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DOI10.1007/s0017000317418ORIGINALARTICLEIntJAdvManufTechnol200424789–793FengXianyingWangAiqunLindaLeeStudyonthedesignprincipleoftheLogiXgeartoothprofileandtheselectionofitsinherentbasicparametersReceived2January2003/Accepted3March2003/Publishedonline3November2004SpringerVerlagLondonLimited2004AbstractThedevelopmentofscientifictechnologyandproductivityhascalledforincreasinglyhigherrequirementsofgeartransmissionperformance.Thekeyfactorinfluencingdynamicgearperformanceistheformofthemeshedgeartoothprofile.Toimproveagearstransmissionperformance,anewtypeofgearcalledtheLogiXgearwasdevelopedintheearly1990s.However,forthisspecialkindofgearthereremainmanyunknowntheoreticalandpracticalproblemstobesolved.Inthispaper,thedesignprincipleofthisnewtypeofgearisfurtherstudiedandthemathematicalmoduleofitstoothprofilededuced.Theinfluenceontheformofthistypeoftoothprofileanditsmeshperformancebyitsinherentbasicparametersisdiscussed,andreasonableselectionsforLogiXgearparametersareprovided.ThusthetheoreticalsysteminformationabouttheLogiXgeararedevelopedandenriched.Thisstudyimpactsmostsignificantlytheimprovementofloadcapacity,miniaturisationanddurabilityofmodernkinetictransmissionproducts.KeywordsBasicparameterDesignprincipleLogiXgearMinuteinvoluteToothprofile1IntroductionInordertoimprovegeartransmissionperformanceandsatisfysomespecialrequirements,anewtypeofgear1wasputforwarditwasnamedLogiXinordertoimprovesomedemeritsofWNWildhaverNovikovandinvolutegears.Besideshavingtheadvantagesofbothkindsofgearsmentionedabove,thenewtypeofgearhassomeotherexcellentF.Xianyinga117W.AiqunSchoolofMechanicalEngineering,ShandongUniversity,P.R.ChinaEmailFXYingsdu.edu.cnTel.8653183958520L.LeeSchoolofMechanicalManufacturingEngineering,SingaporePolytechnic,Singaporecharacteristics.Onthisnewtoothprofile,thecontinuousconcave/convexcontactiscarriedoutfromitsdedendumtoitsaddendum,wheretheengagementswitharelativecurvatureofzeroareassuredatmanypoints.Here,thiskindofpointiscalledthenullpointNP.ThepresenceofmanyNPsduringthemeshprocessofLogiXgearscanresultinasmallerslidingcoefficient,andthemeshtransmissionperformancebecomesalmostrollingfrictionaccordingly.Thusthisnewtypeofgearhasmanyadvantagessuchashighercontactintensity,longerlifeandalargertransmissionratiopowertransferthanthestandardinvolutegear.Experimentalresultsshowedthat,givenacertainnumberofNPsbetweentwomeshedLogiXgears,thecontactfatiguestrengthis3timesandthebendfatiguestrength2.5timeslargerthanthoseofthestandardinvolutegear.Moreover,theminimumtoothnumbercanalsobedecreasedto3,muchsmallerthanthatofthestandardinvolutegear.TheLogiXgear,regardedasanewtypeofgear,stillpresentssomeunsolvedproblems.ThedevelopmentofcomputernumericalcontrollingCNCtechnologymustalsobetakenintoconsiderationnewhighefficiencymethodstocutthisnewtypeofgear.Therefore,furtherstudyofthisnewtypeofgearmostsignificantlyimpactstheaccelerationofitsbroadandpracticalapplication