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DOI101007/S0017000317418ORIGINALARTICLEINTJADVMANUFTECHNOL200424789–793FENGXIANYINGWANGAIQUNLINDALEESTUDYONTHEDESIGNPRINCIPLEOFTHELOGIXGEARTOOTHPROFILEANDTHESELECTIONOFITSINHERENTBASICPARAMETERSRECEIVED2JANUARY2003/ACCEPTED3MARCH2003/PUBLISHEDONLINE3NOVEMBER2004SPRINGERVERLAGLONDONLIMITED2004ABSTRACTTHEDEVELOPMENTOFSCIENTIFICTECHNOLOGYANDPRODUCTIVITYHASCALLEDFORINCREASINGLYHIGHERREQUIREMENTSOFGEARTRANSMISSIONPERFORMANCETHEKEYFACTORINFLUENCINGDYNAMICGEARPERFORMANCEISTHEFORMOFTHEMESHEDGEARTOOTHPROFILETOIMPROVEAGEAR’STRANSMISSIONPERFORMANCE,ANEWTYPEOFGEARCALLEDTHELOGIXGEARWASDEVELOPEDINTHEEARLY1990SHOWEVER,FORTHISSPECIALKINDOFGEARTHEREREMAINMANYUNKNOWNTHEORETICALANDPRACTICALPROBLEMSTOBESOLVEDINTHISPAPER,THEDESIGNPRINCIPLEOFTHISNEWTYPEOFGEARISFURTHERSTUDIEDANDTHEMATHEMATICALMODULEOFITSTOOTHPROFILEDEDUCEDTHEINFLUENCEONTHEFORMOFTHISTYPEOFTOOTHPROFILEANDITSMESHPERFORMANCEBYITSINHERENTBASICPARAMETERSISDISCUSSED,ANDREASONABLESELECTIONSFORLOGIXGEARPARAMETERSAREPROVIDEDTHUSTHETHEORETICALSYSTEMINFORMATIONABOUTTHELOGIXGEARAREDEVELOPEDANDENRICHEDTHISSTUDYIMPACTSMOSTSIGNIFICANTLYTHEIMPROVEMENTOFLOADCAPACITY,MINIATURISATIONANDDURABILITYOFMODERNKINETICTRANSMISSIONPRODUCTSKEYWORDSBASICPARAMETERDESIGNPRINCIPLELOGIXGEARMINUTEINVOLUTETOOTHPROFILE1INTRODUCTIONINORDERTOIMPROVEGEARTRANSMISSIONPERFORMANCEANDSATISFYSOMESPECIALREQUIREMENTS,ANEWTYPEOFGEAR1WASPUTFORWARD;ITWASNAMED“LOGIX”INORDERTOIMPROVESOMEDEMERITSOFWNWILDHAVERNOVIKOVANDINVOLUTEGEARSBESIDESHAVINGTHEADVANTAGESOFBOTHKINDSOFGEARSMENTIONEDABOVE,THENEWTYPEOFGEARHASSOMEOTHEREXCELLENTFXIANYINGA117WAIQUNSCHOOLOFMECHANICALENGINEERING,SHANDONGUNIVERSITY,PRCHINAEMAILFXYINGSDUEDUCNTEL8653183958520LLEESCHOOLOFMECHANICALMANUFACTURINGENGINEERING,SINGAPOREPOLYTECHNIC,SINGAPORECHARACTERISTICSONTHISNEWTOOTHPROFILE,THECONTINUOUSCONCAVE/CONVEXCONTACTISCARRIEDOUTFROMITSDEDENDUMTOITSADDENDUM,WHERETHEENGAGEMENTSWITHARELATIVECURVATUREOFZEROAREASSUREDATMANYPOINTSHERE,THISKINDOFPOINTISCALLEDTHENULLPOINTNPTHEPRESENCEOFMANYNPSDURINGTHEMESHPROCESSOFLOGIXGEARSCANRESULTINASMALLERSLIDINGCOEFFICIENT,ANDTHEMESHTRANSMISSIONPERFORMANCEBECOMESALMOSTROLLINGFRICTIONACCORDINGLYTHUSTHISNEWTYPEOFGEARHASMANYADVANTAGESSUCHASHIGHERCONTACTINTENSITY,LONGERLIFEANDALARGERTRANSMISSIONRATIOPOWERTRANSFERTHANTHESTANDARDINVOLUTEGEAREXPERIMENTALRESULTSSHOWEDTHAT,GIVENACERTAINNUMBEROFNPSBETWEENTWOMESHEDLOGIXGEARS,THECONTACTFATIGUESTRENGTHIS3TIMESANDTHEBENDFATIGUESTRENGTH25TIMESLARGERTHANTHOSEOFTHESTANDARDINVOLUTEGEARMOREOVER,THEMINIMUMTOOTHNUMBERCANALSOBEDECREASEDTO3,MUCHSMALLERTHANTHATOFTHESTANDARDINVOLUTEGEARTHELOGIXGEAR,REGARDEDASANEWTYPEOFGEAR,STILLPRESENTSSOMEUNSOLVEDPROBLEMSTHEDEVELOPMENTOFCOMPUTERNUMERICALCONTROLLINGCNCTECHNOLOGYMUSTALSOBETAKENINTOCONSIDERATIONNEWHIGHEFFICIENCYMETHODSTOCUTTHISNEWTYPEOFGEARTHEREFORE,FURTHERSTUDYOFTHISNEWTYPEOFGEARMOSTSIGNIFICANTLYIMPACTSTHEACCELERATIONOFITSBROADANDPRACTICALAPPLICATIONTHISPAPERHASTHEPOTENTIALTOUSHERINANEWERAINTHEHISTORYOFGEARMESHTHEORYANDAPPLICATION2DESIGNPRINCIPLEOFLOGIXTOOTHPROFILEACCORDINGTOGEARMESHANDMANUFACTURINGTHEORIES,INORDERTOSIMPLIFYPROBLEMANALYSIS,GENERALLYAGEAR’SBASICRACKISBEGUNWITHSOMESTUDIES2SOHERELETUSDISCUSSTHEBASICRACKOFTHELOGIXGEARFIRSTFIGURE1SHOWSTHEDESIGNPRINCIPLEOFDIVIDEDINVOLUTECURVESOFTHELOGIXRACKINFIG1,PLREPRESENTSAPITCHLINEOFTHELOGIXRACKONEPOINTO1ISSELECTEDTOFORMTHEANGLEN0O1N1Α0,PLO1N1THEPOINTSOFINTERSECTIONBYTWORADIALSO1N0ANDO1N1ANDTHEPITCHLINEPLAREN1ANDN0LETO1N0G1,EXTENDO1N0TOOPRIME1,ANDMAKETWOTANGENTBASICCIRCLESWHOSECENTRESAREO1,OPRIME1ANDRADIIAREEQUALTOG1THEPOINTOFINTERSECTIONBETWEENCIRCLEO1ANDPITCHLINE790FIG1DESIGNPRINCIPLEOFLOGIXRACKTOOTHPROFILEPLISN0THEPOINTOFINTERSECTIONBETWEENCIRCLEO2ANDPITCHLINEPLISN1MAKETHECOMMONTANGENTG1S1OFBASICCIRCLEO1ANDOPRIME1,THENGENERATETWOMINUTEINVOLUTECURVESM0S1ANDS1M1WHOSEBASICCIRCLECENTRESAREO1ANDOPRIME1THERADIIOFCURVATUREATPOINTSM0ANDM1ONTHETOOTHPROFILESHOULDBEΡM0M0N0,ΡM1M1N1,ANDTHECENTRESAREMETONTHEPITCHLINEMULTIPLEDIFFERENTMINUTEINVOLUTESCONSISTINGOFALOGIXPROFILESHOULDBEARRANGEDFORAPROPERSEQUENCETHEPRESSUREANGLEOFTHENEXTMINUTEINVOLUTECURVEM1M2SHOULDHAVEANINCREMENTCOMPARABLETOITSLASTSEGMENTM0M1THECENTRESOFCURVATUREATEXTREMEPOINTSM1,M2,ETCSHOULDBEONTHEPITCHLINE,ANDTHERADIUSOFTHEBASICCIRCLEISAFUNCTIONOFPRESSURE1–ITVARIESFROMG1TOG2THECONDITIONFORJOININGFRONTANDREARCURVESISTHATTHERADIUSOFCURVATUREATPOINTM1MUSTBEEQUALTOTHERADIUSOFCURVATUREJUSTAFTERPOINTM1,ANDTHERADIUSOFCURVATUREATPOINTM2MUSTBEEQUALTOTHERADIUSOFCURVATUREJUSTAFTERPOINTM2FIGURE2SHOWSTHECONNECTIONANDPROCESSOFGENERATINGMINUTEINVOLUTECURVESACCORDINGTOTHEABOVEDISCUSSION,THEWHOLETOOTHPROFILECANBEFORMEDFIG2CONNECTIONOFMINUTEINVOLUTECURVES3MATHEMATICMODULEOFLOGIXTOOTHPROFILE31MATHEMATICMODULEOFTHEBASICLOGIXRACKACCORDINGTOTHEABOVEMENTIONEDDESIGNPRINCIPLE,THECURVATURECENTREOFEVERYFINELYDIVIDEDPROFILECURVESHOULDBELOCATEDATTHERACKPITCHLINE,ANDTHEVALUEOFTHERELATIVECURVATUREATEVERYPOINTCONNECTINGDIFFERENTMINUTEINVOLUTECURVESSHOULDBEZEROTHEDESIGNOFTHETOOTHPROFILEISSYMMETRICALWITHRESPECTTOTHEPITCHLINE,ANDTHEADDENDUMISCONVEXWHILETHEDEDENDUMISCONCAVETHUSFORTHEWHOLELOGIXTOOTHPROFILE,ITCANBEDEALTWITHBYDIVIDINGITINTOFOURPARTS,ASSHOWNINFIG3SETUPTHECOORDINATESASSHOWNINFIG4,WHERETHEORIGINOFTHECOORDINATESOCOINCIDESWITHTHEPOINTOFINTERSECTIONM0BETWEENRACKPITCHLINEPLANDTHEINITIALDIVIDEDMINUTEINVOLUTECURVEACCORDINGTOTHECOORDINATESSETUPINFIG4,THEFORMATIONOFINITIALMINUTEINVOLUTECURVEM0M1ISSHOWNINFIG5FIG3LOGIXRACKTOOTHPROFILEFIG4SETUPOFCOORDINATESFIG5FORMATIONPROCESSOFINITIALMINUTEINVOLUTECURVEM0M1791HEREN0NPRIME0O1OPRIME1,N1NPRIME1O1OPRIME1,N1N1N0NPRIME0,ANDTHEPARAMETERSΑ0,Δ,G1ANDΡM0AREGIVENASINITIALCONDITIONSTHECURVATURERADIUSOFTHEINVOLUTECURVEATPOINTS1ISΡS1G1Δ,ORΡS1ΡM1G1Δ1THUSTHECURVATURERADIUSANDPRESSUREANGLEOFTHEMINUTEINVOLUTECURVEATPOINTM1AREASFOLLOWSΡM1ΡS1−G1Δ1G1Δ−Δ11Α1Α0ΔΔ12ACCORDINGTOTHEGEOMETRICALRELATIONSHIP,WECANDEDUCETGΑ0Δ2G1−G1COSΔ−G1COSΔ1G1SINΔ−G1SINΔ12−COSΔCOSΔ1SINΔ−SINΔ13BASEDONEQS1,2AND3ANDTHEFORMINGPROCESSOFTHELOGIXRACKPROFILE,THECURVATURERADIUSFORMULAOFANARBITRARYPOINTONTHEPROFILEISDEDUCEDΡMIΡMI−1GIΔ−ΔIWHENIKANDΡM00,ITISEXPRESSEDASFOLLOWSΡMKG1Δ−Δ1