TheoryofMachinesandMechanismsChapter1Introduction1.

TheConceptofMachinery1.1GeneralInformation(1)MechanismAmechanismisadevice,whichproducesspecificmechanicalmotions.Thus,thefunctionofamechanismistotransmitandmodifyamotion.Weoftenuseadrawingwithsomesimplelinesandsymbolstodescribethemechanism,anditiscalledkinematicaldiagram.

(2)MachineAmachineisadevicewhichproducesspecificmechanicalmotions,anditcantransmitormodifymechanicalenergy,materialsandinformation.Amachinemaybeamechanismoracombinationofmechanismscapableoftransmittingormodifyingmotionandmechanicalenergy.

(3)MachineryIntheviewpointofkinematics,mechanismandmachinehavenodifference，somechanismsandmachinesaregenerallycalledmachineryinmechanicalengineering.Thereisnoenergytransformingandmodifyinginamechanism,forexample,mechanicalwatchisamechanismbutnotamachine,becauseitcannottransformenergy.Fig.1-1Internalcombustionengineandscheme(內燃機及其機構簡圖)

1—piston(活塞)2—coupler(連桿)3—crankshaft(曲軸)4、5、6—gear(齒輪)7—cam(凸輪)

8—follower(推桿)9—lever(杠桿)10—valve(氣門)11—frame(缸體)2.CompositionofMachinesFig.1-2Compositionofmachine(機器組成示意圖)

Amachineconsistsofprimepower,transmission

systemandworkingsystem.Amodernmachinealsocontainsacontrolsystem.Fig.1-2showstheblockdiagramofcompositionofamachine.Fig.1-3Automaticgate(電動大門示意圖)

1—driver(驅動器)2—gate(大門)3—electromotor(電動機)4—reducer(減速器)

5—chaindriver(鏈傳動)6—rollers(滾輪)

Fig.1-3bshowsadiagramofanautomaticgate.Fig.1-3ashowsthecompositionofthedriver.Theprimepoweriselectromotor3whichhasahighspeed.Butgate2needsalowspeedtowork,sowehavetoinstallgearmechanismsandchainmechanismstomodifythespeedofelectromotor3.Thereducer4andchaindriver5aretransmissionsystem.Gate2consistingoftheparallellinkagesiscalledworkingsystem.Fig.1-4Constituteofthecoupler(連桿的組成)

1—bodyofthecoupler(連桿體)2—bearing(小端軸承)3—bearingbush(軸瓦)

4—headofthecoupler(連桿頭)5—bolt(螺栓)6—nut(螺母)7—washer(墊圈)1.2TeachingContentandObjectoftheCourse1.ResearchObjectoftheCourse

Theresearchobjectofthecourseisthetheoryofmachinery,andmachinerycontainsmechanismsandmachines.Therefore,thetheoryofmechanismsandmachinesistheimportantcontentofthiscourse.

Thepiston,coupler,crank,cam,gearandsoonshowninFig.1-1aresmallestmotionunits,andtheyarecalledlinks.Themotionoflinksisanimportantcontentforus.

2.ContentofThisCourse(1)StructuralanalysisofmechanismsItinvolvesthecompositionofmechanism,kinematicaldiagram,calculationofdegreeoffreedomandanalysisofmechanism.

(2)KinematicsofmechanismsItdealswiththerelativemotionsoflinksandforcesactingonthelinks,thusitinvolvesthevelocityanalysis,accelerationanalysis,forceanalysisandsynthesisofmechanism.

