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Analytical and experimental studies in system identification and modeling for structural control and health monitoring
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Analytical and experimental studies in system identification and modeling for structural control and health monitoring
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ANALYTICALANDEXPERIMENTALSTUDIESINSYSTEMIDENTIFICATIONAND MODELINGFORSTRUCTURALCONTROLANDHEALTHMONITORING by MohammadRezaDehghanNayeri ADissertationPresentedtothe FACULTYOFTHEGRADUATESCHOOL UNIVERSITYOFSOUTHERNCALIFORNIA InPartialFulfillmentofthe RequirementsfortheDegree DOCTOROFPHILOSOPHY (MECHANICALENGINEERING) August2007 Copyright 2007 MohammadRezaDehghanNayeri Dedication tomyMother,Maryam,myFatherHossein,andmywifeMojgan ii Acknowledgments I would like to express my deep and sincere gratitude to my advisor, Professor Sami Masri for hisinvaluableguidance,support,andencouragement. Icouldnothaveimaginedhavingabetter advisor and mentor for my PhD. His breath of knowledge and expertise have been invaluable resourcesformeinchoosingthedirectionofmywork. Thisworkwouldnothavebeenpossible withouthiscommon-sense,perceptiveness,andinterdisciplinaryknowledge. I would also like to express my gratitude to Dr. John Caffrey, for helping out with all the experimental set-ups, and his guidance throughout the various parts of this thesis. His deep knowledge, enthusiasticinterest, andincredible patiencearesincerelyappreciated. Iwouldlike tothankmycolleaguesinourresearchgroup,especiallyFarzadTasbihgoo,Hae-Bum(Andrew) Yun,andMiguelG.Hernandezfortheirfriendship,andhoursofusefuldiscussions. Iwouldliketothankmybelovedparents, Maryamand Hossein,fordedicatingtheirlifefor meandcreatinganenvironmentinwhichfollowingthispathseemedsonatural. Andlastbutnot least, I wish to express my love and gratitude to my wife, Mojgan, for the very special person sheis,andfortheincredibleamountofpatienceshehadwithmeinthelastfouryears. Finally,IacknowledgethepartialsupportoftheAirForceOfficeofScientificResearch,the NationalScienceFoundation,andtheNationalAeronauticsandSpaceAdministration. iii TableofContents Dedication ii Acknowledgments iii ListofTables vi ListofFigures viii Abstract xvii Chapter1: Introduction 1 1.1 BackgroundandMotivation . . . . . . . . . . . . . . . . . . . . . . . . . . . 1 1.2 Scope . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4 Chapter 2: A Study of Time-Domain Techniques for Modal Parameter Identification ofaLongSuspensionBridgewithDenseSensorArrays 8 2.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8 2.2 NewCarquinezBridge: CharacteristicsandTestDescription . . . . . . . . . . 11 2.3 FormulationofTheTime-DomainModalParameterIdentificationTechniques . 14 2.3.1 NExtTechnique . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 14 2.3.2 ERAandERA/DCMethods . . . . . . . . . . . . . . . . . . . . . . . 17 2.3.3 LSMethod . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 23 2.3.4 AutonomousSelectionoftheFinalSetofModalParametersUsingMode CondensationAlgorithm . . . . . . . . . . . . . . . . . . . . . . . . . 26 2.4 ImplementationandResults . . . . . . . . . . . . . . . . . . . . . . . . . . . 31 2.4.1 InfluenceofTheDataWindowSizeandTheSizeofTheSVDmatrix . 33 2.4.2 ResultsandComparison . . . . . . . . . . . . . . . . . . . . . . . . . 37 2.5 SummaryandConclusions . . . . . . . . . . . . . . . . . . . . . . . . . . . . 45 Chapter 3: Application of Structural Health Monitoring Techniques to Track Struc- turalChangesinaRetrofittedBuildingBasedonAmbientVibration 46 3.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 46 3.2 BuildingCharacteristicsanditsInstrumentation . . . . . . . . . . . . . . . . . 50 3.3 FiniteElementAnalysisOfBuildingModel . . . . . . . . . . . . . . . . . . . 51 3.3.1 Pre-retrofitmodel . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 51 iv 3.3.2 Post-retrofitmodel . . . . . . . . . . . . . . . . . . . . . . . . . . . . 58 3.4 RetrofitPhase . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 60 3.5 InstrumentationandDataAcquisition . . . . . . . . . . . . . . . . . . . . . . 61 3.6 ImplementationoftheModalParameterIdentificationandItsResults . . . . . 66 3.7 SummaryandConclusions . . . . . . . . . . . . . . . . . . . . . . . . . . . . 79 Chapter 4: Structural Identification and Monitoring of a Full-Scale 17-Story Building BasedonAmbientVibrationMeasurements 81 4.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 81 4.2 ModalParameterIdentificationApproach . . . . . . . . . . . . . . . . . . . . 85 4.3 FormulationofTheChainSystemIdentificationApproach . . . . . . . . . . . 85 4.3.1 GeneralNonlinearCase . . . . . . . . . . . . . . . . . . . . . . . . . 86 4.3.2 TheLinearCasewithUnknownForce . . . . . . . . . . . . . . . . . . 90 4.4 IllustrativeExample . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 93 4.4.1 DescriptionoftheExampleandDamageScenario . . . . . . . . . . . 93 4.4.2 ModalIdentificationusingNExT/ERA . . . . . . . . . . . . . . . . . 95 4.4.3 DamageAssessmentandlocalizationusingChainSystemIdentification 99 4.5 UCLAFactorBuilding: DescriptionandInstrumentation . . . . . . . . . . . . 109 4.6 FactorBuildingIdentificationResults . . . . . . . . . . . . . . . . . . . . . . 111 4.6.1 ModalParameterIdentificationResults . . . . . . . . . . . . . . . . . 112 4.6.2 ChainSystemIdentificationResults . . . . . . . . . . . . . . . . . . . 127 4.7 Discussion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 137 4.8 SummaryandConclusions . . . . . . . . . . . . . . . . . . . . . . . . . . . . 151 Chapter 5: Optimum Strategies for Deploying Passive and Semi-active Multiple-Unit ImpactDampersunderStochasticExcitation 155 5.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 155 5.2 MathematicalModel . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 159 5.3 StationaryRandomExcitationofMulti-UnitImpactDampers . . . . . . . . . . 161 5.4 NonstationaryRandomExcitationofMulti-UnitImpactDampers . . . . . . . 168 5.5 StationaryandNonstationaryRandomExcitationofSemi-ActiveImpactDampers171 5.6 SummaryandConclusions . . . . . . . . . . . . . . . . . . . . . . . . . . . . 181 Chapter6: Conclusions 183 Bibliography 190 v ListofTables 2.1 ComparisonoftheNCBidentificationresultsusingNExT/ERAandNExT/ERA- DC methods. For all cases: window size and overlap=409.6 sec and 75%, respectively, and all available DOFs are used as the reference. For ERA/DC: r = 0,α = 10,β = 70, andp = 2/3 of the available data points. For ERA: r = 25, and p = 2/3 of the available data points, (MPC is the modal phase collinearityandCMIistheconsistentmodeindicator). . . . . . . . . . . . . . 38 2.2 Comparison of the NCB identification results using NExT/LS and NExT/ERA- DCmethods. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 39 3.1 NaturalfrequenciesinHzoftheFEmodelbeforeandafterretrofit. . . . . . . . 59 3.2 RecordDatesandtheCorrespondingIndexNumber. . . . . . . . . . . . . . . 67 4.1 Definition of the symbols used in Fig. 4.3-4.4, and their corresponding values for the firstmode. Superscript(*)indicatestheexactparameter,andsubscripts1and2indicate themodalparametersforreference(undamaged)anddamagecases,respectively. . . . 97 4.2 Values corresponding to the statistical parameters shown in Table 4.1 for the first five modes. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 99 4.3 Statistical mean (μ), dispersion (CV), identification error (e μ ), and detectability ratios (Δμ/μ 1 and Δμ/σ 1 ) corresponding to the distributions of the mass-normalized coeffi- cientofdisplacement(k i /m i ),andvelocity(c i /m i )showninFig. 4.6. Fordefinition ofthesymbolsrefertoTable4.1. Noticethatthefirstdamagelocationisbetweensensor stations2and3;consequentlythecorrespondingdamageeffectsshouldonlybereflected inthepropertiesoftherestoringforcefunctionG 3 (seeFig. 4.6(a 2 )andFig. 4.7) . . . 100 4.4 Effectofthesensorresolution(model-orderreduction)ontheabsoluteerrorandrelative changeoftheestimatedparametersbythechainidentificationmethod. Theseresultsare for the estimatedk eq /m at sensor station 3, which is the closest to the the first damage location. Parameters μ and σ are the mean and standard deviation of the estimated k eq /m at each measurement station, respectively. Superscript (*) indicates the exact parameter,andsubscripts1and2denotetheparametersfortheno-damageanddamage cases,respectively. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 103 vi 4.5 Effectofthenoiseontheabsoluteerrorandrelativechangeoftheestimatedparameters by the chain identification method. These results are for the estimatedk eq /m at sensor station3,whichistheclosesttothethefirstdamagelocation. Themodel-orderreduction ratio is assumed to be Δn n = 1/20. Parameters μ and σ are the mean and standard deviationoftheestimatedk eq /mateachmeasurementstation,respectively. Superscript (*) indicates the exact parameter, and subscripts 1 and 2 denote the parameters for the no-damageanddamagecases,respectively. . . . . . . . . . . . . . . . . . . . . . 103 4.6 Comparisonbetweentheestimatedfrequencyfornight(10PM-2AM)andday(10AM- 2 PM), based on the distributions shown in Fig. 4.30-4.34. In this table, subscripts 1 and 2 correspond to the distributions for night: 10 PM-2 AM and day: 10 AM-2 PM, respectively. The parametersμ 1 andμ 2 represent the mean of frequency, for night(10 PM-2 AM) and day (10 AM-2 PM), respectively; Δμ/μ 1 is 100(μ 2 −μ 1 )/μ 1 , and represent the percentage change of the mean frequency normalized by mean frequency for the night; Δμ/μ 1 is 100(μ 2 −μ 1 )/μ 1 , and represent the percentage change of the meanfrequencynormalizedbythestandarddeviationoffrequency;andtheCVdenotes theCoefficientofVariation. . . . . . . . . . . . . . . . . . . . . . . . . . . . . 146 5.1 Summaryofthenonstationarysimulationresults. . . . . . . . . . . . . . . . . . . 170 vii ListofFigures 2.1 (a)OverviewoftheNewCarquinezbridge,and(b)itsoveralldimensions. . . . 12 2.2 Schematic plot for the accelerometer locations and directions on the New Car- quinezbridge(Conteetal.,2006). . . . . . . . . . . . . . . . . . . . . . . . . 13 2.3 Flowchart of the main steps for the mode condensation algorithm (Pappa et al., 1998). . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 28 2.4 Effect of the window size on the NCB identification results using NExT/ERA method. For all cases: window overlap=75%, r = 24, p = 2/3 of the cross- correlation data points, andT 1 = 6 sec. All available DOFs, one-at-a-time, are usedasthereference. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 34 2.5 Effectofthenumberofblockrows(r+1)intheHankelmatrix(H)ontheNCB identification results using the NExT/ERA method. The value of (r +1) is an indication of the initial model order. For all cases: window size and overlap≈ 410 sec and 75%, respectively, and p = 2/3 of the available data points. All availableDOFsaresequentiallyusedasthereference. . . . . . . . . . . . . . . 35 2.6 Effectofthenumberofblockrows(α+1)inthecorrelationHankelmatrix(H) on the NCB identification results using the NExT/ERA-DC method. The value of(α+1)isanindicationoftheinitialmodelorder. Forallcases: windowsize andoverlap≈ 410secand75%,respectively,r = 0,β = 70,andp = 2/3ofthe availabledatapoints. AllavailableDOFsaresequentiallyusedasthereference. 40 2.7 Comparison of the NCB identified mode shapes using Modal Assurance Cri- terion (MAC): (a) comparison between NExT/ERA and NExT/ERA-DC mode shapes, (b) comparison between NExT/LS and NExT/ERA-DC mode shapes. The size of each rectangle is proportional to the MAC value of the correspond- ing mode pairs. For MAC values greater than 0.7, the corresponding rectangles aredarkened,whichindicatehighcorrelation. . . . . . . . . . . . . . . . . . . 41 2.8 Identified mode shapes and their frequency,f n (NExT/ERA-DC results). Since the mode shapes are typically complex, they are also plotted in the polar plane (figures on the left), where each arrow in the polar plane represents a complex componentofthemodeshapevector. . . . . . . . . . . . . . . . . . . . . . . . 42 viii 2.9 Identifiedmodeshapes(Cont.) . . . . . . . . . . . . . . . . . . . . . . . . . . 43 2.10 Identifiedmodeshapes(Cont.) . . . . . . . . . . . . . . . . . . . . . . . . . . 44 3.1 FiniteElementModelofPre-retrofitLongBeachPublicSafetyBuilding. . . . 53 3.2 GroundaccelerationtimehistoryofNorthridgeearthquake. . . . . . . . . . . . 54 3.3 Critical beams in pre-retrofit building due to the rocking of discontinuous con- cretewalls. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 55 3.4 Criticalbeamsinpre-retrofitbuildingdominatedbylargebendingmoments. . . 55 3.5 Finiteelementmodelofpost-retrofitLongBeachPublicSafetyBuilding. . . . 59 3.6 The1 st modeshape. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 60 3.7 The2 nd modeshape. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 61 3.8 The3 rd modeshape. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 62 3.9 The4 th modeshape. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 62 3.10 ScheduleofShearWallCompletion. . . . . . . . . . . . . . . . . . . . . . . . 63 3.11 Retrofitphaseofthebuilding. . . . . . . . . . . . . . . . . . . . . . . . . . . 63 3.12 Preparationofnewshearwall. . . . . . . . . . . . . . . . . . . . . . . . . . . 64 3.13 SchematicplotofthesensorlocationsinLBPSB. . . . . . . . . . . . . . . . . 64 3.14 Instrumentationlist. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 65 3.15 Instrumentationsystemarchitecture. . . . . . . . . . . . . . . . . . . . . . . . 65 3.16 Identified natural frequencies of one data set (02-Jul-2003) using ERA/DC, for differentchoicesofreferenceDOF.Theotheridentificationparametersareexactly thesame: windowsize=2048points,overlap=50%,r = 0,p = 665,α = 5,and β = 90. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 68 3.17 IdentifiedmodeshapescorrespondingtothefrequenciesinFig. 3.16. . . . . . 69 3.18 Typical semilogarithmic plots of the CPSD between the response of a reference DOF and the response of all available DOFs. For this case, the reference DOF is X 12 , which corresponds to they axis of the sensor on the roof. The window size is 2048 points, and the overlap is 50%. For ease of comparison, identical abscissaandordinatescalesareusedforalldisplayedplots. . . . . . . . . . . . 70 ix 3.19 The CCF corresponding to the CPSD shown in Fig. 3.18. The CCF is equiva- lent to the inverse Fourier transform of CPSD. Note that, for added resolution, differentordinatescalesareusedforeachdisplayedplots. . . . . . . . . . . . . 71 3.20 Timevariationofthenaturalfrequency,dampingratio,andMACvaluechanges for the (a) 1 st mode, and (b) 2 nd mode. The date index corresponds to data sets inTable3.2. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 72 3.21 Timevariationofthenaturalfrequency,dampingratio,andMACvaluechanges for the (a) 3 rd mode, and (b) 4 th mode. The date index corresponds to data sets inTable3.2. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 73 3.22 1 st mode shape in the complex plane and its projection onto thex andy axes. Theplotshowstheresultsforthreedifferentdatasets;30Jun2003,19Sep2003, and5Jan2004. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 75 3.23 2 nd mode shape in the complex plane and its projection onto thex andy axes. Theplotshowstheresultsforthreedifferentdatasets;30Jun2003,19Sep2003, and5Jan2004. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 77 3.24 3 rd mode shape in the complex plane and its projection onto thex andy axes. Theplotshowstheresultsforthreedifferentdatasets;30Jun2003,19Sep2003, and5Jan2004. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 78 3.25 4 th mode shape in the complex plane and its projection onto thex andy axes. Theplotshowstheresultsforthreedifferentdatasets;30Jun2003,19Sep2003, and5Jan2004. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 79 4.1 ModelofaMDOFchain-likesystem. . . . . . . . . . . . . . . . . . . . . . . . . 87 4.2 Schematicplotofthe100DOFlinearchain-likeexampleunderdiscussion. Thereare20 sensorstations,hence Δn n = 1 20 . Damageisintroducedattwolocationsbyreductionof stiffnessoftwodistinctelements(outof100)by10%,whichresultsin2.17%reduction ofequivalentstiffnessbetweentwoconsecutivemeasurementstations. . . . . . . . . 94 x 4.3 Modalparameteridentificationresultsforthechain-likeexampleunderdiscussion,using the NExT/ERA method. In order to quantify the uncertainty in the estimated parame- ters, the statistical averaging was conducted over 100 ensembles, each includes at least 2000 periods of the first fundamental mode of the system. In each figure, dashed and solid lines represent the identification results for reference (undamaged) and damage cases,respectively. Theshortupwardarrowrepresentstheexactparametervalue.μand σ are the mean and standard deviation in each case, and CV denotes the Coefficient of Variation. The ratio Δn n is 1 20 in this case. Table 4.1 summarizes the definition of the symbols used in this figure. (a) projection of the mode shape on the sensor stations for thereferenceanddamagecases(solidanddashedlines,respectively),(b)Histogramand estimatedprobabilitydensityfunctionfortheidentifiedmodalfrequencycorresponding tothereferencestructureandthedamagedone(solidanddashedlines,respectively),and (c)Histogramandestimatedprobabilitydensityfunctionfortheidentifiedmodaldamp- ing corresponding to the reference structure and the damaged one (solid and dashed lines, respectively). Values corresponding to the statistical parameters defined in Table 4.1canbefoundinTable4.2. . . . . . . . . . . . . . . . . . . . . . . . . . . . . 104 4.4 Modalparameteridentificationresultsforthechain-likeexampleunderdiscussion,using theNExT/ERAmethod. Cont. . . . . . . . . . . . . . . . . . . . . . . . . . . . 105 4.5 Comparison between the actual time-histories (Eq. (4.8)), and the cross-correlation time-histories (Eq. (4.10)), for measurement station 11 (Fig. 4.2). It can be seen that the contribution of the excitation force ( ¯ F i (t)) in the restoring force function ( ¯ G (i) ), is fairlysignificantcomparedtotheotherterms(¯ m i ¯ G (i+1) − ¨ x i ),whereasinR xr ¯ G (i),the contributionofthecross-correlationoftheexcitationforceandthedisplacementofaref- erenceDOF(R xr ¯ Fi ),isnegligiblecomparedtotheotherterms(¯ m i R xr ¯ G (i+1)−R xr¨ xi ). Measurement station 20 was used as the reference. For ease of comparison, identical ordinatescalesareusedforeachcase. . . . . . . . . . . . . . . . . . . . . . . . . 106 4.6 Chainsystemidentificationresultsforthechain-likeexampleunderdiscussion. Inorder toquantifytheuncertaintyintheestimatedparameters,thestatisticalaveragingwascon- ductedover100ensembles,eachincludesatleast2000periodsofthefirstfundamental mode of the system. In each figure, dashed and solid lines represent the identifica- tionresultsfortheno-damage(reference)anddamagecases,respectively,andtheshort upward arrow represents the exact parameter value. The abscissa for all the graphs are normalized to the corresponding values for the reference (no-damage) case in order to have identical abscissa scale. The ratio Δn n is 1 20 in this case. (a 1 ), (a 2 ), and (a 3 ) show the mass-normalized coefficient of displacement (k i /m i ) for the sensor stations 2, 3 and4,respectively. (b 1 ),(b 2 ),and(b 3 )showthemass-normalizedcoefficientof velocity (c i /m i )forthesensorstations2,3and4,respectively. Noticethatthefirstdamageloca- tionisbetweensensorstations2and3; consequentlythecorrespondingdamageeffects should only be reflected in the properties of the restoring force function G 3 (see Fig. 4.6(a 2 ) and Fig. 4.7). The corresponding statistical mean (μ), dispersion (CV), iden- tification error (e μ ), and detectability ratios (Δμ/μ 1 and Δμ/σ 1 ) are shown in Table 4.3. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 107 xi 4.7 Sensorstationsanddamagelocationsforthe100DOFchain-likeexampleunderdiscus- sion. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 108 4.8 (a)OverviewoftheFactorBuilding,(b)itsoveralldimensionsandsensorlocations/directions.110 4.9 Schematicplotofthesensorslayoutforeachfloorabovegrade. . . . . . . . . . . . 111 4.10 A typical ambient vibration records of the acceleration time-histories measured at the 14th floor of the Factor Building, and their corresponding velocity and displacement time-histories computed by digital signal processing. Column (a) corresponds to the measured accelerations; column (b) to the processed velocities; and column (c) to the processeddisplacements. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 114 4.11 Typical semi-logarithmic plots of the CPSDs and their associated CCFs for the Factor Building. For this case, the reference DOF is selected to be the x axis of the sensor on the 15th floor, the window size is 163.84 seconds, and the overlap is 50%. For ease of comparison, identical abscissa and ordinate scales are used for all displayed plots. (a)CPSD andR between acceleration of floors 5 and 15; (b)CPSD andR between accelerationoffloors10and15. . . . . . . . . . . . . . . . . . . . . . . . . . . 115 4.12 IdentifiedmodalparametersoftheFactorBuildingusingtheNExT/ERAmethod. These resultsareextractedfromatime-windowoftwohoursoftherecordedaccelerationdata. Figure (a) shows the projection of the first mode in the X, Y, andθ directions; Figure (b) shows the projection of the second mode in the X, Y, and θ directions. It is clear fromthedisplayedmodeshapesthatthefirstmodecorrespondstobendinginprimarily the X-direction, and that the second mode corresponds to bending in primarily the Y- direction. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 116 4.13 Identifiedmodalparameters. Cont. . . . . . . . . . . . . . . . . . . . . . . . . . 117 4.14 Identifiedmodalparameters. Cont. . . . . . . . . . . . . . . . . . . . . . . . . . 118 4.15 Identifiedmodalparameters,Cont. . . . . . . . . . . . . . . . . . . . . . . . . . 119 4.16 Identifiedmodalparameters,Cont. . . . . . . . . . . . . . . . . . . . . . . . . . 120 4.17 Identifiedmodalparameters,Cont. . . . . . . . . . . . . . . . . . . . . . . . . . 121 4.18 ProbabilitydensitiesoftheestimatedmodalfrequenciesfortheFactorBuilding. Atotal of 50 days of data (each 24 hours) were considered in this study. The modal parameter identificationwasconductedovertime-windowsof2hourseach,andwith50%overlap, foratotalnumberof1200statisticalensembles. Theμandσ arethemeanandstandard deviation in each case, and CV denotes the Coefficient of Variation. In each plot panel, a thin line indicates the outline of the histogram of the indicated parameter, and the solid line indicates the estimated Gaussian pdf having a matching mean and standard deviationasthecorrespondinghistogram. . . . . . . . . . . . . . . . . . . . . . . 123 4.19 ProbabilitydensitiesoftheestimatedmodalfrequenciesfortheFactorBuilding. Cont. 124 xii 4.20 Probability densities of the estimated modal damping for the Factor Building. A total of 50 days of data (each 24 hours) were considered in this study. The modal parameter identificationwasconductedovertime-windowintervalsof2hourseach,andwith50% overlap, for a total number of 1200 statistical ensembles. Note that (as seen from the plotted histograms) some of the estimated (identified) damping values had a negative magnitude. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 125 4.21 ProbabilitydensitiesoftheestimatedmodaldampingfortheFactorBuilding. Cont. . . 126 4.22 Representative phase and time-history plots of the restoring force functions associated with the 13th floor of the factor building, in x, y, and θ directions. The first column corresponds to the actual time-histories of the restoring forces by ignoring the force term (see Eq. (4.8)), whereas the second and third columns correspond to the cross- correlation time-histories of the restoring forces (see Eq. (4.12)). In the second and third columns, the solid and dashed lines represent the actual and estimated (identified) restoring force functions, respectively. MSE is the Mean-Squared-Error of the estima- tion. Lackofsmoothnessofsomeofthedisplayedcasesisduetodatapointdecimation, and not a true reflection of the physics of the underlying restoring forces. SymbolR xy standsforE[x(t)y(t+τ)];x r correspondstothereferenceDOFs(X,Y,andθdirections ofthe10thfloor);z =x 13 −x 12 . . . . . . . . . . . . . . . . . . . . . . . . . . . 129 4.23 Sample distributions of the estimated coefficient of displacement term in the interstory restoring functions. Coefficient of displacement is the mass-normalized stiffness term (k i /m i ). For the sake of compactness, only the results for the 2nd, 6th, 9th, and 16th floorsareshown. Otherfloorlocationshavesimilarresults. . . . . . . . . . . . . . 130 4.24 Sample distributions of the estimated coefficient of displacement term in the interstory restoringfunctions. Cont. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 131 4.25 Sampledistributionsoftheestimatedcoefficientofvelocitytermintheinterstoryrestor- ing functions. Coefficient of velocity is the mass-normalized damping term ( c i /m i ). For that sake of compactness, only the results for the 2nd, 6th, 9th, and 16th floors are presented. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 132 4.26 Sampledistributionsoftheestimatedcoefficientofvelocitytermintheinterstoryrestor- ingfunctions. Cont. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 134 4.27 Some representative modal parameters of the Factor Building which were estimated using the reconstructed global matrices,M −1 K andM −1 C. Elements of theM −1 K andM −1 C areestimatedusingtheidentifiedchainsystemlocalparameters(k i /m i and c i /m i )basedontheformulationspresentedinEq. (4.14)and(4.15). . . . . . . . . . 135 4.28 Some representative modal parameters of the Factor Building which were estimated using the reconstructed global matrices(M −1 K andM −1 C) based on the local chain identificationresults. Cont. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 136 xiii 4.29 Comparison between some representative global modes reconstructed from chain iden- tificationresults(solidlineswithcircles)andthemodesidentifiedusingtheNExT/ERA method (dashed lines with squares). MAC (Modal Assurance Criterion) is the measure ofcomparisonbetweenthemodeshapesidentifiedusingtheERAandtheChainmethod, and Δf/f ERA = (f ERA −f chain )/f ERA is the percentage of difference between the identifiednaturalfrequencies. . . . . . . . . . . . . . . . . . . . . . . . . . . . . 137 4.30 The first row shows the time-histories of Some representative modal frequencies of the FactorBuildingina24-hourperiod(oneday),inwhichthecorrespondingfrequenciesat eachhourarecomputedbyaveragingover50days. Thesecondandthirdrowsshowthe comparisonbetweentheestimatedfrequencydistributionsfornight(10PM-2AM)and day (10 AM-2 PM). In the captions, subscripts 1 and 2 correspond to the distributions fornight: 10PM-2AM(solidlines)andday: 10AM-2PM(dashedlines),respectively. The parameters μ 1 and μ 2 represent the mean of frequency, for night(10 PM-2 AM) and day (10 AM-2 PM), respectively; Δμ/μ 1 is 100(μ 2 −μ 1 )/μ 1 , and represent the percentage change of the mean frequency normalized by mean frequency for the night; Δμ/μ 1 is100(μ 2 −μ 1 )/μ 1 ,andrepresentthepercentagechangeofthemeanfrequency normalized by the standard deviation of frequency; and the CV denotes the Coefficient ofVariation. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 142 4.31 Modalfrequencytime-histories,Cont. . . . . . . . . . . . . . . . . . . . . . . . 143 4.32 Modalfrequencytime-histories,Cont. . . . . . . . . . . . . . . . . . . . . . . . 144 4.33 Modalfrequencytime-histories,Cont. . . . . . . . . . . . . . . . . . . . . . . . 145 4.34 Modalfrequencytime-histories,Cont. . . . . . . . . . . . . . . . . . . . . . . . 147 4.35 Temperature time-history in a 24-hour period in Los Angeles area. This time-history was computed by averaging the hourly temperature data over 50 days corresponding to the acceleration recoding dates. Hourly temperature data was obtained from National OceanicandAtmosphericAdministration,U.S.departmentofCommerce(http: //cdo.ncdc. noaa.gov /ulcd /ULCD). . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 148 4.36 Comparisonbetweenthetemperatureandfrequencyvariationsina24-hourperiod. For compactness reasons, the results are only plotted for four representative modes. Please note that the Y axis of these figures shows the normalized percentage of change, which is the percentage change divided by the maximum percentage change. For temperature variation, the profile is multiplied by -1 for improved clarity. It is observed that there is a time delay between frequency and temperature variations. In fact, the frequency variationslaggedbehindthetemperaturevariation. Thiscanbeexplainedbyheatdiffu- sionphenomenon. Thedisplayedtemperatureprofileisfoundusingtheairtemperature records and not using the actual structural materials’ temperature. In reality, it takes a whileformaterialstowarmuporcooldown. Itisinterestingtonoteherethatthedelay time for warm-up period (the first half of day) is around 6 hours, but the delay time for thecool-downperiod(thesecondhalfofday)isaround2hours. . . . . . . . . . . . 149 xiv 4.37 Distributions of the estimated coefficients of the displacement term in the interstory restoring functions. Coefficient of displacement is the mass-normalized stiffness term (k i /m i ). For the sake of compactness, only the results for the 2nd, 9th, and 16th floors are shown. The solid and dashed lines represent the estimated distributions for night (10 PM-2 AM) and day (10 AM-2 PM), respectively. In the captions, subscripts 1 and 2 correspond to the distributions for night: 10 PM-2 AM (solid lines) and day: 10 AM-2 PM (dashed lines), respectively. The parameters μ 1 andμ 2 represent the mean parameter value for night(10 PM-2 AM) and day (10 AM-2 PM), respectively; Δμ/μ 1 is100(μ 2 −μ 1 )/μ 1 ,andrepresentthepercentagechangeofthemeanvaluenormalized by the mean for the night; Δμ/μ 1 is 100(μ 2 −μ 1 )/μ 1 , and represent the percentage changeofthemeanvaluenormalizedbythestandarddeviation;andtheCVdenotesthe CoefficientofVariation. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 150 5.1 Modelofthemultipleunitimpactdamper . . . . . . . . . . . . . . . . . . . . . . 159 5.2 Nonlinearfunction(a)G(z k )and(b)H(z k , ˙ z k ) . . . . . . . . . . . . . . . . . . . 161 5.3 Dependenceofcoefficientofrestitutioneondampingparameterζ 2 . . . . . . . . . 162 5.4 RMS response levels for the primary system withe = 0.75,μ = 0.10, andζ = 0.01. (effectofnumberofparticles) . . . . . . . . . . . . . . . . . . . . . . . . . . . . 163 5.5 RMS response levels for the primary system withμ = 0.10,ζ = 0.01 and 100 particle units. (effectofcoefficientofrestitutione) . . . . . . . . . . . . . . . . . . . . . . 163 5.6 RMS response levels for the primary system withe = 0.75,ζ = 0.01, and 100 particle units. (effectofmassratioμ) . . . . . . . . . . . . . . . . . . . . . . . . . . . . 164 5.7 RMSresponselevelsfortheprimarysystemwithμ = 0.10,e = 0.75, and100particle units. (effectofprimarysystemdampingζ) . . . . . . . . . . . . . . . . . . . . . 164 5.8 Comparing the impact force level for (a) single unit impact damper and (b) MUID, whereeveryotherparameter(μ,e,ζ,andd)remainsthesame. . . . . . . . . . . . . 165 5.9 Effect of viscous damping and mass ratio on the performance of MUID with 100 parti- clesande = 0.75 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 166 5.10 The resulting RMS ratios ( σx σx 0 ) for the uniform distribution of gap clearances between d min andd max ,for50particleunits,e = 0.75,ζ = 0.01,andμ = 0.10,(a)3Dplot,(b) contourplot. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 167 5.11 Exponential envelope functiong 1 (t) = exp(−t)− exp(−1.5t) and the resulting non- stationaryrandomexcitation. . . . . . . . . . . . . . . . . . . . . . . . . . . . . 169 5.12 Transient RMS response of the primary system by averaging over 200 ensembles. Sys- tem parameters: μ = 0.10, ζ = 0.01, e = 0.75, d = d opt , and 100 particle units. Excitationenvelope:g(t) =g 1 (t). . . . . . . . . . . . . . . . . . . . . . . . . . . 169 xv 5.13 Exponential envelope functiong 2 (t) = exp(−0.2t)− exp(−1.5t) and the resulting nonstationaryrandomexcitation. . . . . . . . . . . . . . . . . . . . . . . . . . . 170 5.14 Transient RMS response of the primary system by averaging over 200 ensembles. Sys- tem parameters: μ = 0.10, ζ = 0.01, e = 0.75, d = d opt , and 100 particle units. Excitationenvelope:g(t) =g 2 (t). . . . . . . . . . . . . . . . . . . . . . . . . . . 170 5.15 Exponential envelope functiong 3 (t) = exp(−0.03t)− exp(−0.4t) and the resulting nonstationaryrandomexcitation. . . . . . . . . . . . . . . . . . . . . . . . . . . 171 5.16 Transient RMS response of the primary system by averaging over 200 ensembles. Sys- tem parameters: μ = 0.10, ζ = 0.01, e = 0.75, d = d opt , and 100 particle units. Excitationenvelope:g(t) =g 3 (t). . . . . . . . . . . . . . . . . . . . . . . . . . . 171 5.17 Effect of mass ratio on the performance of MUID in nonstationary random excitation withenvelopefunctiong(t) =g 2 (t) . . . . . . . . . . . . . . . . . . . . . . . . . 172 5.18 TimehistoryofarepresentativesegmentofalinearSDOFsystem,thatisharmonically excited and provided with SAID havingμ = 0.10 ande = 0.75. For clarity, the ampli- tudeofalltheplottedquantitieshavebeennormalized. (a)Absolutedisplacementofthe primary (x 1 ) and the secondary (x 2 ) system. (b) Absolute velocity of the primary ( ˙ x 1 ) and the secondary ( ˙ x 2 ) system. (c) Relative displacement (z) and velocity (˙ z) between theprimaryandsecondarysystems. (d)Totalimpactforce . . . . . . . . . . . . . . 174 5.19 EffectofviscousdampingandmassratioontheefficiencyofSAID(oneunit,e = 0.75) subjectedtostationaryrandomexcitation. . . . . . . . . . . . . . . . . . . . . . . 175 5.20 RMS ratio comparison between SAID (one unit,e = 0.75,ζ = 0.01) and MUID (100 units,e = 0.75,ζ = 0.01)subjectedtostationaryrandomexcitation. . . . . . . . . 176 5.21 Comparison of SAID and MUID in reducing the transient RMS response of a SDOF system subjected to nonstationary random excitations. System parameters: μ = 0.10, ζ = 0.01,e = 0.75. Excitation envelope: (a)g(t) = g 1 (t), (b)g(t) = g 2 (t), and (c) g(t) =g 3 (t). . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 179 5.22 Comparison of SAID and MUID in reducing the area under the RMS time history in nonstationaryrandomexcitationwithenvelopefunctiong(t) =g 2 (t). . . . . . . . . 180 5.23 Comparison of SAID and MUID in reducing the peak RMS ratio in nonstationary ran- domexcitationwithenvelopefunctiong(t) =g 2 (t). . . . . . . . . . . . . . . . . . 180 xvi Abstract Oneoftheissuescomplicatingthereliabilityassessmentofstructuralhealthmonitoring(SHM) methodologiesslatedforimplementationunderfieldconditionsfordamagedetectioninconjunc- tionwithtypicalinfrastructuresystems,isthepaucityofexperimentalmeasurementsfromsuch structures. This study evaluates some promising SHM methodologies based on actual vibration measurements, obtained under realistic field conditions from three different cases of full-scale civilinfrastructures. Thefirstpartofthisstudyprovidesacomprehensiveandcomparativestudy of three time-domain identification algorithms applied to extract the modal parameters of the NewCarquinezBridgewhichisanewly-constructedlong-spanbridgethatwasmonitored,inits virginstate,overarelativelylongperiodoftime,withastate-of-the-artdensesensorarray. The secondpartofthisstudyevaluatestheusefulnessofsomeidentificationtechniquestodetermine the evolution of the modal properties of a full-scale 6-story building which has recently under- takena6-monthseismicretrofitprocess,andtocorrelatethechangesintheidentifiedstructural frequencies with the time that specific structural changes were made. The third part of this study presents the results of two time-domain identification techniques applied to a full-scale 17-story building, based on ambient vibration measurements. The UCLA Factor Building was xvii instrumentedpermanentlywithadensearrayof72-channelaccelerometers. Thefirstidentifica- tion method used in this case is the NExT/ERA, which is regarded as a global (or centralized) approach,andthesecondmethodisachainsystemidentificationtechnique. Sinceinthismethod theidentificationofeachlinkofthechainisperformedindependently,itisregardedaslocal(or decentralized) identification methodology. To have a statistically meaningful result, 50 days of the recorded data are considered in this study. Modal parameter and the chain system iden- tification were successfully implemented using the output-only data acquired form the Factor building. The statistical variability of the estimated parameters due to temperature fluctuations is investigated. The last part of this study deals with a structural control application involving optimum strategies for deploying multiple-unit passive/semi-active impact dampers. This study focuses on the development and evaluation of practical design strategies for maximizing the damping efficiency of multi-unit particle dampers under random excitation, both the stationary andnonstationarytype. xviii Chapter1 Introduction 1.1 BackgroundandMotivation Structural health monitoring (SHM) approaches based on analyzing the vibration signature of targetinfrastructuresystemshavebeenreceivingalotofattention,foralongperiodoftime,by manyinvestigatorsallovertheworld. Whiletherearemanytechniquesandapproachesinvolved inthenondestructiveevaluation(NDE)andconditionassessmentofstructuralsystems,theycan all be broadly categorized as local or global methods, alternatively, decentralized or centralized methods. Thefirstcategoryincludesmethodsdesignedtoprovideinformationaboutarelatively smallregionofthesystemofinterestbyutilizinglocalmeasurements,whilethesecondcategory ofmethodsusesmeasurementsfromadispersedsetofsensorstoobtainglobalinformationabout thecondition ofthesystem. Clearly, thetwoapproaches arecomplementaryto eachother, with the optimum choice of method highly dependent on the scope of the problem at hand and the nature of the sensor network. In this study we consider both of the mentioned categories of the identificationtechniques(thelocalandglobal). Somerepresentativepublicationsthatprovideacomprehensiveoverviewofthebroadinter- disciplinary field of SHM, the main technical challenges, as well as promising approaches that havethepotentialofbeingusefultoolsforconditionassessmentinconjunctionwithfieldimple- mentations related to civil infrastructure systems, are available in the works of Agbabian and 1 Masri (1988), Natke and Yao (1988), Housner and Masri (1990), Natke et al. (1993), Housner etal.