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STRUCTMULTIDISCOPTIM20,76–82SPRINGERVERLAG2000OPTIMALDESIGNOFHYDRAULICSUPPORTMOBLAK,BHARLANDBBUTINARABSTRACTTHISPAPERDESCRIBESAPROCEDUREFOROPTIMALDETERMINATIONOFTWOGROUPSOFPARAMETERSOFAHYDRAULICSUPPORTEMPLOYEDINTHEMININGINDUSTRYTHEPROCEDUREISBASEDONMATHEMATICALPROGRAMMINGMETHODSINTHEfiRSTSTEP,THEOPTIMALVALUESOFSOMEPARAMETERSOFTHELEADINGFOURBARMECHANISMAREFOUNDINORDERTOENSURETHEDESIREDMOTIONOFTHESUPPORTWITHMINIMALTRANSVERSALDISPLACEMENTSINTHESECONDSTEP,MAXIMALTOLERANCESOFTHEOPTIMALVALUESOFTHELEADINGFOURBARMECHANISMARECALCULATED,SOTHERESPONSEOFHYDRAULICSUPPORTWILLBESATISFYINGKEYWORDSFOURBARMECHANISM,OPTIMALDESIGN,MATHEMATICALPROGRAMMING,APPROXIMATIONMETHOD,TOLERANCE1INTRODUCTIONTHEDESIGNERAIMSTOfiNDTHEBESTDESIGNFORTHEMECHANICALSYSTEMCONSIDEREDPARTOFTHISEffORTISTHEOPTIMALCHOICEOFSOMESELECTEDPARAMETERSOFASYSTEMMETHODSOFMATHEMATICALPROGRAMMINGCANBEUSED,IFASUITABLEMATHEMATICALMODELOFTHESYSTEMISMADEOFCOURSE,ITDEPENDSONTHETYPEOFTHESYSTEMWITHTHISFORMULATION,GOODCOMPUTERSUPPORTISASSUREDTOLOOKFOROPTIMALPARAMETERSOFTHESYSTEMTHEHYDRAULICSUPPORTFIG1DESCRIBEDBYHARL1998ISAPARTOFTHEMININGINDUSTRYEQUIPMENTINTHEMINEVELENJESLOVENIA,USEDFORPROTECTIONOFWORKINGENVIRONMENTINTHEGALLERYITCONSISTSOFTWOFOURBARRECEIVEDAPRIL13,1999MOBLAK1,BHARL2ANDBBUTINAR31FACULTYOFMECHANICALENGINEERING,SMETANOVA17,2000MARIBOR,SLOVENIAEMAILMAKSOBLAKUNIMBSI2MPPRAZVOJDOO,PTUJSKA184,2000MARIBOR,SLOVENIAEMAILBOSTJANHARLUNIMBSI3FACULTYOFCHEMISTRYANDCHEMICALENGINEERING,SMETANOVA17,2000MARIBOR,SLOVENIAEMAILBRANKOBUTINARUNIMBSIMECHANISMSFEDGANDAEDBASSHOWNINFIG2THEMECHANISMAEDBDEfiNESTHEPATHOFCOUPLERPOINTCANDTHEMECHANISMFEDGISUSEDTODRIVETHESUPPORTBYAHYDRAULICACTUATORFIG1HYDRAULICSUPPORTITISREQUIREDTHATTHEMOTIONOFTHESUPPORT,MOREPRECISELY,THEMOTIONOFPOINTCINFIG2,ISVERTICALWITHMINIMALTRANSVERSALDISPLACEMENTSIFTHISISNOTTHECASE,THEHYDRAULICSUPPORTWILLNOTWORKPROPERLYBECAUSEITISSTRANDEDONREMOVALOFTHEEARTHMACHINEAPROTOTYPEOFTHEHYDRAULICSUPPORTWASTESTEDINALABORATORYGRM1992THESUPPORTEXHIBITEDLARGETRANSVERSALDISPLACEMENTS,WHICHWOULDREDUCEITSEMPLOYABILITYTHEREFORE,AREDESIGNWASNECESSARYTHEPROJECTSHOULDBEIMPROVEDWITHMINIMALCOSTIFPOS77FIG2TWOFOURBARMECHANISMSSIBLEITWASDECIDEDTOfiNDTHEBESTVALUESFORTHEMOSTPROBLEMATICPARAMETERSA1,A2,A4OFTHELEADINGFOURBARMECHANISMAEDBWITHMETHODSOFMATHEMATICALPROGRAMMINGOTHERWISEITWOULDBENECESSARYTOCHANGETHEPROJECT,ATLEASTMECHANISMAEDBTHESOLUTIONOFABOVEPROBLEMWILLGIVEUSTHERESPONSEOFHYDRAULICSUPPORTFORTHEIDEALSYSTEMREALRESPONSEWILLBEDIffERENTBECAUSEOFTOLERANCESOFVARIOUSPARAMETERSOFTHESYSTEM,WHICHISWHYTHEMAXIMALALLOWEDTOLERANCESOFPARAMETERSA1,A2,A4WILLBECALCULATED,WITHHELPOFMETHODSOFMATHEMATICALPROGRAMMING2THEDETERMINISTICMODELOFTHEHYDRAULICSUPPORTATfiRSTITISNECESSARYTODEVELOPANAPPROPRIATEMECHANICALMODELOFTHEHYDRAULICSUPPORTITCOULDBEBASEDONTHEFOLLOWINGASSUMPTIONS–THELINKSARERIGIDBODIES,–THEMOTIONOFINDIVIDUALLINKSISRELATIVELYSLOWTHEHYDRAULICSUPPORTISAMECHANISMWITHONEDEGREEOFFREEDOMITSKINEMATICSCANBEMODELLEDWITHSYNCHRONOUSMOTIONOFTWOFOURBARMECHANISMSFEDGANDAEDBOBLAKETAL1998THELEADINGFOURBARMECHANISMAEDBHASADECISIVEINflUENCEONTHEMOTIONOFTHEHYDRAULICSUPPORTMECHANISM2ISUSEDTODRIVETHESUPPORTBYAHYDRAULICACTUATORTHEMOTIONOFTHESUPPORTISWELLDESCRIBEDBYTHETRAJECTORYLOFTHECOUPLERPOINTCTHEREFORE,THETASKISTOfiNDTHEOPTIMALVALUESOFLINKLENGTHSOFMECHANISM1BYREQUIRINGTHATTHETRAJECTORYOFTHEPOINTCISASNEARASPOSSIBLETOTHEDESIREDTRAJECTORYKTHESYNTHESISOFTHEFOURBARMECHANISM1HASBEENPERFORMEDWITHHELPOFKINEMATICSEQUATIONSOFMOTIONGIVENBYRAOANDDUKKIPATI1989THEGENERALSITUATIONISDEPICTEDINFIG3FIG3TRAJECTORYLOFTHEPOINTCEQUATIONSOFTRAJECTORYLOFTHEPOINTCWILLBEWRITTENINTHECOORDINATEFRAMECONSIDEREDCOORDINATESXANDYOFTHEPOINTCWILLBEWRITTENWITHTHETYPICALPARAMETERSOFAFOURBARMECHANISMA1,A2,,A6THECOORDINATESOFPOINTSBANDDAREXBX−A5COSΘ,1YBY−A5SINΘ,2XDX−A6COSΘΓ,3YDY−A6SINΘΓ4THEPARAMETERSA1,A2,,A6ARERELATEDTOEACHOTHERBYX2BY2BA22,5XD−A12Y2DA246BYSUBSTITUTING1–4INTO5–6THERESPONSEEQUATIONSOFTHESUPPORTAREOBTAINEDASX−A5COSΘ2Y−A5SINΘ2−A220,7X−A6COSΘΓ−A12Y−A6SINΘΓ2−A2408THISEQUATIONREPRESENTSTHEBASEOFTHEMATHEMATICALMODELFORCALCULATINGTHEOPTIMALVALUESOFPARAMETERSA1,A2,A47821MATHEMATICALMODELTHEMATHEMATICALMODELOFTHESYSTEMWILLBEFORMULATEDINTHEFORMPROPOSEDBYHAUGANDARORA1979MINFU,V,9SUBJECTTOCONSTRAINTSGIU,V≤0,I1,2,,LSCRIPT,10ANDRESPONSEEQUATIONSHJU,V0,J1,2,,M11THEVECTORUU1UNTISCALLEDTHEVECTOROFDESIGNVARIABLES,VV1VMTISTHEVECTOROFRESPONSEVARIABLESANDFIN9ISTHEOBJECTIVEFUNCTIONTOPERFORMTHEOPTIMALDESIGNOFTHELEADINGFOURBARMECHANISMAEDB,THEVECTOROFDESIGNVARIABLESISDEfiNEDASUA1A2A4T,12ANDTHEVECTOROFRESPONSEVARIABLESASVXYT13THEDIMENSIONSA3,A5,A6OFTHECORRESPONDINGLINKSAREKEPTfiXEDTHEOBJECTIVEFUNCTIONISDEfiNEDASSOME“MEASUREOFDIffERENCE”BETWEENTHETRAJECTORYLANDTHEDESIREDTRAJECTORYKASFU,VMAXG0Y−F0Y2,14WHEREXG0YISTHEEQUATIONOFTHECURVEKANDXF0YISTHEEQUATIONOFTHECURVELSUITABLELIMITATIONSFOROURSYSTEMWILLBECHOSENTHESYSTEMMUSTSATISFYTHEWELLKNOWNGRASSHOffCONDITIONSA3A4−A1A2≤0,15A2A3−A1A4≤016INEQUALITIES15AND16EXPRESSTHEPROPERTYOFAFOURBARMECHANISM,WHERETHELINKSA2,A4MAYONLYOSCILLATETHECONDITIONU≤U≤U17PRESCRIBESTHELOWERANDUPPERBOUNDSOFTHEDESIGNVARIABLESTHEPROBLEM9–11ISNOTDIRECTLYSOLVABLEWITHTHEUSUALGRADIENTBASEDOPTIMIZATIONMETHODSTHISCOULDBECIRCUMVENTEDBYINTRODUCINGANARTIfiCIALDESIGNVARIABLEUN1ASPROPOSEDBYHSIEHANDARORA1984THENEWFORMULATIONEXHIBITINGAMORECONVENIENTFORMMAYBEWRITTENASMINUN1,18SUBJECTTOGIU,V≤0,I1,2,,LSCRIPT,19FU,V−UN1≤0,20ANDRESPONSEEQUATIONSHJU,V0,J1,2,,M,21WHEREUU1UNUN1TANDVV1VMTANONLINEARPROGRAMMINGPROBLEMOFTHELEADINGFOURBARMECHANISMAEDBCANTHEREFOREBEDEfiNEDASMINA7,22SUBJECTTOCONSTRAINTSA3A4−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−A24028THISFORMULATIONENABLESTHEMINIMIZATIONOFTHEDIffERENCEBETWEENTHETRANSVERSALDISPLACEMENTOFTHEPOINTCANDTHEPRESCRIBEDTRAJECTORYKTHERESULTISTHEOPTIMALVALUESOFTHEPARAMETERSA1,A2,A4793THESTOCHASTICMODELOFTHEHYDRAULICSUPPORTTHEMATHEMATICALMODEL22–28MAYBEUSEDTOCALCULATESUCHVALUESOFTHEPARAMETERSA1,A2,A4,THATTHE“DIffERENCEBETWEENTRAJECTORIESLANDK”ISMINIMALHOWEVER,THEREALTRAJECTORYLOFTHEPOINTCCOULDDEVIATEFROMTHECALCULATEDVALUESBECAUSEOFDIffERENTINflUENCESTHESUITABLEMATHEMATICALMODELDEVIATIONCOULDBETREATEDDEPENDENTLYONTOLERANCESOFPARAMETERSA1,A2,A4THERESPONSEEQUATIONS27–28ALLOWUSTOCALCULATETHEVECTOROFRESPONSEVARIABLESVINDEPENDENCEONTHEVECTOROFDESIGNVARIABLESUTHISIMPLIESV˜HUTHEFUNCTION˜HISTHEBASEOFTHEMATHEMATICALMODEL22–28,BECAUSEITREPRESENTSTHERELATIONSHIPBETWEENTHEVECTOROFDESIGNVARIABLESUANDRESPONSEVOFOURMECHANICALSYSTEMTHESAMEFUNCTION˜HCANBEUSEDTOCALCULATETHEMAXIMALALLOWEDVALUESOFTHETOLERANCES∆A1,∆A2,∆A4OFPARAMETERSA1,A2,A4INTHESTOCHASTICMODELTHEVECTORUU1UNTOFDESIGNVARIABLESISTREATEDASARANDOMVECTORUU1UNT,MEANINGTHATTHEVECTORVV1VMTOFRESPONSEVARIABLESISALSOARANDOMVECTORVV1VMT,V˜HU29ITISSUPPOSEDTHATTHEDESIGNVARIABLESU1,,UNAREINDEPENDENTFROMTHEPROBABILITYPOINTOFVIEWANDTHATTHEYEXHIBITNORMALDISTRIBUTION,UK∼NK,ΣKK1,2,,NTHEMAINPARAMETERSKANDΣKK1,2,,NCOULDBEBOUNDWITHTECHNOLOGICALNOTIONSSUCHASNOMINALMEASURES,KUKANDTOLERANCES,EG∆UK3ΣK,SOEVENTSK−∆UK≤UK≤K∆UK,K1,2,,N,30WILLOCCURWITHTHECHOSENPROBABILITYTHEPROBABILITYDISTRIBUTIONFUNCTIONOFTHERANDOMVECTORV,THATISSEARCHEDFORDEPENDSONTHEPROBABILITYDISTRIBUTIONFUNCTIONOFTHERANDOMVECTORUANDITISPRACTICALLYIMPOSSIBLETOCALCULATETHEREFORE,THERANDOMVECTORVWILLBEDESCRIBEDWITHHELPOF“NUMBERSCHARACTERISTICS”,THATCANBEESTIMATEDBYTAYLORAPPROXIMATIONOFTHEFUNCTION˜HINTHEPOINTUU1UNTORWITHHELPOFTHEMONTECARLOMETHODINTHEPAPERSBYOBLAK1982ANDHARL199831THEMATHEMATICALMODELTHEMATHEMATICALMODELFORCALCULATINGOPTIMALTOLERANCESOFTHEHYDRAULICSUPPORTWILLBEFORMULATEDASANONLINEARPROGRAMMINGPROBLEMWITHINDEPENDENTVARIABLESW∆A1∆A2∆A4T,31ANDOBJECTIVEFUNCTIONFW1∆A11∆A21∆A432WITHCONDITIONSΣY−E≤0,33∆A1≤∆A1≤∆A1,∆A2≤∆A2≤∆A2,∆A4≤∆A4≤∆A434IN33EISTHEMAXIMALALLOWEDSTANDARDDEVIATIONΣYOFCOORDINATEXOFTHEPOINTCANDΣY1√6RADICALTPRADICALVERTEXRADICALVERTEXRADICALBTSUMMATIONDISPLAYJ∈APARENLEFTBIGG∂G1∂AJ1,2,4PARENRIGHTBIGG2∆AJ,A{1,2,4}35THENONLINEARPROGRAMMINGPROBLEMFORCALCULATINGTHEOPTIMALTOLERANCESCOULDBETHEREFOREDEfiNEDASMINPARENLEFTBIGG1∆A11∆A21∆A4PARENRIGHTBIGG,36SUBJECTTOCONSTRAINTSΣY−E≤0,37∆A1≤∆A1≤∆A1,∆A2≤∆A2≤∆A2,∆A4≤∆A4≤∆A4384NUMERICALEXAMPLETHECARRYINGCAPABILITYOFTHEHYDRAULICSUPPORTIS1600KNBOTHFOURBARMECHANISMSAEDBANDFEDGMUSTFULfiLLTHEFOLLOWINGDEMAND–THEYMUSTALLOWMINIMALTRANSVERSALDISPLACEMENTSOFTHEPOINTC,AND–THEYMUSTPROVIDESUffiCIENTSIDESTABILITYTHEPARAMETERSOFTHEHYDRAULICSUPPORTFIG2AREGIVENINTABLE1THEDRIVEMECHANISMFEDGISSPECIfiEDBYTHEVECTORB1,B2,B3,B4T400,1325D,1251,1310TMM,39ANDTHEMECHANISMAEDBBYA1,A2,A3,A4T674,1360,382,1310TMM40IN39,THEPARAMETERDISAWALKOFTHESUPPORTWITHMAXIMALVALUEOF925MMPARAMETERSFORTHESHAFTOFTHEMECHANISMAEDBAREGIVENINTABLE2
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