土木工程畢業(yè)設(shè)計(jì)外文翻譯4

1356/JOURNALOFSTRUCTURALENGINEERING/NOVEMBER2000Theeigenvalues?iandeigenvectorsviofthecovariancema-trixsatisfy?vi=?ivi(3)Here,aneigenvectorviisalsocalledaprincipalcomponent.Toreducethem-dimensionalvectoru(t)intoad-dimensionalvector,xv(t),whered<m,u(t)isprojectedontotheeigen-vectorscorrespondingto?rstdlargesteigenvaluesxTv(t)=[v1иииvd]u(t)(4)Featureextractionistheprocessofidentifyingdamage-sen-sitivepropertiesderivedfromthemeasuredvibrationresponsethatallowsonetodistinguishbetweentheundamagedanddamagedstructures.Typically,systematicdifferencesbetweentimeseriesfromtheundamagedanddamagedstructuresarenearlyimpossibletodetectbyeye.Therefore,otherfeaturesofthemeasureddatamustbeexaminedfordamagedetection.Inthisstudy,thecoef?cientsofautoregressive(AR)modelsareselectedasdamagesensitivefeatures.Thetimeseriesfromanindividualmeasurementpoint,orthespatiallycompressedtimeseriesobtainedfromPCA,canbeusedtoconstructtheARmodels.IntheAR(n)model,thecurrentpointinatimeseriesismodeledasalinearcombinationofthepreviousnpointsny(t)=??jy(t?y)?e(t)(5)j=1wherey(t)=timehistoryattimet;?j=unknownARcoef?-cient;ande(t)=randomerrorwithzeromeanandconstantTheprecedingsectiondescribedmethodsforobtainingann-dimensionalfeaturespaceofARcoef?cients.Insuchsitu-ationwheremultidimensionalfeaturevectorsexist,severalmonitoringproceduresmaybeemployedforfeaturevectordiscrimination.Forexample,eachARcoef?cientcanbemon-itoredbyavarietyofstatisticalprocedures,orsimultaneousmonitoringofallARcoef?cientscanbedoneusingmultivar-iatestatisticalprocedures.However,forfeaturevectorswithahighdimensionality,the?rstapproachcanresultinalargeamountofdatatobemonitored,andthevisualizationofthemultivariatedatacanbeverydif?cult.Inthisstudythemul-tidimensionalfeaturevectorsareprojectedonto1Dsubspaces,andthestatisticaldiscriminationprocedureisappliedtothe1Dvariable.Twotransformations,linearandquadraticprojec-tions,arepresentedtomaximizetheseparationinfeaturesfromtheundamagedanddamagedstructures.naga(1990)showedthatadecisionboundaryD(x),basedonBayes’theoremminimizestheprobabilityoferror,whichistheprobabilityofmisclassi?cationofassigninganewfeaturetoclassAwhen,infact,itbelongstoclassB,orviceversa.IfclassesAandBhavenormaldistributions,theBayes’decisionruleD(x)canbewritteninaquadraticform(Fuku-naga1990)D(x)=xTQx?Vx(6)whereQ=quadraticprojectionmatrix;andV=linearpro-jection.InthecasewherethecovariancematricesforclassesAandBareidenticalmatrices,theclassi?cationboundarycanbefurthersimpli?edtoalinearformD(x)=FTx(7)TheQ,V,andFmatriceswillbeestimatedlaterinthissec-tion.Thedecisionrulecanalsobeviewedasaprojectionthatmapsmultidimensionalspacexto1DspaceD(x).Thepresentstudyisparticularlyinterestedinde?ningatransformedfea-ture?=D(x)suchthatthemeansoftwoclassesareasfaraspossibleandtheirvariancesarethesmallestpossibleafterei-therquadraticorlinearprojection.TheseprojectionscanbesoughtbymaximizingthefollowingFishercriterion(Bishop1995):f=(mA?mB)2FT(mA?mB)(mA?mB)TF?22=FT(?