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PrecisionEngineering43(2016)468478ContentslistsavailableatScienceDirectPrecisionEngineeringjournalhomepage:/locate/precisionStaticallybalancedbrakesMichielDelftaArticleReceivedReceivedAcceptedAvailableKeywords:BrakeStaticallyLockingClutchaintroducesrequirethatsurfacessetsoftheshowwithforce1.IntroductionConventionalbrakesrequireapowerfulactuatorthatgeneratesaofbraketionsweighttion.requirebereviewbrakesthetherebyofthatment.brakes)tionforcetokeepthebrakedisengaged.Thisissolvedinthethirdcate-gory:bi-stablebrakes11,12.Suchbrakeshaveabi-stableelement(e.g.abi-stablespring),providingthebrakewithtwostablestates:/10.1016/j.precisioneng.2015.09.0130141-6359/normalforcebetweentwofrictionsurfaces1,2.Theamplitudethenormalforce,thefrictioncoefficientandthegeometryofthetogetherdeterminethebrakingtorque.Thereareapplica-inwhichpowerfulactuatorsareundesiredduetosizeandlimitationsortheir(potentiallyhigh)energyconsump-Therefore,researchershaveworkedondesigningbrakesthatlessactuationforce.Researchonthereductionoftherequiredactuationforcecansplitintothreecategories,thatarealsodescribedintherecentpaperonlockingmechanisms3.Firstly,self-engaginghavebeendevelopedthatusetherelativemotionbetweenfrictionsurfacestopullthefrictionsurfacestogetherandreducingtherequiredactuationforce47.Disadvantagessuchbrakesarethattheyonlyworkinonebrakingdirectionandtheycanonlydisengageintheoppositedirectionofengage-Secondly,springbrakes(alsocalledsafetybrakesorparkinguseaspringtokeepthebrakeengagedwithoutactua-force2,810.However,thesebrakesstillrequireanactuationCorrespondingauthor.Tel.:+31152788642.E-mailaddress:m.c.plooijtudelft.nl(M.Plooij).theengagedstateandthedisengagedstate.However,switchingbetweenthesetwostatesstillrequiresahighactuationforce.Otherresearchersfocusedonimplementingactuatorswithahighforcedensityandalowenergyconsumption.Thebestexam-pleofthisispiezo-actuatedbrakes1317.Becauseoftheirhighforcedensityandlowenergyconsumption,theyarepotentiallyveryeffectiveinsolvingtheissuesmentionedabove.However,theyrequirehighvoltages(thatmightnotbeavailable),verypre-cisemanufacturing(sincetheyhaveaverysmallstroke)andareexpensive.Furthermore,thebrakeconstructionhastobeverystiff,otherwisetheconstructionwilldeform,whichreducestheeffec-tivenessofpiezoelectricactuators.Theproblemwiththestate-of-the-artbrakesisthattheactu-atorhastobeabletogenerateaforceequaltothenormalforcebetweenthefrictionplates.Thegoalofthispaperistointroduceabrakeconceptinwhichthenormalforceandtheactuationforcearedecoupled.Thisconceptpotentiallyreducestheactuationforceby100%.Thisnewbrakeconceptisfundamentallydifferentfromcur-rentbrakeconceptsandiscalledstaticallybalancedbrakes(SBBs,seeFig.1).SBBsdonotrequireanactuationforcetoholdacer-tainbrakingtorqueandonlyrequireasmallactuationforceto2015ElsevierInc.Allrightsreserved.Plooij,TomvanderHoeven,GerardDunningUniversityofTechnology,Mekelweg2,2628CDDelft,TheNetherlandsrticleinfohistory:30April2015inrevisedform21August201515September2015online25September2015designbalancedmechanismabstractConventionalbrakesrequiresumingbrakes.Thispaperbrakes(SBBs).SBBsdonotmoveasmallmasstovarySBB,oneofthetwofrictionconnectedthroughamechanismThetotalenergyinthetwobrakingblock.ThepositionfirstsetofspringsandthusthatcanbeusedinSBBsandnegativestiffnessandoneactuationforcecanbereducedthatinSBBs,theactuationsmall,lightweightandenergy,MartijnWissepowerfulactuator,leadingtolarge,heavyandinmostcasesenergycon-afundamentallydifferentbrakeconceptcalledstaticallybalancedanyactuationforcetomaintainabrakingtorqueandonlyhavetotorque