(1994),Chen(1996),Housneretal.(1997b),Chang(2003),CasciatiandMaganotte(2000), Casciati (2002), Balageas (2002), Chen et al. (2002), Liu (2003, 2004), Boller and Staszewski (2004),Smyth(2004),Chang(2005),Sohnetal.(2003),Doeblingetal.(1998),FarrarandJau- regui(1998a),Lusetal.(1999),Becketal.(1994),Fujinoetal.(2005),GaoandSpencer(2002), Stubbs et al. (2000), Lynch et al. (2004), Pei et al. (2006), Masri et al. (2004), and Fraser et al. (2003). There are many challenging technical problems that await solution before the promising aspects of the SHM field can be fully realized in realistic applications under field conditions to detect, locate and quantify the level of changes (damage) in monitored structures. Conse- quently, working groups within the Structural Control community operating under the auspices of the International Association for Structural Control and Monitoring (IASCM) and in collab- oration with the American Society of Civil Engineers (ASCE) have established a sequence of benchmarkproblems(utilizingmainlysyntheticdata),ofincreasingcomplexityandprogressive sophistication, in order to provide the research community with well-planned and documented “experiments” that can be used to assess the utility of various SHM methodologies. Further details about those benchmark problems and the many technical publications arising from their useareavailableintheworkofHousneretal.(1997b). The technical literature that deals with the application of system identification techniques in conjunction with dynamic measurements is very extensive, building on previous studies in the fields of signal processing and system dynamics, with numerous applications in a variety of specializationareasencompassingthebroadfieldsofscienceandengineering. Someillustrative 2 publicationsthatfocusedonspecificapplicationofsystemidentificationapproachesinthefield of civil infrastructure systems are available in the Proceedings of the World Conferences on StructuralControlandMonitoring(Casciati,2002;Housneretal.,1994;IASCM,2006;Kobori, 1998). Some representative publications that deal with identification of bridges, and especially long-span suspension ones, are available in the works of Farrar and James (1997), Farrar and Jauregui(1998b),Qinetal.(2001),PeetersandRoeck(2001b),Smythetal.(2003),Nagayama etal.(2003),Chenetal.(2004),Nagayamaetal.(2005),andSiringoringoandFujino(2006). Thereiscurrentlyfewavailablestudiesthatinvestigatethestatisticalvariabilityofthevibra- tionparametersoffullscalestructuresduetoenvironmentaleffects(Cornwelletal.,1999;Farrar et al., 1997; Peeters and Roeck, 2001a; Sohn et al., 1999). The main reason appears to be the lakeofavailabilityofmeasurementdatafromphysicalstructuresunderrealisticfieldconditions. Inparticular,tohaveastatisticallymeaningfuldiscussion,oneneedstocollectdays(ormonths) of data under various environmental and operational conditions. There are cost related issues thatunderminethefeasibilityofsuchasysteminmostcases. Ontheotherhand,managingand processingthehugeamountofcollecteddatamaybeextremelychallenging. With the above in mind, this study is focused on establishing, comparing, and validating some practical identification techniques to model the dynamic behavior of civil infrastructures. The models are then used to monitor the vibration signature of the target structures for SHM or control purposes. Throughout this study, the proposed identification methodologies are imple- mentedbasedontheactualvibrationmeasurementdataacquiredfromfull-scalestructuresunder realisticfieldconditions. 3 1.2 Scope The first part of this study is concerned with the full-scale dynamic testing which was recently performedbytheauthorsontheNewCarquinezbridge(alsoknownasAlfredZampaMemorial bridge)locatedinthevicinityofSanFrancisco,California. Acomprehensivestudyofthreetime- domain identification algorithms in conjunction with the Natural Excitation Technique (NEXT) was performed. The three methods are: the eigensystem realization algorithm (ERA), the ERA with data correlations (ERA/DC), and the least squares algorithm (LS). Both forced-vibration and ambient vibration measurements, obtained over a relatively long period of time, were mea- suredandanalyzedinordertoconductthepresentstudy. ThesecondpartdealswiththeapplicationofSHMtechniquestotrackstructuralchangesin aretrofittedbuildingbasedonambientvibrationmeasurementdataonly. TheLongBeachPublic Safety Building (LBPSB) is a critical facility in downtown-long-beach, California. As a result of the 1994 Northridge earthquake this facility was found to need significant seismic mitiga- tion measures. LBPSB was instrumented with 14 state-of-the-art strong-motion accelerometers that were placed at various locations and in different orientations throughout the building. The instrumentationnetworkwasusedtoacquireextensivedatasetsatregularintervalsthatcovered the whole construction phase, during which the building evolved from its original condition to theretrofittedstate. ThepaperbyChassiakosetal.(2007a)providesanoverviewoftheambient vibration data collected before, during, and after the structural retrofit. The goal of the present studyistoapplytheNExT/ERAapproachtoidentifythemodalparametersoftheLBPSBbefore, during,andaftertheretrofit,soastomonitorthechangesinthemodalparameters,andcorrelate themwiththeconstructionphases. 4 The third part of this thesis deals with the structural identification and monitoring of a full-scale 17-story building based on ambient vibration measurements. Following the 1994 Northridge earthquake, the Factor Building which is the tallest one on the UCLA campus, was instrumented with 72 state-of-the-art strong-motion accelerometers that were placed at various locationsandindifferentorientationsthroughoutthebuilding. Theinstrumentationnetworkhas beenconstantlyacquiringandstoringextensiveamountofvibrationdata. Thiscollectionofdata provided us with the unique opportunity to evaluate the effectiveness of various identification methodologies on a full-scale structure, and under realistic field conditions. On the other hand, verylongtime-historyrecordingunderdifferentenvironmentalandoperationalconditions,made it possible for us to statistically investigate the variability of the building’s vibration signature. Inthisstudy,twotypesoflocalandglobaltime-domainidentificationmethodsareimplemented. Thefirstmethodistheeigensystemrealizationalgorithm(ERA)(JuangandPappa,1985,1986) in conjunction with the natural excitation technique (NEXT) (James et al., 1993, 1996). It is used to extract the modal parameters (natural frequencies, mode shapes, and modal damping) of the building, based on ambient vibration records. This type of identification methodology is regarded as global (or centralized) approach, since it deals with the global dynamic properties of the structure. The second method is a time-domain identification technique for chain-like MDOFsystemsintroducedbyMasrietal.(1982). Suchaclassofproblemsencompassesmany practicalapplicationsincludingtallbuildingsliketheoneunderdiscussion. Sinceinthismethod the identification of each link of the chain is performed independently, it is regarded as local (or decentralized) identification methodology. For the same reason, this method can be easily adoptedforlargescalesensornetworkarchitecturesinwhichthecentralizedapproachesarenot 5 feasible due to massive storage, power, bandwidth, and computational requirements. Gener- allythismethodcanbeappliedtolinearornonlinearchain-likeMDOFsystems. Nonetheless,it requirestheappliedforcestobeknown. However,forfullscaleinfrastructuresliketheoneunder discussion, providing measurable excitation is very costly, and therefore one needs to just rely ontheavailableambientvibrationrecords. Inordertohandlethecaseswithunmeasurableexter- nalforces, thepresentstudyhasgeneralizedthechainidentificationprocedure, bycombiningit with the idea of the NEXTtechnique. Having 50 days of recorded data in different temperature and ambient conditions provided the opportunity to investigate the effect of environmental and operational conditions on the estimated dynamic properties of the building. Variability of the estimatedparametersduetotemperaturevariationsina24-hourperiodisdiscussed. The last part of this study is concerned with structural vibration suppression using passive and semi-active impact dampers. This class of dampers that exploits “impact damping” phe- nomena for vibration reduction, provides some useful features such as simplicity in design, ruggedness, reliability, and insensitivity to temperature extremes. While there are some appeal- ing vibration-control features of the family of impact dampers, there are also some accompa- nying undesirable characteristics: impulsive loads transmitted during the momentum exchange phase of the coupled system motion, and the attendant noise and potential local deformations accompanying the plastic collisions among the system components. There are still many unre- solved issues needing study in regard to investigating the performance of this class of devices under broad-band excitation, stationary or not. The goal of the present investigation is to per- formfurtheranalyticalandexperimentalstudiestodevelopoptimumstrategiesfordealingwith situations in which vibration attenuation devices that are based on impact damping phenomena 6 can be designed to provide effective and robust damping performance, while simultaneously being relatively insensitive to variations in the spectral characteristics of wide-band excitations, both the stationary and nonstationary type. It will also be shown that, depending on the level of sophistication of the design, a significant improvement in performance can be achieved if an applicationallowstheincorporationofadaptivestiffnesscharacteristicsofthedampers. 7 Chapter2 AStudyofTime-DomainTechniquesfor ModalParameterIdentificationofaLong SuspensionBridgewithDenseSensorArrays 2.1 Introduction BackgroundandMotivation One of the main drivers of growing interest and capabilities in the field of structural health monitoring of civil infrastructure systems is the increasing wide-spread availability of sensor networks that have the potential to collect vast amounts of data hither to fore not feasible to acquire. Simultaneouswiththisincreasingcapabilitytocollectdata,istheparallelinterestinthe applicationofmoresophisticateddataprocessingalgorithmstoidentifythestructuralparameters (indifferentformats)ofthetargetinfrastructuresystem. Whilenumerousstudieshavebeenpublishedconcerningtheapplicationofavarietyofsys- tem identification techniques in conjunction with vibration measurements from civil infrastruc- ture systems, there is a paucity of publications addressing the influence of algorithm-specific controlparametersthat impactthecorrectandefficient applicationoftheselectedidentification scheme. Furthermore, as dense sensor arrays become widely accessible in civil infrastructure 8 applications,voluminousamountsofmulti-channeldatastreamsarebecomingavailableforpro- cessing,thusimposingnewdemandsonidentificationproceduresregardinghigh-dimensionality (inboththespatialaswellasthetemporaldomains)requirementsthatmayrendersomemethods inapplicableifcarefulattentionisnotpaidtopracticalimplementationissues. The technical literature that deals with the application of system identification techniques in conjunction with dynamic measurements is very extensive, building on previous studies in the fields of signal processing and system dynamics, with numerous applications in a variety of specialization areas encompassing the broad fields of science and engineering. Some illus- trative publications focused on specific application of system identification approaches in the fieldofcivilinfrastructuresystemsareavailableintheProceedingsoftheWorldConferenceson StructuralControlandMonitoring(Casciati,2002;Housneretal.,1994;IASCM,2006;Kobori, 1998). Somerepresentativepublicationsthatdealswithidentificationofbridges,andespecially long-span suspension ones, are available in the works of Farrar and James (1997), Farrar and Jauregui(1998b),Qinetal.(2001),PeetersandRoeck(2001b),Smythetal.(2003),Nagayama etal.(2003),Chenetal.(2004),Nagayamaetal.(2005),andSiringoringoandFujino(2006). Whiletherehavebeenmanyapplicationsofsystemidentificationtechniquesinconjunction withstructuraldynamicmeasurementsfrombridgestructures,thenumberofsensorsusedinthe reported studies relied on sensor networks that were deployed before the wide availability of dense sensor networks. Furthermore, while various investigators have applied some promising approaches,bothinthetime-domainaswellasthefrequency-domain,toavailablemeasurement 9 sets,thereiscurrentlyfewavailablepublicationthatcomparesthecapabilitiesofpromisingiden- tificationapproacheswhenappliedtolargedatasetsobtainedfromdensesensorarraysdeployed onlong-spansuspensionbridges. Furthermore,whilestructuralhealthmonitoringcommunityhasbeenapplyingsomepower- fulidentificationtechniquestoidentifythemodalcharacteristicsofsystemswithlargenumbers of degrees of freedom, the influence of the many algorithm-specific control parameters that the user must select before correctly and efficiently implementing the identification procedures has notreceivedmuchattention. With the above in mind, this study is focused on establishing and validating some practical guidelinestoassistintheapplicationofseveralwidely-usedtime-domainidentificationschemes (that originally evolved in the aerospace field) that have been in use for a considerable amount of time, but whose application requires the selection of several control parameters that have significant influence on the computational efficiency and the reliable estimation of pertinent structuralparameters. Specifically, the class of problems of interest in this study corresponds to extended civil structures (such as long suspension bridges) provided with dense sensor arrays to capture, over a very long observation period, high-precision acceleration measurements caused by ambient conditionsaswellasbyapplieddynamicloads. Scope Thisstudyisconcernedwiththefull-scaledynamictestingwhichwasrecentlyperformedbythe authors on the New Carquinez bridge (also known as Alfred Zampa Memorial bridge) located in the vicinity of San Francisco, California. The New Carquinez bridge is a new suspension 10 bridgethatwascompletedinNovember2003. Thisdynamictestingwasthefirstofitstypetobe performedonthisbridgepriortoitsopeningfortraffic. Theresponseofthebridgewasmeasured by utilizing 64 accelerometers which constituted the field instrumentation of NEES@UCLA NetworkofEarthquakeEngineeringSimulation(NEES)site. The NEES@UCLA site has a state-of-the-art mobile laboratory, which can be utilized in performing full-scale structural system testing. The use of this equipment made it possible to developadetailedtestingmatrix, whichcombinedforcedandambientvibrationtests. Utilizing thismobilelaboratory,thedynamicalcharacteristicsofthebridgewereobtained. A companion paper by the authors (Conte et al., 2006) described in detail the instrumenta- tion, data acquisition, vehicle-induced impact tests, and the ambient vibration tests conducted on the subject bridge. In the afore mentioned paper, the authors presented a study of the modal parameter estimation of the bridge using a stochastic subspace identification technique. The presentstudycomplementstheresultsofthepreviouspaperbyprovidingacomprehensivestudy of three time-domain identification algorithms in conjunction with the Natural Excitation Tech- nique (NExT). The three methods are: the eigensystem realization algorithm (ERA), the ERA with data correlations (ERA/DC), and the least squares algorithm (LS). Both forced-vibration and ambient vibration measurements, obtained over a relatively long period of time, were mea- suredandanalyzedinordertoconductthepresentstudy. 2.2 NewCarquinezBridge: CharacteristicsandTestDescription The New Carquinez Bridge is located 32km northeast of San Francisco and it carries highway I-80acrosstheSacramentoriver. Thebridge,whichspanstheCarquinezstrait,islocatedwithin 11 a few miles of several active faults. The bridge was opened for traffic in November, 2003. The full-scaledynamictestreportedhereinwasperformedpriortoopeningthebridgefortraffic. Long-spanbridgeshavenotbeenbuildintheU.S.forthepast36years,sincetheVerrazano Narrows bridge was opened for traffic back in 1964. The New Carquinez long-span suspension bridge with the main span length of 728 meters (2,390 feet) and the side span length of 147 and 181 meters (482 and 593 feet), and the concrete towers 125 meters (410 feet) above the water level, has many first recognitions. It is the first orthotropic steel box girder suspension bridgeevertobebuiltintheUnitedStates. ItisthefirstbridgeintheUnitedStateslocatedina potentiallyhighseismicriskarea,andthefirstbridgeinhighseismicareawithconcretetowers. Figure2.1showsanoverviewofthebridgeanditsoveralldimensions. Figure 2: Overview of the bridge. The bridge consists of main span length of 2390 ft, while the side spans are 482 and 593 ft. The overall span on the bridge is 3465 ft. Figure 3 depicts the overall dimensions of the bridge. (a) Figure 3: Overall dimensions of the Alfred Zampa Memorial Bridge. Concrete Towers The bridge two towers consist of reinforced concrete that rise about 410 ft above the water level. Each tower leg consists of a hollow box section with spirally reinforced corner pilasters about 3.3 ft in diameter connected by 19.7 in thick walls. This hollow cellular section needed to provide superior seismic performance as well as agreeable aesthetics. The tower legs were designed to behave elastically under the most severe design earthquake forces with limited inelasticity being permitted at the tower base. Two reinforced concrete struts were designed to remain elastic. Figure 4 shows the schematic of these towers. 1056m (3465’) 728m (2390’) 147m (482’) 181m (593’) (b) Figure2.1: (a)OverviewoftheNewCarquinezbridge,and(b)itsoveralldimensions. 12 147 728 181 South span North span 7SW 5SW 4SW 3SW 2SW 1SW 0W 1NW 2NW 3NW 4NW 5NW 7NW 7NE 1NE 0E 2NE 5NE 4NE 3NE 1SE 2SE 3SE 4SE 5SE 7SE 54.86 57.00 54.86 66.14 54.86 76.29 54.86 6 NW 6 NE 6 SW 6 SE Center line of main span Figure2.2: SchematicplotfortheaccelerometerlocationsanddirectionsontheNewCarquinez bridge(Conteetal.,2006). The bridge was instrumented with 34 uni-axial and 10 tri-axial EpiSensor force-balance accelerometers. The accelerometers were installed at selected locations as shown in Fig. (2.2), whichincluded14and11stationsalongthewestandeastsidesofthebridgedeck,respectively. Atotalnumberof64accelerometersat25locations(25inthevertical,25inthetransverse,and 14inthelongitudinaldirection)wereusedinthisstudy. Theseaccelerometersarecharacterized by a frequency bandwidth from DC to 200 Hz, a large amplitude range (user scalable from +/- 0.25gto+/-4.0g),andawidedynamicrangeof155dB.Thesesensorswereutilizedinorderto measuretherelativelylow-amplitude,freeandforced-vibrationresponses. The data was recorded from the ambient vibration for several hours. In addition, artificial excitations were also used to excite the higher bridge modes. To this end, two heavy loaded trucks and speed bumps were used. The trucks were driven from one side of the bridge to the other side, and passed over some triangular-shaped steel ramps on their way, and as a result impact-type excitation was generated, which is ideal for exciting higher modes. Different sce- narios for bumper locations and relative truck directions and speeds were utilized. For detailed 13 description of the bridge, its instrumentation, and data acquisition, the reader is referred to a companionpaperbytheauthors(Conteetal.,2006;Heetal.,2006,2005). 