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Chapter3ComplexMotionofParticle(orPoint)
§3.1Basicconceptofcomplexmotionofparticle
§
3.2Velocitycompositiontheoremofparticle§
3.3Accelerationcompositiontheoremwhenthetransportmotionistranslation§
3.4Accelerationcompositiontheoremwhenthetransportmotionisrotation
Maincontents1.
Whatiscomplexmotionofparticle?Motionisrelative.Amotionrelativetoareferenceobjectcanbecomposedofseveralsimplemotionsrelativetootherreferenceobjects.Themotioniscalled
complexmotion.2.ProblemstosolvebytheoryofcomplexmotionofparticleAcomplexmotioncanbedecomposedintotwosimplemotions.Thevaluesofcomplexmotioncanbecomposedbythoseoftwosimplemotions.Therelationsofthemotionofeverycomponentinthemovingmechanism.Therelationoftwomovingobjectswithoutdirectiveconnection.(1)AmovingpointApointintheresearchingobject.(2)Tworeferencesystems(3)Three
kindsof
motionsApoint,tworeferencesystems,andthreekindsofmotionsFixedreferencesystem:Areferencesystemfixedtotheearthground.Movingreferencesystem:
Areferencesystemfixedtoamovingobjectrelativetotheearthground.Absolutemotion:Motionofthemovingpointrelativetothefixedreferencesystem.Relativemotion:
Motionofthemovingpointrelativetothemovingreferencesystem.Transportmotion:Motionofthemovingreferencesystemrelativetothefixedreferencesystem.
3.1BasicconceptofcomplexmotionofparticleAbsolutemotionRelativemotionTransportmotionBothofabsolutemotionandrelativemotionaremotionsofaparticle.Transportmotionismotionofreferenceobject,actuallymotionofarigidbody.
3.1BasicconceptofcomplexmotionofparticleCorrespondingtoabsolutemotion:AbsolutetrajectoryAbsolute
velocityAbsoluteaccelerationCorrespondingtorelativemotion:
RelativetrajectoryRelativevelocityRelativeaccelerationThereisn’ttrajectoryfortransportmotion,becauseitisn’taparticle,butarigidbody.Correspondingtotransportmotion:TransportvelocityTransportaccelerationTransportvelocity
and
transportacceleration
arethevelocityandaccelerationofthepointinthemovingreferencesystemcoincidingwiththemovingpoint(transportpoint)
relativetothefixedreferencesystematanyinstantoftime.
3.1BasicconceptofcomplexmotionofparticleExample
3-1Crankrockermechanism,thecrankOAisconnectedtothesleevebypinA,andthesleeveissetontherockerO1B.WhenthecrankrotatesaroundtheOaxiswithangularvelocityω,therockerO1BisdriventoswingaroundtheO1axisthroughthesleeve.AnalyzethemotionoftheApoint.
3.1BasicconceptofcomplexmotionofparticleSolution:Movingreferencesystem-O1x'y',fixedtorockingbarO1B.2.Motionanalysis.Movingpoint-pin
A
onthesleeve.y'x'1.Choosethemovingpoint,movingreferencesystemandfixedreferencesystem.Fixedreferencesystem-Fixedtotheground.Absolutemotion-CircularmotionwiththecentreO.Relativemotion-ThestraightlinemotionalongO1B.Transportmotion-RotationofrockingbarabouttheaxisO1.
3.1BasicconceptofcomplexmotionofparticleHowtoselectthemovingpointandmovingsystem1.Themovingsystemcanberegardedasaninfiniterigidbody,andthebasicmotionoftherigidbodyistranslationalandfixed-axisrotation.Therefore,themovingsystemisgenerallytakenasthecoordinatesystemoftranslationalmotionorfixed-axisrotation.2.Themovingpointandthemovingreferencecannotbechosenonthesameobject,otherwisetherelativemotionofthemovingpointwithrespecttothemovingreferencewilldisappear.3.Themovingpointmustalwaysbethesamepointinthesystem,andstudyitsmotionatdifferentmoments.Itisnotallowedtotakeapointatoneinstantandanotherpointasthemovingpointatthenextinstant.1.TheoremAtanyinstantoftime,theabsolutevelocityofamovingpointisequaltothegeometricsumofitsrelativevelocityandtransportvelocity.Thisisthe
velocitycompositiontheoremofpoint.
Theabsolutevelocityofamovingpointcanbedeterminedbythediagonallineoftheparallelogramcomposedbyitstransportvelocityandrelativevelocity.
Thisisthe
parallelogramofvelocity.
