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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.F
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