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附录1GEARSpurandhelicalgears.Agearhavingtoothelementsthatarestraightandparalleltoitsaxisisknownasaspurgear.Aspurpaircanbeusedtoconnectparallelshaftsonly.Parallelshafts,however,canalsobeconnectedbygearsofanothertype,andaspurgearcanbematedwithagearofadifferenttype.(Fig.1.1).Topreventjammingasaresultofthermalexpansion,toaidlubrication,andtocompensateforavoidableinaccuraciesinmanufacture,allpower-transmitting,gearsmusthavebacklash.Thismeansthatonthegear,andviceversa.Oninstrumentgears,backlashcaneliminatedbyusingagearsplitdownitsmiddle,onehalfbeingrotatablerelativetotheother.Aspringforcesthesplitgearteethtooccupythefullwidthofthepinionspace.Helicalgearshavecertainadvantages；forexample,whenconnectingparallelshaftstheyhaveahigherloadcarryingthanspurgearswiththesametoothnumbersandcutwiththesamecutter.Becauseoftheoverlappingactionoftheteeth,theyaresmootherinactionandcanoperateathigherpitch-linetotheaxisofrotation,helicalgearscreateanaxialthrust.Ifusedsingly,thisthrustmustbeabsorbedinthesameblank.Dependingonthemethodofmanufacture,thegearmaybeofthecontinuous-toothherringbonevarietyoradouble-helicalgearwithaspacebetweenthetwohalvestopermitthecuttingtooltorunout.Double-helicalgearsarewellsuitedfortheefficienttransmissionofpowerathighspeeds.Helicalgearscanalsobeusedtoconnectnonparallel,non-intersectingshaftsatanyangletooneanother.Ninetydegreesisthecommonestangleatwhichsuchgearsareused.Wormandbevelgears.Inordertoachievelinecontactandimprovetheloadcarryingcapacityofthecrossed-axishelicalgears,thegearcanbemadetocurvepartiallyaroundthepinion,insomewhatthesamewaythatanutenvelopsascrew.Theresultwouldbeacylindricalwormandgear.Wormgearsprovidethesimplestmeansofobtaininglargerationsinasinglepair.Theyareusuallylessefficientthanparallel-shaftgears,however,becauseofanadditionalslidingmovementalongtheteeth.Becauseoftheirsimilarity,theefficiencyofawormandgeardependsonthesamefactorsastheefficiencyofascrew.Single-threadwormsoflargediameterhavesmallleadanglesandlowefficiencies.Multiple-threadwormshavelargerleadanglesandhigherefficiencies(Fig.1.2)Fortransmittingrotarymotionandtorquearoundcorners,bevelgearsarecommonlyused.Theconnectedshafts,whoseaxeswouldintersectifextended,areusuallybutnotnecessarilyatrightanglestooneanother.Whenadaptedforshaftsthatdonotintersect,spiralbevelgearsarecalledhypoidgears.Thepitchsurfacesofthesegearsarenotrollingcones,andtheratiooftheirmeandiametersisnotequaltothespeedConsequently,thepinionmayhavefewteethandbemadeaslargeasnecessarytocarrytheload.Theprofilesoftheteethonbevelgearsarenotinvolutes；theyareofsuchashapethatthetoolsfortheteethareeasiertomakeandmaintainthaninvolutecuttingtools.Sincebevelgearscomein,aslongastheyareconjugatetooneanothertheyneednotbeconjugatetoothergearswithdifferentbothnumbers.1EarlyHistoryofGearingTheearliestwrittendescriptionsofgearsaresaidtohavebeenmadebyAristotleinthefourthcenturyB.C.IthasbeenpointedoutthatthepassageattributedtoAristotlebysomewasactuallyfromthewritingsofhisschool,in“MechanicalProblemsofAristotle”(Ca.280B.C).Inthepassageinquestion,therewasnomentionofgearteethontheparallelwheels,andtheymayjustaswellhavebeensmoothwheelsinfrictionalcontact.Therefore,theattributionofgearingtoAristotleis,mostlikely,incorrect.TherealbeginningofgearingwasprobablywithArchimedeswhoabout250B.C.inventedtheendlessscrewturningatoothedwheel,whichwasusedinenginesofwar.Archimedesalsousedgearstosimu-earlyformsofwagonmileageindicators(odometer)andsurveyinginstruments.Thesedeviceswereprobably“thought”experimentsofHeronofAlexandria(ca.A.D.60),whowroteonthesubjectsoftheoreticalmechanicsandthebasicelementsofmechanism.TheoldestsurvivingreliccontaininggearsistheAntikytheramechanism,sonamedbecauseoftheGreekislandofthatnamenearwhichthemechanismwasdiscoveredinasunkenshipin1900.ProfessorPriceofYaleUniversityhaswrittenanauthoritativeaccountofthismechanism.Themechanismisnotonlytheearliestrelicofgearing,butitalsoisanextremelycomplexarrangementofepicyclicdifferentialgearing.Themechanismisidentifiedasacalendricalcomputingmechanismforthesunandmoon,andhasbeendatedtoabout87B.C.TheartofgearingwascarriedthroughtheEuropeandarkagesafterthefallofRome,appearinginIslamicinstrumentssuchasthegearedastrolabeswhichwereusedtocalculatethepositionsofthecelestialbodies.Perhapstheartwasrelearnedbytheclock-andinstrument-makingartisansoffourteenth-centuryEurope,orperhapssomecrystallizingideasandmechanismswereimportedfromtheEastafterthecrusadesoftheelevenththroughthethirteenthcenturies.ItappearsthattheEnglishabbotofSt.Albansmonastery,bornRichardofWallingford,inA.D.1330,reinventedtheepicyclicgearingconcept.Heappliedittoanastronomicalclock,whichhebegantobuildatthattimeandwhichwascompletedafterhisdeath.AmechanicalclockofaslightlylaterperiodwasconceivedbyGiovannideDondi(1348-1364).Diagramsofthisclock,whichdidnotusedifferentialgearing,appearinthesketchbooksofLeonardodaVinci,whodesignedgearedmechanismshimself.In1967twoofdaVincismanuscripts,lostintheNationalLibraryinMadridsince1830,wererediscovered.Oneofthemanuscripts,writtenbetween1493and1497andknownas“CodexMadridI”,contains382pageswithsome1600sketches.IncludedamongthisdisplayofLenardosartisticskillandengineeringabilityarehisstudiesofgearing.Amongthesearetoothprofiledesignsandgearingarrangementsthatwerecenturiesaheadoftheir“invention”.2BeginningofModernGearTechnologyIntheperiod1450to1750,themathematicsofgear-toothprofilesandtheoriesofgearedmechanismsbecameestablished.AlbrechtDureriscreditedwithdiscoveringtheepicycloidalshape(ca.1525).PhilipdelaHireissaidtohaveworkedouttheanalysisofepicycloidsandrecommendedtheinvolutecurveforgearteeth(ca.1694).LeonardEulerworkedoutthelawofconjugateaction(ca.1754).Gearsdeignedaccordingtothislawhaveasteadyspeedratio.Sincetheindustrialrevolutioninmid-nineteenthcentury,theartofgearingblossomed,andgeardesignssteadilybecamebasedonmorescientificprinciples.In1893WilfredLewispublishedaformulaforcomputingstressingearteeth.Thisformulaisinwideusetodayingeardesign.In1899GeorgeB.Grant,thefounderoffivegearmanufacturingcompanies,published“ATreatiseonGearWheels”.Newinventionsledtonewapplicationsforgearing.Forexample,intheearlypartofthiscentury(1910),parallelshaftgearswereintroducedtoreducethespeedofthenewlydevelopedreactionsteamturbineenoughtoturnthedrivingscrewsofocean-goingvessels.Thisapplicationachievedanoverallincreaseinefficiencyof25percentinseatravel.Theneedformoreaccurateandquiet-runninggearsbecameobviouswiththeadventoftheautomobile.Althoughthehypoidgearwaswithinourmanufacturingcapabilitiesby1916,itwasnotusedpracticallyuntil1926,whenitwasusedinthePackardautomobile.Thehypoidgearmadeitpossibletolowerthedriveshaftandgainmoreusablefloorspace.By1937almostallcarsusedhypoid-gearedrearaxles.Speciallubricantantiwearadditiveswereformulatedinthe1920swhichmadeitpracticaltousehypoidgearing.In1931EarleBuchingham,chairmanofanAmericanSocietyofMechanicalEngineers(ASME)researchcommitteeongearing,publishedamilestonereportongear-toothdynamicloading.Thisledtoabetterunderstandingofwhyfaster-runninggearssometimescouldnotcarryasmuchloadasslower-runninggears.High-strengthalloysteelsforgearingweredevelopedduringthe1920sand1930s.Nitridingandcase-hardeningwasintroducedin1950.Extremelycleansteelsproducedbyvacuummeltingprocessesintroducedin1960haveprovedeffectiveinprolonginggearlife.Sincetheearly1960stherehasbeenincreaseduseofindustrialgasturbinesforelectricpowergeneration.Intherangeof1000to14000hp,epicyclicgearsystemshavebeenusedsuccessfully.Pitch-linevelocitiesareform50to100m/s(10000to20000ft/min).Thesegearsetsmustworkreliablyfor10000to30000hpbetweenoverhaule.In1976bevelgearsproducedtodriveacompressorteststandranstandransuccessfullyfor235hat2984kwand200m/s.formallindicationsthesegearscouldbeusedinanindustrialapplicationifneeded.Areasonablemaximumpitch-linevelocityforcommercialspiral-bevelgearswithcurvedteethis60m/s.Gearsystemdevelopmentmethodshavebeenadvancedinwhichlightweight,