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Analytical and experimental studies of modeling and monitoring uncertain nonlinear systems

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Analytical and experimental studies of modeling and monitoring uncertain nonlinear systems

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Content ANALYTICALANDEXPERIMENTALSTUDIESOFMODELINGANDMONITORING UNCERTAINNONLINEARSYSTEMS by Hae-BumYun ADissertationPresentedtothe FACULTYOFTHEGRADUATESCHOOL UNIVERSITYOFSOUTHERNCALIFORNIA InPartialFulfillmentofthe RequirementsfortheDegree DOCTOROFPHILOSOPHY (CIVILENGINEERING) August2007 Copyright 2007 Hae-BumYun tomyfamily ii Dedication Acknowledgments First, I would like to thank to Professor Sami Masri, my advisor and the chair of my Ph.D. committeeforhissupportandguidanceinmystudy. Inclassrooms, hehelpedmesetupsecure foundationofknowledge. Outofclassrooms, heshowedmehowascholarshouldlivehisdaily life. Iwasreallyhonoredtohaveachancetoworkwithhimclosely. Ialsowouldliketothankto my other Ph.D. committee members, Professors Carter Wellford, Roger Ghanem, Jiin-Jen Lee, andJohnCaffrey. Professor John Caffrey deserves special thanks for his comments and assistance in my lab- oratory works. His expertise and genius on experimental study, which helped me become a more proficient experimentalist. Especially, he gave me many helps in building the magneto- rheological (MR) damper test apparatus (Chapter 5 in this thesis). I also would like to thank to LanceHill,oursupportivelabmanagerincivilandenvironmentalengineeringdepartment. Iwasblessedwithhavingmyexcellentcolleagues,Dr. RaymondWolfe,Dr. MazenWahbeh, Dr. FarzadTasbihgoo,Dr. RezaNayeri,Dr. ElenaKallinikidou,Dr. ElenaKallinikidou,Miguel Hernandez, Mohammadreza Jahanshahi, and Reza Jafarkhani. We worked closely together on many interesting research topics, and the discussion with them was always inspirational and priceless. iii I would like to thank to Professors Robert Nigbor and Ahmed Abdel-Ghaffar, and Li-Hong Shengfortheircommentsandco-authorshiponthepublishedjournalpaperoftheforensicstudy ofcargoshipcollisionwiththeVincentThomasBridge(Chapter6inthisthesis). The assistanceof ProfessorNicos Makris and Dr. Cameron Black at the University of Cali- fornia,Berkeley,andProfessorGianmarioBenzoniattheUniversityofCalifornia,SanDiegoin theexperimentaltestsusingfull-scaleviscousdampersisappreciated. Theyarecoauthorsofthe journalpapersoftheexperimentalstudiesusingfull-scaleviscousdampers(Chapters3and4in thisthesis). The studies reported in this thesis were supported in part by grants from the National Sci- ence Foundation (NSF), the National Aeronautics and Space Administration (NASA), the Air Force Office of Scientific Research (AFOSR), and the California Department of Transportation (Caltrans). Last, but not least, I would like to extend special thanks to my family for their supports and encouragement in completing this work. My parents, Ki-Tae and Byoung-Jin Yun, and my mother-in-law,Il-JeeLee,haveprovidedtheirendlesssupportsthroughmystudy. Withouttheir supports, I should not be able to fulfil this work. My sister and brother, Hea-Bin and Hae- Duk Yun, deserve my special thanks for their words of encouragement, especially when I was staggering with many burdens in this process. I would like to thank to David and Joy Yun, my sonanddaughter. Havingthemduringmygraduatestudywasthebestmemoriesinmylife,and they gave me great pleasures during this long journey of study. Most of all, there is my wife, Myoung-Jae Yun. She has always stood beside me, supported me, and encouraged me. There iv are so many things I have owed her since I met her eighteen years ago, and my doctoral degree isoneofthosemanythings. v TableofContents Dedication ii Acknowledgments iii Abstract xv Chapter1:Introduction 1 1.1 Motivation 1 1.2 Objectives 7 1.3 Approaches 8 1.3.1 Comparison of Modeling Approaches for Full-Scale Nonlinear Viscous Damper 8 1.3.2 Data-Driven Methodologies for Change Detection in Large-Scale Non- linearDamperswithNoisyMeasurements 9 1.3.3 Model-OrderReductionEffectsonChangeDetectioninUncertainNon- linearMagneto-RheologicalDampers 10 1.3.4 MonitoringtheCollisionofaCargoShipwiththeVincentThomasBridge 11 1.4 Scope 11 Chapter2:OverviewofStructuralHealthMonitoring 13 2.1 ComponentsofStructuralHealthMonitoringSystems 13 2.1.1 SensingandInstrumentation 13 2.1.2 DataNetworkingandArchiving 14 2.1.3 AnalysisandInterpretation 15 2.2 DesignoftheStructuralHealthMonitoringSystems 18 Chapter3:ComparisonofModelingApproachesforFull-ScaleNonlinearViscousDampers 21 3.1 Introduction 21 3.1.1 Motivation 21 3.1.2 ViscousDamperTests 24 3.1.3 IdentificationofViscousDampers 26 vi ListofTables ix ListofFigures xi 3.1.4 ObjectivesandScope 26 3.2 ExperimentalStudies 27 3.2.1 TestApparatus 27 3.2.2 TestCases 27 3.2.3 Instrumentation 28 3.2.4 PreliminaryDataProcessing 28 3.3 OverviewofModelingApproaches 29 3.3.1 SimplifiedDesignModel 29 3.3.2 RestoringForceMethod 32 3.3.3 ArtificialNeuralNetworks 33 3.4 IdentificationoftheViscous 34 3.4.1 ParametricIdentificationofSimplifiedDesignModel 34 3.4.2 NonparametricIdentificationUsingRestoringForceMethod 37 3.4.3 NonparametricIdentificationUsingArtificialNeuralNetworks 41 3.5 Discussion 42 3.5.1 ConstitutiveLaw 42 3.5.2 FidelityofIdentifiedModels 43 3.5.3 IdentificationUsingtheDataSetswithConcatenatedSinusoidalExcitation 48 3.5.4 SignificanceofInertiaEffects 51 3.6 SummaryandConclusions 53 Chapter4:Data-DrivenMethodologiesforChangeDetectioninLarge-ScaleNonlinearDampers withNoisyMeasurements 55 4.1 Introduction 55 4.1.1 Motivation 55 4.1.2 Objective 59 4.1.3 Scope 60 4.2 ExperimentalStudies 60 4.2.1 TestApparatus 60 4.2.2 TestProtocolsandPreliminaryDataProcessing 61 4.3 Non-ParametricIdentification 64 4.3.1 OverviewofRestoringForceMethod 64 4.3.2 IdentificationofNonlinearViscousDampers 66 4.4 UncertaintyEstimationofDamperIdentification 72 4.4.1 DataGenerationofNoisyResponse 72 4.4.2 DamperIdentificationwithNoisyResponse 73 4.4.3 StatisticalChangeDetectionofTime-VaryingDamper 74 4.5 BootstrapEstimationoftheIdentificationUncertainty 80 4.5.1 OverviewoftheBootstrapMethod 80 4.5.2 BootstrapResamplingofNoisyResponseData 83 4.6 SummaryandConclusions 88 vii Chapter5:Model-OrderReductionEffectsonChangeDetectioninUncertainNonlinear Magneto-RheologicalDampers 91 5.1 Introduction 91 5.1.1 Motivation 91 5.1.2 Objectives 93 5.1.3 MethodologyandScope 93 5.2 ExperimentalStudy 96 5.2.1 TestApparatus 96 5.2.2 TestProtocols 97 5.3 Non-ParametricIdentificationofMR-Damper 100 5.3.1 OverviewofRestoringForceMethod 100 5.3.2 IdentificationResultsfortheMRDamper 101 5.3.3 PhysicalInterpretationsWithoutAssumingSystemModels 102 5.3.4 StochasticPropertiesoftheIdentifiedRFMCoefficients 109 5.4 StochasticChangeDetectionofMRDamper 116 5.4.1 OverviewofStatisticalClassificationwithPatternRecognitionMethods 117 5.4.2 SupervisedChangeDetectionUsingSupportVectorClassification 120 5.4.3 UnsupervisedChangeDetectionUsingk-MeansClustering 134 5.5 SummaryandConclusion 138 Chapter6:MonitoringtheCollisionofaCargoShipwiththeVincentThomasBridge 139 6.1 Introduction 139 6.1.1 Motivation 139 6.1.2 Objectives 140 6.1.3 Scope 142 6.2 Real-TimeMonitoringoftheBridge 142 6.2.1 BridgeDescription 142 6.2.2 VTBInstrumentation 144 6.2.3 Real-timeBridgeMonitoringSystem 145 6.3 PreliminaryDataProcessing 147 6.4 DescriptionoftheShipCollisionIncident 148 6.4.1 FactualInformationoftheIncident 148 6.4.2 VibrationMonitoringoftheIncident 150 6.4.3 BridgeResponseBeforeandAftertheIncident 151 6.5 SystemIdentificationoftheBridge 153 6.5.1 GlobalSystemIdentificationApproaches 153 6.5.2 LocalSystemIdentificationApproaches 163 6.5.3 ComparisonofGlobalandLocalIdentificationResults 167 6.6 SummaryandConclusions 169 Chapter7:SummaryandConclusion 172 viii ListofTables 3.1 Testspecificationsofthe1112kN(250kip)viscousdamper. 29 3.2 InitialvaluesandboundariesoftheunknownparametersinEquation3.1forthe AdaptiveRandomSearchmethod. 35 3.3 Mass, damping constant, and exponent identified using the simplified design model. 37 3.4 Identificationandvalidationresultsofthesimplifieddesignmodel,therestoring forcemethod,andtheartificialneuralnetworks. 38 3.5 NormalizedChebyshevcoefficientsandde-normalizedpowerseriescoefficients ofthetestedviscousdamperfortherestoringforcemethod. 41 3.6 The averaged normalized mean-square error of the restoring force method and the artificial neural networks identifications using a single and concatenated damperresponsedatasets. 51 3.7 EstimatedsignificanceofinertiaeffectsintheSDM-identification. 52 3.8 A comparison of investigated system identification methods for applications in structuralhealthmonitoring. 54 4.1 Summary of test protocols and preliminary data processing parameters for the threelarge-scalenonlinearviscousdampersusedinthisstudy. 65 4.2 SummaryoftheidentifiedcoefficientsusingtheRestoringForceMethod. 69 4.3 StatisticsoftheidentifiedRFMcoefficientsforthemultipletestsand3000noisy datasets. 75 4.4 Bootstrap estimations of standard errors for the coefficients identified using the RestoringForceMethod. 85 ix 6.1 Examplesofship-bridgecollisionswithfatalitiesindifferentcountries,listedin chronologicalorder(Mastaglio,1997;ProskeandCurback,2003). 140 6.2 Examples of major ship-bridge collision incidents in the U.S.A. during the past 30yearsreportedbyNationalTransportationSafetyBoard. 141 6.3 ComparisonoftheVTBmodalparameteridentificationresultsusingNExT/ERA for three different cases: (1) during accident (impact type excitation), (2) traffic shutdown,and(3)regulartraffic. 159 6.4 Comparisonofthebridgeidentificationresultswithpreviousstudiesfordifferent earthquakes. 161 6.5 Summaryofestimatedlocaldampingratiosofthebridgedeck. 166 6.6 Time lags and dominant frequencies of cross-correlation for different sensor readings. 167 6.7 A comparison of natural frequencies and damping ratios identified with global andlocalidentificationmethods. 170 x 5.1 MRdampertestprotocols. 100 5.2 Summary of the identification results for the MR damper using the Restoring ForceMethod. 103 5.3 Stochasticeffectsofmodel-orderreductiononthecoefficientidentificationwith orthogonalandnon-orthogonalbasisfunctions. 113 5.4 TheprecisionoftheSupportVectorClassificationprocedureforthestatistically independentChebyshevcoefficientsandthestatisticallycorrelatedpowerseries coefficients. 130 5.5 Parametersfork-meansclusteringfortheMRdamperchangedetection. 136 5.6 The results of k-means clustering for the MR damper change detection with differentnumbersoffeaturesandclasses. 137 ListofFigures 2.1 Generalprocedureforperformingstructuralhealthmonitoring. 13 2.2 Componentsandscopeofstructuralhealthmonitoringforcivilinfrastructure. 17 2.3 Preferreddesignapproachforstructuralhealthmonitoringprocedures. 20 3.1 Componentsofaorificedviscousdamper(SoongandDargush,1997). 22 3.2 The1112kN(250kip)viscousdamperinstalledonadampertestingmachineat theUniversityofCalifornia,Berkeley. 28 3.3 Sample time histories of measured damper response after preliminary data pro- cessing. 30 3.4 Sample identification results of the parametric simple design model, the non- parametric restoring force method, and the non-parametric artificial neural net- works. 36 3.5 ThenormalizedmeansquareerrorfordifferentChebyshevpolynomialorders. 39 3.6 An example of normalized Chebyshev coefficients and de-normalized power seriescoefficients. 40 3.7 Relationshipofpeakvelocitiesandpeakforcesatdifferentpeakdisplacements. 43 3.8 Normalized mean-square errors between the measured and the identified forces with parametric simplified design model, and non-parametric restoring force methodandartificialneuralnetworks. 44 3.9 Sample phase plots for the first order damping, the third order damping and the firstorderstiffnesstermsoftheidentifiedforceusingtherestoringforcemethod. 46 3.10 The normalized Chebyshev coefficients of the first and third order damping at differentpeakvelocitiesandpeakdisplacement,respectively. 47 xi 4.7 SampletimehistoriesofnoisyresponseofDamperB. 73 4.8 Sample scatter plots of the normalized Chebyshev coefficients and normalized powerseriescoefficientsforthenoisyresponseofDamperC(Window1inFig- ure4.4). 77 4.9 Histograms and probability density functions (pdf) of the first order damping normalizedChebyshevcoefficient( ¯ C ij )fordifferenttime-historywindows. 79 4.10 BootstrapresamplingproceduresforDampersBandCwithmeasureddisplace- ment(x)andforce(r). 84 4.11 Bootstrap resampling procedures for Damper A with measured acceleration (¨ x) andforce(r). 86 4.12 Time-correlations of the auto-regression (AR) residuals of the identified restor- ingforceresidual(ε e )andthedisplacement(ε x )fordifferentARorders. 88 4.13 AcomparisonoftheoriginalandBootstrap-resampleddatafordifferentnonlin- eardampers. 90 5.1 Themagneto-rheological(MR)dampertestapparatus. 98 5.2 Time histories of the measured and normalized displacements, velocities and forcesoftheMRdampersubjectedtosinusoidalexcitation. 99 xii 3.11 The“static”validationresultsoftheRFM-identification(a)andANN-identification (b)proceduresrandomlyshuffledinitssequentialorder. 50 4.1 Test facilities for large-scale viscous dampers at the University of California, Berkeley(UCB), and the University of California, San Diego (UCSD) used in thisstudy. 61 4.2 Timehistoriesofthemeasuredforcesfordifferentlarge-scalenonlinearviscous damperswithdisplacement-controlledexcitations. 63 4.3 TheidentificationresultsforDampersAandBusingtheRestoringForceMethod. 67 4.4 PartitioningthetimehistoryofthemeasuredforceofDamperCfortheRestoring ForceMethodidentification. 69 4.5 TheidentifiedcoefficientsofDamperCfordifferenttime-historywindows. 70 4.6 The measured and identified forces for the time-varying system of Damper C underthestationarysinusoidalexcitation. 71 xiii 5.3 A sample identification result for the MR damper using the Restoring Force Method. 104 5.4 The identified restoring forces that are dependent on the displacement only and velocity only using the non-orthogonal power series and orthogonal Chebyshev polynomialsfordifferentidentificationmodelorders. 106 5.5 Changesoftheidentifiedrestoringforcesthataredependentonthedisplacement only, velocity only, and coupled with both the displacement and velocity for differentMRdamperinputcurrents. 108 5.6 Changes of the identified normalized Chebyshev coefficients for different MR damperinputcurrents. 108 5.7 Term-wiseidentificationresultswithmodelordersof5and20withthenormal- izedChebyshevpolynomialbasisfunctions. 111 5.8 Bivariate Gaussian distributions of the identified Chebyshev coefficients of two dominant terms in the velocity ( ¯ C 01 ) and displacement ( ¯ C 10 ) for different MR damperinputcurrents. 115 5.9 ThedistributionsoftheidentifiedChebyshevcoefficientsforthefirstorderdamp- ing( ¯ C 01 )fordifferentMRdamperinputcurrents. 116 5.10 The means of the identified normalized Chebyshev coefficients with 1σ error barsfordifferentMRdamperinputcurrents. 117 5.11 SupportVectorClassification. 121 5.12 The classification precision of C-Support Vector Classification for different C andγ values. 128 5.13 TheprecisionsoftheSupportVectorClassificationforthestatisticallyindepen- dentChebyshevcoefficientsandstatisticallycorrelatedpower-seriescoefficients. 129 5.14 Detectionruleswithtwosourcesoferrors(TypeIandTypeIIerrors). 131 5.15 Theprobabilitiesofapparentsuccessfulclassification,TypeIerror,TypeIIerror and the power of test of the Support Vector Classification for the normalized ChebyshevcoefficientsfordifferentnumbersofthenormalizedChebyshevcoef- ficientsintheclassification. 133 6.1 TheVincentThomasBridge. 143 6.2 SensorlocationsanddirectionsontheVincentThomasBridge,SanPedro,CA. 145 xiv 6.3 AschematicoftheVTBreal-timemonitoringsystemarchitecture. 146 6.4 Preprocessed acceleration and displacement of the lateral direction at the mid- spanofthebridgedeck. 148 6.5 Schematicviewoftheincidentarea(courtesyofGoogleInc.) 149 6.6 SchematicviewoftheBeautifulQueen,acargoship,undertheVincentThomas Bridge. 149 6.7 Adamagedmaintenancescaffoldingmemberfromtheship-bridgecollision(Cour- tesyofCaltrans). 151 6.8 Displacements of the bridge deck and column on 27 August 2006 when the cargo-shipincidentoccurred. 152 6.9 Typical weekly root-mean-square displacements of the main span of the bridge deckinverticalandlateraldirectionsbeforeandaftertheship-bridgecollision. 154 6.10 Histograms of the natural frequencies and damping ratios of the first vertical bendingmodeidentifiedusingtheERAmethod. 162 6.11 Local identification of the damping ratio and natural frequency of the bridge deckinlateraldirectionduringtheincidentimpact. 164 6.12 Theverticalandtorsionaldisplacementsatthecenterofthebridgedeck. 165 6.13 Theestimationofdampingratiosfortorsionaldisplacement. 166 6.14 Cross-correlation and its frequency spectrum for the lateral displacements and verticaldisplacementsofthebridgedeck. 168 6.15 TopandlateralviewsofModeAidentifiedwiththeglobalidentificationmethods.168 Abstract The development of effective structural health monitoring (SHM) methodologies is imperative for the efficient maintenance of important structures in aerospace, mechanical and civil engi- neering. Basedonreliablecondition assessment, the ownersofmonitored structures can expect two important benefits: (1) to avoid catastrophic accidents by detecting various types of struc- tural deterioration during operation, and (2) to establish efficient maintenance means and time scheduletoreducemaintenancecosts. Avibration-basedSHMmethodologyisevaluatedforchangedetectioninnonlinearsystems that can be frequently seen in many engineering fields. The proposed methodology is advanta- geous over existing SHM methodologies regarding the following aspects: (1) feasible to detect small changes in complex nonlinear systems, (2) possible to make physical interpretation of detectedchanges, and(3)possibletoquantifytheuncertaintyassociatedwiththechangedetec- tion. A series of analytical and experimental studies was performed to investigate various impor- tant issues in modeling and monitoring of uncertain nonlinear systems. Different parametric and non-parametric identification methods were compared for monitoring purpose using full- scale nonlinear viscous dampers for seismic mitigation in civil structures. Then, the effects xv of uncertainty on change detection performance were investigated. Two types of uncertainty werestudied: measurementuncertainty(ormeasurementnoise)andsystemcharacteristicuncer- tainty (or variation of system parameters). For measurement uncertainty, three different types of full-scale nonlinear viscous dampers were used to validate the proposed SHM methodology when the dampers’ response was polluted with random noise. For system characteristic uncer- tainty,asemi-activemagneto-rheologicaldamperwhosesystemcharacteristicsweredetermined through user controllable input current was used. Statistical pattern recognition methods were studied to detect relatively small changes in nonlinear systems with different uncertainty types. The Bootstrap method, a statistical data resampling technique, was also studied to estimate the uncertainty bounds of change detection when the measurement data are insufficient for reliable statisticalinference. A web-based real-time bridge monitoring system was developed and used for a forensic study involving a cargo ship collision with the Vincent Thomas Bridge, a critical suspension bridgeinthemetropolitanLosAngelesregion. Keywords: structuralhealthmonitoring,systemidentification,RestoringForceMethod,artificial neuralnetworks,Hypothesistest,Bootstrapmethod,statisticalpatternrecognition,supportvec- tor machines, k-mean clustering, error analysis, detection theory, Natural Excitation Technique, Eigensystem Realization Algorithm, full-scale viscous dampers, magneto-rheological dampers, suspensionbridge,web-basedreal-timebridgemonitoringsystem,ship-bridgecollision. xvi Chapter1 Introduction 1.1 Motivation The development of effective structural health monitoring (SHM) methodologies is imperative for the efficient operation and maintenance of important structures in aerospace, mechanical and civil engineering. With the capability of reliable condition assessment using modern sens- ing, data networking and data analysis techniques, the operation and maintenance of monitored structurescanbeimprovedinthefollowingtwoways: 1. Toavoidcatastrophicaccidentsbydetectingvarioustypesofstructuraldeterioration,mod- ificationorchangesduringtheoperation. 2. Toestablishefficientmeansandtimeschedulesforstructuralmaintenanceorrehabilitation forthedetectedorpredictedstructuralchanges. Consequently, the efficiency of SHM methodologies is directly related to the operational costs and safety of monitored systems, and many SHM approaches have been developed for various applications in different science and engineering fields. An example can be found in the Inte- grated Vehicle Health Management (IVHM) program developed by the National Aeronautics and Space Administration (NASA). Using advanced smart sensing, diagnostic and prognostic techniques, and multi-level management and maintenance planning algorithms, the goal of the IVHM systems is to provide both real-time and life-cycle vehicle health information for the 1 second generation Reusable Launch Vehicle (RLV). Consequently, reliable and accurate SHM approaches play critical roles in the development of the IVHM. As shown in the tragedy of the spaceshuttleColumbia,thevehiclesneedtobemonitoredwithanintegratedarrayofonboardin- situ sensing systems rather than periodic, ground based structural integrity inspection (Mancini etal.,2006;Prosseretal.,2004). Thehealthinformationofthevehiclesiscontinuouslyupdated for estimating critical failure modes as well as routinely updated for estimating life cycle con- ditiontrending(NationalAeronauticsandSpaceAdministration,2007). Moreover,inoperating thespaceprograms,thehighprogram’stotalcost,whichislargelyinfluencedbytheefficiencyof theoperationandmaintenanceprocedures,wouldbeoneofthemostsubstantialobstaclestothe progress of space exploration (Schwabacher et al., 2002). The combination of continuous and routineassessmentsofthevehicles’healthinesscouldreducethehighoperationalcoststhrough quickervehicleturn-around(Aaseng,2001). Another example of the motivation for developing effective SHM methodologies can be seen in the maintenance of civil infrastructure system. Current practices of highway bridge inspection are based on the National Bridge Inspection Program (NBIP) (FHWA, 1972). Since 1972,theNBIPhasbeenmanagedbytheFederalHighwayAdministration(FHWA)toassessthe “health”conditionofmajorhighwaybridgesintheU.S.A.However,becausethismethodmainly reliesonvisualinspectionmethodsbyhumaninspectioncrews,theprogram’scostisexpensive, and the inspection results could be subjective and inaccurate. Hence, in order to overcome the limitations of the existing NBIP, the Long-Term Bridge Performance Program (LTBP) was recently proposed by the FHWA and approved by the U.S. Congress in 2006 (FHWA, 2006). 2 The purpose of the LTBP is to develop predictive models for bridge performance and asset- management decision making over 20 years, utilizing powerful sensing, instrumentation, test, monitoringandevaluationtechniques,whichareavailableinthesedays. A number of structural condition assessment approaches have been developed using mod- ern sensing, communication and computing technologies (Housner et al., 1997). Among them, vibration-based structural health monitoring techniques have been employed as promising con- dition assessment approaches. For numerous applications of critical structures in many engi- neering fields, numerous modeling approaches have been proposed worldwide to identify the monitored structures using the structures’ dynamic response measured with advanced sensing and data acquisition techniques (Fujino et al., 2004; Housner et al., 1997; Ou, 2004; Ou and Li,2004;Paiketal.,2004;Rodellar,2004;SpencerandYang,2004;TachibanaandMita,2006; Yun,2006). However,noneoftheproposedmethodscanbecomeuniversallyapplicabletodetect various modes and types of changes in complicated, monitored structures due to many limita- tions. For successful SHM, the developed SHM methodologies should possess the following importantfeatures: 1. (Detectability of system changes): Various types and modes of structural changes should be detectable. The monitored structures are frequently complex nonlinear systems. Due to structural deteriorations or changes, the structural characteristics vary over time (i.e., the structures are time-varying systems). In general, these changes involve not only the changes of system parameter values,butalsothe transformation (evolution) into different classesofnonlinearsystems. Unfortunately,theanalyticalmodelsofthetransformedsys- temsarecommonlyunknown. Ifthemonitoredstructuresarecomplexnonlinearsystems, 3 thenmodel-orderreduction wouldbenecessary,especiallywhentheexactsystemmodels areunknown,orwhentherapidcomputationtimeisasignificantconcern. 2. (Physicalinterpretations): Although the feasibility ofchange detection in nonlineartime- varyingsystemsisveryimportant,itisnottheonlyobjectiveforsuccessfulSHM.Inorder toestablisheffectiveoperationandmaintenancestrategiesforthemonitoredstructures,it is necessary to interpret the physical meanings of the detected changes. Consequently, structuralengineersshouldbeprovidedwithsomeengineering-based guidelinestoeffec- tively deal with the detected changes. The physical interpretations should involve (1) estimating the effects of the detected changes on the structural “healthiness”, at the full- structure level as well as at the component level, (2) characterizing the possible causes of thechanges,and(3)locatingthechanged(ordamaged)partsintheentirestructures. 3. (Uncertainty quantification): The uncertainty quantification of the detected changes should be possible since the dynamic response of monitored structures are usually influ- enced by various sources of uncertainty. In general, there are two types of uncertainty affecting the change detection performance: (1) measurement uncertainty of the system response, and (2) system characteristics uncertainty. Since the measurement uncertainty is due to various types of noise in the data acquisition processes, this uncertainty is often time-uncorrelated(i.e.,whitenoise). Ontheotherhand,the system characteristics uncer- tainty is often periodic and time-correlated (or colored noise) because this type of uncer- tainty is usually caused by structural characteristic changes due to various environmental effects, such as daily and yearly temperature changes. Using SHM techniques, it should 4 bepossibletodistinguishgenuinestructuralchangedetectionfrom“noisy”detection,and toestimatetheuncertaintybounds(orconfidenceintervals)ofthedetectedchanges. As discussed above, developing reliable and practical SHM methodologies is extremely chal- lenging,and,consequently,fewcurrentapproachessatisfythoserequirements. Commonlimita- tionsofcurrentSHMmethodologiesinclude: 1. The system models are over-simplified. The over-simplification is usually made in the following two ways: (1) excessive model-order reduction for nonlinear systems, and (2) lack of knowledge of significant environmental effects. Obviously, these two simplica- tionsmaketheidentificationresultsinaccurate(Peetersetal.,2001;SeberandLee,2003). In the development of current SHM methodologies, however, the effects of model-order reductiononthecorrespondingchangedetectionarerarelystudied. 2. The modeling approaches are not robust enough to identify time-varying structures. In general, two types of modeling approaches are used in SHM applications: (1) paramet- ric system identification methods and (2) non-parametric system identification methods. Because the modeling approaches of the parametric identification methods are based on some physical assumptions of the monitored structures, a priori knowledge of the struc- tures is required. Consequently, if the structures change into other classes of nonlinear systems due to unexpected structural changes, the identification results using the “old” models become no longer accurate. The non-parametric identification methods , how- ever, are more “flexible” than the parametric methods since the modeling processes of the non-parametric methods are data-driven, and no assumptions about the structures’ 5 physicalcharacteristicsarerequiredinitsmodelingprocess. Yunetal.(2007)experimen- tally demonstrated that the non-parametric modeling approaches are more advantageous inmonitoringpurposesthantheparametricapproaches. 3. Although current SHM methodologies adopting non-parametric system identification approaches (e.g., artificial neural networks, principal component analysis, etc.) allow detecting the changes in the structural characteristics, the physical interpretations of the detected changes are rarely possible. For the interpretation of the system changes, the parametric identification methods are more advantageous than the non-parametric meth- ods since the identified parameters are usually directly related to the structures’ physi- cal characteristics (e.g., mass, spring constant and damping constant in linear lumped- mass vibration model). On the other hand, because the identification models of the non- parametric methods usually do not have direct relationships to the structural characteris- tics(e.g., weights and biases of the artificial neural networks), it is difficult to interpret the detected changes. Consequently, there exists a trade-off between parametric and non- parametric modeling approaches, and using current SHM methodologies overcome this trade-off,thetrade-offcanberarelyovercome. 4. Most of current SHM methodologies are deterministic, and the uncertainty bounds of the detectedchangesareseldomestimated. Theestimationofthechangedetectionuncertainty should include various effects of the measurement uncertainty and system characteristics uncertaintyasdiscussedearlier. 6 1.2 Objectives The objective of this study is to develop effective modeling and monitoring methodologies for assessing the healthiness of uncertain, nonlinear, dynamic systems. The developed methodolo- gies are evaluated analytically and experimentally for complex nonlinear systems that can be frequentlyseenintheaerospace,mechanicalandcivilengineeringfields. Different vibration-based system identification methods are compared. For effective SHM, the modeling approaches should be able to identify complex nonlinear systems that change in time due to system deterioration, modification, or changes. Here, the system changes involve transformation into different classes of nonlinearities, as well as changes of system parameter values. IntheSHMpractice,sincethesystemcharacteristicsofthechangedsystemsareusually unknown,themodelingapproachesshouldnotbasedonspecificphenomenologicalmodels. Once the changes are detected, physical interpretation should be made on the detected changes to establish effective strategies to deal with the detected changes. Consequently, the modeling approaches of the developed SHM methodologies should be model-independent, but stillthephysicalinterpretationofthedetectedchangesshouldbepossible. For the uncertain response of the nonlinear systems affected by various types of random- ness,notonlythe“genuine”changesofthesystemcharacteristicsshouldbedetectible,butalso the uncertainty bounds on the change detection should be quantifiable. In addition, the quan- tified detection errors should be analyzed to improve the performance of the developed SHM methodologies. 7 1.3 Approaches In order to achieve these research objectives in a logical fashion, a series of investigations were performedinthisstudy,graduallyintroducingthecomplexitiesoftheproblemsmentionedabove byconductingthefollowingstudies: 1.3.1 Comparison of Modeling Approaches for Full-Scale Nonlinear Viscous Damper Usingafull-scalenonlinearviscousdamperthatisfrequentlyemployedtomitigateseismicand wind-induced vibration in civil structures, the results of a joint study between the University of Southern California (USC) and the University of California, Berkeley (UCB) are presented in this thesis. A series of tests is conducted at UCB with the viscous damper, and the obtained experimentaldataareanalyzedatUSC. Differentparametricandnon-parametricidentificationmethodsarecomparedtoachievethe followingimportantresearchobjectives: (1)toobtainquantitativedataonfull-scaletests,which arerarelyavailableduetothedamper’slargesize,(2)toobtaininformationontheaccuracyand utility of various modeling approaches, (3) to compare the advantages and limitations of para- metricandnon-parametricmodels,(4)tostudynonlinearfeaturesoffull-scaleviscousdampers, and(5)tostudymodeldependencyonthelevelofexcitation. 8 1.3.2 Data-DrivenMethodologiesforChangeDetectioninLarge-ScaleNonlinear DamperswithNoisyMeasurements Once different modeling approaches are compared, the effects of different types of uncertain- ties on the modeling fidelity of complex nonlinear systems must be investigated. Two types of uncertainties can be considered in the development of SHM methodologies: (1) measurement uncertainty,and(2)systemcharacteristicsuncertainty. Amongthesetwouncertaintytypes,theeffectsofmeasurementuncertaintyarefirstlyinves- tigated. Inthisexperimentalstudy,threedifferenttypesoflarge-scalenonlinearviscousdampers are used to understand the various effects of the system nonlinearities on the damper identifica- tion results. Aiming for the model-independent change detection discussed in Section 1.2, the goalofthisstudyistodevelopadata-drivenmethodologyforidentifyingvariousnonlinearvis- cousdampers. A joint study is performed between the University of Southern California (USC), the Uni- versityofCalifornia,SanDiego(UCSD)andtheUniversityofCalifornia,Berkeley(UCB).The experimentalresultsofthelarge-scaleviscousdamperstestedatUCSDandUCBarediscussed. Usingtheexperimentalresults,ananalyticalstudyisperformedatUSC.Inthisanalyticalstudy, the measured data are artificially polluted with random noise to investigate some aspects of the measurement uncertainty effects. Data-driven system identification methods are applied using thenoisydatasets. In general, the uncertainty quantification of the identification results requires multiple tests, which is not usually possible for in-situ monitoring, due to the lack of control of excitation sources. Evenifonehadthecontroloftheexcitation,performingmultipletestswithlarge-scale 9 viscousdamperswouldbeextremelydifficultbecauseoftheenormousamountofheatconverted fromthedissipatedenergy. Consequently,astatisticaldatarecyclingtechniqueisstudiedforthe uncertaintyquantification,evenwithalimitednumberofdatasetsforthestatisticalinference. 1.3.3 Model-OrderReductionEffectsonChangeDetectioninUncertainNonlin- earMagneto-RheologicalDampers Oncetheeffectsofmeasurementuncertaintyareunderstood,theeffectsofsystemcharacteristics uncertainty should be also investigated. The objective of this study is to investigate various effectsonmodelingandmonitoringnonlinearsystemswithuncertainsystemcharacteristics. For achieving this objective, passive type viscous dampers used in previous studies cannot beusedbecauseadirectcontrolofthedampers’physicalcharacteristicsisrequiredforaknown amountof“genuine”(oreffective)systemchangeswithaknownquantityofsystemuncertainty. Consequently, a semi-active magneto-rheological (MR) damper is used in this study. Multiple sets of damper’s response are obtained for Gaussian distributions of MR damper input currents with different means and standard deviations. Here, the mean of the distribution determines the effective system characteristics and the standard deviation of distribution determines the uncertaintyofsystemcharacteristics. In order to identify complex nonlinear systems, the model-order reduction of the identified systems is often necessary when the exact system models are unknown, or when a short com- putation time is an important concern. Hence, the effects of the model-order reduction on the systemchangedetectionshouldalsobealsostudied. Usingpowerfulstatisticalpatternrecognitionandclassificationalgorithms,thedetectability ofthe“genuine”systemchangeswithdifferentlevelsofsystemcharacteristicsuncertaintiesare 10 studied. The classification errors of the change detection are analyzed, and a classifier design approachfortheoptimalchangedetectionisproposed. 1.3.4 MonitoringtheCollisionofaCargoShipwiththeVincentThomasBridge Ingeneral,therearetwodifferentidentificationclassesconsideredinvibration-basedSHM:the component-level and full structure-level SHM. Once various important effects on the identifica- tion in uncertain nonlinear systems are investigated for developing the component-level SHM, thescopeofthisstudyshouldbeexpandedtothefullstructure-levelSHM. The Vincent Thomas Bridge (VTB), an important suspension bridge in southern California, was collided by a cargo ship in 2006. A web-based real-time continuous monitoring system installedonthebridgewassuccessfullymeasuredthebridge’sdynamicresponsebefore,during and after the collision. Using the valuable data sets obtained, a forensic study is performed to assessthechangeofthebridge’sstructuralintegrity,whichisusuallydifficulttodeterminewith human visual inspection. Global and local bridge characteristics are identified for the condition assessmentoftheinitsvibrationsignature. 1.4 Scope Thisthesisisorganizedinthefollowinglayout: anoverviewofthestructuralhealthmonitoring isdescribedinSection2;thecomparisonofmodelingapproachesforfull-scalenonlinearviscous dampers is discussed in Section 3; the data-driven methodologies for change detection in large- scale nonlinear dampers with noisy measurements is discussed in Section 4; an experimental 11 study of model-order reduction effects to change detection in uncertain nonlinear systems is discussedinSection5;andthesummaryandconclusionofthisstudyaregiveninSection7. 12 Chapter2 OverviewofStructuralHealthMonitoring 2.1 ComponentsofStructuralHealthMonitoringSystems Ingeneral,SHMsystemsconsistofthreesubsystemcomponents: (1)thesensingandinstrumen- tation component, (2) the data networking and archiving component, and (3) the analysis and interpretation component. Using those subsystem components, the general procedure involving SHM is performed following the order shown in Figure 2.1. Important issues and roles of each subsystemcomponentarealsosummarizedinFigure2.2. Inthefollowingsection,theobjectives andscopesofeachcomponentofSHMsystemsaredescribed. 2.1.1 SensingandInstrumentation The role of the sensing and instrumentation component is to obtain physical measurements of theresponseofmonitoredstructuresusingvarioustypesofsensorsanddataacquisitionsystems. Inthedesignofthesensingandinstrumentationcomponent,thefollowingimportantissuesneed tobeconsidered: 1. Sensortypes, SENSING & INSTRUMENTATION DATA NETWORKING & ARCHIVING ANALYSIS & INTERPRETATION SENSING & INSTRUMENTATION DATA NETWORKING & ARCHIVING ANALYSIS & INTERPRETATION 1. Determine the objectives and scopes of maintenance policies. 2. Determine the objectives and scopes of the SHM. 3. Design optimal combinations of analysis “toolkits” to meet the objectives. 4. Establish detailed data interpretation strategies using the “toolkits”. Design the data acquisition system based on analysis strategies considering the following aspects: 1. sensor types 2. sensor locations and intensities 3. sampling frequencies and resolutions 4. necessary signal conditioning and preliminary data processing 5. measurement frequencies (continuous, temporal and snap-shot) 6. excitation types (ambient and forced) Design sensor networks based on analysis strategies considering the following aspects: 1. sensor network types (centered and distributed) 2. device network types (GPIB, VISA, VXI, PXI, serial, WI-FI, bluetooth, etc.) 3. remote data communication types (UDP, modified UDP, TCP-IP, etc.) 4. archived data types 5. data archiving frequencies Figure2.1: Generalprocedureforperformingstructuralhealthmonitoring. 