It is shown that perfect balance can be achieved by the use of a closecoiled spring whose free length is e57472ectively zero and whose sti57472ness is chosen appropriately The two degreeoffreedom balancing mechanism commonly seen in desk lamps but u ID: 56214
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Thespring-and-leverbalancingmechanism,GeorgeCarwardineandtheAnglepoiselampMJFrenchMBWiddenEngineeringDepartment,FacultyofAppliedSciences,LancasterUniversity,UKAbstract:Twoshortanddirectmethodsofexactanalysisofthespring-and-leverbalancingmechanismarepresented.Itisshownthatperfectbalancecanbeachievedbytheuseofaclose-coiledspringwhosefreelengthise€ectivelyzeroandwhosesti€nessischosenappropriately.Thetwo-degree-of-freedombalancingmechanism,commonlyseenindesklampsbutusefulinmanyother verticalplane.Theuseofspringsthatdonotlieinaverticalplaneintroducesundesirablesideloadsandrequiresball-jointsinsteadofpinsattheendsofsprings;thusitisdiculttoseewhyanyonewouldwanttousethismethod.2ANALYSISOFTHESINGLE-DEGREE-OF-FREEDOMMECHANISMFigure2aisaschematicdiagramofasimplespring-and-leverbalancingmechanism.AmassisatpointZontheendofanarmCAZ,pivotedatAtoamemberAB.Thewholemechanismisinaverticalplane,andABisheldinaverticalposition.Theaimistosupporttheweightofthemassaccuratelyoverawiderangeofpositions.Twomethodsofsolutionofthisproblemarepresented.2.1Method1Itwouldbepossibletobalancethearmbymeansofaconstant,verticallydownwardforceatCequalto(seeFig.2aforthelengths).ThebalancingspringCBneedstoexertthesamemomentaboutAasthisverticalforce,forallpositionsofthearm.LetaforceinthedirectionACbeaddedtothespringforcesuchthattheresultantofthesetwoforcesisvertical.Then,sincehasnomomentaboutA,thisverticalresultanthasthesamemomentaboutAasforcehas;forexactbalancetheresultantmustalwaysbeequaltoNow,considerABCasatriangleofforcesusedtoobtainthisresultant.Theresultant,representedbyAB,isverticalandwillbeconstantiftheforceinthespringisproportionaltoCB,i.e.tothelengthofthespring.Thusthefreelengthofthespring,i.e.thelengthwhenthereisnotension,mustbezero(seeSection2.3).Finally,tomaketheresultantequalto,thescaleofthetriangleofforcesmustbechosensothatthelengthrepresentstheforce;thenanextensionofthespringwouldgivethisforce,andthesti€nessgivenby 2.2Method2Analternativesolutionisasfollows:AreaoftriangleABC perpendiculardistanceofAfromlineBC Fig.1Anglepoiselampoftheoriginaldesign502MJFRENCHANDMBWIDDENProcInstnMechEngrsVol214PartCIMechE2000 ThereforethearmaboutAoftheforceinthespringBC,inotherwordstheperpendiculardistanceofAfromthelineBC,is areaoftriangleABCHowever,theareaofthetriangleABCisequalto ThusthearmaboutAoftheforce Therequiredmomentis;therefore,forbalance, Fbcsinaˆmgrsin…6†orFˆ asbefore.Inotherwords,thetensioninthespringisproportionaltoitsoveralllength;thespringsti€nessis.Forthetensiontobebroughttozero,mustalsogotozero,i.e.thespringhaszerofreelength.Shouldthereaderbeinanydoubtaboutthis,putequaltotheunstretchedlengthofthespring.ThenTherefore,forexactbalance,substitutingforequation(7), Thelengthclearlyvarieswith,butallthequantitiesinthefractionontheright-handsideofequation(10)areconstants.Theonlywaythatequation(10)canholdforallisiftheright-handsideisindeterminate,i.e.ifandarebothzero.2.3Zero-free-lengthspringsItiswellknownthatapre-tensioninaspringcanbeinducedbyimposingatwistonthewireasthespringiswound[].Thecoilsofthespringthenremaintightlycloseduntiltheappliedforcereachesacertainvalue, Fig.2Simplespring-and-leverbalancingmechanismTHESPRING-AND-LEVERBALANCINGMECHANISM,GEORGECARWARDINEANDTHEANGLEPOISELAMP503IMechE2000ProcInstnMechEngrsVol214PartC