.Thispaperhasthepotentialtousherinanewerainthehistoryofgearmeshtheoryandapplication.2DesignprincipleofLogiXtoothprofileAccordingtogearmeshandmanufacturingtheories,inordertosimplifyproblemanalysis,generallyagearsbasicrackisbegunwithsomestudies2.SohereletusdiscussthebasicrackoftheLogiXgearfirst.Figure1showsthedesignprincipleofdividedinvolutecurvesoftheLogiXrack.InFig.1,P.LrepresentsapitchlineoftheLogiXrack.OnepointO1isselectedtoformtheanglen0O1N1α0,P.LO1N1.ThepointsofintersectionbytworadialsO1n0andO1N1andthepitchlineP.LareN1andn0.LetO1n0G1,extendO1n0toOprime1,andmaketwotangentbasiccircleswhosecentresareO1,Oprime1andradiiareequaltoG1..ThepointofintersectionbetweencircleO1andpitchline790Fig.1.DesignprincipleofLogiXracktoothprofileP.Lisn0.ThepointofintersectionbetweencircleO2andpitchlineP.Lisn1.Makethecommontangentg1s1ofbasiccircleO1andOprime1,thengeneratetwominuteinvolutecurvesm0s1ands1m1whosebasiccirclecentresareO1andOprime1.Theradiiofcurvatureatpointsm0andm1onthetoothprofileshouldbeρm0m0n0,ρm1m1n1,andthecentresaremetonthepitchline.MultipledifferentminuteinvolutesconsistingofaLogiXprofileshouldbearrangedforapropersequence.Thepressureangleofthenextminuteinvolutecurvem1m2shouldhaveanincrementcomparabletoitslastsegmentm0m1.Thecentresofcurvatureatextremepointsm1,m2,etc.shouldbeonthepitchline,andtheradiusofthebasiccircleisafunctionofpressure1–itvariesfromG1toG2.Theconditionforjoiningfrontandrearcurvesisthattheradiusofcurvatureatpointm1mustbeequaltotheradiusofcurvaturejustafterpointm1,andtheradiusofcurvatureatpointm2mustbeequaltotheradiusofcurvaturejustafterpointm2.Figure2showstheconnectionandprocessofgeneratingminuteinvolutecurves.Accordingtotheabovediscussion,thewholetoothprofilecanbeformed.Fig.2.Connectionofminuteinvolutecurves3MathematicmoduleofLogiXtoothprofile3.1MathematicmoduleofthebasicLogiXrackAccordingtotheabovementioneddesignprinciple,thecurvaturecentreofeveryfinelydividedprofilecurveshouldbelocatedattherackpitchline,andthevalueoftherelativecurvatureateverypointconnectingdifferentminuteinvolutecurvesshouldbezero.Thedesignofthetoothprofileissymmetricalwithrespecttothepitchline,andtheaddendumisconvexwhilethededendumisconcave.ThusforthewholeLogiXtoothprofile,itcanbedealtwithbydividingitintofourparts,asshowninFig.3.SetupthecoordinatesasshowninFig.4,wheretheoriginofthecoordinatesOcoincideswiththepointofintersectionm0betweenrackpitchlineP.Landtheinitialdividedminuteinvolutecurve.AccordingtothecoordinatessetupinFig.4,theformationofinitialminuteinvolutecurvem0m1isshowninFig.5.Fig.3.LogiXracktoothprofileFig.4.SetupofcoordinatesFig.5.Formationprocessofinitialminuteinvolutecurvem0m1791Heren0nprime0O1Oprime1,n1nprime1O1Oprime1,n1n1n0nprime0,andtheparametersα0,δ,G1andρm0aregivenasinitialconditions.Thecurvatureradiusoftheinvolutecurveatpoints1isρs1G1δ,orρs1ρm1G1δ1.Thusthecurvatureradiusandpressureangleoftheminuteinvolutecurveatpointm1areasfollowsρm1ρs1−G1δ1G1δ−δ11α1α0δδ1.2Accordingtothegeometricalrelationship,wecandeducetgα0δ2G1−G1cosδ−G1cosδ1G1sinδ−G1sinδ12−cosδcosδ1sinδ−sinδ1.3BasedonEqs.1,2and3andtheformingprocessoftheLogiXrackprofile,thecurvatureradiusformulaofanarbitrarypointontheprofileisdeducedρmiρmi−1Giδ−δi.Whenikandρm00,itisexpressedasfollowsρmkG1δ−δ1G2δ−δ2Gkδ−δkksummationdisplayi1Giδ−δi.4Similarly,thepressureangleonanarbitrarykpointofthetoothprofilecanbededucedasfollowsαkα0δδ1δδ2δδkα0ksummationdisplayi1δδiα0kδksummationdisplayi1δi.5Byni−1niGisinδ−sinδi/cosαi−1δ,Eq.5canbeobtainedn0nkksummationdisplayi1ni−1niksummationdisplayi1Gisinδ−sinδicosαi−1δ.6ThusthemathematicalmodeloftheNo.2portionfortheLogiXrackprofileisasfollowsbraceleftbiggx1n0nk−ρmkcosαky1ρmksinαkNo.2.7Similarly,themathematicalmodelsoftheotherthreesegmentscanalsobeobtainedasfollowsbraceleftbiggx1−n0nk−ρmkcosαky1−ρmksinαkNo.18braceleftbiggx1s−n0nk−ρmkcosαky1ρmksinαkNo.39braceleftbiggx1sn0nk−ρmkcosαky1−ρmksinαkNo.4.10Fig.6.MeshcoordinatesofLogiXgearanditsbasicrack3.2MathematicalmoduleoftheLogiXgearThecoordinatesO1X1Y1,O2X2Y2andPXYaresetupasshowninFig.6toexpressthemeshrelationshipbetweentheLogiXrackandtheLogiXgear.Here,O1X1Y1isfixedontherack,andO1isthepointofintersectionbetweentheracktoothprofileanditspitchline.O2X2Y2isfixedonthemeshedgear,andO2isthegearscentre.PXYisanabsolutecoordinate,andPisthepointofintersectionoftherackspitchlineandthegearspitchcircle.Inaccordancewithgearmeshingtheories3,iftheabovemodeloftheLogiXracktoothprofileischangedfromcoordinateO1X1Y1toOXY,andthenagaintoO2X2Y2,anewtypeofgearprofilemodelcanbededucedasfollowsbraceleftbiggx2−ρmkcosαkcosϕ2−ρmksinαk−r2sinϕ2y2−ρmkcosαksinϕ2ρmksinαk−r2cosϕ2.11Herethepositivedirectionofϕ2isclockwise,andonlythemodeloftheLogiXgeartoothprofileinthefirstquadrantofthecoordinatesisgiven.4EffectontheperformanceoftheLogiXgearbyitsinherentparametersandtheirreasonableselectionBesidesthebasicparametersofthestandardinvoluterack,theLogiXtoothprofilehasinherentbasicparameterssuchasinitialpressureangleα0,relativepressureangleδ,initialbasiccircleradiusG0,etc.TheselectionoftheseparametershasagreatinfluenceontheformoftheLogiXtoothprofile,andtheformdirectlyinfluencesgeartransmissionperformance.Thusthereasonableselectionofthesebasicparametersisveryimportant.4.1Influenceandselectionofinitialpressureangleα0Consideringthehighertransmissionefficiencyinpracticaldesign,theinitialpressureangleα0shouldbeselectedas0◦.ButthefinalcalculationresultshowedthattheLogiXgeartoothprofilecutbytheracktoolwhoseinitialpressureanglewasequaltozerowouldbeovercutonthepitchcirclegenerally.Thustheinitialpressureangleα0cannotbezero.Comparingtherelativedoublecirclearcgear3,wecanalsodeducethatthesmaller792theinitialpressureangleα0,thelargerthegearnumberforproducingtheovercut.Thustheinitialpressureangleα0shouldnotonlynotbezero,butshouldnotbetoosmall,either.FromEqs.3,4and5,theinfluenceofα0ontheLogiXtoothprofilecanbedirectlydescribedbyFig.7.Obviously,increasingtheinitialpressureanglewillcausethecurvatureoftheLogiXracktoothprofiletobecomelarger.Iftherackselectsalargermoduleandtoosmallaninitialpressureangleα0,itsaddendumwillbecometoonarroworevenovercut.ThustheLogiXtoothprofilethatselectsalargermoduleshouldselectasmallerα0,andtheprofilethatselectsasmallermoduleshouldselectalargerα0.Generally,bypracticalcalculationexperience,theselectedα0shouldbelocatedwithinarangeof2◦∼12◦,andthelargertheLogiXgearmodule,thesmallershouldbeitsinitialpressureangleα0.4.2InfluenceandselectionofinitialbasiccircleradiusG0AccordingtotheempiricalformulaGiG0{1−sin0.6αi}1,therearetwoparametersaffectingthebasiccircleradiusGioftheLogiXgearatdifferentpositionsoftoothprofileoneistheG0andtheotheristheinitialpressureangleαi.Figure8showstheinfluenceofG0ontheLogiXtoothprofilewhencertainvaluesofparameterα0andδareselected.Obviously,asG0increases,thecurvatureofthenewtypeofgeartoothprofilewillbecomesmallerandsmaller.Conversely,itwillbecomeincreasinglylargerasG0decreases.ThusthenewtypeofrackwithalargemoduleparametershouldselectalargeG0value,andonehavingasmallmoduleparametershouldselectasmallG0value.4.3InfluenceandselectionofrelativepressureangleδFigure9showsthevariableofthetoothprofileaffectedbytheδparameter.AccordingtotheformingprocessoftheLogiXtoothFig.7.Influenceofα0onLogiXtoothprofileFig.8.InfluenceofG0onLogiXtoothprofileFig.9.InfluenceofδonLogiXtoothprofileprofile,thesmallertheselectedparameterδ,thelargerthenumberofNPsmeshingonthetoothprofileoftwoLogiXgears.FromSect.2.1theformuladescribingtherelativepressureangleδkofanarbitraryNPmkcanbededucedasfollowssinαk−1δcosαk−1δ2−cosδcosδksinδ−sinδk.12ByEqs.5and12,thelargertheδparameterbeingselected,thelargerwillbetheδkparameter,andatcertainselectedvaluesoftheinitialpressureangleandmaximumpressureangle,thelowerwillbethenumberofNPs.Bycontrast,thesmallertheδparameter,thelargerthenumberofNPs.Whileδis0.0006◦,thenumberofzeropointscanexceed46,000.Inthiscase,selectingagearmoduleofm100,thelengthofthemicroinvolutecurvebetweentwoadjoiningNPswillbeonlyafewmicrons.Thatistosay,duringthewholemeshingprocessoftheLogiXgeartransmission,theslidingandrollingmotionshappenalternatelyandlastonlyafewmicrosecondsfromonemotiontoanotherbetweentwomeshedgeartoothprofiles.ThegreaterthenumberofNPs,thelongertherelativerollingtimebetweentwoLogiXgearsandtheshortertherelativeslidingtimebetweentwoLogiXgears.Thusabrasionofthegeardecreasesanditsloadingcapabilityandlifespanareimproved.But,consideringtherestrictionofmemorycapability,interpolationspeed,angularresolution,etc.fortheCNCmachinetoolusedwhilecuttingthistypeofgear,therelativepressureangleselectedshouldnotbeverysmall.δgreaterorequalslant0.0006◦isgenerallysatisfactory.Table1.ParametervaluesselectedforLogiXrackatdifferentmodulesmmmα0δG0mm110◦0.05◦600028.◦0.05◦950046◦0.05◦10000550◦0.05◦1100064.◦0.05◦12000832◦0.05◦12024102.8◦0.05◦14000122.6◦0.05◦16500152.5◦0.05◦20024182.4◦0.05◦30036202.4◦0.05◦35000222.3◦0.05◦38000
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