13 2. Sensorlocationsanddensities, 3. Samplingfrequenciesandresolutions, 4. Necessarysignalconditioningandpreliminarydataprocessingtechniques, 5. Measurementfrequencies,suchascontinuous,temporal,andsingle-timemonitoring, 6. Excitationtypes,suchasambientandforcedvibration. 2.1.2 DataNetworkingandArchiving Thedatanetworkingandarchivingcomponentsinvolvetranceivingandarchivingthemeasured dataforfurtheranalyses. Typicalconsiderationsinthedesignofthedatanetworkingandarchiv- ingsystemcomponentinvolve: 1. Sensornetworktypes,suchascenteredordistributednetworks, 2. Device network types, including General Purpose Interface Bus (GPIB or IEEE-488), Virtual Instrument Software Architecture (VISA), VME eXtensions for Instrumentation (VXI),PCIeXtensionsforInstrumentation(PXI),serial,WI-FI,bluetooth,etc., 3. Remote data communication types, such as User Datagram Protocol (UDP), modified UDP,TransmissionControlProtocolandtheInternetProtocol(TCP/IP),etc., 4. Methodofdataarchiving,and 5. Frequencyofdataarchiving. 14 2.1.3 AnalysisandInterpretation The objectives of the analysis and interpretation component include the following three impor- tanttasksfortheeffectiveSHMsystems: (1)theidentificationofmonitoredsystems, (2)detec- tion of changes in monitored systems, and (3) interpretation of detected damage mechanisms and establishment of maintenance strategies. In designing the analysis and interpretation com- ponents,thefollowingimportantissuesneedtobeconsidered: Systemidentification 1. Scope • regional/system-level/component-level 2. Modeling • parametric/non-parametric • linear/nonlinear • stationary/non-stationary • discrete/continuous • single-input/multiple-input • deterministic/stochastic Systemchangedetection 1. FeasibilityofChangeDetection • Detectionresolution 15 • Estimationofdetectionuncertainty(ordetectionconfidence) 2. PhysicalInterpretationofDetectedDamage • Effectsonthestructuralcharacteristics • Significanceinregardtostructuralhealthiness • Understandingdamagemechanism • Damagelocations Damagemechanismestimationformaintenancestrategies 1. IntegrationofDamageDetectionResultsfromMultipleHeterogeneousCivilStructures 2. DamagePrediction • Predictionoffuturedamagebasedonthetrendsofdetecteddamageintime • Estimationofpredictionuncertainty 3. ReliableMaintenanceStrategy • Reliabledecision-makingbasedonpredicteddamagedevelopment • Planingeffectivebudgetpolicyforinfrastructureoperationandmaintenance 16 STRUCTURAL HEALTH MONITORING SENSING & INSTRUMENTATION DATA NETWORKING & ARCHIVING ANALYSIS & INTERPRETATION EXCITATION - Ambient Vibration - Forced Vibration Impact / sinusoidal / swept-sine / random Narrow-band / broadband SENSING - Transducers Passive / active Point / line / plane Contact / non-contact - Measurands Structure response only (disp, vel, acc, etc.) Structure response and excitation - Sensor Spatial Resolution Lower for lumped mass systems (e.g. buildings) Higher for distributed systems (e.g. bridges) INSTRUMENTATION - Measurement Frequency Continuous / temporal (consecutive) / snap-shot - Sampling Frequency Fast (high resolution, more data, expensive, short-term) Slow (low resolution, less data, economical, long-term) - Device Type Stand-alone / computer-based SIGNAL CONDITIONING - Amplification - Anti-aliasing Filtering / Post-filtering - Differentiation / Integration SENSOR NETWORKING - Network Type centered / distributed electric-wired / optic-wired / wireless - Wireless Sensor Networking DEVICE NETWORKING - Communication between Devices GPIB / VISA / VXI / PXI / serial REMOTE DATA TRANSMITTING / RECEIVING - Internet-Based Communication Protocols UDP / modified UDP / TCP/IP ARCHIVING MEASURED DATA FROM ONE OR MORE CIVIL STRUCTURES - Data Compression and Reduction Techniques SYSTEM IDENTIFICATION - Scope regional / system-level / component-level - Modeling Parametric / non-parametric Linear / nonlinear Stationary / non-stationary Discrete / continuous Single-input / multi-input Deterministic / stochastic DAMAGE (CHANGE) DETECTION - Capability of Change Detection Detection resolution Estimation of detection uncertainty (or confidence) - Physical Interpretation of Detected Damage Effects to the structural characteristics Significance in structural healthiness Understanding damage mechanism Locations DAMAGE PREDICTION & MAINTENANCE STRATEGY (POLICY) - Integration of Damage Detection Results from Multiple Heterogeneous Civil Structures - Damage Prediction Prediction of future damages based on time history of detected damages Estimation of the prediction uncertainty (or confidence) - Maintenance Strategy (Policy) Rational decision making based on predicted damage development Planning effective budget policy for infrastructure operation and maintenance Figure2.2: Componentsandscopeofstructuralhealthmonitoringforcivilinfrastructure. 17 2.2 DesignoftheStructuralHealthMonitoringSystems ThedevelopmentofeffectiveSHMmethodologiesisadesignprocesstoachievevariousdesign objectivesoftheSHMapplicationswithspecified design constraintsand specifications. Conse- quently, establishing clear objectives and scopes of given applications is crucial for successful SHM.Thisinvolvesthefollowingtwoissues: 1. Determiningtheobjectivesandscopesofoperationandmaintenancepolicies, 2. Determiningtheobjectivesandscopesofthemodelingandmonitoringapproaches. Among them, the former issue is the controlling goal of SHM, while the latter involve prac- tical approaches to achieve those ultimate goals. In the design of the analysis/interpretation component, a number of modeling and monitoring approaches could be used simultaneously to achievethevariousdesignobjectives. Consequently,theoptimalcombinationsofmodelingand monitoringapproaches(or“toolkits”)shouldbedetermined. Foreachmodelingandmonitoring approachinthe“toolkits”,detailedidentificationandchangedetectionstrategiesshouldbealso considered. Once the analysis/interpretation component is designed, the sensing/instrumentation and data networking/archiving components need to be designed as the next step. The implemen- tation of those two components should be performed to meet the pre-determined objectives of theanalysis/interpretationcomponent. Theabovediscussionindicatesthat,unliketheprocedureforperformingSHMinFigure2.1, theprocedureofdesigningSHMsystemsshouldconsidertheanalysis/interpretationcomponent first,thentoconsiderthesensing/instrumentationanddatanetworking/archivingcomponentsto 18 meetthegoalsofSHMapplicationsasillustratedinFigure2.3. Inaddition,becausetheperfor- manceoftheanalysis/interpretationcomponentismoredirectlyrelatedtoachievingtheultimate objectives of the SHM system, the analysis/interpretation component needs to be considered morecarefullythanthesensing/instrumentationanddatanetworking/archivingcomponents. Many current SHM methodologies, however, tend to put too much emphasis on the sens- ing/instrumentation and data networking/archiving components. With the advent of modern sensinganddataacquisitiontechnologies,thedevelopmentoftherequiredsensinganddatanet- workingsystemsbecomesmorefeasibleinmanySHMapplications. However,thedevelopment ofeffectiveanalysisandinterpretationcomponentsforcomplexnonlinearstructuresisstillvery challenging,anditconsequentlybecomesoneofthemajorobstaclestodesigningreliableSHM. Hence, in this study, the development of the analysis/interpretation methodologies will be more focused. However, the development of an effective sensing/instrumentation and data network- ing/archiving system for a full-scale suspension bridge is also conducted as a part of the study, andthedescriptionofthedevelopedbridgemonitoringsystemsispresentedinSection6.2.3. 19 SENSING & INSTRUMENTATION DATA NETWORKING & ARCHIVING ANALYSIS & INTERPRETATION SENSING & INSTRUMENTATION DATA NETWORKING & ARCHIVING ANALYSIS & INTERPRETATION 1. Determine the objectives and scope of maintenance policies. 2. Determine the objectives and scope of the SHM. 3. Design optimal combinations of analysis “toolkits” to meet the objectives. 4. Establish detailed data interpretation strategies using the “toolkits”. Design the data acquisition system based on analysis strategies considering the following aspects: 1. sensor types 2. sensor locations and intensities 3. sampling frequencies and resolutions 4. necessary signal conditioning and preliminary data processing 5. measurement frequencies (continuous, temporal and snap-shot) 6. excitation types (ambient and forced) Design sensor networks based on analysis strategies considering the following aspects: 1. sensor network types (centered and distributed) 2. device network types (GPIB, VISA, VXI, PXI, serial, WI-FI, bluetooth, etc.) 3. remote data communication types (UDP, modified UDP, TCP-IP, etc.) 4. archived data types 5. data archiving frequencies Figure2.3: Preferreddesignapproachforstructuralhealthmonitoringprocedures. 20 Chapter3 ComparisonofModelingApproachesfor Full-ScaleNonlinearViscousDampers 3.1 Introduction 3.1.1 Motivation The orifice fluid viscous damper (hereinafter viscous damper) is a passive energy dissipation device that is commonly employed in civil structures to reduce structural vibrations, typically induced by seismic motion or wind. A typical viscous damper consists of a piston rod, seal retainer, acetal resin seal, cylinder, chambers filled with compressible silicon fluid, control valves, rod make-up accumulator, and accumulator housing (Figure 3.1). For effective energy dissipation, the viscous damper employs small orifices on its piston head so that the fluid (usu- ally compressible silicon oil) is forced to pass the orifices, when the piston reciprocates. The relationship between velocity and damping force follows a clear constitutive law at relatively lowfrequencies(Constantinouetal.,1993). Ingeneral,theviscousdamperisutilizedinacivilstructuretocontrolseismic,wind-induced andthermalexpansionmotions,anditisusuallyarrangedinoneofthefollowingconfigurations: (1)adiagonalorchevronbracingelementofsteelorconcretetrusses,(2)apartofthewind/rain cable stays of suspension bridges, (3) a part of a tuned mass damper to reduce the structure 21 Piston Rod Cylinder Chamber 1 Chamber 2 Piston Head with Orifices High-strength Acetal Resin Seal Seal Retainer Accumulator Housing Control Valve Compressible Silicon Fluid Rod Make-up Accumulator Figure3.1: Componentsofaorificedviscousdamper(SoongandDargush,1997). vibration, (4) a part of a base isolation system to add energy dissipation, and (5) as a device for allowing free thermal movement. The viscous damper can be used in the construction of a new building, or the retrofit of an existing structure. In the case of structural retrofit, utilizing a viscous damper is frequently the only measure that will not prolong lane closure and traf- fic interruption (Caltrans, 2003). Thanks to its effective energy dissipation capability and wide rangeofapplication,theimportanceofviscousdampersinvibrationcontrolhasincreased. Var- ious applications of the viscous damper and other passive control devices have been reported worldwide(Aiken,1996;ChenandDuan,2000;Housneretal.,1997;Kareemetal.,1999;Kita- gawaandMidorikawa,1998;OuandLi,2004;ParkandKoh,2001;SpencerJr. andNagarajaiah, 2003;Wolfeetal.,2002). IntheU.S.A.,aftertheLomaPrietaearthquakein1989,theCaliforniaDepartmentofTrans- portation (Caltrans) initiated seismic vulnerability assessment and subsequently retrofitting of all major California toll bridges (Caltrans, 2003). In order to improve the dynamic characteris- tics of a specific bridge, viscous dampers were employed in some retrofit projects (Sheng and 22 Lee, 2003). An example is the massive retrofit effort recently completed on the San Francisco- OaklandBayBridge,WestSpansSuspensionBridge. Morethan100full-scaleviscousdampers (max. force: 3115 kN, max. stroke: ±584 mm) were installed at the truss-to-tower connec- tions of the suspended spans as anti-seismic dissipators. Another example of Caltrans’ retrofit projectsistheVincentThomasBridge,whichemployedfull-scaleviscousdampersforlimiting thedeformationofthesuspendedtrusses(Baker,1998). In Europe, full-scale viscous dampers are also widely used for structural vibration control. ArecentexampleoftheviscousdamperinnewbridgeconstructionistheRion-AntirionBridge project in Greece. This 2252-m multi-span suspension bridge is constructed on a local active seismicfault,whichcauseshighintensityearthquakesandlargetectonicmovements. Anumber offull-scaleviscousdamperswithamaximumforceof3500kNandmaximumstrokeof ±2600 mm were installed between the deck and pier with fuse retainers to reduce the deformation inducedbytheseismicgroundmotion(Infantietal.,2003). U.S.designprovisionsforviscousdampersandseismicisolatorshavebeendevelopedbythe National Earthquake Hazards Reduction Program (NEHRP). The NEHRP is a joint program of theFederalEmergencyManagementAgency(FEMA),NationalInstituteofStandardsandTech- nology (NIST), National Science Foundation (NSF), and the United States Geological Survey (USGS). In the recent update of the NEHRP Recommended Provisions for New Buildings and Other Structures (BSSC, 2004), a new chapter of “Structures with Damping Systems” (FEMA 450 Ch.15) was added. This chapter specifies provisions of designing the damping system and 23 testing damping devices. The MCEER/ATC-49 Recommended LRFD Guidelines for the Seis- mic Design of Highway Bridges also provides the guidelines and design procedures for seismic isolation(Ch.15)(ATC/MCEER,2003). Although the effect of the viscous damper in the design of vibration control is relatively well-known, few studies of structural health monitoring (SHM) techniques for operational and maintenance purposes have been reported on full-scale viscous dampers. The performance of viscousdampersinstalledonimportantcivilstructuresmustbecarefullyassessedtojudgeifthe damperisoperatingasdesigned. Thedevelopmentofconditionassessmenttechniquesincludes testingguidelinesandidentificationmethods. Duetotheirmassivesizeandinherentnonlinearity, specialconsiderationsshouldbegivenintestingandidentificationoffull-scaleviscousdampers. 3.1.2 ViscousDamperTests Currently, testing of viscous dampers is usually conducted as pre-qualification and quality con- trol tests for structural vibration control purposes. Along these lines, NIST developed three classesoftestingguidelines,includingapre-qualificationtest,prototypetest,andqualitycontrol test(Shenton,1994). Thesetestguidelinesprovidetheprovisionsofproject-specificandproject- unspecific testing for both prototype and commercialized seismic control devices. Another pur- poseoftestingviscousdampersistoprovidebasicinformationconcerningthecharacteristicsof thedampingdevicesforthedesignproceduresofvibrationcontrol. The Earthquake Engineering Research Center (EERC) at the University of California, Berkeley has developed testing methods for full-scale viscous dampers and seismic isolation devices (Aiken, 1998; Aiken and Kelly, 1996; Aiken et al., 1993). Aiken classified full-scale 24 testing procedures into quasi-static testing and dynamic testing (Aiken et al., 1993). The quasi- static testing is used when the loading rate and thermodynamic effects are not significant. A typical loading rate in quasi-static testing is 200-300 mm/min (8-12 in/min). The dynamic test- ingincludesdroppingaknownweightontheendofaverticallymountedviscousdamper(drop test), and servo-hydraulic testing (dynamic cyclic test). In the drop test, the relationship of the exciting force and damper response can be obtained. Because no significant energy is input into the damper due to the short impact time, transient temperature effects are not recovered. Thedynamiccyclictestprovidesmoreopportunitiestounderstandthenonlinearityandthermal propertiesoftheviscousdamper. However,becausethedynamiccyclictestrequiresapowerful servo-hydraulic testing facility, the dynamic cyclic test can only be performed at very limited locations (Beck et al., 1994). As a result, there are few available experimental data sets of full- scaleviscousdampers. A monotonic sinusoid excitation is typically used in the dynamic cyclic test. The Highway Innovative Technology Evaluation Center (HITEC) developed nine standardized testing meth- odsforseismicisolatorsandenergydissipationdevices(HITEC,1996,1998a,b,1999). Intheir study, “off-the-shelf” full-scale viscous dampers were tested with sets of monotonic sinusoidal excitation, and the damper response was recorded. For the identification of a nonlinear system, usingamonotonicsinusoidexcitationmaynotrevealthenonlinearityofthesystemcompletely, and for this reason, a broadband random excitation is commonly used. For meaningful identi- fication results, the testing time with a broadband random excitation should be longer than that with a monotonic sinusoidal excitation. In the case of full-scale viscous dampers, however, a long testing time frequently generates an enormous amount of heat that is converted from the 25 dissipated energy. Consequently, the temperature of the silicon fluid inside the damper’s piston chambershouldbecontrolledtoprecludeaddedcomplexities. 3.1.3 IdentificationofViscousDampers Fortheidentificationofviscousdampers,manyanalyticalmodelshavebeendevelopedbasedon Maxwell models (Constantinou and Symans, 1993; Constantinou et al., 1993; Makris and Con- stantinou, 1991; Makris et al., 1993). Current identification techniques for viscous dampers are mostly based on these parametric models. Although parametric identification techniques have beensuccessfullyusedtoidentifyviscousdampers,non-parametricidentificationtechniquesare more suitable in SHM (Soong, 1998). This is because, in the SHM context, the system charac- teristicsmaycontinuouslyvaryovertime,bothquantitativelyaswellasqualitatively. Therefore, the development of a condition assessment methodology for full-scale viscous dampers using non-parametric identification methods will be a critical step towards establishing the operation andmaintenancestrategiesforvibration-controlledstructures. 3.1.4 ObjectivesandScope A joint study between the University of Southern California and the University of California, Berkeley was conducted on a full-scale viscous damper. The research objectives were (1) to obtain quantitative data on full-scale tests, (2) to obtain information on accuracy and utility of variousmodelingapproaches,(3)tocompareadvantagesandlimitationsofbothparametricand non-parametric models, (4) to study nonlinear features of viscous dampers, and (5) to study model dependency on excitation ranges. The 1112 kN (250 kip) full-scale viscous damper was tested using multiple sets of monotonic sinusoidal excitation at the University of California, 26 Berkeley,andthetestdatawereanalyzedattheUniversityofSouthernCalifornia. Theresearch materialisorganizedinthefollowinglayout: TheexperimentalstudiesarediscussedinSection 2; an overview of the modeling approaches is presented in Section 3; the parametric and non- parametric identification approaches are presented in Section 4; and the results are discussed in Section5. 3.2 ExperimentalStudies 3.2.1 TestApparatus The 1112 kN (250 kip) viscous damper was tested in the Earthquake Engineering Research Center (EERC) at the University of California, Berkeley (Figure 3.2). The damper is a sister damper of the eight dampers installed at the 91/5 over-crossing in Orange County, CA. The damper has a mid-stroke length of 1828.8 mm (72 in) and a maximum stroke of ±203.2 mm (8.0 in). The test setup consists of a self-equilibrating reaction frame with a 1335 kN (300 kip) actuator equipped with a 3785 l/min (1000 gpm) proportional valve. The bolted head-piece at the opposite side of the actuator can assume other positions to accommodate dampers with different length. In addition to the load-cell and LVDT, the damper was instrumented with six thermocoupleprobesalongitslength(Figure3.2(b)). 3.2.2 TestCases Atotalof15experimentswereperformedtoobtainthedynamicresponseoftheviscousdamper. The experiments were designed to determine the dynamic performance characteristics of the damper at varying velocities and displacements. The damper was subjected to multiple sets of 27 (a) (b) Figure3.2: The1112kN(250kip)viscousdamperinstalledonadampertestingmachineatthe UniversityofCalifornia,Berkeley. monotonic sinusoidal excitation at peak velocities of±254.0 mm/s (10.0 in/s), ±317.5 mm/s (12.5 in/s),±381.0 mm/s (15.0 in/s), and±444.5 mm/s (17.5 in/s) and peak displacements of ±101.6 mm (4.0 in),±127.0 mm (5.0 in),±152.4 mm (6.0 in), and±177.8 mm (7.0 in). All test cases had a 6-cycle excitation period, except for one having a 10-cycle period. The test specificationsaresummarizedinTable3.1. 3.2.3 Instrumentation The damper displacement and force were measured: the force was measured with an in-line loadcell,andthedisplacementbetweenthereactionframeandthecleviswasmeasuredwithan LVDT.Thetransducermeasurementsweresampledat100Hz. 3.2.4 PreliminaryDataProcessing Oncetheforceanddisplacementofthedamperweremeasured,themeasureddisplacementwas numerically differentiated to obtain the corresponding velocity and acceleration. The records 28 Table3.1: Testspecificationsofthe1112kN(250kip)viscousdamper. Atotalof15experiments wereperformedfordifferentlevelsofpeakvelocityandpeakdisplacement. No Nameofdataset Peakvelocity Peakdisplacement Freqeuncy No. cycles 1 UCB1 10 4 ±254.0mm/s(10.0in/s) ±101.6mm(4.0in) 0.398Hz 6 2 UCB1 10 5 ±254.0mm/s(10.0in/s) ±127.0mm(5.0in) 0.318Hz 6 3 UCB1 10 6 ±254.0mm/s(10.0in/s) ±152.4mm(6.0in) 0.265Hz 6 4 UCB1 12 4 ±317.5mm/s(12.5in/s) ±101.6mm(4.0in) 0.500Hz 6 5 UCB1 12 5 ±317.5mm/s(12.5in/s) ±127.0mm(5.0in) 0.400Hz 6 6 UCB1 12 6 ±317.5mm/s(12.5in/s) ±152.4mm(6.0in) 0.332Hz 6 7 UCB1 12 7 ±317.5mm/s(12.5in/s) ±177.8mm(7.0in) 0.284Hz 6 8 UCB1 15 4 ±381.0mm/s(15.0in/s) ±101.6mm(4.0in) 0.600Hz 6 9 UCB1 15 5 ±381.0mm/s(15.0in/s) ±127.0mm(5.0in) 0.477Hz 6 10 UCB1 15 6 ±381.0mm/s(15.0in/s) ±152.4mm(6.0in) 0.400Hz 10 11 UCB1 15 7 ±381.0mm/s(15.0in/s) ±177.8mm(7.0in) 0.341Hz 6 12 UCB1 17 4 ±444.5mm/s(17.5in/s) ±101.6mm(4.0in) 0.695Hz 6 13 UCB1 17 5 ±444.5mm/s(17.5in/s) ±127.0mm(5.0in) 0.557Hz 6 14 UCB1 17 6 ±444.5mm/s(17.5in/s) ±152.4mm(6.0in) 0.464Hz 6 15 UCB1 17 7 ±444.5mm/s(17.5in/s) ±177.8mm(7.0in) 0.399Hz 6 werebandpass-filteredwithinthefrequencyrange0.1-10Hz. Asampledamperresponseafter dataprocessingisshowninFigure3.3. 3.3 OverviewofModelingApproaches 3.3.1 SimplifiedDesignModel Some analytical models of the orifice viscous damper, hereafter referred to as the Simplified DesignModel(SDM),havebeendevelopedbasedonMaxwellmodelsbyMakrisandConstanti- nou (1991) and Makris et al. (1993). The dynamic performance characteristics of the damper significantly depend on the configuration of the small orifices on the piston head, following a 29 0 2 4 6 8 10 12 14 16 18 −200 0 200 TIME (sec) DSP (mm) (a)Displacement 0 2 4 6 8 10 12 14 16 18 −500 0 500 TIME (sec) VEL (mm/sec) (b)Velocity 0 2 4 6 8 10 12 14 16 18 −2000 0 2000 TIME (sec) ACC (mm/sec 2 ) (c)Acceleration 0 2 4 6 8 10 12 14 16 18 −1000 0 1000 TIME (sec) FRC (kN) (d)Force Figure3.3: Sampletimehistoriesofmeasureddamperresponseafterpreliminarydataprocessing (UCB1 15 6). The figure illustrates the time histories of the (a) displacement, (b) velocity, (c) acceleration,and(d)force. nonlinear constitutive law at relatively low frequency (Constantinou and Symans, 1993; Con- stantinouetal.,1993)asexpressedby ˆ f d (t) =m¨ x+Csgn(˙ x)|˙ x| n (3.1) 30 where ˆ f d (t)isthedesigneddampingforce, ˙ xisthedamperstrokevelocity, ¨ xisthedamperstroke acceleration,misthe“effective”movingmassofthedamper,C isthedampingcoefficient,and n is the exponent (n = 1 for linear;n< 1 for softening; andn> 1 for hardening). The tested viscous damper under discussion was designed forC = 86.03 kN·sec n /mm n (60 kip·sec n /in n ) andn = 0.35. For the optimal values of the parameters (i.e.,m,C, andn), the Adaptive Random Search (ARS)method(Andronikouetal.,1982;Masrietal.,1980)isemployed. UsingtheARSmethod, the optimal values of the parameters are searched within the solution space of a differential equationforminimalnormalizedmeansquareerrorbetweenthemeasuredandidentifieddamper responses. Inordertodeterminetheoptimalvaluesofm,C,andn,thesimplifieddesignmodel (Equation3.1)isreformulatedasafirstorderdifferentialequation     ˙ x ¨ x     | {z } ˙ y =     ˙ x 1 m (f(t)−Csgn(˙ x)|˙ x| n )     | {z } F(t,y) (3.2) The solution of Equation 3.2 can be determined, using standard numerical time-marching tech- niques, if the initial conditions (i.e., x(0) and ˙ x(0)) and the values of the system parameters are specified. The optimal values of the unknown parameters can be found by minimizing the followingcostfunction J() =w 1 × 100 Nσ 2 x N X i=1 (x i − ˆ x i ) 2 +w 2 × 100 Nσ 2 ˙ x N X i=1 (˙ x i − ˆ ˙ x i ) 2 (3.3) 31 wherex and ˙ x are the measured displacement and velocity, ˆ x and ˆ ˙ x are the computed displace- ment and velocity from Equation 3.2,N is the number of data points,σ 2 x andσ 2 ˙ x are variances ofthemeasuredresponses,andw 1 andw 2 arenormalizingweights. 3.3.2 RestoringForceMethod Aconstant-mass,single-degree-of-freedomnonlineardynamicsystemcanberepresentedbythe followingequationofmotion m¨ x(t)+r(x(t), ˙ x(t)) =f(t) (3.4) wheref(t)istheexcitingforce,misthemass,andr istherestoringforce,whichisanonlinear function of the displacement (x) and velocity (˙ x). Using the restoring force method (RFM), the restoring force surface can be approximated by a series of two-dimensional Chebyshev polyno- mials(MasriandCaughey,1979) r(x, ˙ x)≈ ˆ r(x, ˙ x) = MX X i=0 NY X j=0 ¯ C ij T i (¯ x)T j ( ¯ ˙ x) (3.5) where ¯ x and ¯ ˙ x are the normalized displacement and velocity in the range [-1, 1], ¯ C ij is the normalized Chebyshev coefficient, and T n is the n th order Chebyshev polynomial. For the testedviscousdamper,iftheexcitationfrequencyislow(i.e.,nearlyquasi-staticconditions),the inertiaterminEquation3.4becomesnegligible. Then,thedampercanbemodeledas ˆ f r (t)≈ MX X i=0 NY X j=0 ¯ C ij T i (¯ x)T j ( ¯ ˙ x) (3.6) 32 where ˆ f r (t) is the damping force identified with the RFM. One of the advantages of using the RFM is that the orthogonality is preserved, using the Chebyshev polynomials. That is, with the normalizeddomainof[-1,1],theidentifiedChebyshevcoefficientofeachtermisnotaffectedby theotherterms. ThisfeatureoftheRFMcanbeveryattractive,especiallyinSHMapplications, where the proper order of the expansion is often unknown. In identification, the Chebyshev polynomial is more appropriate to the polynomial type nonlinearity than the piece-wise orthog- onal functions. It is also known that the orthogonal polynomials with limited bounds, such as the Chebyshev polynomial, is more accurate than those with unlimited bounds, such as Her- mite polynomials. According to de Moivre’s Theorem, a Chebyshev series has the following relationshipwithapowerseries(MasonandHandscomb,2003) T 0 (x) = 1, T 1 (x) =x, T 2 (x) = 2x 2 −1,..., T n+1 (x) = 2xT n (x)−T n−1 (x),... (3.7) UsingtherelationshipinEquation3.7,theviscousdamperrestoringforcecanbemodeled ˆ f p (t) = MX X i=0 NY X j=0 a ij x i ˙ x j (3.8) wherea ij is a de-normalized power series coefficient, and ˆ f p (t) is the damping force identified withpowerseries. 3.3.3 ArtificialNeuralNetworks The artificial neural networks (ANN) identification technique has been applied successfully to broad classes of nonlinear systems. For the tested viscous damper, a three-layer feedforward 33 neural network (the last layer is the output layer) was constructed and trained with the ARS. Descriptions of the ANN and ARS are given in Masri et al. (1993, 2000, 1999). Using the followingthree-layerANN,theviscousdamperforcecanbeexpressed: ˆ f a (t) = Γ    N2 X j=1 " w j Γ N1 X k=1 v j,k y k +b v,j ! +b w #    (3.9) where w and v are weights, b is bias, N1 is the number of nodes in the first layer, N2 is the number of nodes in the second layer, Γ is a tangent-sigmoid function, and ˆ f a (t) is the damping forceidentifiedwiththeANNmethod. Here,thethirdlayerisusedasanoutputlayer. 3.4 IdentificationoftheViscousDamper 3.4.1 ParametricIdentificationofSimplifiedDesignModel Identification For the optimal values of the unknown parameters in Equation 3.1, the parametric identifica- tion using a time-marching technique (SDM) was applied. In the ARS, the design values of C = 86.03 kN·sec n /mm n (60 kip·sec n /in n ) and n = 0.35 were used as the initial values of the unknown parameters. The solution space of the optimization was constrained for positive valuesoftheparameters. Theinitialvaluesandtheboundsofthesolutionspacearesummarized in Table 3.2. For each optimization process, 300 global searches and 25 local searches were performed. Three identical optimization processes were conducted with randomly generated initial parameters, and the identified parameters with the least normalized mean-square-error wereselected. 34 Table 3.2: Initial values and boundaries of the unknown parameters in Equation 3.1 for the AdaptiveRandomSearchmethod. Parametermisthe“effective”movingmass,C isthedamp- ingcoefficient,andnistheexponent. Parameter Initialvalue Lowerbound Upperbound 14,593.9kg 0.1459kg 1,459,390kg m (1.0slug) (1×10 −5 slug) (100.0slug) 86.03kN·sec n /mm n 0.14kN·sec n /mm n 143.38kN·sec n /mm n C (60kip·sec n /in n ) (1×10 −1 kip·sec n /in n ) (100kip·sec n /in n ) n 0.35 1×10 −1 2.0 AsampleidentificationresultoftheSDMisillustratedinFigures3.4(a)and(d). Inthefig- ure, the identified damper response was obtained, using Equation 3.1 for the identified optimal values of the parameters. Them, C, andn were determined for all test cases, and the identi- fied values are summarized in Table 3.3. The mean of the identified m is 735.64 kg with the coefficient of variance of 0.3185. The mean of the identifiedC is 4.357 kN·sec n /mm n , and its coefficientofthevarianceis0.0512. Themeanoftheidentifiednis0.391,anditscoefficientof varianceis0.0345. The normalized mean-square-error (NMSE) of the SDM-identification was calculated to evaluatetheaccuracyoftheidentificationresults. TheNMSEwascalculatedas NMSE( ˆ f) = 100 Nσ 2 f N X i=1 (f i − ˆ f i ) 2 (3.10) wheref is the measured force, ˆ f is the identified force,N is the number of data points, andσ 2 f is the variance of the measured force (Worden and Tomlinson, 2001). The identification errors 35 −200 −100 0 100 200 −800 −600 −400 −200 0 200 400 600 800 DISPLACEMENT (mm) FORCE (kN) −200 −100 0 100 200 −800 −600 −400 −200 0 200 400 600 800 DISPLACEMENT (mm) FORCE (kN) −200 −100 0 100 200 −800 −600 −400 −200 0 200 400 600 800 DISPLACEMENT (mm) FORCE (kN) (a)SDM (b)RFM (c)ANN −500 0 500 −800 −600 −400 −200 0 200 400 600 800 VELOCITY (mm/sec) FORCE (kN) −500 0 500 −800 −600 −400 −200 0 200 400 600 800 VELOCITY (mm/sec) FORCE (kN) −500 0 500 −800 −600 −400 −200 0 200 400 600 800 VELOCITY (mm/sec) FORCE (kN) (d)SDM (e)RFM (f)ANN Figure 3.4: Sample identification results of the parametric simple design model (SDM), the non-parametricrestoringforcemethod(RFM),andthenon-parametricartificialneuralnetworks (ANN) for the data set UCB1 15 6. The phase plots show approximately a one-cycle period of thedamperresponse(solidlineformeasured,anddashedforidentifiedforces). Inthefigure,the first row shows the relationship between displacement and force, and the second row shows the relationshipbetweenvelocityandforceforeachinvestigatedidentificationmethod. foralltestcasesaresummarizedinthegraycellsinTable3.4(a). Themeanoftheidentification erroris3.75%withthecoefficientofcovarianceof0.28. Validation OncetheviscousdamperwasidentifiedusingtheSDM,theresultswerevalidatedwiththedata sets, which had not been used in the identification phase. A total of 196(=14×14) validation processes were performed, and the NMSE’s are summarized in Table 3.4 (a). In the table, the identification errors are shown in the grayed cells, and the validation errors are summarized in thewhitecells. Thevaluesinthesamecolumnshowtheerrorswithrespecttothesameidentified 36 Table 3.3: Mass, damping constant (C), and exponent (n) identified using the simplified design model(SDM).Thesystemparametersweredetermined,usingtheadaptiverandomsearchopti- mization, and for the determined parameters, the governing differential equation of motion was directlysolvedthroughtheuseofconventionaltime-marchingtechniques. No Nameofdataset Mass(kg) C (kN·sec n /mm n ) n 1 UCB1 10 4 567.7 78.5085 0.3773 2 UCB1 10 5 1199.6 70.1302 0.4055 3 UCB1 10 6 1208.4 70.1866 0.4108 4 UCB1 12 4 685.9 74.6388 0.3896 5 UCB1 12 5 777.9 70.7238 0.4030 6 UCB1 12 6 869.8 70.5987 0.4036 7 UCB1 12 7 974.9 71.4950 0.3997 8 UCB1 15 4 567.7 78.5085 0.3773 9 UCB1 15 5 594.0 75.4005 0.3862 10 UCB1 15 6 785.2 69.6316 0.4076 11 UCB1 15 7 645.1 72.1798 0.3967 12 UCB1 17 4 494.7 79.7295 0.3732 13 UCB1 17 5 522.5 79.4230 0.3747 14 UCB1 17 6 519.5 77.2283 0.3789 15 UCB1 17 7 621.7 76.9658 0.3799 avg 735.64 74.366 0.3909 cov 0.3185 0.0512 0.0345 parameters (i.e., C andn), and the values in the same row show the errors with respect to the same data set. The mean of the averaged validation errors for the same identified parameters is 3.79% with the coefficient of variance of 0.098. The mean of the averaged validation errors for thesamedatasetis3.79%withthecoefficientofvarianceof0.31. 3.4.2 NonparametricIdentificationUsingRestoringForceMethod Identification AnoptimalChebyshevorderoftheRFM-identificationwasdeterminedusingthetesteddamper response. The damper was identified with the order ofn = 1,2,3,...,15, and the determined NMSE’s are plotted in Figure 3.5 in semi-log scale. The NMSE decreases for 1 < n≤ 9, and increases for 9 < n ≤ 15 due to over-fitting. Because Figure 3.5 is plotted in semi-log scale, 37 Table3.4: Identificationandvalidationresultsofthesimplifieddesignmodel(SDM),therestor- ing force method (RFM), and the artificial neural networks (ANN). The values are the normal- ized mean-squared errors (NMSE) of the estimated force versus the measured force. For each identificationmethod,theidentificationerrorsareshowningrayedcells,andthevalidationerrors are shown in white cells. The values in the same row show the errors with respect to the same data set, while the values in the same column show the errors with respect to the same identi- fied coefficients (i.e., the damping constant (C) and exponent (n) for the SDM, the normalized ChebyshevcoefficientsfortheRFM,andthetrainedweightsandbiasesfortheANN). (a)SimplifiedDesignModel(Parametric) Data DataNo. (w.r.t. SameIdentifiedCandn) No. 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 avg cov 1 4.60 4.99 5.65 4.71 4.91 4.93 4.81 4.60 4.62 5.05 4.72 4.56 4.59 4.49 4.51 4.78 0.063 2 3.15 3.55 4.17 3.27 3.47 3.49 3.37 3.15 3.18 3.61 3.29 3.11 3.15 3.07 3.08 3.34 0.087 3 2.62 2.77 3.13 2.64 2.72 2.73 2.68 2.62 2.61 2.80 2.64 2.62 2.62 2.60 2.60 2.69 0.051 4 5.32 5.79 6.50 5.45 5.69 5.72 5.58 5.32 5.35 5.86 5.48 5.27 5.31 5.22 5.23 5.54 0.061 5 3.60 4.01 4.64 3.71 3.92 3.95 3.83 3.60 3.63 4.07 3.74 3.57 3.59 3.53 3.54 3.80 0.078 6 2.94 3.31 3.90 3.03 3.23 3.26 3.15 2.94 2.96 3.37 3.06 2.91 2.93 2.87 2.88 3.12 0.088 7 1.94 2.42 3.06 2.08 2.33 2.35 2.22 1.94 1.99 2.49 2.13 1.90 1.93 1.88 1.89 2.17 0.149 8 5.35 6.01 6.85 5.55 5.88 5.92 5.75 5.35 5.42 6.10 5.61 5.29 5.33 5.24 5.26 5.66 0.078 9 3.86 4.42 5.16 4.02 4.31 4.34 4.19 3.86 3.91 4.50 4.07 3.81 3.84 3.77 3.79 4.12 0.092 10 3.03 3.79 4.66 3.26 3.66 3.69 3.50 3.03 3.13 3.89 3.35 2.95 3.00 2.92 2.94 3.39 0.144 11 2.23 2.74 3.39 2.37 2.64 2.67 2.53 2.23 2.29 2.81 2.43 2.20 2.22 2.18 2.19 2.48 0.135 12 5.38 6.22 7.17 5.63 6.05 6.10 5.89 5.38 5.47 6.31 5.72 5.29 5.35 5.25 5.28 5.77 0.093 13 3.78 4.52 5.38 4.00 4.38 4.42 4.23 3.78 3.87 4.62 4.08 3.71 3.76 3.68 3.71 4.13 0.115 14 2.82 3.55 4.37 3.04 3.41 3.45 3.27 2.82 2.91 3.64 3.13 2.76 2.80 2.74 2.76 3.17 0.145 15 2.38 3.18 4.06 2.63 3.04 3.08 2.88 2.38 2.49 3.29 2.73 2.31 2.36 2.29 2.76 2.76 0.181 avg 3.53 4.09 4.81 3.69 3.98 4.01 3.86 3.53 3.59 4.16 3.75 3.48 3.52 3.45 3.47 cov 0.33 0.30 0.27 0.32 0.30 0.30 0.31 0.33 0.33 0.29 0.31 0.33 0.33 0.33 0.33 (b)RestoringForceMethodIdentification(Non-parametric) Data DataNo. (w.r.t. SameIdentifiedCandn) No. 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 avg cov 1 1.50 2.11 2.39 2.38 1.47 1.67 2.76 3.18 2.37 2.72 4.49 3.69 3.94 4.11 4.08 2.86 0.35 2 1.86 1.61 1.57 2.24 1.24 1.09 1.92 3.19 2.13 2.23 3.51 3.98 4.01 3.77 3.60 2.53 0.41 3 2.37 2.00 1.56 2.21 1.04 0.78 1.58 2.71 1.60 1.43 2.52 3.28 3.08 2.69 2.47 2.09 0.35 4 2.55 3.00 2.58 1.68 1.06 1.35 2.39 1.30 0.94 1.25 2.54 1.40 1.50 1.78 2.11 1.83 0.36 5 2.91 3.41 2.77 2.37 1.07 1.25 2.16 1.80 1.07 0.89 2.13 1.57 1.32 1.30 1.30 1.83 0.42 6 3.19 3.11 2.32 2.40 1.03 0.91 1.70 2.01 1.06 0.72 1.72 1.96 1.55 1.26 1.13 1.74 0.44 7 3.08 3.13 2.34 2.87 1.12 0.92 1.65 2.59 1.35 0.75 1.83 2.44 1.91 1.46 1.07 1.90 0.42 8 3.66 4.51 3.82 2.45 1.60 2.01 3.09 1.24 1.00 1.19 2.46 0.72 0.69 1.07 1.43 2.06 0.59 9 3.94 4.44 3.52 2.66 1.52 1.76 2.63 1.52 1.02 0.87 1.90 1.05 0.77 0.81 0.98 1.96 0.61 10 3.83 4.26 3.28 3.02 1.47 1.52 2.25 2.06 1.23 0.74 1.68 1.59 1.10 0.88 0.75 1.98 0.62 11 4.69 4.83 3.61 3.48 1.86 1.78 2.48 2.37 1.45 0.81 1.63 1.85 1.22 0.87 0.68 2.24 0.60 12 4.47 5.54 4.75 3.21 2.17 2.66 3.76 1.61 1.40 1.51 2.86 0.74 0.63 1.08 1.40 2.52 0.61 13 4.91 5.80 4.78 3.68 2.25 2.61 3.55 2.00 1.54 1.29 2.48 1.04 0.66 0.83 0.96 2.56 0.64 14 5.24 5.76 4.58 3.85 2.28 2.44 3.28 2.26 1.59 1.11 2.12 1.38 0.83 0.73 0.73 2.54 0.65 15 5.31 5.82 4.59 4.26 2.39 2.43 3.20 2.77 1.86 1.16 2.16 1.84 1.15 0.89 0.68 2.70 0.60 avg 3.57 3.95 3.23 2.85 1.57 1.68 2.56 2.17 1.44 1.25 2.40 1.90 1.62 1.57 1.56 cov 0.34 0.37 0.35 0.25 0.32 0.38 0.27 0.29 0.30 0.46 0.32 0.54 0.70 0.69 0.68 (c)ArtificialNeuralNetworksIdentification(Non-parametric) Data DataNo. (w.r.t. SameIdentifiedCandn) No. 