Chapter5SynthesisofPlanarLinkages1.TheCharacteristicsofPlanarLinkages1)Alinkagehasasimplestructureanditiseasytomanufacture,soithasalowcost.2)Thepressureinalowerpairwithsurfacecontactislower,sothelinkagehasalargeloadcapacity.3)Alinkagecanachievevariousmotionsbydesigningitsdimensionsofthelinks.4)Alinkagecantransmitmotionwithinalongdistance.5)Itisdifficulttoperformprecisionmotion;thisisadisadvantageoflinkages.5.1CharacteristicsandTypesofPlanarLinkagesFig.5-1Typesoffour-barlinkages(鉸鏈四桿機構的類型)2.TypesofPlanarLinkagesFig.5-2Typesoffour-barlinkageswithaslidingpair(含有一個移動副的四桿機構類型)(1)CrankrockerlinkageIfafourbarlinkageshowninFig5-1aisdesignedsothatlink1canrotatecontinuouslywhilelink3onlyoscillatesthroughanangle,itiscalledacrankrockerlinkage.(2)DoublecranklinkageBothlink2andlink4inFig5-1bcanrotatecontinuouslyrelativetotheframe1;thislinkageiscalledadoublecranklinkage.(3)DoublerockerlinkageBothlink2andlink4inFig5-1ccannotrotatecompletelyrelativetotheframe3;thislinkageiscalledadoublerockerlinkage.(4)ParallelcrankfourbarlinkageIfinafourbarlinkage,twooppositelinksareparallelandequalinlength,asshowninFig5-1d,cranks2and4alwayshavethesameangularvelocity.Thismechanismiscalledaparallelcrankfourbarlinkage.Itiswidelyused,suchasinthecouplingoflocomotivewheels,inpantograph,etc.(5)IsoscelestrapeziumlinkageIftworockersofadoublerockerlinkageareequalinlength,seetheFig5-1e,thislinkageiscalledanisoscelestrapeziumlinkage;itcanbeusedinthesteeringmechanisminautomobiles.(6)SlidercranklinkageThislinkageisshowninFig5-2a,inwhichsidelink1isacrank,andtheothersidelink3isaslider.TurningpairsAandBarefullturningpairs,andCisoscillatingturningpair.(7)RotatingguidebarlinkageIfboththesidelink2andsidelink4canrotatecompletely,thislinkageiscalledarotatingguidebarlinkageshowninFig5-2b,inwhichthecouplerbecomesasliderinshape.(8)RockingblocklinkageThecrankrotatescompletelywhiletheblockonlyoscillatesaboutitspivotcenterC,thislinkageiscalledrockblocklinkageshowninFig5-2c.Itiswidelyusedinhydrauliccylindermechanism.(9)SlidingguidebarlinkageIftheguidelink4reciprocatesalongtheaxisofthefixedblock,andtheothersidelink2oscillatesaboutthepivotC,thelinkageiscalledaslidingguidebarlinkageshowninFig5-2d.Itcanbeusedinsomewaterpumps.(10)RockingguidebarlinkageIfthelink2canrotatesaboutpivotBcompletely,andthelink4oscillatesaboutthepivotAshowninFig5-2e,thislinkageiscalledarockingguidebarlinkage.Itisusedinmachinetoolssuchasshapers.(11)DoubleslidercranklinkageInafourbarlinkage,iftwosidelinks1and3becomesliderswhicharereciprocatingalongeachaxisofcrossframe,andthetwopairsofthesamekindareadjacent,itisknownasadoubleslidercranklinkageshowninFig5-3a.Thislinkagecandrawellipses.(12)DoublerotatingblocklinkageIftwoslidersarerotatingcompletelyabouttheirpivotsAandBrelativetotheframe2,thislinkageiscalledadoublerotatingblockmechanismshowninFig5-3b.ItcanalsobecalledOldhamlinkage,anditisusedtoconnecttwoshaftshavingparallelmisalignment.(13)SinelinkageThislinkagecanalsobecalledaScotchyokelinkage,inwhichthecrank2rotatesaboutitspivot.Acompletely,andlink4reciprocatesinthefixedlink1toproduceasimpleharmonicmotion.ItisshowninFig5-3c.

