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CHINESEJOURNALOFMECHANICALENGINEERINGVol.22,aNo.4,a2009594DOI:10.3901/CJME.2009.04.594,；ReliabilitySimulationandDesignOptimizationforMechanicalMaintenanceLIUDeshun*,HUANGLiangpei,YUEWenhui,andXUXiaoyanHunanProvincialKeyLaboratoryofHealthMaintenanceforMechanicalEquipmentHunanUniversityofScienceandTechnology,Xiangtan411201,ChinaReceivedSeptember8,2008；revisedApril16,2009；acceptedApril30,2009；publishedelectronicallyMay5,2009Abstract:Reliabilitymodelofamechanicalproductsystemwillbenewlyreconstructedandmaintenancecostwillincreasebecausefailedpartscanbereplacedwithnewcomponentsduringservice,whichshouldbeaccountedforinsystemdesign.Inthispaper,areliabilitymodelandreliability-baseddesignoptimizationmethodologyformaintenancearepresented.First,basedonthetime-to-failuredensityfunctionofthepartofthesystem,theagedistributionsofallpartsofthesystemduringserviceareinvestigated,areliabilitymodelofthemechanicalsystemformaintenanceisdeveloped.Then,reliabilitysimulationsofthesystemswithWeibullprobabilitydensityfunctionsareperformed,thesystemminimumreliabilityandsteadyreliabilityformaintenancearedefinedbasedonreliabilitysimulationduringthelifecycleofthesystem.Thirdly,amaintenancecostmodelisdevelopedbasedonreplacementratesoftheparts,areliability-baseddesignoptimizationmodelformaintenanceispresented,inwhichtotallifecyclecostisconsideredasdesignobjectiveandsystemreliabilityasdesignconstrain.Finally,thereliability-baseddesignoptimizationmethodologyformaintenanceisusedtodesignofalinkringforthechainconveyor,whichshowsthatoptimaldesignwiththelowestmaintenancecostcanbeobtained,andminimumreliabilityandsteadyreliabilityofthesystemcansatisfyrequirementofsystemreliabilityduringserviceofthechainconveyor.Keywords:maintenance,reliability,simulation,designoptimization1IntroductionDuringthelifecycleofamechanicalproduct,maintenance,whichisimplementedonthejudgmentofpracticalstates,preservationandreconstructionofsomecertainstatesfortheproduct,isveryimportanttokeeptheproductavailableandprolongitslife.Studiesonmaintenanceformechanicalproductsareroughlyclassifiedintothefollowingthreecatalogs.(1)Howtoformulatemaintenancepolicyor(and)howtooptimizemaintenanceperiodsconsideringsystemreliabilityandmaintenancecost,e.g.,whensystemreliabilityissubjectedtosomecertainconditions,maintenancepolicyandoptimalmaintenanceintervalaredeterminedtomakemaintenancecostlowest14.(2)Todevelopmaintenancemethodsandtoolstoensuresystemmaintenancetobothlowcostandshortrepairtime,suchasspecialmaintenancetoolboxesdeveloped59.(3)Todesignformaintenance(DFM),namelyduringdesignprocedure,systemmaintainabilityisevaluatedand*Correspondingauthor.E-mail:ThisprojectissupportedbyNationalBasicResearchProgramofChina(973Program,GrantNo.2003CB317001),ScientificResearchFundofHunanProvincialEducationDepartmentofChina(GrantNo.07A018),HunanProvincialNaturalScienceFoundationofChina(GrantNo.07JJ5074),andNationalNaturalScienceFoundationofChina(GrantNo.50875082)isimproved1014.Maintenancestartsatdesign.Obviously,designmethodologyformaintenance,whichisoneofbesteffectivemaintenancemeansinthelife-cycleofaproduct,attractsmanyresearchersinterests.However,researchondesignformaintenanceismainlycentralizedontwofields.Oneismaintainabilityevaluationonproductdesignalternatives,theotherissomepeculiarstructuresofpartsdesignedforconvenientmaintenance.Forexample,computer-aidedmaintainabilityevaluationtoolsforproductdesign11,productassemblyanddisassemblysimulationprogramsformaintenance12,airplanedesignformaintenance13,andsoon.Butstudiesondesignmethodologiesconsideringproductreliability,maintenancecostandmaintenancepolicyareseldomreported.SHUandFLOWERoncepointedoutthatreckoninginlaborcostandproductionintervalcost,designdecisionofalternativesofthepartwouldbeinfluenced.However,subsequentresearchreportshavenotbeenpresented15.Inthispaper,basedonthetime-to-failuredensityfunctionofthepart,distributionsofserviceageofpartsforamechanicalsystemthatundergoesmaintenanceareinvestigated.Thenthereliabilitymodelofthemechanicalsystemisreconstructedandsimulated.Finally,anoveldesignoptimizationmethodologyformaintenanceisdevelopedandillustratedbymeansofdesignofalinkringforthechainconveyor.CHINESEJOURNALOFMECHANICALENGINEERING5952ReconstructionofReliabilityModelofMechanicalSystemforMaintenance2.1ModelassumptionsAfteramechanicalsystemrunssometime,duetoreplacementoffailparts,primaryreliabilitymodelisinapplicabletochangedsystem,thusthereliabilitymodelshouldbereconstructed.Themechanicalsystemdiscussedinthispaperhasfollowingcharacteristics.