2.3 FormulationofTheTime-DomainModalParameterIdentifica- tionTechniques Thissectiondealswiththebasicformationofsomeoftheleadingtime-domainsystemidentifi- cationapproachesthatarebeingcommonlyappliedintheSHMfield: theNExttechnique,ERA, ERA/DCandtheLSalgorithms. 2.3.1 NExtTechnique Providing measurable and dominant excitations (using hammer or shaker) for civil infrastruc- tures such as bridges and buildings, is very difficult, costly, and in some cases infeasible. On the other hand, ambient excitation (from wind, traffic, ground motion, etc.) is always available; however its source is usually immeasurable. These facts show the importance of output-only modal parameter identification methods. The NExt technique introduced by James et al. (1993, 1996), hasbeensuccessfullyusedforidentificationofstructuresbasedonoutput-onlyinforma- tion(Caicedoetal.,2004). Herewebrieflyexplainthemainideasofthetechnique. Consider the forced vibration of an n DOF linear, time-invariant system where motion is governedbythefollowingequationofmotion: M¨ x(t)+D˙ x(t)+Kx(t) =f(t) (2.1) 14 wherex(t)andf(t)arethen×1displacementandexternalexcitationvectors,respectively,and M,D,andK arethen×nmass,damping,andstiffnessmatrices,respectively. The basic idea behind the NExT method is that the cross-correlation function between the responsevectorandtheresponseofaselectedreferenceDOFsatisfiesthehomogeneousequation of motion, provided that the excitation and responses are weakly stationary random processes, whichisnormallythecaseforambientvibration. Equation(3.1)canberewrittenas ME[ ¨ X(t)X ref (t−τ)]+DE[ ˙ X(t)X ref (t−τ)]+KE[X(t)X ref (t−τ)] =E[F(t)X ref (t−τ)] (2.2) whereX, ˙ X, ¨ X, andF are displacement, velocity, acceleration, and excitation stochastic vector processes, respectively, andE[.]denotestheexpectationoperator. Consideringthedefinitionof thecorrelationfunctionR(.),onecanrewriteEq. (2.2)as MR X ref ¨ X (τ)+DR X ref ˙ X (τ)+KR X ref X (τ) =R X ref F (τ) (2.3) Since the excitation and the system responses are weakly stationary random processes, they are uncorrelated, thus renderingR X ref F (τ) = 0. On the other hand, it can be proven that (Bendat andPiersol,2000) R X ref ˙ X (τ) = ˙ R X ref X (τ) (2.4) and R X ref ¨ X (τ) = ¨ R X ref X (τ) (2.5) 15 Usingtheabovementionedresults,Eq. (2.3)canberewrittenas M ¨ R X ref X (τ)+D ˙ R X ref X (τ)+KR X ref X (τ) =0 (2.6) Equation (2.6) signifies that the cross correlation function between the displacement process vector and the displacement of a reference DOF, satisfies the homogeneous (or free vibration) equation of motion. It can be similarly shown that the acceleration cross correlation function, alsosatisfiesthehomogeneous(orfreevibration)equationofmotion: M ¨ R ¨ X ref ¨ X (τ)+D ˙ R ¨ X ref ¨ X (τ)+KR ¨ X ref ¨ X (τ) =0 (2.7) Previous experience has shown that one cannot rely on a single reference DOF for identi- fication of all modes. The optimum accuracy for the identification of different modes typically occurs at different selection choices of the reference DOFs. With this in mind, we are going to formulatehereintheidentificationalgorithmforusingmultipleDOFsasthereference. The importance of Eq. (2.7) is that: (1) the stationary random excitation (ambient noise) is eliminated from the equation of motion, and that (2) only the acceleration records are needed to implement the identification technique. Once the homogeneous equation of motion (2.7) is formed using the NExt technique, ERA, ERA/DC or LS can be used to extract the modal parametersofthehomogeneousmodel. 16 2.3.2 ERAandERA/DCMethods JuangandPappa(1985,1986)proposedanEigensystemRealizationAlgorithm(ERA)formodal parameteridentificationandmodelreductionoflineardynamicalsystems. Inalaterstudy(Juang etal.,1988)thealgorithmwasmodifiedbyproposingtheERAwithdatacorrelations(ERA/DC). ERA/DCreducesbiaserrorsduetonoisecorruptionwithouttheneedforlargemodeloverspec- ification. Here, we briefly present the fundamental principles of the ERA and ERA/DC algo- rithms. Thefirstfundamentalstepistoformthen(r+1)×m(p+1)Hankelblockdatamatrix asfollows: 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) (2.8) wheren andm are the number of measurement stations, and reference DOFs, respectively; r andp are integers corresponding to the number of block rows and columns, respectively. Y(k) isthen×mmatrixofthecross-correlationfunctionswhichsatisfiesthehomogeneousequation ofmotion(Eq. (2.6,2.7)),andcanbewrittenas Y(k) = y 1,1 (k) y 1,2 (k) ... y 1,m (k) y 2,1 (k) y 2,2 (k) ... y 2,m (k) . . . . . . . . . . . . y n,1 (k) y n,2 (k) ... y n,m (k) (2.9) 17 Originally, y i,j (k) was meant to be the impulse response of the ith DOF, at time step k, due to an impulse at the jth DOF. Here, since the impulse responses are not available, they are replaced by the cross-correlation function R ¨ X ref ¨ X (k) of DOFi, at time stepk, due to the selection of reference DOF j. The formulation represented by Eq. (2.9), enables one to use multiple reference DOFs simultaneously. A similar approach for simultaneous use of multiple referenceDOFscanbefoundintheworkofHeetal.(2006). TheERAprocessstartswithfactorizationoftheHankelblockdatamatrix,fork = 1,using singularvaluedecomposition 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 (2.10) whereDisthediagonalmatrixofmonotonicallynon-increasingsingularvalues. D 1 isN×N (N ≤ p) diagonal matrix formed by truncating the relatively small singular values, where N is the final system order. It is worth noting that the selection of the final model order it not a trivial task. In the next section, an algorithm for automatic selection of the final model order is presented. P 1 andQ 1 aren(r+1)×N,andm(p+1)×N matrices,thatincludethefirstN columnsof the originalP andQ matrices, respectively. The discrete-time state-space realization matrices forthestructuralmodelcanbeestimated(JuangandPappa,1985)as ˆ A =D −1/2 1 P T 1 H(1)Q 1 D −1/2 1 (2.11) ˆ C =E T m P 1 D 1/2 1 (2.12) 18 where E T m = I 0 , and its size is determined accordingly. The control influence matrix cannot be estimated using the output-only information. The estimated discrete-time realization, needs to be transformed to the continuous-time domain format. To this end, let us consider the eigenvalueproblemfor ˆ A ˆ A = ˆ Ψ ˆ Λ ˆ Ψ −1 (2.13) where ˆ Λ and ˆ Ψ are the eigenvalue and eigenvector matrices, respectively. The natural frequen- cies,ω i , damping ratios,ζ i , and the mode shapes, ˆ Φ i , of the continuous-time structural model canbefoundasfollows: ω i = q σ 2 i +Ω 2 i ζ i =−cos[tan −1 (Ω i /σ i )] ˆ Φ i = ˆ C ˆ Ψ i (2.14) where ˆ Λ =diag(σ i ±jΩ i ) σ i ±jΩ i = ln(σ i ) Δt ±j ln(Ω i ) Δt (2.15) where Δt is the sampling period of data records. In order to reduce the bias error due to noise corruptioninthemeasurements,analternativeformoftheERA,namedERA/DCwasdeveloped (Juang et al., 1988). Lew et al. (1993) presented the comparison of four system identification algorithmsincludingERAandERA/DCforabenchmarkproblemattheNASALangleyresearch 19 center. The study showed that for the example considered, ERA/DC gives the best results; that ERA/DCisalwaysatleastasgoodasERA,andcanbeshowntobeaespecialcaseofERA. TopresenttheERA/DC,inaconciseform,let’sdefinetheblockcorrelationmatrixR hh (k) as R hh (k) =H(k)H T (0) (2.16) AblockcorrelationHankelmatrix,H(k),canbeformedas H(k) = R hh (k) R hh (k+τ) ... R hh (k+βτ) R hh (k+τ) R hh (k+2τ) ... R hh (k+(1+β)τ) . . . . . . . . . . . . R hh (k+ατ) R hh (k+(α+1)τ) ... R hh (k+(α+β)τ) (2.17) where τ is an integer chosen to prevent significant overlap of adjacentR hh blocks, and the integers α and β define how many correlation lags are included in the analysis (Juang et al., 1988). Similar to ERA, the first step in the processing is the singular value decomposition of H(0) 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 (2.18) TherestoftheprocedureisexactlythesameaswhatwasexplainedfortheERAmethod. Sincethesingularvaluedecompositionisthemosttime-consumingstepinboththeERAand ERA/DCmethods,itisworthwhiletocomparethecomputationalburdenofERAandERA/DC. The size of Hankel matrix, H(0), used in ERA, is n(r + 1)×m(p + 1), whereas the size of the block correlation Hankel matrix,H(0), used in ERA/DC, is (1 +α)n(r + 1)× (1 + 20 β)n(r +1). Therefore, for the case where the number of block columnsp is much larger than the number of rowsr,H(0) would be smaller in size thanH(0), assuming bothα, andβ are small. Consequently, unlike ERA, in ERA/DC, one can increase the number of columns in the Hankel Matrix, to include more data points, without increasing the dimension of the singular value decomposition matrix (Lew et al., 1993). This computational advantage becomes crucial whenalongrecordofdataforalargenumberofchannelsisavailable(asisthecaseintheclass ofproblemsunderconsiderationinthisstudy). Oneissueintheabovementionedalgorithmsisthechoosingoftherightvaluesfortheuser- selectable parameters. Experience with experimental data has shown that the results fluctuate with the changes in the algorithm parameters. In ERA, the user needs to select the number of rows and columns in the Hankel matrix. In addition, in ERA/DC, the user should also select τ, α, and β. In this paper, some useful guidelines for the proper selection of the mentioned parameterswillbepresented. InfluenceofSVDmatrixsizeinERAandERA/DCmethods The smaller dimension of the SVD matrix (in this study, the row dimension), is an indication of the initial model order (Cooper and Wright, 1992). The model order is twice the number of modestobeidentified. Consequently,fortheidentificationofM modes,themodelorderneeds to be at least 2M. However, in practice, when noise is present, the initial model order needs to beoverspecified. Asaruleofthumb, theinitialmodelordershouldbeselectedtobemorethan 10 times the true model order (20 times the expected number of modes,M). Therefore, in the 21 ERAmethod,thenumberoftheHankelmatrixblockrows(r+1),canbedeterminedfromthe followinginequality: (r+1)> 10(2M/n) (2.19) wherenandM arethenumberofmeasurementstationsandexpectednumberofmodes,respec- tively. Typically, the number of modes of interest can not be greater than the number of mea- surementstations. UnlikeERA,theERA/DCmethodrequireslessinitialmodeloverspecification(Juangetal., 1988). However, some model overspecification in the presence of noise is preferable (Cooper andWright,1992). Consequently,inERA/DC,withmorethan4timesinitialoverspecification, onecanselectr andαbasedonthefollowinginequality: (1+r)(1+α)> 4(2M/n) (2.20) Onesimplechoiceistoselectr = 0,sothatHbecomesarowofblockmatrices,andthenselect αbasedonEq. (2.20). The greater dimension of the SVD matrix (here, the number of columns), is an indication of the number of data points to be included in the identification process (Cooper and Wright, 1992). This dimension must be selected such that the significant part of the correlation time history (typically more than half of the available cross-correlation data points) is included in the analysis. The number of SVD matrix block columns, in both ERA and ERA/DC, can be computedbasedonEq. (2.21),and(2.22),respectively: ERA : p = (2/3)N p −r−2 (2.21) 22 ERA/DC : p+β = (2/3)N p −r−α−2 (2.22) where N p is the number of data points, and p is the number of block columns in the Hankel matrix. Forsimplicity,τ maybechosentobeone. Asmentionedbefore,unliketheERAcase,p hasnoeffectonthesizeoftheSVDmatrixinERA/DC.Onecanusethiscomputationaladvan- tageofERA/DC, andincreasep toincludeallavailable datapoints, andthendefineβ fromEq. (2.22). This fact, along with the fact that ERA/DC requires less initial model overspecification, canmakeitcomputationallymuchfasterthanERA,especiallyforlargescaleproblems. Itmust be noted that, since the second half of the correlation time history in one frame is the mirror of the first half, the maximum available data points for processing is equal to half of the window size. 2.3.3 LSMethod Theleat-squarestime-domainmethod(LS)usedinthisstudyisbasedonthemethodologyintro- duced by Masri et al. (1987a,b), for the identification of general nonlinear vibrating structures; however,inthisstudy,weconsideralinearizedversionofthemethod. Thehomogeneousequa- tionofmotion(Eq. 2.7)canberewrittenas 1 A ˙ R ref(j) (k)+ 2 AR ref(j) (k) =− ¨ R ref(j) (k) (2.23) where 1 A =M −1 C, 2 A =M −1 K, ¨ R ref(j) (k) =R ¨ X ref (j) ¨ X (k), ˙ R ref(j) (k) =R ¨ X ref (j) ˙ X (k), R ref(j) (k) = R ¨ X ref (j)X (k), j = 1,...,N r , and k = 1,2,...,N t . It is assumed that N r 23 reference DOFs andN t data points have been selected for the identification. It is worth noting thatdimensionof 1 A,and 2 Aisn×n,wherenisthenumberofmeasurementstations. Lettheresponsevectorr ref(j) (k)oforder2nbedefinedas r ref(j) (k) = ( ˙ R T ref(j) (k),R T ref(j) (k)) T (2.24) Let< j A i >= i th row of a generic matrix j A, and introduce the parameter vectorα i of order 2n α i = (< 1 A i >,< 2 A i >) T (2.25) IntroducingmatricesΦ ref(j) andb ref(j) Φ ref(j) = r T ref(j) (1) r T ref(j) (2) . . . r T ref(j) (N t ) (Nt×2n) (2.26) b ref(j) = − ¨ R T ref(j) (1) − ¨ R T ref(j) (2) . . . − ¨ R T ref(j) (N t ) (Nt×n) (2.27) 24 andusingthenotationabove,theequationsofmotionforeachsingleselectionoftheref(j)can beexpressedconciselyas Φ ref(j) α i =b ref(j) (i) i = 1,...,n (2.28) whereb ref(j) (i) is thei th column ofb ref(j) . One can extend the formulation for the multiple referencecaseas Φ ref(1) . . . Φ ref(Nr) (N t Nr×2n) α i = b ref(1) (i) . . . b ref(Nr) (i) (N t Nr×1) i = 1,...,n (2.29) Least-squares procedure can then be used to solve for all the system parameters that constitute theentriesinα i . α i = Φ ref(1) . . . Φ ref(Nr) † b ref(1) (i) . . . b ref(Nr) (i) i = 1,...,n (2.30) wherethesuperscript†denotesthepseudoinverse. Oncetheunknownvectorα i isidentified,one can extract the rows ofM −1 K, andM −1 D. Continuous-time state-space realization matrices (A,C)canthenbeformedas A = 0 I −M −1 K −M −1 D (2.31) 25 C = I 0 (2.32) The structural modal parameters are estimated from eigenvalues (σ i ±jΩ i ) and eigenvectors ( ˆ Ψ i )ofAbasedonEq. (2.14). Since we do not have any control on the model order in this approach, the number of iden- tifiedmodesisnecessarilyequaltothenumberofmeasurementDOFs,n. However,inpractice, notallnmodesobtainedusingthisapproacharevalidstructuralmodesduetonoise,nonlinear- ity, etc. Consequently, the actual number of modes contributing to the system’s response might be much smaller than the number of measurement DOFs. In the next section, an “autonomous” algorithm is presented to distinguish the fictitious noise modes from genuine structural modes, basedonselectedaccuracyindicators. 2.3.4 Autonomous Selection of the Final Set of Modal Parameters Using Mode CondensationAlgorithm As mentioned earlier, in order to reduce the bias error due to noise, the initial model order needs to be overspecified. Consequently, some spurious computational or noise modes will appearintheidentificationresultswhichdonotrepresentthephysicalbehaviorofthestructure. The question requiring answer is, what the true model order is; in another words, how can one extract the final set of modal parameters from a combination of the identified genuine and spurious modes. Answering this question is not trivial, especially when we are dealing with experimental (as opposed to synthetic) data. Previous experience with experimental data has shownthattheresultsfluctuatetosomedegreewiththechangesinthefinalmodelorder. Onthe otherhand,theoptimumselectionofthemodelorderdiffersfrommodetomode,especially,for 26 theidentificationofweaklyexcitedmodes,wherehighermodelorderisusuallyrequired(Pappa andElliott,1993). Aspreviouslymentioned,thefinalmodelorderisequivalenttothenumberofSingularVal- ues (SVs) retained in the SVD process. Theoretically, the true model order can be estimated by inspecting the SVs. When there is a relatively large gap between two successive SVs, the relativelysmall(closetozero)SVsafterthegap,mustbeeliminated,andthenumberoftheSVs before the gap determines the final model order. In practice, however, it is very hard to find the mentioned“gap”;consequentlythatmethodcannotbereliedon. Onewaytoachievethisgoal(selectionofthefinalmodelorder,andeliminationofthespu- riousnoisemodes)istouseaso-calledstabilizationdiagram(PeetersandRoeck,2001b). Such adiagramshowstheevolutionofmodalparametersasafunctionofthefinalmodelorder(twice theassumednumberofmodes). Thegenuinestructuralmodesstabilizethroughtheprocess,but the spurious modes will not stabilize at all. Pappa et al. (1998) introduced a more automated algorithm which not only eliminates the spurious modes, but also converges to the best set of genuine modes (with the highest confidence level). To determine the confidence level for each mode,anaccuracyindicatorknownastheConsistentModeIndicator(CMI)foreachmodei,is definedasfollows CMI i = EMAC i ·MPC i (2.33) where EMAC (Extended Modal Amplitude Coherence) is a number between 0 and 1, which quantifiestheconsistencyofeachidentifiedmodethroughtime. Anoisemodeisnotconsistent through time and its EMAC value is close to zero. MPC (Modal Phase Collinearity) is also a numberbetween0and1,whichquantifiesthemonophasebehaviorofeachidentifiedmode. For 27 Identify mode shapes, natural frequencies, damping factors, and their corresponding accuracy indicators (CMI) for the new set k+1, using NExT/ERA, NExT/ERA-DC, or NExT/LS. Eliminate modes with CMI<0.70, or damping factor<0, or damping factor>0.30, or frequency=0 (rigid body modes). Add mode i in set k to set k+1 as a new mode. No For those modes in set k+1 (m modes) which are within 20% in frequency of mode i in set k, compute the MAC value. Yes 70 . 0 )) , , 1 ( mode , ) ( mode ( 1 k k > + m i MAC K No Yes )) ( mode ( C )) ( mode ( C k 1 k i MI j MI < + Replace mode j in set k+1 with mode i in set k. Yes Change the final model order (twice the assumed number of modes), and/or the reference DOFs, and/or data sets, then repeat until convergence. Iteration k+1 For each mode in the old set k, compare its frequency ( ) with the frequencies of all modes in the new set ( ). ) (i f k ) , , 1 ( 1 1 + + k k n f K For those modes in set k+1 which are within 20% in frequency of mode i in set k, and their corresponding MAC values are greater than 0.7, select the mode in set k+1 with the highest MAC value ( ), then compare and . ) ( mode 1 k j + )) ( mode ( C 1 k j MI + )) ( mode ( C k i MI } Thresholding Consolidation } Identification k k k k k n i i f n f i f , , 1 % 20 ) ( ) , , 1 ( ) ( 1 1 K K = < − + + Figure 2.3: Flowchart of the main steps for the mode condensation algorithm (Pappa et al., 1998). a classical normal mode, the components of the identified complex mode shape are collinear (monophase) in the complex plane, hence the corresponding MPC is close to one. For detail mathematical description of EMAC and MPC, the reader is referred to the work of Pappa and Elliott(1993). 