54883《機(jī)械原理（英語）（第2版）》基本課件Chapter 3 Kinematic Analysis of Planar Mechanisms

Chapter3KinematicAnalysisofPlanarMechanisms1.PurposeofKinematicAnalysis(1)TheworkspaceofamechanismisnecessarybymeansofanalysisofpositionsortracingpathFig.3-1ashowsainternalcombustionengine,inwhichthestrokeofthepistoncanbeusedtodesignthelengthofthecylinder,andpathofthecouplercanbeusedtodesigntheinternaldimensionsoftheengineblock.(2)DeterminethevelocitiesandaccelerationsoflinkstoinvestigatetheworkingcharacteristicsofamachineFig.3-1bshowsashapermechanism.Theramintheworkingstrokedemandsconstantvelocityapproximatelyandthevariationofaccelerationsisaslittleaspossible.Sothevelocityanalysisisveryimportanttodesignashapermechanism.(3)MotionanalysisisneededforthedynamicforcecalculationOnceapositionanalysisisdone,thenextstepistodeterminethevelocitiesofdrivenlinksortracingpointsofinterestinthemechanism.3.1IntroductionFig.3-1Kinematicanalysisofmechanisms(機(jī)構(gòu)的運(yùn)動(dòng)分析)2.MethodsofMotionAnalysis(1)Graphicalmethod

(2)Analyticalmethod

Wefirstestablishavectorloop（orloops）aroundthemechanism,inwhichthelinksarerepresentedaspositionvectors,thenwecantakethederivativeswithrespecttotimetofindthevelocityandacceleration.

(3)Experimentalmethod

Wemayinstallthedisplacementsensor,speedsensororaccelerationsensorinthemachinetomeasurethedisplacements,speedsandaccelerations,whicharerequired.Theexperimentalmethodisaconventionalmethodtoanalyzetheperformanceofmachines.1)Instantaneouscentermethod

2)Relativemotionmethod1.ConceptofInstantaneousCenterofVelocity(1)InstantaneouscenterofvelocityAninstantaneouscenterofvelocityisacenterofrotationofamovinglinkrelativetoanotherlink.Ifalinkisinmotionrelativetoafixedlink,thecenteriscalledasanabsolutecenter;otherwiseitiscalledasarelativecenter.

3.2VelocityAnalysiswithInstantaneousCentersFig.3-2Instantaneouscenter(速度瞬心)(2)Thenumberofinstantaneouscenters(3)Locatinginstantaneouscenters

Thefollowingrulesareusedwhenlocatinginstantaneouscenters.Fig.3-3Primaryinstantaneouscentersoftwolinkslinkedbykinematicpair

(兩構(gòu)件用運(yùn)動(dòng)副連接時(shí)的瞬心位置)1)Twolinksareconnectedbyakinematicpair.Iftwolinksareconnectedbyapivotjoint,thecenterofthepivotistheinstantaneouscenter;seetheFig.3-3a,b.Iftwolinkshaveslidingcontact,theinstantaneouscenterliesatinfinityinadirectionperpendiculartothepathofthemotionoftheslider;seetheFig.3-3c.Iftwolinkshavepurerollingcontact,theinstantaneouscenteristhepointofcontact,thisisbecausethetwopointsofcontactonthetwobodieshavethesamelinearvelocityandthereisnorelativemotionatthecontactpoint.SeetheFig.3-3d.Iftwolinkshaverollingandslidingcontact,theinstantaneouscenterliessomewhereonthecommonnormalofthecontactpoint.SeetheFig.3-3e.

2)Twolinkshavingrelativemotionarenotconnectedbykinematicpair.ThisinstantaneouscentercanbedeterminedbyKennedytheorem.Anythreebodiesinplanemotionwillhaveexactlythreeinstantaneouscenters,andtheywillbeonthesamestraightline.ThisisknownasKennedytheorem.Fig.3-4Kennedytheorem(三心定理)Fig.3-5Instantaneouscentersforfour-barmechanisms(四桿機(jī)構(gòu)的瞬心)Example3-1Fig.3-5showsafourbarlinkageandaslidercrankmechanism.Findalltheinstantaneouscentersbygraphicalmethod.

2.VelocityAnalysiswithInstantaneousCentersExample3-2Fig.3-6showsafourbarlinkage.Theangularvelocityω1oflink1isknown,asshowninthefigure.Findtheangularvelocitiesω

2andω3.

Fig.3-6Applicationofinstantaneouscentersforfour-barmechanism

(瞬心法在鉸鏈四桿機(jī)構(gòu)速度分析中的應(yīng)用)Fig.3-7Applicationofinstantaneouscentersforcam

mechanism(瞬心法在凸輪機(jī)構(gòu)中的應(yīng)用)Example3-3Fig.3-7showsacammechanismwithaflatfollower.Theangularvelocityofcam1isknownanditisrequiredtofindthevelocityofthefollower3.

1.PrinciplesofRelativeMotions(1)Relativemotion(velocityandacceleration)oftwopointsonthesamelinkLetusconsiderabodywhichhasplanemotion;seetheFig.3-8.