(3)DynamicsofmachineryItinvolvestheinertiaforcesbalancingandthefluctuationinspeedandregulation.1.Purpose

Themachineryiscloselyrelatedtohumanlife,economicconstruction,andnationaldefenseconstruction,somechanicalindustrycanreflectaindustrylevelofacountryandalevelofscienceandtechnology.Althoughtherearealotofmachines,themechanismsconsistingmachinesarefinite.Soonthebasisoffinitemechanismstodesignsomenewmachineisacreativework.Itisalsothepurposeofthiscourse.Thiscourseisthebasisofsomeothersubsequentcourses,suchasMachineryDesign,AdvancedKinematicsandDynamicsofMechanisms,MechanicalManufacture,etc.Sothepurposeofthiscourseisalsotolaythefoundationforthefurthermachinedesign.1.3PurposeofThisCourse2.Conclusion1)Machineryisveryimportanttodeveloptheeconomy.2)TheoryofMechanismandMachinesisveryusefultodesignmachines.3)Mechanismisadevicewhichcantransmitandmodifymotion.4)Thekinematicaldiagramisasimpleformtodescribethemechanisms.5)Amachinemaybeamechanismorcombinationofmechanismscapableoftransmittingormodifyingmotionandmechanicalenergy.6)Machineryisthegenerictermofmechanismsandmachines.7)Alinkisasmallestkinematicalunitofamachinefromthepointofviewofthemovement.8)Anelementisasmallestmanufacturingunitofamachine.9)TheoryofMachinesandMechanismsisabasiccourse,anditisapartofmachinerydesign.Chapter2StructuralAnalysisofPlanarMechanisms1.Link2.1KinematicChainandMechanisms

Apartisaminimumunitinamachinefromthestandpointofmanufacture,suchasarigidlink,nut,bolt,gearandsoon.Fig.2-1ashowsasinglecylinderfourstrokecycleengine,andFig.2-1bistheschematicdiagram.Fig.2-1cisacoupler.Thecouplerconsistsofseveralpartswhichareconnectedrigidly.Fig.2-1Coupleroftheinternalcombustionengine(內燃機中的連桿)2.KinematicPair

(1)KinematicpairsaccordingtonatureofrelativemotionAccordingtothefeatureofrelativemotionbetweenthetwolinks,kinematicpairsmaybeclassifiedasturningpairsandslidingpairs.

1)Turningpair.Whenonelinkhasaturningorrevolvingmotionrelativetotheotherlink,thesetwolinksconstituteaturningpairorarevolvingpair.

Akinematicpairisajointthatpermitsrelativemotion.Fig.2-2Turningpairs(轉動副)2)Slidingpair.Iftwolinkshaveaslidingmotionrelativetoeachother,theyformaslidingpairorprismaticpair.Fig.2-3showssomeslidingpairsandtheirsymbols.Fig.2-3Slidingpairs(移動副)(2)KinematicpairsaccordingtonatureofcontactAccordingtothefeatureofcontactbetweenthetwolinks,kinematicpairsmaybeclassifiedaslowerpairsandhigherpairs.1)Lowerpair.Whenapairhassurfaceorareacontactbetweentwolinks,itisknownasalowerpair.

2)Higherpair.Whenapairhasapointorlinecontactbetweentwolinks,itisknownasahigherpair.

Fig.2-4Higherpairs(高副)(3)KinematicpairelementThegeometricformsofcontactinapair,suchaspoint,lineorsurface,areknownaspairelements.

Apairismadeupoftwoelements,oneoneachlinkbeingjoined,suchastheoutsidecylindricalsurfaceofshaft1inFig.2-2andinnercylindricalsurfaceofbearing2inFig.2-2arepairelementsrespectively.。3.KinematicChainFig.2-5Closedkinematicchains(閉鏈)

Akinematicchainisanassemblyoflinksinwhichtherelativemotionsofthelinksarepossible.Kinematicchainscanbeclassifiedasclosedchainsandunclosedchains.Ifeverylinkinakinematicchainhasatleasttwopairelements,andlinksformaclosedloop,thiskinematicchainiscalledaclosedchain.Aclosedchainatleasthasoneloop.Fig.2-5showssomeclosedkinematicchains.Fig.2-6Unclosedkinematicchains(開鏈)Fig.2-6showssomeunclosedkinematicchains.Thefirstandthelastlinkhaveonlyonepairelementintheunclosedchain.Fig.2-7Structures(桁架)

