COMMONDEFINITIONFOREND-SURFACECONTACTRATIOOFGEARSANDITSAPPLICATIONSAbstract:Commondefinitionandcalculatingexpressionsofend-surfacecontactratioforalltypeofgearsareputforward,andwithcalculationexpressionsforinvolutegears,micro-segmentsprofilegears,andsine-curvedprofilegearsbeingdiscussed.Theend-surfacecontactratioofgearsisdefinedastheratiooftheactionangle(therotationangleofgearfromgear-intogear-outforonepairofteeth)totherotationangleperpitch(orcentralanglepertooth).Accordingtothetheoryofgearing,equationofthemeshinglinecanbededucedfromthetoothprofilesofbasicrack.Havingobtainedtheequationofthemeshingline,andbeinggiventheaddendumoutlineofthegears,thecontactratiocanbecalculatedwiththecalculationexpressions.Fortheinvolutegears,thisdefinitionhassameeffectasthewell-knowndefinition:ratioofthecontactlinetothebasepitch.Thisdefinitionofcontactratioisalsosuitabletoothernon-involutegears,suchasmicro-segmentsprofilegears,sine-curvedprofilegears,andcangivemorereliableresults.Keywords:GearsContactratioActionangleRotationangleperpitchMicro-segmentsgearsSine-curvedprofilegears0INTRODUCTIONContactratioisoneofthemostimportantitemsforevaluatingthetransmissioncontinuityandloaddistributionofgears-pairs.Itisoftencalculatedduringdesigningapairofgears,andmustbediscussedwhileanewtypeoftooth-profileoranewpitch-curveofgearsisbeingsearched.Gearswithinvolutetooth-profilesaremostwidelyusedinmachineryindustry,andtheirevaluationitemsandcalculations,includingthecontactratio,arewelldesigned.Duetotherelativelylowerloadingcapacityoftheinvolutegears,anumberofgearswithnon-involutetooth-profileshavebeensearched.Thegearswithdouble-circular-arcprofilesorwithpoint-linemeshingprofileshavealreadybeenused,andthosewithmicro-segmentprofilesorsine-curvedprofileshavetheirgreatpotentialvalues.Theend-surfacecontactratioofinvolutegearsisdefinedastheratioofmeshinglinetothebase-pitch.Thisdefinitionisnotsuitableforgearswithnon-involutetooth-profiles,astherearenorealbase-pitchesinthegearswithnon-involuteprofiles.So,itisnecessarytosearchanewcommondefinitionoftheend-surfacecontactratioapplicabletogearswithanytypeoftooth-profiles,andtoworkoutcalculationexpressionsunderthenewdefinition.Infact,thecontactratiocanbeconsideredasanindicationofaverageteeth-pairsinmeshofagear-pair,andnaturallyisoughttobedetSnedaccordingtotherotationangleofagearfromgear-intogear-outofapairofteeth.Underthisconsideration,thecalculationexpressionsfortheend-surfacecontactratioofinvolutegears,micro-segmentgearsandsine-curvedgearsareputforward.Becauseoftakingnoaccountofaxialcontactratio,theend-surfacecontactratioissimplycalledcontactratiohere.1DEFINITIONANDCALCULATION1.1DefinitionIncommonwords,thecontactratioisanaveragenumberofteeth-pairinmeshatanytime.Basedonthisconsideration,itisdefinedastheratiobetweentherotationangleofagearfromgear-intogear-outofonepairofteethtotherotationangleperpitch(rotationanglepertooth)ofthesamegear.Therotationangleofagearduringtheperiodfromgear-intogear-outofonepairofteethiscalledactionangleofthegear,andthekey-pointtocalculatethecontactratioistoobtainthevalueoftheactionangle.Accordingtothetheoryofgearing,whenapairofgearsisinmesh,thecommonnormallineatcontactpoint(meshpoint)ofthetwoconjugatetoothprofilesalwayspassesthroughthepitchpoint.Locusofthemeshpointrelativetothepitchpointiscalledthemeshingline,anditisdeterminedbytooth-profilesofthebasicrack.Alltoothprofilesinapplicationhavesomegeometricalsymmetry,themeshinglineisanoddfunctionpassingthoughthepitchpoint.Intersectingpointofthemeshinglineandtheaddendumoutlineofgearbecomesthegear-inpointorgear-outpoint,whenthemeshpointismovingfromtheintersectingpointtothepitchpoint,orfrompitchpointtotheintersectingpoint,therotationangleofthegearcanbedeterminedbyusingofthemeshingline.Alltoothprofilesdependontheirbasicracks,andsodotheirmeshinglines.1.2CalculationexpressionFig.1showsgearingoftoothprofiles1and2.InthebasicrackcoordinatesystemPxy,parametricequationofthemeshingline3isasfollows=rrxx=yrryFig.1SketchdiagramofgearingRotatingthegearslightlywithmicroangled,makingthemeshingpointmovecorrespondinglywithmicrosegmentdl,then000=cos+=cos+=cos=cos=cosdlvdtrdtrdtrdtrd2=cos/2cosdldldmzz2221=2/cos2cosrrdxdyddlzdzz22112222/11=coscosrrrrdxdydxddyddTakeunitvalueofmodule,i.e.m=l,andsodobelow.Becausethebasicrackisgeometricallysymmetricusually,itsmeshinglinemeetsoddfunctioncondition,makingtheintegrationbeanevenfunction.Consequently212212/1=+cosrrdxddydd12210/1=cosrrdxddydd22220/1=cosrrdxddyddwhere,1andcarecalledsingle-sidecontactratioofgears,andgeneralcontactratioisequaltothesumofthetwosingle-sidecontactratios.Integrationlimits1and2aredeterminedwiththeintersectingpointsoftheaddendumoutlinesofthetwogearsandthemeshinglineofbasicrack,bythefollowingequation222011101+=+rraryxrh222022202+=+rraryxrhBeinggiventhemeshinglineofthebasicrackandtheaddendumoutlineofthegear,thecontactratiocanbecalculatedwiththecalculationexpressionabove.2APPLICATIONTOINVOLUTEGEAR