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翻译部分英文原文GEARMECHANISMSGEARMECHANISMSAREUSEDFORTRANSMITTINGMOTIONANDPOWERFROMONESHAFTTOANOTHERBYMEANSOFTHEPOSITIVECONTACTOFSUCCESSIVELYENGAGINGTEETHINABOUT2,600BC,CHINESEAREKNOWNTOHAVEUSEDACHARIOTINCORPORATINGACOMPLEXSERIESOFGEARSLIKETHOSEILLUSTRATEDINFIG27ARISTOTLE,INTHEFOURTHCENTURYBCWROTEOFGEARSASIFTHEYWERECOMMONPLACEINTHEFIFTEENTHCENTURYAD,LEONARDODAVINCIDESIGNEDAMULTITUDEOFDEVICESINCORPORATINGMANYKINDSOFGEARSINCOMPARISONWITHBELTANDCHAINDRIVES,GEARDRIVESAREMORECOMPACT,CANOPERATEATHIGHSPEEDS,ANDCANBEUSEDWHEREPRECISETIMINGISDESIREDTHETRANSMISSIONEFFICIENCYOFGEARSISASHIGHAS98PERCENTONTHEOTHERHAND,GEARSAREUSUALLYMORECOSTLYANDREQUIREMOREATTENTIONTOLUBRICATION,CLEANLINESS,SHAFTALIGNMENT,ETC,ANDUSUALLYOPERATEINACLOSEDCASEWITHPROVISIONFORPROPERLUBRICATIONGEARMECHANISMSCANBEDIVIDEDINTOPLANARGEARMECHANISMSANDSPATIALGEARMECHANISMSPLANARGEARMECHANISMSAREUSEDTOTRANSMITMOTIONANDSPATIALGEARMECHANISMSPLANARGEARMECHANISMSAREUSEDTOTRANSMITMOTIONANDPOWERBETWEENPARALLELSHAFTS,ANDSPATIALGEARMECHANISMSBETWEENNONPARALLELSHAFTSTYPESOFGEARS1SPURGEARSTHESPURGEARHASACYLINDRICALPITCHSURFACEANDHASSTRAIGHTTEETHPARALLELTOITSAXISASSHOWNINFIG28THEYAREUSEDTOTRANSMITMOTIONANDPOWERBETWEENPARALLELSHAFTSTHETOOTHSURFACESOFSPURGEARSCONTACTONASTRAIGHTLINEPARALLELTOTHEAXESOFGEARSTHISIMPLIESTHATTOOTHPROFILESGOINTOANDOUTOFCONTACTALONGTHEWHOLEFACEWIDTHATTHESAMETIMETHISWILLTHEREFORERESULTINTHESUDDENLOADINGANDSUDDENUNLOADINGONTEETHASPROFILESGOINTOANDOUTOFCONTACTASARESULT,VIBRATIONANDNOISEAREPRODUCED2HELICALGEARSTHESEGEARSHAVETHEIRTOOTHELEMENTSATANANGLEORHELIXTOTHEAXISOFTHEGEARFIG29THETOOTHSURFACESOFTWOENGAGINGHELICALGEARSINNPLANARGEARMECHANISMSCONTACTONASTRAIGHTLINEINCLINEDTOTHEAXESOFTHEGEARSTHELENGTHOFTHECONTACTLINECHANGESGRADUALLYFROMZEROTOMAXIMUMANDTHENFROMMAXIMUMTOZEROTHELOADINGANDUNLOADINGOFTHETEETHBECOMEGRADUALANDSMOOTHHELICALGEARSMAYBEUSEDTOTRANSMITMOTIONANDPOWERBETWEENPARALLELSHAFTSFIG29AORSHAFTSATANANGLETOEACHOTHERFIG29DAHERRINGBONEGEARFIG29CISEQUIVALENTTOARIGHTHANDANDALEFTHANDHELICALGEARPLACEDSIDEBYSIDEBECAUSEOFTHEANGLEOFTHETOOTH,HELICALGEARSCREATECONSIDERABLESIDETHRUSTONTHESHAFTAHERRINGBONEGEARCORRECTSTHISTHRUSTBYNEUTRALIZINGIT,ALLOWINGTHEUSEOFASMALLTHRUSTBEARINGINSTEADOFALARGEONEANDPERHAPSELIMINATINGONEALTOGETHEROFTENACENTRALGROOVEISMADEAROUNDTHEGEARFOREASEINMACHINING3BEVELGARSTHETEETHOFABEVELGEARAREDISTRIBUTEDONTHEFRUSTUMOFACONETHECORRESPONDINGPITCHCYLINDERINCYLINDRICALGEARSBECOMESPITCHCONETHEDIMENSIONSOFTEETHONDIFFERENTTRANSVERSEPLANESAREDIFFERENTFORCONVENIENCE,PARAMETERSANDDIMENSIONSATTHELARGEENDARETAKENTOBESTANDARDVALUESBEVELGEARSAREUSEDTOCONNECTSHAFTSWHICHARENOTPARALLELTOEACHOTHERUSUALLYTHESHAFTSARE90DEGTOEACHOTHER,BUTMAYBEMOREORLESSTHAN90DEGTHETWOMATINGGEARSMAYHAVETHESAMENUMBEROFTEETHFORTHEPURPOSEOFCHANGINGDIRECTIONOFMOTIONONLY,ORTHEYMAYHAVEADIFFERENTNUMBEROFTEETHFORTHEPURPOSEOFCHANGINGBOTHSPEEDANDDIRECTIONTHETOOTHELEMENTSMAYBESTRAIGHTORSPIRAL,SOTHATWEHAVEPLAINANDSPIRALBEVELGEARSHYPOIDCOMESFROMTHEWORDHYPERBOLOIDANDINDICATESTHESURFACEONWHICHTHETOOTHFACELIESHYPOIDGEARSARESIMILARTOBEVELGEARS,BUTTHETWOSHAFTSDONOTINTERSECTTHETEETHARECURVED,ANDBECAUSEOFTHENONINTERSECTIONOFTHESHAFTS,BEARINGSCANBEPLACEDONEACHSIDEOFEACHGEARTHEPRINCIPALUSEOFTHIDTYPEOFGEARISINAUTOMOBILEREARENDSFORTHEPURPOSEOFLOWERINGTHEDRIVESHAFT,ANDTHUSTHECARFLOOR4WORMANDWORMGEARSWORMGEARDRIVESAREUSEDTOTRANSMITMOTIONANDOWERBETWEENNONINTERSECTINGANDNONPARALLELSHAFTS,USUALLYCROSSINGATARIGHTANGLE,ESPECIALLYWHEREITISDESIREDTOOBTAINHIGHGEARREDUCTIONINALIMITEDSPACEWORMSAREAKINDOFSCREW,USUALLYRIGHTHANDEDFORCONVENIENCEOFCUTTING,ORLEFTHANDEDITNECESSARYACCORDINGTOTHEENVELOPINGTYPE,WORMSCANBEDIVIDEDINTOSINGLEANDDOUBLEENVELOPINGWORMSAREUSUALLYDRIVERSTOREDUCETHESPEEDIFNOTSELFLOCKING,AWORMGEARCANALSOBETHEDRIVERINASOCALLEDBACKDRIVINGMECHANISMTOINCREASETHESPEEDTWOTHINGSCHARACTERIZEWORMGEARINGALARGEVELOCITYRATIOS,ANDBHIGHSLIDINGVELOCITIESTHELATTERMEANSTHATHEATGENERATIONANDPOWERTRANSMISSIONEFFICIENCYAREOFGREATERCONCERNTHANWITHOTHERTYPESOFGEARS5RACKSARACKISAGEARWITHANINFINITERADIUS,ORAGEARWITHITSPERIMETERSTRETCHEDOUTINTOASTRAIGHTLINEITISUSEDTOCHANGERECIPROCATINGMOTIONTOROTARYMOTIONORVICEVERSAALATHERACKANDPINIONISGOODEXAMPLEOFTHISMECHANISMGEOMETRYOFGEARTOOTHTHEBASICREQUIREMENTOFGEARTOOTHGEOMETRYISTHEPROVISIONOFANGULARVELOCITYRATIONSTHATAREEXACTLYCONSTANTOFCOURSE,MANUFACTURINGINACCURACIESANDTOOTHDEFLECTIONSWELLCAUSESLIGHTDEVIATIONSINVELOCITYRATIO；BUTACCEPTABLETOOTHPROFILESAREBASEDONTHEORETICALCURVESTHATMEETTHISCRITERIONTHEACTIONOFAPAIROFGEARTEETHSATISFYINGTHISREQUIREMENTISTERMEDCONJUGATEGEARTOOTHACTION,ANDISILLUSTRATEDINFIG212THEBASICLAWOFCONJUGATEGEARTOOTHACTIONSTATESTHATASTHEGEARSROTATE,THECOMMONNORMALTOTHESURFACESATTHEPOINTOFCONTACTMUSTALWAYSINTERSECTTHELINEOFCENTERSATTHESAMEPOINTPCALLEDTHEPITCHPOINTTHELAWOFCONJUGATEGEARTOOTHCANBESATISFIEDBYVARIOUSTOOTHSHAPES,BUTTHEONLYONEOFCURRENTIMPORTANCEISTHEINVOLUTE,OR,MOREPRECISELY,THEINVOLUTEOFTHECIRCLEITSLASTIMPORTANTCOMPETITORWASTHECYCLOIDALSHAPE,USEDINTHEGEARSOFMODELTFORDTRANSMISSIONSANINVOLUTEOFTHECIRCLEISTHECURVEGENERATEDBYANYPOINTONATAUTTHREADASITUNWINDSFROMACIRCLE,CALLEDTHEBASECIRCLETHEGENERATIONOFTWOINVOLUTESISSHOWNINFIG213THEDOTTEDLINESSHOWHOWTHESECOULDCORRESPONDTOTHEOUTERPORTIONOFTHERIGHTSIDESOFADJACENTGEARTEETHCORRESPONDINGLY,INVOLUTESGENERATEDBYUNWINDINGATHREADWRAPPEDCOUNTERCLOCKWISEAROUNDTHEBASECIRCLEWOULDFORTHEOUTERPORTIONSOFTHELEFTSIDESOFTHETEETHNOTETHATATEVERYPOINT,THEINVOLUTEISPERPENDICULARTOTHETAUTTHREAD,SINCETHEINVOLUTEISACIRCULARARCWITHEVERINCREASINGRADIUS,ANDARADIUSISALWAYSPERPENDICULARTOITSCIRCULARARCITISIMPORTANTTONOTETHATANINVOLUTECANBEDEVELOPEDASFARASDESIREDOUTSIDETHEBASECIRCLE,BUTANINVOLUTECANNOTEXISTINSIDEITSBASECIRCLELETUSNOWDEVELOPAMATINGPAIROFINVOLUTEGEARTEETHINTHREESTEPSFRICTIONDRIVE,BELTDRIVE,ANDFINALLY,INVOLUTEGEARTOOTHDRIVEFIGURE214SHOWSTWOPITCHCIRCLESIMAGINETHATTHEYREPRESENTTWOCYLINDERSPRESSEDTOGETHERIFSLIPPAGEDOESNOTOCCUR,ROTATIONOFONECYLINDERPITCHCIRCLEWILLCAUSEROTATIONOFTHEOTHERATANANGULARVELOCITYRATIOINVERSELYPROPORTIONALTOTHEIRDIAMETERSINANYPAIROFMATINGGEARS,THESMALLEROFTHETWOISCALLEDTHEPINIONANDTHELARGERONETHEGEARTHETERM“GEAR”ISUSEDINAGENERALSENSETOINDICATEEITHEROFTHEMEMBERS,ANDALSOINASPECIFICSENSETOINDICATETHELARGEROFTHETWOUSINGSUBSCRIPTSPANDGTODENOTEPINIONANDGEAR,RESPECTIVELYINORDERTOTRANSMITMORETORQUETHANISPOSSIBLEWITHFRICTIONDRIVEALONE,WENOWADDABELTDRIVERUNNINGBETWEENPULLEYSREPRESENTINGTHEBASECIRCLES,ASINFIG215IFTHEPINIONISTURNEDCOUNTERCLOCKWISEAFEWDEGREES,THEBELTWILLCAUSETHEGEARTOROTATEINACCORDANCEWITHCORRECTVELOCITYRATIOINGEARPARLANCE,ANGLEΦISCALLEDTHEPRESSUREANGLEFROMSIMILARTRIANGLES,THEBASECIRCLESHAVETHESAMERATIOASTHEPITCH；THUS,THEVELOCITYRATIOPROVIDEDBYTHEFRICTIONANDBELTDRIVESARETHESAMEINFIG216THEBELTISCUTATPOINTC,ANDTHETWOENDSAREUSEDTOGENERATEINVOLUTEPROFILESDEANDFGFORTHEPINIONANDGEAR,RESPECTIVELYITSHOULDNOWBECLEARWHYΦISCALLEDTHEPRESSUREANGLENEGLECTINGSLIDINGFRICTION,THEFORCEOFONEINVOLUTETOOTHPUSHINGAGAINSTTHEOTHERISALWAYSATANANGLEEQUALTOTHEPRESSUREANGLEACOMPARISONOFFIG216ANDFIG212SHOWSTHATTHEINVOLUTEPROFILESDOINDEEDSATISFYTHEFUNDAMENTALLAWOFCONJUGATEGEARTOOTHACTIONINCIDENTALLY,THEINVOLUTEISTHEONLYGEOMETRICPROFILESATISFYINGTHISLAWTHATMAINTAINSACONSTANTPRESSUREANGLEASTHEGEARSROTATENOTEESPECIALLYTHATCONJUGATEINVOLUTEACTIONCANTAKEPLACEONLYOUTSIDEOFBOTHBASECIRCLESNOMENCLATUREOFSPURGEARTHENOMENCLATUREOFSPURGEARFIG217ISMOSTLYAPPLICABLETOALLOTHERTYPEOFGEARSTHEDIAMETEROFEACHOFTHEORIGINALROLLINGCYLINDERSOFTWOMATINGGEARSISCALLEDTHEPITCHDIAMETER,ANDTHECYLINDER’SSECTIONALOUTLINEISCALLEDTHEPITCHCIRCLETHEPITCHCIRCLESARETANGENTTOEACHOTHERATPITCHPOINTTHECIRCLEFROMWHICHTHEINVOLUTEISGENERATEDISCALLEDTHEBASECIRCLETHECIRCLEWHERETHETOPSOFTHETEETHLIEISCALLEDTHEDEDENDUMCIRCLESIMILARLY,THECIRCLEWHERETHEROOTSOFTHETEETHLIEISCALLEDTHEDEDENDUMCIRCLEBETWEENTHEADDENDUMCIRCLEANDTHEDEDENDUMCIRCLE,THEREISANIMPORTANTCIRCLEWHICHISCALLEDTHEREFERENCECIRCLEPARAMETERSONTHEREFERENCECIRCLEARESTANDARDIZEDTHEMODULEMOFAGEARISINTRODUCEDONTHEREFERENCECIRCLEASABASICPARAMETER,WHICHISDEFINEDASMP/ΠSIZESOFTHETEETHANDGEARAREPROPORTIONALTOTHEMODULEMTHEADDENDUMISTHERADIALDISTANCEFROMTHEREFERENCECIRCLETOTHEADDENDUMCIRCLETHEDEDENDUMISTHERADIALDISTANCEFROMTHEREFERENCECIRCLETOTHEDEDENDUMCIRCLECLEARANCEISTHEDIFFERENCEBETWEENADDENDUMANDDEDENDUMINMATINGGEARSCLEARANCEPREVENTSBINDINGCAUSEDBYANYPOSSIBLEECCENTRICITYTHECIRCULARPITCHPISTHEDISTANCEBETWEENCORRESPONDINGSIDEOFNEIGHBORINGTEETH,MEASUREDALONGTHEREFERENCECIRCLETHEBASEPITCHISSIMILARTOTHECIRCULARPITCHISMEASUREDALONGTHEBASECIRCLEINSTEADOFALONGTHEREFERENCECIRCLEITCANEASILYBESEENTHATTHEBASERADIUSEQUALSTHEREFERENCERADIUSTIMESTHECOSINEOFTHEPRESSUREANGLESINCE,FORAGIVENANGLE,THERATIOBETWEENANYSUBTENDEDARCANDITSRADIUSISCONSTANT,ITISALSOTRUETHATTHEBASEPITCHEQUALSTHECIRCULARPITCHTIMESTHECOSINEOFTHEPRESSUREANGLETHEPRESSUREANGLEISTHEANGLEBETWEENTHENORMALANDTHECIRCUMFERENTIALVELOCITYOFTHEPOINTONASPECIFICCIRCLETHEPRESSUREANGLEONTHEREFERENCECIRCLEISALSOSTANDARDIZEDITISMOSTCOMMONLY20SOMETIMES15THELINEOFCENTERSISALINEPASSINGTHROUGHTHECENTERSOFTWOMATINGGEARSTHECENTERDISTANCEMEASUREDALONGTHELINEOFCENTERSEQUALSTHESUMOFTHEPITCHRADIIOFPINIONANDGEARTOOTHTHICKNESSISTHEWIDTHOFTHETOOTH,MEASUREDALONGTHEREFERENCECIRCLE,ISALSOREFERREDTOASTOOTHTHICKNESSWIDTHOFSPACEISTHEDISTANCEBETWEENFACINGSIDEOFADJACENTTEETH,MEASUREDALONGTHEREFERENCECIRCLETOOTHTHICKNESSPLUSWIDTHOFSPACEEQUALSTHECIRCULARPITCHBACKLASHISTHEWIDTHOFSPACEMINUSTHETOOTHTHICKNESSFACEWIDTHMEASURESTOOTHWIDTHINANAXIALDIRECTIONTHEFACEOFTHETOOTHISTHEACTIVESURFACEOFTHETOOTHOUTSIDETHEPITCHCYLINDERTHEFLANKOFTHETOOTHISTHEACTIVESURFACEINSIDETHEPITCHCYLINDERTHEFILLETISTHEROUNDEDCORNERATTHEBASEOFTHETOOTHTHEWORKINGDEPTHISTHESUMOFTHEADDENDUMOFAGEARANDTHEADDENDUMOFITSMATINGGEARINORDERTOMATEPROPERLY,GEARSRUNNINGTOGETHERMUSTHAVEATHESAMEMODULE；BTHESAMEPRESSUREANGLE；CTHESAMEADDENDUMANDDEDENDUMTHELASTREQUIREMENTISVALIDFORSTANDARDGEARSONLYGEARTOOTHSTRENGTHHAVINGDEALTWITHGEARGEOMETRY,WENOWTURNTOTHEQUESTIONOFHOWMUCHPOWERORTORQUEAPAIROFGEARSWILLTRANSMITWITHOUTTOOTHFAILUREFIGURE218SHOWSAPHOTOELASTICPATTERNOFGEARTOOTHSTRESSESTHEDETAILSOFTHISEXPERIMENTALSTRESSANALYSISPROCEDUREAREBEYONDTHESCOPEOFTHISBOOK,ANDITISS