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 avg cov 1 0.31 0.48 0.54 2.49 0.47 0.57 0.67 2.82 1.13 0.86 0.79 1.37 2.39 1.23 1.57 1.18 0.69 2 0.97 0.28 0.29 5.57 0.73 0.73 0.57 5.65 1.96 1.34 1.01 2.79 4.01 1.78 2.40 2.01 0.89 3 0.89 0.39 0.27 6.54 0.85 0.97 0.43 7.19 2.00 1.29 1.03 3.32 4.25 1.73 2.42 2.24 0.98 4 0.47 0.70 0.71 0.39 0.39 0.51 1.10 0.52 0.55 1.02 0.84 0.97 1.20 1.13 1.67 0.81 0.46 5 0.54 0.55 0.40 1.47 0.26 0.41 0.50 1.94 0.52 0.46 0.44 0.96 1.25 0.53 0.66 0.73 0.64 6 0.84 0.48 0.36 2.46 0.40 0.24 0.37 3.29 0.77 0.40 0.38 1.43 1.92 0.64 0.81 0.99 0.91 7 0.98 0.37 0.41 6.26 0.72 0.78 0.19 7.40 1.43 1.17 0.72 2.89 3.48 1.29 1.50 1.97 1.11 8 0.52 0.99 0.51 0.44 0.35 0.43 1.19 0.22 0.32 1.33 0.71 0.48 0.71 0.88 1.54 0.71 0.56 9 0.49 0.73 0.45 0.58 0.32 0.42 0.75 0.56 0.27 0.74 0.45 0.66 0.57 0.42 0.72 0.54 0.30 10 0.83 0.56 0.43 2.42 0.46 0.40 0.42 3.28 0.92 0.31 0.44 1.44 2.02 0.78 0.78 1.03 0.85 11 1.01 0.64 0.55 2.05 0.43 0.33 0.43 2.86 0.77 0.47 0.29 1.28 1.51 0.46 0.50 0.91 0.81 12 0.51 1.49 0.67 1.70 0.57 0.75 1.18 0.66 0.43 1.76 0.63 0.20 0.30 0.42 1.39 0.84 0.62 13 0.50 1.28 0.82 1.43 0.48 0.67 0.91 0.92 0.45 1.37 0.41 0.41 0.19 0.29 0.75 0.73 0.53 14 0.68 0.94 0.61 1.75 0.37 0.39 0.62 2.13 0.48 0.57 0.30 1.00 0.70 0.17 0.31 0.74 0.74 15 1.00 0.83 0.68 2.08 0.51 0.44 0.52 2.03 0.79 0.53 0.28 1.14 1.02 0.33 0.20 0.83 0.69 avg 0.70 0.71 0.51 2.51 0.49 0.54 0.66 2.77 0.85 0.91 0.58 1.36 1.70 0.81 1.15 cov 0.34 0.48 0.31 1.26 0.35 0.37 0.47 0.84 0.65 0.50 0.45 0.69 0.77 0.64 0.61 38 0 5 10 15 10 −1 10 0 10 1 CHEBYSHEV POLYNOMIAL ORDER NORMALIZED MSE Order NMSE(%) Order NMSE(%) 1 5.95 9 0.40 2 3.84 10 0.44 3 1.69 11 0.42 4 1.28 12 0.49 5 0.74 13 0.49 6 0.50 14 0.53 7 0.45 15 0.53 8 0.41 Figure 3.5: The normalized mean square error (NMSE) for different Chebyshev polynomial orders for the data set UCB1 15 6. The LHS-figure shows the NMSE in semi-log scale. The RHS-tableshowstheNMSEvaluesinthefigure. the accuracy of the identification results is negligible with NMSE of 0.34% for 5 ≤ n ≤ 9. Therefore,theChebyshevordern = 5wasrepeatedlyusedforallanalyzeddatasets. The RFM-identification was performed for all test cases. A sample result of the RFM- identificationisshowninFigures3.4(b)and(e). ThemeanandstandarddeviationoftheNMSE are(1.16±0.41)%,andtheNMSE’sforalltestcasesaresummarizedinthegraycellsinTable3.4 (b). The normalized Chebyshev coefficients were determined for all test cases (Table 3.5). The coefficients associated with the first order damping, the third order damping, and the first order stiffness are the most dominant terms in the RFM-identification, and the other terms are either negligibleorcanceleachotherout. ThemeansandstandarddeviationsofthedominantCheby- shevcoefficientsare(716.3±53.5)kNforthefirstorderdamping,(−45.05±4.801)kNforthe third order damping, and (88.29±38.90) kN for the first order stiffness. A sample normalized ChebyshevcoefficientsisillustratedinFigure3.6(a). Thede-normalizedpowerseriescoefficientswerealsocalculatedusingthedeMoivre’sThe- orem (Table 3.5). The means and the standard deviations of the power series coefficients are 39 0 1 2 3 4 5 0 1 2 3 4 5 0 50 100 150 200 VELOCITY DISPLACEMENT NORMALIZED CHEBYSHEV COEFFICIENT 0 1 2 3 4 5 0 1 2 3 4 5 0 5 10 15 VELOCITY DISPLACEMENT DE−NORMALIZED PS COEFFICIENT (a)normalizedChebyshevcoefficients (b)de-normalizedpowerseriescoefficients Figure 3.6: An example of normalized Chebyshev coefficients ( ¯ C ij ) and de-normalized power seriescoefficients(a ij )forthetestedviscousdamper(UCB1 15 6). (1.782± 0.234) kN sec/mm for the first order damping, (4.047± 2.822) kN sec 3 /mm 3 for the thirdorderdamping,and(10.72±5.641)kN/mmforthefirstorderstiffness. UnliketheCheby- shevpolynomialexpansion,theorthogonalitypropertyisnotvalidamongthepowerseriesterms. Asamplede-normalizedpowerseriescoefficientsisshowninFigure3.6(b). Validation The RFM-identification results were validated using the data sets, which had not been used in the identification phase. A total of 196 (=14×14) validation processes were performed, and the normalized mean-square errors between the measured and estimated forces are summarized in Table 3.4 (b). In the table, the values in one row are the validation errors with respect to one data set, while the values in one column are the validation errors with respect to the same iden- tified normalized Chebyshev coefficients. The means and standard deviations of the averaged 40 Table3.5: NormalizedChebyshevcoefficients( ¯ C ij )andde-normalizedpowerseriescoefficients (a ij )ofthetestedviscousdamperfortherestoringforcemethod(RFM). No Nameofdataset ¯ C 01 ¯ C 03 ¯ C 10 a 01 a 03 (10 −6 ) a 10 (10 −1 ) 1 UCB1 10 4 630.3 −37.89 184.0 2.186 −0.403 25.32 2 UCB1 10 5 621.5 −39.36 85.99 2.000 7.960 9.887 3 UCB1 10 6 669.8 −44.62 95.09 2.310 8.881 9.364 4 UCB1 12 4 669.9 −39.51 122.0 1.716 5.318 17.30 5 UCB1 12 5 707.1 −45.69 81.75 1.810 8.574 9.454 6 UCB1 12 6 683.3 −47.97 40.93 1.865 1.973 5.946 7 UCB1 12 7 684.9 −46.83 99.65 1.884 2.292 7.423 8 UCB1 15 4 739.4 −41.75 120.2 1.651 5.166 17.82 9 UCB1 15 5 727.5 −44.27 63.87 1.578 4.428 8.669 10 UCB1 15 6 761.8 −50.33 58.64 1.710 5.586 6.242 11 UCB1 15 7 756.6 −52.88 128.5 1.755 3.442 9.914 12 UCB1 17 4 767.7 −39.19 88.35 1.610 1.492 14.46 13 UCB1 17 5 770.2 −45.11 63.35 1.538 1.715 7.784 14 UCB1 17 6 781.7 −48.11 53.22 1.564 2.431 6.090 15 UCB1 17 7 772.3 −52.24 38.75 1.554 1.854 5.074 avg 716.3 -45.05 88.29 1.782 4.047 10.72 cov 0.075 0.107 0.434 0.131 0.697 0.526 validationerrorsvaryfrom(1.44±0.43)%to(3.95±1.45)%forthesameidentifiednormalized Chebyshevcoefficientset,andfrom(1.74±0.77)%to(2.86±1.01)%forthesamedataset. 3.4.3 NonparametricIdentificationUsingArtificialNeuralNetworks Training The artificial neural network (ANN) identification was performed with the measured viscous damper response. Three-layer feedforward neural networks were constructed with 15 nodes in the first layer and 10 nodes in the second layer. The third layer was used as the output layer. The adaptive random search method was employed to train the neural networks with 10 global searches and 500 local searches. Three identical training processes were performed with randomlygeneratedinitialweights,andthebestidentificationresult(i.e.,theresultwithminimal NMSE)waschosen. 41 A sample ANN-identification result is illustrated in Figures 3.4 (c) and (f). The mean and standard deviation of the ANN-identification are ( 0.25± 0.06)%. The NMSE of the ANN- identificationforalltestcasesaresummarizedinthegrayedcellsinTable3.4(c). Validation The trained neural networks using the ANN identification method were validated with the data sets, which had not been used in the training phase. A total of 196 validation processes were performed,andthenormalizedmean-squareerrors(NMSE)aresummarizedinTable3.4(c). In the table, the values in one row are the validation error for one data set, and the values in one column are the validation errors for the same trained neural networks. The means and standard deviationsoftheNMSEvaryfrom(0.49±0.17)%to(2.77±2.33)%forthesametrainedneural networks,andfrom(0.54±0.16)%to(2.24±2.19)%forthesamedataset. 3.5 Discussion 3.5.1 ConstitutiveLaw Thefrequency-dependentdampingpropertiesofthetestedviscousdamperwerestudiedtoinves- tigate the dependence of the estimated peak damper force with the corresponding displacement and velocity range. Figure 3.7 shows that the relationship between the peak force and the peak velocityfollowstheconstitutivelawatalltestedpeakdisplacements. Theidentifiedpeakforces are also plotted in the table. The peak forces identified with the SDM ( ) are less than the measured peak forces by 3.0% on average. The peak forces identified with the RFM () are less than the measured peak forces by 5.1% on average, and the peak forces identified with the 42 250 300 350 400 450 600 620 640 660 680 700 720 740 760 780 800 250 300 350 400 450 600 620 640 660 680 700 720 740 760 780 800 250 300 350 400 450 600 620 640 660 680 700 720 740 760 780 800 250 300 350 400 450 600 620 640 660 680 700 720 740 760 780 800 (a)±101.6mm(4.0in) (b)±127.0mm(5.0in) (c)±152.4mm(6.0in) (d)±177.8mm(7.0in) Figure3.7: Relationshipofpeakvelocitiesandpeakforcesatdifferentpeakdisplacements. ( : measured,4: SDM,: RFM,∗: ANN).Foreachpeakdisplacementlevel(i.e.,(a),(b),(c),and (d)),thex-axisshowsthepeakvelocityandthey-axisshowstheforce. ANN(∗)arelessthanthemeasuredpeakforcesbyanaverageof4.8%. Consequently,thepeak force of the tested damper can be identified successfully by using all investigated identification methods. 3.5.2 FidelityofIdentifiedModels ParametricSimplifiedDesignModel TheparametricSDM-identificationresultsshowthatthehysteresisinthevelocity-forcephaseis notmodeledaccurately(Figure3.4(d)),whilethedampingenergycanbemodeledaccuratelyin thedisplacement-forcephaseplot(Figure3.4(a)). Table 3.3 shows that the identified damping constant (C) and exponent (n) are reasonably close to the design values; the mean of the identified damping constant is 74.37 kN·sec n /mm n , which is about 86% of the design value 86.03 kN·sec n /mm n ; the mean of the exponent is 0.39, whichisabout112%ofthedesignvalue0.35. 43 254.0(10.0) 317.5(12.5) 381.0(15.0) 444.5(17.5) 101.6(4) 127.0(5) 152.4(6) 177.8(7) 0 1 2 3 4 5 6 7 VELOCITY DISPLACEMENT NMSE (%) 254.0(10.0) 317.5(12.5) 381.0(15.0) 444.5(17.5) 101.6(4) 127.0(5) 152.4(6) 177.8(7) 0 1 2 3 4 5 6 7 VELOCITY DISPLACEMENT AVG. NMSE (%) 254.0(10.0) 317.5(12.5) 381.0(15.0) 444.5(17.5) 101.6(4) 127.0(5) 152.4(6) 177.8(7) 0 1 2 3 4 5 6 7 VELOCITY DISPLACEMENT AVG. NMSE (%) (a)SDM (b)RFM (c)ANN Figure3.8: Normalizedmean-squareerrorsbetweenthemeasuredandtheidentifiedforceswith parametric simplified design model (SDM), and non-parametric restoring force method (RFM) and artificial neural networks (ANN). In the figure, the data set forx = 254.0 mm/sec and ˙ x = 177.8mmismissing. Figure3.8(a)showsthattheNMSEincreasesasthepeakvelocityorthepeakdisplacement increases. AccordingtoSoongandDargush(1997),thesimplifieddesignmodelismoreappro- priate for a lower excitation frequency because of the smaller contribution of relaxation time in theMaxwellequation. Therefore,theNMSEoftheSDM-identificationincreaseswhenthepeak velocityincreases. Inaddition,usingthesimplifieddesignmodel,becausethedamperresponse is modeled as a function of velocity only, the displacement-related nonlinearity of the damper cannotbemodeledproperly. Non-parametricRestoringForceMethod Inthenon-parametricrestoringforcemethod(RFM)identification,theresponseofthedamperis modeledrelativelyaccurately(Figures3.4(b)and(e)). TheNMSEoftheRFM-identificationis 2.22%,whichislessthanthe3.75%NMSEoftheSDM-identification. Thus,thetesteddamper is modeled more accurately with the RFM than the SDM. In contrast to the SDM-identification case,theNMSEoftheRFM-identificationbecomessmallerwithlargerpeakvelocity(Figure3.8 (b)). That is, the accuracy of the RFM-identification increases, when the viscous damper is 44 identified using data sets with higher frequency excitations. This result implies that, in the RFM-identification,thetestedviscousdamperismodeledmoreaccuratelywithahighfrequency excitation. ThenormalizedChebyshevcoefficientsshowthatthefirstorderdamping,thirdorderdamp- ing and the first order stiffness terms are the dominant terms in the RFM-identification. The identifiedforcecomponentscorrespondingtothesedominanttermsareillustratedinFigure3.9. Figure 3.9 (a) shows that the first order damping coefficient ( ¯ C 01 ) is the most important one in modelingthedampingenergy. Figure3.9(d)showsthatthe“effective”dampingconstantofthe viscous damper is modeled by the ¯ C 01 . Therefore, the damping characteristics of the viscous damperaremainlygovernedbythefirstorderdampingcoefficient. The plot in Figure 3.9 (e) clearly shows the nonlinear characteristic of the viscous damper. As summarized in Table 3.5, the third order Chebyshev coefficient ( ¯ C 03 ) has a negative value for all test cases. The negative value is due to the “softening” of the damping force (α < 1). Hadthetestedviscousdamperhadlinearor“hardening”characteristics,thethirdorderdamping coefficientwouldhavebeenclosetozero,orhavetakenapositive. Figure3.9(c)showsthatthecontributionofthe ¯ C 10 tothetotaldampingenergyisnegligible, as indicated by the slight slope. However, Figure 3.9 (e) reveals that the first order stiffness coefficientisimportanttomodelthedamperhysteresisinthevelocity-forcephaseplot. Figure 3.10 shows the ¯ C 01 and ¯ C 03 coefficients at different peak velocities and peak dis- placements. The ¯ C 01 increasesby20.8%onaverage,asthepeakvelocityincreases(Figure3.10 (a)),whilethe ¯ C 01 remainsconstantfordifferentdisplacements(Figure3.10(b)). Theseresults imply that the nonlinearity of the first order damping depends on the peak velocity, rather than 45 ¯ C 01 ¯ C 03 ¯ C 10 −200 −100 0 100 200 −800 −600 −400 −200 0 200 400 600 800 DISPLACEMENT (mm) FORCE (kN) −200 −100 0 100 200 −800 −600 −400 −200 0 200 400 600 800 DISPLACEMENT (mm) FORCE (kN) −200 −100 0 100 200 −800 −600 −400 −200 0 200 400 600 800 DISPLACEMENT (mm) FORCE (kN) (a)Firstorderdamping (b)Thirdorderdamping (c)Firstorderstiffness −400 −200 0 200 400 −800 −600 −400 −200 0 200 400 600 800 VELOCITY (mm/sec) FORCE (kN) −400 −200 0 200 400 −800 −600 −400 −200 0 200 400 600 800 VELOCITY (mm/sec) FORCE (kN) −400 −200 0 200 400 −800 −600 −400 −200 0 200 400 600 800 VELOCITY (mm/sec) FORCE (kN) (d)Firstorderdamping (e)Thirdorderdamping (f)Firstorderstiffness Figure 3.9: Sample phase plots for the first order damping, the third order damping and the firstorderstiffnesstermsoftheidentifiedforceusingtherestoringforcemethod. Thesolidline representsthemeasured(total)forceandthedashedlinerepresentsthetermwiseidentifiedforce. Thefirstrowshowstherelationshipbetweendisplacementandforce,andthesecondrowshows therelationshipbetweenvelocityandforce. the peak displacement. As was previously shown, in the RFM-identification, the “effective” damping constant of the damper is mainly modeled by the first order damping term. Conse- quently,theseresultsalsoimplythatthe“effective”dampingconstantvarieswithdifferentpeak velocities,ratherthanwithdifferentpeakdisplacements. The ¯ C 03 decreasesabout20.7%onaverageasthepeakdisplacementincreases(Figure3.10 (d)), while the ¯ C 03 remains constant at different peak velocities (Figure 3.10 (c)). Therefore, unlikethe ¯ C 01 ,thenonlinearityofthethirdorderdampingdependsonpeakdisplacement,rather than peak velocity. Previously, it was also shown that the softening of the damper is modeled 46 ◦ : 101.6mm 4 : 127.0mm : 152.4mm ∗ : 177.8mm ◦ : 254.0mm/sec 4 : 317.5mm/sec : 381.0mm/sec ∗ : 444.5mm/sec 250 300 350 400 450 600 650 700 750 800 VELOCITY (mm/sec) FIRST ORDER DAMPING 100 120 140 160 180 600 650 700 750 800 DISPLACEMENT (mm) FIRST ORDER DAMPING (a)Peakvelvs. 1 st orderdampingcoef. (b)Peakdispvs. 1 st orderdampingcoef. 250 300 350 400 450 −55 −50 −45 −40 −35 VELOCITY (mm/sec) THIRD ORDER DAMPING 100 120 140 160 180 −55 −50 −45 −40 −35 DISPLACEMENT (mm) THIRD ORDER DAMPING (c)Peakvelvs. 3 rd orderdampingcoef. (d)Peakdispvs. thirdorderdampingcoef. Figure 3.10: The normalized Chebyshev coefficients of the first and third order damping at dif- ferent peak velocities and peak displacement, respectively. (a) and (c) show the relationship between peak velocity and the normalized Chebyshev coefficient for different peak displace- ments. (b)and(d)showtherelationshipbetweenpeakdisplacementandthenormalizedCheby- shevcoefficientfordifferentpeakvelocities. with the ¯ C 03 in the RFM-identification. Therefore, the decrease of the ¯ C 03 at a larger peak displacement would imply that the magnitude of the damper’s softening increases as the peak displacementincreases. Non-parametricArtificialNeuralNetworks In the non-parametric artificial neural networks (ANN) identification, the response of the tested viscous damper is modeled fairly accurately (Figures 3.4 (c) and (f)). With the given identifi- cation parameters, the NMSE of the ANN-identification is the smallest among the investigated identification methods. In the training phase, the averaged NMSE of the ANN identification is 47 0.23%, which is approximately five times less than the averaged NMSE of the RFM in identifi- cationphase,1.16%. TheratiooftheNMSEinthevalidationphasetotheNMSEinthetraining phase (NNMSE) of the ANN is not as that of the RFM. The averaged NNMSE of the ANN is 4.45,andtheaveragedNNMSEoftheRFMis2.20. Thisresultshowsthattherelativeaccuracy oftheANNinthevalidationphaseislessthanthecorrespondingaccuracyoftheRFM. ComputationTime A fair comparison of computation times of the investigated methods is difficult, because the computationtimeisacomplicatedfunctionofmanyparametersusedineachmethod. However, it was noticed that the optimization-based methods (the SDM and the ANN) require a longer computation time than the quadrature-based method (the RFM) due to the fundamental differ- ences in the identification methods. In general, the optimization-based methods would require moretime,whentheerrorsurfaceismorecomplex. 3.5.3 IdentificationUsingtheDataSetswithConcatenatedSinusoidalExcitation Using a single-frequency sinusoidal excitation, the dynamic response of a linear system can be fully identified. However, for a nonlinear system, such as the nonlinear viscous damper under discussion, the dynamic characteristics, in general, can not be accurately identified with a single-frequency sinusoidal excitation, since the dynamic response depends on the amplitude and frequency of the excitation. On the other hand, for a full-scale viscous damper, using a broadbandrandomexcitation(whichismuchbetterfromtheidentificationpointofview)could generateanexcessiveamountofheatforanextendedtestingperiod. 48 An alternative way to identify a nonlinear viscous damper is to use combined data sets of thedamperresponsesformultiplesingle-frequencysinusoidalexcitation. Thecombineddataset canbeobtainedbyconcatenatingmultiplesetsofdamperresponsewithdifferentfrequencyand amplitudecharacteristics. Becausethetimesequentialorderofthecombineddatasetshouldnot affect the identification results, the combined data are randomly shuffled in time. Identification approachesthatuseerrorfunctionsbasedonthesolutionofthegoverningdifferentialequations requiresequentialorderingintimeofthemeasureddataandsimulatedones. However,if“static” data comparison is used (i.e., by directly using the assumed model form), then the elements of thereferencedatasetcanbearbitrarilyshuffledintime. The influence on the identification results when using the time-shuffled concatenated data was studied. A concatenated data set was prepared by adding 14 data sets (among a total of 15 data sets) in series, and the time-order of the concatenated data set was randomly shuffled. TheRFM-andANN-identificationswereperformedusingtheconcatenateddataset. TheRFM- and ANN-identification was validated with the data set, which had not been used in the iden- tification (or training) phase. The same procedures were repeated to validate the RFM- and ANN-identificationresultsusingall15tested datasets. Asamplevalidation resultfortheRFM andtheANNisshowninFigure3.11. In the RFM, the mean and standard deviation of the averaged NMSE is (1.39± 0.28)% fortheidentificationand(2.46±1.42)%forthevalidation. IntheANN,themeanandstandard deviationofNMSEis(0.43±0.06)%forthetrainingand(0.64±0.23)%forthevalidation. Inthe identification(ortraining)phase,theaveragedNMSEwithconcatenateddatasetisgreaterthan thatwiththesingledataset: fortheRFM,(1.16±0.41)%withsingledatasetand(1.39±0.28)% 49 10 10.2 10.4 10.6 10.8 11 11.2 11.4 11.6 11.8 12 −1000 −500 0 500 1000 TIME (SEC) FORCE (kN) (a)Restoringforcemethod(RFM) 10 10.2 10.4 10.6 10.8 11 11.2 11.4 11.6 11.8 12 −1000 −500 0 500 1000 TIME (SEC) FORCE (kN) (b)Artificialneuralnetworks(ANN) Figure3.11: The“static”validationresultsoftheRFM-identification(a)andANN-identification (b) procedures for the data set UCB1 15 6 randomly shuffled in its sequential order. The solid lineisforthemeasuredforce,andthedashedlineisfortheidentifiedforce. withtheconcatenateddataset,andfortheANN,(0.25±0.06)%withsingledatasetand(0.43± 0.06)%withthecombineddatasets. Inthevalidationphase,however,theaveragedNMSEwith the concatenated data set for the ANN is less than that with single data set, while the averaged NMSE’swithsingleandconcatenateddatasetsfortheRFMareapproximatelythesame: forthe RFM, (2.22±0.85)% with single data set and (2.46±1.42)% with concatenated data set, and fortheANN,(1.08±0.72)%withsingledatasetand(0.64±0.23)%withconcatenateddataset. TheaveragedNMSEswithsingleandconcatenateddatasetsaresummarizedinTable3.6. Note that the accuracy of the ANN identification model to generalize; increases with the combined dataset,whiletheaccuracyoftheRFMidentificationisnotimprovedwiththeconcatenateddata set. 50 Table 3.6: The averaged normalized mean-square error of the restoring force method (RFM) andtheartificialneuralnetworks(ANN)identificationsusingasingleandconcatenateddamper responsedatasets. Thetableshowsthemeanandstandarddeviationoftheaveragednormalized mean-squareerror. Restoringforcemethod Artificialneuralnetworks Single Concat. Single Concat. Phase mean stdv mean stdv mean stdv mean stdv Identification 1.16 0.41 1.39 0.28 0.25 0.06 0.43 0.06 Validation 2.22 0.85 2.46 1.42 1.08 0.72 0.64 0.23 3.5.4 SignificanceofInertiaEffects Thequasi-staticapproximationinEquation3.6isvalidwhentheinertiaeffectisnegligibleinthe dynamicresponseofthedamper. Thepeakvelocityoftheexcitationusedinthisstudyiswithin therangeof254.0mm/sto444.5mm/s(10.0in/sto17.5in/s),whichisgreaterthanthetypical excitationvelocityinaquasi-statictesting,3mm/sto5mm/s(0.12in/sto0.20in/s). Therefore, thesignificanceoftheinertiaeffecttothedamperresponseshouldbeevaluated. The significance of the inertia effects was assessed, using the SDM-identification method; a formal parametric identification procedure was performed in which the governing differential equation of motion was directly solved through the use of conventional time-marching tech- niques. The mass term was explicitly included in the governing equations, and the damper restoringforcewasmodeledbytheexpressionofEquation3.1. Table3.7showsthesignificance oftheinertiaeffectsatdifferentpeakvelocitiesandpeakdisplacements. Forall15datasets,the variability of the identified mass around the mean value of 735.6 kg is 31.9%. It is important to note that the “effective” mass in the actual tests includes not only the weight of the moving internal components of the damper and a portion of the squeezed fluid, but also a part of the 51 Table3.7: EstimatedsignificanceofinertiaeffectsintheSDM-identification. Inthetable, || ˆ f||is thenormoftheSDM-identifiedforce, || ˆ f ¨ x ||isthenormoftheinertiatermoftheSDM-identified force,and||f||isthenormofthemeasuredforce. No Nameofdataset Mass(kg) || ˆ f ¨ x ||/|| ˆ f||(%) || ˆ f ¨ x ||/||f||(%) 1 UCB1 10 4 567.7 0.0504 0.0524 2 UCB1 10 5 1199.6 0.0841 0.0902 3 UCB1 10 6 1208.4 0.0685 0.0730 4 UCB1 12 4 685.9 0.0835 0.0877 5 UCB1 12 5 777.9 0.0763 0.0811 6 UCB1 12 6 869.8 0.0704 0.0746 7 UCB1 12 7 974.9 0.0698 0.0743 8 UCB1 15 4 567.7 0.0951 0.0990 9 UCB1 15 5 594.0 0.0786 0.0819 10 UCB1 15 6 785.2 0.0852 0.0942 11 UCB1 15 7 645.1 0.0611 0.0642 12 UCB1 17 4 494.7 0.1060 0.1104 13 UCB1 17 5 522.5 0.0895 0.0930 14 UCB1 17 6 519.5 0.0759 0.0784 15 UCB1 17 7 621.7 0.0776 0.0810 avg 735.6 0.0781 0.0824 cov 0.319 0.17 0.17 externalhardwareattachmentsutilizedinperformingthetest. Oncethe“effective”movingmass wasidentified,thesignificanceoftheinertiatermwasmeasuredas || ˆ f ¨ x || || ˆ f|| ×100(%), || ˆ f ¨ x || ||f|| ×100(%) (3.11) where|| ˆ f ¨ x ||isthenormoftheinertiatermoftheidentifiedforce,|| ˆ f||isthenormoftheidentified force, and||f|| is the norm of the measured force. The significance of the inertia effects is also summarized in Table 3.11 for all test cases. The mean of the|| ˆ f ¨ x ||/|| ˆ f|| is 0.0781% with the coefficient of variance of 0.17. The mean of || ˆ f ¨ x ||/||f|| is 0.0824% with the coefficient of variance of 0.17. Therefore, the induced inertia forces are negligible, confirming the earlier conclusiontoignoreinertiaterms. 52 3.6 SummaryandConclusions The goal of this study was to investigate the applicability of a set of parametric and non- parametricidentificationmethodstomeasurementsobtainedfromafull-scalenonlinearviscous damper. Suchmodelscanbeusedforstructuralhealthmonitoringpurposes. Twonon-parametric identification methods, the restoring force method and the artificial neural networks, were stud- ied and the results were compared with the parametric simplified design model of the full-scale viscous damper. The nonlinear full-scale viscous damper used in this study was successfully identified with the simplified parametric model, as well as with the restoring force method and the artificial neural networks. The identification results show that the normalized Chebyshev coefficients can be used to interpret the nature and relative contribution of the linear and non- linear characteristics of the viscous damper. A comparison of the investigated identification methodsisshowninTable3.8. 53 Table3.8: Acomparisonofinvestigatedsystemidentificationmethodsforapplicationsinstruc- turalhealthmonitoring. IdentificationMethods Advantages Disadvantages -Mostaccurateiftheexactsystemmodel - Aprioriknowledgeofthesystemisrequired. SimplifiedModel isknown. -Theidentifiedparametersbecomesignificantly (parametric) -Directphysicalinterpretationis biasedwhentheinitialmodelisincorrect. possibleusingtheidentified parameters. -No aprioriknowledgeofthesystemis -Theidentificationyieldsanapproximating required. model. -Thesamemodelcanbeusedwhenthe -Onlylimitedphysicalinterpretationof RestoringForceMethod systemchangesintodifferent identificationresultsispossible. (non-parametric) nonlinearclasses. -Itisapplicabletoawiderangeof nonlinearities. -BothChebyshevandpowerseries coefficientscanbeidentified. -Physicalinterpretationofsomeofthe identificationresultsispossiblewith identifiedcoefficients. -No aprioriknowledgeofthesystemis -Changedetectionispossible,butphysical required. interpretationofthedetectedchangesarenot ArtificialNeuralNetworks -Itisapplicabletoawiderangeof generallypossible. (non-parametric) nonlinearities. -Changedetectionofthesystemispossible throughmonitoringtheregression errorofthetrainednetworks. 54 Chapter4 Data-DrivenMethodologiesforChange DetectioninLarge-ScaleNonlinearDampers withNoisyMeasurements 4.1 Introduction 4.1.1 Motivation Large-scale orifice viscous dampers are frequently used in modern civil structures to mitigate seismic or wind-induced vibration. Among various types of dampers, orifice viscous dampers (hereinafter viscous dampers) provide excellent efficiency of energy dissipation — the orifice damper employs small orifices on its piston head, so that the silicon fluid sealed inside the damperchamberisforcedtopassthroughtheorificeswhenthedamperpistonreciprocates. Con- sequently, thedynamicproperties of an orifice viscous damper largely depend on the geometric characteristics of the orifice design. Soong and Constantinou (1994) and Soong and Dargush (1997)providedetaileddescriptionsoforificeviscousdampers. Due to their importance in applications involving civil structures, many government agen- cies require a series of quality assurance tests for large-scale dampers before the dampers are 55 installed in actual civil structures (HITEC, 1996, 1998a,b, 1999). After installation, the condi- tionassessmentoftheinstalleddampersiscommonlyperformedintwoways: visualinspection, and monitoring the internal pressure of the damper’s silicon fluid. First, visual inspection is usually conducted by trained inspectors, searching for noticeable damage on the damper sur- face, often evident by fluid leakage. The second method employs a pressure gauge to measure the internal pressure levels of the dampers. Thus, with a pressure change, the inspectors can presume that the damper has changed during the operation. If the pressure change were signifi- cant, the damper would be removed from the structure and delivered to testing facilities to find possible causes of the change. However, none of the current practices are adequate for reliable conditionassessment. Thevisualinspectionisoftensubjective. Althoughpressuremonitoringis obviously a more advanced method than visual inspection, the direct relationships between the pressure level and engineering characteristics of the nonlinear dampers are difficult to identify. Moreover,nocurrentpracticesofdampermonitoringareappropriatewhenanumberofdampers are employed in a structure. For example, after a major seismic retrofit of the west spans of the San Francisco Oakland Bay Bridge in 2004, more than 100 large-scale viscous dampers are employed. In this case, more systematic and efficient condition assessment methodologies are required. As an alternative approach for damper condition assessment, a vibration-based structural health monitoring technique is proposed in this study. Yun et al. (2007) demonstrated that the non-parametric Restoring Force Method (RFM) is a very promising tool for the condition assessment of large-scale nonlinear viscous dampers. Comparing one parametric (the simpli- fied damper design model) and two non-parametric identification methods (the Restoring Force 56 MethodandArtificialNeuralNetworks),theydemonstratedthattheRFMhassignificantadvan- tages than other methods because (1) no apriori knowledge of the system is needed, (2) the same non-parametric model is applicable to a wide-range of nonlinearities, and (3) the physical interpretation of the identification results is possible, which is generally impossible with other non-parametricidentificationmethods,suchasArtificialNeuralNetworks. Recent progress in sensing and internet-based data communication technologies allow the development of real-time remote monitoring systems for civil infrastructure system. Yun et al. (2007)havedevelopedareliablereal-timeweb-basedcontinuousbridgemonitoringsystemthat has been applied to a critical bridge (the Vincent Thomas Bridge) in the Los Angeles, Califor- nia, metropolitan region to perform forensic studies of various earthquakes, as well as a recent ship-bridge collision. Therefore, by combining the technology of a web-based monitoring sys- tem with the Restoring Force Method, a feasible methodology can be developed for a real-time remoteconditionassessmentoflarge-scalenonlinearviscousdampers. In the development of the monitoring system, the following practical and challenging prob- lemsmustbeconsidered: First,theeffectsofmeasurementnoiseontheresultsofchangedetec- tion must be considered, since sensor readings can be more significantly affected by noise in the in-situ measurements than in laboratory testing, due to various sources of noise. In many cases of in-situ monitoring, only the displacement or acceleration is measured, depending on the measurement feasibility, and then other necessary response states are numerically obtained through digital signal processing techniques using the measured response. In such cases, the effects of measurement noise are not simply additive, and propagate throughout the response states, which are numerically obtained from noisy measurements. Consequently, the developed 57 methodologyshouldbeabletodealwiththosecomplicatednoiseeffects. Second,theresultsof thechangedetectionwillbeaffectedbythemeasurementuncertainty. Therefore,theuncertainty of the detected change due to the measurement noise must be quantified for reliable condition assessment. However, the uncertainty quantification requires multiple tests, which is not usu- ally possible for the in-situ monitoring due to lack of control of excitation sources. Even if one had the control of the excitation, performing multiple tests with full-scale viscous dampers is extremelydifficultbecauseofanenormousamountofheatconvertedfromthedissipatedenergy. Having the proposed condition assessment methodology will provide contributions in the followingthreeways: 1. Enablingtheinterpretationofphysicalsignificanceofdetectedchanges,onecanquantify thesignificanceofthechangesatthefull-structurelevelaswellasatthecomponentlevel. This attribute remains even when the dampers’ evolving properties change into different classesofnonlinearity,duetovarioustypesofdeterioration. 2. With more reliable condition assessment methodologies, one can minimize unnecessary removalofundamageddampers. Damperremovalfromcivilstructuresistime-consuming andexpensiveduetotheirlargephysicalsize. 3. Since the methodology proposed in this study is data-driven and model independent, the same approach is applicable to other types of nonlinear components, such as different typesofenergydissipatingdevices,baseisolators,andnonlinearjoints. 58 4.1.2 Objective The objective of this study is to develop a data-driven methodology for change detection in large-scale nonlinear viscous dampers. A joint study was performed between the University of SouthernCalifornia(USC),theUniversityofCaliforniaatSanDiego(UCSD)andtheUniversity of California at Berkeley (UCB). Three different large-scale nonlinear viscous dampers were tested at UCB and UCSD. The damper experiments were designed to introduce different types ofnonlinearityinasystematicway. Threelarge-scaleviscousdampersusedintheexperimental study involved different nonlinear features. In the experiments, two different excitation types weretested,includingmonotonicsinusoidalandbroadbandrandomexcitations. Using the experimental results, an analytical study was performed at USC. A data-driven change detection methodology for the tested large-scale dampers was investigated using the non-parametric Restoring Force Method. In order to study the effects of measurement uncer- tainty, the damper data were intentionally polluted with random noise. As a statistical data recycling technique, the Bootstrap method was investigated for uncertainty quantification, even withinsufficientdataformeaningfulstatisticalinferences. Usingthedevelopedchangedetection methodology,theaimwastoachievethefollowing: 1. Abilitytodetectevensmall(genuine)changesinthenonlineardampers; 2. Abilitytointerpretthephysicalmeaningofdetectedchanges;and 3. Abilitytoquantifytheuncertaintyassociatedwiththedetectedchanges. 59 4.1.3 Scope This chapter is organized as follows: the experimental studies using three large-scale nonlinear dampersarediscussedinSection4.2;thedata-drivenidentificationapproachusingtheRestoring ForceMethodisdiscussedinSection4.3;theuncertaintyestimationandstatisticalchangedetec- tion of the large-scale viscous dampers are discussed in Section 4.4; and the Bootstrap method asadatarecyclingtechniqueanditsuncertaintyestimationarediscussedinSection4.5. 4.2 ExperimentalStudies 4.2.1 TestApparatus Threedifferentlarge-scalenonlinearviscousdampersweretestedattwodifferenttestfacilities: the66.7kN(15kip)viscousdamperwastestedattheEarthquakeEngineeringResearchCenter (EERC) of the University of California, Berkeley (Figure 4.1 (a)), and the 2001.6 kN (450 kip) and 2891.3 kN (650 kip) viscous dampers were tested at the Seismic Response Modification Device(SRMD)facilityoftheUniversityofCalifornia,SanDiego(Figure4.1(b)). The 66.7 kN damper with the maximum velocity of 431.8 mm/sec (Damper A) has the smallest size among the tested dampers in this study. The damper was designed using a simpli- fied Maxwell model (Constantinou and Symans, 1993; Constantinou et al., 1993; Den Hartog, 1956;MakrisandConstantinou,1991;Makrisetal.,1993)as r(x, ˙ x) =C sgn(˙ x)|˙ x| n (4.1) 60 (a)TestattheUniversityof (b)TestattheUniversityof California,Berkeley(UCB) California,SanDiego(UCSD) Figure4.1: Testfacilitiesforlarge-scaleviscousdampersattheUniversityofCalifornia,Berke- ley(UCB),andtheUniversityofCalifornia,SanDiego(UCSD)usedinthisstudy. wherer istherestoringforce,C isthedampingconstant,andnisthenonlineardampingexpo- nent. This simplified design model is valid when the excitation frequency is low. In this case, theinertiatermofthedamperresponsebecomesinsignificant,andconsequently,f(t)≈r(x, ˙ x), wheref isthemeasuredforce. Yunetal.(2007)demonstratedthattheinertiatermofthelarge- scaledamperresponsewouldbenegligible atalowvelocity. The designparametersofDamper A are C = 1.12 kN· sec n /mm n and n = 1.0, which makes the damper response approxi- matelylinear. The2001.6kNdamperatthemaximumvelocityof215.9cm/sec(DamperB)was designedwiththeparametersC = 398.93 kN·sec n /cm n andn = 0.3. The2891.3kNdamperat themaximumvelocityof40.6cm/sec(DamperC)wasdesignedwithC = 957.44 kN·sec n /cm n andn = 0.3. Hence,therestoringforceofDampersBandCwillbe“softening”withn< 1.0. 4.2.2 TestProtocolsandPreliminaryDataProcessing TestwithDamperA DamperAwassubjectedtobroadbandrandomexcitationwithalowpasscutofffrequencyof5.0 61 Hz. Duringtheexperiment,theacceleration(¨ x)andforce(f)ofthedamperweremeasuredwith asamplingfrequencyof1kHz. ThemeasuredforceofDamperAunderbroadbandrandomexci- tationisshowninFigure4.2(a). Once ¨ xandf weremeasured,preliminarydataprocessingwas performedtoobtainthedisplacement(x)andvelocity(˙ x)requiredforthedamperidentification. Thedataprocessingwasperformedinaccordancewiththefollowingprocedures: 1. Themeasured ¨ xandf werede-trendedandzero-phasefilteredwiththecutofffrequencies of0.1∼10.0Hz,andacosine-taperedwindowwasappliedtothetime-historiesof ¨ xand r. 2. The filtered ¨ x was integrated to obtain the corresponding velocity ˙ x. The same filter and time-historywindowwereappliedto ˙ x. 3. The processed ˙ x was numerically integrated to obtain the corresponding displacementx. Thesamefilterandtime-historywindowwerealsoappliedto x. The test protocols, preliminary data processing and phase plots of the resulting Damper A responsearesummarizedinTable4.1. TestwithDampersBandC DampersBandCweresubjectedtomonotonicsinusoidalexcitationwithanexcitationfrequency of 0.2 Hz for both dampers. Unlike Damper A,x andf (but not the ¨ x) were measured during the experiments. The sampling frequency of the measurement was 100 Hz. Figures 4.2 (b) and (c) show the measured force of Dampers B and C, respectively. In the figures, notice that the forceamplitudeofDamperBisconstant,whilethatofDamperCdecreases. Bothdamperswere subjected to the sinusoidal excitation with a constant frequency and constant peak amplitudes overtime. 62 0 10 20 30 40 50 60 70 80 90 100 110 −20 −10 0 10 20 TIME (sec) FORCE (KN) (a)DamperA(closetolinearandtime-invariant) 0 50 100 150 200 −600 −300 0 300 600 TIME (sec) FORCE (KN) (b)DamperB(damping“softening”andtime-invariant) 0 50 100 150 200 250 300 350 400 −1600 −800 0 800 1600 TIME (sec) FORCE (KN) (c)DamperC(damping“softening”andtime-varying) Figure 4.2: Time histories of the measured forces for different large-scale nonlinear viscous dampers with displacement-controlled excitations. (a) The force of Damper A was measured underbroadbandrandomexcitation. (b)TheforceofDamperBwasmeasuredundermonotonic sinusoidalexcitationwithaconstantfrequencyof0.2Hzandconstantpeakamplitudesof±50.8 mm. (c) The force of Damper C was measured under monotonic sinusoidal excitation with a constantfrequencyof0.2Hzandconstantpeakamplitudesof±25.4mm. Once x and f are measured for Dampers B and C, preliminary data processing was per- formedtoobtainthevelocity(˙ x)usingthefollowingprocedures: 1. Themeasuredxandf werede-trendedandzero-phasefilteredwiththecutofffrequencies of0.05∼ 5.0Hz. Then, a cosine-tapered window was applied to the time histories of the filteredresponse,xandf. 63 2. The displacementx was differentiated to obtain the corresponding ˙ x. The same filter and time-historywindowwereappliedtotheobtained ˙ x. Thetestprotocols,preliminarydataprocessingandphaseplotsofDampersBandCaresumma- rizedinTable4.1. 