(1)Systemconsistsofalargenumberofsametypeparts,inwhichthenumberofpartsisconstantduringthewholelifecycleofthesystem.(2)Thetime-to-failuredensitydistributionfunctionsofallpartsarethesame,also,replacementpartshavethesamefailuredistributionfunctionsastheoriginalparts(3)Failureofeachpartisarandomindependentevent,i.e.,failureofonepartdoesnotaffectfailureofotherpartsinthesystem.Forexample,achainconveyorwidelyusedinmanyindustriesconsistsofalargenumberofsameroundrings,samelinksheetsandsamescrapeboards.Theirrespectivenumbersareconstantafterthechainconveyorisputintotheservice.Also,eachpart,beingsubjectedtosimilarworkconditionsandsimilarfailurestates,hasthesameoridenticaldensitydistributionoftimetofailure.Moreover,replacementpartshavefailuretimedensityfunctionsameoridenticaltotheoriginalpartsduringtheserviceofthechainconveyor.2.2ReliabilitymodelingformaintenanceReliabilityofamechanicalsystemdependsonitsparts,yetreliabilityandfailureprobabilityofwhichrestontheirserviceages.Herein,accordingtothedensitydistributionfunctionoftimetofailureofthepart,partserviceagedistributionofthemechanicalsystemiscalculated,thenreliabilitymodelofthemechanicalsystemformaintenanceisdeveloped.Duringtheserviceofamechanicalsystem,somepartsthatfailrequiretobereplacedintime,henceagedistributionofpartsofthemechanicalsystemundergoingmaintenancehasbeenchanged.Supposedthatafterthemechanicalsystemrunssometimentn=,whereistimebetweenmaintenanceactivities,i.e.,maintenanceinterval,theunitofcanbehours,days,months,oryears.If()inptrepresentsageproportionofpartsatntwithagei,thusagedistributionofpartsattimentdenotesmatrix01(),(),nnptpt(),inpt()nnpt.Thefailuredensityfunctionofpartsandcurrentagedistributionofpartsinthesystemdetermineagedistributionatnexttime,ortheportionofthecontentsofeachbinthatsurvivetothenexttimestep.Anagedistributionobtainedateachtimestepforeachpartpopulationdeterminesfailurerateforthefollowingtimestep.Tofindfailureprobabilityofpartsthefailuredensityfunctionisintegratedfromzerotont.Theportionofthepopulationthatsurvivesadvancestothenextagebox,andtheportionthatfailisreplacedbynewpartstobecomezeroagetoreenterthefirstbox.Initially,allpartsarenewandzeroageinthefirstbox.Thatis,at00t=,theportioninthefirstboxis00()1pt=.(1)At1t=,agefractionsofthefirstboxandthesecondboxarerepresentedas1100001000()()1()d,()()()d.ptptfxxptptfxx=(2)Portionsofbothageboxessurviveandadvancetothenextagebox,andportionsoffailedpartsfrombothboxesreplacedbynewpartsappearinthefirstbox.At22t=,theproportionsofthefirstthreeboxesarecalculatedasfollows:22211012010202110100()()1()d,()()1()d,()()()d()()d,ptptfxxptptfxxptptfxxptfxx=+#(3)So,atntn=,portionsofpartsineachboxarecalculatedbyusingthefollowingequations:110(1)1210(2)23103321022110()()1()d,()()1()d,()()1()d,()()1()d,()()1()d,nnnnnnnnnnnnnnnnnnnptptfxxptptfxxptptfxxptptfxxptptfxxp=#10101(1)0100()()1()d,()()()d.nnnininitptfxxptptfxx+=(4)Where0()nptisthefractionofpopulationofpartswithage0atnt,representingpartsthathavejustbeenputintoservice.Itmeansthat0()nptisfailurerateofparts,orreplacementrateoffailedparts.Inotherword，thefractionsofpartsinthefirstboxat01,ntttarenewpartsthatreplacethesefailedparts.AseriessystemconsistsofNpartsthathavethesamefailuredensitydistribution,eachpartisjustaseriesunit,andeachunitisrelativelyindependent.InseriessystemtheYLIUDeshun,etal:ReliabilitySimulationandDesignOptimizationforMechanicalMaintenanceY596failureofanyoneunitresultsinsystemfailure,inaccordingtotheprincipleofprobabilitymultiplication,thereliabilityofseriessystemsbecomes()00()1()d.inptNniniRtfxx=(5)Sincethenumberofpartsthatcomprisethesystemisconstant,here,thesystemreliabilityofthemechanicalsystemformaintenanceisdefinedas()00()()1()dinNnnptNniNiRtRtfxx=()001()d.inptniifxx=(6)Fromabovetosee,aslongasthetime-to-failuredensityfunctionandmaintenanceintervalaregiven,serviceagedistributionsofpartsandsystemreliabilitycouldbeobtainedbysimulation.3ReplacementRateandReliabilitySimulationforMaintenance3.1WeibulldistributionoftimetofailureTheWeibullprobabilitydensityfunctioniswidelyusedinfailuremodelinginmechanicalpartsandelectroniccomponents.HeretheWeibulldistributionwithtwoparametersisusedtosimulatereliabilityofthesystemthatisundergoingmaintenance,thatis,thetime-to-failuredensityfunctionofsystemsconstitutedpartsis1()exp,0xxfxx=.