(14)TangentlinkageInatangentmechanism,thesidelink2oscillatesaboutitspivotA,theotherlink4reciprocatesalongitsaxis,andthelink3isasliderinshape.ItisalsocalledtheRapsonsslidelinkage.Thedisplacementoflink4hasatangentmotion.Fig.5-3Typesoffour-barlinkageswithtwoslidingpairs

(含有兩個移動副的四桿機構類型)3.EvolutionandMutationofPlanarLinkages(1)InversionofafourbarlinkageDifferentmechanismscanbeobtainedbyfixingdifferentlinksofamechanism.Thisisknownasinversion.Theprincipleisthattherelativemotionbetweenlinksofafourbarlinkagedoesnotchangeindifferentinversion.

Fig.5-4Evolutionfromturningpairtoslidingpair(轉動副向移動副的演化)(2)ConvertingaturningpairintoaslidingpairIftheradiusofthepinDisincreasedtoalengthoflDC,andtherocker3ismadeasliderwithacurvewhichradiusislDC,thenthismechanismbecomesacurveslidercrankmechanismshowninFig5-4b.IfthelDCweremadeinfiniteinlength,thenpointCwouldhaverectilinearmotionandlink3couldbereplacedbyaslider,asshowninFig5-4c.Fig.5-5Eccentricdiskmechanism(偏心盤機構)(3)ExpansionofpinsizeinaturningpairIfweincreasethesizeofthecrankpinBshowninFig5-5auntilitislargerthanthelengthofthecrank,thisenlargedcrankpiniscalledaneccentricdiskandcanbeusedtoreplacethecrankshowninFig5-5a.ThecrankconsistsofacirculardiskwithcenterB,whichispivotedoffcenteratAtotheframe.Thediskrotatesinsidetheringendofcoupler2.4.ApplicationsofPlanarLinkages(1)ApplicationsoflinkagesconsistedofturningpairsFig5-6aisajawcrusher.Itconsistsofacrankrockerlinkage.Theconnectingrodisthemovingjawusedtocrushtherocks.ItskinematicdiagramisshowninFig5-6b.Fig5-6cshowsadoughmixingmachine;itconsistsofacrankrockerlinkagealso.Fig.5-6Applicationsofcrank-rockerlinkage(曲柄搖桿機構的應用)

1—eccentricdisk(偏心盤)2—beltwheel(帶輪)3—movingjaw(動鄂)4—movingjawboard(動鄂板)

5—fixedjawboard(靜鄂板)6—rocker(擺桿)7—spring(彈簧)Fig.5-7Applicationsofdouble-rockerlinkage(雙搖桿機構的應用)Fig5-7ashowsacranewhichisusedinports.Itconsistsofadoublerockerlinkage.ThepointEoncouplercanmovealongastraightline.ornearlyalongastraightline,thereforetheliftedbodymaymovesteadilyalongthispath.Fig.5-8Applicationsofdouble-cranklinkage(雙曲柄機構的應用)Fig5-8ashowsainertiavibratingscreen.Itisadoublecranklinkage,andisalsocalledadraglinklinkage.IfthedriverACrotatescounterclockwisewithconstantangularvelocity,theslidermakesaslowstrokeandreturnswithaquickstroke.Theinertiaforcescancausevibrationofthescreentoperformseparationofthematerials.Fig5-8bshowsaparalleldoublecranklinkageusedinthecoupleroflocomotivewheels,andFig5-8cshowsahydraulicliftwithtwosetsofparalleldoublecranklinkages.(2)ApplicationsofslidercranklinkageSlidercranklinkagesareusedwidelyinvariousmulticylinderengines.Fig5-9aisaVenginewitheightcylinders.Thisengineconsistsofeightslidercranklinkages.Fig5-9bshowsashearingmachine.Itconsistsofoneslidercranklinkage.Fig.5-9Applicationsofslider-cranklinkage(曲柄滑塊機構的應用)(3)ApplicationofcranksliderquickreturnlinkageFig5-10aisashaper;itconsistsofgearmechanismandslidingguidebarlinkage.Fig5-10bisashapertoo;itconsistsofrotatingguidebarlinkage.Fig.5-10Applicationsrockingguide-barlinkageandrotatingguide-barlinkage

(擺動導桿機構和轉動導桿機構)Fig.5-11Applicationsofslidingguide-barlinkageandrocking-blocklinkage

(移動導桿機構和曲柄搖塊機構的應用)

1—handle(手柄)2—link(連桿)3—cylinder(簡體)4—plunger(活塞)Fig5-11ashowsahandpump,inwhichlink4ismadeintheformofacylinderandaplungerfixedtothelink3reciprocatesinit；itconsistsofslidingguidebarlinkage.Fig5-11bshowsaselfloadingtruck,inwhicharockingblocklinkageisusedtoperformselfloadingwork.

