AGMA-90FTM1-1990.pdf
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1、90 FTM 1 ContactStresses in Gear Teeth by: JohnR. Colbourne,Universityof Alberta AmericanGear ManufacturersAssociation IIIll!Illl I TECHNICALPAPER Copyright American Gear Manufacturers Association Provided by IHS under license with AGMA Licensee=IHS Employees/1111111001, User=Wing, Bernie Not for Re
2、sale, 04/18/2007 11:38:21 MDTNo reproduction or networking permitted without license from IHS -,-,- ContactStressesin GearTeeth JohnR. Colbourne, Universityof Alberta The Statementsand opinionscontainedhereinare thoseof the authorand shouldnot be construedas an officialactionor opinionof the America
3、nGearManufacturersAssociation. ABSTRACT: It is shownthatneitherHertzsline contacttheorynor his pointcontacttheoryare entirelyadequatefor the accuratecalculationof contactstressesin gear teeth.A numericalprocedureis described,whichcan beusedto findthecontactstressin caseswheretherelativecurvaturesin
4、thecontactregionare not constant. Copyright 1990 AmericanGearManufacturersAssociation 1500 King Street,Suite201 Alexandria,Virginia,22314 October,1990 ISBN:1-55589-553-0 Copyright American Gear Manufacturers Association Provided by IHS under license with AGMA Licensee=IHS Employees/1111111001, User=
5、Wing, Bernie Not for Resale, 04/18/2007 11:38:21 MDTNo reproduction or networking permitted without license from IHS -,-,- CONTACTSTRESSESIN GEARTEETH J.R.Colbourne, DepartmentofMechanicalEngineering, UniversityofAlberta, Edmonton,AlbertaT6G2G8,Canada. Introductionactualcontactareasencounteredingear
6、ing oftendifferfromtheellipticalshapes Withtheuseofmismatchinwormandpredictedbyconventionalpointcontact bevelgears,andcrowninginhelicalgears,theory. thereisnolongerlinecontactbetweenthe teeth,anditisevidentthatthecontactInthispaper,wewilluseexamples stresscannotbefoundaccuratelybyHertzsfoundinwormge
7、aring,butthemethod linecontacttheory.Hispointcontactdescribedforcalculatingthecontactstress theoryisalsosometimesunsuitable,becauseisequallyapplicabletobevelgears.Itmay theassumptionofconstantcurvaturewithinalsobeusefulforhelicalgearswith thecontactareaisnotrealistic,andthecrowning,wheretherealstres
8、sdistribution isprobablysimilartolinecontactnearthe 00050middleoftheteeth,andtopointcontact rneartheends. 0000 _TheNeedForaNewContactStressTheory Atraditionallinecontactdiagramfor iawormandwormgearisshowninFigureI. Thisisthecontactdiagramweobtainwhen thereisnomismatch,sothatthegearis I-|oo0o_exactly
9、conjugatewiththeworm.Thethree IIlinesshowninthediagramrepresentthe ooo5ocontactlinesonthreeadjacentteeth.To FigureI.obtainthetoothforcesandthe GearpairNumberI.ContactLines.correspondingcontactstresses,weassume Copyright American Gear Manufacturers Association Provided by IHS under license with AGMA
10、Licensee=IHS Employees/1111111001, User=Wing, Bernie Not for Resale, 04/18/2007 11:38:21 MDTNo reproduction or networking permitted without license from IHS -,-,- thattheloadintensityisconstantalongthegearpairof FigureI, whenthereis some eachcontactline.Fromthespecifiedmismatchbetweenthe wormandtheg
11、ear.The torque,wecalculatethetotaltoothforce,wormisthread-milledwithfivethreads,and Wethendividethetotaltoothforcebythehasaleadangleof21.31degrees.The totallengthof the contactlines,to obtainmismatchisobtainedbymodifyingthe theloadintensity,andtofindtheprofileofthehobthatisusedtocutthe individualtoo
12、thforceswe multiplytheloadgear,bymakingitsleadslightlyshorter intensitybythelengthsoftheindividualthanthatofthe worm,andbyswivellingthe contactlines.Thecontactstressateachdirectionofitsaxiswhenthegeariscut, pointisfoundbyHertzs.linecontactsothatitisnotexactlyperpendicularto theory,usingtherelativecu
