescribingtherelativepressureanglekofanarbitraryN-Pmkcanbededucedasfollows:sin(k1+)cos(k1+)=2(cos+cosk)sinsink.(12)ByEqs.5and12,thelargertheparameterbeingselected,thelargerwillbethekparameter,andatcertainselectedvaluesoftheinitialpre