3.2Velocitycompositiontheoremofparticle
moveto
2.Provement
3.2VelocitycompositiontheoremofparticleExample
3-2
Thequick-returnmechanismofplanerisshowninthefigure.TheendAofacrankOAisarticulatedwithaslideblock.ThecrankOArotatesaroundthefixedaxisOwiththeuniformangularvelocityω.Theslideblockslidesontherockingbar,whichisdriventoswingaboutthefixedaxisO1.ThelengthofthecrankOA=r,OO1=l.Findtheangularvelocityω1oftherockingbarwhenthecrankmovestothehorizontalposition.
3.2VelocitycompositiontheoremofparticleSolution:Movingreferencesystem-O1x'y',fixedtorockingbarO1B.2.Motionanalysis.Movingpoint-pin
A
onthesleeve.y'x'1.Choosethemovingpoint,movingreferencesystemandfixedreferencesystem.Fixedreferencesystem-Fixedtotheground.Absolutemotion-CircularmotionwiththecentreO.Relativemotion-ThestraightlinemotionalongO1B.Transportmotion-RotationofrockingbarabouttheaxisO1.
3.2Velocitycompositiontheoremofparticle3.VelocityanalysisvavevrAbsolutevelocityva:va=OA·ω
=rω,
Direction:verticaltoOA,plumbedupwardsTransportvelocity
ve:ve
istheunknownquantity,andneedtobesolvedDirection:verticaltoO1BRelativevelocityvr:themagnitudeisunknownDirection:alongtherockingbarO1B
Accordingtothevelocitycompositiontheoremofapoint
3.2Velocitycompositiontheoremofparticle∵∴Supposetheangularvelocityoftherockingbaratthemomentisω1,yieldsSovavevr
3.2Velocitycompositiontheoremofparticle1.Relativeandabsolutederivativeofvector●MOxyzisafixedcoordinatesystem,andO1x1y1z1isamotioncoordinatesystem,theradiusvectorofthemovingpointMinthemotionsystemisWetakethetimederivativeinthefixedsystemtoobtainThisistheabsoluterateofchangeofthevectorr1Takethederivativeofr1withrespecttotimeinthemotionsystemtoobtainThisistherelativerateofchangeofthevectorr13.3Accelerationcompositiontheoremwhenthetransportmotionistranslation2.Threekindsofaccelerations(1)Absoluteacceleration(2)Relativeacceleration3.3Accelerationcompositiontheoremwhenthetransportmotionistranslation●M2.Threekindsofaccelerations(3)Transportacceleration3.3Accelerationcompositiontheoremwhenthetransportmotionistranslation●M3.AccelerationcompositiontheoremWhenthemotionsystemistranslatingmotion,andi1,j1,k1
areconstantvectors,andtheirmagnitudesanddirectionsareconstant,sotheirtimederivativesareallzero,wecangetAccelerationcompositiontheoremwhenthetransportmotionistranslation3.3Accelerationcompositiontheoremwhenthetransportmotionistranslation●MExample
3-3
Aplanemechanismshowninthefigure,thecrankOA=r,rotatesuniformlywithangularvelocityω0.SleeveAcanslidsalongthebarBC.BC=DE,且BD=CE=l.FindtheangularvelocityandangularaccelerationofBDatthemomentshowninthefigure.ABCDEOω0ωαSolution:Choosethemovingpoint,movingreferencesystemandfixedreferencesystemMovingreferencesystem-Cx´y´,fixedtothebar
BC.2.MotionanalysisTransportmotion-translationMovingpoint-slideblock
A.Fixedreferencesystem-
fixedtothebase.ABCDEOω0ωαx'y'Absolutemotion-CircularmotionwithcentreORelativemotion-straightlinemotionalongBCABCDEOω0ωαvBvevavr3.VelocityanalysisyieldsSotheangularvelocityof
BDAbsolute
velocity
va:va=ω0r,verticalto
OA
downwards.
Transportvelocity
ve:ve=
vB,verticalto
BDrightdownwands.