13、rvatureinthethegearaxis.Thecontactdiagramshowsthe directionperpendicularto thecontactline.contoursofconstantseparationbetweenthe wormthread,andtheenvelopeofthegear Theassumptionofconstantloadtoothpositionsrelativetotheworm.In intensityisquestionable,evenwhentheotherwords,itshowshowcloselythegear wor
14、mandthegearareexactlyconjugate,andteethapproachthewormthreadduringthe thecontactlinesextendtotheedgesoftheentiremeshingcycle,assumingthatbothare teeth.Iftheloadintensityisconstantrigid andthattheangularvelocityratiois alongeachcontactline,itmustbeexactlyconstant.Thetwocurvesinthe independentof there
15、lativecurvature,whichdiagramshowthecontourswherethe doesvaryalongthecontactlines,andthisseparationis0.0005inchesand0.0010 independenceseemsunlikely.However,in theinches.Ifwehadincludedthecurvewhere casewherethereis somemismatchthecontacttheseparationis0.00025inches,thiswould areasdonotreachtheedgeso
16、ftheteeth,correspondapproximatelytotheareathat anditisquitecertainthattheloadwouldbe colouredby the markingcompoundin intensityisnotuniform.Insteadoflinea bearingtest. contactwehavepointcontact,thoughthe contactareasmaybeveryelongated,andwewenowconsiderthegearpairinthe thereforehaveto considertheuse
17、of HertzssamepositionasinFigureI,andwe pointcontacttheory,calculatethemagnitudeoftheseparation betweentheteethin eachofthethreetooth Figure2showsthecontactdiagramforpairs.Figure3 showsthecontourswherethe 0separationbetweentheteethis0.0010 I iiii_inches.Itisobviousthatwehavepoint _-_0contactinsteadof
18、linecontact,whichis whatwewouldexpectwhenweintroducethe mismatch.Italsoappearsthatanystress .“_“-“-_.calculationbasedonlinecontactwillbe 4_0000s,.extremelyinaccurate,particularlyinthe _%_j_/caseoftheuppertoothpair,wherethe possiblecontactareaisnowmuchshorter thanthelinein FigureI. encounteredwhenwet
19、ryto useHertzspoint 0 theory.First,thetheorywasdevelopedto Figure2.calculatethecontactstresswhenthe GearPairNumberI. ContactDiagram.contactforceisknown.Inourcase,weknow Copyright American Gear Manufacturers Association Provided by IHS under license with AGMA Licensee=IHS Employees/1111111001, User=W
20、ing, Bernie Not for Resale, 04/18/2007 11:38:21 MDTNo reproduction or networking permitted without license from IHS -,-,- vO 0050 C-ooooow i_i iEquivalentSpringSystem. q threespringshaveinitialgaps,andthe _-_00000_relationbetweenforceand compression for _ooo5oeachspringis non-linear,asin allcontact
21、Figure 3.problems. GearPairNumberI.SeparationContours. Theseconddiffultywefindinusing HertzspointcontacttheoryisthatHertz thetotalcontactforce,butwedonotknowdealtonlywithcontactingbodieswhichhave howitissharedbetweenthetoothpairs,constantcurvaturesinthecontactregion. Thisproblemismademoredifficultby
22、theThisconditionimpliesthatthebody factthatthereisinitiallynocontactatsurfacesneartheinitialpointofcontact theupperandlowertoothpairs,andthatcanberepresentedbyellipsoids,andthat thiscontactonlyoccurswhenthereissomecontoursofconstantseparationarethen flexibityinthemiddletoothpair.Theconcentricellipse
23、s.Thecontoursin minimumseparationsinthedirectionoftheFigure3arealmostelliptical,soit appears wormaxisbetweenthethreetoothpairswerethatHertzsconditionmaybesatisfiedfor foundtobe0.00033,0.00003and0.00083thisgearpair.However,Figure5showstwo inches.Thesewerecalculatedassumingtheseparationcontoursfortheu
24、ppertooth teetharerigid,andthattheangularpair,theinneroneat0.0010inchesandthe velocityratioisconstant.Tounderstandouteroneat0.0020inches.Underlightload theeffectoftheseseparationswewillthecontactwilllooklikeHertzianpoint considerthegearasfixed.First,thewormcontact,withintheellipticalcontour,but canm
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