Engineering Basic Mechanics Ⅱ Dynamics 工程基礎力學 Ⅱ 動力學 課件 Chapter 3 Complex Motion of Particle

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