(4)ApplicationoffourbarlinkageswithtwoslidingpairsFig5-12ashowsanellipticaltrammelinwhichthefixedlink1isintheformofguidesforsliders2and4.Fig5-12bshowsanOldhamscoupling,anditconsistsofadoublerotatingblocklinkageshowninFig5-3b.Fig5-12cshowsamechanismwhichisusedtodrivetheneedleofasewingmachine;itisasinelinkage.Fig.5-12Applicationsoffour-barlinkageswithtwoslidingpairs(含有兩個移動副機構的應用)1.GrashofCriteria5.2FundamentalFeaturesofFourbarLinkagesFig.5-13Grashoflaw(曲柄存在條件)Fig5-13showsafourbarlinkageABCD,andifthelinkABcanrotatethroughafullrevolutionwithanangularvelocityωclockwise,itmustpassthroughthepositionsAB1andAB2.Thelink1whichcanrotatethroughafullrevolutionmustbetheshortestlink;thesumofthelengthofshortestlinkandthelongestlink(theremustbealongestlinkamongb,c,d)cannotbegreaterthanthesumoftheremainingtwolinks.

TheGrashofcriteriacanbestatedasfollowing:

1)Alinkage,inwhichthesumofthelengthoftheshortestandlongestlinksislessthanthesumofthelengthoftheothertwolinks,musthaveacrank.2)Thecrankmustbetheshortestlink.2.QuickreturnMotion

Whenthecrankrotatesthroughafullrevolutionwithanangularvelocityωclockwise,andtherockeronlyoscillatesthroughanangleψ,thecrankandcouplerwillformastraightlineateachextremeposition.Fig-14showsacrankrockerlinkageinthetwoextremepositionsofthedrivenlink.Fig.5-14Quick-returncharacteristics(急回特性)3.PressureAngleandTransmissionAngleFig.5-15Transmissionangleandpressureangle(傳動角和壓力角)Thepressureangleαistheacuteanglebetweenthedirectionofthestaticforcetransferredthroughthecouplerandtheabsolutevelocityoftheoutputlinkattheconnectedpoint,suchasatC.Sincethecouplerisatwoforcelink,thedirectionofthestaticforceofthecouplerisalongthelineofitspinjoints.

TheacuteangleγbetweenthecouplerandoutputlinkisknownastransmissionangleshowninFig5-15.Noticethatinthiscase,α+γ=90.Fig.5-16Transmissionangleinaslider-cranklinkage(曲柄滑塊機構的傳動角)ThetransmissionangleinaslidercranklinkageisshowninFig5-16.4.DeadPointFig5-17showsacrankrockerlinkage.Whenthecrankrotatescompletely,thereisnodangerofthelinkagelocking,becausethetransmissionanglecannotbezeroatanypositions.However,iftherockerbecomesadrivinglink,thecrankisadrivenlink,andthentherockermovestotheextremepositionDC1andDC2,thetransmissionanglesbecomezero,thelinkagelockingwilloccur.Wecallthesepositionsasdeadpointsortogglepositions.Therockerwillthenhavetobedriventogetthelinkageoutofthetoggleposition.Fig.5-17Deadpoints(死點)Fig.5-18Overcomedeadpointbyusingflywheel

(利用飛輪克服死點)

1—flywheel(飛輪)2—crankshaft(曲軸)

3—camshaft(凸輪軸)4—valve(氣門)