andthee€ectiveunstretchedlengthislessthantheinitialclose-woundlength.Itisquitepossibletoreducethee€ectiveunstretchedlengthtozero,oreventoanegative2.4AlternativearrangementwiththespringaboveAIfdesired,CmaybeplacedonthesamesideofAasZ,butthenBmustbeaboveA(Fig.2b).Itcanbeshownbysimilarmethodsthat,forperfectbalance,therequirementsarethesame,namelythattheunstretchedlengthofthespringmustbezero,andthesti€nessofthespringmustbegivenby3TWO-DEGREE-OF-FREEDOMDESIGNS3.1Con®gurationwithspringsonthearmsAtwo-degree-of-freedomsystemmaybemadeusingaparallelogram(orequivalent)mechanismtocarrytheverticalreferencefromtheinnertotheouterjoint.OnesucharrangementisshowninFig.3a,wherethebasicschemeisthatofFig.2b,i.e.withConthesamesideofAastheloadinbothcases.AandAcongruentrigidlinks,andtheparallelogrammaticfour-barchainLmaintainsAparalleltoAi.e.vertical.ThespringBfortheoutersystemneedstobedesignedaccordingtotherequirementssetoutinSection2above.Asfarastheinnersystemisconcerned,theweightatZcanbereplacedbyadownwardforceatAtogetherwithaclockwisecouple.TheverticalatAissupportedbythespringB,whichmusthavezerofreelengthandsti€nesscalculatedassetoutinequation(1).ThecoupleiscarriedentirelybytensionandcompressioninthearmsLandArespectively,andimposesnoloadonthespringB.ThisisevidentsincethelineAremainsvertical;therefore,thecoupledoesnoworkastheangleisvaried.Hence,oncethesti€nessoftheinnerspringhasbeencalculatedtobringtheforceatAequilibrium,thelinkageisbalancedforallvaluesofthe3.2Con®gurationwithallthespringsatthebaseTheAnglepoisedesignhasthemoreeleganttwo-degree-of-freedomsystemshowninFig.3b,withallthespringsatthebaseandonlyonelinkAB.Thefour-barchainHJEDisparallelogrammaticandthelengthsCDandAEareequal,sothatthepointsCandA,althoughnotdirectlylinked,lieonalineparalleltoDEandHJZ.Thelengthsand*andthereforethetensionsinthetwospringsthendependonlyonrespectively;thusthetwoarmsCAEJandHJZaree€ectivelydecoupledasfarastheirstaticequilibriumisconcerned.Thisisborneoutbytheargumentbelow.Bymethod2ofSection2.2,thearmsofthetwospringsaboutpointAarerespectively b1c1sin1a1and BymomentsaboutpointA,forbalance, b1c1sin1a1‡F2 whereandarethetensionsinsprings1and2respectively.Equation(12)istoholdforallvaluesofand,butthiscanonlybesoifthetermsintheequationinvolvingalwaysbalanceandthetermsinvolvingalsoalwaysbalance.Therefore mgr1b1c1a1;F2ˆ (Thereareotherwaysofapproachingthiswhichareequallygood,e.g.usingenergymethods.)AsinSection2.2,thetensionineachspringispro-portionaltoitsoveralllength,i.e.itmusthavezerounstretchedlength.Thesti€nessesofthetwospringsare mgr1b1c1;k2ˆ 4DISTRIBUTEDMASSSofartheweightsofthearmshavebeenneglected.Inpracticetheseareoftenrelativelysmall,buttheymaynotbeinsigni®cant.Fordesignofthesingle-degree-of-freedommechan-ism,referringagaintoFig.2a,thecentreofmass,G,ofthearmandloadcombinedshouldbeusedinplaceoftheendofthearm,Z.PointsC,AandGshouldbeinastraightline,andisnowthedistanceAG.Thesti€nessofthespring,,isequalto,anditsfreelengthistobezero,asbefore.Figure4showsthetwo-degree-of-freedommechanismwiththeweightsoftheprincipalpartsincluded.ForexactbalancetheouterpartofthemechanismmusthavepointsH,JandGinastraightline,justasforthesingle-degree-of-freedommechanism.Inthe®gureithasbeenassumed(asislikely)thatthecentreofgravity,G,ofarmCAEJliesonthestraightlinethroughthesefourpoints.Similarly,GhasbeenassumedtolieonthelineChavesimilarmeaningstoinFig.2a.504MJFRENCHANDMBWIDDENProcInstnMechEngrsVol214PartCIMechE2000 Fig.3Two-degree-of-freedombalancingmechanismsTHESPRING-AND-LEVERBALANCINGMECHANISM,GEORGECARWARDINEANDTHEANGLEPOISELAMP505IMechE2000ProcInstnMechEngrsVol214PartC MomentscanbetakenaboutthepointA,asinSec-tion3.2.ThemomentsofthethreeweightforcesareSummingtheseandequatingtothemomentsofthetwospringtensionsgive b1c1sin1a1‡F2 BythesameargumentaswasusedinSection3.2,thetermsinvolvinginthisequationmustbalance,andthoseinvolvingmustalsobalance.Hence …m1R1‡m2r1‡m3R3†gb1c11F2ˆ Onceagain,eachspringistoexertaforceproportionaltoitslength,i.e.thefreelengthofthespringistobezero.Thesti€nessesofthetwospringsaregivenbytheexpressionsinsquarebrackets.IfthecentresofgravityofthearmsCAEJandCweretolieo€thestraightlinesthroughthepivots,itcanreadilybeshownthatatermincoswouldbeintro-ducedintoequation(16);thentheequationwouldbeexactlysatis®edonlyforasingleposition Fig.4Two-degree-of-freedommechanismwithweightsofthearmsincluded506MJFRENCHANDMBWIDDENProcInstnMechEngrsVol214PartCIMechE2000 InmostlatercopiesoftheAnglepoiselamp,itseemsthatprecisionhasbeensacri®cedtocheapnessofman-ufacture,sincethepositionofthecentroidGvariesasthelampisrotated.IntheoriginalAnglepoise,however,thelampismountedintrunnionsonayokesothatthepositionofitsowncentroidremainsunchanged,andexactbalanceismaintained.5CONCLUSIONSThetwo-degree-of-freedomspringbalancingmechanismpatentedbyCarwardinein1933andmadefamousintheclassicAnglepoiselampdesignhasbeenanalysed,includingthee€ectsoftheweightofthemechanismitself,andtheconditionsforexactbalanceestablished.Thesimplicityofthedesignofthismechanismandtheperfectionofitsactionrecommenditforuseinmanyapplications.REFERENCES1French,M.J.EngineeringDesign:theConceptualStage1971(Heinemann,London);2ndedition,1985(DesignCouncilBooks±SpringerVerlag,London).2Hain,K.Springmechanisms.1:forceanalysis;2:pointbalancing;3:continuousbalancing.InSpringDesignand(Ed.N.Chironis),1961,pp.268±275(McGraw-Hill,NewYork).3Nathan,R.H.Aconstantforcegenerationmechanism.Trans.ASME,J.Mechanisms,TransmissionsandAutoma-tionDes.,1985,(4),508±512.4Streit,D.A.Gilmore,B.J.`Perfect'springequili-bratorsforrotatablebodies.InProceedingsoftheASME20thBiennialMechanismsConference,1988,Vol.15(2),pp.487±496(AmericanSocietyofMechanicalEngineers,NewYork).5Walsh,G.J.,Streit,D.A.Gilmore,B.J.Spatialspringequilibratortheory.MechanismsMachineTheory,1991,(2),155±170.6Carwardine,G.Improvementsinelasticequipoisingmechanisms.UKPat.404615(®led4July1932,granted4January1934).7Wahl,A.M.MechanicalSprings,2ndedition,1963(McGraw-Hill,NewYork).8Carwardine,G.Improvementsinelasticforcemechanisms.UKPat.379680(®led21March1931,granted22August9Carwardine,G.Improvementsinequipoisingmechanism.UKPat.417970(®led7October1933,granted16October10Carwardine,G.Improvementsinequipoisingmechanism.UKPat.433617(®led10Februaryand7March1934,granted12August1935).APPENDIXGeorgeCarwardineandtheAnglepoiselampGeorgeCarwardine(pronounced`Car--deen')wasborninBathon4April1887.Thesecondyoungestof12survivingchildren,heattendedBathBluecoatSchoolonascholarshipbutleftwhenaged14(R.Raven,1995,personalcommunication).From1901to1905Carwardineservedhisappren-ticeshipattheWhitingAutoWorksinBath.Followingthishewasemployedinasuccessionofthetown'sengineeringworkshops,allthewhilegainingformalquali®cationsthroughstudyathome.Asayoungman,healsostudiedfortheMinistrywiththeintentionofjoininghiselderbrotherCharlesasamissionaryinChina,butaboutofillnesslefthimun®ttofollowthisvocation.In1912hejoinedtheHorstmannCarCompanyinBathaschargehand,risingby1916tothepositionofbothWorksManagerandChiefDesigner.SecondonlytotheentrepreneurialinventorSidneyHorstmann,Carwardinewaslargelyresponsibleforthedesignofallcarsmanufacturedthere,travellingonoccasiontotheBrooklandsracetrackfortrials.Inabout1924hesetuphisownbusiness,CardineAccessories,inLocksbrookRoad,Bath.Herehedesignedandmanufacturedvariousitemsandcompo-nents,mostnotablyautomobilesuspensionsystems.Ithasbeenalleged(butneverproved)thathissuccessfuldesignforindependentfrontsuspensionwaspiratedbyGeneralMotors.Thelate1920ssawCardineAcces-soriesceaseoperations,nodoubtavictimoftheDepression,andCarwardine'sbriefreturntoHorst-mann.In1931hebecameafreelanceconsultingengineerandinventor.Hehadalreadydisplayedinterestindevelopingver-satilecounterbalancingdevices,intendedtosupportaweightinanypositioninthreedimensions.Thestreamofpatentsregisteredinhisnamebetween1931and1934demonstratesclearlyhowhisideasdeveloped.The®rstpatentinthisseries[]thatrelatesdirectlytothespring-and-leverbalancingmechanismdescribes,asonepossibleembodiment,aspecialformofthemechanisminwhich(referringtothelettersusedinFig.2)thelengthsABandACareequal.AswehaveseeninSection2,thisisanunnecessaryrequirement.Thezero-free-lengthspringwastobeachievedbyarrangingapivotedslidesothat(a)thecentre-lineofthespringalwayspassedthroughthepivotpointand(b)thefreeendofthespringwhenrelaxedwasoppositethepivot.Carwardinedoesnotseemtohavebeenawarewhenhe®ledthepatentthatABandACdonotneedtobeequal,northataspringcouldbewoundsothatithadatensioninitintheclose-coiledcondition.Thenextpatent[],®ledon4July1932,hastheessentialfeaturesoftheAnglepoisemechanism:twoTHESPRING-AND-LEVERBALANCINGMECHANISM,GEORGECARWARDINEANDTHEANGLEPOISELAMP507IMechE2000ProcInstnMechEngrsVol214PartC cranks,notnecessarilyofequallength,connectedbyaspringofzerofreelength.Thepatentbrie¯ydescribesthemeansofproducingsuchsprings.Atwo-degree-of-freedomlinkagetosupportanelectriclampisshown,withthespringsonthearmsasinFig.3a.UKPatent417970[],®ledinOctober1933,relatestoameansofsupportingtablemirrors,picturesandthelikesothattheycanbetiltedatanydesiredangleandstillremaininequilibrium.Theinventionincludestheclose-woundspringwhosetensionisproportionaltoitsoveralllengthandissimilarinessencetothearrange-mentshowninFig.2b.Adicultyinthiscaseistheabsenceofa®xedverticalreference,whichCarwardineovercomesbyaddingathirdmemberpivotedtothebackofthemirrorandslidingonapininthesupportingstrutsothat,althoughnot®xed,itremainsvertical.TheAnglepoisecon®gurationasshowninFig.1,withallthespringsatthebase,wasintroducedinUKPatent433617[].Thedrawingsthatillustratethepatentareclearlyofthe®naldesignofthelamp.Theessentialfeaturesoftheequipoisingmechanism,namelytheclose-woundspringsofzerofreelengthandtheparal-lelogramlinkage,areclearlydescribed,andCarwardinenotedthatthelampitselfismountedinaforksothattheaxispassesthroughthecentreofgravityofthelampandthebalanceismaintainedasthelampisrotated.Healsonotesthatfrictioncannotbeentirelyeliminatedfromthemechanism,sothatthelinksmayremaininpositioneveninpresenceofsmallerrorssuchasdevia-tionsfromlinearityinthesprings.Carwardine'sdesignshavebeenandcontinuetobecopiedinmanyproductsthroughouttheworld.508MJFRENCHANDMBWIDDENProcInstnMechEngrsVol214PartCIMechE2000