4.3 Non-ParametricIdentification 4.3.1 OverviewofRestoringForceMethod TheRestoringForceMethod(RFM)isanon-parametricidentificationmethodfornonlinearsys- tems,usingaseriesexpansionoftwo-dimensionalChebyshevpolynomials(MasriandCaughey, 1979). Using the RFM, the restoring force of a single-degree-of-freedom (SDOF) nonlinear dynamicsystemcanbemodeledas r(x, ˙ x) = P X i=0 Q X j=0 ¯ C ij T i (¯ x)T j ( ¯ ˙ x) (4.2) where r(x, ˙ x) is the restoring force of the nonlinear dynamic system, ¯ C ij is the normalized Chebyshev coefficient, T i (•) is the i th order Chebyshev polynomial, P and Q are the high- est orders of the Chebyshev polynomial of the normalized displacement (¯ x) and velocity ( ¯ ˙ x), respectively,withintherangeof[-1,1]. Oncethe ¯ C ij areidentified,the ¯ C ij canbeconvertedintotheequivalentpowerseriescoeffi- cientsusingthefollowingrelationship(MasonandHandscomb,2003): T 0 (y) = 1, T 1 (y) =y, T 2 (y) = 2y 2 −1,..., T k+1 (y) = 2yT k (y)−T k−1 (y),... (4.3) 64 Table 4.1: Summary of test protocols and preliminary data processing parameters for the three large-scalenonlinearviscousdampersusedinthisstudy. Parameters DamperA DamperB DamperC Nominaloutputforce 66.7(15) 2001.6(450) 2891.3(650) kN(kips) Max. velocityrating 43.2(17) 215.9(85) 40.6(16) cm/s(ips) Designedparameters C = 1.12,n = 1.0 C = 398.93,n = 0.3 C = 957.44,n = 0.3 fordamping, kN (sec/cm) n Excitationtype Broadbandrandom Monotonicsinusoidal Monotonicsinusoidal Excitationfrequency ≤5.0Hz 0.2Hz 0.2Hz Nonlinearity Closetolinear Polynomial,hysteretic Polynomial,hysteretic Time-invariancy Time-invariant Time-invariant Time-varying Measuredresponse ¨ x,f x,f x,f Performed Integrationfor ˙ x Dataprocessing Doubleintegrationforx Differentiationfor ˙ x Differentiationfor ˙ x xvs.f −10 −5 0 5 10 −20 −10 0 10 20 DISPLACEMENT (mm) FORCE (KN) −60 −30 0 30 60 −300 −150 0 150 300 DISPLACEMENT (mm) FORCE (KN) −30 −15 0 15 30 −1600 −800 0 800 1600 DISPLACEMENT (mm) FORCE (KN) ˙ xvs.f −200 −100 0 100 200 −20 −10 0 10 20 VELOCITY (mm/sec) FORCE (KN) −80 −40 0 40 80 −300 −150 0 150 300 VELOCITY (mm/sec) FORCE (KN) −40 −20 0 20 40 −1600 −800 0 800 1600 VELOCITY (mm/sec) FORCE (KN) 65 Theconvertedpowerseriescoefficientsarecalledthenormalizedpowerseriescoefficients(¯ a ij ). With the de-normalization of ¯ x and ¯ ˙ x, the de-normalized power series coefficients ( a ij ) can be obtained. Usingthesecoefficients,Equation4.2canbealsoexpressedas r(x, ˙ x) = P X i=0 Q X j=0 ¯ C ij T i (¯ x)T j ( ¯ ˙ x) = P X i=0 Q X j=0 ¯ a ij ¯ x i ¯ ˙ x j = P X i=0 Q X j=0 a ij x i ˙ x j (4.4) 4.3.2 IdentificationofNonlinearViscousDampers ItwasknownthattheforcecharacteristicsforDampersAandBdonotchangeovertimeunder stationarydisplacement-controlledexcitation. Forexample,asshowninFigure4.2(b),themea- sured force of Damper B is stationary over time under the stationary sinusoidal excitation with aconstantfrequencyof0.2Hzandconstantpeakamplitudesof±50.8mm. Consequently,since the outputs (i.e., measured force) of Dampers A and B do not depend explicitly on time, the dampers are time-invariant systems under stationary excitation. On the other hand, as shown in Figure4.2(c),themeasuredforceofDamperCdecreasesovertimealthoughthesinusoidalexci- tation has a constant frequency of 0.2 Hz and constant peak amplitudes of±25.4 mm. Hence, Damper C is a time-varying system since the output of Damper C depends on time under sta- tionaryexcitation. Forthesetwoclassesofnonlinearsystems(time-invariantandtime-varying), different procedures were applied in the damper identification. Detailed identification proce- duresforeachclassaredescribedbelow. Identificationresultsoftime-invariantsystems Using the time-invariant systems of Dampers A and B, the RFM identification was applied for theentiredomainofthemeasuredtimehistories. Inbothcases,theorderoftheseriesexpansion 66 −10 −5 0 5 10 −20 −10 0 10 20 DISPLACEMENT (mm) FORCE (KN) −10 −5 0 5 10 −20 −10 0 10 20 DISPLACEMENT (mm) FIT (KN) −200−100 0 100 200 −20 −10 0 10 20 VELOCITY (mm/sec) FORCE (KN) −200−100 0 100 200 −20 −10 0 10 20 VELOCITY (mm/sec) FIT (KN) (measured) (identified) (measured) (identified) (a)DamperA −60 −30 0 30 60 −300 −150 0 150 300 DISPLACEMENT (cm) FORCE (KN) −60 −30 0 30 60 −300 −150 0 150 300 DISPLACEMENT (cm) FIT (KN) −80 −40 0 40 80 −300 −150 0 150 300 VELOCITY (cm/sec) FORCE (KN) −80 −40 0 40 80 −300 −150 0 150 300 VELOCITY (cm/sec) FIT (KN) (measured) (identified) (measured) (identified) (b)DamperB Figure4.3: TheidentificationresultsforDampersAandBusingtheRestoringForceMethod. wasfive. TheidentificationresultsforDampersAandBareshowninFigure4.3. Thequalityof theRFMidentificationwasmeasuredwiththenormalizedmean-squareerrors(NMSE)as NMSE = 1 nσ 2 f n X i=1 (f i − ˆ f i ) 2 (4.5) wheren is the number of data points,f is the measured force, ˆ f is the identified force, andσ f is the standard deviation of the measured force (Worden and Tomlinson, 2001). Considering DamperA,excellentidentificationresultswereobtainedwiththeNMSEof0.82%asillustrated inFigure4.3(a). ForDamperB,“softening”hysteresisweresuccessfullyidentified(Figure4.3 (b)). However, the identification failed to accurately model the nonlinearity near the damper’s neutralposition(i.e.,x≈ 0and ˙ x≈ 0). TheNMSEfortheDamperBidentificationwas3.0%. 67 TheidentifiedRFMcoefficientsforDampersAandBaresummarizedinTable4.2. Forthe normalized Chebyshev coefficients ( ¯ C ij ), the first order damping coefficient ( ¯ C 01 ) is dominant forbothDampersAandB:27.99forDamperAand257.00forDamperB.NoticethatDamper B is designed for a larger damping capacity than Damper A (refer Table 4.1). The third order damping coefficient of Damper B ( ¯ C 03 = −13.17) is negative because the designed damping exponent is less than one (n = 0.3), while the ¯ C 03 of Damper A is close to zero ( ¯ C 03 = 0.30) because Damper A was designed forn = 1.0 (Equation 4.1). The stiffness-related coefficients ( ¯ C 10 for the linear stiffness and ¯ C 30 for the cubic stiffness) are relatively small compared to the dampingcoefficient( ¯ C 01 )forbothdampers,whichindicatesthatthecontributionofthestiffness terms is less significant in the identification than the damping terms (i.e., ¯ C 01 and ¯ C 03 ). These resultsarereasonableforviscousdampers. The identified power series coefficients (¯ a ij and a ij ) also show the damper nonlinearity without a priori knowledge of the dampers. For Damper A, the cubic damping coefficient (¯ a 03 = 4.21) is ignorable, compared to the linear damping (¯ a 01 = 25.82). This result indi- catesthatthedampingcharacteristicofDamperAisclosedtolinearratherthan“softening”. On the other hand, the significance of the cubic damping coefficient (¯ a 03 = 148.20) with respect to the linear damping coefficient (¯ a 01 = 203.10) becomes larger for Damper B. However, since the ¯ a 03 is still smaller than the ¯ a 01 , the force of Damper B is “softening”. The identified RFM coefficientsforDampersAandBaresummarizedinTable4.2. Identificationresultsoftime-varyingsystem In order to identify a time-varying nonlinear system of Damper C, the time histories of the damper data (i.e.,x, ˙ x andf) were partitioned into eight windows as illustrated in Figure 4.4. 68 Table4.2: SummaryoftheidentifiedcoefficientsusingtheRestoringForceMethod. Coefficients DamperA DamperB DamperC mean stdv max min entire ˆ ¯ C 10 1.24E-2 40.27 37.59 21.01 55.79 -1.48 51.27 ˆ ¯ C 01 27.99 257.00 516.48 64.64 625.80 436.66 526.6 ˆ ¯ C 30 0.63 2.80 3.75 2.06 7.04 0.98 10.90 ˆ ¯ C 03 0.30 -13.17 -26.70 3.47 -21.67 -30.63 -52.09 ˆ ¯ a 10 -1.41 66.32 31.02 56.63 100.30 -59.67 72.60 ˆ ¯ a 01 25.82 203.10 430.40 41.33 490.90 373.30 507.9 ˆ ¯ a 30 -0.55 10.33 101.20 48.11 155.90 38.70 17.38 ˆ ¯ a 03 4.21 148.20 244.66 115.71 387.80 89.29 117.0 ˆ a 10 -0.19 1.33 1.33 2.37 4.18 -2.47 3.02 ˆ a 01 0.27 3.12 13.17 1.24 14.92 11.38 15.37 ˆ a 30 2.11E-5 7.38E-5 7.08E-3 3.38E-3 1.08E-2 2.90E-3 1.37E-3 ˆ a 03 3.68E-6 5.34E-4 6.94E-3 3.37E-3 1.08E-2 2.35E-3 3.04E-3 NMSE(%) 0.82 5.03 0.50 0.09 0.66 0.39 1.92 0 100 200 300 400 −1000 −500 0 500 1000 TIME (sec) FORCE (KN) window 1 window 2 window 3 window 7 window 8 ........ Figure 4.4: Partitioning the time history of the measured force of Damper C for the Restoring ForceMethodidentification. The time history partition was designed to have ten cycles per window. Then, the RFM identi- fication was performed for each time-history window. Damper C was accurately identified, and the mean and standard deviation of the NMSE for the eight windows were 0.50% and 0.09%, respectively. The identified normalized Chebyshev coefficients and normalized power-series coefficients for the eight windows are illustrated in Figure 4.5. For the normalized Chebyshev coefficients( ¯ C ij ),thelineardamping( ¯ C 01 )isdominantwiththemeanvalueof516.48,whilethe 69 THENORMALIZEDCHEBYSHEVCOEFFICIENTS( ¯ C ij ) 1 2 3 4 5 6 7 8 −200 −100 0 100 200 WINDOW NO. ¯ C 10 1 2 3 4 5 6 7 8 400 450 500 550 600 650 WINDOW NO. ¯ C 01 1 2 3 4 5 6 7 8 0 2 4 6 8 WINDOW NO. ¯ C 30 1 2 3 4 5 6 7 8 −35 −30 −25 −20 WINDOW NO. ¯ C 03 (a)1 st stiffness (b)1 st damping (c)3 rd stiffness (d)3 rd damping THENORMALIZEDPOWERSERIESCOEFFICIENTS(¯ a ij ) 1 2 3 4 5 6 7 8 −200 −100 0 100 200 WINDOW NO. ¯ a 10 1 2 3 4 5 6 7 8 300 350 400 450 500 550 WINDOW NO. ¯ a 01 1 2 3 4 5 6 7 8 −100 0 100 200 300 WINDOW NO. ¯ a 30 1 2 3 4 5 6 7 8 0 100 200 300 400 500 WINDOW NO. ¯ a 03 (e)1 st stiffness (f)1 st damping (g)3 rd stiffness (h)3 rd damping Figure4.5: TheidentifiedcoefficientsofDamperCfordifferenttime-historywindows. cubic stiffness ( ¯ C 30 ) is negligible with the mean value of 3.75. The linear damping coefficient ( ¯ C 01 )decreasesasthemeasuredforcedecreases(Figure4.5(b)),whilethelinearstiffness( ¯ C 10 ) remainsconstant(Figure4.5(a)). Thecubicdampingcoefficient( ¯ C 03 )decreasesasthedamper reciprocates. For the normalized power-series coefficients ( ¯ a ij ), the first order damping (¯ a 01 ) and third order damping (¯ a 03 ) decrease (Figures 4.5 (f) and (h)), while the first order stiffness (¯ a 10 ) and third order stiffness remain constant (Figures 4.5 (e) and (g)). These results indicate thatthedegradingforceofDamperCisduetothechangeofdampingcharacteristicsratherthan stiffnesscharacteristicsovertime. TheidentifiedRFMcoefficientsforDamperCaresummarizedinTable4.2. Inthetable,the mean,standarddeviation,maximumandminimumvaluesoftheRFMcoefficientsidentifiedfor the eight identification windows in Figure 4.4 are shown. For a comparison purpose, Damper 70 0 50 100 150 200 250 300 350 400 −600 −400 −200 0 200 400 600 TIME (SEC) FORCE, FIT (KN) Figure 4.6: The measured and identified forces for the time-varying system of Damper C under the stationary sinusoidal excitation with a constant frequency of 0.2 Hz and constant peak dis- placements of±25.4 mm using the entire domain of measured time histories of displacement, velocity and force. In the figure, the measured force is in the solid line, and the identified force isinthedashedline. C was also identified using the entire domain of measured time histories, and the correspond- ing identified RFM coefficients for the entire time domain are also shown in the last column of Table 4.2. The table shows that the dominant coefficients for the entire time-history data are within the range of the minimum and maximum for the partitioned time-history data (e.g., −1.48≤ 51.27≤ 55.79forthe ¯ C 10 and436.66≤ 526.60≤ 625.80). TheNMSEoftheformer isalsoabout3.5timesgreaterthanthelatter. Themeasuredandidentifiedforcesusingtheentire timedomainarecomparedinFigure4.6. Thefigureillustratesthattheidentifiedforceestimates theaverageofthedegradingmeasuredforceovertime. Findingsfromtheidentificationresults Based on above results, several important conclusions can be drawn. First, two different types of nonlinear dampers were accurately identified without using apriori knowledge about the identified dampers. This is because the identification procedures of the RFM are data-driven and model-independent. Although no apriori knowledge was used in the identification, the identified Chebyshev and power series coefficients still contain the information concerning the 71 dominant physical characteristics of the identified dampers. Consequently, in the development of the change detection methodology, these coefficients can be used as “change indicators” (or “features”inpatternrecognitionsense). Moreover,knowingwhichcoefficientsthechangeswere observedin,onecaninterpretthephysicalmeaningsofthedetectedchanges. Hence,guidelines todealwiththedetectedchangescanbeestablishedforfieldapplications. Anexcellentexample canbefoundintheidentificationresultsofDamperC.Again,withoutaprioriknowledgeofthe time-varying damper, the identified coefficients show that the decreasing measured force is due to degradation of the damping efficiency (decreasing damping coefficients) in time rather than thechangesofdamperstiffness(constantstiffnesscoefficients). NoticethatalthoughtheidentifiednormalizedChebyshevcoefficients( ¯ C ij )arerelatedtothe dampers’stiffnessordampingcharacteristics,theyarenotexactlyequivalenttotheactualspring ordampingconstantsofthedampers. Forphysicalinterpretationpurposes,thenormalizedpower series coefficients (¯ a ij ) and de-normalized power series coefficients ( a ij ) can be used as more convenient indices. However, the ¯ C ij have many advantages over ¯ a ij and a ij , because of the orthogonal property of the Chebyshev polynomials. The orthogonal property of ¯ C ij can reduce the complexity of the uncertainty quantification of change detection with noisy measurements. DetaileddiscussionofthisissueisprovidedinSection4.4.3. 4.4 UncertaintyEstimationofDamperIdentification 4.4.1 DataGenerationofNoisyResponse Inordertostudytheeffectsofmeasurementnoiseonthedamperidentification,thesensormea- surements of Dampers A, B and C were polluted with 5% additive zero-mean Gaussian noise 72 100 105 110 −60 −30 0 30 60 TIME (sec) DISPLACEMENT (cm) 100 105 110 −80 −40 0 40 80 TIME (sec) VELOCITY (cm/sec) 100 105 110 −300 −150 0 150 300 TIME (sec) FORCE (KN) Figure 4.7: Sample time histories of noisy response of Damper B. The displacement and force werepollutedwith5%additivezero-meanGaussiannoisewithrespecttothemeasuredresponse states,andthenthevelocitywasobtainedwithnumericaldifferentiationfollowingthedatapro- cessingproceduresdiscussedinSection4.2.2. withrespecttotheroot-mean-square(RMS)ofthemeasurementstates: theacceleration( ¨ x)and force (f) for Damper A, and the displacement (x) and force (f) for Dampers B and C. Once the measurement states were polluted, the necessary damper response for the RFM identifica- tion was obtained numerically with the noisy measurements: x and ˙ x for Damper A, and ˙ x for DampersBandC.Hence,theuncertaintyofthenoisymeasurementspropagatedthroughoutthe numericallyobtainedresponse. Thedetaileddataprocessingprocedureswerethesameasthose describedinSection4.2.2. Atotalof3000noisydatasetsweregeneratedforalltesteddampers. SampletimehistoriesofnoisydatasetsforDamperBareshowninFigure4.7. 4.4.2 DamperIdentificationwithNoisyResponse Once the 3000 noisy data sets were obtained for each damper, the RFM identification was per- formed, and the corresponding Chebyshev coefficients ( ¯ C ij ) and power series coefficients (¯ a ij anda ij ) were identified. The NMSE of the RFM identification was relatively low for all tested dampers: themeanandstandarddeviationoftheNMSEforDamperAwere1.33%and0.79%, 73 respectively;thoseforDamperBwere3.91%and2.54%,respectively;andthoseforDamperC were4.59%and2.98%,respectively. Forthelineardamping( ¯ C 01 ),whichwasthedominanttermintheidentification,themeanof ¯ C 01 forDamperAwas16.84,whichwas61.16%comparedtotheidentified ¯ C 01 of27.99using the “clean” data set, while the means of ¯ C 01 for Dampers B and C were 278.06 and 562.48, respectively, which were 108.19% and 108.91%, compared to the identified ¯ C 01 of 257.00 and 516.48, respectively, using the “clean” data set. Hence, the discrepancy between the identified ¯ C 01 for“clean”and“noisy”datawaslargerwithDamperAthanwithDampersBandC. The statistics of identified coefficients for Dampers A, B and C using the RFM are summa- rized in Table 4.3. The table shows that the coefficients of variance (cv) of ¯ C 01 and the cubic damping ( ¯ C 03 ) for Dampers B and C are almost identical: the cv of ¯ C 01 for Dampers B and C were 0.03 and 0.03, respectively, and the cv of ¯ C 03 were -0.15 and -0.18, respectively. This result is expected since the “softening” characteristics of Dampers B and C are similar with the same designed damping exponent (n = 0.3). On the other hand, Damper A has a different cv (cv of ¯ C 03 = −5.00) because the designed damping exponent for Damper A was n = 1.0. Hence,physicalinterpretationusingtheidentifiedcoefficientwasstillvalidevenwiththenoisy measurements. 4.4.3 StatisticalChangeDetectionofTime-VaryingDamper StatisticalindependenceoftheRFMcoefficients In Section 4.3.2, it was shown that the identified RFM coefficients can be used as excellent “change indicators”. A question is left: among three kinds of RFM coefficients, which one 74 Table4.3: StatisticsoftheidentifiedRFMcoefficientsforthemultipletestsand3000noisydata sets. The mean, standard deviation and coefficient of variation are shown. In this table, only significant coefficients are shown, including the linear stiffness ( ¯ C 10 , ¯ a 10 ,a 10 ), linear damping ( ¯ C 01 , ¯ a 01 ,a 01 ), cubic stiffness ( ¯ C 30 , ¯ a 30 ,a 30 ), and cubic damping ( ¯ C 03 , ¯ a 03 ,a 03 ). The * indi- catesthecoefficientsofDamperCaretheaveragedvaluesforeighttime-historywindowsshown inFigure4.4. (a)NormalizedChebyshevCoefficients( ˆ ¯ C ij ) DamperA DamperB DamperC ∗ Coefficients mean stdv cv mean stdv cv mean stdv cv ˆ ¯ C 10 0.67 0.19 0.28 26.23 14.26 0.54 55.19 32.17 0.58 ˆ ¯ C 01 16.84 0.16 0.01 278.06 8.60 0.03 562.48 17.17 0.03 ˆ ¯ C 30 0.14 0.14 1.00 -6.05 14.09 -2.33 -6.17 28.40 -4.60 ˆ ¯ C 03 -0.02 0.10 -5.00 -62.46 9.14 -0.15 -112.19 20.71 -0.18 (b)NormalizedPowerSeriesCoefficients( ˆ ¯ a ij ) DamperA DamperB DamperC ∗ Coefficients mean stdv cv mean stdv cv mean stdv cv ˆ ¯ a 10 1.55 0.69 0.45 110.49 115.28 1.04 286.46 245.46 0.86 ˆ ¯ a 01 18.23 0.74 0.04 642.43 86.03 0.13 1093.50 198.68 0.18 ˆ ¯ a 30 -1.20 2.32 -1.93 -190.07 360.35 -1.90 -473.24 786.05 -1.66 ˆ ¯ a 03 -2.63 2.26 -0.86 -747.97 322.88 -0.43 -982.85 669.23 -0.68 (c)De-normalizedPowerSeriesCoefficients( ˆ a ij ) DamperA DamperB DamperC ∗ Coefficients mean stdv cv mean stdv cv mean stdv cv ˆ a 10 1.42E-1 6.27E-2 0.44 2.00 2.09 1.05 10.98 7.85 0.71 ˆ a 01 1.17E-1 5.56E-3 0.05 7.25 0.90 0.12 26.25 3.63 0.14 ˆ a 30 -6.04E-4 2.36E-3 -3.91 -1.10E-3 2.12E-3 -1.93 -2.56E-2 4.24E-2 -1.66 ˆ a 03 -8.17E-7 6.69E-7 -0.82 -1.05E-3 4.35E-4 -0.41 -1.31E-2 8.32E-3 -0.64 (d)NormalizedRoot-Mean-SquareofIdentificationErrors DamperA DamperB DamperC ∗ Coefficients mean stdv cv mean stdv cv mean stdv cv NMSE 1.33E-2 7.90E-3 0.59 3.91E-2 2.54E-2 0.65 4.59E-2 2.98E-2 0.65 75 is most useful for change detection in a probabilistic sense. The advantage of using the de- normalizedpowerseriescoefficients(a ij )isthatdirectphysicalinterpretationispossiblebecause thea ij preservesthephysicalunits(e.g.,theunitofa 10 forDamperBiskN/mm,thatisthesame as the linear spring constant). The advantage of using the normalized power series coefficients (¯ a ij ) is that although the direct physical interpretation is not convenient due to using the nor- malizeddisplacement(¯ x)andvelocity( ¯ ˙ x), ¯ a ij measurestherelativecontributionofeachpower seriestermtotheidentifiedrestoringforce. However,whenmeasurementuncertaintyexists,the identifieda ij and ¯ a ij are not statistically independent because the basis functions of the power series expansion (i.e., x i ˙ x j and ¯ x i ¯ ˙ x j ) are not orthogonal. Consequently, for the uncertainty quantification of the system changes, the testing dimension of the statistical Hypothesis Test (HT)becomestoohighbecausethea ij and¯ a ij aremultivariatecoefficients. Forexample,inthis study,thereare36identifiedcoefficientswiththehighestseriesorderoffiveforthedisplacement and velocity. Fora ij and ¯ a ij , because each of the coefficients are not statistically independent, theHTshouldbeperformedwiththetestingdimensionof36(maximum). InFigure4.8(a),the scatter plot between the first order damping (¯ a 01 ) and linear stiffness (¯ a 10 ) shows no significant statistical correlation. However, a strong correlation is observed between the linear damping (¯ a 01 )andcubicdamping(¯ a 03 ). On the other hand, the normalized Chebyshev coefficients ( ¯ C ij ) preserves the statistical independence because the basis function of Chebyshev polynomials are orthonormal (Mason and Handscomb, 2003). In Figure 4.8 (b), both scatter plots illustrate that no significant statis- tical correlations are observed between the identified Chebyshev coefficients. With the statisti- cal independence property, the testing dimension of the HT dramatically reduces to one. That 76 Coefficienttype Scatterplots (a)Normalizedpowerseries −500 500 1500 2500 3500 −2000 −1000 0 1000 2000 3000 ¯ a01 ¯ a10 −500 500 1500 2500 3500 −10000 −4000 2000 8000 ¯ a01 ¯ a03 Coefficients(¯ a ij ) (ρ =−0.04) (ρ =−0.96) (b)NormalizedChebyshev 500 600 700 800 900 −100 0 100 200 300 ¯ C01 ¯ C10 500 600 700 800 900 −300 −200 −100 0 100 ¯ C01 ¯ C03 Coefficients( ¯ C ij ) (ρ =−0.08) (ρ = 0.05) Figure4.8: SamplescatterplotsofthenormalizedChebyshevcoefficientsandnormalizedpower seriescoefficientsforthenoisyresponseofDamperC(Window1inFigure4.4). Themagnitude of the linear correlation coefficient (ρ) between two identified coefficients is also shown in the table. is, the HT can be performed for each individual Chebyshev coefficient to detect possible sys- tem changes. Hence, the normalized Chebyshev coefficients were used in the statistical change detectioninthisstudy. Statisticalchangedetectionusingidentifiedcoefficients Usingthe3000setsoftheidentified,normalizedChebyshevcoefficients( ¯ C ij ),thedistributions of the identified ¯ C ij were obtained. The histograms of the identified first order damping coef- ficient ( ¯ C 01 ), the dominant coefficient in the Damper C identification, for different time-history 77 windowsareshowninFigure4.9. Thebinwidthofthehistogramswasdeterminedusingthenor- mal reference rule (or Scott’s rule) (Scott, 1979, 1992), optimized for the Gaussian distribution as h = 3.5S X N −1/3 (4.6) wherehisthebinwidth(orsmoothingfactor),S x isthesamplestandarddeviationofastatistic of interest X, and N is the sample size. The probability density functions (pdf) of ¯ C 01 were estimated with the Gaussian distribution assumption and are shown in Figure 4.9. In the figure, themeanofthedistributionsdecrease in time, while the standard deviations of thedistributions remain approximately constant. The pseudo-constant deviation is the justification as to why the noiseamplitudeswerefixedat5%RMSwithrespecttothemeasurementstatesamongthewin- dows(Section4.4.1). Afterobtainingthedistributionsofidentifiedcoefficients,onecanachieve thethreeobjectivesofthisstudythatwerediscussedinSection4.1.2. First,withthemeanofthe distribution, one can accurately check if the damper has had a genuine system change. Second, one can interpret the physical meaning of the detected changes. In Sections 4.3.2 and 4.4.2, it was shown that the actual changes in Damper C are due to the degradation of the damping efficiencyratherthanstiffnessefficiency. Third,withthestandarddeviationsofthedistributions determined, one can quantify the uncertainty of the detected changes. Using the RFM identi- fication procedure, these objectives can be achieved without knowing the underlying physical characteristicsoftheidentifiedsystem. 78 350 400 450 500 550 600 650 700 750 800 0 0.01 0.02 0.03 0.04 FIRSTORDER DAMPING, C 0 1  ¯ PROBABILITY WINDOW 8 WINDOW 4 WINDOW 1 Figure4.9: Histogramsandprobabilitydensityfunctions(pdf)ofthefirstorderdampingnormal- izedChebyshevcoefficient( ¯ C ij )fordifferenttime-historywindows. Thebinwidth(orsmooth- ing factor) of the histogram was determined using the normal reference rule (or Scott’s rule). Thepdf’swereestimatedwiththeassumptionofthenormaldistribution. Using the extracted coefficient distributions, the statistical HT was performed to detect the changesinthedistributionmeans. Thistestcanbeperformedwiththeteststatisticsoftwo-tailed T-distribution(HoggandTanis,1997;MendenhallandSincich,1995): H 0 : (μ 1 −μ 2 ) = 0, z = ¯ y 1 − ¯ y 2 σ (¯ y 1 −¯ y 2 ) ≈ ¯ y 1 − ¯ y 2 q s 2 1 n 1 + s 2 2 n 2 (4.7) where H 0 is the null-hypothesis, ¯ y 1 and ¯ y 2 are the identified Chebyshev coefficients for two different identification windows, μ 1 andμ 2 are the means of the coefficient distributions from twoidentificationwindows,σ 1 andσ 2 arethestandarddeviationsofthecoefficientdistributions fromtwoidentificationwindows,ands 1 ands 2 thesamplestandarddeviationsofthecoefficient distributions from two identification windows. In the HT’s, the change of the distribution mean wasobservedwithallwindows(Windows1to8)witha95%confidencelevel. 79 4.5 BootstrapEstimationoftheIdentificationUncertainty The uncertainty quantification usually requires many data sets — in Section 4.4, 3000 data sets were used to measure the identification uncertainty. However, collecting sufficient data sets of large-scale viscous dampers for reliable statistical estimation is very difficult and expensive. Statisticaldatarecyclingtechniqueshavebeenappliedsuccessfullyinmanyfieldsofengineering andsciencefortheerrorgeneralizationofidentificationresultsusinginsufficientdatasets. Inthis section,theBootstrapmethodisusedtomeasuretheuncertaintyofthedamperchangedetection with a single data set. The Bootstrap estimates of the identification uncertainty with a single datasetwillbecomparedwiththeuncertaintyestimateswiththemultipledatasetsdiscussedin Section4.4. 4.5.1 OverviewoftheBootstrapMethod The Bootstrap method is a statistical data recycling technique for the uncertainty estimation of any kind of identification parameters. This method is commonly used where the estimation of parameteruncertaintyisneeded,butaninsufficientamountofdataisavailableforastatistically reliable uncertainty quantification. Excellent introductory literature on the Bootstrap method can be found in the work of Efron (1979); Efron and Tibshirani (1993), Davison and Hinkley (1997),andMartinezandMartinez(2002). 80 The Bootstrap method starts with a very simple assumption. An arbitrary parameter (θ) identified using an independently and identically distributed (i.i.d) random data set, y = (y 1 ,y 2 ,··· ,y n ) T withtheunderlyingtruedistribution(F)canbemodeledas θ =t(F) (4.8) wheret(•) is a nonlinear function ofF. Without knowingF, the uncertainty ofθ is commonly determinedwithmultipledatasets,{y 1 ,y 2 ,··· ,y M },drawnfromthesamedistributionF as s θ = v u u t 1 M−1 M X i=1 (θ i −m θ ) 2 (4.9) wheres θ isthesamplestandarddeviationofθ,m θ isthesamplemeanofθ,M isthenumberof multipletests,andθ i istheparameteridentifiedinthei th test. Insteadofperformingmultipletestsfortheuncertaintyquantification,theBootstrapmethod recyclesasingledataset,ywiththeempiricaldistribution( ˆ F). Thedatarecyclingisperformed with the random selection of a sample (y k , where 1≤k≤n) fromy forn times with replace- ment. Withreplacement, theprobability ofeachsample tobe selectedis 1/n. Performing these proceduresB times, one can obtain multiple Bootstrap replicates,{y ∗ 1 ,y ∗ 2 ,··· ,y ∗ B }. The Boot- strapestimateoftheparameteruncertaintyisdeterminedas s ∗ θ = v u u t 1 B−1 B X i=1 (θ ∗ i −m ∗ θ ) 2 (4.10) 81 wheres ∗ θ is the Bootstrap standard error ofθ,θ ∗ i is the parameter identified in thei th Bootstrap replicate of the data set, B is the number of the Bootstrap replicates, andm ∗ θ is the Bootstrap estimateofθ definedas m ∗ θ = 1 B B X i=1 θ ∗ i (4.11) In order that s θ ≈ s ∗ θ , the empirical distribution ˆ F should be close to the true distribution F. Therefore, the following two conditions should be satisfied for the Bootstrap estimation of the standarderror: 1. Therandomdatayisi.i.d. 2. Theempiricaldistribution ˆ F isclosetothetruedistributionF. In the context of the damper identification problem under discussion, however, since the noisy measurement states (¨ x,f) for Damper A and (x,f) for Dampers B and C are time- correlated (i.e., the data are not i.i.d), the standard Bootstrap method described above needs to be modified to deal with the time-dependency. Many modified algorithms have been devel- oped and introduced: model-based resampling (Efron and Tibshirani, 1986; Kreiss and Franke, 1992), block resampling (Carlstein, 1986; Hall, 1985; Shi, 1991), phase scrambling (Theiler etal.,1992;Timmer,1998),andperiodogramresampling(DavisonandHinkley,1997). Detailed descriptions of each of these methods can be found in Davison and Hinkley (1997), and H¨ ardle et al. (2003). Among these methods, one of the most widely used method is the model-based resampling,becauseofitssimpleprocedureandgoodtheoreticalbehaviorwhenthetime-series modeliscorrect. Consequently,inthisstudy,themodel-basedresamplingmethodwasemployed fortheuncertaintyestimationofthetime-dependentdata. InSection4.5.2, adetailedBootstrap 82 resamplingprocedureisproposedanddescribedindetailforthecasesthatthedisplacementand forceweremeasured(DamperA),andthattheaccelerationandforceweremeasured(Dampers BandC). 4.5.2 BootstrapResamplingofNoisyResponseData Single data sets of Dampers A, B and C were recycled with the Bootstrap method using the followingprocedures: Approachwhendisplacementismeasured A single data set of noisy (5% RMS) displacement (x) and force (f) for Dampers B and C was resampledwiththeBootstrapmethodasfollows: 1. The same data processing procedures for Dampers B and C in Section 4.2.2 were per- formedtoobtainthetriplet(x, ˙ x,f). 2. TheRFMidentificationwasperformedwiththenoisy(x, ˙ x,f). Theidentificationresidual (e)wasobtainedase =f− ˆ f,where ˆ f istheidentifiedforceusingtheRFM. 3. Theauto-regression(AR)wasperformedforthetimehistoriesof xande. Thecorrespond- ingARestimateofxis ˆ x. The ARordersweredeterminedsoastosatisfythe conditions thatε x andε e become i.i.d, whereε x is the AR residual ofx, andε e is the AR residual ofe. The detailed procedure for determining the optimal AR orders for theε x andε e is describedlaterinthissection. 4. TheBootstrapresamplingwasperformedwiththeε x andε e toobtaintheBootstraprepli- catesoftheε x andε e (ε ∗ x andε ∗ e ,respectively). 83 Single differentiation for velocity x & DSP detrending, filtering, etc. ) , , ( f x x & RFM non-parametric identification f f e ˆ − = Auto Regression for time histories x x x ε + = ˆ e e e ε + = ˆ Bootstrapping for residuals * x x ε ε → * e e ε ε → Sample Reconstruction * * ˆ x x x ε + = * * ˆ ˆ e e f f ε + + = Single differentiation for velocity * x & DSP detrending, filtering, etc. ) , , ( * * * f x x & Figure4.10: BootstrapresamplingproceduresforDampersBandCwithmeasureddisplacement (x)andforce(r). 5. The Bootstrap replicates of the displacement (x ∗ ) and force (f ∗ ) were obtained with the samplereconstructionasx ∗ = ˆ x+ε ∗ x andf ∗ = ˆ f + ˆ e+ε ∗ e . 6. The Bootstrap version of the velocity (˙ x ∗ ) was obtained through the differentiation of x ∗ . In this procedure, the same filter and time-history window as those discussed in Sec- tion4.2.2wereapplied. A total of 3000 Bootstrap replicates (x ∗ , ˙ x ∗ ,f ∗ ) were obtained. The Bootstrap resampling pro- ceduresforDampersBandCarealsoillustratedschematicallyinFigure4.10. AsamplecomparisonoftheoriginalandBootstrap-resampleddataisshowninFigures4.13. TheBootstrapresampleddatashowslightlylargerdispersionthantheoriginaldatainthephase plots. The RFM identification was performed with the 3000 Bootstrap replicates, and the cor- responding RFM coefficients were identified. The Bootstrap standard errors of 3000 identified coefficient sets were estimated using Equation 4.10, and compared to the standard deviations of multiple tests. Table 4.4 shows a comparison of the error estimations of the RFM identified coefficients with multiple tests. In the table, the error estimates with the Bootstrap method are larger than those with multiple tests: 7%∼ 42% for Damper B and -0.2% ∼ 53% for Damper 84 Table 4.4: Bootstrap estimations of standard errors for the coefficients identified using the Restoring Force Method. The Bootstrap estimates are compared with the standard deviations through the multiple tests shown in Table 4.3. The sample size is 3000 for both the Bootstrap andmultipletestestimates. DamperA DamperB DamperC Coefficients Multiple Bootstrap Ratio Multiple Bootstrap Ratio Multiple Bootstrap Ratio ¯ C10 0.19 0.32 1.68 14.26 17.35 1.22 32.17 42.14 1.31 ¯ C01 0.16 0.31 1.94 8.60 12.22 1.42 17.17 26.21 1.53 ¯ C30 0.14 0.32 2.29 14.09 18.63 1.32 28.40 37.81 1.33 ¯ C03 0.10 0.20 2.00 9.14 12.49 1.37 20.71 27.47 1.33 ¯ a10 0.69 1.44 2.09 115.28 149.10 1.29 245.46 311.77 1.27 ¯ a01 0.74 1.64 2.22 86.03 104.55 1.22 198.68 235.09 1.18 ¯ a30 2.32 4.76 2.05 360.35 474.97 1.32 786.05 983.70 1.25 ¯ a03 2.26 5.67 2.48 322.88 395.84 1.23 669.23 823.17 1.23 a10 6.27E-2 7.13E-2 1.14 2.09 2.67 1.28 7.85 11.60 1.48 a01 5.56E-3 1.01E-2 1.82 0.90 1.06 1.18 3.63 4.58 1.26 a30 2.36E-3 7.03E-4 0.30 2.12E-3 2.71E-3 1.28 4.24E-2 5.14E-2 1.21 a03 6.69E-7 1.30E-6 1.94 4.35E-4 4.66E-4 1.07 8.32E-3 8.13E-3 0.98 NMSE(%) 7.90E-3 2.05E-2 2.59 2.54E-2 3.02E-2 1.19 2.98E-2 2.98E-2 1.00 C. Hence, it can be seen that the Bootstrap estimation of the identification error is more conser- vative than the results obtained through estimation with multiple tests. In addition, the results indicatethattheBootstrapmethodisapplicabletothetime-varyingsystem(DamperC),aswell asthetime-invariantsystem(DamperB). Approachwhenaccelerationismeasured Using a single data set of noisy (5% RMS) measurements of the acceleration (¨ x) and force (f) forDamperA,theBootstrapmethodwasappliedasfollows: 1. The same data processing procedures for Damper A in Section 4.2.2 were performed to obtainthetriplet(x, ˙ x,f). 2. TheRFMidentificationwasperformedwiththenoisy(x, ˙ x,f). Theidentificationresidual (e)wasobtainedase =f− ˆ f,where ˆ f istheidentifiedforceusingtheRFM. 85 Single and double Integrations for velocity and displacement x x & , DSP detrending, filtering, etc. ) , , ( f x x & RFM non-parametric identification f f e ˆ − = Auto Regression for time histories x x x & & & & & & ε + = ˆ e e e ε + = ˆ Bootstrapping for residuals * x x & & & & ε ε → * e e ε ε → Sample Reconstruction * * ˆ x x x & & & & & & ε + = * * ˆ ˆ e e f f ε + + = Single and double Integrations for velocity and displacement * * ,x x & DSP detrending, filtering, etc. ) , , ( * * * f x x & Figure4.11: BootstrapresamplingproceduresforDamperAwithmeasuredacceleration(¨ x)and force(r). 3. The AR was performed for the time-histories of ¨ x and e. The AR estimate of ¨ x is ˆ ¨ x. The AR orders of ˆ ¨ x and ˆ e were determined so as to satisfy the conditions thatε ¨ x andε r becomei.i.d,whereε ¨ x istheARresidualof ¨ x,andε e istheARresidualofe. Thedetailed procedurefordeterminingtheoptimalARordersisdescribedbelow. 4. TheBootstrapresamplingwasperformedwithε ¨ x andε e toobtaintheBootstrapreplicates ofε ¨ x andε e (ε ∗ ¨ x andε ∗ e ,respectively). 5. The Bootstrap replicates of the acceleration (¨ x ∗ ) and force (f ∗ ) were obtained with the samplereconstructionas ¨ x ∗ = ˆ ¨ x+ε ∗ ¨ x andf ∗ = ˆ f + ˆ e+ε ∗ e . 6. The ¨ x ∗ wasintegratedandthendouble-integratedfortheBootstrapversionofthevelocity (˙ x ∗ ) and displacement (x ∗ ), respectively. The same filter and time-history window were appliedto ˙ x ∗ andx ∗ asdescribedinSection4.2.2. A total of 3000 Bootstrap replicates (x ∗ , ˙ x ∗ ,f ∗ ) were generated. The Bootstrap resampling proceduresforDamperAarealsoillustratedinFigure4.11. A sample comparison between the original and Bootstrap-resampled data for Damper A is shown in Figure 4.13 (a). Unlike Dampers B and C, the range of the Bootstrap-resampled 86 displacement is approximately twice larger than that of the original displacement in the displacement-forceplot,whilethevelocity-forceplotsoftwodatasetsarealmostidentical. Issuesinvolvingtheauto-regressionprocedure Theaboveresultsindicatethatusingthemeasuredacceleration,theBootstrappingfortheveloc- itythroughsingle-integrationwassuccessful,buttheBootstrappingforthedisplacementthrough double-integration failed. In the model-based Bootstrap method, the resampling results are largelydependentontheperformanceoftheARidentification. TheARisperformedtoremove the trends of the time-series data, and with successful AR, the corresponding AR residuals ( ε ¨ x andε e for Damper A, andε x andε e for Dampers B and C) becomei.i.d. Figure 4.12 shows the significance of the time-correlation for different AR orders. The significance of the time- correlation is commonly measured with the correlation coefficient (ρ) in a lag plot. Here, the lag is defined as a fixed time distance. For example, for the vector ε e = {ε e 1 ,ε e 2 ,...,ε en } for Damper B, theε e 2 andε e 5 have a lag with order three. Hence, in the lag plot (usually with order one), which has the x-axis of ε e i and the y-axis of ε e i−1 (i = 2,3,...,n), the correlation coefficientρ(ε e i ,ε e i−1 )measurestheserialcorrelationsoftheε e intime. InFigure4.12(a),the ρ(ε e i ,ε e i−1 )asymptoticallyapproachestozeroastheARorderincreases. However,Figure4.12 (b)illustratesthattheρ(ε x i ,ε x i−1 )approachestozeroastheARorderapproachesfrom1to40. Then, theρ(ε x i ,ε x i−1 ) increases as the AR order increases more than 40. This result indicates that the AR regression for the identification residual (e) becomes overfitted when the AR order is greater than 40. Consequently, the AR order of 40 was used in the Bootstrap resampling for Damper B. The same procedure of determining the optimal AR order was applied for Dampers AandC. 87 10 0 10 1 10 2 10 3 −0.3 −0.25 −0.2 −0.15 −0.1 −0.05 0 0.05 AUTO−REGRESSION ORDER ρ(ε ei ,ε ei−1 ) 10 0 10 1 10 2 10 3 −0.5 −0.4 −0.3 −0.2 −0.1 0 0.1 0.2 0.3 AUTO−REGRESSION ORDER ρ(ε xi ,ε xi−1 ) (a)ResidualoftheIdentifiedforceresidual(ε e ) (b)Displacementresidual(ε x ) Figure 4.12: Time-correlations of the auto-regression (AR) residuals of the identified restoring forceresidual(ε e )andthedisplacement(ε x )fordifferentARorders. Thetime-correlationswere measured with the correlation coefficients of the order-one lags for ε e andε x . The definition of theorder-onelagsisexplainedinthetext. AlthoughtheserialcorrelationsintimewerecarefullyremovedwiththeoptimalARorders, however, perfect removal of the time correlations is almost impossible. Consequently, a slight amount of time-correlation will affect the results of differentiation or integration. In this study, the results indicate that the unremoved trend does not significantly affect to the results of the single differentiation (x ∗ → ˙ x ∗ ) and integration (¨ x ∗ → ˙ x ∗ ). However, the unremoved trend significantlyinfluencestheresultsofthesecondintegration(˙ x ∗ →x ∗ )astheexampleofDamper A. Consequently, the Bootstrapping for the displacement becomes unsuccessful. Therefore, in theapplicationoftheBootstrapmethodtonoisymeasurements,itisrecommendedthattheforce aswellasthedisplacementofthedamperbedirectlymeasured. 4.6 SummaryandConclusions An experimental study was conducted to develop a probabilistic change detection methodology forin-situ monitoringofnonlinearviscousdamperswithmeasurementuncertainty. Itwasfound 88 that the coefficients identified using the Restoring Force Method can be used as excellent indi- cators (or features) (1) to detect the changes of nonlinear systems, (2) to interpret the physical meaning of the detected changes, and (3) to quantify the uncertainty of the detected system changes. The Bootstrap method was also investigated for uncertainty quantification of the detected changes when the measurement data are insufficient for reliable statistical inference. Using the Bootstrapmethod,theuncertaintyoftheidentificationwasestimatedreasonablyaccuratelyeven withasingledatasetwhenthedisplacementandforceweremeasured. 89 Type (a)DamperA (b)DamperB (c)DamperC Original −20 −10 0 10 20 −20 −10 0 10 20 DISPLACEMENT (mm) FORCE (KN) −60 −30 0 30 60 −300 −150 0 150 300 DISPLACEMENT (mm) FORCE (KN) −30 −15 0 15 30 −700 −350 0 350 700 DISPLACEMENT (mm) FORCE (KN) Bootstrap −20 −10 0 10 20 −20 −10 0 10 20 DISPLACEMENT (mm) FORCE (KN) −60 −30 0 30 60 −300 −150 0 150 300 DISPLACEMENT (mm) FORCE (KN) −30 −15 0 15 30 −700 −350 0 350 700 DISPLACEMENT (mm) FORCE (KN) Original −200 −100 0 100 200 −20 −10 0 10 20 VELOCITY (mm/sec) FORCE (KN) −100 −50 0 50 100 −300 −150 0 150 300 VELOCITY (mm/sec) FORCE (KN) −50 −25 0 25 50 −700 −350 0 350 700 VELOCITY (mm/sec) FORCE (KN) Bootstrap −200 −100 0 100 200 −20 −10 0 10 20 VELOCITY (mm/sec) FORCE (KN) −100 −50 0 50 100 −300 −150 0 150 300 VELOCITY (mm/sec) FORCE (KN) −50 −25 0 25 50 −700 −350 0 350 700 VELOCITY (mm/sec) FORCE (KN) Figure 4.13: A comparison of the original and Bootstrap-resampled data for different nonlinear dampers. The upper half of the figure shows displacement-force plots, while the lower half showsthevelocity-forceplots. 90 Chapter5 Model-OrderReductionEffectsonChange DetectioninUncertainNonlinear Magneto-RheologicalDampers 5.1 Introduction 5.1.1 Motivation Thedevelopmentofaneffectivestructuralhealthmonitoring(SHM)methodologyisimperative fortwomajorpurposes: (1)toavoidcatastrophicstructuralfailurebydetectingvarioustypesof structuraldeterioration,modificationorchangesduringtheoperation,and(2)toreducemainte- nancecostbyestablishingeffectivemeansandtimeschedulesforstructuralmaintenanceorreha- bilitationforthedetectedorpredictedchanges. However,thedevelopmentofaneffectiveSHM methodology is very challenging, especially when monitored structures are complex nonlinear systems and their system characteristics are uncertain. The system characteristics uncertainty canbefrequentlyfoundduetouncertainsystemparametersorvariousenvironmentaleffectson systemcharacteristics. Current SHM approaches, however, have the following limitations for the condition assess- mentofnonlinearstructureswithsystemcharacteristicsuncertainty: 91 1. The system models are commonly over-simplified. The over-simplification can be con- ducted in two ways: (1) excessive model-order reduction of complex nonlinear systems and (2) ignorance of significant environmental effects. The excessive model-order reduc- tion makes the identification results inaccurate. Moreover, the effects of model-order reduction to the change detection are rarely studied. Detected structural changes could be also significantly biased with ignoring the changes of environmental effects (Peeters etal.,2001). 