5—piston(活塞)6—connectlink(連桿)Fig5-18showsasinglecylinderenginewithfourstrokecycle.Theflywheelcancarrythepistonthroughtheenginesexhauststroke,intakestrokeandcompressionstroke.Fig5-19isadrivingmechanismofthelocomotivewheels;thephaseangleofthecranksis90°.Ifoneofthemisatthedeadposition,theothercanbedriveneasily.Fig.5-19Overcomedeadpointsbyusingdifferentphase(錯位排列克服死點)Fig.5-20Self-lockingclamp(自鎖夾具)Fig5-20showsaclampingdevice,inwhichafterasmallforceFappliedonthecouplerhasbeenremoved,thereactionforceoftheclampedworkpiecetothelinkABcannotdrivethelinkagetomoveatall.Wecanalsodesignmanydevicesusingthedeadpointposition.1.Introduction（1）MotiongenerationFig5-21ashowsaoverturningdeviceusedinacastingworkshop.ThisdeviceisdesignedtoguidethesandboxwhichisfixedtothecouplerintwopositionsB1C1andB2C2.Fig5-21bshowsaspeedchangedevicewhichiscontrolledbyhand.Itisdesignedtoactualizetheinputandoutputlinksthroughthethreepositions.

（2）PathgenerationAcoupleristhemostinterestinglinkinamechanism.5.3SynthesisofFourbarLinkagesFig.5-21Problemsofsynthesisofafour-barlinkage1(四桿機構設計基本問題1)Fig5-22ashowsafourbarlinkagewhichisontheextremepositions,andfromthisfigurewehave:a=（AC1－AC2）/2Thisformulaforcalculatingthelengthofthecrankisveryusefultodesignafourbarlinkage.

Fig.5-22Problemsofsynthesisoffour-barlinkages2(四桿機構設計基本問題2)Fig.5-23Guidingabodythroughtwocouplerpositions(按連桿的兩個位置設計四桿機構)1)Twoprescribedlinkpositions.Fig5-23ashowstwopositionsofacouplerBC.Itisdesiredtodesignamechanismwhichwillcarrythelinkthroughthesetwopositions.(1)DesignofafourbarmechanismforspecifiedcouplerpositionsGuidingabodythroughtwoorthreelinkpositionsareoftenusedtodesignafourbarlinkage.2.GraphicalSynthesisofFourbarLinkage

2)Threeprescribedlinkpositions.Theprocedureissimilartotheaboveprocesses(Fig5-24).Fig.5-24Guidingabodythroughthreecouplerpositions(按連桿的三個位置設計四桿機構)Fig.5-25Guidingabodythroughanumberoflinkplanepositions

(按連桿平面位置設計四桿機構)3)Twoorthreeprescribedpositionsofthecouplerplane.ThreeprescribedpositionsofacouplerplaneareshowninFig5-25.

Fig.5-26Principleofinversion(反轉法的原理)（2）Designofafourbarlinkageforcorrelatedangularpositionsofinputandoutputlinks1)Inversionmethod.Fig5-26showsafourbarlinkage;themotionoftheinputlinkABfromtheφ1toφ2causesamotionoftheoutputlinkDCfromψ1toψ2.Fig.5-27Coordinationofthepositionsoftheinputandoutputlinks

(按兩連架桿的三組對應位置設計四桿機構)2)Threeprescribedangularpositionsoftheinputandoutputlinks.Fig5-27illustratesaprobleminwhichitisdesiredtodeterminethedimensionsofalinkageinwhichtheoutputlinkistooccupythreeangularpositionsψ1,ψ2andψ3correspondingtothethreegivenpositionsφ1,φ2andφ3oftheinputlink.

（3）DesignofafourbarlinkageforquickreturnmotionSomeexternalworkisbeingdonebythelinkageontheforwardstroke,andthereturnstrokeneedstobeaccomplishedasrapidlyaspossiblesothatamaximumtimewillbeavailableforworkingstroke.1)Crankrockerlinkage.SynthesizingacrankrockerlinkageforspecifiedvalueofK,thelengthofrockerDCandtheangleψbetweenthetwoextremepositionsoftherockerareknown.TheproceduresareinFig5-28.

Fig.5-28Synthesisofacrank-rocker

linkageforagivenK(已知K設計

曲柄搖桿機構)Fig.5-29Synthesisofaslider-crank

linkageforagivenK(已知K值設計

曲柄滑塊機構)2)