mechanicalengineeringAnintroductionto1Lecturer:LiuJu-rongCHAPTER7MotionofMachineryOVERVIEW1ROTATZONALMOTION23SPEED,TORQUE,ANDPOWERINGEARSETS

2SIMPLEANDCOMPOUNDGEARTRAINS4PLANETARYGEARTRAINS5Vocabulary3dashedlines虛線tachometer

n.轉(zhuǎn)速表radiansn.弧度derivationn.導(dǎo)出;公式推導(dǎo)instantaneouspowern.瞬時(shí)功率Throttlen.節(jié)流閥;油門;風(fēng)門shiftforkn.調(diào)檔叉;拔叉splinedshaft

n.花鍵軸intrinsicnature內(nèi)在性質(zhì)recursivelyadv.遞推;遞歸sungearn.中心齒輪planetgearn.行星齒輪ringgearn.環(huán)形齒輪;齒圈differential差動(dòng)light-duty

n.輕型helicoptern.直升飛機(jī)formfactor

形狀系數(shù)crank

n.曲柄1OVERVIEW4Afterstudy,whatshouldyoubeabletohave?Understandthedifferentchoicesofunitsforangularvelocity,andmakeappropriateconversionsbetweenthem.Calculatethespeedsofgears,thetorquesactingontheirshafts,andthepowertransferredinsimpleandcompoundgeartrains.

Determinetheoverallvelocityandtorqueratiosforageartrain.Sketchaplanetarygeartrainandexplainhowitoperates.Relatethemotionsofthepistonandcrankshaftinaninternalcombustionengineorcompressor.2ROTATZONALMOTION---RotationalVelocityAngularvelocity5AsProtatesalongacircleofradiusr,thedistancethatitmovesisthearclength△s=r△θ.Fordimensionalconsistency,since△shastheunitsoflength,theangle△θisexpressedintheunitsofradians(andnotdegrees).ThevelocityofpointPisthedistancetraveledduringthetimeinterval,orvp=△s/△t=r(△θ/△t).Instandardform,wedefineω(thelowercaseGreekcharacteromega)astherotationalorangularvelocityofthelink,andthevelocityis

vp=rωTheappropriateunitsforωarechosenbasedoncontext.itiscustomarytousetheunitsofrevolutionsperminute(rpm),whichatachometerwouldmeasure.Iftherotationspeedisveryhigh,theunitsofrevolutionspersecond(rps)mightbeusedinstead.2ROTATZONALMOTION---RotationalWorkandPowerWorkofatorque6torqueworkiscalculatedfrom

△W=T△θwheretheangle△θhastheunitofradians.InstantaneouspowertheinstantaneouspowerP=FvorP=Tωisoftenmoreusefulwhenanalyzingmachinery.Inshort,instantaneouspoweristheproductofforceandvelocity,ortorqueandangularvelocity.3SPEED,TORQUE,ANDPOWERINGEARSETSSpeedvelocityratio7thespeedofapointonitspitchcircleisVp=rpωp.

thevelocityofpointsincontactarethesame,andrgωg=rpωp.Withtheangularvelocityofthepinion'sshaftbeingknownastheinputtothegearset,thespeedoftheoutputshaftis

Thevelocityratioofthegearsetisdefinedasintermsoftheoutputandinputspeeds.

3SPEED,TORQUE,ANDPOWERINGEARSETSTorqueratio8thetorqueratioofthegearsetisdefinedas

powerByeliminatingtheunknownforceF,weobtainanexpressionfortheoutputtorqueofthegearset

4SIMPLEANDCOMPOUNDGEARTRAINS9Compoundgeartrainsareoftenusedingearboxesandtransmissionswhenlargervelocityortorqueratiosareneededorwhenthesystemmustbecompact.Suchasystem,ithasthecharacteristicthateachshaftcarriesasinglegear.Theintermediategearsprovidenospeedortorquemodificationsforthegeartrainasawhole.simplegeartrainIdlergearCompoundgeartrains4SIMPLEANDCOMPOUNDGEARTRAINS10FINGURE7.1Asimplegeartrainformedasaserialconnectionoffourgearsonseparateshafts.Thegearteethareomittedforclarity.4SIMPLEANDCOMPOUNDGEARTRAINS11FINGURE7.2Acompoundgeartraininwhichthesecondandthirdgearsaremountedonthesameshaftandrotatetogether.Thegearteethareomittedforclarity.5PLANETARYGEARTRAINS12thecenteroftheplanetgearcanorbitaroundthesungear.Itisalsopossible,anddesirable,toconstructgeartrainsinwhichthecentersthemselvesofcertaingearsmove.thecenterpointsofgearsinthesimplegearsetasbeingconnectedbyamotionlesslinkthatisfixedtoground.thecenterofthesungearisstationary.planetarygeartrainsSungearPlanetgearthecenteroftheplanetgearcanorbitaroundthesungear.RinggearThelinkthatconnectsthecentersofthetwogearsiscalledthecarrier,anditcanrotateaswellaboutoneofitsends.Ringgear5PLANETARYGEARTRAINS13Planetarygeartrainsareusuallydesignedwithmorethanoneplanetgearinordertoreducenoise,vibration,andtheforcesappliedtogearteeth.multipleplanetgearsarepresent,thecarrierissometimescalledthespider(支架),totheextent,thatithasseveral(althoughperhapsnotasmanyaseight)legs,whichevenlyspacetheplanetgearsaroundtheringgearcircumference.BalancedgeartrainSpiderOurintuitionisgenerallynotsufficienttodeterminethedirectionthateachgearandshaftinaplanetarygeartrainwillrotate.Forthatreason,weinsteadapplyadesignequationthatrelatestherotationalvelocitiesofthesungear(denotedωs),carrier(ωc),andringgear(ωr)asfollows:(2+n)ωr+nωs-2(1+n)ωc=05PLANETARYGEARTRAINS14Anautomobiledifferentialisaspecialtypeofplanetarygeartrain.weapplytheconventionthatrotationsofthegearsandcarrierarepositivewhendirectedclockwise,andnegativewhencounterclockwise.WiththenumbersofteethonthesunandplanetgearsdenotedbyNsandNp,theformfactorninisdefinedasSignconventionFormfactorDifferential

SUMMARY15Intheprevioussections,weexaminedthemotionofmachineryasafifthelementofmechanicalengineering.

Gearsets,simplegeartrains,compoundgeartrains,andplanetarygeartrainsareusedforthepurposesoftransmittingpower,changingtherotationspeedofashaft,andmodifyingthetorquethatisappliedtoashaft.Atahigherlevel,theapplicationofcalculusfordeterminingthevelocityandaccelerationofapistonshouldgiveyouaperspectiveontheimpactofmathematicsintheengineer