(8)A??BA??B)FwheremAandmB=meanvectorsoftheclassesAandBdistributions;?Aand?B=covariancematricesofeachclass;mAandmB=meansoftheprojectedfeatureinclassesAandB;and?Aand?B=correspondingstandarddeviationsofthetransformedfeatures,respectively.Furthermore,themomentsoftheprojectedfeaturearerelatedtothoseofthemultidi-mensionalfeaturevectorxasfollows:mi=FTmi;?2=FTi?iFfori=AorB(9a,b)TakingderivativesoffwithrespecttoFandsettingthisquan-tityequaltozero,yieldsthefollowinglinearprojection(Bishop1995):F=2(?A??B)?1(mA?mB)(10)Itisimportanttomentionthattheperformanceofthelinearclassi?erwillnotbeoptimalunless?Aand?Barethesame.Itisonlyundertheassumptionofequalcovariancematricesthatthedecisionrolereducestoalinearone.ForthetestdataemployedintheApplicationtoConcreteColumnssectionbe-low,accelerationdatafromundamagedanddamagedclassesareobservedtohaveunequalcovariancematrices.BecausetheBayesiandecisionboundaryisquadraticunderthemoregen-eralcircumstanceofunequalcovariancematricesbetweenclasses,thequadratictransformationyieldsthebestdiscrimi-nationpower.ThecalculationofthequadratictermQandlineartermVin(6)iscomputationallymoreintensivethanthelinearcase.However,introducinganewvariableyi,whichrepresentstheproductoftwoxis,(6)canbelinearizedinthefollowingform(Fukunaga1990):1)/2nD(x)=??nnn(n?qijxixj?vixi=i=1j=1?ni=1?aiyi?i(11)i=1?vixi=1whereqijandvi=componentsofQandV,respectively;yirepresentstheproductofthexjsandai=correspondingentryintheQmatrix.Inaddition,nistheorderoftheARmodelorthedimensionofARcoef?cientsde?nedin(5).JOURNALOFSTRUCTURALENGINEERING/NOVEMBER2000/1357LetYandXdenotecolumnvectorsofyisandxjs,respec-tively.Now,thefollowingequationanalogoustothelinearcasecanbesolvedforQandVbyintroducinganewvariablevectorZ=[YTXT]TandlettingEandSbetheexpectedvectorandcovariancematrixofZ,respectively[a?11иииan(n?1)/2v1иииvn]T=2[SA?SB](EA?EB)(12)ThenaisandvjscanberearrangedtoformtheQmatrixandVvector.Notethattheprojectiontechniquespresentedhereareusedforadimensionalityreductionpurposeaswellasforconstructionofadiscriminantfunction.Thatis,then-dimensionalARcoef?cientspaceisprojectedontoasinglescalarspacemaximizingthemeandifferencesbetweentwoclasses.Damagediagnosisisconductedonthetransformedfeatureusingthestatisticalprocesscontrol(SPC)techniquedescribedinthefollowingsection.STATISTICALMODELING—SPCStatisticalmodeldevelopmentisconcernedwiththeimple-mentationofthealgorithmsthatanalyzethedistributionofextractedfeaturestodeterminethedamagestateofthestruc-ture.Thealgorithmsusedinstatisticalmodeldevelopmentfallintothethreegeneralcategories:(1)Groupclassi?cation;(2)regressionanalysis;and(3)outlierdetection.Theappropriatealgorithmtousewilldependontheabilitytoperformsuper-visedorunsupervisedlearning.Here,supervisedlearningre-ferstothecasewhereexamplesofdatafromdamagedandundamagedstructuresareavailable.Unsupervisedlearningre-ferstothecasewheredataareonlyavailablefromtheundam-agedstructure.Thispaperfocusesonunsupervisedlearningmethods.Inthisstudy,controlchartanalysis,whichisthemostcom-monlyusedSPCtechniqueandverysuitableforautomatedcontinuoussystemmonitoring,isappliedtotheselectedfea-turestoinvestigatetheexistenceofdamageinthestructureofinterest.Whenthesystemofinterestexperiencesabnormalconditions,themeanand/orvarianceoftheextractedfeaturesareexpectedtochange.HereanX-barcontrolchartisem-ployedtomonitorthechangesoftheselectedfeaturemeansandtoidentifysamplesthatareinconsistentwiththepastdatasets.ApplicationoftheScontrolchart,whichmeasuresthevariabilityofthestructureovertime,tothecurrentteststruc-tureispresentedinFugateetal.