.Therefore,theirenergyconsumptionispotentiallyverylow.Inanisconnectedthroughspringstoabrakingblock.Thisbrakingblockistoasecondsetofsprings,theothersideofwhichconnectstotheground.ofspringsisconstant,whichresultsinazero-forcecharacteristicatthethisstaticallybalancedbrakingblockdeterminesthedisplacementofthenormalforcebetweenthefrictionsurfaces.Wecategorizemechanismstwoembodiments:onewithleafspringswitharangeofpositionswithtorsionspringsandanon-linearcammechanism.Resultsshowthatthebyapproximately9597%incomparisontoregularbrakes.ThisshowscanbealmosteliminatedandthusshowingthepotentialofSBBstobeefficient.2015ElsevierInc.Allrightsreserved.M.Plooijetal./PrecisionEngineering43(2016)468478469Fig.withwithvarybetieswhilemechanismssafemicroexplainsgorizesandtwoshow9597%.a2.First,springs.springs.andTheeachCoulomb|whereabsolutenormalofbrakingofspringsthethe|Thisamplitudesprings,Fig.2.Aschematicdrawingofastaticallybalancedbrake.Therightfrictionsurfaceisconnectedtoajointthathastobebraked.Theleftfrictionsurfaceisconnectedthroughthenormalforcespringswiththebrakingblock.Amechanismconnectsthebrakingblockwiththecompensationspringsandtheothersideofthecompensationspringsconnectstotheground.Thismechanismisvisualizedbythemechanismequationxc=H(xn),thatgivestherelationshipbetweenthepositionsofthetwosidesofthemechanism.Thepositionofthebrakingblockdeterminesthenormalforcebetweenthefrictionsurfaces.thebrakingblockinacertainposition.Inordertodecouplethisnor-malforcefromtheactuationforce,asecondspringsystemisused:thecompensationsprings.Thecompensationspringsareplacedbetweenthegroundandamechanismthatalsoconnectstothebrakingblock.ThismechanismisdepictedinFig.2asacloudwiththemechanismequationxc=H(xn).Thisequationassumesthattheoverallmechanismhasonedegreeoffreedom(DOF).InSection3,wewillzoominonthispartanddiscusspossiblemechanisms.Herewewillanalyzethestaticbalanceofmechanismsfromanenergy1.Apictureofthetwoprototypesofstaticallybalancedbrakes.(a)Aprototypeleafspringswitharangeofpositionswithnegativestiffness.(b)Aprototypetorsionspringsandarotationalcammechanism.thattorque.Furthermore,withsmalladjustments,SBBscanchangedtoincorporateanyofthethreedifferentfunctionali-mentionedabove(i.e.regular-,spring-andbi-stablebehavior),stillonlyrequiringasmallactuationforce.Staticallybalancedhavealsobeenusedamongstothersforintrinsicallyroboticarms18,19,exoskeletons20,prostheses21andandprecisionmechanisms22,23.Therestofthispaperisstructuredasfollows.First,Section2theconceptofSBBsinmoredetail.Then,Section3cate-allpossibleembodimentsofSBBsthatarerelativelysimplethereforesmallandlightweight.Sections4and5thenshowprototypesofSBBsandtheirperformance.ThoseresultswillthattheactuationforcesintheprototypesarereducedbyFinally,thepaperendswithadiscussioninSection6andconclusioninSection7.TheconceptofstaticallybalancedbrakesInthissectionweexplaintheconceptofSBBsinmoredetail.wegiveageneralformulationwithoutassuminglinearThen,weworkouttheequationsforasystemwithlinearFig.2showsaschematicdrawingoftheconceptofaSBBFig.3showstheworkingprincipleofaSBBforlinearsprings.brakeisengagedbypushingthetwofrictionsurfacesagainstother.Thefrictionbetweenthesurfacesisassumedtobeatypefriction:Ff|max=SYNFn(1)SYNistheCoulombfrictioncoefficient,|Ff|maxisthemaximumfrictionforcebeforethesurfacesstarttoslipandFnistheforce.ThebrakeinFig.2isstaticallybalancedbytwogroupssprings.Thegroupofnormalforcespringsisplacedbetweentheblockandtheleftfrictionsurface.TheenergyinthisgroupspringsisequaltoEn(xn),withxnbeingthedisplacementoftheasshowninFig.2.Theforceinthesesprings(Fn)isequaltonormalforcebetweenthefrictionsurfaces