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附录1GEARSpurandhelicalgears.Agearhavingtoothelementsthatarestraightandparalleltoitsaxisisknownasaspurgear.Aspurpaircanbeusedtoconnectparallelshaftsonly.Parallelshafts,however,canalsobeconnectedbygearsofanothertype,andaspurgearcanbematedwithagearofadifferenttype.Fig.1.1.Topreventjammingasaresultofthermalexpansion,toaidlubrication,andtocompensateforavoidableinaccuraciesinmanufacture,allpowertransmitting,gearsmusthavebacklash.Thismeansthatonthegear,andviceversa.Oninstrumentgears,backlashcaneliminatedbyusingagearsplitdownitsmiddle,onehalfbeingrotatablerelativetotheother.Aspringforcesthesplitgearteethtooccupythefullwidthofthepinionspace.Helicalgearshavecertainadvantagesforexample,whenconnectingparallelshaftstheyhaveahigherloadcarryingthanspurgearswiththesametoothnumbersandcutwiththesamecutter.Becauseoftheoverlappingactionoftheteeth,theyaresmootherinactionandcanoperateathigherpitchlinetotheaxisofrotation,helicalgearscreateanaxialthrust.Ifusedsingly,thisthrustmustbeabsorbedinthesameblank.Dependingonthemethodofmanufacture,thegearmaybeofthecontinuoustoothherringbonevarietyoradoublehelicalgearwithaspacebetweenthetwohalvestopermitthecuttingtooltorunout.Doublehelicalgearsarewellsuitedfortheefficienttransmissionofpowerathighspeeds.Helicalgearscanalsobeusedtoconnectnonparallel,nonintersectingshaftsatanyangletooneanother.Ninetydegreesisthecommonestangleatwhichsuchgearsareused.Wormandbevelgears.Inordertoachievelinecontactandimprovetheloadcarryingcapacityofthecrossedaxishelicalgears,thegearcanbemadetocurvepartiallyaroundthepinion,insomewhatthesamewaythatanutenvelopsascrew.Theresultwouldbeacylindricalwormandgear.Wormgearsprovidethesimplestmeansofobtaininglargerationsinasinglepair.Theyareusuallylessefficientthanparallelshaftgears,however,becauseofanadditionalslidingmovementalongtheteeth.Becauseoftheirsimilarity,theefficiencyofawormandgeardependsonthesamefactorsastheefficiencyofascrew.Singlethreadwormsoflargediameterhavesmallleadanglesandlowefficiencies.MultiplethreadwormshavelargerleadanglesandhigherefficienciesFig.1.2Fortransmittingrotarymotionandtorquearoundcorners,bevelgearsarecommonlyused.Theconnectedshafts,whoseaxeswouldintersectifextended,areusuallybutnotnecessarilyatrightanglestooneanother.Whenadaptedforshaftsthatdonotintersect,spiralbevelgearsarecalledhypoidgears.Thepitchsurfacesofthesegearsarenotrollingcones,andtheratiooftheirmeandiametersisnotequaltothespeedConsequently,thepinionmayhavefewteethandbemadeaslargeasnecessarytocarrytheload.Theprofilesoftheteethonbevelgearsarenotinvolutestheyareofsuchashapethatthetoolsfortheteethareeasiertomakeandmaintainthaninvolutecuttingtools.Sincebevelgearscomein,aslongastheyareconjugatetooneanothertheyneednotbeconjugatetoothergearswithdifferentbothnumbers.1EarlyHistoryofGearingTheearliestwrittendescriptionsofgearsaresaidtohavebeenmadebyAristotleinthefourthcenturyB.C.IthasbeenpointedoutthatthepassageattributedtoAristotlebysomewasactuallyfromthewritingsofhisschool,inMechanicalProblemsofAristotleCa.280B.C.Inthepassageinquestion,therewasnomentionofgearteethontheparallelwheels,andtheymayjustaswellhavebeensmoothwheelsinfrictionalcontact.Therefore,theattributionofgearingtoAristotleis,mostlikely,incorrect.TherealbeginningofgearingwasprobablywithArchimedeswhoabout250B.C.inventedtheendlessscrewturningatoothedwheel,whichwasusedinenginesofwar.Archimedesalsousedgearstosimuearlyformsofwagonmileageindicatorsodometerandsurveyinginstruments.ThesedeviceswereprobablythoughtexperimentsofHeronofAlexandriaca.A.D.60,whowroteonthesubjectsoftheoreticalmechanicsandthebasicelementsofmechanism.TheoldestsurvivingreliccontaininggearsistheAntikytheramechanism,sonamedbecauseoftheGreekislandofthatnamenearwhichthemechanismwasdiscoveredinasunkenshipin1900.ProfessorPriceofYaleUniversityhaswrittenanauthoritativeaccountofthismechanism.Themechanismisnotonlytheearliestrelicofgearing,butitalsoisanextremelycomplexarrangementofepicyclicdifferentialgearing.Themechanismisidentifiedasacalendricalcomputingmechanismforthesunandmoon,andhasbeendatedtoabout87B.C.TheartofgearingwascarriedthroughtheEuropeandarkagesafterthefallofRome,appearinginIslamicinstrumentssuchasthegearedastrolabeswhichwereusedtocalculatethepositionsofthecelestialbodies.PerhapstheartwasrelearnedbytheclockandinstrumentmakingartisansoffourteenthcenturyEurope,orperhapssomecrystallizingideasandmechanismswereimportedfromtheEastafterthecrusadesoftheelevenththroughthethirteenthcenturies.ItappearsthattheEnglishabbotofSt.Albansmonastery,bornRichardofWallingford,inA.D.1330,reinventedtheepicyclicgearingconcept.Heappliedittoanastronomicalclock,whichhebegantobuildatthattimeandwhichwascompletedafterhisdeath.AmechanicalclockofaslightlylaterperiodwasconceivedbyGiovannideDondi13481364.Diagramsofthisclock,whichdidnotusedifferentialgearing,appearinthesketchbooksofLeonardodaVinci,whodesignedgearedmechanismshimself.In1967twoofdaVincismanuscripts,lostintheNationalLibraryinMadridsince1830,wererediscovered.Oneofthemanuscripts,writtenbetween1493and1497andknownasCodexMadridI,contains382pageswithsome1600sketches.IncludedamongthisdisplayofLenardosartisticskillandengineeringabilityarehisstudiesofgearing.Amongthesearetoothprofiledesignsandgearingarrangementsthatwerecenturiesaheadoftheirinvention.2BeginningofModernGearTechnologyIntheperiod1450to1750,themathematicsofgeartoothprofilesandtheoriesofgearedmechanismsbecameestablished.AlbrechtDureriscreditedwithdiscoveringtheepicycloidalshapeca.1525.PhilipdelaHireissaidtohaveworkedouttheanalysisofepicycloidsandrecommendedtheinvolutecurveforgearteethca.1694.LeonardEulerworkedoutthelawofconjugateactionca.1754.Gearsdeignedaccordingtothislawhaveasteadyspeedratio.Sincetheindustrialrevolutioninmidnineteenthcentury,theartofgearingblossomed,andgeardesignssteadilybecamebasedonmorescientificprinciples.In1893WilfredLewispublishedaformulaforcomputingstressingearteeth.Thisformulaisinwideusetodayingeardesign.In1899GeorgeB.Grant,thefounderoffivegearmanufacturingcompanies,publishedATreatiseonGearWheels.Newinventionsledtonewapplicationsforgearing.Forexample,intheearlypartofthiscentury1910,parallelshaftgearswereintroducedtoreducethespeedofthenewlydevelopedreactionsteamturbineenoughtoturnthedrivingscrewsofoceangoingvessels.Thisapplicationachievedanoverallincreaseinefficiencyof25percentinseatravel.Theneedformoreaccurateandquietrunninggearsbecameobviouswiththeadventoftheautomobile.Althoughthehypoidgearwaswithinourmanufacturingcapabilitiesby1916,itwasnotusedpracticallyuntil1926,whenitwasusedinthePackardautomobile.Thehypoidgearmadeitpossibletolowerthedriveshaftandgainmoreusablefloorspace.By1937almostallcarsusedhypoidgearedrearaxles.Speciallubricantantiwearadditiveswereformulatedinthe1920swhichmadeitpracticaltousehypoidgearing.In1931EarleBuchingham,chairmanofanAmericanSocietyofMechanicalEngineersASMEresearchcommitteeongearing,publishedamilestonereportongeartoothdynamicloading.Thisledtoabetterunderstandingofwhyfasterrunninggearssometimescouldnotcarryasmuchloadasslowerrunninggears.Highstrengthalloysteelsforgearingweredevelopedduringthe1920sand1930s.Nitridingandcasehardeningwasintroducedin1950.Extremelycleansteelsproducedbyvacuummeltingprocessesintroducedin1960haveprovedeffectiveinprolonginggearlife.Sincetheearly1960stherehasbeenincreaseduseofindustrialgasturbinesforelectricpowergeneration.Intherangeof1000to14000hp,epicyclicgearsystemshavebeenusedsuccessfully.Pitchlinevelocitiesareform50to100m/s10000to20000ft/min.Thesegearsetsmustworkreliablyfor10000to30000hpbetweenoverhaule.In1976bevelgearsproducedtodriveacompressorteststandranstandransuccessfullyfor235hat2984kwand200m/s.formallindicationsthesegearscouldbeusedinanindustrialapplicationifneeded.Areasonablemaximumpitchlinevelocityforcommercialspiralbevelgearswithcurvedteethis60m/s.Gearsystemdevelopmentmethodshavebeenadvancedinwhichlightweight,