(7)InEq.(7),istheshapeparameter,isthescaleparameter.xistime,whoseunitecanbehours,days,oryears.FivefailuredensityfunctionswiththeirWeibullparameters10,1,2,3,4,5=aredescribedinFig.1.Itisshownthatislarge,beforeserviceageofpartsarrivesattheexpectedvalue,failureprobabilityofpartsisextremelylow.Whereas,issmall,manypartsfailsinshorttimeofservice.3.2ReliabilitysimulationDifferentmaintenanceintervalofthemechanicalsystemanddifferenttime-to-failuredensityfunctionofitspartsareselectedtosimulatereliabilityofthesystemshownasFig.2Fig.4.Fig.2showshowsimulationtimestep(maintenanceinterval)affectssystemreliability,theplotsshowncorrespondtomaintenanceinterval0.5,1,2=,andwithWeibulldistributionparameters4,10=.Fig.3plotstheinfluenceofthescaleparameterofWeibulldistributiononsystemreliability,andfourcurvesrepresentfourdifferenttypepartscorrespondingtoaconstantvalueofequalto4pairedwithvalueof8,10,12,15respectively.Fig.4revealshowtheshapeparameterofWeibulldistributionaffectssystemreliability,andWeibulldistributionparametersoffivecurvesare10,=1,2,3,4,5=.Correspondingly,theirreplacementratecurvesofsystemspartsforthesetime-to-failuredensitydistributionfunctionsareplottedinFig.5.Additionally,inFig.3Fig.5,maintenanceintervalis1=.Fig.1.WeibullprobabilitydistributionsFig.2.SystemreliabilityR(t)withFig.3.SystemreliabilityR(t)withSeveralcharacteristicsofthesefiguresareofinterest.First,thereliabilityandreplacementrateeventuallyreachessteadystate.ThisagreeswithDrenicksTheorem,whichCHINESEJOURNALOFMECHANICALENGINEERING597statesthesuperpositionofaninfinitenumberofindependentFig.4.SystemreliabilityR(t)withFig.5.Partreplacementratep0(t)equilibriumrenewalprocessishomogeneousPoissonprocess.Duringtheinitialstageofsystemservice,partsofthesystemare“new”,then,become“old”.Theportionofpartsthatfailgraduallyincreases,thusthepartreplacementrateincreasesandsystemreliabilitywilldropmonotonically.Withthereplacementofasignificantportionofthepopulation,portionofpartsthatfailwilldecrease,thusthepartreplacementratewilldropandthesystemreliabilitywillriseuntilthisoscillationisoverandnextoscillationbegins.Aftersomeoscillations,thepopulationbecomesmoreage-diversifiedwitheachoscillation,andtheagedistributionapproachessteady.Atthattime,theoscillationsinreplacementrateandsystemreliabilitydiminish.ComparedFig.4withFig.5,itisshownthatthetrendofreplacementrateiscontrarytothechangeofsystemreliability.Whensystemreliabilityincreases,partreplacementratereduces.Otherwise,assystemreliabilityreduces,partreplacementrateincreases.Secondly,thesteadystatevalueandthedegreeofoscillationofthesystemreliabilitydependonmaintenanceinterval.AsFig.2shows,thereliabilityrisesasmaintenanceintervaldecreasessincepartsthatfailarebeingreplacedmorequickly.Theshorterthemaintenanceintervalis,thehigherreliabilityis,andthesmalleroscillationsare.However,frequentrepairswillresultinhighermaintenancecost.Thirdly,thesteadystatevalueofthesystemreliabilitydependsontheparametersofWeibulldistribution.Thedependenceonisnotsurprising,highervaluesofforagivensetofyieldhighervaluesforexpecttimetofailureandthuslowerreplacementrateandhigherreliability.Moreinterestingly,withtheincreaseofthevalueof,thesteadyvaluesofreplacementratedecreaseandthesteadyvaluesofreliabilityincrease.Fourth,thedegreeofoscillationofsystemreliabilitydependsontheparametersofWeibulldistribution.Althoughtheinfluenceofonoscillationscanbeneglected,theinfluenceofonoscillationsshouldbepaidspecialattentionto.Biggervalueofdenotesthatfailurerateofpartsislowerbeforeserv