Relativevelocity
vr:magnitudeunknown,along
BCleftEmployingthetheoremofcompositionofvelocities4.AccelerationanalysisAbsoluteacceleration
aa:aa=ωor
,along
OA,pointtoOTransportaccelerationae:tangentialcomponentaet:sametoaBt,magnitude
unknown,verticaltoDB,
supposedownwardsRelativeacceleration
ar:magnitude
unknown,along
BC,
supposetoleftnormalcomponentaen:aen
=aBn=
ω2l
=ωo2r2
/l,alongDB,
pointtoDaaarABCDEOω0ωα
Projecttoaxisy,
yieldsyieldsApplyingthecompositiontheoremofaccelerationsSotheangularaccelerationof
BD:
aaarABCDEOωαyAfixedcoordinatesystemOxyzandmotioncoordinatesystemOx1y1z1,letthemovingpointMmoveinthemotionsystemOx1y1z1,andthemotionsystemOx1y1z1rotatesaboutthez-axisofthefixedsystemwithangularvelocityωandangularaccelerationε●MBasedonthepreviousproofofthevelocitycompositiontheorem,wehave
TherelativevelocityandrelativeaccelerationofthemovingpointM3.4AccelerationcompositiontheoremwhenthetransportmotionisrotationAndthen
Basedonthevelocitycompositiontheorem:AccordingtothePoissonformula:3.4Accelerationcompositiontheoremwhenthetransportmotionisrotation
Coriolisacceleration:Thisistheaccelerationcompositiontheoremwhenthetransportmotionisrotation.3.4AccelerationcompositiontheoremwhenthetransportmotionisrotationExample
3-4Thequick-returnmechanismofplanerisshowninthefigure.TheendAofacrankOAisarticulatedwithaslideblock.ThecrankOArotatesaroundthefixedaxisOwiththeuniformangularvelocityω.Theslideblockslidesontherockingbar,whichisdriventoswingaboutthefixedaxisO1.ThelengthofthecrankOA=r,OO1=l.Findtheangularaccelerationα1oftherockingbarwhenthecrankmovestothehorizontalposition.
Basedonthe
velocityanalysisobtainedfromlastclass,weknowthatSolution:Choosethemovingpoint,movingreferencesystemandfixedreferencesystem.Movingreferencesystem-O1x1y1,fixedtorockingbarO1B.Movingpoint-slideblock
A.vavevry1x1Fixedreferencesystem-Fixedtothe
base2.AccelerationanalysisAbsoluteacceleration
aa:
aa
=ω2r
,along
OA,pointto
O.Relative
acceleration
ar:magnitude
is
unknown
,suppose
it
is
along
O1B
upwards.
Tangential
component
aet:magnitude
is
unknown,
vertical
to
O1B,supposerightdownwardsTransport
acceleration:Normal
component
:
along
O1A,point
to
O1Coriolis
acceleration
aC:verticaltoO1B,showninthefigurex'y'O1Oφωω1ABaaaraCProjectitto
O1x'yieldsTheangularaccelerationofrockingbar:α1Applyingtheaccelerationcompositiontheoremx'y'O1Oφωω1ABaaaraCor
TheEnd
Chapter4
PlanarMotionofaRigidBody§
4.1Basicconceptanddecompositionofrigidbodyplanarmotion
Maincontents§4.2
Velocityofanypointinaplanarmotion§4.3
Accelerationofanypointinaplanarmotion1.Whatisplanarmotionofarigidbody?Thedistancebetweenanypointinarigidbodyandafixedplanealwayskeepsunchangedduringitsmotion.Thismotionofrigidbodyiscalled
planarmotionofarigidbody.4.1Basicconceptanddecompositionofrigidbodyplanarmotion2.SimplificationofaplanarmotionTheplanarmotionofarigidbodycanbesimplifiedtoamotionofaplanegraphintheplaneitselfwithoutconsideringitsthickness.