Iftherelativemotionofthelinksintheassemblyisimpossible,theassemblyoflinksiscalledastructureorsuperstructure.Fig.2-7showssomestructures.Astructuremaybeconsideredasalink.Fig.2-8Mechanismsinwhichallthepairsarelowerpairs(低副機構)4.MechanismIfonelinkofakinematicchainisfixedtotheground,thekinematicchainbecomesamechanism.Ifallthepairsinamechanismarelowerpairs,themechanismiscalledalowerpairmechanism.Ifamechanismhasoneormorehigherpairs,themechanismiscalledahigherpairmechanism.Fig.2-9Mechanismincludinghigherpair(高副機構)

Fig.2-9showsahigherpairmechanisminwhichlinks1and2areconnectedbyahigherpairatpointC.2.2SchematicDiagramofMechanisms1.SchematicDiagram

Asimplediagraminwhichthelinksandpairsarerepresentedbysomesimplelinesandpairsymbolstodescribethecompositionofamechanismiscalledaschematicdiagramofmechanisms.Theschematicdiagramtakesoneortwoforms:aschematicdiagramandscaledschematicdiagram.Aschematicdiagramisproportionalbutnotexactlytoscale,whileascaledschematicdiagramrequiresa“strippeddown”stickdiagram,“strippeddown”stickdiagramisusuallyusedforfurthermotionanalysisandforceanalysis.2.SymbolsofCommonUsedLinksandPairs

ThespecialsymbolsusedinaschematicdiagramofmechanismsarelistedinTab.2-1.3.TheProcedureofDrawingaSchematicDiagramofaMechanism

(1)Mechanismnomenclature1)Frame.Linkwhichisfixedinamechanism.2)Drivinglink.Linkactedbythedrivingforceinamechanism.3)Drivenlink.Alltheothermovinglinksexcepttheframeandthedrivinglinksinamechanism.4)Couplerorconnectedrod.Linkswhicharenotconnectedwiththeframeinamechanism.

Fig.2-10Mechanismnomenclature(機構術語)

1、3—linkconnectedframe(連架桿)2—coupler(連桿)4—frame(機架)(2)Theprocedureofdrawingaschematicdiagram1)Findoutthedrivinglinksandthedrivenlinks.2)Runthemechanismslowlyforawhile,thenstopitatasuitableposition,andobserveitscomposition.3)Findoutthenumberoflinksandthenumberofpairs,anddeterminethetypeofpairsfrominputlinktotheoutputlink.4)Theplaneonwhichmostlinksmovecanbeselectedasadrawingplane.5)Thedimensionsbetweentwopairsandtheotherkinematicdimensionsmustbemeasured,thenselectproperscaletodrawthesketch.Example2-1Fig.2-11ashowsapump.Drawaschematicdiagramofthepump.

Fig.2-11Schematicdiagramofthepump(泵的機構運動簡圖)

1—eccentricdisk(偏心輪)2—coupler(連桿)3—slider(滑塊)4—frame(機架)Example2-2Fig.2-12ashowsashaper.Drawaschematicdiagramoftheshaper.

Fig.2-12Shaperanditsschematicdiagram(牛頭刨床及其機構運動簡圖)

1、2—gear(齒輪)3—block(滑塊)4—rocker(擺桿)

5—link(連桿)6—slidebar(滑枕)7—frame(機架)1.GrueblersEquation(1)DegreeoffreedomofalinkDegreeoffreedomisalsocalledthemobility,anditcanbedefinedasthenumberofindependentcoordinatesrequiredtodetermineitsposition.

(2)ConstraintsofakinematicpairTheturningpairshowninFig.213bhastwoconstraints.

(3)DegreeoffreedomofakinematicpairItcanbedefinedasthenumberoftheindependentrelativemotion.

