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INTJADVMANUFTECHNOL2001171041132001SPRINGERVERLAGLONDONLIMITEDFIXTURECLAMPINGFORCEOPTIMISATIONANDITSIMPACTONWORKPIECELOCATIONACCURACYBLIANDSNMELKOTEGEORGEWWOODRUFFSCHOOLOFMECHANICALENGINEERING,GEORGIAINSTITUTEOFTECHNOLOGY,GEORGIA,USAWORKPIECEMOTIONARISINGFROMLOCALISEDELASTICDEFORMATIONATFIXTUREWORKPIECECONTACTSOWINGTOCLAMPINGANDMACHININGFORCESISKNOWNTOAFFECTSIGNIFICANTLYTHEWORKPIECELOCATIONACCURACYAND,HENCE,THEFINALPARTQUALITYTHISEFFECTCANBEMINIMISEDTHROUGHFIXTUREDESIGNOPTIMISATIONTHECLAMPINGFORCEISACRITICALDESIGNVARIABLETHATCANBEOPTIMISEDTOREDUCETHEWORKPIECEMOTIONTHISPAPERPRESENTSANEWMETHODFORDETERMININGTHEOPTIMUMCLAMPINGFORCESFORAMULTIPLECLAMPFIXTURESUBJECTEDTOQUASISTATICMACHININGFORCESTHEMETHODUSESELASTICCONTACTMECHANICSMODELSTOREPRESENTTHEFIXTUREWORKPIECECONTACTANDINVOLVESTHEFORMULATIONANDSOLUTIONOFAMULTIOBJECTIVECONSTRAINEDOPTIMISATIONMODELTHEIMPACTOFCLAMPINGFORCEOPTIMISATIONONWORKPIECELOCATIONACCURACYISANALYSEDTHROUGHEXAMPLESINVOLVINGA321TYPEMILLINGFIXTUREKEYWORDSELASTICCONTACTMODELLINGFIXTURECLAMPINGFORCEOPTIMISATION1INTRODUCTIONTHELOCATIONANDIMMOBILISATIONOFTHEWORKPIECEARETWOCRITICALFACTORSINMACHININGAMACHININGFIXTUREACHIEVESTHESEFUNCTIONSBYLOCATINGTHEWORKPIECEWITHRESPECTTOASUITABLEDATUM,ANDCLAMPINGTHEWORKPIECEAGAINSTITTHECLAMPINGFORCEAPPLIEDMUSTBELARGEENOUGHTORESTRAINTHEWORKPIECEMOTIONCOMPLETELYDURINGMACHININGHOWEVER,EXCESSIVECLAMPINGFORCECANINDUCEUNACCEPTABLELEVELOFWORKPIECEELASTICDISTORTION,WHICHWILLADVERSELYAFFECTITSLOCATIONAND,INTURN,THEPARTQUALITYHENCE,ITISNECESSARYTODETERMINETHEOPTIMUMCLAMPINGFORCESTHATMINIMISETHEWORKPIECELOCATIONERRORDUETOELASTICDEFORMATIONWHILESATISFYINGTHETOTALRESTRAINTREQUIREMENTPREVIOUSRESEARCHERSINTHEFIXTUREANALYSISANDSYNTHESISAREAHAVEUSEDTHEFINITEELEMENTFEMODELLINGAPPROACHORCORRESPONDENCEANDOFFPRINTREQUESTSTODRSNMELKOTE,GEORGEWWOODRUFFSCHOOLOFMECHANICALENGINEERING,GEORGIAINSTITUTEOFTECHNOLOGY,ATLANTA,GEORGIA303320405,USAEMAILSHREYESMELKOTEMEGATECHEDUTHERIGIDBODYMODELLINGAPPROACHEXTENSIVEWORKBASEDONTHEFEAPPROACHHASBEENREPORTED18WITHTHEEXCEPTIONOFDEMETER8,ACOMMONLIMITATIONOFTHISAPPROACHISTHELARGEMODELSIZEANDCOMPUTATIONCOSTALSO,MOSTOFTHEFEBASEDRESEARCHHASFOCUSEDONFIXTURELAYOUTOPTIMISATION,ANDCLAMPINGFORCEOPTIMISATIONHASNOTBEENADDRESSEDADEQUATELYSEVERALRESEARCHERSHAVEADDRESSEDFIXTURECLAMPINGFORCEOPTIMISATIONBASEDONTHERIGIDBODYMODEL911THERIGIDBODYMODELLINGAPPROACHTREATSTHEFIXTUREELEMENTANDWORKPIECEASPERFECTLYRIGIDSOLIDSDEMETER12,13USEDSCREWTHEORYTOSOLVEFORTHEMINIMUMCLAMPINGFORCETHEOVERALLPROBLEMWASFORMULATEDASALINEARPROGRAMWHOSEOBJECTIVEWASTOMINIMISETHENORMALCONTACTFORCEATEACHLOCATINGPOINTBYADJUSTINGTHECLAMPINGFORCEINTENSITYTHEEFFECTOFTHECONTACTFRICTIONFORCEWASNEGLECTEDBECAUSEOFITSRELATIVELYSMALLMAGNITUDECOMPAREDWITHTHENORMALCONTACTFORCESINCETHISAPPROACHISBASEDONTHERIGIDBODYASSUMPTION,ITCANUNIQUELYONLYHANDLE3DFIXTURINGSCHEMESTHATINVOLVENOMORETHAN6UNKNOWNSFUHANDNEE14ALSOPRESENTEDANITERATIVESEARCHBASEDMETHODTHATCOMPUTESTHEMINIMUMCLAMPINGFORCEBYASSUMINGTHATTHEFRICTIONFORCEDIRECTIONSAREKNOWNAPRIORITHEPRIMARYLIMITATIONOFTHERIGIDBODYANALYSISISTHATITISSTATICALLYINDETERMINATEWHENMORETHANSIXCONTACTFORCESAREUNKNOWNASARESULT,WORKPIECEDISPLACEMENTSCANNOTBEDETERMINEDUNIQUELYBYTHISMETHODTHISLIMITATIONMAYBEOVERCOMEBYACCOUNTINGFORTHEELASTICITYOFTHEFIXTUREWORKPIECESYSTEM15FORARELATIVELYRIGIDWORKPIECE,THELOCATIONOFTHEWORKPIECEINTHEMACHININGFIXTUREISSTRONGLYINFLUENCEDBYTHELOCALISEDELASTICDEFORMATIONATTHEFIXTURINGPOINTSHOCKENBERGERANDDEMETER16USEDEMPIRICALCONTACTFORCEDEFORMATIONRELATIONSCALLEDMETAFUNCTIONSTOSOLVEFORTHEWORKPIECERIGIDBODYDISPLACEMENTSDUETOCLAMPINGANDQUASISTATICMACHININGFORCESTHESAMEAUTHORSALSOINVESTIGATEDTHEEFFECTOFMACHININGFIXTUREDESIGNPARAMETERSONWORKPIECEDISPLACEMENT17GUIETAL18REPORTEDANELASTICCONTACTMODELFORIMPROVINGWORKPIECELOCATIONACCURACYTHROUGHOPTIMISATIONOFTHECLAMPINGFORCEHOWEVER,THEYDIDNOTADDRESSMETHODSFORCALCULATINGTHEFIXTUREWORKPIECECONTACTSTIFFNESSINADDITION,THEAPPLICATIONOFTHEIRALGORITHMFORASEQUENCEOFMACHININGLOADSREPRESENTINGAFINITETOOLPATHWASNOTDISCUSSEDLIANDMELKOTE19ANDHURTADOANDMELKOTE20USEDCONTACTMECHANICSTOFIXTURECLAMPINGFORCEOPTIMISATION105SOLVEFORTHECONTACTFORCESANDWORKPIECEDISPLACEMENTPRODUCEDBYTHEELASTICDEFORMATIONATTHEFIXTURINGPOINTSOWINGTOCLAMPINGLOADSTHEYALSODEVELOPEDMETHODSFOROPTIMISINGTHEFIXTURELAYOUT21ANDCLAMPINGFORCEUSINGTHISMETHOD22HOWEVER,CLAMPINGFORCEOPTIMISATIONFORAMULTICLAMPSYSTEMANDITSIMPACTONWORKPIECEACCURACYWERENOTCOVEREDINTHESEPAPERSTHISPAPERPRESENTSANEWALGORITHMBASEDONTHECONTACTELASTICITYMETHODFORDETERMININGTHEOPTIMUMCLAMPINGFORCESFORAMULTICLAMPFIXTUREWORKPIECESYSTEMSUBJECTEDTOQUASISTATICLOADSTHEMETHODSEEKSTOMINIMISETHEIMPACTOFWORKPIECEMOTIONDUETOCLAMPINGANDMACHININGLOADSONTHEPARTLOCATIONACCURACYBYSYSTEMATICALLYOPTIMISINGTHECLAMPINGFORCESACONTACTMECHANICSMODELISUSEDTODETERMINEASETOFCONTACTFORCESANDDISPLACEMENTS,WHICHARETHENUSEDFORTHECLAMPINGFORCEOPTIMISATIONTHECOMPLETEPROBLEMISFORMULATEDANDSOLVEDASAMULTIOBJECTIVECONSTRAINEDOPTIMISATIONPROBLEMTHEIMPACTOFCLAMPINGFORCEOPTIMISATIONONWORKPIECELOCATIONACCURACYISANALYSEDVIATWOEXAMPLESINVOLVINGA321FIXTURELAYOUTFORAMILLINGOPERATION2FIXTUREWORKPIECECONTACTMODELLING21MODELLINGASSUMPTIONSTHEMACHININGFIXTURECONSISTSOFLLOCATORSANDCCLAMPSWITHSPHERICALTIPSTHEWORKPIECEANDFIXTUREMATERIALSARELINEARLYELASTICINTHECONTACTREGION,ANDPERFECTLYRIGIDELSEWHERETHEWORKPIECEFIXTURESYSTEMISSUBJECTEDTOQUASISTATICLOADSDUETOCLAMPINGANDMACHININGTHECLAMPINGFORCEISASSUMEDTOBECONSTANTDURINGMACHININGTHISASSUMPTIONISVALIDWHENHYDRAULICORPNEUMATICCLAMPSAREUSEDINREALITY,THEELASTICITYOFTHEFIXTUREWORKPIECECONTACTREGIONISDISTRIBUTEDHOWEVER,INTHISMODELDEVELOPMENT,LUMPEDCONTACTSTIFFNESSISASSUMEDSEEFIG1THEREFORE,THECONTACTFORCEANDLOCALISEDDEFORMATIONATTHEITHFIXTURINGPOINTCANBERELATEDASFOLLOWSFIJKIJDIJ1WHEREKIJJX,Y,ZDENOTESTHECONTACTSTIFFNESSINTHETANGENTIALANDNORMALDIRECTIONSOFTHELOCALXI,YI,ZICOORDINATEFRAME,DIJFIG1ALUMPEDSPRINGFIXTUREWORKPIECECONTACTMODELXI,YI,ZI,DENOTETHELOCALCOORDINATEFRAMEATTHEITHCONTACTJX,Y,ZARETHECORRESPONDINGLOCALISEDELASTICDEFORMATIONSALONGTHEXI,YI,ANDZIAXES,RESPECTIVELY,FIJJX,J,ZREPRESENTSTHELOCALCONTACTFORCECOMPONENTSWITHFIXANDFIYBEINGTHELOCALXIANDYICOMPONENTSOFTHETANGENTIALFORCE,ANDFIZTHENORMALFORCE22WORKPIECEFIXTURECONTACTSTIFFNESSMODELTHELUMPEDCOMPLIANCEATASPHERICALTIPLOCATOR/CLAMPANDWORKPIECECONTACTISNOTLINEARBECAUSETHECONTACTRADIUSVARIESNONLINEARLYWITHTHENORMALFORCE23THECONTACTDEFORMATIONDUETOTHENORMALFORCEPIACTINGBETWEENASPHERICALTIPPEDFIXTUREELEMENTOFRADIUSRIANDAPLANARWORKPIECESURFACECANBEOBTAINEDFROMTHECLOSEDFORMHERTZIANSOLUTIONTOTHEPROBLEMOFASPHEREINDENTINGANELASTICHALFSPACEFORTHISPROBLEM,THENORMALDEFORMATIONDINISGIVENAS23,P93DINS9PI216RIE2D1/32WHERE1E1N2WEW1N2FEFEWANDEFAREYOUNGSMODULIFORTHEWORKPIECEANDFIXTUREMATERIALS,RESPECTIVELY,ANDNWANDNFAREPOISSONRATIOSFORTHEWORKPIECEANDFIXTUREMATERIALS,RESPECTIVELYTHETANGENTIALDEFORMATIONDITDITXORDITYINTHELOCALXIANDYITANGENTIALDIRECTIONS,RESPECTIVELYDUETOATANGENTIALFORCEQIQIXORQIYHASTHEFOLLOWINGFORM23,P217DTITQI8AIS2NFGF2NWGWD3WHEREAIS3PIRI4S1NFEF1NWEWDD1/3ANDGWANDGFARESHEARMODULIFORTHEWORKPIECEANDFIXTUREMATERIALS,RESPECTIVELYAREASONABLELINEARAPPROXIMATIONOFTHECONTACTSTIFFNESSCANBEOBTAINEDFROMALEASTSQUARESFITTOEQ2THISYIELDSTHEFOLLOWINGLINEARISEDCONTACTSTIFFNESSVALUESKIZ882S16RIE29D1/34KIXKIY4ES2NJGF2NWGWD1KIZ5INDERIVINGTHEABOVELINEARAPPROXIMATION,THENORMALFORCEPIWASASSUMEDTOVARYFROM0TO1000N,ANDTHECORRESPONDINGR2VALUEOFTHELEASTSQUARESFITWASFOUNDTOBE0943CLAMPINGFORCEOPTIMISATIONTHEGOALISTODETERMINETHESETOFOPTIMALCLAMPINGFORCESTHATWILLMINIMISETHEWORKPIECERIGIDBODYMOTIONDUETO106BLIANDSNMELKOTELOCALISEDELASTICDEFORMATIONINDUCEDBYTHECLAMPINGANDMACHININGLOADS,WHILEMAINTAININGTHEFIXTUREWORKPIECESYSTEMINQUASISTATICEQUILIBRIUMDURINGMACHININGMINIMISATIONOFTHEWORKPIECEMOTIONWILL,INTURN,REDUCETHELOCATIONERRORTHISGOALISACHIEVEDBYFORMULATINGTHEPROBLEMASAMULTIOBJECTIVECONSTRAINEDOPTIMISATIONPROBLEM,ASDESCRIBEDNEXT31OBJECTIVEFUNCTIONFORMULATIONSINCETHEWORKPIECEROTATIONDUETOFIXTURINGFORCESISOFTENQUITESMALL17THEWORKPIECELOCATIONERRORISASSUMEDTOBEDETERMINEDLARGELYBYITSRIGIDBODYTRANSLATIONDDWDXWDYWDZWT,WHEREDXW,DYW,ANDDZWARETHETHREEORTHOGONALCOMPONENTSOFDDWALONGTHEXG,YG,ANDZGAXESSEEFIG2THEWORKPIECELOCATIONERRORDUETOTHEFIXTURINGFORCESCANTHENBECALCULATEDINTERMSOFTHEL2NORMOFTHERIGIDBODYDISPLACEMENTASFOLLOWSIDDWIDXW2DYW2DZW26WHEREIIDENOTESTHEL2NORMOFAVECTORINPARTICULAR,THERESULTANTCLAMPINGFORCEACTINGONTHEWORKPIECEWILLADVERSELYAFFECTTHELOCATIONERRORWHENMULTIPLECLAMPINGFORCESAREAPPLIEDTOTHEWORKPIECE,THERESULTANTCLAMPINGFORCE,PRCPRXPRYPRZT,HASTHEFORMPRCRCPC7WHEREPCPL1PLCTISTHECLAMPINGFORCEVECTOR,RCNL1NLCTISTHECLAMPINGFORCEDIRECTIONMATRIX,NLICOSALICOSBLICOSGLITISTHECLAMPINGFORCEDIRECTIONCOSINEVECTOR,ANDALI,BLI,ANDGLIAREANGLESMADEBYTHECLAMPINGFORCEVECTORATTHEITHCLAMPINGPOINTWITHRESPECTTOTHEXG,YG,ZGCOORDINATEAXESI1,2,CINTHISPAPER,THEWORKPIECELOCATIONERRORDUETOCONTACTREGIONDEFORMATIONISASSUMEDTOBEINFLUENCEDONLYBYTHENORMALFORCEACTINGATTHELOCATORWORKPIECECONTACTSTHEFRICTIONALFORCEATTHECONTACTSISRELATIVELYSMALLANDISNEGLECTEDWHENANALYSINGTHEIMPACTOFTHECLAMPINGFORCEONTHEWORKPIECELOCATIONERRORDENOTINGTHERATIOOFTHENORMALCONTACTSTIFFNESS,KIZ,TOTHESMALLESTNORMALSTIFFNESSAMONGALLLOCATORS,KSZ,BYJII1,L,ANDASSUMINGTHATTHEWORKPIECERESTSONNX,NY,ANDNZNUMBEROFLOCATORSORIENTEDINTHEXG,FIG2WORKPIECERIGIDBODYTRANSLATIONANDROTATIONYG,ANDZGDIRECTIONS,THEEQUIVALENTCONTACTSTIFFNESSINTHEXG,YG,ANDZGDIRECTIONSCANBECALCULATEDASKSZSONXI1JID,KSZSONYI1JID,ANDKSZSONZI1JIDRESPECTIVELYSEEFIG3THEWORKPIECERIGIDBODYMOTION,DDW,DUETOCLAMPINGACTIONCANNOWBEWRITTENASDDW3PRXKSZSONXI1JIDPRYKSZSONYI1JIDPRZKSZSONZI1JID4T8THEWORKPIECEMOTION,ANDHENCETHELOCATIONERRORCANBEREDUCEDBYMINIMISINGTHEWEIGHTEDL2NORMOFTHERESULTANTCLAMPINGFORCEVECTORTHEREFORE,THEFIRSTOBJECTIVEFUNCTIONCANBEWRITTENASMINIMIZEIPRCIW11PRXONXI1JI221PRYONYI1JI221PRZONZI1JI2229NOTETHATTHEWEIGHTINGFACTORSAREPROPORTIONALTOTHEEQUIVALENTCONTACTSTIFFNESSESINTHEXG,YG,ANDZGDIRECTIONSTHECOMPONENTSOFPRCAREUNIQUELYDETERMINEDBYSOLVINGTHECONTACTELASTICITYPROBLEMUSINGTHEPRINCIPLEOFMINIMUMTOTALCOMPLEMENTARYENERGY15,23THISENSURESTHATTHECLAMPINGFORCESANDTHECORRESPONDINGLOCATORREACTIONSARE“TRUE”SOL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