28 Roughly speaking, CMI values greater than 80% indicate a high confidence in the modal identification results (Pappa and Elliott, 1993). Spurious computational and noise modes have CMI values close to zero. In this study, the mode condensation algorithm (Pappa et al., 1998) was generalized for not only the autonomous selection of the final model order, but also the application of multiple reference DOFs, one-at-a-time . In this way, there are two main iteration loops; the outer loop changes the reference DOF, and the inner loop changes the model order for each choice of the reference DOF. Consequently, the autonomous selection of the model order and the application of multiple references (one-at-a-time ) are being done using the same algorithm. Inthethresholdingprocedure,thespuriousnoisemodesareeliminatedandthen,the resultinggenuinemodesareclassifiedandconsolidatedwiththeexistingsetofmodalparameters fromthepreviousstep. TheflowchartofthealgorithmmainstepsisshowninFig. 2.3. This algorithm can also be very useful when we are dealing with multiple data sets for the samestructure. Inthealgorithm,theModalAssuranceCriterion(MAC),isusedtoquantifythe correlation between complex mode shapes. The MAC value between two mode shapes ˆ Φ i , and ˆ Φ j isdefinedas MAC i,j = | ˆ Φ H i ˆ Φ j | 2 ( ˆ Φ H i ˆ Φ i )( ˆ Φ H j ˆ Φ j ) (2.34) wherethesuperscriptH,denotestheHermitianofamatrix. TheMACvaluerangeisbetween0 and1;0fororthogonaland1foridenticalmodeshapes. 29 The same algorithm (Fig. 2.3) can be implemented for the NExT/LS method. However, since EMAC is specifically designed for ERA (ERA/DC), an alternative accuracy indicator is needed. Tothisend,CMI i isdefinedas CMI i = MCF i ·MPC i (2.35) where MCF i is the Modal Confidence Factor (Ibrahim, 1978) for mode i identified using NExT/LS,andiscalculatedasfollows. SupposethatthestatematrixidentifiedusingNExT/LSisa2n×2nmatrix A. Letthetime history(displacement,velocity,andacceleration)oftheDOFj isdelayedintimeforτ seconds, whereτ is 10 times the sampling period, by default. Then, the new state matrix e A is identified for the delayed system. It can be shown that for a noise-free system, the eigenvalues of A and e A (λ and e λ, respectively) are identical, and the eigenvectors of A and e A (ψ and e ψ, respectively) havethefollowingrelationship: e ψ i =D i ψ i (2.36) where D i =diag(1,...,1,e −τλ i → j th row,1,...,1,e −τλ i → (n+j) th row,1,...,1) 2n×2n (2.37) 30 The MCF’s definition in this study is slightly different than it is in Ibrahim (1978). Here, the MCF for each modei is calculated by comparing the estimated and identified mode shapes of thedelayedsystemsusingtheMACasfollow MCF i = MAC(D i ψ i , e ψ i ) (2.38) This procedure can be easily expanded for computing the MCF by delaying multiple measure- mentDOFssimultaneously. 2.4 ImplementationandResults ThissectionreportstheresultsoftheapplicationofthealgorithmsunderdiscussiontotheNew CarquinezBridge. Aspreviouslymentioned,theNCBwasinstrumentedwith64accelerometers at25locations;25inthevertical,25inthetransverse,and14inthelongitudinaldirection. Data was recorded at the sampling rate of 200 Hz. Two sets of recorded data were collected, one for ambient vibration and the other one for the forced vibration using loaded trucks, for the total record length of about 4 hours. Using the algorithm in Fig. 2.3, the identification results for thesetwosetsareconsolidatedintoonefinalset. Itshouldbenotedthattheverticalacceleration response at station 5SE (see Fig. 2.2) was not recorded properly, and consequently was not consideredintheanalysis. To implement the identification methodologies under discussion, the first step is to com- pute the Cross-Correlation Functions (CCF) between the response of the preselected reference DOF (or DOFs) and the response of all available DOFs. As mentioned earlier, one can not rely 31 on just one single reference DOF to reliably identify all modes. One single reference that is a proper selection for some modes, might not be proper for other modes. For this reason, it is recommended to use multiple reference DOFs, as opposed to a single reference DOF. There are two ways for implementing multiple references: (a) simultaneously, and (b) one at a time. Simultaneous implementation of multiple references can be done efficiently using the formula- tion presented in Eq. (2.9) and (2.29) for the NExT/ERA (ERA/DC) and NExT/LS methods, respectively. However, using many references at a time may cause some computer memory (storage) problems. Therefore, it is recommended to use multiple references, but one at a time, andthenconsolidatetheresultingmodalparametersfromdifferentsetsintoasinglesetusingthe algorithm shown in Fig. 2.3. In this way, one can virtually use all available DOFs (or as many of them as reasonable) as the reference, without worrying about the memory restrictions. This method is precisely what we used in this study. It is worth noting that, since the spurious noise modesareeliminatedinthethresholdingprocedure,oneshouldnotworryabouttheselectionof the reference DOFs. At the end, the algorithm will converge to the best set of genuine modes (withthehighestpossibleconfidencelevel). The CCF can be estimated by the inverse Fourier transform of the Cross-Power-Spectral Density(CPSD),wheretheCPSDiscomputeddirectlyfromthedata. Randomerrorsassociated with the CPSD can be minimized by windowing and averaging. To implement the NExT/LS method,wealsoneedthevelocityanddisplacementtimehistoriesforallmeasurementstations. Tothisend,theaccelerationrecordsneedtobeintegratednumerically. Windowing,detrending, and band-pass filtering is required for that process. Since the frequency range of interest is less than5Hz,thedata(afterintegration)wasdown-sampledto20Hz. 32 2.4.1 InfluenceofTheDataWindowSizeandTheSizeofTheSVDmatrix As mentioned earlier, there are some algorithmic parameters that must be selected based on the user’s judgment. The window size required for the computation of the CPSD, is one of those parameters. InapreviousstudyonthefiniteelementmodeloftheIASC-ASCEbenchmarkprob- lem,Caicedoetal.(2004)concluded(onthebasisofsyntheticdata)thattheNExT/ERAresults are not sensitive to the window size used in the computation of the CPSD. However, it may not be true in general, especially when one deals with experimental data. Generally speaking, the windowsizemustbeselectedbasedontwofactors: (a)howfastthesystemresponseis,and(b) howlongtherecordlengthis. Thelowerlimitofthewindowsizeisdeterminedbythesystem’s fundamentalperiod. Basedonexperience,thewindowsizeshouldbelargeenoughtoincludeat least 50 periods of the fundamental mode. On the other hand, the upper limit is determined by therecordlength. Ifthewindowsizeisverylargecomparedtotherecordlength,itwillresultin lessstatisticalaveraging,andconsequentlymorenoisycorrelationtimehistories. Toconfirmtheabovementionedstatement,aparameterstudyoftheinfluenceofthewindow size was performed. To this end, the NExT/ERA method was implemented for a Hamming windowwithdifferentsizes,rangingfromT 1 to75T 1 ,whiletheotheruser-selectableparameters were kept unchanged. T 1 is the first natural period of the system, whose value for the NCB is approximately6seconds. TheresultsofthementionedparameterstudyfortheNCBdatasetare showninFig. 2.4. TheplotsintheFig. 2.4showtheinfluenceofthedatawindowsizeontheidentifiednatural frequencies, and the first two modal ratios of critical damping. It is clear from Fig. 2.4 that the 33 0.2 0.3 0.4 0.5 0.6 0.7 0 10 20 30 40 50 60 70 80 Identified natural frequencies [Hz] Window size T 1 (a) 0 5 10 15 20 0 10 20 30 40 50 60 70 80 Identified modal damping (ζ 1 ) % Window size T 1 0 5 10 15 20 0 10 20 30 40 50 60 70 80 Identified modal damping (ζ 2 ) % Window size T 1 (b) (c) Figure 2.4: Effect of the window size on the NCB identification results using NExT/ERA method. For all cases: window overlap=75%, r = 24, p = 2/3 of the cross-correlation data points,andT 1 = 6sec. AllavailableDOFs,one-at-a-time,areusedasthereference. resultsaresensitive,tosomedegree,tothechangesinthewindowsize. However,nofurthersig- nificant changes is observed, when the window size is increased beyond 50T 1 . Figures 2.4(b,c) showtheeffectofthewindowsizeontheidentifiedmodaldampingforthefirsttwomodes. Itis worth noting that, although the natural frequency of the first two modes converged to their final valueforawindowsizegreaterthan10T 1 ,themodaldampingdidnotconvergeuntilthewindow size increased to25T 1 . Overall, the window size of about50T 1 would be an appropriate choice for both the natural frequencies and modal damping values. Another important user-selectable 34 0.2 0.3 0.4 0.5 0.6 0.7 0 5 10 15 20 25 Identified natural frequencies [Hz] No. of block rows (r+1) in the ERA Hankel matrix (H) (a) 0 5 10 15 20 0 5 10 15 20 25 Identified modal damping (ζ 1 ) % No. of block rows (r+1) in the ERA Hankel matrix (H) 0 5 10 15 20 0 5 10 15 20 25 Identified modal damping (ζ 2 ) % No. of block rows (r+1) in the ERA Hankel matrix (H) (b) (c) Figure 2.5: Effect of the number of block rows (r +1) in the Hankel matrix (H) on the NCB identification results using the NExT/ERA method. The value of(r+1) is an indication of the initialmodelorder. Forallcases: windowsizeandoverlap≈ 410secand75%,respectively,and p = 2/3oftheavailabledatapoints. AllavailableDOFsaresequentiallyusedasthereference. parameter is the smaller dimension of the SVD matrix, which is equivalent to the number of block rows in theH andH matrices for ERA and ERA/DC, respectively. once the number of block rows is selected, the number of block columns is determined based on the available data points(N p )usingEq. (2.21)and(2.22). Asdiscussedearlier,thesmallerdimensionoftheSVD matrix (here the number of rows) is an indication of the model order. However, the final model orderisnotknownapriori,andtheinitialmodelorderneedstobeoverspecified,tosomedegree, 35 inordertoreducethebiaserrorduetonoise. Intheprevioussection,someguidelineswerepre- sented to determine the degree of the initial model order overspecification for both ERA and ERA/DC (see Eq. (2.19) and (2.20)). To verify the aforementioned guidelines in conjunction with the NCB experimental data, a parameter study was performed. Figures 2.5 and 2.6 show theeffectofincreasingthenumberofSVDmatrixblockrowsontheNCBidentificationresults fortheNExT/ERAandNExT/ERA-DCmethods,respectively. FromFig.2.5,itisclearthattherearenosignificantchangesintheidentifiednaturalfrequen- ciesanddampingfactorsfor(r+1)> 20,whichconfirmsthevalidityofEq. (2.19),assuming M = n. Relatively small fluctuations in the corresponding damping factors can be neglected, keeping in mind that damping estimation is always less accurate than the frequency estimation. As expected, the initial model order required for the ERA/DC is much lower, compared to the ERA. Figure 2.6 shows that, for the same data set, ERA/DC results (frequency and damping) convergedtotheirfinalvalueforamuchsmallernumberofblockrows((α+1)> 9). However, for the third mode, it requires higher initial model order. The reason is due to the fact that the second and third modes are very closely-spaced in frequency. Since ERA/DC requires a lower initialmodelorder(smallernumberofrows),itsSVDmatrixwillbesmallerinsize,andconse- quently its computation is faster. It is worth noting that the overspecified model order is finally reducedtoitstruevalueusingthealgorithmshowninFig. 2.3. 36 2.4.2 ResultsandComparison Tables 2.1 and 2.2 summarize and compare the NCB modal parameter identification results obtained using NExT/ERA, NExT/ERA-DC, and NExT/LS methods. It also compares the cor- relation between the identified mode shapes using the modal assurance criterion. In the afore- mentionedtables,highCMIvaluesshowhighconfidence(forCMI > 80%)orextremelyhigh confidence(forCMI > 99%)intheidentificationresults. Ontheotherhand,highMPCvalues for the identified mode shapes, indicate the monophase behavior of the mode shapes’ complex components;inanotherwords,itindicatesthattheidentifiedmodescanbeconsideredasclassi- cal(proportionallydamped)normalmodes. It is seen from Table 2 that while the least-squares approach results match the results of the othertwomethodsformostofthedominantmodes,thereareseveralmodesthatarenotdetected, andforseveraloftheidentifiedmodes,theLSidentificationresultedinrelativelylowvaluesfor theMPCandCMIindices. Figure 2.7 shows the correlation (MAC value) between the mode shapes identified using NExT/ERA, NExT/ERA-DC, and NExT/LS methods. In this plot, the size of each rectangle is proportional to the MAC value of the corresponding mode pairs. For MAC values greater than 0.7,thecorrespondingrectanglesaredarkened,whichindicatehighcorrelation. From Tables 2.1 and 2.2 and Fig. 2.7, one can observe that, for the majority of modes, thereisaverygoodagreementbetweentheidentificationresultsofthethreepresentedmethods, especially between NExT/ERA and NExT/ERA-DC results. However, there are some modes that did not show up in the NExT/LS results. On the other hand, the modal damping values identified using different methods are occasionally significantly different. The main reason is 37 that, typically the identified modal damping values are not as accurate as the corresponding identifiedmodalfrequencies. However,fortheNCB,themodaldampingvaluesobtainedbythe presentedmethodsareallpositiveandwithinareasonablerange. Figures 2.8 to 2.10 show the final identified mode shapes and their corresponding frequen- cies. Since the mode shapes are typically complex, they are also plotted in the polar plane (fig- uresontheleft-hand-side),whereeacharrowinthepolarplanerepresentsacomplexcomponent ofthemodeshapevector. Table 2.1: Comparison of the NCB identification results using NExT/ERA and NExT/ERA- DC methods. For all cases: window size and overlap=409.6 sec and 75%, respectively, and all available DOFs are used as the reference. For ERA/DC:r = 0,α = 10,β = 70, andp = 2/3 of the available data points. For ERA:r = 25, andp = 2/3 of the available data points, (MPC isthemodalphasecollinearityandCMIistheconsistentmodeindicator). Identificationresults Identificationresults ModeShape usingNExT/ERA-DCmethod usingNExT/ERAmethod Comparison mode freq ζ MPC CMI mode freq ζ MPC CMI MAC No. Hz % % % No. Hz % % % % 1 0.165 0.59 99.79 99.81 1 0.165 1.47 99.67 99.72 99.99 2 0.193 1.21 99.62 99.95 2 0.193 1.28 99.41 99.93 99.85 3 0.202 4.00 97.66 99.10 3 0.197 5.99 92.13 97.06 97.85 4 0.257 0.46 99.96 99.99 4 0.257 0.49 99.89 99.99 100.00 5 0.349 0.31 99.89 99.95 5 0.349 0.35 99.72 99.95 100.00 6 0.364 0.61 99.94 99.97 6 0.366 0.55 99.60 99.62 99.74 7 0.476 0.19 99.14 99.43 7 0.474 0.43 99.15 99.29 99.48 8 0.559 0.79 98.07 99.36 8 0.559 1.09 96.09 98.72 99.46 9 0.644 0.30 99.22 99.66 9 0.644 0.47 98.84 99.56 99.96 10 0.685 1.47 97.01 98.03 10 0.688 1.47 97.49 98.16 97.76 11 0.733 0.27 98.04 98.74 11 0.733 0.08 98.55 99.53 99.80 12 0.782 1.50 87.72 88.41 12 0.774 0.27 86.38 88.05 98.03 13 0.795 0.27 99.62 99.77 13 0.795 0.29 99.18 99.77 99.99 - - - - - 14 0.954 1.10 95.93 96.46 - 14 0.958 0.22 99.69 99.82 15 0.958 0.17 98.94 99.69 99.36 15 1.034 0.82 92.38 94.44 16 1.036 0.51 92.43 95.25 98.93 16 1.154 0.41 99.69 99.78 17 1.154 0.42 99.37 99.81 100.00 17 1.333 0.72 99.09 99.23 18 1.335 0.75 98.56 99.12 99.94 18 1.361 0.53 92.46 93.88 19 1.364 0.66 88.90 92.93 91.78 19 1.564 0.69 99.57 99.64 20 1.562 0.68 99.48 99.70 99.99 20 1.686 0.39 97.61 97.83 21 1.686 0.44 97.03 98.12 99.42 38 Table 2.2: Comparison of the NCB identification results using NExT/LS and NExT/ERA-DC methods. Identificationresults Identificationresults ModeShape usingNExT/LSmethod usingNExT/ERA-DCmethod Comparison mode freq ζ MPC CMI mode freq ζ MPC CMI MAC No. Hz % % % No. Hz % % % % 1 0.167 1.04 99.69 99.86 1 0.165 0.59 99.79 99.81 99.58 2 0.194 0.75 93.92 94.07 2 0.193 1.21 99.62 99.95 94.38 - - - - - 3 0.202 4.00 97.66 99.10 - 3 0.258 0.31 99.91 99.92 4 0.257 0.46 99.96 99.99 99.96 4 0.351 0.16 99.92 99.94 5 0.349 0.31 99.89 99.95 99.93 5 0.367 0.51 99.96 99.99 6 0.364 0.61 99.94 99.97 99.74 6 0.474 0.20 99.58 99.68 7 0.476 0.19 99.14 99.43 99.34 - - - - - 8 0.559 0.79 98.07 99.36 - 7 0.619 1.02 58.30 68.73 9 0.644 0.30 99.22 99.66 75.20 8 0.673 0.59 63.94 97.50 10 0.685 1.47 97.01 98.03 95.85 9 0.732 0.21 72.05 77.64 11 0.733 0.27 98.04 98.74 90.83 - - - - - 12 0.782 1.50 87.72 88.41 - 10 0.811 0.55 84.89 89.84 13 0.795 0.27 99.62 99.77 96.04 11 0.971 0.25 94.63 95.84 14 0.958 0.22 99.69 99.82 70.54 - - - - - 15 1.034 0.82 92.38 94.44 - 12 1.147 0.40 75.94 86.79 16 1.154 0.41 99.69 99.78 89.06 13 1.346 0.68 95.99 97.19 17 1.333 0.72 99.09 99.23 97.88 - - - - - 18 1.361 0.53 92.46 93.88 - 14 1.562 0.47 98.18 99.11 19 1.564 0.69 99.57 99.64 99.56 15 1.618 0.39 90.66 93.85 20 1.686 0.39 97.61 97.83 93.52 39 0.2 0.3 0.4 0.5 0.6 0.7 0 5 10 15 Identified natural frequencies [Hz] No. of block rows (α+ 1) in the ERA/DC correlation Hankel matrix (H) (a) 0 5 10 15 20 0 5 10 15 Identified modal damping (ζ 1 ) % No. of block rows (α+ 1) in the ERA/DC correlation Hankel matrix (H) 0 5 10 15 20 0 5 10 15 Identified modal damping (ζ 2 ) % No. of block rows (α+ 1) in the ERA/DC correlation Hankel matrix (H) (b) (c) Figure 2.6: Effect of the number of block rows (α + 1) in the correlation Hankel matrix (H) ontheNCBidentificationresultsusingtheNExT/ERA-DCmethod. Thevalueof (α+1)isan indicationoftheinitialmodelorder. Forallcases: windowsizeandoverlap≈ 410secand75%, respectively,r = 0,β = 70, andp = 2/3 of the available data points. All available DOFs are sequentiallyusedasthereference. 40 5 10 15 20 2 4 6 8 10 12 14 16 18 20 NExT/ERA mode number NExT/ERA−DC mode number Modal Assurance Criterion (MAC) 2 4 6 8 10 12 14 2 4 6 8 10 12 14 16 18 20 NExT/LS mode number NExT/ERA−DC mode number Modal Assurance Criterion (MAC) (a) (b) Figure 2.7: Comparison of the NCB identified mode shapes using Modal Assurance Criterion (MAC):(a)comparisonbetweenNExT/ERAandNExT/ERA-DCmodeshapes,(b)comparison betweenNExT/LSandNExT/ERA-DCmodeshapes. Thesizeofeachrectangleisproportional to the MAC value of the corresponding mode pairs. For MAC values greater than 0.7, the correspondingrectanglesaredarkened,whichindicatehighcorrelation. 41 30 210 60 240 90 270 120 300 150 330 180 0 (a)1stbendinginlateraldirection,f n = 0.165Hz. Note: thisistopview. 30 210 60 240 90 270 120 300 150 330 180 0 (b)1stbendinginverticaldirection,f n = 0.193Hz. 30 210 60 240 90 270 120 300 150 330 180 0 (c)2ndbendinginverticaldirection,f n = 0.202Hz. 30 210 60 240 90 270 120 300 150 330 180 0 (d)3rdbendinginverticaldirection,f n = 0.257Hz. Figure 2.8: Identified mode shapes and their frequency,f n (NExT/ERA-DC results). Since the mode shapes are typically complex, they are also plotted in the polar plane (figures on the left), whereeacharrowinthepolarplanerepresentsacomplexcomponentofthemodeshapevector. 42 30 210 60 240 90 270 120 300 150 330 180 0 (e)4thbendinginverticaldirection,f n = 0.349Hz. 30 210 60 240 90 270 120 300 150 330 180 0 (f)2ndbendinginlateraldirection,f n = 0.364Hz. Note: thisistopview. 