2.3DegreeofFreedomofPlanarMechanismsFig.2-13Constrainsofpair(運動副的約束)(4)DegreeoffreedomofaplanarmechanismInaplanarmechanism,thefixedlinkhaszerodegreeoffreedom;eachmovinglinkhas3d.o.f.,eachlowerpairhas2constraintsandeachhigherpairhas1constraint.Wesupposethattherearemovinglinksn,lowerpairspl,higherpairsph,thenthedegreeoffreedominaplanarmechanismisasfollows:F=3n-2pl-phExample2-3DeterminethedegreesoffreedomofthemechanismsshowninFig.2-14.

Fig.2-14Thecaculationofdegreeoffreedom(自由度計算)2.ConditionsHavingPredictableMotioninaMechanismFig.2-15Conditionsofcausingdefiniteandpredictablemotions(機構具有確定運動的條件)Thedegreeoffreedomofamechanismisthenumberofindependentcoordinatestodefineitsposition,andisalsothenumberofinputlinkswhichneedtobeprovidedinordertocreateapredictableoutputmotion3.PointsforAttentionWhenCalculatingDegreeofFreedom(1)RedundantdegreeoffreedomSometimes,oneormorelinksofamechanismmaybemovedwithoutcausinganymotiontotheotherlinksofthemechanism.

Fig.2-16Partialdegreeoffreedom(局部自由度)(2)MultiplepinjointsTwolinksareconnectedtogetherbyonlyoneturningpair,whichisillustratedinFig.2-17&2-18.Fig.2-17Multiplepinjoints(復合鉸鏈)Fig.2-18Examplesofmultiplepinjoints(復合鉸鏈的示例)(3)RedundantconstraintsSometimes,amechanismmayhaveoneormoreredundantconstraintswhichdonoteffectthemovementoflinks,oramechanismmayhaveoneormorelinkswhichdonotintroduceanyextraconstraint.Fig.2-19Redundantconstrainintheparallel-crankmechanism(平行四邊形機構的虛約束)1)Twolinksareconnectedbyseveralturningpairsandtheiraxesarecoincident.

2)Twolinksareconnectedbyseveralslidingpairsandtheirguidelinesareparallel.

3)Twolinksareconnectedbyseveralhigherpairsandtheircommonnormallinesarecoincident.

4)Redundantlinks.

theredundantconstraintsandlinks:Fig.2-20Redundantconstrainsofturningpairs(轉動副的虛約束)Fig.2-21RedundantconstrainsofSlidingpairs(移動副的虛約束)Fig.2-22RedundantconstrainsofHigherpairs(高副機構的虛約束)Fig.2-23Redundantconstraints(虛約束)Fig.2-24Redundantconstrainproducedbyconnectingtwoequidistancepoints

(連接等距點產生的虛約束)Example2-4CalculatethedegreeoffreedomofthemechanismshowninFig.2-25.Fig.2-25Degreeoffreedomofthecomplexmechanism(復雜機構的自由度)Example2-5CalculatethedegreeoffreedomoftheshearingmechanismshowninFig.2-26a.Fig.2-26Degreeoffreedomoftheshearingmechanism(剪床機構的自由度)1.LinkGroupAnalysis(1)DrivinglinkThedrivinglinkmayrotateaboutitsaxisortranslatealongaguideline,andithasonedegreeoffreedom.Fig.2-27showstwokindsofdrivinglinks.

