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StructMultidiscOptim20,76–82SpringerVerlag2000OptimaldesignofhydraulicsupportM.Oblak,B.HarlandB.ButinarAbstractThispaperdescribesaprocedureforoptimaldeterminationoftwogroupsofparametersofahydraulicsupportemployedintheminingindustry.Theprocedureisbasedonmathematicalprogrammingmethods.Inthefirststep,theoptimalvaluesofsomeparametersoftheleadingfourbarmechanismarefoundinordertoensurethedesiredmotionofthesupportwithminimaltransversaldisplacements.Inthesecondstep,maximaltolerancesoftheoptimalvaluesoftheleadingfourbarmechanismarecalculated,sotheresponseofhydraulicsupportwillbesatisfying.Keywordsfourbarmechanism,optimaldesign,mathematicalprogramming,approximationmethod,tolerance1IntroductionThedesigneraimstofindthebestdesignforthemechanicalsystemconsidered.Partofthiseffortistheoptimalchoiceofsomeselectedparametersofasystem.Methodsofmathematicalprogrammingcanbeused,ifasuitablemathematicalmodelofthesystemismade.Ofcourse,itdependsonthetypeofthesystem.Withthisformulation,goodcomputersupportisassuredtolookforoptimalparametersofthesystem.ThehydraulicsupportFig.1describedbyHarl1998isapartoftheminingindustryequipmentinthemineVelenjeSlovenia,usedforprotectionofworkingenvironmentinthegallery.ItconsistsoftwofourbarReceivedApril13,1999M.Oblak1,B.Harl2andB.Butinar31FacultyofMechanicalEngineering,Smetanova17,2000Maribor,Sloveniaemailmaks.oblakunimb.si2M.P.P.Razvojd.o.o.,Ptujska184,2000Maribor,Sloveniaemailbostjan.harlunimb.si3FacultyofChemistryandChemicalEngineering,Smetanova17,2000Maribor,Sloveniaemailbranko.butinarunimb.simechanismsFEDGandAEDBasshowninFig.2.ThemechanismAEDBdefinesthepathofcouplerpointCandthemechanismFEDGisusedtodrivethesupportbyahydraulicactuator.Fig.1HydraulicsupportItisrequiredthatthemotionofthesupport,moreprecisely,themotionofpointCinFig.2,isverticalwithminimaltransversaldisplacements.Ifthisisnotthecase,thehydraulicsupportwillnotworkproperlybecauseitisstrandedonremovaloftheearthmachine.AprototypeofthehydraulicsupportwastestedinalaboratoryGrm1992.Thesupportexhibitedlargetransversaldisplacements,whichwouldreduceitsemployability.Therefore,aredesignwasnecessary.Theprojectshouldbeimprovedwithminimalcostifpos77Fig.2Twofourbarmechanismssible.Itwasdecidedtofindthebestvaluesforthemostproblematicparametersa1,a2,a4oftheleadingfourbarmechanismAEDBwithmethodsofmathematicalprogramming.Otherwiseitwouldbenecessarytochangetheproject,atleastmechanismAEDB.Thesolutionofaboveproblemwillgiveustheresponseofhydraulicsupportfortheidealsystem.Realresponsewillbedifferentbecauseoftolerancesofvariousparametersofthesystem,whichiswhythemaximalallowedtolerancesofparametersa1,a2,a4willbecalculated,withhelpofmethodsofmathematicalprogramming.2ThedeterministicmodelofthehydraulicsupportAtfirstitisnecessarytodevelopanappropriatemechanicalmodelofthehydraulicsupport.Itcouldbebasedonthefollowingassumptions–thelinksarerigidbodies,–themotionofindividuallinksisrelativelyslow.Thehydraulicsupportisamechanismwithonedegreeoffreedom.ItskinematicscanbemodelledwithsynchronousmotionoftwofourbarmechanismsFEDGandAEDBOblaketal.1998.TheleadingfourbarmechanismAEDBhasadecisiveinfluenceonthemotionofthehydraulicsupport.Mechanism2isusedtodrivethesupportbyahydraulicactuator.ThemotionofthesupportiswelldescribedbythetrajectoryLofthecouplerpointC.Therefore,thetaskistofindtheoptimalvaluesoflinklengthsofmechanism1byrequiringthatthetrajectoryofthepointCisasnearaspossibletothedesiredtrajectoryK.Thesynthesisofthefourbarmechanism1hasbeenperformedwithhelpofkinematicsequationsofmotiongivenbyRaoandDukkipati1989.ThegeneralsituationisdepictedinFig.3.Fig.3TrajectoryLofthepointCEquationsoftrajectoryLofthepointCwillbewritteninthecoordinateframeconsidered.CoordinatesxandyofthepointCwillbewrittenwiththetypicalparametersofafourbarmechanisma1,a2,...,a6.ThecoordinatesofpointsBandDarexBx−a5cosΘ,1yBy−a5sinΘ,2xDx−a6cosΘγ,3yDy−a6sinΘγ.4Theparametersa1,a2,...,a6arerelatedtoeachotherbyx2By2Ba22,5xD−a12y2Da24.6Bysubstituting1–4into5–6theresponseequationsofthesupportareobtainedasx−a5cosΘ2y−a5sinΘ2−a220,7x−a6cosΘγ−a12y−a6sinΘγ2−a240.8Thisequationrepresentsthebaseofthemathematicalmodelforcalculatingtheoptimalvaluesofparametersa1,a2,a4.782.1MathematicalmodelThemathematicalmodelofthesystemwillbeformulatedintheformproposedbyHaugandArora1979minfu,v,9subjecttoconstraintsgiu,v≤0,i1,2,...,lscript,10andresponseequationshju,v0,j1,2,...,m.11Thevectoruu1...unTiscalledthevectorofdesignvariables,vv1...vmTisthevectorofresponsevariablesandfin9istheobjectivefunction.ToperformtheoptimaldesignoftheleadingfourbarmechanismAEDB,thevectorofdesignvariablesisdefinedasua1a2a4T,12andthevectorofresponsevariablesasvxyT.13Thedimensionsa3,a5,a6ofthecorrespondinglinksarekeptfixed.TheobjectivefunctionisdefinedassomemeasureofdifferencebetweenthetrajectoryLandthedesiredtrajectoryKasfu,vmaxg0y−f0y2,14wherexg0yistheequationofthecurveKandxf0yistheequationofthecurveL.Suitablelimitationsforoursystemwillbechosen.ThesystemmustsatisfythewellknownGrasshoffconditionsa3a4−a1a2≤0,15a2a3−a1a4≤0.16Inequalities15and16expressthepropertyofafourbarmechanism,wherethelinksa2,a4mayonlyoscillate.Theconditionu≤u≤u17prescribesthelowerandupperboundsofthedesignvariables.Theproblem9–11isnotdirectlysolvablewiththeusualgradientbasedoptimizationmethods.Thiscouldbecircumventedbyintroducinganartificialdesignvariableun1asproposedbyHsiehandArora1984.Thenewformulationexhibitingamoreconvenientformmaybewrittenasminun1,18subjecttogiu,v≤0,i1,2,...,lscript,19fu,v−un1≤0,20andresponseequationshju,v0,j1,2,...,m,21whereuu1...unun1Tandvv1...vmT.AnonlinearprogrammingproblemoftheleadingfourbarmechanismAEDBcanthereforebedefinedasmina7,22subjecttoconstraintsa3a4−a1a2≤0,23a2a3−a1a4≤0,24a1≤a1≤a1,a2≤a2≤a2,a4≤a4≤a4,25g0y−f0y2−a7≤0,y∈vextendsinglevextendsingley,yvextendsinglevextendsingle,26andresponseequationsx−a5cosΘ2y−a5sinΘ2−a220,27x−a6cosΘγ−a12y−a6sinΘγ2−a240.28ThisformulationenablestheminimizationofthedifferencebetweenthetransversaldisplacementofthepointCandtheprescribedtrajectoryK.Theresultistheoptimalvaluesoftheparametersa1,a2,a4.793ThestochasticmodelofthehydraulicsupportThemathematicalmodel22–28maybeusedtocalculatesuchvaluesoftheparametersa1,a2,a4,thatthedifferencebetweentrajectoriesLandKisminimal.However,therealtrajectoryLofthepointCcoulddeviatefromthecalculatedvaluesbecauseofdifferentinfluences.Thesuitablemathematicalmodeldeviationcouldbetreateddependentlyontolerancesofparametersa1,a2,a4.Theresponseequations27–28allowustocalculatethevectorofresponsevariablesvindependenceonthevectorofdesignvariablesu.Thisimpliesv˜hu.Thefunction˜histhebaseofthemathematicalmodel22–28,becauseitrepresentstherelationshipbetweenthevectorofdesignvariablesuandresponsevofourmechanicalsystem.Thesamefunction˜hcanbeusedtocalculatethemaximalallowedvaluesofthetolerances∆a1,∆a2,∆a4ofparametersa1,a2,a4.Inthestochasticmodelthevectoruu1...unTofdesignvariablesistreatedasarandomvectorUU1...UnT,meaningthatthevectorvv1...vmTofresponsevariablesisalsoarandomvectorVV1...VmT,V˜hU.29ItissupposedthatthedesignvariablesU1,...,Unareindependentfromtheprobabilitypointofviewandthattheyexhibitnormaldistribution,Uk∼Nµk,σkk1,2,...,n.Themainparametersµkandσkk1,2,...,ncouldbeboundwithtechnologicalnotionssuchasnominalmeasures,µkukandtolerances,e.g.∆uk3σk,soeventsµk−∆uk≤Uk≤µk∆uk,k1,2,...,n,30willoccurwiththechosenprobability.TheprobabilitydistributionfunctionoftherandomvectorV,thatissearchedfordependsontheprobabilitydistributionfunctionoftherandomvectorUanditispracticallyimpossibletocalculate.Therefore,therandomvectorVwillbedescribedwithhelpofnumberscharacteristics,thatcanbeestimatedbyTaylorapproximationofthefunction˜hinthepointuu1...unTorwithhelpoftheMonteCarlomethodinthepapersbyOblak1982andHarl1998.3.1ThemathematicalmodelThemathematicalmodelforcalculatingoptimaltolerancesofthehydraulicsupportwillbeformulatedasanonlinearprogrammingproblemwithindependentvariablesw∆a1∆a2∆a4T,31andobjectivefunctionfw1∆a11∆a21∆a432withconditionsσY−E≤0,33∆a1≤∆a1≤∆a1,∆a2≤∆a2≤∆a2,∆a4≤∆a4≤∆a4.34In33EisthemaximalallowedstandarddeviationσYofcoordinatexofthepointCandσY1√6radicaltpradicalvertexradicalvertexradicalbtsummationdisplayj∈Aparenleftbigg∂g1∂ajµ1,µ2,µ4parenrightbigg2∆aj,A{1,2,4}.35Thenonlinearprogrammingproblemforcalculatingtheoptimaltolerancescouldbethereforedefinedasminparenleftbigg1∆a11∆a21∆a4parenrightbigg,36subjecttoconstraintsσY−E≤0,37∆a1≤∆a1≤∆a1,∆a2≤∆a2≤∆a2,∆a4≤∆a4≤∆a4.384NumericalexampleThecarryingcapabilityofthehydraulicsupportis1600kN.BothfourbarmechanismsAEDBandFEDGmustfulfillthefollowingdemand–theymustallowminimaltransversaldisplacementsofthepointC,and–theymustprovidesufficientsidestability.TheparametersofthehydraulicsupportFig.2aregiveninTable1.ThedrivemechanismFEDGisspecifiedbythevectorb1,b2,b3,b4T400,1325d,1251,1310Tmm,39andthemechanismAEDBbya1,a2,a3,a4T674,1360,382,1310Tmm.40In39,theparameterdisawalkofthesupportwithmaximalvalueof925mm.ParametersfortheshaftofthemechanismAEDBaregiveninTable2.
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