(a)Connectingrodmotion(b)Simplificationofconnectingrodmotion4.1Basicconceptanddecompositionofrigidbodyplanarmotion3.EquationsofplanarmotionSTodeterminethemotionofaplanegraph,choosethefixedreferencesystemOxy,anarbitrarypointO'intheplanegraphS,anarbitrarylinesegmentO'M.Todeterminetheplanarmotionofarigidbody,onlythepositionofthelinesegmentO'Minthisgraphisneededtobedetermined.EquationsofplanarmotionAplanemotioncanberegardedasthecompositionofa
translation
androtation.4.1Basicconceptanddecompositionofrigidbodyplanarmotion4.Planarmotioncanbedecomposedintotranslationandrotation
Aplanemotionofarigidbodycanbedecomposedintoa
translationwithabasicpointanda
rotation
aboutanaxisthatpassesthroughthebasicpoint.Thevelocityandaccelerationofthe
translation
withabasicpoint
intheplanegraphdependson
theselectionof
thebasicpoint,however,theangularvelocityandaccelerationoftherotationabouttheselectedbasicpoint
doesn’tdependon
thechoiceofthebasicpoint.4.1BasicconceptanddecompositionofrigidbodyplanarmotionAThevelocityofpointAintheplanegraphSis,andtherotationalvelocityoftheplanegraphis.SelectAasthebasicpoint;ThemovingreferencesystemattachedtopointA;Thetransportmotionistranslationwiththebasicpoint
A;Therelativemotionisrotationaboutthebasicpoint
A.(1)Basicpointmethod
·BDeterminethevelocityofpointBintheplanegraph.4.2VelocityofanypointinaplanarmotionABTheorem:Forplanarmotionofarigidbody,thevelocityofanypointinthegraphcanbeobtainedasthevectorsumofthevelocityofthebasicpointandtherelativerotationalvelocitywithrespecttothebasicpoint.4.2Velocityofanypointinaplanarmotion
isverticaltothelinkofABallthetime,sotheprojectionofonABisvanish.Thevelocityprojectiontheorem:thevelocityprojectionsofanytwopointinaplanegraphonthelinelinkingthesetwopointsareidentical.(2)VelocityprojectiontheoremAB4.2Velocityofanypointinaplanarmotiona.Background
Ifapointwhosevelocityiszeroisselectedasthebasicpoint,theprocessoffindingthevelocityofanypointwillbegreatlysimplified.Therefore,itisnaturaltoaskifsuchapointexistsinanyinstant.Ifitdoesexist,howtofindsuchapoint?b.InstantaneouscenterofvelocityAtanyinstant,itmustexistasolepointwhosevelocityiszerointheplanegraphoritsexpandingarea,whichiscalledtheinstantaneousvelocitycenterofthisplanegraphatthisinstant.Foraplanegraph,itsinstantaneousvelocitycenteralwaysexistsuniquely.
(3)Instantaneouscenterofvelocitymethod4.2Velocityofanypointinaplanarmotionc.InstantaneouscenterofvelocitymethodConsideraplanegraph.TheinstantaneousvelocitycenterisP,andtheangularvelocityoftheplanegraphis.SelectinstantaneousvelocitycenterPisabasicpoint,thevelocityofanarbitrarypointAintheplanegraph:4.2Velocityofanypointinaplanarmotiond.MethodstodeterminetheinstantaneousvelocitycenterPA(1)Whenthevelocityofapointandtheangularvelocity
oftheplanegraphareknown,theinstantaneousvelocitycenter(pointP)canbedetermined,
pointPisinthedirectionofthelineformedbyrotatingthethrough90ºinthedirectionof
aroundpointA.4.2Velocityofanypointinaplanarmotion(2)Whenaplanegraphrollsalongafixedsurfacewithoutslipping,thecontactpointPbetweenthegraphandthefixedsurfacewillbetheinstantaneousvelocitycenter.
(3)WhenthedirectionsofthevelocitiesattwopointsAandBinagraphareknown,andisnotparallelto,drawlinesfromAandBperpendiculartorespectively,andthecrosspointPofthesetwolineswillbetheinstantaneousvelocitycenter.ABP4.2Velocityofanypointinaplanarmotion(4)WhenthevelocitiesoftwopointsAandBaregivenatanyinstant,and.Therearethreecases:ABP
Whenandpointtothesamedirection,but.DrawtheextensionlineofAB,thelinkinglineoftheendingsofand,thecrosspointofthesetwolineswillbetheinstantaneousvelocitycenter.Therotationdirectionofcanbedetermined,anditsmagnitudeis:◆ω
◆Whenandhaveoppositedirections,drawthelinkinglineoftheendingsofand,andthelineconnectingAB.Thecrosspointofthesetwolineswillbetheinstantaneousvelocitycenter.Therotationdirectionofcanbedetermined,anditsmagnitudeis:
ω4.2VelocityofanypointinaplanarmotionBPAB(5)ThevelocitiesoftwopointsAandBpointtothesamedirectionatanyinstant,,,buttheyarenotperpendiculartolineAB.Inthiscase,theinstantaneousvelocitycenterisindefinitelyfaraway,andtheangularvelocity
=0,i.e.allpointinthefigurehavethesamevelocityatthisinstantoftime.Suchamotioniscalledinstantaneoustranslation,buttheiraccelerationsarenotequal.
When,
theinstantaneousvelocitycenterisindefinitelyfaraway.Theplanegraphhasinstantaneoustranslation,=0,allpointsinthegraphhavethesamevelocityatthisinstantoftime,buttheiraccelerationsarenotequal.◆ω4.2VelocityofanypointinaplanarmotionAABAAttheinstant,theangularvelocityofthegraphis,angularaccelerationis,accelerationofapointAis
.DeterminetheaccelerationofanarbitrarypointBinthegraph.·
4.3
AccelerationofanypointinaplanarmotionBA
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