(2000).SeveralvariationsofthecontrolchartscanbefoundinMontgomery(1997).Tomonitorthemeanvariationofthefeatures,thefeatures(i.e.,theARcoef?cientsorthetransformedfeatureafterlinearorquadraticprojection)are?rstarrangedinsubgroupsofsizep.Thevariable?ijisthejthfeaturefromtheithsubgroup.Thesubgroupsizepisoftentakentobe4or5(Montgomery1997).Ifpischosentoolarge,adriftpresentintheindividualsubgroupmeanmaybeobscured,oraveragedout.Anaddi-tionalmotivationforusingsubgroups,asopposedtoindivid-ualobservations,isthatthedistributionofthesubgroupmeanvaluescanbereasonablyapproximatedbyanormaldistribu-tionasaresultofcentrallimitNext,thesubgroupmeanXˉtheorem.iandstandarddeviationSiofthefeaturesarecomputedforeachsubgroup(i=1,...,q,whereqisthenumberofsubgroups)Xˉi=mean(?ij);Si=std(?ij)(13a,b)Here,themeanandstandarddeviationarewithrespecttopobservationsineachsubgroup.Finally,anX-barcontrolchartisconstructedbydrawingacenterline(CL)atthesubgroupmeanandtwoadditionalhorizontallinescorrespondingtotheupperandlowercontrollimits(UCLandLCL)versussub-groupnumbers(orwithrespecttotime).Thecenterlineandtwocontrollimitsarede?nedasfollows:1358/JOURNALOFSTRUCTURALENGINEERING/NOVEMBER2000UCL,LCL=CL?ZS?/2?n;CL=mean(Xˉi)(14a,b)wherethecalculationofmeaniswithrespecttoallsubgroupsbyaveragingtheiofNotethat,ifXˉsubgroups:S=mean(Si).monitoringofdamageoccurrenceisperformedbyplottingXˉivaluesobtainedfromthenewdatasetalongwiththeprevi-ouslyconstructedcontrollimits.APPLICATIONTOCONCRETECOLUMNSAhydraulicactuatorwasusedtoapplylateralloadstothetopofthecolumninaquasi-staticcyclicmanner.Theloadswere?rstappliedinaforce-controlledmannertoproducelat-eraldeformationsatthetopofthecolumncorrespondingto0.25?yT,0.5?yT,0.75?yT,and?yT.Here,?yTisthelateraldeformationatthetopofthecolumncorrespondingtothetheoretical?rstyieldofthelongitudinalreinforcement.Thestructurewascycledthreetimesateachoftheseloadlevels.Next,alateraldeformationcorrespondingtotheactual?rstyield?ywasestimatedbasedontheobservedresponse.Loadswerethenappliedinadisplacement-controlledmanner,againinsetsofthreecycles,atdisplacementscorrespondingto1.5?y,2.0?y,2.5?y,etc.,untiltheultimatecapacityofthecolumnwasreached.Vibrationtestswereconductedonthecolumninitsundam-agedstate,andaftercyclicloadingatthesubsequentdisplace-Operationalevaluationbeginstosetthelimitationsonwhatwillbemonitoredandhowtoperformthemonitoringaswellastailoringthemonitoringtouniqueaspectsofthesystemanduniquefeaturesofthedamagethatistobedetected.Becausetheteststructurewasalaboratoryspecimen,operationaleval-uationwasnotconductedinamannerthatwouldtypicallybeappliedtoaninsitustructure.However,becausethevibrationtestswerenottheprimarypurposeofthisinvestigation,com-promiseshadtobemaderegardingthemannerinwhichthevibrationtestswereconducted.Theprimarycompromisewasassociatedwiththemountingoftheshaker,whereitwouldhavebeenpreferabletosuspendtheshakerfromsoftsupportsandtoapplytheinputatapointlocationusingastinger.Thesecompromisesareanalogoustooperationalconstraintsthatmayoccurwithinsitustructures.Environmentalvariabilitywasnotconsideredanissuebecausethetestswereconductedinalab-oratorysetting.Theavailabledynamicmeasurementhardwareandsoftwareplacedtheonlyconstraintsonthedataacquisi-tionprocess.DataAcquisitionandCleansingFortyaccelerometersweremountedonthestructureasshowninFig.1.Theselocationswereselectedbasedontheinitialdesiretomeasuretheglobalbendingandaxialandtor-sionalmodesofthecolumn.Notethatatlocations2,39,and40theaccelerometershadanominalsensitivityof10mV/gandwerenotsensitiveenoughforthemeasurementsbeingmade.Atlocations33–37theaccelerometershadanominalsensitivityof100mV/g.Allotherchannelshadaccelerometerswithanominalsensitivityof1V/g.Anaccelerometerontheslidingmassoftheshakerprovidedameasureoftheinputforceappliedtothecolumn.Analogsignalsfromtheaccel-erometersweresampledanddigitizedwithacommercialdy-namicdataacquisitionsystem.Dataacquisitionparameterswerespeci?edsuchthat8-stimehistoriesdiscretizedwith8,192pointswereacquired.Nowindowingfunctionwasap-pliedtothesetimehistories.ThePCA,SPC,andprojectiontechniquesareillustratedus-ingthevibrationtestdataobtainedfromthetestcolumnshowninFig.1.First,theapplicabilityofSPCtothedamagediag-nosisproblemisdemonstratedusingasingleARcoef?cientobtainedfromindividualmeasurementpoint.Here,theARcoef?cientsarede?nedasdamaged-sensitivefeatures,andthesubsequentcontrolchartanalysisisconductedusingtheARcoef?cient(seeStatisticalModeling—X-BarControlChartUsingSingleARCoef?cientsectionbelow).Next,theadvan-tageofprojectiontechniquesisinvestigated.Linearandquad-raticprojectionsareintroducedtomapmultidimensionalARcoef?cientspaceinto1Dspacetomaximizethemeandiffer-encesbetweenthedatasetsobtainedfromtheundamagedanddamagedclasses(seeFalse-PositiveAlarmTesting).SPCanal-ysesarethenconductedonthetransformedsingle-scalefea-ture.Finally,PCAiscarriedouttoallresponsetimeseriesforspatialdimensionalityreductionpriortofeatureselectionandSPCanalysis(seePCAsectionbelow).Thatis,alltimeseriesfrom39responsepointsareprojectedontothe?rstprincipalJOURNALOFSTRUCTURALENGINEERING/NOVEMBER2000/1359FIG.2.TABLE1.X-BarControlChartUsingFirstARCoef?cientOutlierNumbersofX-BarControlChartUsingDifferentARCoef?cientsDamageLevelARcoef?cient?1?2?3Totalnumberofoutliers00/1280/1282/128a10/1280/1281/1281/384(0.26)26/1286/12812/12824/384(6.25)36/12810/12831/12847/384(12.24)42/12830/12877/128109/384(28.39)51/12823/12888/128112/384(29.17)2/384(0.52)1/128ponentofthecovariancematrixofthetimeseries.ThesubsequentfeatureselectionandSPCanalysesareperformedbasedonthissingletimeseries,whichisalinearcombinationofthe39measuredtimeseries.StatisticalModeling—X-BarControlChartUsingSingleARCoef?cientThe8,192-pointmeasuredtimeseriesare?rstdividedinto51216-pointtimewindows,andAR(3)is?ttoanindividualwindowresultingin512setsofARcoef?cients.Then,usingsubgroupsize4,128(=512/4)subgroupmeansareobtained.Fig.2showsthedamagediagnosisresultsusingthe?rstco-ef?cientoftheAR(3)model.Timehistoriesfrommeasurementpoint1showninFig.1areusedfortheconstructionofthecontrolchart.UCL,LCL,andCLdenotetheupperandlowercontrollimits,andcenterlineobtainedfromthetimeseriesoftheundamagedstructure.Thecontrollimitscorrespondingtoa99%con?denceintervalareconstructedbysetting?=0.01in(14).Aftertheconstructionofthecontrollimits,damagediagnosesusingtheX-barchartareperformedforsubsequentdamagelevels1–5.Notethattheextractedfeature?(the?rstARcoef?cientin1360/JOURNALOFSTRUCTURALENGINEERING/NOVEMBER2000thiscase)isstandardizedpriortotheconstructionoftheX-barcontrolchart:Themeanissubtractedfromthefeatureandthefeatureisnormalizedbythestandarddeviation.Therefore,CLforall?guresinthispapercorrespondstozero.Afteres-tablishingthecontrollimitsandcenterline,featuresobtainedateachdamagelevelareplottedrelativetothecontrollimitsandcenterlineobtainedfromtheundamageddata.Theoutliers,whicharesamplesoutsidethecontrollimits,areindicatedbya‘‘?’’inFig.2.Thefeaturesextractedateachdamagelevelarealsostandardizedinthesamefashionasbefore.Notethatthemeanandstandarddeviationestimatedfromdamagelevel0areusedtonormalizedatafromallofthesubsequentdamagelevels.ThediagnosisresultsusingtheotherARcoef?cientsarealsosummarizedinTable1.Forthisparticularexample,thethirdARcoef?cientseemsmostindicativeofdamage,andthe?rstcoef?cientisveryinsensitivetodamage.Fordamagelev-els0and1,thenumbersoftotaloutliersoutof384samplesare2and1,respectively.