G2Δ−Δ2GKΔ−ΔKKSUMMATIONDISPLAYI1GIΔ−ΔI4SIMILARLY,THEPRESSUREANGLEONANARBITRARYKPOINTOFTHETOOTHPROFILECANBEDEDUCEDASFOLLOWSΑKΑ0ΔΔ1ΔΔ2ΔΔKΑ0KSUMMATIONDISPLAYI1ΔΔIΑ0KΔKSUMMATIONDISPLAYI1ΔI5BYNI−1NIGISINΔ−SINΔI/COSΑI−1Δ,EQ5CANBEOBTAINEDN0NKKSUMMATIONDISPLAYI1NI−1NIKSUMMATIONDISPLAYI1GISINΔ−SINΔICOSΑI−1Δ6THUSTHEMATHEMATICALMODELOFTHENO2PORTIONFORTHELOGIXRACKPROFILEISASFOLLOWSBRACELEFTBIGGX1N0NK−ΡMKCOSΑKY1ΡMKSINΑKNO27SIMILARLY,THEMATHEMATICALMODELSOFTHEOTHERTHREESEGMENTSCANALSOBEOBTAINEDASFOLLOWSBRACELEFTBIGGX1−N0NK−ΡMKCOSΑKY1−ΡMKSINΑKNO18BRACELEFTBIGGX1S−N0NK−ΡMKCOSΑKY1ΡMKSINΑKNO39BRACELEFTBIGGX1SN0NK−ΡMKCOSΑKY1−ΡMKSINΑKNO410FIG6MESHCOORDINATESOFLOGIXGEARANDITSBASICRACK32MATHEMATICALMODULEOFTHELOGIXGEARTHECOORDINATESO1X1Y1,O2X2Y2ANDPXYARESETUPASSHOWNINFIG6TOEXPRESSTHEMESHRELATIONSHIPBETWEENTHELOGIXRACKANDTHELOGIXGEARHERE,O1X1Y1ISFIXEDONTHERACK,ANDO1ISTHEPOINTOFINTERSECTIONBETWEENTHERACKTOOTHPROFILEANDITSPITCHLINEO2X2Y2ISFIXEDONTHEMESHEDGEAR,ANDO2ISTHEGEAR’SCENTREPXYISANABSOLUTECOORDINATE,ANDPISTHEPOINTOFINTERSECTIONOFTHERACK’SPITCHLINEANDTHEGEAR’SPITCHCIRCLEINACCORDANCEWITHGEARMESHINGTHEORIES3,IFTHEABOVEMODELOFTHELOGIXRACKTOOTHPROFILEISCHANGEDFROMCOORDINATEO1X1Y1TOOXY,ANDTHENAGAINTOO2X2Y2,ANEWTYPEOFGEARPROFILEMODELCANBEDEDUCEDASFOLLOWSBRACELEFTBIGGX2−ΡMKCOSΑKCOSΦ2−ΡMKSINΑK−R2SINΦ2Y2−ΡMKCOSΑKSINΦ2ΡMKSINΑK−R2COSΦ211HERETHEPOSITIVEDIRECTIONOFΦ2ISCLOCKWISE,ANDONLYTHEMODELOFTHELOGIXGEARTOOTHPROFILEINTHEFIRSTQUADRANTOFTHECOORDINATESISGIVEN4EFFECTONTHEPERFORMANCEOFTHELOGIXGEARBYITSINHERENTPARAMETERSANDTHEIRREASONABLESELECTIONBESIDESTHEBASICPARAMETERSOFTHESTANDARDINVOLUTERACK,THELOGIXTOOTHPROFILEHASINHERENTBASICPARAMETERSSUCHASINITIALPRESSUREANGLEΑ0,RELATIVEPRESSUREANGLEΔ,INITIALBASICCIRCLERADIUSG0,ETCTHESELECTIONOFTHESEPARAMETERSHASAGREATINFLUENCEONTHEFORMOFTHELOGIXTOOTHPROFILE,ANDTHEFORMDIRECTLYINFLUENCESGEARTRANSMISSIONPERFORMANCETHUSTHEREASONABLESELECTIONOFTHESEBASICPARAMETERSISVERYIMPORTANT41INFLUENCEANDSELECTIONOFINITIALPRESSUREANGLEΑ0CONSIDERINGTHEHIGHERTRANSMISSIONEFFICIENCYINPRACTICALDESIGN,THEINITIALPRESSUREANGLEΑ0SHOULDBESELECTEDAS0◦BUTTHEFINALCALCULATIONRESULTSHOWEDTHATTHELOGIXGEARTOOTHPROFILECUTBYTHERACKTOOLWHOSEINITIALPRESSUREANGLEWASEQUALTOZEROWOULDBEOVERCUTONTHEPITCHCIRCLEGENERALLYTHUSTHEINITIALPRESSUREANGLEΑ0CANNOTBEZEROCOMPARINGTHERELATIVEDOUBLECIRCLEARCGEAR3,WECANALSODEDUCETHATTHESMALLER792THEINITIALPRESSUREANGLEΑ0,THELARGERTHEGEARNUMBERFORPRODUCINGTHEOVERCUTTHUSTHEINITIALPRESSUREANGLEΑ0SHOULDNOTONLYNOTBEZERO,BUTSHOULDNOTBETOOSMALL,EITHERFROMEQS3,4AND5,THEINFLUENCEOFΑ0ONTHELOGIXTOOTHPROFILECANBEDIRECTLYDESCRIBEDBYFIG7OBVIOUSLY,INCREASINGTHEINITIALPRESSUREANGLEWILLCAUSETHECURVATUREOFTHELOGIXRACKTOOTHPROFILETOBECOMELARGERIFTHERACKSELECTSALARGERMODULEANDTOOSMALLANINITIALPRESSUREANGLEΑ0,ITSADDENDUMWILLBECOMETOONARROWOREVENOVERCUTTHUSTHELOGIXTOOTHPROFILETHATSELECTSALARGERMODULESHOULDSELECTASMALLERΑ0,ANDTHEPROFILETHATSELECTSASMALLERMODULESHOULDSELECTALARGERΑ0GENERALLY,BYPRACTICALCALCULATIONEXPERIENCE,THESELECTEDΑ0SHOULDBELOCATEDWITHINARANGEOF2◦∼12◦,ANDTHELARGERTHELOGIXGEARMODULE,THESMALLERSHOULDBEITSINITIALPRESSUREANGLEΑ042INFLUENCEANDSELECTIONOFINITIALBASICCIRCLERADIUSG0ACCORDINGTOTHEEMPIRICALFORMULAGIG0{1−SIN06ΑI}1,THEREARETWOPARAMETERSAFFECTINGTHEBASICCIRCLERADIUSGIOFTHELOGIXGEARATDIFFERENTPOSITIONSOFTOOTHPROFILEONEISTHEG0ANDTHEOTHERISTHEINITIALPRESSUREANGLEΑIFIGURE8SHOWSTHEINFLUENCEOFG0ONTHELOGIXTOOTHPROFILEWHENCERTAINVALUESOFPARAMETERΑ0ANDΔARESELECTEDOBVIOUSLY,ASG0INCREASES,THECURVATUREOFTHENEWTYPEOFGEARTOOTHPROFILEWILLBECOMESMALLERANDSMALLERCONVERSELY,ITWILLBECOMEINCREASINGLYLARGERASG0DECREASESTHUSTHENEWTYPEOFRACKWITHALARGEMODULEPARAMETERSHOULDSELECTALARGEG0VALUE,ANDONEHAVINGASMALLMODULEPARAMETERSHOULDSELECTASMALLG0VALUE43INFLUENCEANDSELECTIONOFRELATIVEPRESSUREANGLEΔFIGURE9SHOWSTHEVARIABLEOFTHETOOTHPROFILEAFFECTEDBYTHEΔPARAMETERACCORDINGTOTHEFORMINGPROCESSOFTHELOGIXTOOTHFIG7INFLUENCEOFΑ0ONLOGIXTOOTHPROFILEFIG8INFLUENCEOFG0ONLOGIXTOOTHPROFILEFIG9INFLUENCEOFΔONLOGIXTOOTHPROFILEPROFILE,THESMALLERTHESELECTEDPARAMET
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