30 210 60 240 90 270 120 300 150 330 180 0 (g)1sttorsion,f n = 0.476Hz. 30 210 60 240 90 270 120 300 150 330 180 0 (h)5thbendinginverticaldirection,f n = 0.559Hz. Figure2.9: Identifiedmodeshapes(Cont.) . 43 30 210 60 240 90 270 120 300 150 330 180 0 (i)6thbendinginverticaldirection,f n = 0.644Hz. 30 210 60 240 90 270 120 300 150 330 180 0 (j)3rdbendinginlateraldirection,f n = 0.685Hz. Note: thisistopview. 30 210 60 240 90 270 120 300 150 330 180 0 (k)2ndtorsion,f n = 0.733Hz. 30 210 60 240 90 270 120 300 150 330 180 0 (isometricview) (topview) (l)combinationoftorsionandbendinginlateraldirection,f n = 0.782Hz. Figure2.10: Identifiedmodeshapes(Cont.) . 44 2.5 SummaryandConclusions In this study, three time-domain techniques were considered for the modal parameter identifi- cation of the new Carquinez bridge, a modern long suspension bridge, based on ambient and forced vibration measurements collected before this new bridge was opened for traffic. These three methods are: the eigensystem realization algorithm (ERA), the ERA with data correla- tions (ERA/DC), and the least squares (LS) algorithm. In order to implement these methods using output-only information, the natural excitation technique (NExT) was first used to con- vertthenonhomogeneousequationofmotiontoahomogeneousone. Anautonomousalgorithm waspresentedtodistinguishthegenuinestructuralmodesfromspuriousnoiseorcomputational modes. OneimportantissueintheNExTtechniqueistheselectionofaproperreferenceDOF.Since theoptimumaccuracyfordifferentmodesoccursfordifferentchoicesofthereferenceDOFs,itis preferabletousemultiplereferencesasopposedtoasinglereference. Identificationformulations weremodifiedtoincludemanyreferencepointssimultaneously,oroneatatime. The study shows that the ERA/DC method requires less initial model overspecification in the presence of noise, and it can be computationally faster than the ERA. Another issue in the implementation of the mentioned techniques in real experimental applications is choosing the right values for user-selectable parameters. Some useful guidelines for the selection of critical parameterswerepresented. Toverifythoseguidelines,someparameterstudieswereperformed. It was shown that these identification techniques are capable of being used in online structural healthmonitoringschemesandincalibratingandvalidatingthefiniteelementmodels. 45 Chapter3 ApplicationofStructuralHealthMonitoring TechniquestoTrackStructuralChangesina RetrofittedBuildingBasedonAmbient Vibration 3.1 Introduction Background Structural health monitoring (SHM) approaches based on analyzing the vibration signature of targetinfrastructuresystemshavebeenreceivingalotofattention,foralongperiodoftime,by manyinvestigatorsallovertheworld. Somerepresentativepublicationsthatprovideacomprehensiveoverviewofthebroadinter- disciplinary field of SHM, the main technical challenges, as well as promising approaches that havethepotentialofbeingusefultoolsforconditionassessmentinconjunctionwithfieldimple- mentations related to civil infrastructure systems, are available in the works of Agbabian and Masri (1988), Natke and Yao (1988), Housner and Masri (1990), Natke et al. (1993), Housner etal.(1994),Chen(1996),Housneretal.(1997b),Chang(2003),CasciatiandMaganotte(2000), 46 Casciati (2002), Balageas (2002), Chen et al. (2002), Liu (2003, 2004), Boller and Staszewski (2004),Smyth(2004),Chang(2005),Sohnetal.(2003),Doeblingetal.(1998),FarrarandJau- regui(1998a),Lusetal.(1999),Becketal.(1994),Fujinoetal.(2005),GaoandSpencer(2002), Stubbs et al. (2000), Lynch et al. (2004), Pei et al. (2006), Masri et al. (2004), and Fraser et al. (2003). Whiletheabovelistedpublicationscontainnumerousstudiesandtechnicalmaterialaddress- ing all aspects of the broad field of structural health monitoring, there are virtually no publica- tions in the open literature that report on studies involving SHM of full-scale civil structures where the dynamic response was monitored before, during, and after significant, quantifiable structuralchangesintroducedintothem. There are many challenging technical problems that await solution before the promising aspects of the SHM field can be fully realized in realistic applications under field conditions to detect, locate and quantify the level of changes (damage) in monitored structures. Conse- quently, working groups within the Structural Control community operating under the auspices of the International Association for Structural Control and Monitoring (IASCM) and in collab- oration with the American Society of Civil Engineers (ASCE) have established a sequence of benchmarkproblems(utilizingmainlysyntheticdata),ofincreasingcomplexityandprogressive sophistication, in order to provide the research community with well-planned and documented “experiments” that can be used to assess the utility of various SHM methodologies. Further details about those benchmark problems and the many technical publications arising from their useareavailableintheworkofHousneretal.(1997b). 47 Given the importance of having well-calibrated damage-detection data sets, as well as the paucityofstudiesthatevaluatemethodologiesbasedonactualvibrationmeasurementsobtained under realistic field conditions, it would be quite useful to have a sequence of data sets from the same physical structure in which quantifiable levels of structural modifications (acting as surrogates of “damage”) are introduced at various locations in the dispersed structure. That is preciselythefocusofthispaper. OneimportantissueinSHMistomonitorthestructuralchangesusingtheresponsedataonly. The importance of this issue comes from the fact that for identification of large civil structures, applying known excitation forces is costly, if not infeasible. Moreover, output only algorithms are the best option for an online monitoring system which can monitor the vibration signature ofastructureconstantly,withouttheapplicationofameasurableinputtothesystem. Theiden- tification approaches which use both well-defined inputs as well as output records may not be applicableforthecasewheretheexcitationisfromambientsourcesandisgenerallyunmeasur- able. The Natural Excitation Technique (NExT) developed by James et al. (1993, 1996); Farrar and James (1997), has been successfully used for identification of structures based on response dataonly(Caicedoetal.,2004). UsingtheNExT,onecanconverttheforcedvibrationequation of motion to the free vibration (homogeneous) one. The Eigensystem Realization Algorithm (ERA) (Juang and Pappa, 1985, 1986) can then be used to identify the modal parameters of the system, from free-decay data. ERA is a multi-input/multi-output, time domain technique that can efficiently identify “minimum order” realization. It is also very efficient in identification of closely spaced and weakly excited modes. Other studies used similar identification approaches 48 forestimatingthemodalpropertiesofstructuresbasedonoutput-onlytimehistories,suchasQ- Markov cover algorithm (Lew et al., 1993), stochastic subspace identification method (Peeters and Roeck, 2001b; Van Overschee and De Moor, 1996), and least-squares based algorithms (Smythetal.,2003). AnapplicationtocivilengineeringstructuresoftheNExTmethodologyin combinationwiththeERAispresentedin(Brownjohn,2003),whereambientvibrationdataare usedforidentificationoftallbuildings. Scope Asaresultofthe1994Northridgeearthquake,acriticalfacilityinthemetropolitanLosAngeles regionwasfoundtoneedsignificantseismicmitigationmeasures. TheLongBeachPublicSafety Building (LBPSB) was instrumented with 14 state-of-the-art strong-motion accelerometers that wereplacedatvariouslocationsandindifferentorientationsthroughoutthebuilding. Theinstru- mentation network was used to acquire extensive data sets at regular intervals that covered the whole construction phase, during which the building evolved from its original condition to the retrofitted state. The paper by Chassiakos et al. (2007a) provides an overview of the ambient vibration data collected before, during, and after the structural retrofit. The goal of the present studyistoapplytheNExT/ERAapproachtoidentifythemodalparametersoftheLBPSBbefore, during,andaftertheretrofit,soastomonitorthechangesinthemodalparameters,andcorrelate themwiththeconstructionphases. 49 3.2 BuildingCharacteristicsanditsInstrumentation The Long Beach Public Safety Building (LBPSB) , built in the 1950’s, is a six-story, rectangu- lar, steel frame building, with one of its dimensions being considerably longer than the other one(dimensions270ftx70ft;equivalentto82.3x21.3meters). TheLBPSBisacriticalfacil- ity located in downtown Long Beach, California. The facility houses the city police, the office of emergency services, and the city jail. Attached to the LBPSB is the downtown fire station. Duetothecriticalrolethatthebuildingisrequiredtoplayduringanearthquakeemergency,itis importantthatnotonlyitremainssafeforitsoccupants,butthatitalsoremainscompletelyfunc- tional. As a result of the 1994 Northridge earthquake, the mentioned facility was found to need significantseismicmitigationmeasures. Forthisreason,a$30millionretrofitprojectwasunder- taken,tostiffenthebuildingbyconstructingshearwallsatselectedlocations. Partoftheretrofit project is the instrumentation of the building with different sensors, such as accelerometers, strain gages, fiberoptic strain gages, acoustic emission sensors, and laser-based displacement sensors. Theaccelerometerswereinstalledbeforethebeginningoftheretrofitconstruction,and havebeenremainedoperationalduringtheconstructionphase. Thisprovidedtheopportunityto measurethechangingcharacteristicsofthebuildingastheconstructionprogressed. In a recent paper (Chassiakos et al., 2007a) the authors presented the preliminary data col- lected from the accelerometers during the construction phase, and correlated the changes in fundamental frequencies with the construction progress. A detailed finite element model of the pre-retrofit and post-retrofit building was developed and static, dynamic, linear, and nonlinear FEanalyseswereperformed. 50 The structural changes recommended on the basis of the structural dynamic studies using the two FE models discussed in (Chassiakos et al., 2007a), were incorporated in the retrofitted building. Two main options were initially considered for the retrofit of the LBPSB: the first optionincludedbaseisolation,whereasthesecondoptionincludedstiffeningofthebuildingvia shear walls. The option of constructing additional shear walls was eventually chosen for the retrofit. The construction phase involved the introduction of significant structural modifications which required an extended period of time to incorporate. During the 6-month construction period, several shear walls were added at various locations and in different orientations in the structure. The fact that the structural modifications under discussion were introduced in an incremen- tal fashion, and in a distributed form, makes the collected data sets quite useful in offering the opportunitytocapturevariouslevelsofphysicalstructuralchanges,thushelpingtoestablishthe “detectibility”thresholdofpotentialstructuralhealthmonitoringanddamage-detectionmethod- ologies. 3.3 FiniteElementAnalysisOfBuildingModel 3.3.1 Pre-retrofitmodel In the numerical model of the pre-retrofit building, six floors above the ground level plus the warehouseontheroofweremodeledindetailwithfiniteelements(asshowninFigure3.1). The model included beam elements for columns and girders, truss elements for bracing members, and plate-shell elements for slabs and shear walls. Lumped mass elements were used to model 51 the exterior concrete panels as well as concrete partition walls, which were considered as non- structural components. However, their weight was an important factor in the dynamic analysis. Otherbuilding’sdeadweightswereestimatedbasedonthefloorusageandmodeledbyapplying nonstructuralweightdensitiestoplate-shellelements. Twoconcreteslabthicknesses,i.e.,4to4.5inches(10.16to11.43centimeters),wereusedin thebuilding. Becausetheslabswererelativelystiffcomparedwithbeamsanddidnotcarrymajor seismic loads, the materials of the slabs were assumed to remain in the linear range throughout theanalysis. Theshearwallswereprimarilylocatedonthefirstfloor. Theconcretewallsforthe stairwayswerenotshearwalls,sincetheywerenotbuiltcontinuouslytothegroundlevel. They were,however,includedinthemodeltosimulatethelocalreinforcement. Theconcretestructure of the roof warehouse was relatively heavy and stiff, and contributed noticeable inertia loads to thebuildingduringtheseismicmotions. Moreover,thecoordinatereferencesofthemodelwere defined at the floor levels, which did not align with the centerlines of the beams. In order to match the model definition level, each end of the beam elements was specified with an offset distancefunctioninglikearigidjointattheendsofthebeams. Sincethebuildingwasprimarily a steel frame construction, the possible locations of failure were defined at the joints of beams andcolumns. ThesectionpropertiesofbeamsandcolumnswereextractedfromtheManualofSteelCon- struction published by American Institute of Steel Construction. Since the grade of steel mem- bers were not available from the as-built plan, a most commonly used material Grade 36 ( F y = 36ksi)wasassumedforallmembers. 52 Figure3.1: FiniteElementModelofPre-retrofitLongBeachPublicSafetyBuilding. Thesoilswereproperlyrepresentedinthemodaltransientresponseanalysis. Sincethelarge nonlinearsoilbehaviorwasnotexpectedduringearthquakes,areasonableassumptionwasmade bymodelingthesoilsasasinglelayeredhalf-space. Thestiffnessandinertiaofsupportingsoils was simulated by using the real part of the impedance of the foundation at the fundamental frequency of the building. The soil damping is considered in the modal damping ratio selected for the analysis. In addition, a modal damping ratio of 0.05 was used in the modal transient responseanalysis. Toinvestigatetheseismicresistancestrengthofthepre-retrofitbuilding,thestructuralmem- ber forces are computed and checked against their individual failure measure defined by the yieldingbendingmomentsinthebeamsandthecolumnsandthebucklingforcesofthecolumns. In theory, the stress measure of the beams or columns should be determined by an interactive equationbetweentheircorrespondingaxialforceandbendingmoments. However,foramedium heightstructureliketheLBPSbuildingtheaxialforcesinthecolumnsandbeamsarerelatively smallcomparedwiththemomentsgeneratedduringmajorearthquakes. Tosimplifythestrength 53 evaluation,onlythebendingmomentsinthebeamsandcolumnsareexaminedagainsttheindi- vidual yield moment obtained by multiplying the yield stress by the members section modulus. ThiscanbeexpressedasM/M y ≤ 1.0,whereM y istheyieldmomentofabeam/columncross- section in the load carrying direction. The structural response for the Northridge earthquake input, as shown in Figure 3.2, was computed with a 0.02-second time increment. The peak acceleration of the record was 0.35g. The bending moments in the beams and columns at each time step were examined. However, results indicated that some members yielded for the case witha50%reducedearthquakeinput. Thiswasrelativelylowcomparedwithordinarybuildings withevenlydistributedstiffness. Acceleration Record of Northridge Earthquake (Arleta 90 Deg.) -0.3 -0.2 -0.1 0 0.1 0.2 0.3 0.4 0 5 10 15 time (second) acceleration (g) Figure3.2: GroundaccelerationtimehistoryofNorthridgeearthquake. Byobservingtheresults,itwasfoundthatcriticalbeamsandcolumnsidentifiedwithayield strength ratio exceeding 100% were mostly located on the second and third floors, with some on the 5th and 6th floors. The large bending moments in those members resulted from large lateralfloordisplacementsduetotheweakstiffnessofthefirstandsecondfloorsandtherocking 54 motion of heavy concrete walls such as stairways or partition walls from above, which were not built continuously from the first floor of the building, during earthquakes. The locations of criticalstructuralmembersareshowninFigures 3.3and 3.4. 2nd & 3rd flrs. 5th & 6th flrs. Fig. A-2. Critical beams in pre-retrofit building due to the rocking of discontinuous concrete walls. N Figure 3.3: Critical beams in pre-retrofit building due to the rocking of discontinuous concrete walls. 2nd flr. 3rd flr. Fig. A-3. Critical columns in Pre-retrofit Building dominated by large bending moments. N Figure3.4: Criticalbeamsinpre-retrofitbuildingdominatedbylargebendingmoments. The finite element model of the pre-retrofit building was also enhanced by adding concrete basement floors and surrounding soils. The purpose of this enhancement was to simulate the 55 soil-structureinteractioneffects. Whenthebuildingwassubjectedtoearthquakeexcitations,the surroundingsoilwouldrespondtotheseismicforcesalongwiththebuilding. Insuchaway,the structure-soilsystemvibratedatlowerfrequencies,andaportionoftheseismicenergywouldbe dissipatedbackintothesoil. Theenergydissipationaccountsforasizeableamountofdamping in the dynamic system, and is important to the prediction of accurate structural behavior. Two levels of the basement, with stiffness primarily from retaining walls, were modeled. The soils were represented by stiffness and damping elements. The basement levels were relatively stiff in comparison with the upper structures. Therefore, no structural member in the basement was expected to deform into the nonlinear range. As shown in Figure 3.1, there are 1250 nodal pointsand3891finiteelementsinthecomputationalmodelofheLBPSB.Figure 3.5showsthe post-retrofitmodel. Modalanalysisofpre-retrofitbuilding The modal analysis of the finite element model was performed using the computer software NASTRAN. The natural frequencies of the first three modes were found to be 0.94, 1.20 and 1.47 Hz, respectively. The fundamental mode is the first bending mode of the building in the N-Sdirection,thesecondisthefirstbendingmodeintheE-Wdirection,thethirdthefirsttorsion mode, andthefourththesecondbendingintheN-Sdirectionofthebuilding. Themodeshapes are shown in Figures 3.6 to 3.9. Because every beam joint is included in the finite element model, the motions such as in-plane bending of floor and pitch of stairways and warehouse can be simulated. Compared to general building structures, the fundamental period of the Public SafetyBuildingisslightlylonger,whichindicatesthatthestructureismoreflexiblethantypical buildings. 