2. Themodelingapproachesarenot“flexible”enoughtoidentifytimelychanging(ordeteri- orating) structures. Because parametric identification approaches require a priori knowl- edge of the monitored structures, if the structures change into another classes of nonlin- ear systems, the system identification using the “old” models are no longer valid. Non- parametric approaches, however, are more “flexible” than the parametric approaches by identifying the time-varying systems with no assumption about the structures’ physical characteristics. Yun et al. (2007) experimentally demonstrated that non-parametric mod- elingapproachesaremoreadvantageousinmonitoringpurposes 3. Although many current methodologies can detect changes of structural characteris- tics, physical interpretations of detected changes are rarely possible with current non- parametric approaches. Some necessary physical interpretations for effective SHM are discussedearlier. 92 4. MostofcurrentSHMmethodologiesarebasedondeterministicmodels,anduncertaintyof detectedstructuralchangesisrarelyestimated. Theestimationofchangedetectionuncer- taintyshouldincludetheeffectsofthemeasurementandsystemparameteruncertaintyas discussedearlier. 5.1.2 Objectives The objectives of this part of study was to develop a reliable change detection methodology for uncertain nonlinear systems. An experimental study was conducted to test the validity of the developed methodology. A complex nonlinear system with system parameter uncertainty was usedinthisexperimentalstudy. Theeffectsofmodel-orderreductionofthenonlinearsystemon theperformanceofthechangedetectionmethodologywereinvestigated. 5.1.3 MethodologyandScope Approachofexperimentalstudy For the experimental study of change detection in uncertain nonlinear systems, a single degree- of-freedom (SDOF)magneto-rheological (MR) damper was used. MR dampers are semi-active energy dissipating devices (Dyke et al., 1996; Ehrgott and Masri, 1992, 1994; Spencer et al., 1997, 1998; Yang et al., 2004). The MR dampers typically consist of a piston rod, electromag- net, damper cylinder filled with MR fluid, accumulator, bearing and seal. The magnetic field generated with the electromagnet changes the characteristics of the MR fluid, which consists of smallmagneticparticlesandfluidbase. Consequently,thestrengthoftheelectromagnet’sinput currentdeterminesthephysicalcharacteristicsofMRdampers. 93 Inthisstudy,aseriesoftestswasconductedwithrandomMRdamperinputcurrents: under deterministic broadband random excitation, the MR damper was characterized with a constant inputcurrent,whichrandomlyvariesbetweentests. Hence,theaverage(effective)characteristics oftheMRdamperaredeterminedbythemeanoftherandominputcurrents,andtheuncertainty (variability) of the damper characteristics are controlled by the standard deviation of the input currents. Identificationapproach Therestoringforceofanonlinearsystemcanbeexpressedas r(t) =g(x, ˙ x,p) (5.1) where r(t) is the restoring force, g(•) is a nonlinear function, x is the displacement, ˙ x is the velocity,andpisthesystemparametervector. Thenonlinearrestoringforcecanbemodeledas r(t) = ˆ r(t)+e(t), ˆ r(t) =h(x, ˙ x,q) (5.2) where ˆ r istheidentifiedrestoringforce,e(t)isthemodelingerror,h(•)isanonlinearfunction, andq is the postulated model parameter vectors. In parametric modeling approaches,q usually is postulated with assumptions of physical characteristics of the system so that the characteris- tics should be known a priori. The physical interpretation, using the parametric approaches, is usually straightforward because the identified q is directly related to the assumed system char- acteristics based on the assumptions concerning the phenomenological model. On the other hand, the non-parametric models do not require the assumption of system characteristics, but 94 their identification processes are model-independent and data-driven. Consequently, the non- parametricmodelsremainvalidevenifthesystemistransformedintoanothertypeofnonlinear- ity. An example of the non-parametric approach can be found in the artificial neural networks (ANN) (Masri et al., 1993, 2000, 1999). However, the physical interpretation with the non- parametric approaches is not straightforward because there are no direct relationships between the identified q and system characteristics, and as can be seen with ANN, the q is not uniquely definedevenevenwith“successful”identification(Masrietal.,2000). Inthiscase,theRestoringForceMethod(RFM)wouldprovideanexcellentsolution,taking both advantages of the parametric and non-parametric approaches: no a priori knowledge of the system is required and physical interpretation of some of the identification results with the identified coefficients (Wolfe et al., 2002; Yun et al., 2007). Therefore, in this study, the RFM wasextensivelyinvestigatedfordevelopinganeffectiveSHMmethodology. Effectsofmodel-orderreduction Thesignificanceofthemodelingerrorcanbedeterminedbytherelationshipofthesystemcom- plexity,O(p) and the model complexity,O(q). If the true system parameters, p are uncertain, therelationshipbetweenO(p)andO(q)canbedefinedas 1. O(p)>O(q): thesystemcomplexityisgreaterthanthemodelcomplexity(underfitting). 2. O(p) =O(q): thesystemcomplexityisequaltothemodelcomplexity(perfectfitting). 3. O(p)<O(q): themodelcomplexityisgreaterthanthesystemcomplexity(overfitting). Somestudiesofstochasticnon-parametricmodelsofuncertainnonlinearsystemswerereported byMasrietal.(2006). Inpractice,thesystemidentificationisrarelyperfectfittingsincethepare 95 oftenunknown. Whenthesystemidentificationiseitherunderfittingoroverfitting,theidentified qisgenerallybiased(Mendel,1995;SeberandLee,2003). Consequently,thedamagedetection with the identified q becomes inaccurate. Hence, in this study, the effects of the model-order reduction on the change detection were investigated with the identified coefficients using the RFM. Scope This chapter is organized as follows: the experimental studies using the nonlinear MR damper are discussed in Section 5.2; the non-parametric RFM identification for the MR damper is discussed in Section 5.3; and statistical change detection for the MR damper using pattern- recognition-basedclassificationmethodsisdemonstratedinSection5.4. 5.2 ExperimentalStudy 5.2.1 TestApparatus AnMRdamperwastestedintheStructuralDynamicsLaboratoryattheUniversityofSouthern California. In order to investigate the effects of system parameter uncertainty, performing a numerous series of tests is necessary. For the successful experimental study, controlling the damper tem- peratureiscriticalbecausetheinternaltemperatureoftheMRdamperthatisconvertedfromthe dissipated energy increases significantly during the series of tests. Consequently, an effective watercoolingsystemwasdevelopedtominimizethetemperatureeffectsonthedamper’sphysi- calcharacteristics(Figure5.1(a)). TheMRdamperwasmountedontheactuator,controllingthe 96 damper displacement with a PID controller (Figure 5.1 (b)). The MR damper was fully instru- mentedwithvarioussensors,includinganLVDT(displacement),LVT(velocity),accelerometer (acceleration),loadcell(force),andtemperature(dampersurfacetemperature). Inordertocon- ductanumberoftestsinthisstudy,adata-acquisition(DAQ)softwarewasalsodeveloped. The roleofthedevelopedDAQsoftwarewastoautomatethetestprocedurebycontrollingtheactu- ator, controllingtheMRdamperinputcurrent, andmeasuringthesensorreadings. Aschematic figureofthearchitectureoftheMRdampertestapparatusisillustratedinFigure5.1(c). TheMRdamperusedinthisstudyhadverycomplicatednonlinearities: ahystereticnonlin- earity due to the viscous action of the MR fluid combined with a dead-space nonlinearity due to a mechanical gap in the damper. Figure 5.2 illustrates the time histories of the measured dis- placement, velocity and force under sinusoidal excitation. For the given displacement that was controlled by the actuator controller, the measured force shows the combination of the dead- space nonlinearity due to a mechanical gap near the damper’s neutral position (i.e.,x≈ 0) and theviscousnonlinearityduetotheMRdampercharacteristicswithintheremainingdisplacement range. InFigure5.1,themechanicalgapcanbeseentobeapproximately0.9mm. 5.2.2 TestProtocols A series of tests was performed with different statistics of the damper input current. A total of eight test sets was conducted, with four different mean values (μ I ) and two different values of standard deviation (σ I ), for the MR damper input current (I): μ I = 1.0 A, 0.8 A, 0.6 A and 0.4 A, andσ I = 0.1 A and 0.15 A. Consequently, for each data set, the input current had a Gaussian distribution ofN ∼ (μ I ,σ I ). Therefore, the effective (nominal) characteristics of the MR damper are determined by means of μ I , and the uncertainty of the MR damper with 97 (a)TheMRdampermounted (b)TheMRdamperinstalled withthewatercooler ontheactuator PC SCXI MR Damper Actuator (Shake Table) LabVIEW - actuator command voltage displacement-controlled actuation magnetic field strength (I) - damper velocity (LVT) - damper acceleration (MEMS-based accel.) - damper force (load cell) - damper surface temperature (thermocouple) - fixture acceleration - damper displacement (LVDT) - actuator command voltage measured data test parameters (c)Aschematicoftheinstrumentationsystemarchitecture Figure5.1: Themagneto-rheological(MR)dampertestapparatus. σ I . For each test set, 500 experiments were conducted. Consequently, a total of 4000 tests was performed in this study. The MR damper was subjected to broadband random excitation with cutoff frequencies of 0.1∼ 3.0 Hz. The test protocols using the MR damper are summarized in Table5.1. 98 8 8.5 9 9.5 10 10.5 −0.4 −0.2 0 0.2 0.4 TIME (SEC) x (a)Displacement(cm) 8 8.5 9 9.5 10 10.5 −4 −2 0 2 4 TIME (SEC) ˙ x (b)Velocity(cm/sec) 8 8.5 9 9.5 10 10.5 −400 −200 0 200 400 TIME (SEC) f (c)Force(N) 8 8.5 9 9.5 10 10.5 −1 −0.5 0 0.5 1 TIME (SEC) ¯ x, ¯ ˙ x, ¯ f (d)Normalizeddisplacement,velocityandforce Figure 5.2: Time histories of the measured and normalized displacements, velocities and forces oftheMRdampersubjectedtosinusoidalexcitation. 99 Table5.1: MRdampertestprotocols. Testno. Inputcurrent(A) Inputcurrent Samplesize Excitation mean stdv distribution A 1 1.0 0.10 A 2 1.0 0.15 B 1 0.8 0.10 Deterministic B 2 0.8 0.15 Gaussian 500 broadband-random C 1 0.6 0.10 (0.1∼3.0Hz) C 2 0.6 0.15 D 1 0.4 0.10 D 2 0.4 0.15 5.3 Non-ParametricIdentificationofMR-Damper 5.3.1 OverviewofRestoringForceMethod The Restoring Force Method (RFM) is a non-parametric identification technique for nonlinear dynamicsystems(MasriandCaughey,1979;WordenandTomlinson,2001). ASDOFnonlinear systemcanbemodeledusingatwo-dimensionalseriesexpansionoftheChebyshevpolynomials: r(x, ˙ x) = P X i=0 Q X j=0 ¯ C ij T i (¯ x)T j ( ¯ ˙ x) (5.3) wherer(x, ˙ x) is the restoring force of the nonlinear dynamic system, the ¯ C ij is the normalized Chebyshev coefficient, T i (•) is the i th order Chebyshev polynomial, P and Q are the highest ordersoftheChebyshevpolynomialofthenormalizeddisplacement(¯ x)andvelocity( ¯ ˙ x)within therangeof[-1,1]. Forgivenmeasuredvectorsof x, ˙ x,andr,the ¯ C ij canbeidentifiedas ¯ C ij = <r(t),T i (¯ x)T j ( ¯ ˙ x)> <T i (¯ x)T j ( ¯ ˙ x),T i (¯ x)T j ( ¯ ˙ x)> = RR w(¯ x)w( ¯ ˙ x)r(t)T i (¯ x)T j ( ¯ ˙ x)d¯ xd ¯ ˙ x RR T i (¯ x) 2 T j ( ¯ ˙ x) 2 d¯ xd ¯ ˙ x , (5.4) 100 where w(·) is the weighting function. Once the ¯ C ij is identified, the normalized and de- normalizedpowerseriescoefficients(¯ a ij anda ij ,respectively)canbeidentifiedas r(x, ˙ x) = P X i=0 Q X j=0 ¯ C ij T i (¯ x)T j ( ¯ ˙ x) = P X i=0 Q X j=0 ¯ a ij ¯ x i ¯ ˙ x j = P X i=0 Q X j=0 a ij x i ˙ x j (5.5) usingthefollowingrelationships(MasonandHandscomb,2003): T 0 (y) = 1, T 1 (y) =y, T 2 (y) = 2y 2 −1,..., T n+1 (y) = 2yT n (y)−T n−1 (y),... (5.6) 5.3.2 IdentificationResultsfortheMRDamper TheMRdamperwasidentifiedusingtheRFMforthemeasureddatasetsinTable5.1. Inorderto understand the model-order reduction effects on the identification results, the normalized mean squareerrors(NMSE)oftheRFMidentificationfordifferentmodel-ordersweremeasuredas NMSE = 1 mσ 2 f m X i=1 (f i − ˆ f i ) 2 (5.7) wherem is the number of data points,f is the measured force, ˆ f is the identified force, andσ f is the standard deviation of the measured force (Worden and Tomlinson, 2001). Figure 5.3 (a) shows the relationship between the series order and the NMSE. The NMSE decreases rapidly from order 1 to 5, and the slope becomes gradually saturated from order 6 to 20. Hence, the MRdamperwasidentifiedwiththemodelordersof5and20(O(5)andO(20),respectively)to investigate the effects of the model-order reduction. For O(5) andO(20), a total of 8000 RFM identifications was performed (i.e., 2 model complexities× 4 input current means× 2 input 101 current standard deviations× 500 observations). In the RFM identification, the contribution of each Chebyshev polynomial term can be measured using the following normalized weighting equation: ¯ w ij = ¯ C 2 ij P P p=0 P Q q=0 ¯ C 2 pq (5.8) The identification results showed that the three most significant terms in the identification were the linear damping ( ¯ C 01 ), linear stiffness ( ¯ C 10 ), cubic damping ( ¯ C 03 ) and cubic stiffness ( ¯ C 30 ). Thecumulativeweightforthesetermswasgreaterthan90%forbothmodelingorders. Table5.2 summarizesthestatisticsof ¯ C 01 , ¯ C 10 , ¯ C 03 ,thecorrespondingpowerseriescoefficients(¯ a 01 ,¯ a 10 , ¯ a 03 ),andtheNMSEoftheRFMidentificationforO(5)andO(20). 5.3.3 PhysicalInterpretationsWithoutAssumingSystemModels Figure 5.3 (b)-(d) show the velocity-force plot of the measured and identified response with O(5)andO(20). Figure5.3(c)showsthatwiththeO(5),althoughthemajorityoftracesofthe velocity-force plot can be identified, the details of the traces largely due to the dead-space non- linearityfailtobeidentified. UsingthebasisfunctionsoftheChebyshevpolynomials,“smooth” (or continuous) nonlinearities can be identified using a relatively small number of the series expansion terms. However, for discontinuous nonlinearity, such as the dead-space nonlinearity, arelativelylargenumberoftheseriesexpansiontermsisusuallyneededofthesamebasisfunc- tions. Figure5.3(C)showsthatwiththehigherO(20),thediscontinuousnonlinearityarefairly accuratelyidentified. 102 Table 5.2: Summary of the identification results for the MR damper using the Restoring Force Method. (a)NormalizedChebyshevCoefficients( ¯ C ij ) Order Testno. 1 st stiffness( ¯ C 10 ) 1 st damping( ¯ C 01 ) 3 rd stiffness( ¯ C 03 ) mean stdv cv mean stdv cv mean stdv cv A 1 337.88 80.46 0.2381 1038.40 180.30 0.1736 -255.60 49.99 -0.1956 A 2 330.07 110.38 0.3344 1018.04 257.94 0.2534 -249.14 71.99 -0.2889 B 1 502.14 60.83 0.1211 1348.91 99.05 0.0734 -324.02 24.96 -0.0770 B 2 491.39 85.59 0.1742 1328.49 162.36 0.1222 -318.98 41.20 -0.1292 5 C 1 573.79 54.61 0.0952 1511.32 67.97 0.0450 -357.11 17.73 -0.0497 C 2 571.06 67.76 0.1187 1502.65 103.44 0.0688 -354.69 25.79 -0.0707 D 1 624.05 43.21 0.0692 1618.85 53.48 0.0330 -386.94 17.29 -0.0447 D 2 619.77 50.65 0.0817 1612.89 75.61 0.0469 -385.18 21.13 -0.0549 A 1 360.56 86.72 0.2405 1048.99 183.05 0.1754 -270.00 53.09 -0.1966 A 2 352.44 117.60 0.3373 1028.54 264.32 0.2570 -263.08 77.02 -0.2928 B 1 541.28 65.98 0.1219 1364.32 100.87 0.0739 -343.60 26.80 -0.0780 B 2 529.02 93.41 0.1766 1342.24 166.03 0.0124 -337.73 44.35 -0.0131 20 C 1 619.45 59.27 0.0957 1530.89 69.27 0.0452 -380.29 18.98 -0.0499 C 2 615.86 73.37 0.1191 1522.08 105.47 0.0693 -377.95 27.99 -0.0741 D 1 668.16 48.02 0.0719 1637.19 54.24 0.0331 -412.16 17.16 -0.0416 D 2 664.76 56.02 0.0843 1632.14 77.78 0.0477 -410.72 22.80 -0.0555 (b)NormalizedPowerSeriesCoefficients(¯ a ij ) Order Testno. 1 st stiffness(¯ a 10 ) 1 st damping(¯ a 01 ) 3 rd stiffness(¯ a 03 ) mean stdv cv mean stdv cv mean stdv cv A 1 647.78 170.41 0.2631 2340.12 430.94 0.1842 -3179.03 644.04 -0.2026 A 2 881.37 229.41 0.3658 2283.33 612.69 0.2683 -3089.39 881.37 -0.2853 B 1 942.21 137.39 0.1458 3034.01 243.71 0.0803 -4314.57 470.03 -0.1089 B 2 922.13 172.21 0.1867 2983.92 385.27 0.1291 -4233.05 650.00 -0.1536 5 C 1 1076.99 139.31 0.1293 3383.90 193.39 0.0571 -4822.62 472.61 -0.0980 C 2 1072.01 153.27 0.1430 3363.73 265.94 0.0791 -4792.21 548.46 -0.0114 D 1 1210.75 115.62 0.0954 3612.91 167.06 0.0462 -5026.61 467.50 -0.0930 D 2 1189.28 141.32 0.1188 3603.26 201.03 0.0558 -5019.45 480.98 -0.0958 A 1 430.95 549.97 1.2762 3205.08 658.45 0.2054 -18432.98 19792.93 -1.0738 A 2 500.18 526.79 1.0532 3084.24 815.13 0.2643 -16998.43 20179.07 -1.1871 B 1 777.02 840.00 1.0810 4498.72 929.77 0.2067 -41326.02 33137.60 -0.8019 B 2 811.18 737.51 0.9092 4369.22 981.22 0.2246 -38265.26 31812.22 -0.8314 20 C 1 1036.13 921.20 0.8891 4913.95 1009.55 0.2054 -43620.36 37791.24 -0.8664 C 2 1054.86 979.76 0.9288 4916.29 1007.44 0.2049 -44503.36 35387.04 -0.7952 D 1 1163.74 968.06 0.8318 4615.94 1036.27 0.2245 -24603.48 39224.80 -1.5943 D 2 1167.03 963.62 0.8257 4712.44 1037.13 0.2201 -28390.98 39049.60 -1.3754 (c)NormalizedMeanSquareError(NMSE) Order Testno. mean stdv cv Order Testno. mean stdv cv A 1 0.1532 0.0118 0.0769 A 1 0.1173 0.0196 0.1669 A 2 0.1516 0.0138 0.0908 A 2 0.1150 0.0242 0.2106 B 1 0.1733 0.0195 0.1128 B 1 0.1419 0.0218 0.1538 B 2 0.1720 0.0203 0.1179 B 2 0.1406 0.0235 0.1672 5 C 1 0.1769 0.0187 0.1056 20 C 1 0.1484 0.0210 0.1414 C 2 0.1768 0.0184 0.1039 C 2 0.1494 0.0213 0.1422 D 1 0.1828 0.0108 0.0593 D 1 0.1579 0.0161 0.1018 D 2 0.1822 0.0115 0.0629 D 2 0.1579 0.0172 0.1090 103 0 5 10 15 20 0.1 0.2 0.3 0.4 ORDER NMSE −22 −11 0 11 22 −2000 −1000 0 1000 2000 VELOCITY (cm/sec) FORCE (N) (a)Ordervs. NMSE (b)Measured −22 −11 0 11 22 −2000 −1000 0 1000 2000 VELOCITY (cm/sec) FIT (N) −22 −11 0 11 22 −2000 −1000 0 1000 2000 VELOCITY (cm/sec) FIT (N) (c)Identified(order5) (d)Identified(order20) Figure5.3: AsampleidentificationresultfortheMRdamperusingtheRestoringForceMethod. Physicalinterpretationswithoutassumingsystemmodels Asdiscussedearlier,theRFMcanbeusedtoidentifynonlinearsystemswithoutaprioriknowl- edge of the systems because the method is model-independent and data-driven, like other non- parametricidentificationmethods. UsingtheRFM,somephysicalinterpretationsarealsopossi- blewiththeidentifiedcoefficients. Forexample,therestoringforceofnonlinearsystemscanbe modeledas r(x, ˙ x) =r x +r ˙ x +r x,˙ x , (5.9) where r x is the restoring force component that is dependent on the displacement only, r ˙ x is the restoring force component dependent on the velocity only, and r x˙ x is the restoring force 104 componentthatdependsonbothdisplacementandvelocity. IntheRFM,thecomponentsr x ,r ˙ x , andr x˙ x canbeexpressed,bygroupingthetermsoftheseriesexpansionas r x = P X i=0 ¯ C i0 T i (¯ x) = P X i=0 ¯ a i0 ¯ x i (5.10) r ˙ x = Q X j=0 ¯ C 0j T j ( ¯ ˙ x) = P X i=0 ¯ a 0j ¯ ˙ x j (5.11) r x˙ x = P X i=1 Q X j=1 ¯ C ij T i (¯ x)T j ( ¯ ˙ x) = P X i=1 Q X j=1 ¯ a ij ¯ x i ¯ ˙ x j (5.12) First, using ther x ,r ˙ x , andr x˙ x , some physical interpretations can be made with the identi- fied RFM coefficients without assuming any system models. The effects of different modeling complexitiesforthesameinputcurrentwerestudied. Figure5.4showsthephaseplotsofther x andr ˙ x usingtheorthogonalChebyshevandnon-orthogonalpowerseriesbasisfunctions. Inthe figure, the identified forces are shown as the solid lines, and the measured forces as the dashed lines. Thefirstrowshowsthedisplacement-forceplotsof r x forthepowerseriesandChebyshev polynomials with the model complexities of O(5) and O(20), and the second row shows the correspondingvelocity-forceplotsof r ˙ x . ForthesamemodelcomplexityofO(5),theidentified r x andr ˙ x withthepowerseriesandChebyshevpolynomialsaredifferentbecauseofemploying different basis functions (Figures 5.4 (a) and (b), and Figures 5.4 (d) and (e)). The identifiedr x andr ˙ x forboth polynomialstrace the slopes of the displacement-force and velocity-force plots, respectively. The identified r x with the power series polynomials, however, is approximately 50% less than the measured force at the peak displacements of approximately±1.3 cm, while theidentifiedr x withtheChebyshevpolynomialsisalmostthesameasthemeasuredforceatthe peakdisplacements. InthecomparisonofO(5)andO(20)usingtheChebyshevpolynomialsas 105 −1.5 −0.75 0 0.75 1.5 −2000 −1000 0 1000 2000 DISPLACEMENT (cm) FORCE (N) −1.5 −0.75 0 0.75 1.5 −2000 −1000 0 1000 2000 DISPLACEMENT (cm) FORCE (N) −1.5 −0.75 0 0.75 1.5 −2000 −1000 0 1000 2000 DISPLACEMENT (cm) FORCE (N) (a)powerseries,O(5) (b)Chebyshev,O(5) (c)Chebyshev,O(20) −22 −11 0 11 22 −2000 −1000 0 1000 2000 VELOCITY (cm/sec) FORCE (N) −22 −11 0 11 22 −2000 −1000 0 1000 2000 VELOCITY (cm/sec) FORCE (N) −22 −11 0 11 22 −2000 −1000 0 1000 2000 VELOCITY (cm/sec) FORCE (N) (d)powerseries,O(5) (e)Chebyshev,O(5) (f)Chebyshev,O(20) Figure 5.4: The identified restoring forces that are dependent on the displacement only (r x ) and velocity only (r ˙ x ) using the non-orthogonal power series ( ¯ a ij ) and orthogonal Chebyshev ( ¯ C ij ) polynomials for different identification model orders (O(N)). The solid lines are of the identified force, and the dashed lines are of the measured force. In the top row, (a) ther x with the power series polynomials for O(5), (b) the r x with the Chebyshev polynomials for O(5), and(c)ther x withtheChebyshevpolynomialsforO(20). Inthesecondrow,(d)ther ˙ x withthe powerseriespolynomialsforO(5),(e)ther ˙ x withtheChebyshevpolynomialsforO(5),and(f) ther ˙ x withtheChebyshevpolynomialsforO(20). showninFigures5.4(b)and(c),andFigures5.4(e)and(f),theslopesofther x andr ˙ x become moreaccuratewiththehigherO(20),buttheimprovementisnotsignificant. Second, the effects of the system changes on the identified coefficients were investigated. In general, the stiffness and damping characteristics of dynamic systems can be determined with the slopes of the phase plots. That is, if the system stiffness is large, the identified ¯ a i0 and ¯ C i0 become large, and vice versa. Similarly, if the system damping is large, the ¯ a 0j and ¯ C 0j terms become large, and vice versa. Figure 5.5 illustrates the changes of the identifiedr x , 106 r ˙ x , and r x˙ x for different MR damper input currents. With the O(20), the first row shows the phase plots of ther x , r ˙ x , andr x˙ x forI = 1.0 A (Figures 5.5 (a) to (c), respectively), and the second row shows the phase plots for I = 0.4 A (Figures 5.5 (d) to (f), respectively). In the identification, the Chebyshev polynomials were used. The identifiedr x ,r ˙ x , andr x˙ x are shown assolidlinesandthemeasuredforceasdashedlinesinthefigure. WhentheI changes,boththe stiffness and damping characteristics of the MR damper change. The exact relationships of the stiffness and damping characteristics on the currentI are very complicated nonlinear functions influenced also by the MR fluid properties, electro-magnet design, and damper cylinder and orificedesign. VariousmechanicalmodelsoftheMRandcontrollabledamperscanbefoundin the works by Ehrgott and Masri (1992). Although no damper models are assumed in the RFM identification, the data-driven technique automatically adjusts its coefficients to obtain the best fit for given data sets. Consequently, in the figure, the slopes ofr x andr ˙ x for the displacement and velocity decrease as theI decreases (Figures 5.5 (a) and (d), and Figures 5.5 (b) and (e)), andtheareaofr x˙ x alsodecreasesasI decreases(Figures5.5(c)and(f)). Figure5.6alsoshows the changes of the Chebyshev coefficients for the different input currents. The changes of the MRdampercharacteristicsarereflectedintheidentifiedChebyshevcoefficientsasshowninthe figure. Hence, the above identification results indicate that the identified RFM coefficients could be used as excellent indicators for change detection of a complicated nonlinear system. Using the information of system characteristics contained in the identified coefficients, some physical interpretation of the detected change would be possible. A question is left: among three kinds oftheRFMcoefficients, ¯ C ij , ¯ a ij ,anda ij ,whichkindismoreusefultodetectthechangesinthe 107 −1.5 −0.75 0 0.75 1.5 −2000 −1000 0 1000 2000 DISPLACEMENT (cm) FORCE (N) −22 −11 0 11 22 −2000 −1000 0 1000 2000 VELOCITY (cm/sec) FORCE (N) −22 −11 0 11 22 −2000 −1000 0 1000 2000 VELOCITY (cm/sec) FORCE (N) (a)Disp. only(I = 1.0A) (b)Vel. only(I = 1.0A) (c)Coupled(I = 1.0A) −1.5 −0.75 0 0.75 1.5 −2000 −1000 0 1000 2000 DISPLACEMENT (cm) FORCE (N) −22 −11 0 11 22 −2000 −1000 0 1000 2000 VELOCITY (cm/sec) FORCE (N) −22 −11 0 11 22 −2000 −1000 0 1000 2000 VELOCITY (cm/sec) FORCE (N) (d)Disp. only(I = 0.4A) (e)Vel. only(I = 0.4A) (f)Coupled(I = 0.4A) Figure5.5: Changesoftheidentifiedrestoringforcesthataredependentonthedisplacementonly (r x ),velocityonly(r ˙ x ),andcoupledwithbothdisplacementandvelocity(r x˙ x )fordifferentMR damper input current (I) of 1.0 A and 0.4 A. In the identification, the Chebyshev polynomials were used as the basis functions, and the model complexity (O(N)) was fixed at 20 for both inputcurrentcases. Thesolidlinesareoftheidentifiedrestoringforces,andthedashedlinesare of the measured forces. The top row shows the phase plots of the displacement and force, and thesecondrowshowsthephaseplotsofthevelocityandforce. 0 2 4 6 8 10 12 14 16 18 20 0 2 4 6 8 10 12 14 16 18 20 −1000 0 1000 2000 Order ˙ x ( j ) Order x ( i ) ¯ C i j Order ˙ x ( j ) Order x ( i ) ¯ C i j −1000 0 1000 2000 0 2 4 6 8 10 12 14 16 18 20 0 2 4 6 8 10 12 14 16 18 20 Figure 5.6: Changes of the identified normalized Chebyshev coefficients for different MR damperinputcurrents(I)of1.0Aand0.4A.TheidentificationwasperformedusingtheRestor- ingForceMethodwiththemodelcomplexityofO(20)forbothinputcurrents. 108 uncertainnonlinearsystem? Inthenextsection,thestochasticpropertiesoftheRFMcoefficients arediscussed. 5.3.4 StochasticPropertiesoftheIdentifiedRFMCoefficients AsshowninEquations5.3and5.5,threekindsofequivalentcoefficientsareavailableusingthe RFM: the normalized Chebyshev coefficients ( ¯ C ij ), normalized power series coefficients (¯ a ij ), andde-normalizedpowerseriescoefficients( a ij ). Thebasisfunctionsforthe ¯ C ij areorthogonal, while the basis functions for the ¯ a ij anda ij are non-orthogonal. The orthogonality of the basis functions significantly influences the stochastic properties of the identified coefficients, and as wellastheperformancesofthesystemchangedetectioncapability. Ananalyticaldescriptionof thestochasticeffectsoftheorthogonalityontheidentifiedcoefficientsisprovidedbelow. Biasnessoftheidentifiedcoefficients Three kinds of orthogonal basis functions are generally used in system identification: (1) poly- nomialorthogonalfunctions,(2)piecewiseconstantorthogonalfunctions,and(3)Fourier(sine- cosine) functions. An excellent overview of using orthogonal basis functions for system identi- fication and control application can be found in Datta and Mohan (1995). In system identifica- tion,thepolynomialorthogonalfunctionsareadvantageoustoidentifycontinuousnonlinearities, whilethepiecewiseconstantorthogonalfunctionsareadvantageousfordiscontinuousnonlinear- itiesbyusingasmallernumberoftermsintheidentificationforeachfunctionkind. Ingeneral,theidentifiedrestoringforcesforthereduced-orderandhigher-ordermodelscan beexpressedas ˆ r h = ˆ r l + ˆ r e = ˆ φ l ψ l + ˆ φ e ψ e = ˆ φ h ψ h (5.13) 109 where ˆ r l and ˆ r h are the restoring force components identified with a reduced-order model and higher-order model, respectively, ˆ r e is the residual between ˆ r h and ˆ r l , ˆ φ h and ˆ φ l are the iden- tified model parameters for the higher-order and reduced-order models, respectively, ˆ φ e is the identifiedmodelparametersfortheresidual,ψ h andψ l arethebasisfunctionsforthehigherand reduced-ordermodels,and ψ e isthebasisfunctionfortherestoringforceresidual. Theidentified modelparameterscanbeestimatedas E[ ˆ φ h ] = < ˆ r h ,ψ l > <ψ l ,ψ l > = < ˆ φ l ψ l + ˆ φ e ψ e ,ψ l > <ψ l ,ψ l > = ˆ φ l <ψ l ,ψ l > <ψ l ,ψ l > + ˆ φ e <ψ e ,ψ l > <ψ l ,ψ l > = ˆ φ l + ˆ φ e Ψ e (5.14) where < • > is the inner product of two functions. Therefore, the biasness of the reduced- order model parameter ( ˆ φ h ) depends on the significance of the term ˆ φ e Ψ e . For the RFM, the orthogonality of the Chebyshev polynomial basis functions is guaranteed with the normalized displacement (¯ x) and velocity ( ¯ ˙ x) within the range of [-1, 1]. Consequently, the identified ¯ C ij becomeunbiasedbecauseψ l andψ e areorthogonal,and ˆ φ e Ψ e = 0. Consequently, E[ ˆ φ h ] = ˆ φ l (unbiased), whenψ l andψ e areorthogonal. (5.15) Figure 5.7 shows a comparison of term-wise identification results, with different modeling orders, for the normalized Chebyshev polynomial basis functions. In the figure, the first row showstheterm-wiseidentificationresultswiththe O(5)forthelineardamping(a),cubicdamp- ing (b) and linear stiffness terms (c), and the second row shows the same term-wise identified restoring forces with theO(20) since the term-wise identification results with O(5) andO(20) 110 Order5 −22 −11 0 11 22 −2000 −1000 0 1000 2000 VELOCITY (cm/sec) FORCE (N) −22 −11 0 11 22 −2000 −1000 0 1000 2000 VELOCITY (cm/sec) FORCE (N) −22 −11 0 11 22 −2000 −1000 0 1000 2000 VELOCITY (cm/sec) FORCE (N) (a)1 st dampingterm(ˆ r 01 ) (b)3 rd dampingterm(ˆ r 03 ) (c)1 st stiffnessterm(ˆ r 10 ) Order20 −22 −11 0 11 22 −2000 −1000 0 1000 2000 VELOCITY (cm/sec) FORCE (N) −22 −11 0 11 22 −2000 −1000 0 1000 2000 VELOCITY (cm/sec) FORCE (N) −22 −11 0 11 22 −2000 −1000 0 1000 2000 VELOCITY (cm/sec) FORCE (N) (d)1 st dampingterm(ˆ r 01 ) (e)3 rd dampingterm(ˆ r 03 ) (f)1 st stiffnessterm(ˆ r 10 ) Figure 5.7: Term-wise identification results with model orders of 5 and 20 with the normalized Chebyshevpolynomialbasisfunctions. are identical. The comparison shows clearly that the identified Chebyshev coefficients are not biasedwithrespecttothemodelcomplexity. Theunbiasnessisnotgenerallytruefornon-orthogonalbasisfunctions. Table5.3showsthe stochasticeffectsofthemodel-orderreductionontheidentifiedcoefficientswiththeChebyshev and power series polynomials. For the Chebyshev polynomials, both the means and standard deviations of the identified coefficients for different model orders are approximately the same, whichindicatesthestatisticalunbiasnessoftheidentifiedcoefficients. Forthepowerseriespoly- nomials, however, significant biasness is observed with the mean of the identified coefficients: themeanratiowithO(5)andO(20)variesfrom20.4%to104.0%. Thestandarddeviationofthe 111 identifiedcoefficientswerealsosignificantlyvarieswithdifferentmodelcomplexities: theratio oftheO(5)andO(20)varieswithintherangeof1.19%to16.1%. Theresultsdemonstratethat for different levels of modeling complexity, the identified coefficients with the orthogonal basis functions are statistically unbiased, while the identified coefficients with the non-orthogonal basisfunctionsaresignificantlystatisticallybiased. 112 Table5.3: Stochasticeffectsofmodel-orderreductiononthecoefficientidentificationwithorthogonalandnon-orthogonalbasisfunctions. The(%)columnshowsthepercentagefractionofO(5)/O(20). Chebyshevpolynomials Powerseriespolynomials (orthogonal) (non-orthogonal) term mean stdv mean stdv (i,j) O(5) O(20) (%) O(5) O(20) (%) O(5) O(20) (%) O(5) O(20) (%) (0,1) 1618.9 1637.2 98.9 53.5 54.2 98.6 3612.9 4615.9 78.3 167.7 1036.3 16.1 (1,0) 624.1 668.2 93.4 43.2 48.0 90.0 1210.8 1163.7 104.0 115.6 968.1 11.9 (0,3) 14.6 15.6 93.9 17.3 17.2 100.7 -5026.6 -24603.5 20.4 467.5 39224.8 1.19 113 DistributionsofIdentifiedCoefficients The unbiasness of the identified coefficients using the orthogonal basis functions is critical for implementing the change detection in uncertain nonlinear systems: the the probability of the identified coefficients should be a function of the system uncertainty, not a function of the model complexity. When the unbiasness is guaranteed, the identified coefficients of a reduced- order model can be safely used for change detection. Consequently, change detection could be observedevenwithafewdominanttermsoftheidentifiedcoefficients. Forexample,theO(20) model has a total of 441 coefficients. Using the orthogonality property, the testing procedure forchangedetectioncouldbedramaticallysimplifiedbyusingasmallernumberofcoefficients. Figure 5.8 shows the bivariate Gaussian probability density functions (pdfs) of the two domi- nantidentifiedChebyshevcoefficientsinthedisplacement( ¯ C 10 )andvelocity( ¯ C 01 )fordifferent MR damper input currents. The figure illustrates that, even with two dominant coefficients, the bivariatepdfscanstillaccuratelyrepresentthephysicalchangesintheMRdamper. Figure 5.9 shows the pdfs of the first-order damping coefficient ( ¯ C 01 ) for different MR damper input currents. In the figure, the mean of pdfs decreases as the input current decreases. Since the damping force of the MR damper is proportional to the input current (Dyke et al., 1996),itisobservedthatthemeanof ¯ C 01 byitselfproperlyrepresentstheactualchangesinthe damper properties. Figure 5.10 illustrates the current-dependence of the means of the identi- fied Chebyshev coefficients with one standard deviation error bars for different means (μ I ) and standard deviations (σ I ) of input currents. Forσ I = 0.1 A, as the input current increases, both thefirstorderdampingandstiffnesscoefficients( ¯ C 01 and ¯ C 10 ,respectively)increase, whilethe third order damping coefficient ( ¯ C 03 ) decreases (Figure 5.10 (a)). For the sameμ I but different 114 200 600 1000 1400 1800 0 200 400 600 800 0 0.5 1 x 10 −4 ¯ C 0 1 ¯ C 1 0 PROBABILITY 1.0 A 0.4 A 0.8 A 0.6 A Figure5.8: BivariateGaussiandistributionsoftheidentifiedChebyshevcoefficientsoftwodom- inanttermsinthevelocity( ¯ C 01 )anddisplacement( ¯ C 10 ),fordifferentMRdamperinputcurrents (Testno. A 1 ,B 1 ,C 1 ,D 1 ). σ I of0.15A,themeansofthecoefficientsarealmostidenticaltothemeansforσ I = 0.1A,but thestandarddeviationoftheidentifiedcoefficientsincreases42.2%onaverage(Figure5.10(b)). This result indicates that the change ofσ I is also properly reflected in the standard deviation of theidentifiedcoefficients. Basedontheaboveresults,theexperimentalstudyhasdemonstratedthefollowingimportant factstodetectchangesinuncertainnonlinearsystems: 1. BecausetheidentificationprocedureoftheRFMisdata-driven,no a prioriknowledgeof themonitorednonlinearsystemsisrequired. 115 100 400 700 1000 1300 1600 1900 2200 0 2 4 6 8 x 10 −3 ¯ C 0 1 PROBABILITY 0.4 A 0.6 A 0.8 A 1.0 A Figure5.9: ThedistributionsoftheidentifiedChebyshevcoefficientsforthefirstorderdamping ( ¯ C 01 ) for different MR damper input currents (Test no. A 1 , B 1 , C 1 and D 1 ). In the figure, the smooth factor (bin width) of the histograms were determined with the normal reference rule (or Scott’s rule) (Martinez and Martinez, 2002). The pdf’s were estimated with the Gaussian distributionassumption. 2. The identified coefficients with the orthogonal basis functions are statistically unbiased withrespecttothemodelcomplexity. 3. Duetotheirstatisticalunbiasness,theidentifiedcoefficients(usingorthogonalbasisfunc- tions)withareduced-ordermodelcanbesafelyusedtodetectchangesinthesystems. 4. Using the distributions of the identified coefficients, not only detecting changes in the monitored systems, but also quantifying the detection uncertainty is possible: the mean changesofthedistributionsmeasurethegenuinesystemchanges,andthestandarddevia- tionsofthedistributionsmeasurethedetectionuncertainty. 5.4 StochasticChangeDetectionofMRDamper In the previous sections, it was analytically and experimentally demonstrated that the identified coefficients with the orthogonal basis functions are statistically unbiased with respect to the modelcomplexity. Inthissection,someexamplesareprovidedtodemonstratehowtheunbiased 116 0.4 0.6 0.8 1 −500 0 500 1000 1500 2000 INPUT CURRENT (A) ¯ C 01 , ¯ C 10 , ¯ C 03 0.4 0.6 0.8 1 −500 0 500 1000 1500 2000 INPUT CURRENT (A) ¯ C 01 , ¯ C 10 , ¯ C 03 (a)σ I = 0.1A (b)σ I = 0.15A Figure 5.10: The means of the identified normalized Chebyshev coefficients with 1σ error bars for different MR damper input currents (I). (a) The input current standard deviations of 0.1 A (Test no. A 1 , B 1 , C 1 and D 1 ). (b) The input current standard deviations of 0.15 A (Test no. A 2 , B 2 , C 2 and D 2 ). In the figures, the solid lines are for ¯ C 01 , dashed lines for ¯ C 10 , and dash-dot linesfor ¯ C 03 . coefficients can be used for the change detection in uncertain nonlinear systems. Two pattern recognition algorithms are used, as supervised and and unsupervised classification methods, in this demonstration. Brief description of both classification approaches will be followed in the nextsection. 