.MultiplyingFnbyeffectiveradiusrofthebrake,givesthebrakingtorque:T|max=SYNFnr=SYNEn(xn)xnr(2)meansthatthepositionofthebrakingblockdeterminestheofthebrakingtorque.NowifthisweretheonlygroupofanactuatorwouldstillhavetogeneratetheforceFntoholdperspective.TheenergyinthegroupofcompensationspringsisEc(xc)withxcbeingthedisplacementofthespringsasshowninFig.2.NowthesystemisstaticallybalancedwhenE=En+Eciscon-stantforallpositionsofthesystem.ThetransferratiohfromtheFig.3.Theworkingprincipleofastaticallybalancedbrake.Thecompensationspringshaveanegativestiffnesswhenmeasuredatthebrakingblockandthenor-malforcespringshaveapositivestiffness.Sincethestiffnessescanceloutandtheequilibriumpositionscoincide,theoverallcharacteristichasarangeofzeroforce,whichistheactuationstroke.Asmallactuatorcanpositionthebrakeatanypositioninthisrange,controllingthenormalforceandthusthebrakingtorque.470M.Plooijetal./PrecisionEngineering43(2016)468478normalforcespringstothecompensationspringsatpositionxnisequalto:h(xn)=H(xn)xn=xcxn(3)WecannowwritetheconditionforstaticbalanceasNowbalancedhThecanFItwhichthatareEEwherethementhFromanismclutchthesprings,xUsinghFig.anismshowsthetion.toNoteisbetweendisengagemovespushesbrakingtodeterminesThisanalysisresultsinalistoffeasibleconceptsandadescriptionofhowtoconstructthem.3.1.Rigidbody:LinkagesThefirstclassofrigidbodymechanismsforSBBSislinkagemechanisms.InordertocategorizeoneDOFlinkagesfurther,wehavetorealizethatthemechanismshouldatleastpossessonesingularposition.ThisfollowsfromEq.(10),wherethetransferfunctionbecomeszeroatpositionxn=0.Thispositionshouldbereachabletofullyunloadthenormalforcespringtoobtainazerobrakingtorque.Therefore,wecategorizebothlinkageandcammechanismsfurtherbycategorizingsingularmechanisms.ThereexistsliteratureonsingularmechanismsandhowtocategorizeExn=0(4)En(xn)xn+Ec(xc)xch(xn)=0(5)giventhetwospringcharacteristics,thissystemisstaticallyforallxnforwhichitholdsthat(xn)=xcEc(xc)En(xn)xn(6)forcethatthecompensationspringappliesonthebrakingblockbeexpressedas:c=Ecxn=Ec(xc)xch(xn)=En(xn)xn=Fn(7)islogicalthatFc=Fnbecausethisresultsinforceequilibrium,isanotherwaytoconsiderstaticbalancing.Nowassumeboththenormalforcespringsandthecompensationspringslinear:n=12knmax(xn,0)2(8)c=12kcx2c(9)knandkcarespringstiffnessesandthemaxoperatorreturnsmaximumvalueofthetwoinputsandmodelsthedisengage-ofthefrictionsurfaces.Eq.(6)nowbecomes:(xn)=knmax(xn,0)kcxc(10)Eq.(10)itfollowsthath(xn0)=0.Thismeansthatthemech-isinasingularpositionorthatthemechanismcontainsathatdecouplesthetwomotions.FromEqs.(8)and(9)andrequirementthatE=En+Ecisconstant,itfollowsthatforlinearthemechanismshouldsatisfyc=radicalbigg2Eknmax(xn,0)2kc(11)Eq.(3),thetransferfunctionbecomes:(xn)=knkcxnradicalbig(2Eknx2n)/kc(12)3showsaschematicexplanationoftheforcesinsuchamech-asfunctionofthepositionofthebrakingblock.Thisfigurethattheoverallcharacteristicisequaltozeroforxn0,whilenormalforceatthosepositionsdependslinearlyontheposi-Thismeansthattheactuatordoesnothavetoapplyanyforcemaintainacertainnormalforcebetweenthefrictionsurfaces.thatinthisexample,h(xn0)/=0,meaningthatthesystemnotstaticallybalancedforxn0.TheconceptofSBBsdependsonadecouplingofthenormalforcetwofrictionsurfacesandtheforcerequiredtoengageorthebrake.Withoutthestaticbalancing,theactuatorthatthebrakingblockwouldalsohavetodelivertheforcethatthefrictionsurfacestogether.Withthestaticbalancing,theblockcanbemovedbyanactuatorthatdoesnothavecounteractanyspringforce(Eq.(7).Thiscontrolledpositionthebrakingtorque(Eq.