2.4MechanismAnalysisandInnovationFig.2-27Drivinglinks(原動件)Fig.2-28Diridingoflinkgroups(拆分桿組)(2)LinkgroupAnymechanismconsistsofdrivinglink,drivenlinkandaframe.Thenumberofdrivinglinksisequaltothenumberofdegreesoffreedom.Forexample,Fig.2-28showsaonedegreeoffreedommechanism,andthedrivinglinkiscrankAB.AfterthedrivinglinkABandtheframehavebeenremovedfromthemechanism,thedegreeoffreedomofthelinkgroupBCDEFiszero.Fig.2-29ClassⅡlinkgroups(Ⅱ級桿組)

Whentherearetwolinksandthreelowerpairsinthelinkgroup,thislinkgroupiscalledclassⅡlinkgroup.Thereisonepairwhichconnectstwolinksinthelinkgroupandtwopairswhichwillconnecttheotherlinks.TheclassⅡlinkgroupsareillustratedinFig.2-29.Fig.2-30ClassⅢlinkgroups(Ⅲ級桿組)

Whentherearefourlinksandsixlowerpairsinthelinkgroup,thelinkgroupiscalledclassⅢlinkgroup.Therearethreepairswhichconnectlinksinthelinkgroupandthreepairswhichwillconnecttheotherlinks.ThecommonclassⅢlinkgroupsareillustratedinFig.2-30.Fig.2-31ClassⅣlinkgroup(Ⅳ級桿組)AnotherlinkgroupinwhichtherearefourlinksandsixlowerpairsisillustratedinFig.2-31.Therearefourpairswhichconnectlinksinthelinkgroupandtwopairswhichwillconnecttheotherlinks.WecalledthislinkgroupasclassIVlinkgroup.Thiskindoflinkgroupisusedrarely.Fig.2-32Shapermechanismdesign(牛頭刨床的組合過程)2.PrincipleofMechanismComposition

Anymechanismcanbedesignedbyconnectingbasiclinkgroupinwhichthedegreeoffreedomiszerowiththedrivinglinkandtheframe.ThisisillustratedinFig.2-32.

3.ReplacementofHigherPairbyLowerPairsWhenamechanismincludingahigherpairmustbeanalyzed,wecanreplacethehigherpairbylowerpairs.Therefore,wecanusetheprincipleoflinkgrouptoanalyzethemechanismconnectedwithlowerpairs.Asweknowthatahigherpairhasoneconstraint,andalowerpairhastwoconstraints,sothatwecanusealinkwithtwoturningpairstoreplacethehigherpair.Fig.2-33Replacementofhigherpairbylowerpairs(高副低代)Fig.2-33showssomehigherpairmechanisms;thehigherpaircanbereplacedbyonebinarylinkwithtwoturningpairs.ThecentersofcurvaturesatthecontactpointPofthetwoprofileslieatC1andC2;thelinkC1C2withturningpairsatC1andC2replacesthehigherpair.4.StructuralAnalysisofPlanarMechanism

Whendeterminingaclassofamechanism,thefollowingproceduremustbenoticed.1)Removetheredundantdegreeoffreedomandredundantconstraints.2)Thehigherpairsarereplacedbylowerpairs.3)Calculatethedegreeoffreedom,anddeterminethedrivinglinks.4)FindouttheclassIIlinkgroupsfirstandremovethemfromthemechanism.IfthereisnotanyclassIIlinkgroup,theclassIIIlinkgroupmustbeconsidered.5)Thelastlinkswhichhavebeenleftmustbethedrivinglinks,andtheyareequaltothenumberofthedegreesoffreedom.Fig.2-34Mechanismanalysis(機構的分析)Example2-6DeterminetheclassoftheshearingmechanismasshowninFig.2-26.

Example2-7DeterminetheclassoftheshapermechanismshowninFig.2-35.

Fig.2-35Shapermechanismanalysis(牛頭刨床機構的分析)Fig.2-36SeriesmechanismconsistedbyclassⅡlinkgroups(Ⅱ級桿組組成的串聯機構)5.InnovativeDesignofMechanism(1)DesignatandemmechanismWhenaddingaclassⅡlinkgroupshowninFig.2-36btothedrivinglinkshowninFig.2-36aandtheframe,wecanobtainafourbarlinkageshowninFig.2-36c.IfaddinganotherclassⅡlinkgroupshowninFig.2-36dtothelinkDCandtheframe,wecanobtainasixbarmechanismshowninFig.2-36e.