5.4.1 OverviewofStatisticalClassificationwithPatternRecognitionMethods Classificationmethods Statisticalclassificationmethodsarepatternrecognitionproceduresinwhichthepatterndata(or observations)areplacedintotwoormorelabeledgroups(orclasses)basedononeormorechar- acteristics(or features). A classifier isanonlinearfunctionmappinganobservationinafeature space to a class label. In general, there exist two types of classifiers: supervised classifiers and unsupervised classifiers. The supervised and unsupervised classification are briefly described 117 below. More detailed and formal descriptions of these methods can be found in Duda and Hart (1973). The supervised classifiers require the training pattern data consisting of pairs of feature inputs and desired class labels. Using the training data, the optimal relationships between the feature inputs and class labels can be obtained, minimizing the prediction errors of the desired class labels for the given feature input data. In this study, Support Vector Machines are used as an example of supervised classifications for detecting changes in the MR damper. Detailed descriptionofSupportVectorMachineswillbeprovidedlaterinSection5.4.2. Theunsupervisedclassifiersaredistinguishedfromthesupervisedclassifiersbythefactthat theunsupervisedclassifiersdonotrequirethedesiredclasslabels. Consideringtheinputfeatures asrandomvariables,thismethodfindstheprobabilisticrelationshipsbetweentheinputfeatures, commonly employingBayesian inference to obtain the conditional probabilities of the features. Without this a priori information for the classification, the unsupervised classification is often more challenging than the supervised classification. The k-means clustering is one of most widely used unsupervised classification method due to its simple procedure and relatively good classificationresults(Kanungoetal.,2002). Hence,inthisstudy,thek-meansclusteringisused asanunsupervisedclassifierforthechangedetectionintheMRdamper. Detaileddescriptionof thek-meansclusteringisprovidedinSection5.4.3. Thepurposeoftheclassificationdemonstrationistoillustratetheadvantageofusingthesta- tisticallyunbiasedcoefficients(i.e., ¯ C ij )forthechangedetectioninuncertainnonlinearsystems. Consequently,thisstudyfocusesondemonstratingtheimportanceofselectingeffectivefeatures 118 for the system change detection rather than evaluating the performance of different statistical classifiers. Modelselectionanderrorgeneralizationtechniques Accuracy estimation and error generalization are critical steps to select good classifier models andevaluatetheperformanceoftheselectedmodelsforfuturedatasets(Kohavi,1995). Inboth cases, low bias and low variance of the classification results are desirable. If data sets are suf- ficient, the given data sets are generally partitioned into three groups to perform the following steps: (1) model training, (2) model validation, and (3) model assessment. The model training andmodelvalidationareperformedtochoosethemodelwiththebestperformanceonnewdata. For these, the first-group data are used to train different classification models, and the second- group data, which were not used in the model training, are used to select the model with the bestperformance. Oncethebestperformancemodelisselected,modelassessment isperformed withthethird-groupofdatasetstoestimatethegeneralizationerroronfuturedata. However,the given dataare oftennot sufficient to partition into three groups and to perform all the necessary evaluations. In this case, statistical data partitioning and resampling techniques provide effec- tive approaches to maximize the use of a limited amount of data (Efron and Tibshirani, 1993; MartinezandMartinez,2002). Thecross-validationmethod (CV)isastatisticaldata-partitioningtechniqueformodelselec- tion with an insufficient amount of data. Common types of the CV include: thek-fold cross- validation,andleave-one-outcross-validation. The k-foldcross-validation partitionstheoriginal data intok subsamples. Ofk partitioned subsamples, one subsample is retained for model val- idation, and the remaining (k− 1) subsamples are used for model training. Then, the same 119 process is repeated with the next subsample to “cross-validate” trained classifier models. Con- sequently,k accuracy estimates are obtained from the folds, and the averaged accuracy is used as the final estimation. The leave-one-out cross-validation has similar procedures to thek-fold cross-validation, but this method retains a single observation (or data point) for model valida- tion, and the remaining observations are used for model training. Consequently, if the original datahavemobservations,theleave-one-outand k-foldcross-validationsbecomeidenticalwhen k =m. 5.4.2 SupervisedChangeDetectionUsingSupportVectorClassification OverviewofSupportVectorMachineclassification TheSupportVector(SV)algorithms are statistical learning techniquesfor variousclassification andregressionproblems(Boseretal.,1992;Burges,1998;SmolaandSch¨ olkopf,1998;Vapnik, 1995, 1998). For the classification problems, the Support Vector Classifiers (SVC) have been successfully used for various system-identification and damage-detection-related applications (Gao et al., 2002; Mita and Hagiwara, 2003; Oh and Beck, 2006; Park et al., 2005; Worden and Lane, 2001; Yun et al., 2006; Zhang et al., 2006). Here, a brief description of the SVC is provided. More complete information on this method can be found in the work by Sch¨ olkopf andSmola(2002). Suppose thatm sets are available of training pattern vectorsx 1 ,x 2 ,...,x m in a dot product spaceH as illustrated in Figure 5.11. If the given training data set is separable, the goal is to find the hyperplane (H) with the maximal geometrical marginρ. The purpose of maximizingρ isobvious: theclassifierbecomesrobust withtheρwhenthe trainingvectorsxarenoisy. Ifthe noise of x is bounded inr > 0, the separation margin should beρ > r, so that the separating 120 + + + + + + H H H 1 H 2 w x k {C 1 } {C 2 } } 1 , | { − < + > < b x w x } 1 , | { + > + > < b x w x k ρ ρ ρ + + + + + + H H w {C 2 } {C 1 } r ρ (a) (b) Figure5.11: SupportVectorClassification. hyperplane (H) correctly classifies the noisy data (Figure 5.11 (b)). Therefore, for a givenr of the training pattern vectors, ρ is optimized with the maximal value ρ ∗ (Sch¨ olkopf and Smola, 2002). AnyhyperplaneHinHcanbeexpressedas {x∈H|< w,x> +b = 0}, (5.16) where w is a vector orthogonal to the hyperplane (w ∈ H), b is a threshold (and b ∈ <), and <• > is a dot product. Because w andb are arbitrary, we can imagine two linear hyperplanes as H 1 :={x|< w,x> +b = +1}, H 2 :={x|< w,x> +b =−1}. (5.17) Thentheseparationconditionsforanarbitraryvectorx k intotwoclassesbecome < w,x> +b> +1 for x k ∈{C 1 }, < w,x> +b<−1 for x k ∈{C 2 }. (5.18) 121 Ormoreconcisely, y k < w,x> +b> +1, wherey k = sgn(< w,x> +b>). (5.19) Sincethedistancebetweenx k andH(ρ k )isgivenby y k (< w,x> +b) ||w|| ≥ρ, (5.20) the separating hyperplane H can be obtained by solving the following constrained quadratic optimizationproblemwith||w||ρ = 1: minimize Q(w,b) = 1 2 ||w|| 2 , (5.21) subjectto y k (< w,x k > +b)≥ 1. (5.22) This is called the primal optimization problem. Introducing the Lagrangian, the optimization becomes minimize L(w,b,λ) = 1 2 ||w|| 2 − m X i=1 λ i (y i (< x i ,w> +b)−1) (5.23) whereλ i areLagrangemultipliers. Sincethequadraticequationisconvex,thisleadsto ∂ ∂b L(w,b,λ) = 0 ⇒ m X i=1 λ i y i = 0, (5.24) ∂ ∂w L(w,b,λ) = 0 ⇒ w = m X i=1 λ i y i x i . 122 attheoptimum. TheKarush-Kuhn-Tucker(KKT)theoremthatassertstheexistenceofnon-zero Lagrangemultipliers(i.e.,λ i > 0)attheoptimum(Bertsekas,1999)leadsto λ i (y i (< x i ,w> +b)−1) = 0 (5.25) Thetrainingvectorsx i withλ i > 0arecalled SupportVectors(SVs)locatedonthegeometrical margins(i.e.,H 1 andH 2 ),andtherestofthetrainingvectorsx j areirrelevanttotheoptimization procedures becauseλ j = 0. Substituting Equations 5.24 and 5.25 into Equation 5.23, the dual formationoftheprimaloptimizationproblemcanbeobtainedas maximize P(λ) = m X i=1 λ i − 1 2 m X i,j=1 λ i λ j y i y j < x i ,x j >, (5.26) subjectto λ i ≥ 0 ∀i = 1,2,...,m, and (5.27) m X i=1 λ i y i = 0. Oncetheoptimalλ i ’sarefound,theclassificationfunctioncanbesolvedas f(x) = sgn m X i=1 λ i y i < x,x i > +b ! (5.28) When the training pattern vectors are not linearly separable, one can make H 1 and H 2 soft marginhyperplaneswithso-calledslackvariables( ξ i ): y i (<x i ,w> +b)≥ 1−ξ i , where ξ i ≥ 0 (i = 1,2,...,m). (5.29) 123 Two approaches are commonly used for soft margin hyperplanes: theC-Support Vector Classi- fication,andν-SupportVectorClassification. CortesandVapnik(1995)proposedaSVclassifier byintroducingslackvariablesandapenaltyparameterC totheprimaloptimizationfunctionin Equation5.22as: maximize Q(w,ξ) = 1 2 ||w|| 2 + C m m X i=1 ξ i , (5.30) subjectto ξ i ≥ 0, and (5.31) y i (<x i ,w> +b)≥ 1−ξ i , ∀i = 1,2,...,m. Thismodifiedprimalproblemis calledtheC-Support VectorClassification ( C-SVC). Thedual formoftheC-SVCis maximize P(λ) = m X i=1 λ i − 1 2 m X i,j=1 λ i λ j y i y j < x i ,x j >, (5.32) subjectto 0≤λ i ≤ C m ∀ i = 1,2,...,m, and (5.33) m X i=1 λ i y i = 0. The penalty parameterC determines the trade-off between maximizing the geometrical margin and minimizing the training error. One practical drawback of the C-SVC is that there is no guidelinestochooseareasonablevalueofC,becauseC isratherunintuitive. Inordertoaddress 124 this problem, Sch¨ olkopf et al. (2000) proposed the ν-Support Vector Classification ( ν-SVC) replacingC withanotherparameterν as: maximize Q(w,ξ) = 1 2 ||w|| 2 −νρ+ 1 m m X i=1 ξ i , (5.34) subjectedto ξ i ≥ 0, ρ≥ 0, and (5.35) y i (<x i ,w> +b)≥ρ−ξ i , ∀i = 1,2,...,m. Thedualformoftheν-SVCis maximize P(λ) =− 1 2 m X i,j=1 λ i λ j y i y j < x i ,x j >, (5.36) subjectto 0≤λ i ≤ 1 m ∀ i = 1,2,...,m, (5.37) m X i=1 λ i y i = 0, and m X i=1 λ i ≥ν. Sofar,thetrainingpatternvectorshavebeenassumedtobelinearlyseparable,andtheSVC algorithm can be extended to the nonlinear classification using so-called kernel-trick technique. Using a kernel function Φ, the training vectorsx 1 ,x 2 ,...,x m ∈H are nonlinearly transformed into a higher feature space, and the linear SVC is performed in the higher order space. Cover (1965)foundtherelationshipbetweenthenumberofpossiblelinearseparationsandmnumbers oftrainingvectorsingeneralpositioninanN-dimensionalspace. Thenumberofpossiblelinear separationis 2 m when m≤ (N +1) (5.38) 125 and 2 N X i=0     m−1 i     when m> (N +1) (5.39) Consequently, since m > N + 1 in this study, the more N increases, the larger the number of possible linear separation that exists. However, in Cover’s theory, the training vectors are required to be in a general position. Sch¨ olkopf and Smola (2002) pointed out that the Cover’s theory “does not strictly make a statement about the separability of a given data set in a given feature space. E.g., the feature map might be such that all points lie on a rather restrictive lower-dimensional manifold, which could prevent us from finding points in general position. ” This issue becomes very important in the classification of the reduced-order model, and more detailed discussion will be provided later in this section with actual experimental results from the MR damper. Solving the dual form of the optimization using the kernel-trick approach, the classificationdecisionfunctioninEquation5.28canbeconvertedto f(x) = sgn m X i=1 y i λ i < Φ(x),Φ(x i )> ! , (5.40) and then solving the dual form of the quadratic optimization function with kernel for separable trainingvector(referEquation5.28)as maximize P(λ) = m X i=1 λ i − 1 2 m X i,j=1 λ i λ j y i y j < Φ(x i ),Φ(x j )>, (5.41) subjectto λ i ≥ 0 ∀i = 1,2,...,m, and (5.42) m X i=1 λ i y i = 0. 126 FourkernelfunctionsarecommonlyusedintheSVC: Linearclassifiers : < x,x i > (5.43) Polynomialclassifiersoforderd : < x,x i > d (5.44) Radialbasisfunctionclassifiers : exp(−||x−x i || 2 /γ) (5.45) Sigmoidneuralnetworksclassifiers : tanh(α< x,x i > +β) (5.46) ClassifiermodelselectionfortheMRdamperchangedetection A supervised change detection was performed using the C-SV radial basis function classifier (Equations 5.32, 5.33 and 5.45) to monitor the changes of the MR damper using the orthogonal Chebyshev coefficients ( ¯ C ij ) and non-orthogonal power series coefficients ( ¯ a ij ). In order to find the optimal parametersC andγ, a grid search method (Fan et al., 2005; Hsu et al., 2007) was used within the ranges of 2 −9 ≤ C ≤ 2 15 and 2 −15 ≤ γ ≤ 2 5 . The SV classifier was trainedusingthedatasetsof ¯ C ij and¯ a ij . Thecross-validationof5-foldswasusedforthemodel selection of the SV classifier with the training data sets consisting of (441 features)× (2000 observations). Theclassifiermodelselectionwasperformedusingthe3-foldcrossvalidationofthetraining data sets for differentC andγ. Figure 5.12 (a) shows the classification precisions for the ¯ C ij , and Figure 5.12 (b) for the ¯ a ij . From the grid search with 3-fold cross validation, the optimal values of the C and γ were found: C opt = 8192 and γ opt = 3.91× 10 −3 for the orthogonal Chebyshev coefficients, andC opt = 2.0 andγ opt = 1.95×10 −3 for the non-orthogonal power- seriescoefficients. 127 0.2 0.2 0.2 0.22 0.22 0.22 0.22 0.22 0.22 0.24 0.24 0.24 0.24 0.24 0.24 0.26 0.26 0.26 0.28 0.28 0.28 0.3 0.3 0.3 0.32 0.32 0.32 0.34 0.34 0.34 0.36 0.36 0.36 0.38 0.38 0.38 0.4 0.4 0.4 0.42 0.42 0.42 0.44 0.44 0.44 0.46 0.46 0.46 0.48 0.48 0.48 0.5 0.5 0.5 0.52 0.52 0.52 0.54 0.54 0.54 0.56 0.56 0.56 0.58 0.58 0.58 0.6 0.6 0.6 0.6 0.62 0.62 0.62 0.62 0.64 0.64 0.64 0.64 0.66 0.66 0.66 0.66 0.68 0.68 0.68 0.68 0.68 0.7 0.7 0.7 0.7 0.7 0.7 0.72 0.72 0.72 0.72 0.72 0.72 0.74 0.74 0.74 0.74 0.74 0.74 0.76 0.76 0.76 0.76 0.76 0.76 0.76 0.78 0.78 0.78 0.78 0.78 0.78 0.78 0.8 0.8 0.8 0.8 0.8 0.8 0.8 0.82 0.82 0.82 0.82 0.82 l o g 2 C l o g 2 γ −5 0 5 10 15 −15 −10 −5 0 5 0.26 0.26 0.26 0.26 0.28 0.28 0.28 0.28 0.3 0.3 0.3 0.32 0.32 0.32 0.34 0.34 0.34 0.36 0.36 0.36 0.38 0.38 0.38 0.4 0.4 0.4 0.42 0.42 0.42 0.44 0.44 0.44 0.46 0.46 0.46 0.48 0.48 0.48 0.5 0.5 0.5 0.52 0.52 0.52 0.54 0.54 0.54 0.54 0.56 0.56 0.56 0.56 0.58 0.58 0.58 0.58 0.6 0.6 0.6 0.6 0.6 0.62 0.62 0.62 0.62 0.62 0.64 0.64 0.64 0.64 0.64 0.66 0.66 0.66 0.66 0.66 0.66 0.68 0.68 0.68 0.68 0.68 0.68 0.68 0.7 0.7 0.7 0.7 0.7 0.7 0.7 0.72 0.72 0.72 0.72 0.72 0.72 0.72 0.74 0.74 0.74 0.74 0.74 0.74 0.74 0.76 0.76 0.76 0.76 0.76 0.76 0.78 0.78 0.78 0.78 0.78 0.78 l o g 2 C l o g 2 γ −5 0 5 10 15 −15 −10 −5 0 5 (a)Chebyshevcoefficients (b)Powerseriescoefficients Figure 5.12: Theclassification precision ofC-Support Vector Classification for different C and γ values. ThecontoursshowtheclassificationprecisionfordifferentparametervaluesofC and γ. ClassifierprecisionassessmentsfortheMRdamperchangedetection Once the SV classifier model was selected with the optimal C and γ for the ¯ C ij and ¯ a ij , the classification precision was assessed using 50% of the data for training and 50% of the data for the precision assessment. In order to understand the model-order reduction effects, the number of features (m) in the classifications increased from one to 441 for theO(20) models, and the correspondingclassificationprecisionsweremeasured. For1≤m≤ 5,thefivemostsignificant terms were determined using Equation 5.8, and the coefficients were used in the classification cumulativelyintheorderofthe ¯ C 01 , ¯ C 01 , ¯ C 10 , ¯ C 03 , ¯ C 30 and ¯ C 34 ,andthesametermsandorder ofthe ¯ a ij . Effectsofmodelorderreductionontheclassificationresults Figure 5.13 shows the classification precisions with the unbiased ¯ C ij and the biased ¯ a ij for differentnumberoffeatures. Inthefigure,theclassificationprecisionsforthedatasetsofσ I = 0.1 A ( ) and σ I = 0.15 A (4) are also compared. Figures 5.13 (a) and (b) illustrate that 128 10 0 10 1 10 2 10 3 0.35 0.45 0.55 0.65 0.75 0.85 NO. OF FEATURES (log) CLASSIFICATION PRECISION (%) 10 0 10 1 10 2 10 3 0.35 0.45 0.55 0.65 0.75 0.85 NO. OF FEATURES (log) CLASSIFICATION PRECISION (%) 10 0 10 1 10 2 10 3 10 0 10 1 10 2 NO. OF FEATURES (log) NORMAL. COMPUTATION TIME (log) (a)Orthogonal( ¯ C ij ) (b)Non-orthogonal( ¯ a ij ) (c)Normalizedcomput. time Figure5.13: TheprecisionsoftheSupportVectorClassification(SVC)forthestatisticallyinde- pendentChebyshevcoefficients( ¯ C ij )andstatisticallycorrelatedpower-seriescoefficients( ¯ a ij ). The SVC was performed for different number of features (i.e., identified coefficients) to study theeffectsofmodel-orderreduction. TwodifferentstandarddeviationsoftheMRdamperinput current (σ I ) of 0.1 A ( ) and 0.15 A (4) are compared for each type of coefficients. The computationtimes,normalizedwithrespecttothesmallesttime,arealsoshowninthefigure. the classification precision increases as m increases for both ¯ C ij and ¯ a ij . The semi-log plots also show that the precision improvement becomes saturated for a largem. In the figures, the precisionswiththe ¯ C ij arelargerthanthosewiththe¯ a ij . Fordifferentsystemuncertaintylevels, the precisions with σ I = 0.1 A are greater than those with σ I = 0.15 A. Figure 5.13 (c) shows the computation times, normalized with respect to the smallest time, for different m. The computation time with 441 features is about 15 times larger than with one feature. The classificationresultsinFigure5.13isalsosummarizedinTable5.4. Theaboveresultsindicatethattheclassificationwiththeunbiased ¯ C ij ismoreefficientthan with the statistically biased ¯ a ij due to many advantageous properties of the orthogonal basis functions discussed in Sections 5.3.3 and 5.3.4. For the change detection in the MR damper, with the unbiased ¯ C ij , using reduced-order models would be more efficient, especially when a short computation time is a critical concern. For example, in order to improve the classification 129 Table 5.4: The precision of the Support Vector Classification (SVC) procedure for the statis- tically independent Chebyshev coefficients ( ¯ C ij ) and the statistically correlated power series coefficients(¯ a ij ). TheSVCwasperformedfordifferentnumberoffeatures(i.e.,identifiedcoef- ficients) to study the effects of model-order reduction. Two different standard deviations of the MR damper input current (σ I ) of 0.1 A ( ) and 0.15 A (4) are compared for each type of coefficients. The computation times, normalized with respect to the smallest time, are also summarizedinthetable. Numberof Numberof 0.1A(%) 0.15A(%) Normalized classes features computationtime ¯ C ij ¯ a ij ¯ C ij ¯ a ij 1 62.4 38.8 55.1 40.7 1.0 2 68.9 44.8 56.7 45.1 1.2 3 69.7 53.0 60.4 51.2 1.2 4 71.6 58.1 61.4 53.3 1.1 4 5 72.1 60.7 62.0 54.6 1.1 36 79.4 64.5 70.0 60.5 1.6 121 81.2 65.1 74.0 62.9 4.3 256 81.1 65.9 74.4 64.3 8.4 441 82.8 68.9 75.6 66.3 14.6 precision from 80% to 85% forσ I = 0.1 A, them should be increased from 36 to 441 (Fig- ure5.13(a)). However,thecorrespondingcomputationtimeincreasesapproximatelyninetimes (Figure 5.13 (c)). In the comparison of data sets withσ I = 0.1 A andσ I = 0.15 A, because the classification precision is inversely proportional to the system uncertainty, the classification withhighersystemuncertaintyshouldinvolvemorefeaturesinordertohavethesameprecision aswithasmallersystemuncertainty. ErroranalysisoftheSVclassication In general, there exist two sources of classification error: Type I and Type II errors (Hogg and Tanis, 1997; Mendenhall and Sincich, 1995). For a given null hypothesis (H 0 ), Type I error is defined as H 0 is rejected when H 0 is true. For the same H 0 , Type II error is defined as H 0 is 130 (R,F) Correct (A,F) “Missed ” (R,T) “False alarm ” (A,T) Correct (Type II error) (Type I error) H 0 : The MR damper does NOT belong to this class. (A,F): We accept H 0 when H 0 is false (Type II error). (R,T): We reject H 0 when H 0 is true (Type I error). Figure5.14: Detectionruleswithtwosourcesoferrors(TypeIandTypeIIerrors). acceptedwhenH 0 isfalse. Inthisstudy,theH 0 anditsalternativehypothesis(H a )foreachclass aredefinedas(Figure5.14) H 0 : TheMRdamperdoesNOTbelongtothisclass. (5.47) H a : TheMRdamperbelongstothisclass. (5.48) Consequently,inthisstudy,TypeIerrorisofa“false-alarmed”error,whileTypeIIerrorisofa “missed-classification” error. The power of a test (or probability of detection) is defined as the probabilityofrejectingH 0 whenH a istrue. Thepoweroftestcanbeexpressedas Poweroftest = 1−p(TypeIIerror), (5.49) where p(TypeIIerror) is the probability of Type II error. Therefore, the power of test is the probabilitythatthetestwilldeclareH a truewheninfactH a istrue. Using those definitions, the Type I and Type II errors of the SVC for the ¯ C ij were assessed. The probabilities of the apparent successful classification (correct), Type I error (false-alarm), 131 Type II error (missed) and the power of test for each class are shown in Figure 5.15. The prob- abilities were measured for different m, so that the effects of model-order reduction on the classificationprecisionsanderrorscanbeunderstood. In Figure 5.15 (a), the probabilities of the apparent successful classification were measured with the number of observations that belong to (R,F) and (A,T) in Figure 5.14 divided by total number of observations for different m. The highest probability of apparent successful classification is observed with Class D 1 (×), and the lowest with Class B 1 (4) (refer Table 5.1 fortheclasslabels). TheprobabilitiesoftheTypeIandTypeIIerrorsweremeasuredwiththenumberofobserva- tionsthatbelongto(R,T)and(A,F),respectively,dividedbythetotalnumberofobservations (Figures5.15(b)and(c)). ForbothTypeIandTypeIIerrors, thehighesterrorprobabilitiesare observed with Class B 1 (4), and the lowest error probabilities with Class D 1 (×). This result indicates that the classifier performance varies with different types of classes. The probabilities of both Type I and Type II errors decrease asm increases. In Figures 5.15 (b) and (c), it is also observed that there exist trade-offs between Type I and Type II errors for all classes. That is, if theTypeIerrordecreases,theTypeIIerrorincreases,and vice versa. Betweenthesetrade-offs, minimizingtheTypeIIerrorwouldbemoreappropriateforthepurposeoftheSHM.Forexam- ple, let us assume that there occurs significant damage in a monitored system. In the damage detection of the system, the chances of “false alarms” of the change detection increase with a largerTypeIerror. Inthiscase, although the performance of the classifier becomes worse, hav- ing “false alarms” is more conservative to prevent the failure of the system. On the other hand, the chances of “missed” damage detection increases with a larger Type II error. In this case, 132 10 0 10 1 10 2 10 3 65 70 75 80 85 90 95 100 NO. OF FEATURES APPARENT SUCCESSFUL CLASS.(%) 10 0 10 1 10 2 10 3 0 5 10 15 20 25 30 NO. OF FEATURES TYPE I ERROR (FALSE−ALARM) (%) (a)Apparentsuccessfulclassifications(correct) (b)TypeIerrors(false-alarmed) 10 0 10 1 10 2 10 3 0 5 10 15 20 25 30 NO. OF FEATURES TYPE II ERROR (MISSED) (%) 10 0 10 1 10 2 10 3 75 80 85 90 95 100 NO. OF FEATURES POWER OF TEST (%) (c)TypeIIerrors(missed) (d)Powersoftest Figure 5.15: The probabilities of apparent successful classification, Type I error, Type II error and the power of test of the Support Vector Classification (SVC) for different numbers of the normalized Chebyshev coefficients (features) in the classification. In the SVC, four classes of dataareclassified: Testno. A 1 ( ),B 1 (4),C 1 ()andD 1 (×). the monitored system could be in a dangerous condition as results of failing to detect serious structuraldamage. Figure5.15(d)showsthepowersoftestfordifferentm. Thepowersoftests increase asm increases, especially whenm > 10. Similar to the probabilities of the apparent successfulclassification,thelargestpoweroftestisobservedwithClassD 1 (×),andthelowest withClassB 1 (4). 133 Classifierdesignfortheoptimalnumberoffeatures Basedontheaboveresults,anoptimalclassifierdesignstrategycanbeproposedbyminimizing the number of features in the identification subjected to chosen thresholds of the “false alarm” (orTypeIerror),“missed”(orTypeIIerror)andcomputationtime. Twosimpledesignexamples areshownbelow: (i)Designspecifications1 • Powersoftest≤90%(orTypeIIerrors≤10%), • TypeIerrors≤10%,and • Thenormalizedcomputationtime≤2.0 (ii)Designspecifications2 • Powersoftest≤90%(orTypeIIerrors≤10%), • TypeIerror≤15%,and • Thenormalizedcomputationtime≤2.0 The figures necessary for these design examples can be found in Figure 5.15 (d) for the powers oftest,inFigure5.15(c)fortheTypeIerror,andFigure5.13(c)forthenormalizedcomputation time. Theoptimalnumberoffeaturesforthedesignexamplesis 36 forthespecifications1,and 5forthespecifications2(withouttheinterpolationsbetweendifferentm). 5.4.3 UnsupervisedChangeDetectionUsingk-MeansClustering Overviewofthek-meansclustering Thek-means clustering is an unsupervised algorithm to cluster the pattern vectors in a feature 134 spaceintok partitions. Withaprioriinformationaboutthenumberofclusters(notdesiredclass labels),thealgorithmdefinesk centroids,andoneforeachcluster. Forthegivenpatternvectors x 1 ,x 2 ,...,x m (m≥k), letc i be the geometrical centroid of thei th cluster. Then, thek-means classifiercanbeexpressedasthefollowingoptimizationfunction: minimize J(c 1 ,c 2 ,...,c k ) = k X i=1 X x j ∈S i |x j −c i | 2 (5.50) whereS i ={x| xassignedinthei th cluster}. Simpleproceduresofthek-meansclusteringwere proposedbyMacQueen(1967): 1. Randomlygeneratek pointsastheinitialcentroidsc i ,wherei = 1,2,...,k. 2. Assignx j tothenearestclustercentroid,wherej = 1,2,...,m. 3. Onceallxareassignedtothecentroids,recalculatethepositionofthecentroids. 4. Repeatabovetwostepsuntilthelocationsofthecentroidsareconverged. In general, however, the solution of Equation 5.50 is not necessarily lead to a global mini- mum(BottouandBengio,1995;Mangasarian,1997;Pollard,1982;SelimandIsmail,1984). ClassificationresultsfortheMRdamper Aunsupervisedchangedetectionwasperformedusingthek-meanclusteringalgorithmwiththe identified RFM coefficients. The distances between the centroid clusters are measured with the squared Euclidean distance. The maximum iteration of each clustering was set to be 5000. In order to avoid the local minimum problem discussed earlier, and the procedures were statisti- cally averaged over 100 sets. The parameters of thek-mean clustering used are summarized in Table5.5. 135 Table5.5: Parametersfork-meansclusteringfortheMRdamperchangedetection. Parameters Values Distancemeasurement SquaredEuclideandistance Maximumiteration 5000 Numberofstatisticalaveragings 600 Table 5.6 summarizes the results of unsupervised k-mean clustering for the MR damper change detection. The table shows the powers of test (Equation 5.49) for different numbers of features(m)andclasses(M). Thepoweroftestisalsoreferredtoastheprobabilityofdetection that declares that the MR damper belongs to a class (H 0 ) when H 0 is actually true. The results showthat, unliketheSVC,thereis no noticeableimprovement of thepowers oftest withthek- meansclusteringbyaddingmorefeaturesintheclassificationforboth ¯ C ij and¯ a ij . However,the poweroftestisslightlylargerwiththe ¯ C ij thanwiththe ¯ a ij forapproximately9%onaverage. 136 Table5.6: Theresultsofk-meansclusteringfortheMRdamperchangedetectionwithdifferentnumbersoffeaturesandclasses. Thetable showspowersoftest(Equation5.49)foreachcase. Orthogonal( ¯ C ij )(%) Non-orthogonalcoefficients( ¯ a ij )(%) Classes No. ofobservations 2features 5features 441features 2features 5features 441features A 1 56.56 94.34 95.94 81.54 81.22 82.87 B 1 1000 66.28 86.98 51.44 78.34 73.11 83.24 A 1 96.95 97.09 70.34 81.15 80.11 86.56 B 1 1500 88.50 88.43 90.85 79.90 73.39 88.50 C 1 85.72 85.03 94.46 82.74 71.31 66.87 A 1 99.18 75.89 75.89 84.04 86.61 82.25 B 1 76.04 88.25 76.04 78.96 79.53 77.83 C 1 2000 90.30 74.45 92.05 83.58 86.04 74.55 D 1 86.15 86.20 74.04 74.19 75.68 77.12 Averages 82.85 86.30 80.12 80.49 78.56 79.98 137 5.5 SummaryandConclusion Aneffectiveandreliablestructuralhealthmonitoringmethodologyisproposedforchangedetec- tioninuncertainnonlineardynamicsystems. AnexperimentalstudywasperformedusinganMR dampertoevaluatetherangeofvalidityoftheproposedmethodology. Theexperimentalresults demonstratedthattheproposedmethodologycansuccessfullyassesstheconditionsofuncertain nonlinear systems by: (1) detecting (small) genuine system changes, (2) interpreting physical meaning of the detected changes without a priori knowledge of system characteristics, and (3) quantifyingtheuncertaintyboundsofthedetectedchanges. Intheproposedmethodology,theRestoringForceMethodwasusedasanon-parametricsys- temidentificationapproach. ItwasdemonstratedthattheRestoringForceMethodismoreuseful than other modeling approaches in structural health monitoring applications, taking advantage of features from both the parametric and non-parametric modeling approaches: the identifica- tion procedure is data-driven and some physical interpretation is possible, using the identified coefficients. Supervisedandunsupervisedstatisticalclassificationmethodswereappliedtodetectgenuine system changes with different levels of system uncertainty. The classification results demon- stratedthattheidentifiedcoefficientsusingtheRestoringForceMethodcanbeusedasexcellent features to detect system changes in uncertain nonlinear systems. With statistical unbiasness of theidentifiedcoefficients,itwasshownthatthechangedetectionprocedurecanbedramatically simplifiedusingreduced-ordermodels. 138 Chapter6 MonitoringtheCollisionofaCargoShipwith theVincentThomasBridge 6.1 Introduction 6.1.1 Motivation Thedemandonadvancedtransportationinfrastructureincreasesineveryregionoftheworld. In the United States and across the world, more highways and bridges are being built than in the past. With new construction technologies and materials to link lands and islands, bridges have becomelongerandmorereliable. Asmorebridgeshavebeenconstructed,however,thechances of collisions with ships have also increased. In fact, ship-bridge collisions (with potentially serious consequences) happen relatively frequently. Some examples of these, with fatalities, in differentcountriesareshowninTable6.1. IntheUnitedStates,manysignificantship-bridgecol- lisionshaveoccurred,andmanyoftheminvolvedhumanfatalities. Somemajorship-bridgecol- lisionincidentsintheUnitedStates,reportedbyNationalTransportationSafetyBoard(NTSB), aresummarizedinTable6.2. When a ship-bridge collision occurs, accurate and rapid condition assessment of the bridge iscritical. Suchanassessmentshouldincludetheestimationofpotentialdamage,aswellasthat of direct damage in order to prevent secondary disasters that could be induced by the collision. 139 Table 6.1: Examples of ship-bridge collisions with fatalities in different countries, listed in chronologicalorder(Mastaglio,1997;ProskeandCurback,2003). Bridgename Year Fatalities SevernRiverRailwayBridge,UK 1960 5 LakePonchartain,USA 1964 6 SidneyLanierBridge,USA 1972 10 LakePonchartainBridge,USA 1974 3 TasmanBridge,Australia 1975 15 PassManchacBridge,USA 1976 1 TjornBridge,Sweden 1980 8 SunshineSkywayBridge,USA 1980 35 LorrainePipelineBridge,France 1982 7 SentosaAerialTramway,China 1983 7 VolgaRiverRailroadBridge,Russia 1983 176 ClaibornAvenueBridge,USA 1993 1 CSX/AmtrakRailroadBridge,USA 2001 47 PortIsabel,USA 2001 8 Webber-Falls,USA 2002 12 Sincecurrentpracticesofdamageestimationmainlyrelyonhumanvisualinspections,accurate and reliable condition assessment of a target bridge is often infeasible, as damage may not be visible. In such a case, vibration-based structural health monitoring approaches can augment traditionaldamageinspectionmethods. Thankstothemulti-disciplinaryadvancedtechnologies of sensor networks, data acquisition, communication, computation powers and system iden- tification techniques, this approach has the potential to provide a useful and reliable damage quantification,whichmightbedifficultwithtraditionalvisualinspectionapproaches. 6.1.2 Objectives Thischapterpresentsaforensicstudyofthefirst-evercollisionofacargoshipwiththeVincent Thomas Bridge (VTB), a critical 1850-m suspension bridge located in the larger metropolitan LosAngeles,Californiaregion. 140 Table 6.2: Examples of major ship-bridge collision incidents in the U.S.A. during the past 30 yearsreportedbyNationalTransportationSafetyBoard. Date Location Accidentdescription U.S.TankshipSSMarineFloridanCollisionwiththe 1977-02-24 Hopewell,Virginia BenjaminHarrisonMemorialBridge. CollisionofM/VstudwiththeSouthernPacific 1978-04-01 BerwickBay,Louisiana RailroadBridgeovertheAtchafalayaRiver. RammingoftheSunshineSkywayBridgebythe 1980-05-09 TampaBay,Florida LiberianbulkcarrierSummitVenture. RammingofthePopularStreetBridgebythetow 1983-04-02 St. Louis,Missouri boatM/VCityofGreenvilleanditsfour-bargetow. CollisionofthePanamainiacementcarrierM/V 1983-11-23 NewOrleans,Louisiana AmparoPaolawiththeDanzigerBridgeInnerHarbor Nav. Canal. RammingofthePoplarStreetBridgebythetowboat 1984-04-26 St. Louis,Missouri M/VErinMarieanditstwelve-bargetow. RammingoftheSidneyLanierBridgebythePolish 1987-05-03 Brunswick,Georgia bulkcarrierZiemiaBialostocka. RammingoftheCSXTRailroadBridgebytheCyprian 1988-05-06 Chicago,Illinois BulkcarrierM/VPontokratisCalumetRiver. U.S.TowboatChriscollisionwiththeJudgeWilliam 1993-05-28 NewOrleans,Louisiana SeeberBridge. RammingoftheEadsBridgebybargesintowofthe Merchant/MotorVessel(M/V)AnneHollywith 1998-04-04 St. LouisHarbor,Missouri subsequentrammingandnearbreakawayofthe PresidentCasinoontheAdmiral. U.S.towboatRobertY.LoveallisionwithInterstate 2002-05-26 Oklahoma 40HighwayBridgenearWebbersFalls. Using advanced structural health monitoring technologies, the main objective of this study was to demonstrate various analysis and interpretation capabilities of the bridge’s global con- dition after the collision. The dynamic response of the VTB was successfully measured (with a real-time monitoring system installed on the bridge) before and after the incident, as well as during the impact process. Using these valuable data, various system identification approaches, 141 includingglobal(multi-sensor)andlocal(single-sensor)identificationmethods,wereperformed independently to detect the potential occurrence of significant changes in the bridge’s vibration signature. 6.1.3 Scope The contents of this chapter are organized as follows. The description of the VTB and its real- time monitoring system are presented in Section 6.2. The procedure for the preliminary data processing and its results are discussed in Section 6.3. In Section 6.4, detailed information andsensormeasurementsoftheship-bridgecollisionincidentarepresented. Variousglobaland localidentificationapproachesusedinthisstudyareexplained,andtheiridentificationresultsare showninSection6.5. ThesummaryandconclusionsofthechapterareprovidedinSection6.6. 6.2 Real-TimeMonitoringoftheBridge 6.2.1 BridgeDescription TheVTBislocatedinthemetropolitanLosAngelesregion. Thisbridgewasoneoftollbridges before 2000, and itis still considered as a major bridge in California. It connects two mainhar- borsinthisregion,thePortofLosAngelesandthePortofLongBeach(SeeFigure6.1). These two ports are among the busiest ports in the U.S. The bridge handles approximately 39000 cars andtrucksdaily. TheVTBisacable-suspensionbridge,approximately1850-mlong,consisting ofamainspanof457m,twosuspendedsidespansof154meach,andtwoten-spancast-in-place concrete approaches of 545-m length on both ends. The roadway is 16-m wide and accommo- dates four lanes of traffic. The bridge was completed in 1964 with 92000 tons of Portland 142 cement, 13000 tons of light weight concrete, 14100 tons of steel and 1270 tons of suspension cables. The bridge was designed to withstand winds of up to 145 kilometer per hour. A major seismicretrofitwasperformedduringtheperiod1996-2000,includingavarietyofstrengthening measures,andtheincorporationofabout48large-scalenonlinearpassiveviscousdampers. (a) A photo of the Vincent Thomas Bridge (Courtesy of Port of Los Angeles). In the photo, the leftistheEasttowertowardTerminalIsland,andtherightistheWesttowertowardSanPedro. Total Spans (1847.70 m) Suspended Spans (765.96 m) Main Span (457.20 m) East Tower West Tower Cable Anchorage Terminal Island Approach Spans (519.84 m) San Pedro Approach Spans (561.29 m) Cable Anchorage (b)AschematicviewoftheVincentThomasBridgewithspandimensions. Figure6.1: TheVincentThomasBridge. 143 6.2.2 VTBInstrumentation The VTB has been instrumented by the California Strong Motion Instrumentation Program (CSMIP) of the California Geology Services (CGS), formerly known as the Division of Mines and Geology (CDMG), for more than twenty years. The strong-motion recording system con- sists of twenty-six accelerometers mounted on the bridge and an original analog recording sys- tem (later converted to a digital recording system) located in the east anchor block. Figure 6.2 showsthesensorlocationsforthissystem. Significant motions have been recorded for the 1987 Whittier, 1994 Northridge, and sev- eral other earthquakes. Analysis of these recordings has provided much information about the dynamicresponseoflargesuspensionbridges. Thepreviousanalogfilmrecordingsystem,(used untilthemid1990s)hasproventobeveryreliable,buttherecordeddatawerelimitedindynamic rangeanddifficulttoconverttodigitalformatappropriateforcomputeranalysis. Modern digital recording technology certainly can provide superior data quality and ease of analysis. To demonstrate this, a temporary digital monitoring system with remote commu- nications capability was installed in parallel with the existing analog recording system for the Vincent Thomas Bridge strong motion instrumentation between November 3 and December 5, 1995. Duringthisshorttimeperiod,alargeamountofambientvibrationdatawasrecorded. The capabilityofremotereal-timedatamonitoringwasalsodemonstrated. Abdel-Ghaffar et al. (1995) includes examples of preliminary analysis in the appendices, showing these measurements and the large amount of high-quality digital data obtained during the monitoring period. Examples of preliminary analyses are included in the appendices. In 144 24 25 26 9 13 19 20 1 23 14 3 4 15 16 5 17 18 6 12 7 21 22 10 11 8 2 North 457.2 m Figure XX. Vincent Thomas Bridge sensor location. Figure6.2: SensorlocationsanddirectionsontheVincentThomasBridge,SanPedro,CA. addition to successfully demonstrating this application of modern structural monitoring instru- mentation, the recorded ambient vibration data provided a baseline for evaluating the effects of theseismicretrofitonthebridge’sdynamicbehavior,occurringfrom1999to2000. More information concerning instrumentation and analysis of the VTB can be found in Abdel-Ghaffar and Housner (1978); Abdel-Ghaffar et al. (1992); He et al. (2004); Ingham etal.(1997);Masrietal.(2004);Smythetal.(2003);Wahbehetal.(2003). 6.2.3 Real-timeBridgeMonitoringSystem The VTB has been monitored with a web-based real-time bridge monitoring system developed bytheauthorssince2005(Wahbehetal.,2005). Themonitoringsystemconsistsoffoursubsys- tems, including: (1) sensor networks; (2) publisher; (3) server; and (4) clients (see Figure 6.3). 145 SERVER USC Storage USC Server USC FTP PUBLISHER Data Transmission Data Acquisition SENSOR NETWORKS Sensor 1 Sensor 2 Sensor 26 Sensor 3 ... CLIENTS Client 1 Client 2 Client N Client 3 ... LOCATION: BRIDGE SITE LOCATION: USC LOCATION: ANY PLACE Wired Cables TCP/IP UDP Figure6.3: AschematicoftheVTBreal-timemonitoringsystemarchitecture. 1. For the sensor network subsystem, twenty-six strong-motion accelerometers are used to measure the bridge’s ambient and earthquake vibrations. The sensor locations and mea- surementdirectionsareillustratedinFigure6.2. Noticethattheeasternhalfofthebridge ismoredenselyinstrumentedthanthewesternhalf,becausethedataacquisitionsystemis housedintheeasterncableanchorage. 2. Bridgemotionissensedbytheaccelerometers,thenthesensorsignalsareconveyedtothe publishersubsystem,whichconsistsofthedataacquisitionmoduleanddatatransmission module. Theaccelerometersarephysicallyconnectedtothedataacquisitionmodulewith wirecables,andthesensorsignalsaresampledat100Hz. 3. Using the data transmission module, the digitized signals are transmitted to the server subsystem accessed via the Internet. The TCP/IP protocol is used for reliable data com- munication between the publisher and server subsystems (Stevens, 1998; Stevens et al., 2002). 4. The acquired data can be downloaded using the FTP server located in the University of Southern California (USC), the USC FTP module, for further analysis. The data are also 146 senttotheUSCservermoduletodistributethedatasimultaneouslytomultipleauthorized clients,suchasCDMGandCaltrans. 5. In the server-to-multiclient communication, the data transmition rate often becomes a “bottle-neck” for successful data communication. Therefore, a faster and less reliable communicationprotocol,UDP,isused(Stevens,1998;Stevensetal.,2002). 6.3 PreliminaryDataProcessing Once the bridge accelerations were measured, the raw data were processed to obtain the corre- spondingvelocitiesanddisplacementsusingthefollowingprocedure: 1. TheDCandlineartrendweresubtractedfromtherawaccelerations,andacosine-tapered windowwasappliedtotheaccelerationtimehistoriestopreventfrequencyleakage. 2. A bandpass filter was designed with the cutoff frequencies of 0.1 to 30 Hz and applied to theaccelerationtimehistories. 3. Standard numerical integration procedures were subsequently used to obtain the corre- spondingvelocityanddisplacementtimehistories. A sample processed acceleration and displacement time history record at the moment of the cargo-shipcollisionisshowninFigure6.4. 147 0 50 100 150 200 −0.2 −0.1 0 0.1 0.2 TIME (sec) ACCELERATION (g) 0 50 100 150 200 −40 −20 0 20 40 TIME (sec) DISPLACEMENT (cm) (a)Processedacceleration (b)Processeddisplacement Figure 6.4: Preprocessed acceleration and displacement of Channel 4 (lateral direction at the mid-span of the bridge deck). The acceleration was numerically double-integrated to obtain the displacementwiththecutofffrequenciesof0.1to30Hz. 6.4 DescriptionoftheShipCollisionIncident 6.4.1 FactualInformationoftheIncident The Beautiful Queen is a 189-m (620-ft) 32000-ton cargo ship, owned by Pasha Hawaii Trans- portation Line. The cargo ship is a bulk carrier, not a container ship, commonly hauling rolled steel,coalorgrain. Theshipisequippedwithonboardcranesforfreightloading. On Sunday, 27 August 2006, the ship departed from the Los Angeles harbor via one of the channelsintheharbordistrict. At16:40(PacificDaylightTime),theshipwaspassingunderthe Vincent Thomas Bridge, linking San Pedro and Terminal Island as shown in Figure 6.5. When the ship was passing under the bridge, the ship operators miscalculated the tide, and one of the onboardcranesscrapedaguiderailofamaintenancescaffoldsecuredatthebridgecenterspan, which was about 56 m (185 ft) above water. No injuries were reported during the incident. A schematic view of the ship-bridge collision is illustrated in Figure 6.6, and the damaged guide railofthemaintenancescaffoldisshowninFigure6.7. 