3.3KinematicAnalysisbyGraphicalMethodFig.3-8Relativevelocityoftwopointsonalink

(同一構(gòu)件上兩點(diǎn)之間的速度關(guān)系)Fig.3-9Relativevelocityofcoincidentpointon

separatelinks(兩構(gòu)件重合點(diǎn)處的運(yùn)動(dòng)關(guān)系)(2)Relativemotion(velocityandacceleration)oftwocoincidentpointsondifferentlinksInmanymechanisms,suchasinFig.3-9,constraintofrelativemotionisachievedbyguidingtheslider2ontheguiderbar1alongitspath.Theslider2isreciprocatedalongtheguiderbar1,andtheyrotateaboutthepivotOtogetherwithanangularvelocityω1.

(3)VelocityimageandaccelerationimageWhenweknowthevelocitiesoraccelerationsattwodifferentpointsonalink,thevelocityoraccelerationofthethirdpointcanbedeterminedbydrawingtheirimages.Whiledrawingtheimages,thefollowingpointsshouldbekeptinmind:1)Thevelocityimageoraccelerationimageofalinkisascaledreproductionofthelinkshapeinthevelocitydiagramoraccelerationdiagram.2)Theorderofthelettersinthevelocityimageoraccelerationimageisthesameasinthelinkconfiguration.2.GraphicalMethodofRelativeMotionsTheprocedureofkinematicanalysisofplanarmechanismisasfollows:1)Drawthescaledkinematicdiagram.2)Writethevelocityvectorequationanddrawthevelocitydiagram,thenfindouttheunknownvelocitiesorangularvelocities.3)Writetheaccelerationvectorequationanddrawtheaccelerationdiagram,thenfindouttheunknownaccelerationsorangularaccelerations.

Fig.3-10Kinematicanalysisofafour-barlinkage(鉸鏈四桿機(jī)構(gòu)的運(yùn)動(dòng)分析)Example3-4Fig.3-10ashowsafourbarlinkage;allthedimensionsofthelinksandangularpositionofthedrivinglinkABareknown.WhenthecrankABrotatescounterclockwisewithanangularvelocityω

1,determinetheangularvelocitiesω2,ω3,thevelocityofthepointEonthelink2andangularaccelerationsα

2,α

3.

Fig.3-11Kinematicanalysisofaguide-barmechanism(導(dǎo)桿機(jī)構(gòu)的運(yùn)動(dòng)分析)Example3-5Fig.3-11ashowsaguiderbarmechanism;allthedimensionsofthelinksandangularpositionofthedrivinglinkABareknown.WhenthecrankABrotatescounterclockwisewithanangularvelocityω1,determinetheangularvelocitiesω

2,ω3andtheangularaccelerationα2andα3.

3.SomeKeyPointsofMotionAnalysis1)TheCoriolisaccelerationofcoincidentpointsontwodifferentlinksmustbediscriminatedcorrectly.

2)Whenwewouldestablishthevelocityequationoraccelerationequation,thevelocityandaccelerationofthebasepointmustbeknown.Ifwewanttofindouttheangularvelocityofthelink3inFig.3-12,thepointBmustbeconsideredtoestablishthevelocityequation,becausethevelocityofthepointBonthelink1isknown.

Fig.3-12Expandedlink(構(gòu)件的擴(kuò)大)3)Whenthemechanismisatitslimitedpositions,thevelocitypolygonoraccelerationpolygonbecomessimple,butsometimesitisdifficulttoanalysis.Fig.3-13ashowsafourbarlinkageinwhichthecrankandcouplerareincollinear.IntheguiderbarmechanismshowninFig.3-13b,thecrankisperpendiculartotherocker.Theirvelocitypolygonandaccelerationpolygonareverysimple.

Fig.3-13Kinematicanalysisinlimitedpositions(特殊位置的運(yùn)動(dòng)分析)4)Hydraulicmechanismcanbetransformedintoarelatingguiderbarmechanism.Fig.3-14ashowsahydraulicmechanism.ItcanbetransformedintoaguiderbarmechanismshowninFig.3-14b;theyareequivalentmechanisms.Fig.3-14Kinematicanalysisofhydraulicmechanism(擺動(dòng)液壓缸機(jī)構(gòu)運(yùn)動(dòng)分析)1.FundamentalLawofAnalyticalMethod

Theprocedureofmotionanalysisisasfollows:

1)Establishacoordinatesysteminwhichtheoriginofthecoordinatesystemiscoincidentwiththerotatingcenterofthedrivinglinkandthexaxisisalongtheframeofthemechanism.

2)Establishpositionequation.

3)ThevectorloopequationwhichrepresentsthepositionequationcanbewrittenastwoprojectiveequationsintheCartesiancoordinate.

4)Differentiatethepositionequationwithrespecttotime.

5)Thevelocityequationisdifferentiatedwithrespecttotimeagain.3.4KinematicAnalysisbyAlgebraicMethod2.KinematicAnalysisbyAnalyticalMethodExample3-6Fig.3