(TherearethreeARcoef?cientsand128samplesforeachARcoef?cient.Therefore,atotalof384samplesareobtained.)Theseareequivalentto0.52and0.26%ofoutliers.Consideringthefactthattheconstructedcontrollimitscorrespondtoa99%con?denceinterval,featuresex-TABLE2.OutlierNumbersofX-BarControlChartUsingLinearorQuadraticProjectionDamageLevelProjectionLinearQuadratica01/128a3/12815/1283/128224/12834/1283125/128128/1284121/128127/1285127/128128/1281/128indicatesthatasingleoutlierexistsoutof128sampledatapoints.tractedfromthein-controlsystemcanstillproduceapproxi-mately1%oftheoutlierswithoutindicatinganydamage.Therefore,itisnotclearifthesystemexperiencedanysignif-icantdamageatdamagelevel1basedontheanalysisoftheX-barcontrolchartusingtheindividualARcoef?cient.StatisticalModeling—ControlChartAnalysisafterLinearorQuadraticProjectionNext,theprojectiontechniquesareincorporatedintotheX-barcontrolchart.Asshowninthepreviousexample,someARcoef?cientsaremoresensitivetodamagethanothers.Fur-thermore,constructingseparatecontrolchartsforeachARco-ef?cientwouldbetimeconsuming.Toovercomethesedif?-culties,theconstructionofmultiplecontrolchartsusinganindividualARcoef?cientissimpli?edintoasinglecontrolchartusinga1Dtransformedfeature.Inthefollowingexam-ples,the3DARcoef?cientsare?rstprojectedontoa1Dspace,andtheX-barchartisconstructedbasedonthetrans-formedfeature.Ingeneral,theprojectionontoa1Dspaceleadstoalossofinformation,andclasseswellseparatedintheoriginalmultidimensionalspacemaybestronglyover-lappedintheprojectedspace.However,byusingtheFishercriterionin(8),theprojectionsaredeterminedtomaximizetheclassseparation.Table2showstheresultsofprocessmonitoringafteralin-earprojection.ComparisonofTable1andTable2clearlyre-vealstheimprovementofdiagnosisperformance.Again,thediagnosesinTable2areperformedusingthetimeseriesfrommeasurementpoint1.Diagnosisresultsusingtheothermea-surementpointsareconducted,andsimilarperformanceim-provementisobserved.However,thediagnosisresultsarenotpresentedbecauseofspacelimitations.Asmentionedearlier,thelinearprojectionmaynotbetheoptimalprojectioninthisexamplebecausetheordersoftwoclasscovariancematrices(onefromtheundamagedcaseandtheotherfromeachdam-agelevel)arequitedifferent.Intheory,thequadraticprojec-tionistheoptimaloneinasenseofminimizingtheerrorofmisclassi?cation.However,nosigni?cantperformancediffer-encebetweenlinearandquadraticprojectionsisobservedinthisexample(Table2).False-PositiveAlarmTestingWhileitisdesirabletohavefeaturessensitivetodamageoccurrence,themonitoringsystemalsoneedstoberobustagainstafalse-positiveindicationofdamage.Afalse-positiveindicationofdamagemeansthatthemonitoringsystemindi-catesdamagealthoughnodamageispresent.ToinvestigatetherobustnessoftheproposedX-barcontrolchartagainstafalse-positivewarningofdamage,twoseparatetestsarede-signed.Inthe?rsttest,thetimehistoriesobtainedfromtheundam-agedstateoftheteststructurearedividedintotwoparts.The?rsthalfofthetimeseriesisemployedtoconstructthecontrollimits,andthefalse-positivetestingiscarriedoutusingthesecondhalfofthetimeseries.Notethattheoriginaltimeseriesare8-slongwith8,192timepoints,andeachhalfofthetimeseriesis4-slongwith4,096points.Halfofthetimeseriesisfurtherdividedinto256setsof16-pointtimewindows,andAR(3)isagain?ttoeachtimewindowproducing256setsofARcoef?cients.Next,asmentionedbefore,fourconsecutiveARcoef?cientsaregroupedtogetherresultingin64sampleswithsubgroupsize4.Fig.3(a)showstheconstructionofthecontrollimitsusingthe?rsthalfofthetimeseries,andthe?uctuationofthefeaturesextractedfromthe?rsthalftimeseriesareplottedtogether.Fig.3(b)presentsthefalse-positivetestingusingthesecondhalfofthetimeseries.JOURNALOFSTRUCTURALENGINEERING/NOVEMBER2000/1361ofTable2andmuchbetterthanthoseofTable1.Thatis,PCAcondensesalltimeseriesinformationthatisspatiallydistrib-utedalongthecolumnandsuccessfullyidenti?esall?vedam-agecases.SUMMARYANDDISCUSSIONFIG.5.X-BarControlChartofARCoef?cientsafterPCAofAllMeasurementPointsandLinearProjectionTABLE3.DamageDiagnosisResultsafterPCAandLinear/QuadraticProjectionsDamageLevelProjectionLinearQuadratic01/128a(0.78)1/128(0.78)17/128(5.47)7/128(5.47)2127/128(99.22)126/128(98.44)3128/128(100.0)127/128(99.22)4120/128(93.75)121/128(94.53)5120/128(93.75)124/128(96.88)transformed1Dfeaturedata.Third,therobustnessofthepro-posedapproachagainstafalse-positiveindicationofdamageisdemonstratedusingtwoseparatetimehistoriesobtainedfromtheinitialteststructure.Finally,PCAiscarriedoutonallresponsetimeseriesforspatialdimensionalityreductionpriortofeatureextraction.Thatis,alltimeseriesfrommultiplemeasurementpointsareprojectedontothe?rstprincipalcom-ponentofthetimeseriescovariancematrix,andthesubse-quentfeatureselectionisperformedusingthiscompressedtimeseries.TheprojectiontechniquesimprovedtheperformanceofcontrolchartanalysiscomparedtothedamagediagnosisusingtheindividualARcoef?cient.WhentheprojectiontechniquesandPCAarecombined,thecontrolchartssuccessfullyindi-catedthesystemresponseanomalyforallinvestigateddamagelevelsbyshowingastatisticallysigni?cantnumberofoutliersoutsidethecontrollimits.Itshouldalsobenotedthatthisstudyiscarriedoutinanunsupervisedlearningmode.Al-thoughtheprojectiontechniquesrequiretwoseparatedatasets,noclaimismadethattheyarefromtwodifferentclasses.Itisonlyassumedthatthereisonedatasetfromtheundam-agedclassandthattheotherdatasetisfromanunknownclass.Theabilitytoapplyunsuperviseddamagedetectiontechniquestocivilengineeringstructuresisveryimportantbecausere-sponsedatafromasimilardamagedsystemarerarelyavail-able.Ingeneral,theobservationofalargenumberofoutliersinthecontrolchartdoesnotnecessarilyindicatethatthestructureisdamaged,butonlythatthesystemhasvaried,causingastatisticallysigni?cantchangeinitsvibrationresponse.Thisvariabilitycanbecausedbyavarietyofenvironmentalandoperationalconditionsthatthesystemissubjectedto.Becausethein?uenceofoperationalandenvironmentalfactorsonthedynamiccharacteristicsoftheteststructureisminimalforthepresentedlaboratorytest,thedeteriorationofthestructurewasassumedtobethemaincauseoftheabnormalchangesofthesystem.However,operationalandenvironmentalconditionssuchaswind,humidity,intensity,andfrequencyoftraf?cloadingshouldbetakenintoaccountforapplicationstoinsitucivilengineeringinfrastructures.Anovelapproachtodatanor-malization,combiningARandARwithexogenousinputs(ARX)techniques,isdevelopedtoexplicitlyincorporatetheenvironmentalandoperationalconditionsintothestatisticalpatternrecognitionparadigmsothattheeffectofdamageonthevibrationresponsecouldbediscriminatedfromtheseef-fectsandtopreventtheoperationalandenvironmentalvaria-bilityfromcausingf