IfaddingclassⅢlinkgrouptothedrivinglinkandtheframe,wecanobtainaclassⅢmechanism;seetheFig.2-37.Fig.2-37SeriesmechanismconsistedbyclassⅢlinkgroups(Ⅲ級桿組組成的串聯機構)(2)DesignaparallelmechanismWhenconnectingaclassⅡlinkgroupshowninFig.2-38btotwodrivinglinksshowninFig.2-38a,wecanobtainafivebarlinkageshowninFig.2-38c.Thefivebarlinkageisaparallelmechanism.Fig.2-38ParallelmechanismconsistedbyclassⅡlinkgroups(Ⅱ級桿組組成的并聯機構)Fig.2-39ParallelmechanismconsistedbyclassⅢlinkgroups(Ⅲ級桿組組成的并聯機構)IfconnectingaclassⅢlinkgrouptothreedrivinglinks,wecanobtainanotherparallelmechanism;seetheFig.2-39.Thiskindofmechanismcanbeusedtoparallelrobot.

Chapter3KinematicAnalysisofPlanarMechanisms1.PurposeofKinematicAnalysis(1)TheworkspaceofamechanismisnecessarybymeansofanalysisofpositionsortracingpathFig.3-1ashowsainternalcombustionengine,inwhichthestrokeofthepistoncanbeusedtodesignthelengthofthecylinder,andpathofthecouplercanbeusedtodesigntheinternaldimensionsoftheengineblock.(2)DeterminethevelocitiesandaccelerationsoflinkstoinvestigatetheworkingcharacteristicsofamachineFig.3-1bshowsashapermechanism.Theramintheworkingstrokedemandsconstantvelocityapproximatelyandthevariationofaccelerationsisaslittleaspossible.Sothevelocityanalysisisveryimportanttodesignashapermechanism.(3)MotionanalysisisneededforthedynamicforcecalculationOnceapositionanalysisisdone,thenextstepistodeterminethevelocitiesofdrivenlinksortracingpointsofinterestinthemechanism.3.1IntroductionFig.3-1Kinematicanalysisofmechanisms(機構的運動分析)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

(兩構件用運動副連接時的瞬心位置)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(四桿機構的瞬心)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構速度分析中的應用)Fig.3-7Applicationofinstantaneouscentersforcam

mechanism(瞬心法在凸輪機構中的應用)Example3-3Fig.3-7showsacammechanismwithaflatfollower.Theangularvelocityofcam1isknownanditisrequiredtofindthevelocityofthefollower3.

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

3.3KinematicAnalysisbyGraphicalMethodFig.3-8Relativevelocityoftwopointsonalink

(同一構件上兩點之間的速度關系)Fig.3-9Relativevelocityofcoincidentpointon

separatelinks(兩構件重合點處的運動關系)(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(鉸鏈四桿機構的運動分析)Example3-4Fig.3-10ashowsafourbarlinkage;allthedimensionsofthelinksandangularpositionofthedrivinglinkABareknown.WhenthecrankABrotatescounterclockwisewithanangularvelocityω

1,determinetheangularvelocitiesω2,ω3,thevelocityofthepointEonthelink2andangularaccelerationsα

2,α

3.

Fig.3-11Kinematicanalysisofaguide-barmechanism(導桿機構的運動分析)Example3-5Fig.3-11ashowsaguiderbarmechanism;allthedimensionsofthelinksandangularpositionofthedrivinglinkABareknown.WhenthecrankABrotatescounterclockwisewithanangularvelocityω1,determinetheangularvelocitiesω

2,ω3andtheangularaccelerationα2andα3.

3.SomeKeyPointsofMotionAnalysis1)TheCoriolisaccelerationofcoincidentpointsontwodifferentlinksmustbediscriminatedcorrectly.

2)Whenwewouldestab