148 Vincent Thomas Bridge Los Angeles Harbor Terminal Island San Pedro Figure6.5: Schematicviewoftheincidentarea(courtesyofGoogleInc.) Main Span (457.20 m) San Pedro Terminal Island Beautiful Queen with crane Center Spans of the Vincent Thomas Bridge East Tower West Tower Clearance (56.39 m) Figure 6.6: Schematic view of the Beautiful Queen, a cargo ship, under the Vincent Thomas Bridge. Thefigureispresentedforillustrationpurpose,anddoesnotshowtheactualpathtaken bytheshipduringthecollision. 149 About thirty minutes after the collision, the vehicular traffic across the bridge was stopped by Caltrans to investigate potential damage. Vessel traffic was also stopped under the bridge by the Los Angeles Port Police and Coast Guard. Two incoming cargo ships were delayed due to the vessel traffic shut-down. After investigating the incident for a period of about two hours, Caltrans engineers declared that the bridge was sound and that the damage was limited to the maintenance scaffolding. Both vehicle and vessel traffic were re-opened at 18:55 the same day. AnindependentinvestigationwasalsoconductedbytheCoastGuardonthecollidingcargoship. 6.4.2 VibrationMonitoringoftheIncident The VTB vibration during the cargo-ship incident, and two-hour traffic shut-down afterward, weresuccessfullycapturedbythereal-timemonitoringsystem. Sampleaccelerationtimehistory dataareillustratedinFigure6.8. Thefigureshowsatime-windowof24-hours(frommidnightto midnight) corresponding to the displacement measurements at the mid-span of the bridge deck (Figure 6.8 (a)) and at the east column (Figure 6.8 (b)) in the lateral and vertical directions. According to the measurements, the incident occurred at 16:41 and resulted in approximately two minutes of superstructure vibration. At 17:12, thirty-one minutes after the incident, the displacement RMS reduced dramatically for a 1:45 hour duration, corresponding to the post- incidenttrafficshut-downbyCaltrans. Theimpactbythecargo-shipwasmorenoticeableinthe lateral displacements than in the vertical displacements, for both the bridge deck and columns, sincethebridgewasrammedbytheshipinthelateraldirection. 150 Figure 6.7: A damaged maintenance scaffolding member from the ship-bridge collision (Cour- tesy of Caltrans). This figure is presented for illustration purpose. The location of the damaged memberinthebridgecoulddifferfromtheexactlocation. 6.4.3 BridgeResponseBeforeandAftertheIncident The bridge response is largely influenced by various environmental conditions, such as traffic intensityandtemperature,andthebridgecharacteristicsdeterminedwithidentificationmethods couldbealsoaffectedbytheseconditions. Therefore,itisworthytoinvestigatethetrendsofthe bridgeresponseovercertainperiods. 151 −40 −20 0 20 40 LATERAL DSP (cm) Normal Traffic Normal Traffic Collision & Traffic Shut-down 0 3 6 9 12 15 18 21 24 −40 −20 0 20 40 TIME (hour) VERTICAL DSP (cm) (a)Displacementsatthemid-spanofthebridgedeck—Channel3(top)andChannel16(bottom). Normal Traffic Normal Traffic Collision & Traffic Shut-down −3 −1.5 0 1.5 3 LATERAL DSP (cm) 0 3 6 9 12 15 18 21 24 −3 −1.5 0 1.5 3 LONGITUDINAL DSP (cm) TIME (hour) (b)Displacementsatthetopofthebridgecolumn—Channel8(top)andChannel10(bottom). Figure 6.8: Displacements of the bridge deck and column on 27 August 2006 when the cargo- ship incident occurred. The top figure shows the lateral displacement, and the bottom figure showstheverticaldirection. TypicalweeklyRMSdisplacementsofthemainspanofthebridgedeckbeforeandafterthe incidentareshowninFigure6.9. Inthefigure,◦indicatesanhourlyRMSdisplacement,and4 indicates a daily average of the hourly RMS displacements. One standard deviation (σ) of the hourlyRMSdisplacementsforonedayisshownasthegrayregioninthefigure. TheRMSlevels ofthedisplacementduringtheincidentimpactandtrafficshut-downaftertheincidentwerealso 152 determined and shown as a dash line and dash-dot line, respectively. The figure shows that no significant difference was observed in the bridge response before and after the incident. For the vertical displacement, the hourly RMS displacement is less than 2.5 cm during the particular weeks before and after the incident (Figures 6.9 (a) and (c)). A daily cycle was observed for a week starting on Monday and ending Sunday – smaller displacements were noted at night and larger displacements during the day due to traffic. It is also shown that the daily average of the hourly RMS displacement is relatively high during weekdays, while much lower during weekends. Similartrendswerefoundinthelateraldisplacements(Figures6.9(b)and(d)),while itsamplitudeisaboutonethirdoftheverticaldisplacementsbeforeandaftertheincident. 6.5 SystemIdentificationoftheBridge 6.5.1 GlobalSystemIdentificationApproaches This section deals with the basic formulation of the Natural Excitation Technique (NExT) in conjunction with the Eigensystem Realization Algorithm (ERA), which was used to extract the modal parameter information of the VTB. For more detailed formulation and discussion, the readerisreferredtootherpapersbytheauthors(Nayerietal.,2007,2006). Formulationofthetime-domainmodalparameteridentificationtechniques Providing knowninput excitations for large civil structures is very difficult, costly, and inmany cases infeasible. On the other hand, ambient excitation (from wind, traffic, ground motion, etc.) is always available. However, ambient vibrations are output-only, as the inputs cannot be measuredorquantifiedwithanycertainty. Thesefactsshowtheimportanceofoutput-onlymodal 153 BEFOREACCIDENT MON TUE WED THR FRI SAT SUN −0.5 0 0.5 1 1.5 2 2.5 TIME (DAY) DISPLACEMENT (cm) MON TUE WED THR FRI SAT SUN −0.5 0 0.5 1 7 7.5 TIME (DAY) DISPLACEMENT (cm) (a)Verticaldisplacement(sensor15) (b)Lateraldisplacement(sensor3) AFTERACCIDENT MON TUE WED THR FRI SAT SUN −0.5 0 0.5 1 1.5 2 2.5 TIME (DAY) DISPLACEMENT (cm) MON TUE WED THR FRI SAT SUN −0.5 0 0.5 1 7 7.5 TIME (DAY) DISPLACEMENT (cm) (c)Verticaldisplacement(sensor15) (d)Lateraldisplacement(sensor3) Figure 6.9: Typical weekly root-mean-square (RMS) displacements of the main span of the bridge deck in vertical and lateral directions before and after the ship-bridge collision. In the figure,◦ indicates an hourly RMS displacement, and4 indicates a daily average of the hourly RMSdisplacements. Onestandarddeviation(1σ)ofthehourlyRMSdisplacementsforoneday isshownasthegrayregioninthefigure. TheRMSlevelsofthedisplacementduringtheincident impact and traffic shut-down after the incident were also determined and shown as a dash line anddash-dotline,respectively. parameteridentificationmethods. TheNExTapproach,introducedbyJamesetal.(1993,1996), has been successfully used for the identification of structures based on output-only information (Caicedo et al., 2004). The basic idea behind the NExT method is that the cross-correlation functionbetweentheresponsevectorandtheresponseofaselectedreferenceDOFsatisfiesthe homogeneous equation of motion, provided the excitation and responses are weakly stationary 154 randomprocesses. Weakstationaritycanusuallybeassumedforambientvibrationsovertypical analysistimedurationsofminutestotensofminutes. Using NExT, it can be also shown that the cross correlation function between the accelera- tion process vector and the acceleration of a reference DOF satisfies the homogeneous (or free vibration)equationofmotionpertheequation: M ¨ R ¨ X ref ¨ X (τ)+D ˙ R ¨ X ref ¨ X (τ)+KR ¨ X ref ¨ X (τ) =0 (6.1) where ¨ X and ¨ X ref arethen×1accelerationvector,andthereferenceDOFacceleration,respec- tively,M,D,andK arethen×nmass,damping,andstiffnessmatricesrespectively,andR(.) denotesthecorrelationfunction. Previous experience (Nayeri et al., 2006) has shown that one cannot rely one a single ref- erence DOF for identification of all modes. Optimum accuracy for different modes typically occurs at different choices of the reference DOFs. The importance of Equation (6.1) is that: (a) thestationaryrandomexcitation(ambientnoise)iseliminatedfromtheequationofmotion,and (b)onlytheaccelerationrecordsareneededtoimplementthetechnique. Once the homogeneous equation of motion is formed using the NExT, the ERA (Juang and Pappa, 1985, 1986) can be used to extract the modal parameters of the homogeneous model. 155 Here, we briefly present the fundamental principles of ERA. The first fundamental step is to formthen(r+1)×m(p+1)Hankelblockdatamatrixasfollows: H(k−1) =             Y(k) Y(k +1) ... Y(k +p) Y(k +1) Y(k +2) ... Y(k +p+1) . . . . . . . . . . . . Y(k +r) Y(k +r+1) ... Y(k +p+r)             (6.2) wherenandmarethenumberofmeasurementstations,andthereferenceDOFs,respectively;r andp are integers corresponding to the number of block rows and columns, respectively.Y(k) isthen×mmatrixofthecross-correlationfunctionswhichsatisfiesthehomogeneousequation of motion (Equation 6.1). The ERA process starts with factorization of the Hankel block data matrix,fork = 1,usingthesingularvaluedecompositionprocedure: H(0) =PDQ T = P 1 P 2     D 1 0 0 0         Q T 1 Q T 2     =P 1 D 1 Q T 1 (6.3) whereDisthediagonalmatrixofmonotonicallynon-increasingsingularvalues. D 1 isanN×N (N ≤p)diagonalmatrixformedbytruncatingtherelativelysmallsingularvalues.N isthefinal system order. It is worth noting that the selection of the final model order it not a trivial task (Nayeri et al., 2006). The discrete-time state-space realization matrices for the structural model canbeestimatedas(JuangandPappa,1985) ˆ A =D −1/2 1 P T 1 H(1)Q 1 D −1/2 1 (6.4) 156 ˆ C =E T m P 1 D 1/2 1 (6.5) whereE T m = I 0 , and its size is determined accordingly. The control influence matrix can not be estimated using the output-only information. The estimated discrete-time realization needs to be transformed to the continuous-time domain version, and the modal parameters can thenbeextractedfromtheidentifiedcontinuous-timesystem(Nayerietal.,2007). Therearelotsofissueswiththeimplementationofthesetechniques,including: selectionof user-selectable parameters such as the size of the SVD matrix, the reference DOF (or DOFs), windowsize,finalmodelorder,andmoreimportant,recognizingandeliminatingspuriousmodes whichwillappearduetonoiseandmodeloverspecification. Nayerietal.(2006)addressedthese problemsindetail. Implementationandresults ThissectionreportstheresultsoftheapplicationoftheproposedalgorithmstotheVTBrecorded data. Aswasmentionedearlier, forimplementingtheNExT/ERAalgorithm, onlyoutputaccel- eration records are needed. In this study, three distinct time windows of data were considered. The first window captures data during the accident (impact type excitation), which lasted about twenty minutes, the second one corresponds to the traffic shut-down period which lasted about two hours, and the third window corresponds to regular traffic conditions for eight hours. Rela- tivelylongtime-historyrecordswereusedtoenhancethestationarityoftheanalysisdata. AsshowninFigure6.2,theVTBwasinstrumentedwithtwenty-sixaccelerometers,however, only the acceleration measurements on the main deck, and towers (six vertical, six lateral, and 157 three longitudinal directions) were used in this study. Sensors at the base of the VTB recorded negligible levels of response. Data were recorded at a sampling rate of 100 Hz. Since the frequencyrangeofinterestislessthan5Hz,thedata(afterpre-processing)weredown-sampled to50Hz. After pre-processing (filtering, detrending, etc.), the next step was to compute the Cross- Correlation Functions (CCF) between the response of the preselected reference DOF (or, for more reliability, multiple DOFs) and the response of all available DOFs. As mentioned earlier, one cannot rely on just one single reference DOF for all modes. One single reference that is a proper selection for some modes, might not be proper for other modes. Consequently, it is recommended to use multiple reference DOFs, as opposed to a single reference DOF (Nayeri etal.,2006). Inthisstudy,inordertoimprovetheidentificationresults,allavailableDOFswere included (in sequential order) as the reference ones. The CCF can be estimated by the inverse Fourier transform of the Cross-Power-Spectral Density (CPSD), where the CPSD is computed directlyfromthedata. RandomerrorsassociatedwiththeCPSDcanbeminimizedbywindow- ingandaveraging(BendatandPiersol,2000). Three different time windows were considered in this study: (1) during accident (impact type excitation), (2) traffic shut down, and (3) regular traffic. Table 6.3 summarizes the modal parameter identification results for the three above mentioned time windows. A total of five dominantmodeswereidentified: thefirstlateralbending(ModeA),firstverticalbending(Mode B), first torsion (Mode C), second vertical bending (Mode D), and second torsion (Mode E). One interesting observation from this table is that the first lateral mode only appeared during the accident. This makes sense, since the traffic can barely excite that pure lateral mode. Mode 158 Table6.3: ComparisonoftheVTBmodalparameteridentificationresultsusingNExT/ERAfor three different cases: (1) during accident (impact type excitation), (2) traffic shut down, and (3) regulartraffic. Forallcases: windowsizeandoverlap=327.68secand75%,respectively,andall availableDOFsareusedasthereference. ForERA:r = 30,andp = 2/3ofthecorrelationdata points. Naturalfrequency(Hz) Modeshaperatiow/MAC(%) Mode anddampingratio(%) andfrequencydifference(%) No. Modeshape impact w/otraffic w/traffic (1) (2) (3) (1)&(2) (1)&(3) (2)&(3) A 0.1496 – – – – – (topview) 4.0% – – – – – B 0.2327 0.2441 0.2353 99.7% 99.9% 99.8% 2.7% 2.5% 1.9% 4.87% 1.11% 3.61% C 0.5357 0.5430 0.5339 99.8% 99.7% 99.4% 0.8% 0.6% 0.6% 1.36% 0.67% 0.68% D 1.3938 1.3920 1.4004 99.2% 99.7% 98.9% 1.5% 1.3% 1.7% 0.13% 0.47% 0.60% E 1.8685 1.8930 1.8668 98.9% 98.7% 99.3% 1.3% 1.2% 1.9% 1.31% 0.09% 1.38% shape and frequency comparisons between the results of these three time windows indicate that the mode shapes are virtually identical; however, there is up to a 5% change in frequency. That isnotasurpriseinviewoftheuncertaintyissuesrelatedtoenvironmentalconditions. Comparisonwithpreviousidentificationstudiesfordifferentearthquakes The identification results in this study are compared with previous identification works by Lus ¸ et al. (1999) and Smyth et al. (2003). In their identification work, Lus ¸ et al. (1999) employed the ERA method with the Observer/Kalman filter Identification (OKID) approach to extract the 159 modal parameters of the Vincent Thomas Bridge, based on the data obtained during the 1987 Whittier and 1994 Northridge earthquakes. Using the same earthquake data sets, Smyth et al. (2003) applied a linear least-squares method to identify the bridge. The three identification results are summarized in Table 6.4. Obviously, the number of identified modes in this study is smaller than those in the previous studies. As mentioned earlier, we used an autonomous algorithm to eliminate the spurious modes and include only genuine modes of the bridge. This autonomous algorithm works based on some accuracy indicators, which are used to perform a validationtest(Nayerietal.,2006). Thus,themodesnotsatisfyingthetestcriteriaareregarded as non-genuine modes and automatically eliminated from the process. The mode shapes in Table6.3clearlyindicatethatnon-genuinemodesweresuccessfullyrejected. Moreover,thegen- uinemodesidentifiedinthisstudyrepeatedlyappearedindifferentidentificationtime-windows underboththeimpactandambientvibrationconditions. Therepeatabilityoftheidentifiedmodes invariousexcitationconditionsiscriticalforreliablestructuralhealthmonitoringapplications. Uncertaintystudyofthebridgeidentification A statistical study was performed to estimate the identification uncertainty. Because the struc- tural conditions of the bridge characterized with the identification methods used in this study couldvarysignificantlywithdifferentexcitationandenvironmentalconditions(e.g. trafficinten- sity and temperature), it is important to estimate the bounds of uncertainty in the identification results. Forthree-monthduration(July,2007 ∼September,2007),thestatisticsoftheidentified natural frequencies and damping ratios were obtained. The statistics were obtained separately forweekdaysandweekendsbecauseasignificantdifferenceofthebridgeresponsewasobserved betweenweekdaysandweekends,asshowninFigure6.9. Sampledistributionsoftheidentified 160 Table 6.4: Comparison of the bridge identification results with previous studies for different earthquakes. The previous studies in the comparison include Smyth et al. (2003) and Lus ¸ et al. (1999) for the 1987 Whittier and 1994 Northridge earthquakes. In the table, f is the natural frequency(Hz),andζ isthedampingratio(%). Smythetal.(2003) Lus ¸ etal.(1999) Yunetal. Verticaldirection Alldirections Alldirections Whittier Northridge Whittier Northridge Impact w/otraffic w/traffic f ζ f ζ f ζ f ζ f ζ f ζ f ζ 0.212 1.2 0.225 0.1 0.234 1.5 0.225 1.7 0.150 4.0 – – – – 0.242 1.7 0.240 8.2 0.388 38.2 0.304 28.6 0.233 2.7 0.244 2.5 0.235 1.9 0.317 -4.3 0.358 -4.7 0.464 9.7 0.459 1.8 0.536 0.8 0.543 0.6 0.534 0.6 0.531 10.2 0.390 4.2 0.576 9.9 0.533 4.0 1.394 1.5 1.392 1.3 1.400 1.7 0.570 0.6 0.448 -0.7 0.617 14.5 0.600 26.2 1.869 1.3 1.893 1.2 1.867 1.9 0.636 4.2 0.478 1.3 0.617 76.8 0.632 13.7 0.672 0.1 0.522 1.4 0.769 29.7 0.791 15.6 0.734 2.4 0.587 -0.1 0.804 1.4 0.811 1.0 0.818 1.9 0.625 7.4 0.857 11.6 0.974 2.7 0.958 2.9 0.733 1.2 0.947 4.3 1.110 0.6 1.027 -1.9 0.837 5.0 1.111 1.3 0.935 -1.8 1.159 1.7 1.036 1.6 1.391 2.3 1.110 1.7 1.554 -1.3 1.136 1.4 natural frequencies and damping ratios for Mode B (the first vertical bending) are illustrated in Figure6.10. The average of natural frequencies and damping ratios for four different identified modes (Modes B∼ E) were determined with the averaged sample sizes of 288 for weekdays and 123 for weekends. The sample size of each mode varied because not all modes were identifiable with ambient vibration in the NExT-ERA identification. The mean of the averaged natural fre- quency for weekdays and weekend differed from 0.39% to 1.47%. The coefficient of variance of the averaged natural frequency was determined between 0.35% and 1.98% for weekdays, and between 0.41% and 1.85% for weekend. Thus, no significant difference for weekdays and weekendwasobservedintheaveragednaturalfrequencies. Fortheaverageddampingratio,the differenceofitsmeanrangedfrom7.13%to19.64%,andthecoefficientofvariancerangedfrom 0.98%to45.64%. Therefore,thediscrepancybetweentheaverageddampingratioforweekdays 161 0.22 0.23 0.24 0.25 0 50 100 150 200 250 FREQUENCY (Hz) PROBABILITY DENSITY 0.22 0.23 0.24 0.25 0 50 100 150 200 250 FREQUENCY (Hz) PROBABILITY DENSITY (a)weekday (b)weekend 0 5 10 15 0 0.1 0.2 0.3 0.4 DAMPING RATIO (%) PROBABILITY DENSITY 0 5 10 15 0 0.1 0.2 0.3 0.4 DAMPING RATIO (%) PROBABILITY DENSITY (c)weekday (d)weekend Figure6.10: Histogramsofthenaturalfrequenciesanddampingratiosofthefirstverticalbend- ingmode(ModeB)identifiedusingtheERAmethod. and weekend is greater than that of the averaged natural frequency. Therefore, it was observed that the uncertainty of identifying the damping ratio was greater than that of identifying the naturalfrequency. Effectsoftemperaturevariation It is well known that the effects of temperature variations are very significant to the dynamic response of bridges, and in many cases, the genuine changes of bridge modal properties could be overwhelmed by the temperature-induced changes. Unfortunately, because no temperature measurements were conducted in this study due to the limitation of the current monitoring sys- tem configuration, more rigorous studies of this important temperature effects could not be performed. However, this chapter is designed to demonstrate the practical applications of the 162 SHMapproachesforforensicinvestigations,andadvancedissuesofthebridgeidentificationare beyondthescopeofthisstudy. 6.5.2 LocalSystemIdentificationApproaches Identificationofnaturalfrequencyanddampingratio Once the global (multiple-sensor) system identification was performed, local (single-sensor) identification approaches were also applied independently for comparison purposes. Modal frequencies and damping ratios of the lateral displacement modes of the bridge deck were estimated. The logarithmic decrement (δ) method was used to estimate modal damping ratio as(Meirovitch,1986): δ j = 1 j ln x 1 x j+1 , ¯ δ = 1 n n X j=1 δ j (6.6) ¯ ζ = ¯ δ p (2π) 2 + ¯ δ 2 ∼ = ¯ δ 2π (6.7) wherex j isthej th peakdisplacement,δ j isthelogarithmicdecrementbetweenx 1 andx j+1 , ¯ δ is theaveragedlogarithmicdecrement,and ¯ ζ istheaverageddampingratio. Theaverageddamping ratios of sensors 3 and 5 were calculated with the peaks and valleys of the oscillation as shown inFigure6.11(a). Aslightdiscrepancyof ¯ ζ betweenthepeaksandvalleyswasobserved. The ¯ ζ ofpeaksandvalleysweremeasuredat6.40and4.78forsensor3,and6.40and4.60forsensor5, respectively. Notice that the averaged values of ¯ ζ of peaks and valleys for sensors 3 and 5 were 5.59 and 5.50. Second, the natural frequencies of the bridge deck at sensors 3 and 5 were also 163 estimatedfromitspowerspectraldensityplotasshowninFigure6.11(b). Theestimatedlateral natural frequencies (ω n ) of the bridge were 0.138 Hz for sensor 3, and 0.142 Hz for sensor 5. Theidentificationresultsofthelateraldampingratiosandnaturalfrequenciesofthebridgedeck aresummarizedinTable6.5. 0 50 100 150 200 −40 −30 −20 −10 0 10 20 30 40 RELATIVE TIME (SEC) DISPLACEMENT (cm) 0 0.2 0.4 0.6 0.8 1 0 500 1000 1500 2000 2500 FREQUENCY (Hz) PSD DISPLACEMENT (a)Timehistory (b)Powerspectraldensity Figure 6.11: Local identification of the damping ratio and natural frequency of the bridge deck inlateraldirection(sensor3)duringtheincidentimpact. The natural frequencies and damping ratios of the vertical and torsional bridge response were also estimated. First, the vertical displacements at the center of the main span (sensors 15 and 16) and their frequency spectra are shown in Figures 6.12 (a)-(d). From the frequency spectra, two identical natural frequencies were identified at 0.232 Hz and 0.537 Hz for both sensors 15 and 16. The torsional displacement was obtained from the subtraction between the time histories of sensors 15 and 16 as shown in Figure 6.12 (e). The natural frequencies of the torsional displacement were identified at 0.147 Hz and 0.537 Hz for sensors 15 and 16, and at 0.147 Hz, 0.537 Hz, and 0.717 Hz for sensors 17 and 18. The slight variations in the single- sensorfrequencyestimatesareprimarilyattributabletomode-order-reductioneffects. In order to estimate the damping ratios for the identified natural frequencies, a bandpass filter was applied to the torsional displacement, and Equations 6.6 and 6.7 were used for the 164 0 20 40 60 80 100 120 140 −10 −5 0 5 10 DISPLACEMENT (cm) RELATIVE TIME (SEC) 0 0.2 0.4 0.6 0.8 1 0 0.5 1 1.5 2 FFT DISPLACEMENT FREQUENCY (Hz) (a)Timehistoryofsensor15 (b)Frequencyspectrumofsensor15 0 20 40 60 80 100 120 140 −10 −5 0 5 10 DISPLACEMENT (cm) RELATIVE TIME (SEC) 0 0.2 0.4 0.6 0.8 1 0 0.5 1 1.5 2 FFT DISPLACEMENT FREQUENCY (Hz) (c)Timehistoryofsensor16 (d)Frequencyspectrumofsensor16 0 20 40 60 80 100 120 140 −10 −5 0 5 10 DISPLACEMENT (cm) RELATIVE TIME (SEC) 0 0.2 0.4 0.6 0.8 1 0 0.5 1 1.5 2 FFT DISPLACEMENT FREQUENCY (Hz) (e)Timehistoryofthedifference (f)Frequencyspectrumofthedifference ofsensors15and16 ofsensors15and16 Figure6.12: Theverticalandtorsionaldisplacementsatthecenterofthebridgedeck(sensors15 and16). Thetorsionaldisplacementwasobtainedwiththesubtractionbetweenthetimehistories ofsensors15and16. filtered signal. An example of the damping ratio estimation for the torsional displacement is shown in Figure 6.13. The damping ratios based on sensors 15 and 16 for peaks and valleys were identified at 4.8% (peak) and 6.9% (valley) for the natural frequency of 0.147 Hz, and 0.5% (peak) and 0.6% (valley) for the natural frequency of 0.537 Hz. The damping ratios for sensors 17 and 18 were identified at 5.0% (peak) and 7.3% (valley) for the natural frequency of 165 0 20 40 60 80 100 120 140 −3 −2 −1 0 1 2 3 TIME (SEC) DISPLACEMENT (cm) 0 20 40 60 80 100 120 140 −3 −2 −1 0 1 2 3 TIME (SEC) DISPLACEMENT (cm) (a)Cutofffrequencies: 0.13∼0.2Hz (b)Cutofffrequencies: 0.5∼0.6Hz Figure 6.13: The estimation of damping ratios for torsional displacement. The damping ratios wereestimatedwiththebandpass-filteredsignalofthetorsionaldisplacementillustratedinFig- ure6.12(e). Table6.5: Summaryofestimatedlocaldampingratiosofthebridgedeck. Averageddampingratio(%) Direction SensorNo. Locations peaks valleys average Frequency(Hz) 3 center 6.4 4.8 5.6 0.138 Lateral 5 eastquarter 6.4 4.6 5.5 0.142 – – – 0.232 15 center – – – 0.537 Vertical – – – 0.232 16 eastquarter – – – 0.537 – – – 0.720 4.8 6.9 5.8 0.147 4(15−16) center 0.5 0.6 0.6 0.537 Torsional 5.0 7.3 6.1 0.147 4(17−18) eastquarter 0.5 0.7 0.6 0.537 1.0 1.0 1.0 0.717 0.147Hz,0.5%(peak)and0.7%(valley)forthenaturalfrequencyof0.537Hz,and1.0%(peak) and 1.0% (valley) for the natural frequency of 0.717 Hz. The identified damping ratios of the bridgedeckaresummarizedinTable6.5. Phaseoftwodifferentsensorreadings The cross-correlation of bridge displacements was measured to determine the phase lag of two 166 Table6.6: Timelagsanddominantfrequenciesofcross-correlationfordifferentsensorreadings. Direction Sensorno. Timelag(sec) Dominantfrequency(Hz) Lateral 3and4 0 0.147 Vertical 15and16 approx. 8 0.232and0.537 differentsensorreadings. Figure 6.14(a)illustratesthecross-correlation ofthelateraldisplace- ments at the main span of the bridge deck (sensors 3 and 5). The cross-correlation shows that the time lag between sensors 15 and 16 is zero, which implies the oscillation phases of the sensors are identical. The frequency spectrum of the cross-correlation shows that the dominant frequency is placed at 0.147 Hz, which is almost identical to the identified natural frequencies ofthelateraldisplacements,0.138Hzforsensor3and0.142Hzforsensor5showninTable6.5 (Figure 6.14 (b)). The cross-correlation of the vertical displacements, sensors 15 and 16, was also determined, and its time lag was measured at approximately 8 seconds; that is, the period of relative vertical displacement between sensors 15 and 16. The dominant frequencies of the cross-correlationspectrumwereobservedat0.232Hzand0.537Hz. Thetimelagsanddominant frequenciesofthecross-correlationaresummarizedinTable6.6. 6.5.3 ComparisonofGlobalandLocalIdentificationResults Oncethelocalidentificationwasperformed,theresultsofthelocalidentificationwerecompared with those of the global identification for validation purpose. A comparison of the global and localidentificationresultsisshowninTable6.7. Asdepictedinthetable,onlyminordiscrepan- cies were observed between the global and local identification results for Modes A through C. Notice that the same natural frequency and damping ratio of Mode A, a strong lateral motion, were observed from the torsional displacements of sensors 15 and 16, and sensors 17 and 18, 167 −50 −40 −30 −20 −10 0 10 20 30 40 50 −1.2 −1 −0.8 −0.6 −0.4 −0.2 0 0.2 0.4 0.6 0.8 1 1.2 TIME LAG (SEC) CROSS CORRELATION 0 0.2 0.4 0.6 0.8 1 0 0.02 0.04 0.06 0.08 0.1 FREQUENCY (Hz) CROSS CORRELATION FFT (a)Cross-correlationoflateraldisplacement (b)FFTofcross-correlationoflateraldisplacement −50 −40 −30 −20 −10 0 10 20 30 40 50 −1.2 −1 −0.8 −0.6 −0.4 −0.2 0 0.2 0.4 0.6 0.8 1 1.2 TIME LAG (SEC) CROSS CORRELATION 0 0.2 0.4 0.6 0.8 1 0 0.02 0.04 0.06 0.08 0.1 FREQUENCY (Hz) CROSS CORRELATION FFT (c)Cross-correlationofverticaldisplacement (d)FFTofcross-correlationofverticaldisplacement Figure6.14: Cross-correlationanditsfrequencyspectrumforthelateraldisplacements(sensors 3and5)andverticaldisplacements(sensors15and16)ofthebridgedeck. which are the differences of those two-vertical displacements. Figure 6.15 shows a top view and lateral view of Mode A identified with the global identification method. Although a lateral motion is dominant in this mode (Figure 6.15 (a)), there also exists a minor torsional motion (Figure6.15(b)). Thus,thenaturalfrequencyanddampingratioofModeAwerealsoobserved inthetorsionaldisplacementcalculatedfromsensors15and16,andfromsensors17and18. (a)Topview (b)Lateralview Figure6.15: TopandlateralviewsofModeAidentifiedwiththeglobalidentificationmethods. 168 Modes D and E, which are relatively higher modes identified with the global identification method, were not successfully observed with the local identification. It should be noticed that usingtheglobalidentificationmethod,thesehighermodeswereaccuratelydeterminedwiththe ambientexcitationaswellastheimpactexcitationduringtheincident. Usingthelocalidentificationmethod,thenaturalfrequencyof0.717Hzanditscorrespond- ing damping ratio of 1.0% were identified with the torsional displacement calculated from sen- sors 17 and 18. This is approximately the quarter-point of the bridge main span. The same natural frequency and damping ratio were not observed with the torsional displacement calcu- latedfromsensors15and16,locatedatthecenterofthemainspan. Thisisasymmetrictorsional mode, and so the center of the main span is an antinode for the mode shape. Using the global identificationmethod,thismodewasdetectable. However,thecorrespondingmodeshapecould notbedeterminedduetoalowsensordensity. 6.6 SummaryandConclusions Thischapterreportsonastudyoftheanalysisofmulti-channeltime-historyaccelerationrecords captured by the digital instrumentation network installed on the Vincent Thomas Bridge, near thePortofLosAngeles,California,andcausedbyanaccidentthatoccurredon27August2006 betweenalargecargoshipandthebridge. Relatively long time history records of the bridge oscillations before, during, and after the accident, were used to analyze its nearly stationary response by applying multi-sensor system identificationapproaches,utilizingtheNaturalExcitationTechniquewiththeEigensystemReal- ization Algorithm. Modal parameter estimates for the bridge based on analysis of single-sensor 169 Table 6.7: A comparison of natural frequencies and damping ratios identified with global and localidentificationmethods. Global(Multi-Sensor)Identification Local(Single-Sensor)Identification Comparison Modeno. Modeshape f G (Hz) ζ G (%) sensorno. f L (Hz) ζ L (%) f L f G ζ L ζ G 3 0.138 5.6 0.92 1.40 5 0.142 5.5 0.95 1.38 Δ(15−16) 0.147 5.8 0.98 1.45 Δ(17−18) 0.147 6.1 0.98 1.53 A 0.150 4.0 xcorr(3,4) 0.147 – 0.98 – 15 0.232 – 1.00 – 16 0.232 – 1.00 – xcorr(15,16) 0.232 – 1.00 – B 0.233 4.0 16 0.537 – – Δ(15−16) 0.537 0.6 – Δ(17−18) 0.537 0.6 – xcorr(15,16) 0.537 – – C 0.536 0.8 – – – – – D 1.394 1.5 – – – – – E 1.869 1.3 Δ(17−18) 0.712 1.0 – – 16 0.720 – – – – – – – measurementsatselectedlocationswerealsousedtodemonstratetherangeofvalidityofcrude estimates of selected modal parameters when drastic reduction in the identified model-order is used. Byutilizingaweb-enabledstructuralhealthmonitoringsystemthatisinstalledonthebridge, it is shown that analysis of the acquired sensor measurements, using various levels of sophisti- cation in the digital signal processing of the captured data, can provide the owners of critical infrastructure systems with forensic tools that enable reliable and rapid assessment to analyze thecircumstancesandconsequencesofextremeeventstowhichthetargetsystemissubjected. 170 The power of the results reported in this chapter is that it provides maintenance engineers with the ability to quickly determine the need for, or order of, visual inspection required after an event, such as an earthquake. Thus, assuming a number of large structures are appropriately instrumented,maintenanceinspectionengineersareabletoreviewthedamagepotentialateach location and schedule visual inspections, or investigations utilizing more sophisticated means, accordingly. 171 Chapter7 SummaryandConclusion The objective of this study was to develop effective modeling and monitoring methodologies for assessing the “health” of uncertain, nonlinear, dynamic systems. The SHM methodology proposed in this study is more advantageous than existing methodologies with the following three aspects: (1) its feasibility to detect (small) changes in complex nonlinear systems, (2) the possibilitytomakephysicalinterpretationofdetectedchanges,and(3)thepossibilitytoquantify the uncertainty of change detection, which is usually influenced by various uncertainty sources. A series of investigations was performed by gradually introducing the complexities of various problemsinmodelingandmonitoringuncertainnonlinearsystemsinalogicalfashion. Fromthe investigationsreportedinthisthesis,thefollowingimportantfactscanbeconcluded: Comparisonofmodelingapproachesforfull-scalenonlinearviscousdamper Oneparametric(simplifieddesignmodel)andtwonon-parametric(RestoringForceMethodand artificialneuralnetworks)identificationmethodswerecomparedusingafull-scalenonlinearvis- cousdamperofthetypethatisfrequentlyemployedtomitigateseismicandwind-inducedvibra- tion in civil structures. A series of experimental studies was conducted on the viscous damper. The viscous damper was successfully identified with the parametric as well as non-parametric identification methods. Among the modeling approaches investigated in this study, the Restor- ing Force Method was more advantageous than other methods for monitoring purposes due to 172 the following aspects: (1) no a priori knowledge of the system being monitored is required; (2) the same model can be used when the system evolves into different types of nonlinearity; (3) themethodisapplicabletoawiderangeofnonlinearities;(4)bothChebyshevandpowerseries coefficientscanbeidentified;and(5)physicalinterpretationofsomeoftheidentificationresults ispossiblewithidentifiedcoefficients. Data-drivenmethodologiesforchangedetectioninlarge-scalenonlineardamperswithnoisy measurements Therearetwotypesofuncertaintiesaffectingmodelingandmonitoringresultsofuncertainnon- linearsystems: (1)measurementuncertainty(ormeasurementnoise),and(2)systemcharacteris- ticuncertainty(orvariationofsystemparameters). Amongthem,theeffectsofthemeasurement uncertaintyonthechangedetectionperformancewerefirstlyinvestigated. An experimental study was conducted using three different types of large-scale nonlinear viscousdampers,andmultiplesetsoftesteddampers’responsepollutedwithrandomnoisewere produced to investigate the stochastic effects of the measurement noise on the change detection performance. Using the Restoring Force Method, the viscous dampers were identified with the noisymeasurements,andthecorrespondingcoefficientswereobtained. It was found that the coefficients identified using the Restoring Force Method can be used as excellent features for (1) detecting the changes of nonlinear systems, (2) interpreting the physical meaning of the detected changes, and (3) quantifying the uncertainty of the detected systemchanges. TheBootstrapmethodwasalsostudiedtoestimatetheuncertaintyboundsonthecoefficient identification,whenthemeasurementdataareinsufficientforreliablestatisticalinference. Using 173 the Bootstrap method, the uncertainty in the identification was estimated reasonably accurately even with a single data set when the displacement and force were measured, rather than when theaccelerationandforceweremeasuredwithrandomnoise. Model-orderreductioneffectsonchangedetectioninuncertainnonlinearmagneto-rheological dampers Once the effects of measurement uncertainty were understood, the effects of system character- istic uncertainty were investigated. In order to study the system characteristic uncertainty, a semi-active magneto-rheological (MR) damper was employed. Multiple sets of the damper’s response were obtained for Gaussian distributions of MR damper input currents with different means and standard deviations. Here, the mean of the distribution determines the effective sys- tem characteristics and the standard deviation of distribution determines the uncertainty of the systemcharacteristics. A series of experimental studies was performed with the MR damper, and the MR damper was identified using the Restoring Force Method. Using the distributions of the corresponding identifiedcoefficients,itwasdemonstratedthatthedevelopedchangedetectionmethodologycan successfully assess the conditions of uncertain nonlinear systems by (1) detecting the effective (nominal)systemchangeswiththemeanchangesofthecoefficientdistributions,(2)quantifying the uncertainty bounds of the detected changes with the standard deviation of the coefficient distributions, and (3) interpreting the physical meaning of the detected changes without a priori knowledgeofthesystemcharacteristics. Supervised and unsupervised statistical classification methods were applied to detect effec- tive system changes with different levels of system uncertainty. Among the three coefficients 174 available in the Restoring Force Method (the normalized Chebyshev coefficients, normalized power series coefficients, and de-normalized power series coefficients) the normalized Cheby- shev coefficients with orthogonal basis functions demonstrated many advantageous aspects for change detection purposes, due to the statistical unbiasness of the identified coefficients, which was not observed with non-orthogonal power series coefficients identified with non-orthogonal basisfunctions. Inaddition,thechangedetectionwithstatisticallyunbiasedcoefficientsshowed higheraccuracyandlesscomputationtimewithreduced-ordermodels. MonitoringthecollisionofacargoshipwiththeVincentThomasBridge Once various important effects of uncertain nonlinear systems for the development of the component-level structural health monitoring were investigated, the scope of this study was expandedtothefull-system-levelstructuralhealthmonitoring. On 27 August 2006, the Vincent Thomas Bridge, an important suspension bridge in south- ern California, had a collision with a cargo ship. An investigation was performed on the multi- channeltime-historyaccelerationrecordscapturedbytheweb-baseddigitalinstrumentationnet- workinstalledonthebridge. Relatively long time history records of the bridge oscillation before, during and after the accident, were used to analyze its nearly stationary response by applying multi-sensor system identificationapproaches,utilizingtheNaturalExcitationTechniquewiththeEigensystemReal- ization Algorithm. Modal parameter estimates for the bridge based on analysis of single-sensor measurementsatselectedlocationswerealsousedtodemonstratetherangeofvalidityofcrude estimates of selected modal parameters, when drastic reduction in the identified model-order is used. 175 Byutilizingaweb-enabledstructuralhealthmonitoringsystemthatisinstalledonthebridge, it is shown that analysis of the acquired sensor measurements, using various levels of sophisti- cation in the digital signal processing of the captured data, can provide the owners of critical infrastructure systems with forensic tools that enable reliable and rapid assessment to analyze thecircumstancesandconsequencesofextremeeventstowhichthetargetsystemwassubjected. The power of this study is that it allows maintenance engineers the ability to quickly deter- mine the need for, or order of, visual inspection required after an event, such as an earth- quake. 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Abstract (if available)

Abstract The development of effective structural health monitoring (SHM) methodologies is imperative for the efficient maintenance of important structures in aerospace, mechanical and civil engineering. Based on reliable condition assessment, the owners of monitored structures can expect two important benefits: to avoid catastrophic accidents by detecting various types of structural deterioration during operation, and to establish efficient maintenance means and time schedule to reduce maintenance costs.

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Asset Metadata

Creator Yun, Hae-Bum (author)

Core Title Analytical and experimental studies of modeling and monitoring uncertain nonlinear systems

School Viterbi School of Engineering

Degree Doctor of Philosophy

Degree Program Civil Engineering (Structural Mechanics)

Publication Date 07/24/2007

Defense Date 06/22/2007

Publisher University of Southern California (original), University of Southern California. Libraries (digital)

Tag artificial neural networks,Bootstrap Method,detection theory,Eigensystem Realization Algorithm,error analysis,full-scale viscous dampers,Hypothesis Test,k-means clustering,magneto-rheological dampers,Natural Excitation Technique,OAI-PMH Harvest,Restoring Force Method,ship-bridge collision,statistical pattern recognition,Structural health monitoring,support vector machines,suspension bridge,system identification,web-based real-time bridge monitoring system

Language English

Advisor Sami F. Masri (committee chair), Carter Wellford (committee member), Jiin-Jen Lee (committee member), John P. Caffrey (committee member), Roger G. Ghanem (committee member)

Creator Email haebum@usc.edu

Permanent Link (DOI) https://doi.org/10.25549/usctheses-m648

Unique identifier UC1128958

Identifier etd-Yun-20070724 (filename),usctheses-m40 (legacy collection record id),usctheses-c127-528810 (legacy record id),usctheses-m648 (legacy record id)

Legacy Identifier etd-Yun-20070724.pdf

Dmrecord 528810

Document Type Dissertation

Rights Yun, Hae-Bum

Type texts

Source University of Southern California (contributing entity), University of Southern California Dissertations and Theses (collection)

Repository Name Libraries, University of Southern California

Repository Location Los Angeles, California

Repository Email cisadmin@lib.usc.edu