(2).Fig.4.Avisualizationofthecategorizationofmechanismsforstaticallybalancedbrakes.Afirstdivisionismadebetweenrigidbodymechanismsandcompliantmechanisms.Rigidbodymechanismsaresplitintolinkagesandcammechanisms.Linkagesarecategorizedbasedonthenatureoftheirinputandoutput(rotationalortranslational).Onecategoryoflinkagesleadstoafeasibleconcept.Cammechanismsarecategorizedonthenatureoftheirinput,outputandcammovement(rotationalortranslational).Inthegreycategoriesitispossibletoobtainperfectstaticbalance.Fromthedark-greycategoriesweshowaprototypeinthispaper.3.PossibleembodimentsTheprevioussectionpresentedrequirementsonmechanismsforSBBs.Theoretically,whensatisfyingthoserequirements,areductionintheactuationforceof100%canbeachieved.Inordertodesignmechanismsthatmeetthoserequirements,inthissectionwecategorizemechanismsthataresuitabletobeusedinSBBs.Twoofthoseconceptswerebuiltandtheresultswillbeshowninthenexttwosections.Thecategorizationislimitedbythreeconstraintsthatwillleadtomechanismsthathavethepotentialtobesmallandlightweight.First,weonlyconsideroneDOFmechanisms.MoreDOFmecha-nismsforSBBsarealsopossible,butthisrequiresextracomponentsandextraactuators,increasingthesizeandmassofthebrake.Secondly,thetypeofspringthatisusedasnormalforcespringorcompensationspringshouldmatchthetypeofDOFthatitisattachedto.Forinstance,wedonotconsidermechanismsinwhichatranslationalspringisconnectedtoarotatinglink.Suchacon-structiondoesnotallowforalignmentofspringandtheDOFandwillthereforeleadtoanincreaseinsize.Thirdly,weonlyconsidermechanismswiththeleastamountofcomponents.Forinstance,afourbarmechanismisconsidered,butaneightbarmechanismwithoneDOFisnotconsidered.Ingeneral,mechanismscanbedividedintorigidbodymecha-nismsandcompliantmechanisms.Inrigidbodymechanisms,allthepartsarerigidexceptforthespringsthatareeithertransla-tionalorrotational.Forthepurposeofthispaper,wesplitrigidbodymechanismsintolinkagemechanismsandcammechanisms.TheoverallcategorizationofmechanismsisshowninFig.4,thatalsoalreadyshowswhichcategoriesarefeasible.Thissectionfirstanalysesrigidbodymechanismsandthencompliantmechanisms.M.Plooijetal./PrecisionEngineering43(2016)468478471Fig.nism(b)withthementlistprovidesnisms,ofgularIneitherfour.Allplacementsxortively.andanglesBeforetranslationalareFig.nectatzero.3.1.1.ItbetweensliderslengththeOwithcanE=12knx2n+12kcx2cE=C1+12(kn1)x2i+12(kc1)x2o+xixocos(DC23)knl1xikcl2xocos(DC22)(14)whereC1istheconstantterm:C=l2+1kl2+1kl2(15)5.Thecollectionofpossiblesimplesingularlinkagemechanisms.(a)Amecha-withtranslationalinputandoutput.Thismechanismcanbestaticallybalanced.Amechanismwithtranslationalinputandrotationaloutput.(c)Amechanismrotationalinputandoutput.(d)Astaticallybalancedversionof(a).2426.However,thosecategorizationsarebasedondiffer-typesofmechanicalsingularitiesanddonotleadtoacompleteofmechanisms.Hereweintroduceanewcategorizationthatsuchalistandonlyincorporatessimplesingularmecha-leadingtosmallandlightweightdesigns.OurcategorizationsingularmechanismsisbasedonthenotionthatalloneDOFsin-mechanismshaveoneinputmotionandoneoutputmotion.simplesingularmechanisms,theseinputandoutputmotionsaretranslationalorrotationalmotions.Inlinkagesthisleadstocategories:translationalinputtranslationaloutput(seeFig.5a),translationalinputrotationaloutput(seeFig.5b),rotationalinputtranslationaloutput(seeFig.5b),rotationalinputrotationaloutput(seeFig.5c).mechanismsusethesamenotation.xnandxcdenotethedis-ofthespringsandcanberotationalortranslational.iandxodenotethepositionoftheinputandoutputtranslationsrotations.landDC2refertoconstantdistancesandangles,respec-knandkcdenotethestiffnessesofthenormalforcespringscompensationsprings.AndfinallydandCRrefertodistancesandthatchangewhenthepositionofthemechanismchanges.Thefourcategoriesofmechanismswillbediscussedbelow.discussingthem,itshouldbenotedthattheplacementofnormalforcespringsbecomesimpracticalwhentheybothrotatingandtranslating;seeforexampletheleftspringin5a.Therefore,translationalnormalforcespringscanonlycon-toasliderthatisinlinewiththespring.Thisalsoensuresthatacertainposition,theforceinthenormalforcespringsbecomesTranslationalinputtranslationaloutputAgeneralizedversionofthismechanismisshowninFig.5a.consistsoftwoslidersthatintersectinO.DC23denotestheanglethetwoslidersandxiandxodenotethepositionsofthemeasuredfromO.Thelinkbetweenthetwoslidershasl3andeachsliderconnectstoaspring.Theothersidesofspringsareconnectedtothegroundatdistancesl1andl2fromunderanglesofDC21andDC22.Sincetheleftspringshouldbeinlinetheleftslider,itisgiventhatDC21=0.Theenergyinthesystembeobtainedbyapplyingcosinerules:x2n=l21+x2i2l1xix2c=l22+x2o2l2xocos(DC22)l23=x2i+x2o2xixocos(DC23)(13)132n12c2WecanderivexoasfunctionofxifromEq.(13)andfillitintoEq.(14).Nowforstaticbalance,Eq.(4)shouldholdforallxi,whichisonlytruewhenDC22=DC23=0.5EM,l1=0andkn=kc.SuchamechanismisdepictedinFig.5d,wherethegroundcanbemovedfreelyalongthedashedline.Thenormalforcetensionspringischangedintoacompressionspringinthisexampleandisconnectedtofrictionplates.Thefactthatthiscategoryleadstoafeasiblesolutionisindi-catedinFig.4.Thefeasiblemechanisminthiscategoryisnotanewmechanism27.However,hereweprovedthatthismechanismisinfacttheonlyfeasiblemechanisminthiscategory.3.1.2.TranslationalinputrotationaloutputAgeneralizedversionofthismechanismisshowninFig.5b.Itconsistsofonesliderwithazeropositionxi=0atOandacrankmechanismwithlinksoflengthsl1andl2.Onespringisplacedbetweenthesliderandthegroundandtheotherisplacedbetweenthecrankandtheground.Sincetheleftspringshouldbeinlinewiththeleftslider,itisgiventhatDC2=0.Theenergyinthissystemcanbederivedasfollows.First,wedefined2asthedistancebetweenthejointthatconnectsthetwobarsofthecrankandthelined1:d2=l1cos(CR)d1=radicalbigl21d22+radicalbigl22d22xi=radicalbigd21l24xn=(xil3)E=12knx2n+12kc(xcxc,0)2(16)wherexc,0istheequilibriumpositionoftherotationalspring.Againforstaticbalance,Eq.(4)shouldholdforallxi,whichisonlythecasewhenkn=kc=0.Sincethestiffnessesshouldbelargerthanzero,itisimpossibletousethismechanismforaperfectlystaticallybal-ancedbrake,asindicatedinFig.4.TheuseofimperfectlystaticallybalancedmechanismswillbediscussedinSection.3.RotationalinputtranslationaloutputThissystemisthesameasthesystemwithtranslationalinputandrotationaloutputinFig.5b,withthedifferencethattheinputandoutputareswitched.Theenergyinthesystemcannowbecalculatedby:xi=radicalbigd21l24x2n=l23+x2i2l3xicos(DC2)E=12knx2n+12kc(xcxc,0)2(17)SinceitisimpossibletosatisfyEq.(4)withnon-zerostiffnesses,itisimpossibletoperfectlystaticallybalancethissystem,asindicatedinFig..RotationalinputrotationaloutputAgeneralizedversionofthismechanismisshowninFig.5c.Theinputandoutputlinksandthelinkinbetweenformafourbarmechanismwithlengthsl1,l2,l3andl4.Onerotationalspringisplacedbetweenlink1andtheground.Thesecondrotational472M.Plooijetal./PrecisionEngineering43(2016)468478Fig.toandcategories:c-c.springcalculatedEAgain,asmechanism3.2.nisms.usedtheever,translational,discussstructedcategoriessomeofspringcomponents.InwetheSecondly,connectedcamblethatspringsunderananglearealsopossible,butwouldmaketheanal-ysisunnecessarilycomplicated.Thenotationthatisusedisasfollows.Thelengthsofthespringsaredenotedbyxnforthenormalforcespringsandxcforthecom-pensationsprings.xn,0andxc,0denotetheequilibriumpositionsofthesprings,meaningthatthedisplacementsofthespringsarexnxnan