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CHINESEJOURNALOFMECHANICALENGINEERINGVol.22,aNo.4,a2009594DOI10.3901/CJME.2009.04.594,availableonlineatwww.cjmenet.comwww.cjmenet.com.cnReliabilitySimulationandDesignOptimizationforMechanicalMaintenanceLIUDeshun,HUANGLiangpei,YUEWenhui,andXUXiaoyanHunanProvincialKeyLaboratoryofHealthMaintenanceforMechanicalEquipmentHunanUniversityofScienceandTechnology,Xiangtan411201,ChinaReceivedSeptember8,2008revisedApril16,2009acceptedApril30,2009publishedelectronicallyMay5,2009AbstractReliabilitymodelofamechanicalproductsystemwillbenewlyreconstructedandmaintenancecostwillincreasebecausefailedpartscanbereplacedwithnewcomponentsduringservice,whichshouldbeaccountedforinsystemdesign.Inthispaper,areliabilitymodelandreliabilitybaseddesignoptimizationmethodologyformaintenancearepresented.First,basedonthetimetofailuredensityfunctionofthepartofthesystem,theagedistributionsofallpartsofthesystemduringserviceareinvestigated,areliabilitymodelofthemechanicalsystemformaintenanceisdeveloped.Then,reliabilitysimulationsofthesystemswithWeibullprobabilitydensityfunctionsareperformed,thesystemminimumreliabilityandsteadyreliabilityformaintenancearedefinedbasedonreliabilitysimulationduringthelifecycleofthesystem.Thirdly,amaintenancecostmodelisdevelopedbasedonreplacementratesoftheparts,areliabilitybaseddesignoptimizationmodelformaintenanceispresented,inwhichtotallifecyclecostisconsideredasdesignobjectiveandsystemreliabilityasdesignconstrain.Finally,thereliabilitybaseddesignoptimizationmethodologyformaintenanceisusedtodesignofalinkringforthechainconveyor,whichshowsthatoptimaldesignwiththelowestmaintenancecostcanbeobtained,andminimumreliabilityandsteadyreliabilityofthesystemcansatisfyrequirementofsystemreliabilityduringserviceofthechainconveyor.Keywordsmaintenance,reliability,simulation,designoptimization1Introduction∗Duringthelifecycleofamechanicalproduct,maintenance,whichisimplementedonthejudgmentofpracticalstates,preservationandreconstructionofsomecertainstatesfortheproduct,isveryimportanttokeeptheproductavailableandprolongitslife.Studiesonmaintenanceformechanicalproductsareroughlyclassifiedintothefollowingthreecatalogs.1Howtoformulatemaintenancepolicyorandhowtooptimizemaintenanceperiodsconsideringsystemreliabilityandmaintenancecost,e.g.,whensystemreliabilityissubjectedtosomecertainconditions,maintenancepolicyandoptimalmaintenanceintervalaredeterminedtomakemaintenancecostlowest1−4.2Todevelopmaintenancemethodsandtoolstoensuresystemmaintenancetobothlowcostandshortrepairtime,suchasspecialmaintenancetoolboxesdeveloped5−9.3TodesignformaintenanceDFM,namelyduringdesignprocedure,systemmaintainabilityisevaluatedandCorrespondingauthor.Emailliudeshunhnust.edu.cnThisprojectissupportedbyNationalBasicResearchProgramofChina973Program,GrantNo.2003CB317001,ScientificResearchFundofHunanProvincialEducationDepartmentofChinaGrantNo.07A018,HunanProvincialNaturalScienceFoundationofChinaGrantNo.07JJ5074,andNationalNaturalScienceFoundationofChinaGrantNo.50875082isimproved10−14.Maintenancestartsatdesign.Obviously,designmethodologyformaintenance,whichisoneofbesteffectivemaintenancemeansinthelifecycleofaproduct,attractsmanyresearchersinterests.However,researchondesignformaintenanceismainlycentralizedontwofields.Oneismaintainabilityevaluationonproductdesignalternatives,theotherissomepeculiarstructuresofpartsdesignedforconvenientmaintenance.Forexample,computeraidedmaintainabilityevaluationtoolsforproductdesign11,productassemblyanddisassemblysimulationprogramsformaintenance12,airplanedesignformaintenance13,andsoon.Butstudiesondesignmethodologiesconsideringproductreliability,maintenancecostandmaintenancepolicyareseldomreported.SHUandFLOWERoncepointedoutthatreckoninginlaborcostandproductionintervalcost,designdecisionofalternativesofthepartwouldbeinfluenced.However,subsequentresearchreportshavenotbeenpresented15.Inthispaper,basedonthetimetofailuredensityfunctionofthepart,distributionsofserviceageofpartsforamechanicalsystemthatundergoesmaintenanceareinvestigated.Thenthereliabilitymodelofthemechanicalsystemisreconstructedandsimulated.Finally,anoveldesignoptimizationmethodologyformaintenanceisdevelopedandillustratedbymeansofdesignofalinkringforthechainconveyor.CHINESEJOURNALOFMECHANICALENGINEERING5952ReconstructionofReliabilityModelofMechanicalSystemforMaintenance2.1ModelassumptionsAfteramechanicalsystemrunssometime,duetoreplacementoffailparts,primaryreliabilitymodelisinapplicabletochangedsystem,thusthereliabilitymodelshouldbereconstructed.Themechanicalsystemdiscussedinthispaperhasfollowingcharacteristics.1Systemconsistsofalargenumberofsametypeparts,inwhichthenumberofpartsisconstantduringthewholelifecycleofthesystem.2Thetimetofailuredensitydistributionfunctionsofallpartsarethesame,also,replacementpartshavethesamefailuredistributionfunctionsastheoriginalparts3Failureofeachpartisarandomindependentevent,i.e.,failureofonepartdoesnotaffectfailureofotherpartsinthesystem.Forexample,achainconveyorwidelyusedinmanyindustriesconsistsofalargenumberofsameroundrings,samelinksheetsandsamescrapeboards.Theirrespectivenumbersareconstantafterthechainconveyorisputintotheservice.Also,eachpart,beingsubjectedtosimilarworkconditionsandsimilarfailurestates,hasthesameoridenticaldensitydistributionoftimetofailure.Moreover,replacementpartshavefailuretimedensityfunctionsameoridenticaltotheoriginalpartsduringtheserviceofthechainconveyor.2.2ReliabilitymodelingformaintenanceReliabilityofamechanicalsystemdependsonitsparts,yetreliabilityandfailureprobabilityofwhichrestontheirserviceages.Herein,accordingtothedensitydistributionfunctionoftimetofailureofthepart,partserviceagedistributionofthemechanicalsystemiscalculated,thenreliabilitymodelofthemechanicalsystemformaintenanceisdeveloped.Duringtheserviceofamechanicalsystem,somepartsthatfailrequiretobereplacedintime,henceagedistributionofpartsofthemechanicalsystemundergoingmaintenancehasbeenchanged.Supposedthatafterthemechanicalsystemrunssometimentnτ,whereτistimebetweenmaintenanceactivities,i.e.,maintenanceinterval,theunitofτcanbehours,days,months,oryears.Ifinptrepresentsageproportionofpartsatntwithageiτ,thusagedistributionofpartsattimentdenotesmatrix01{,,,nnptpt,,inpt}nnpt.Thefailuredensityfunctionofpartsandcurrentagedistributionofpartsinthesystemdetermineagedistributionatnexttime,ortheportionofthecontentsofeachbinthatsurvivetothenexttimestep.Anagedistributionobtainedateachtimestepforeachpartpopulationdeterminesfailurerateforthefollowingtimestep.Tofindfailureprobabilityofpartsthefailuredensityfunctionisintegratedfromzerotont.Theportionofthepopulationthatsurvivesadvancestothenextagebox,andtheportionthatfailisreplacedbynewpartstobecomezeroagetoreenterthefirstbox.Initially,allpartsarenewandzeroageinthefirstbox.Thatis,at00t,theportioninthefirstboxis001pt.1At1tτ,agefractionsofthefirstboxandthesecondboxarerepresentedas11000010001d,d.ptptfxxptptfxxττ⎧⎡⎤−⎪⎢⎥⎣⎦⎨⎪⎩∫∫2Portionsofbothageboxessurviveandadvancetothenextagebox,andportionsoffailedpartsfrombothboxesreplacedbynewpartsappearinthefirstbox.At22tτ,theproportionsofthefirstthreeboxesarecalculatedasfollows222110120102021101001d,1d,dd,ptptfxxptptfxxptptfxxptfxxττττ⎧⎡⎤−⎪⎢⎥⎣⎦⎪⎡⎤⎪−⎪⎢⎥⎨⎣⎦⎪⎪⎪⎪⎩∫∫∫∫3So,atntnτ,portionsofpartsineachboxarecalculatedbyusingthefollowingequations110112102231033210221101d,1d,1d,1d,1d,nnnnnnnnnnnnnnnnnnnptptfxxptptfxxptptfxxptptfxxptptfxxpτττττ−−−−−−−−−−−−⎡⎤−⎢⎥⎣⎦⎡⎤−⎢⎥⎣⎦⎡⎤−⎢⎥⎣⎦⎡⎤−⎢⎥⎣⎦⎡⎤−⎢⎥⎣⎦∫∫∫∫∫10101101001d,d.nnnininitptfxxptptfxxττ−−−⎧⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎨⎪⎪⎪⎪⎪⎡⎤⎪−⎢⎥⎣⎦⎪⎪⎪⎪⎩∫∑∫4Where0nptisthefractionofpopulationofpartswithage0atnt,representingpartsthathavejustbeenputintoservice.Itmeansthat0nptisfailurerateofparts,orreplacementrateoffailedparts.Inotherword,thefractionsofpartsinthefirstboxat01,,,ntttarenewpartsthatreplacethesefailedparts.AseriessystemconsistsofNpartsthathavethesamefailuredensitydistribution,eachpartisjustaseriesunit,andeachunitisrelativelyindependent.InseriessystemtheYLIUDeshun,etalReliabilitySimulationandDesignOptimizationforMechanicalMaintenanceY596failureofanyoneunitresultsinsystemfailure,inaccordingtotheprincipleofprobabilitymultiplication,thereliabilityofseriessystemsbecomes001d.inptNniniRtfxxτ⎡⎤′−⎢⎥⎣⎦∏∫5Sincethenumberofpartsthatcomprisethesystemisconstant,here,thesystemreliabilityofthemechanicalsystemformaintenanceisdefinedas001dinNnnptNniNiRtRtfxxτ′⎡⎤−⎢⎥⎣⎦∏∫001d.inptniifxxτ⎡⎤−⎢⎥⎣⎦∏∫6Fromabovetosee,aslongasthetimetofailuredensityfunctionandmaintenanceintervalaregiven,serviceagedistributionsofpartsandsystemreliabilitycouldbeobtainedbysimulation.3ReplacementRateandReliabilitySimulationforMaintenance3.1WeibulldistributionoftimetofailureTheWeibullprobabilitydensityfunctioniswidelyusedinfailuremodelinginmechanicalpartsandelectroniccomponents.HeretheWeibulldistributionwithtwoparametersisusedtosimulatereliabilityofthesystemthatisundergoingmaintenance,thatis,thetimetofailuredensityfunctionofsystemsconstitutedpartsis1exp,0xxfxxααααββ−⎛⎞⎛⎞⎜⎟−≤∞⎜⎟⎝⎠.7InEq.7,αistheshapeparameter,βisthescaleparameter.xistime,whoseunitecanbehours,days,oryears.FivefailuredensityfunctionswiththeirWeibullparameters10,1,2,3,4,5βαaredescribedinFig.1.Itisshownthatαislarge,beforeserviceageofpartsarrivesattheexpectedvalue,failureprobabilityofpartsisextremelylow.Whereas,αissmall,manypartsfailsinshorttimeofservice.3.2ReliabilitysimulationDifferentmaintenanceintervalofthemechanicalsystemanddifferenttimetofailuredensityfunctionofitspartsareselectedtosimulatereliabilityofthesystemshownasFig.2−Fig.4.Fig.2showshowsimulationtimestepmaintenanceintervalaffectssystemreliability,theplotsshowncorrespondtomaintenanceinterval0.5,1,2τ,andwithWeibulldistributionparameters4,10αβ.Fig.3plotstheinfluenceofthescaleparameterβofWeibulldistributiononsystemreliability,andfourcurvesrepresentfourdifferenttypepartscorrespondingtoaconstantvalueofαequalto4pairedwithβvalueof8,10,12,15respectively.Fig.4revealshowtheshapeparameterαofWeibulldistributionaffectssystemreliability,andWeibulldistributionparametersoffivecurvesare10,β1,2,3,4,5α.Correspondingly,theirreplacementratecurvesofsystemspartsforthesetimetofailuredensitydistributionfunctionsareplottedinFig.5.Additionally,inFig.3−Fig.5,maintenanceintervalis1τ.Fig.1.WeibullprobabilitydistributionsFig.2.SystemreliabilityRtwithτFig.3.SystemreliabilityRtwithβSeveralcharacteristicsofthesefiguresareofinterest.First,thereliabilityandreplacementrateeventuallyreachessteadystate.ThisagreeswithDrenicksTheorem,whichCHINESEJOURNALOFMECHANICALENGINEERING597statesthesuperpositionofaninfinitenumberofindependentFig.4.SystemreliabilityRtwithαFig.5.Partreplacementratep0tequilibriumrenewalprocessishomogeneousPoissonprocess.Duringtheinitialstageofsystemservice,partsofthesystemarenew,then,becomeold.Theportionofpartsthatfailgraduallyincreases,thusthepartreplacementrateincreasesandsystemreliabilitywilldropmonotonically.Withthereplacementofasignificantportionofthepopulation,portionofpartsthatfailwilldecrease,thusthepartreplacementratewilldropandthesystemreliabilitywillriseuntilthisoscillationisoverandnextoscillationbegins.Aftersomeoscillations,thepopulationbecomesmoreagediversifiedwitheachoscillation,andtheagedistributionapproachessteady.Atthattime,theoscillationsinreplacementrateandsystemreliabilitydiminish.ComparedFig.4withFig.5,itisshownthatthetrendofreplacementrateiscontrarytothechangeofsystemreliability.Whensystemreliabilityincreases,partreplacementratereduces.Otherwise,assystemreliabilityreduces,partreplacementrateincreases.Secondly,thesteadystatevalueandthedegreeofoscillationofthesystemreliabilitydependonmaintenanceinterval.AsFig.2shows,thereliabilityrisesasmaintenanceintervaldecreasessincepartsthatfailarebeingreplacedmorequickly.Theshorterthemaintenanceintervalis,thehigherreliabilityis,andthesmalleroscillationsare.However,frequentrepairswillresultinhighermaintenancecost.Thirdly,thesteadystatevalueofthesystemreliabilitydependsontheparametersofWeibulldistribution.Thedependenceonβisnotsurprising,highervaluesofβforagivensetofβyieldhighervaluesforexpecttimetofailureandthuslowerreplacementrateandhigherreliability.Moreinterestingly,withtheincreaseofthevalueofα,thesteadyvaluesofreplacementratedecreaseandthesteadyvaluesofreliabilityincrease.Fourth,thedegreeofoscillationofsystemreliabilitydependsontheparametersofWeibulldistribution.Althoughtheinfluenceofβonoscillationscanbeneglected,theinfluenceofαonoscillationsshouldbepaidspecialattentionto.Biggervalueofαdenotesthatfailurerateofpartsislowerbeforeservicetimeofpartsreachesexpectedlifetime,andthemajorityofpartsprolongusetime,thus,thesteadyvalueofsystemreliabilitybecomeshigher.However,inthiscase,themajorityofpartsfailatquitecentralizedtime,sominimumvalueofsystemreliabilityislower.Itissuggestedthatα,denotingconcentrativedegreeoffailuretimedistribution,isasensitiveparameter.Theinfluenceofαonsteadyvalueofreliabilityisdifferentfromandcontrarytothatofαonminimumvalueofreliability.Therefore,selectionofappropriateαshouldbepaidspecialattentiontoindesign,becausebothsteadyvalueandminimumreliabilitycoincidentallymeetdesignrequirements.3.3DefinitionsSimulationresultsshowthatsystemreliabilityvariesduringservice.Thereliabilityofasystemexperiencesseveraloscillations,sometimesismaximumvalueandthenminimumvalue,finallyreachessteadyvalue.Oscillationsofsystemreliabilityperiodicallydecay,andtheperiodisabouttheexpectedlifetimeofpartsμforWeibulldistribution,theparameterβapproximatesexpectedlifeatbigα.Fordesignandmaintenanceofmechanicalsystems,minimumvalueandsteadyvalueofsystemreliabilityareofimportance.Minimumreliabilityofthesystemappearsatbeginningstage,butsteadyreliabilityvalueofthesystemappearsafterrunningalongtime.Here,toconvenientlydiscusslater,minimumreliabilityandsteadyreliabilityofthesystemformaintenancearedefinedbasedonsimulationresultsofsystemreliabilityshownasinFig.6.Fig.6.Systemreliabilityparametersdefinition
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