56 Modaltransientresponseanalysisofpre-retrofitbuildingusingNASTRAN Linear transient analysis of the pre-retrofit building was performed using time history data of sample earthquakes. The bending moments at beam joints and axial forces of truss members were checked against their corresponding yielding and buckling strength to identify the critical structuralmembersinthefiniteelementmodel. Theanalysisequationcanbewrittenas M¨ u(t)+C˙ u(t)+Ku(t) =p(t) (3.1) Where M, C, and K are the mass matrix, damping matrix, and stiffness matrix of the building respectively; ¨ u(t), ˙ u(t), and u(t) are the acceleration, velocity and displacement vectors of the building structure respectively; p(t) is the time dependent loading, which in this case, it is the earthquake input and can be written as−M¨ u g , where ¨ u g is the ground acceleration at the base ofthebuilding. Inthemodalapproach,thedisplacementcanbewrittenas u =Φq (3.2) in whichΦ is the matrix of mode shapes, q is the vector of modal amplitude. Premultiplying equation(3.1)byΦ T leadsto m¨ q(t)+c˙ q(t)+kq(t) =f(t) (3.3) 57 Where m =Φ T MΦ, c =Φ T CΦ, and k =Φ T KΦ are the modal mass, modal damping, and modalstiffnessmatricesrespectively. f(t)isthemodalloadingvector,Φ T p(t). Because NASTRAN does not allow a fix-based earthquake input, an approach using very large masses at the base of the first floor was adopted. In this approach, the earthquake input is applied to the masses with a factor proportional to the magnitude of the masses in order to generateproperresponseattheupperstructure. Theearthquakeinputrecordsincludedthetime history of 1994 Northridge earthquake recorded at Arelta in the three different directions, the records of 1989 Loma Prieta earthquake and a local earthquake recorded at the Long Beach Public Utilities Building. It was found that a significant number of major structural members approachedorexceededtheiryieldlevel. 3.3.2 Post-retrofitmodel The finite element model of the post-retrofit building was developed. The model was generated based on engineering drawings provided by the structural engineering company responsible for the retrofit design of the building. In the design, the lateral stiffness of the building was signifi- cantlyenhancedbyintroducingadditionalshearwallsandextendingtheconcretestairwaycases tothebasements. Thenewshearwalllayoutwassymmetricwithrespecttothecenterlineofthe building, and the shear walls were relatively stiff. Compared with the pre-retrofit model in Fig- ure 3.1, the additional shear walls in the post-retrofit model can be clearly visualized. It should bementionedthatthebasementisincludedonlyinthepost-retrofitmodel,whichhoweverdoes nothaveanynoticeablecontributiontothefrequencychanges. Using the two FE models of the building discussed above, the first five natural frequencies, before and after retrofit are listed in table 3.1, and the the first four natural mode shapes of 58 Figure3.5: Finiteelementmodelofpost-retrofitLongBeachPublicSafetyBuilding. the building are illustrated in Figures 3.6 to 3.9. Because of different layouts of shear walls, the vibration mode sequences of the two models are somewhat different. For example, the first torsionofthebuildingisthethirdmodeinthepre-retrofitmodel,butbecomesthesecondmode in the post-retrofit model. An interesting behavior of the post-retrofit model is that the in-plane bending of floors becomes a lower mode, i.e., the fourth mode, compared to the eighth mode in the pre-retrofit model. This also indicates the significant increase of lateral stiffness in the new building. Table3.1: NaturalfrequenciesinHzoftheFEmodelbeforeandafterretrofit. Mode FundamentalFrequency FundamentalFrequency Percentage beforeretrofit afterretrofit Change 1 0.94 2.09 122.34% 2 1.20 2.52 110.00% 3 1.47 2.87 95.24% 4 3.00 5.21 73.67% 5 4.25 7.60 78.82% 59 (a)beforeretrofit,f 1 = 0.94Hz (b)afterretrofit,f 1 = 2.09Hz Figure3.6: The1 st modeshape. Thebuildingplanswereslightlymodified,soonecanexpectasmalldifferencebetweenthe model and the post-retrofit measurements, after analyzing the data from March 2004, when all theshearwallswerecompleted. 3.4 RetrofitPhase The structural changes recommended on the basis of the structural dynamics studies based on the two computational models discussed above, were incorporated in the retrofitted build- ing. This construction phase involved the introduction of significant structural modifications whichrequiredanextendedperiodoftimetoincorporate. Duringtheconstructionperiodwhich requiredsixmonths,severalshearwallswereaddedatvariouslocationsandindifferentorienta- tionsinthestructure. Figure 3.10showsthetimescheduleofshearwallcompletion,wherethe dash and solid lines denote partially and complectly built respectively. Figures 3.11 and 3.12 60 (a)beforeretrofit,f 2 = 1.20Hz (b)afterretrofit,f 2 = 2.52Hz Figure3.7: The2 nd modeshape. showsomesamplestructuralmodificationsunderconstruction. Thefactthatthestructuralmod- ificationsunderdiscussionwereintroducedinanincrementalfashion,andinadistributedform, makes the collected data sets quite useful in offering the opportunity to capture various levels ofphysicalstructuralchanges,thushelpingtoestablishthe“detectibility”thresholdofpotential structuralhealthmonitoringmethodologies. 3.5 InstrumentationandDataAcquisition Before the retrofit, the LBPSB was instrumented with strong-motion accelerometers, placed on each floor. The 2 nd and 5 th floors were instrumented with 3-axis accelerometers (X,Y on hori- zontal plane, along the major axes of the building; and Z along the vertical). The remaining 4 floors were instrumented with 2-axis accelerometers, for a total of 14 channels of acceleration 61 (a)beforeretrofit,f 3 = 1.47Hz (b)afterretrofit,f 3 = 2.87Hz Figure3.8: The3 rd modeshape. (a)beforeretrofit,f 4 = 4.25Hz (b)afterretrofit,f 4 = 7.60Hz Figure3.9: The4 th modeshape. measurements. Figure3.13showsaschematicplotoftheaccelerometers’locationsinthebuild- ing. The high dynamic range accelerometers (120 db) recorded the data at the sampling rate of 100Hz. TheyalsoweresupportedbyGPStimingsynchronization. 62 Schedule of Shear Wall Completion M6 M7 M8 M9 M10 M11 M12 Time (Months) Floor 2 3 4 5 Completed Completed Completed Completed Dashed line = Partially built Solid line = Completed Figure3.10: ScheduleofShearWallCompletion. Figure3.11: Retrofitphaseofthebuilding. Inadditiontotheaccelerometers,theLBPSBwasalsoinstrumentedwithstraingages,fiber- opticstrainsensors,laserbaseddisplacementsensors,andacousticemissionsensors. Theinstru- mentationlististabulatedinfigure 3.14andthecorrespondinginstrumentationsystemarchitec- tureisshowninfigure 3.15. 63 Figure3.12: Preparationofnewshearwall. X Z Y 1 x 2 x 3 x 4 x 5 x 6 x 2 y 5 y 1 y 2 z 3 y 4 y 5 z 6 y Figure3.13: SchematicplotofthesensorlocationsinLBPSB. 64 2-axis EPI Sensor STRONG MOTION ACCELEROMETERS. Installed pre-retrofit K2 recorder DATA ACQUISITION . Pre- and Post-retrofit installation. SG STRAIN GAGES. Installation during retrofit. AE ACOUSTIC EMISSION SENSORS. Post-retrofit installation. FO SG FIBER-OPTIC STRAIN GAGES. Post-retrofit installation. LFD LASER-BASED FLOOR DISPLACEMENT. Install during retrofit. EPI Sensor STRONG MOTION ACCELEROMETERS. Post-retrofit installation. Figure3.14: Instrumentationlist. 2-axis EPI Sensor 6 th Floor 3-axis EPI Sensor 5 th Floor EPI Sensor EPI Sensor 2-axis EPI Sensor 4 th Floor EPI Sensor 2-axis EPI Sensor 3 rd Floor EPI Sensor 3-axis EPI Sensor 2 nd Floor EPI Sensor 2-axis EPI Sensor 1st Floor EPI Sensor K2 recorder K2 recorder K2 recorder PC RS232 RS232 Basement Instrument cabinet on 1 st floor SG AE SG SG SG SG SG FO SG LFD LFD LFD LFD LFD LFD AE Figure3.15: Instrumentationsystemarchitecture. 65 3.6 Implementation of the Modal Parameter Identification and Its Results This section reports the results of implementation of the proposed modal parameter identifica- tionalgorithmsinconjunctionwiththedatarecordedfromLBPSB,before,during,andafterthe retrofit phase. The modal parameter identification procedure used in this study consists of to mainstages. Thefirstistoeliminatetheeffectofunknownforcefromthegoverningequationof motion using the Natural Excitation Technique (NExT), and the second is to extract the modal parametersofthehomogeneousmodelusingEigensystemRealizationAlgorithm. Detailedfor- mulation and discussion about this procedure can be found in the chapter 2 and in the paper publishedbytheauthors(Nayerietal.,2006). As mentioned earlier, the LBPSB was instrumented with 14 accelerometers. The 2 nd and 5 th floorswereinstrumentedwith3-axisaccelerometers,andtheremaining4floorswereinstru- mented with 2-axis accelerometers (shown in Fig. 3.13). In this study, the records from 12 channelswereused,whichincludes6sensorsinthexand6sensorsinthey direction. Thedata were synchronized among all channels, using GPS-based timing. Before, during, and after the retrofitphase,thedatawereacquiredalmosttwiceaweek,forall14channelsattherateof100 samplespersecond. Totally,42setsoftherecordeddataareusedinthisstudy. Table3.2shows the recording dates and their corresponding index number. As one can see from Fig. 3.10, the retrofit phase was completed in December, 2003, but the recording process was continued for another month up to January, 2004. It should be mentioned that for the first two months (June and July of 2003) the record length was about 400 seconds, and after that it was increased to 66 1214 seconds. It is worth noting that, the record lengths were arbitrary chosen by the building ownersandthedatawasthenprovidedtotheauthors. Table3.2: RecordDatesandtheCorrespondingIndexNumber. indexnumber recorddate indexnumber recorddate indexnumber recorddate 1 12-Jun-2003 15 06-Aug-2003 29 31-Oct-2003 2 16-Jun-2003 16 11-Aug-2003 30 05-Nov-2003 3 19-Jun-2003 17 13-Aug-2003 31 12-Nov-2003 4 23-Jun-2003 18 17-Sep-2003 32 14-Nov-2003 5 30-Jun-2003 19 19-Sep-2003 33 19-Nov-2003 6 02-Jul-2003 20 24-Sep-2003 34 03-Dec-2003 7 07-Jul-2003 21 26-Sep-2003 35 05-Dec-2003 8 09-Jul-2003 22 01-Oct-2003 36 10-Dec-2003 9 14-Jul-2003 23 03-Oct-2003 37 12-Dec-2003 10 16-Jul-2003 24 08-Oct-2003 38 19-Dec-2003 11 25-Jul-2003 25 10-Oct-2003 39 22-Dec-2003 12 28-Jul-2003 26 15-Oct-2003 40 29-Dec-2003 13 30-Jul-2003 27 17-Oct-2003 41 05-Jan-2004 14 04-Aug-2003 28 29-Oct-2003 42 09-Jan-2004 In order to track the modal parameter changes in the 42 data sets, we need to compare the identified mode shapes, set by set. The Modal Assurance Criterion (MAC), is calculated to quantify the correlation between complex mode shapes, identified from different data sets. The MACvaluebetweentwomodeshapes ˆ Φ i and ˆ Φ j isdefinedas MAC i,j = | ˆ Φ H i ˆ Φ j | 2 ( ˆ Φ H i ˆ Φ i )( ˆ Φ H j ˆ Φ j ) (3.4) where the superscriptH, denotes the Hermitian of a matrix. The MAC value ranges between 0 and1;0fororthogonaland1foridenticalmodeshapes. Giventwosetsofnmodeshapesfrom twodifferentdatasets,onecanforman(n×n)matrixofMACvalues,whichcanthenbeused torelatethemodeshapesofthetwodatasets. 67 0 2 4 6 8 10 12 14 X 1st floor Y 1st floor X 2nd floor Y 2nd floor X 3rd floor Y 3rd floor X 4th floor Y 4th floor X 5th floor Y 5th floor X 6th floor Y 6th floor All 12 DOF simultaneously Ref. DOF Identified natural frequencies [Hz] 1st mode 2nd mode 3rd mode 4th mode 5th mode 6th mode Figure 3.16: Identified natural frequencies of one data set (02-Jul-2003) using ERA/DC, for different choices of reference DOF. The other identification parameters are exactly the same: windowsize=2048points,overlap=50%,r = 0,p = 665,α = 5,andβ = 90. 68 0 1 2 3 4 5 6 projection of the 1st mode on X axis normalized relative disp X DOF 0 1 2 3 4 5 6 projection of the 1st mode on Y axis normalized relative disp Y DOF 0 1 2 3 4 5 6 projection of the 2nd mode on X axis normalized relative disp X DOF 0 1 2 3 4 5 6 projection of the 2nd mode on Y axis normalized relative disp Y DOF (a)1 st mode,f 1 = 1.15Hz,ζ 1 = 3.7% (b)2 nd mode,f 2 = 1.99Hz,ζ 2 = 2.3% 0 1 2 3 4 5 6 projection of the 3rd mode on X axis normalized relative disp X DOF 0 1 2 3 4 5 6 projection of the 3rd mode on Y axis normalized relative disp Y DOF 0 1 2 3 4 5 6 projection of the 4th mode on X axis normalized relative disp X DOF 0 1 2 3 4 5 6 projection of the 4th mode on Y axis normalized relative disp Y DOF (c)3 rd mode,f 3 = 4.19Hz,ζ 3 = 3.4% (d)4 th mode,f 4 = 5.50Hz,ζ 4 = 6.5% 0 1 2 3 4 5 6 projection of the 5th mode on X axis normalized relative disp X DOF 0 1 2 3 4 5 6 projection of the 5th mode on Y axis normalized relative disp Y DOF 0 1 2 3 4 5 6 projection of the 6th mode on X axis normalized relative disp X DOF 0 1 2 3 4 5 6 projection of the 6th mode on Y axis normalized relative disp Y DOF (e)5 th mode,f 5 = 9.68Hz,ζ 5 = 5.3% (f)6 th mode,f 6 = 13.51Hz,ζ 6 = 2.9% Figure3.17: IdentifiedmodeshapescorrespondingtothefrequenciesinFig. 3.16. 69 2 4 6 8 10 −120 −100 −80 −60 −40 −20 0 Freq [Hz] CPSD¨ X 1 ¨ X 12 (f)[dB] 2 4 6 8 10 −120 −100 −80 −60 −40 −20 0 Freq [Hz] CPSD¨ X 2 ¨ X 12 (f)[dB] 2 4 6 8 10 −120 −100 −80 −60 −40 −20 0 Freq [Hz] CPSD¨ X 3 ¨ X 12 (f)[dB] 2 4 6 8 10 −120 −100 −80 −60 −40 −20 0 Freq [Hz] CPSD¨ X 4 ¨ X 12 (f)[dB] 2 4 6 8 10 −120 −100 −80 −60 −40 −20 0 Freq [Hz] CPSD¨ X 5 ¨ X 12 (f)[dB] 2 4 6 8 10 −120 −100 −80 −60 −40 −20 0 Freq [Hz] CPSD¨ X 6 ¨ X 12 (f)[dB] 2 4 6 8 10 −120 −100 −80 −60 −40 −20 0 Freq [Hz] CPSD¨ X 7 ¨ X 12 (f)[dB] 2 4 6 8 10 −120 −100 −80 −60 −40 −20 0 Freq [Hz] CPSD¨ X 8 ¨ X 12 (f)[dB] 2 4 6 8 10 −120 −100 −80 −60 −40 −20 0 Freq [Hz] CPSD¨ X 9 ¨ X 12 (f)[dB] 2 4 6 8 10 −120 −100 −80 −60 −40 −20 0 Freq [Hz] CPSD¨ X 10 ¨ X 12 (f)[dB] 2 4 6 8 10 −120 −100 −80 −60 −40 −20 0 Freq [Hz] CPSD¨ X 11 ¨ X 12 (f)[dB] 2 4 6 8 10 −120 −100 −80 −60 −40 −20 0 Freq [Hz] CPSD¨ X 12 ¨ X 12 (f)[dB] Figure 3.18: Typical semilogarithmic plots of the CPSD between the response of a reference DOF and the response of all available DOFs. For this case, the reference DOF is X 12 , which corresponds to the y axis of the sensor on the roof. The window size is 2048 points, and the overlap is 50%. For ease of comparison, identical abscissa and ordinate scales are used for all displayedplots. 70 0 2 4 6 8 10 −1 −0.5 0 0.5 1 x 10 −4 time [sec] R ¨ X 1 ¨ X 12 (τ) 0 2 4 6 8 10 −1 −0.5 0 0.5 1 x 10 −4 time [sec] R ¨ X 2 ¨ X 12 (τ) 0 2 4 6 8 10 −1.5 −1 −0.5 0 0.5 1 1.5 x 10 −4 time [sec] R ¨ X 3 ¨ X 12 (τ) 0 2 4 6 8 10 −2 −1 0 1 2 x 10 −4 time [sec] R ¨ X 4 ¨ X 12 (τ) 0 2 4 6 8 10 −2 −1 0 1 2 x 10 −4 time [sec] R ¨ X 5 ¨ X 12 (τ) 0 2 4 6 8 10 −4 −2 0 2 4 x 10 −4 time [sec] R ¨ X 6 ¨ X 12 (τ) 0 2 4 6 8 10 −1 −0.5 0 0.5 1 1.5 2 x 10 −4 time [sec] R ¨ X 7 ¨ X 12 (τ) 0 2 4 6 8 10 −6 −4 −2 0 2 4 x 10 −4 time [sec] R ¨ X 8 ¨ X 12 (τ) 0 2 4 6 8 10 −1 −0.5 0 0.5 1 x 10 −3 time [sec] R ¨ X 9 ¨ X 12 (τ) 0 2 4 6 8 10 −1.5 −1 −0.5 0 0.5 1 1.5 x 10 −3 time [sec] R ¨ X 10 ¨ X 12 (τ) 0 2 4 6 8 10 −1.5 −1 −0.5 0 0.5 1 1.5 x 10 −3 time [sec] R ¨ X 11 ¨ X 12 (τ) 0 2 4 6 8 10 −4 −2 0 2 4 x 10 −3 time [sec] R ¨ X 12 ¨ X 12 (τ) Figure3.19: TheCCFcorrespondingtotheCPSDshowninFig. 3.18. TheCCFisequivalentto theinverseFouriertransformofCPSD.Notethat,foraddedresolution,differentordinatescales areusedforeachdisplayedplots. 71 5 10 15 20 25 30 35 40 1 1.5 2 ω 1 [Hz] date index 10 20 30 40 0 0.02 0.04 0.06 0.08 0.1 ζ 1 date index 10 20 30 40 0 0.2 0.4 0.6 0.8 1 MAC change for the 1st mode date index 10 20 30 40 1.8 2 2.2 2.4 2.6 ω 2 [Hz] date index 10 20 30 40 0 0.02 0.04 0.06 0.08 0.1 ζ 2 date index 5 10 15 20 25 30 35 40 0 0.2 0.4 0.6 0.8 1 MAC change for the 2nd mode date index (a) (b) Figure3.20: Timevariationofthenaturalfrequency,dampingratio,andMACvaluechangesfor the(a)1 st mode,and(b)2 nd mode. ThedateindexcorrespondstodatasetsinTable3.2. 72 10 20 30 40 4 4.5 5 5.5 ω 3 [Hz] date index 10 20 30 40 0 0.02 0.04 0.06 0.08 0.1 ζ 3 date index 5 10 15 20 25 30 35 40 0 0.2 0.4 0.6 0.8 1 MAC change for the 3rd mode date index 5 10 15 20 25 30 35 40 5 5.5 6 6.5 7 7.5 8 8.5 ω 4 [Hz] date index 10 20 30 40 0 0.02 0.04 0.06 0.08 0.1 ζ 4 date index 5 10 15 20 25 30 35 40 0 0.2 0.4 0.6 0.8 1 MAC change for the 4th mode date index (a) (b) Figure3.21: Timevariationofthenaturalfrequency,dampingratio,andMACvaluechangesfor the(a)3 rd mode,and(b)4 th mode. ThedateindexcorrespondstodatasetsinTable3.2. Asexplainedintheprevioussection,selectionofthereferenceDOFmightbeachallenging task, and, in most cases, one can not rely on just one reference. To clarify the challenge, a parameterstudyoverthereferenceDOF,wasconducted. Foronesetofdata,allpossiblechoices ofthereferenceDOFwereimplementedwhiletherestoftheidentificationalgorithmparameters werekeptthesame. TheresultingnaturalfrequenciesandmodesshapesareshowninFig. 3.16 and3.17,respectively. InFig. 3.16,theverticalaxisshowsthedifferentchoicesofthereference DOF, and the horizonal axis shows the resulting identified natural frequencies using ERA/DC. 73 Itclearlyillustratestheadvantageofusingall12DOFsasthereferenceatthesametime. Itcan be also concluded that some modes might not show up when a certain DOF is selected as the reference,forexample: • Iftheyaxisofthe6 th floorisselectedasreference,onlythe1 st ,3 rd ,and4 th modeswillbe identified. • If the x axis of the 6 th floor is selected as reference, only the 2 nd , 3 rd , and 4 th modes will beidentified. • Iftheyaxisofthe5 th floorisselectedasreference,onlythe1 st ,3 rd ,and5 th modeswillbe identified. • The 2 nd mode, which is dominantly a bending mode in the x direction, will not show up, iftheyaxisofanyfloorisselectedasthereference. However,whenall12DOFsaresimultaneouslyusedasthereference,asignificantimprovement isobserved,andallsixmodesshowup. Therefore,forthisstudy,wewillusesimultaneouslyall availableDOFsasreference. To implement the identification methodology, the first step is to compute the Cross- Correlation Function (CCF) between the response of the preselected reference DOF and the response of all available DOFs. The CCF can be estimated using the inverse Fourier transform of the Cross-Power-Spectral Density (CPSD). The CPSD is computed directly from the data. Random errors associated with the CPSD can be minimized by windowing and averaging. For this study, a Hanning window of 2048 points, with 50% overlap was used for the computation oftheCPSD.Sincethefirstfundamentalfrequencyofthebuildingisaround1Hz,aframewith 74 30 210 60 240 90 270 120 300 150 330 180 0 1st complex mode shape 0 0.5 1 0 1 2 3 4 5 6 normalized relative disp X DOF projection of the 1st mode on X axis 0 0.5 1 0 1 2 3 4 5 6 normalized relative disp Y DOF projection of the 1st mode on Y axis 30−Jun−2003 19−Sep−2003 05−Jan−2004 Figure 3.22: 1 st mode shape in the complex plane and its projection onto thex andy axes. The plotshowstheresultsforthreedifferentdatasets;30Jun2003,19Sep2003,and5Jan2004. 2048points,thatcontainsatleast20cyclesofthefirstfundamentalmode,isareasonablechoice. Figure3.18showstypicalCPSDsbetweentheresponseofareferenceDOFandtheresponseof all available DOFs. The corresponding inverse Fourier transforms which are equivalent to the CCFareshowninFig. 3.19. UsingthecalculatedCCF,onecanformtheHankelmatrix(Eq. 2.8) to implement the ERA or ERA/DC algorithm. As mentioned earlier, ERA/DC has better noise rejection properties and less computational burden. Consequently, we just present the ERA/DC results in this paper. However, it must be mentioned that for the present study, both the ERA and ERA/DC were implemented, and the results show that for the data sets used, the estimated modalparametersfromthetwoalgorithmsarequitsimilar. TheparametersofERA/DC,wereselectedbasedontheguidelinespresentedintheprevious section. Since there are 12 measurement DOFs, N was chosen to be 12, and then the initial modelorderwasoverspecifiedto3times(3×(2×12) = 72). BasedonEq. (2.20)and(2.22), r,p,α,andβ wereselectedtobe0,665,5,and90,respectively. Asdiscussedearlier,ERA/DC 75 has computational advantage when the number of columns in the Hankel matrix is much larger thanthenumberofrows,andthatisthereasontheHankelmatrixof1blockrow,and666block columnswereselected. During the structural retrofit, the dominant frequencies of the building increased consid- erably, reflecting a stiffer building. These changes are quantified through frequency evolution plotsinFigs3.20and3.21. Figures3.20and3.21showthetimehistoryoftheidentifiednatural frequencies, damping ratios and the MAC value changes for the 1 st , 2 nd , 3 rd , and 4 th modes, respectively. Thedateindex(abscissa)inthesefiguresisrelatedtotherecordingdatesbasedon Table3.2. Itmustbenotedthat: • DuringthemonthofJuly2003,partoftheshearwallsonthe3 rd and4 th floorswerebuilt. This resulted in changes of about 20%, 7%, 14%, and 12% for the natural frequencies of the1 st ,2 nd ,3 rd ,and4 th modes,respectively. • DuringAugust2003,the2 nd and3 rd floorshearwallswerecompleted,resultinginchanges of about 17%, 4%, 3%, and 8% for the natural frequencies of the 1 st , 2 nd , 3 rd , and 4 th modes,respectively. • During September and October 2003, very little retrofit activity took place, resulting in smallchangesintheidentifiednaturalfrequencies. • During November 2003, the shear walls for the whole 5 th floor were built, resulting in about20%, 11%, 6%, and12%changesinthe naturalfrequenciesofthe1 st , 2 nd , 3 rd , and 4 th modes,respectively. 76 • After November 2003 (data set index 34), no further retrofit activities were conducted; hencethere mustbe noconsiderable changesafter that. The timehistoryof theidentified frequenciesconfirmsthat. 30 210 60 240 90 270 120 300 150 330 180 0 2nd complex mode shape 0 0.5 1 0 1 2 3 4 5 6 normalized relative disp X DOF projection of the 2nd mode on X axis 0 0.5 1 0 1 2 3 4 5 6 normalized relative disp Y DOF projection of the 2nd mode on Y axis 30−Jun−2003 19−Sep−2003 05−Jan−2004 Figure3.23: 2 nd modeshapeinthecomplexplaneanditsprojectionontothexandy axes. The plotshowstheresultsforthreedifferentdatasets;30Jun2003,19Sep2003,and5Jan2004. The identified complex mode shapes and the projection of their real parts onto thex andy axesareshowninFig. 3.22,3.23,3.24,and3.25,forthe1 st ,2 nd ,3 rd ,and4 th modes,respectively. The complex mode shapes are plotted in the polar plane, where each arrow in the polar plane representsacomplexcomponentofthemodeshapevector. ItisclearfromFig. 3.22to3.25that the first and third modes are dominantly bending in the Y-Z plane, and the second and fourth modesare dominantlybending inthe X-Zplane. Thehigher modesareformed fromprojection of other torsional and bending modes on the sensor locations. It should be noted that, since the sensorlocationsareselectedtobecollinear,theycannotcaptureallthetorsionalmodes. 77 30 210 60 240 90 270 120 300 150 330 180 0 3rd complex mode shape −0.4 −0.2 0 0.2 0.4 0.6 0.8 0 1 2 3 4 5 6 normalized relative disp X DOF projection of the 3rd mode on X axis −0.4 −0.2 0 0.2 0.4 0.6 0.8 0 1 2 3 4 5 6 normalized relative disp Y DOF projection of the 3rd mode on Y axis 30−Jun−2003 19−Sep−2003 05−Jan−2004 Figure3.24: 3 rd modeshapeinthecomplexplaneanditsprojectionontothexandy axes. The plotshowstheresultsforthreedifferentdatasets;30Jun2003,19Sep2003,and5Jan2004. In general, the identified mode shapes using the ERA or ERA/DC algorithms are complex vectors. However, for a classically (proportionally) damped mode, all the complex components ofthemodeshapearecollinearinthecomplexplane(phaseangledifferencesareeither0or180 degrees). Inotherwords,ifthecomponentsofamodeshapearescatteredinthecomplexplane, the mode is not proportionally damped. From figures 3.22, 3.23, one can see that the first two modes are completely real (collinear in the complex plane), and hence they can be regarded as classically damped modes. On the other hand, the 3 rd and 4 th modes in the complex plane (Fig. 3.24 and 3.25) show that some relatively small elements deviated from the collinearity assump- tion, although the main (larger) elements are still collinear. The deviation from collinearity in the higher modes may be due to violation of modeling assumptions and processing errors. As a result, the estimated modal damping factors for these two modes are not as accurate as the first two modes, and that might be the reason why the estimated damping values for the 3 rd and 4 th modesarescatteredfarapartoverawiderange. 78 30 210 60 240 90 270 120 300 150 330 180 0 4th complex mode shape −0.5 0 0.5 0 1 2 3 4 5 6 normalized relative disp X DOF projection of the 4th mode on X axis −0.5 0 0.5 0 1 2 3 4 5 6 normalized relative disp Y DOF projection of the 4th mode on Y axis 30−Jun−2003 19−Sep−2003 05−Jan−2004 Figure3.25: 4 th modeshape inthe complexplaneand itsprojection ontothexandy axes. The plotshowstheresultsforthreedifferentdatasets;30Jun2003,19Sep2003,and5Jan2004. 3.7 SummaryandConclusions The background and identification results from a unique opportunity to apply Structural Health Monitoring approaches to a critical facility in Southern California are presented. As a result of the 1994 Northridge earthquake, a critical facility in the metropolitan Los Angeles region was found to need significant seismic mitigation measures. The facility consisted of a six-story buildingthathousedessentialemergencyservices. Thebuildingwasinstrumentedwith14state- of-the-art strong motion accelerometers that were placed at various locations and in different orientationsthroughoutthebuilding. Theinstrumentationnetworkwasusedtoacquireextensive datasetsatregularintervalsthatcoveredthewholeconstructionphase,duringwhichthebuilding evolvedfromitsoriginalconditiontotheretrofittedstatus. Afinite-elementmodelofthebuilding wasdevelopedandusedtoestimatethedominantfrequenciesandmodeshapesbeforeandafter theretrofit. 79 In this study, two multi-input/multi-output state space methods that are suitable for exper- imental modal parameter identification of structures were considered. These two methods are the eigensystem realization algorithm (ERA) and the ERA with data correlations (ERA/DC). The study shows that the ERA/DC requires less initial model overspecification in the presence of noise, and it can be computationally faster than the ERA. In order to implement the ERA (ERA/DC) using output-only information, the natural excitation technique (NExT) was used, whichcanconvertthenonhomogeneousequationofmotiontoahomogeneousone. Oneissuein theimplementationofthementionedtechniquesinrealexperimentalapplicationsischoosingthe right values for user-selectable parameters. Some useful guidelines for the selection of critical parameterswerepresented. Theidentificationmethodologywasimplementedtotrackstructural changesofLongBeachPublicSafetyBuilding(LBPSB)duringitsretrofit. Themainchallenges with this unique experimental study were: (a) low resolution sensor placement, which results in high model order reduction, (b) the ambient excitation was so small that the higher modal displacements were in the noise level, hence not adequately excited. It was shown that these identificationtechniquesareextremelycapabletobeusedinonlinestructuralhealthmonitoring schemes. 80 Chapter4 StructuralIdentificationandMonitoringofa Full-Scale17-StoryBuildingBasedonAmbient VibrationMeasurements 4.1 Introduction BackgroundandMotivation Structural health monitoring (SHM) approaches based on analyzing the vibration signature of targetinfrastructuresystemshavebeenreceivingalotofattention,foralongperiodoftime,by manyinvestigatorsallovertheworld. Whiletherearemanytechniquesandapproachesinvolved inthenondestructiveevaluation(NDE)andconditionassessmentofstructuralsystems,theycan all be broadly categorized as local or global methods, alternatively, decentralized or centralized methods. Thefirstcategoryincludesmethodsdesignedtoprovideinformationaboutarelatively smallregionofthesystemofinterestbyutilizinglocalmeasurements,whilethesecondcategory ofmethodsusesmeasurementsfromadispersedsetofsensorstoobtainglobalinformationabout thecondition ofthesystem. Clearly, thetwoapproaches arecomplementaryto eachother, with the optimum choice of method highly dependent on the scope of the problem at hand and the 81 nature of the sensor network. In this study we consider both of the mentioned categories (the localandglobal)oftheidentificationtechniques. One of the main drivers of growing interest and capabilities in the field of SHM of civil infrastructuresystemsistheincreasingwide-spreadavailabilityofsensornetworksthathavethe potentialtocollectvastamountsofdatahither-to-forenotfeasibletoacquire. Simultaneouswith thisincreasingcapabilitytocollectdata,istheparallelinterestintheapplicationofmoresophis- ticated data processing algorithms to identify the structural parameters (in different formats) of thetargetinfrastructuresystem. Somerepresentativepublicationsthatprovideacomprehensive overview of the broad interdisciplinary field of SHM, the main technical challenges, as well as promising approaches that have the potential of being useful tools for condition assessment in conjunctionwithfieldimplementationsrelatedtocivilinfrastructuresystems,areavailableinthe worksofAgbabianandMasri(1988), NatkeandYao(1988), HousnerandMasri(1990), Natke etal.(1993),Housneretal.(1994),Chen(1996),Housneretal.(1997b),Chang(2003),Casciati and Maganotte (2000), Casciati (2002), Balageas (2002), Chen et al. (2002), Liu (2003, 2004), Boller and Staszewski (2004), Smyth (2004), Chang (2005), Sohn et al. (2003), Doebling et al. (1998), Farrar and Jauregui (1998a), Lus et al. (1999), Beck et al. (1994), Fujino et al. (2005), GaoandSpencer(2002),Stubbsetal.(2000),Lynchetal.(2004),Peietal.(2006),Masrietal. (2004),Fraseretal.(2003),andSPIE(2007). Beforeusingtheidentifiedvibrationsignatureofastructureforhealthmonitoringpurposes, the associated statistical variability must be investigated. There are many sources other than damagethatcancausesomevariationsinthedynamiccharacteristicsofastructure,suchasnon- linearity in the system, temperature variations, moisture absorption, soil structure interactions, 82 nonstationarities in the observed system, and measurement noise and hysteresis. Hence, it is worthwhile to use statistical averaging to extract a distribution of the identified parameters, and then use those distributions to assess the variability of the identified parameters under normal conditionsaswellashavingameasureofconfidenceintervals. There are currently few available studies that investigate the statistical variability of the vibrationparametersoffullscalestructuresduetoenvironmentaleffectsorrandomerrorsasso- ciatedwithmeasurementandsignalprocessing(Cornwelletal.,1999;Farraretal.,1997;Peeters and Roeck, 2001a; Sohn et al., 1999). The main reason for this situation appears to be the lack of availability of measurement data from physical structures under realistic field conditions. In particular,tohaveastatisticallymeaningfuldiscussion,oneneedstocollectdays(ormonths)of data under various environmental and operational conditions. There are cost related issues that undermine the feasibility of such a system in most cases. In addition, managing and processing thehugeamountofcollecteddataisachallenginghurdletoovercome. Scope Followingthe1994Northridgeearthquake,theFactorBuildingwhichisacriticalfacilitylocated at the University of California, Los Angeles (UCLA) campus, was instrumented with 72 state- of-the-art strong-motion accelerometers that were placed at various locations and in different orientations throughout the building. The instrumentation network has been constantly acquir- ing and storing extensive amounts of vibration data. This collection of data provides a unique opportunity to evaluate the effectiveness of various identification methodologies on a full-scale structure,andunderrealisticfieldconditions. Inaddition,verylongtime-historyrecordingunder 83 different environmental and operational conditions, makes it possible to statistically investigate thevariabilityofthebuilding’svibrationsignature. In this study, two types of local and global time-domain identification methods are imple- mented. The first method is the eigensystem realization algorithm (ERA) (Juang and Pappa, 1985, 1986) in conjunction with the natural excitation technique (NExT) (James et al., 1993, 1996). It is used to extract the modal parameters (natural frequencies, mode shapes, and modal damping) of the building, based on ambient vibration records. This type of identifica- tion methodology is regarded as global (or centralized) approach, since it deals with the global dynamic properties of the structure. The second method is a time-domain identification tech- niqueforchain-likeMDOFsystemsintroducedbyMasrietal.(1982). Suchaclassofproblems encompasses many practical applications including tall buildings like the one under discussion. Since in this method the identification of each link of the chain is performed independently, it is regarded as local (or decentralized) identification methodology. For the same reason, this method can be easily adopted for large scale sensor network architectures in which the central- ized approaches are not feasible due to massive storage, power, bandwidth, and computational requirements. Generally this method can be applied to linear or nonlinear chain-like MDOF systems. Nonetheless,itrequirestheappliedforcestobeknown. However,forfullscaleinfras- tructuresliketheoneunderdiscussion,providingmeasurableexcitation(i.e.,beingabletomea- sure the excitation forces) is very costly, and therefore one needs to just rely on the available ambient vibration records. In order to handle the cases with unmeasurable external forces, the chain identification procedure is generalized by combining it with the idea of the NExT tech- nique. Having 50 days of recorded data under different temperature and ambient conditions 84 provided the opportunity to investigate the effects of environmental and operational conditions ontheestimateddynamicpropertiesofthebuilding. Variabilityoftheestimatedparametersdue totemperaturevariationswithina24-hourperiodisdiscussed. 4.2 ModalParameterIdentificationApproach Themodalparameteridentificationprocedureusedinthisstudyconsistsoftwomainstages: The firstistoeliminatetheeffectoftheunknownforcefromthegoverningequationofmotionusing the Natural Excitation Technique (NExT), and the second is to extract the modal parameters of the homogeneous model using the Eigensystem Realization Algorithm. Detailed formulation anddiscussionaboutthisprocedurecanbefoundinthechapter2andinthepaperpublishedby theauthors(Nayerietal.,2006). 4.3 FormulationofTheChainSystemIdentificationApproach Modal parameters are global vibration signature of a structure. However, it is well-known that damageistypicallyalocalphenomenon,andconsequentlyitmaynotimposenoticeablechanges in the global dynamic characteristics of a structure (Farrar et al., 2001; Farrar and Jauregui, 1998a). In another words, typical modal parameters (natural frequencies, modal damping, and modeshapes)arenotsensitiveenoughtolowlevelsofdamage. Furthermore,assumingthatthe damageissevereenoughtobedetectedbythechangesintheglobalmodalparametersofastruc- ture,theystillcannotprovideenoughinformationtopinpointthelocationofthedamage(Farrar et al., 2001). On the other hand, future generation of smart sensor networks with dense array of hundreds or thousands of sensors require the development of new identification techniques 85 thatworkbasedonlocal(asopposedtoglobal)processing. Centralizedstructuralmonitoringin real-timeusingrawtime-historiesoflargenumberofsensorsisnotfeasibleduetomassivestor- age,power,bandwidth,andcomputationalrequirements(SpencerandNagayama,2006). These factsmotivatetheinvestigationinthisthesisofmoresophisticatedidentificationtechniquesthat especially relies on local behavior of structure, in order to detect and locate possible structural damage. Inthefollowingsubsections,adecentralizedidentificationtechniqueispresentedforchain- like Multi-Degree-Of-Freedom (MDOF) linear or non-linear dynamic systems. The method uses information about the state variables of each element of the chain-like system to express the system characteristics in terms of some polynomial basis functions. Since in this method, the identification of each element of the chain is done individually, it is amenable to a decen- tralized structural health monitoring procedure. The present study extends the work of Masri et al.(1982) bygeneralizingthe approachto handlethe caseof chain-likeMDOF dynamicsys- tems with unknown (immeasurable) external force. It is worth noting that chain-like structures encompassesmanypracticalengineeringapplicationssuchastallbuildings, turbineblades, and airplanewings. 4.3.1 GeneralNonlinearCase ConsideraMDOFchain-likestructureshowninFig. 4.1whichconsistsof nelements,eachwith alumpedmassm i ,andanarbitrary(unknown)non-linearrestoringfunction G (i) . Thestructure may be subjected to a base excitationx 0 (t), and/or directly applied forces,F i (t). In context of civil structures, this system would be analogous to a n-story building under the ambient forces and ground motion. Let’s assume the absolute acceleration at each element, ¨ x i (t), is directly 86 availablefrommeasurement. Theotherstatevariables,x i and ˙ x i (t),canbecomputedbydigital signal processing of the acceleration records. At this stage, we also need to assume that the applied forceF i (t) and the magnitude of the lumped mass at each elementm i are measurable. Therelativemotionbetweentwoconsecutiveelementscanbecomputedasfollows z i (t) =x i (t)−x i−1 (t), i = 1,2,...,n (4.1) i x n m ) (n G 2 m ) 2 ( G 1 m ) 1 ( G ) 3 ( G ) (t F n ) ( 2 t F ) ( 1 t F Figure4.1: ModelofaMDOFchain-likesystem. Areasonableassumptioninthefieldofstructuraldynamicsisthattherestoringforceateach elementis onlydependent onthe relativedisplacement andvelocity acrossthe terminals ofthat elementas: G (i) =G (i) (p (i) ,z i , ˙ z i ) (4.2) 87 wherep i is a vector whose parameters characterize the nature of the nonlinear element. The equationsofmotionforsuchasystemcanbewrittenas m n ¨ x n = F n (t)−G (n) (p (n) ,z n , ˙ z n ) m i ¨ x i = F i (t)−G (i) (p (i) ,z i , ˙ z i )+G (i+1) (p (i+1) ,z i+1 , ˙ z i+1 ) for i =n−1,n−2,...,1 (4.3) OnecanrewritetheEq. (4.3)toexpresstheunknownrestoringforcefunctionsas G (n) (p (n) ,z n , ˙ z n ) = F n (t)−m n ¨ x n G (i) (p (i) ,z i , ˙ z i ) = F i (t)−m i ¨ x i +G (i+1) (p (i+1) ,z i+1 , ˙ z i+1 ) for i =n−1,n−2,...,1 (4.4) Thus, starting from the tip of the chain, one can sequentially determine the time histories of all the “interstory” restoring forces within the chain. The advantage of this formulation is that the identification problem of a MDOF system is converted to a set of decoupled SDOF problems. For the very top element, the restoring force is directly computed by subtracting the inertiaforcefromtheexternalforcemeasuredatthetopelement. Then,startingfromtheelement right before the very last, the restoring force can be calculated by subtracting the inertia force from the external force at that element, plus the restoring force of the previous element which hasbeenalreadycomputedinthepreviousstep. Once the time-history of the restoring functions for all the elements are computed, one can use suitable basis functions to approximate a non-parametric model for each element. Based 88 on the work of Masri et al. (2004) a suitable choice of basis functions would be a power series expansioninthedoublyindexedseriesasfollows: Basis = Φ = qmax X q=0 rmax X r=0 z q ˙ z r (4.5) A third order expansion seems to be enough for most of the practical applications dealing withstructuraldynamics. Thus,forq max =r max = 3,thesetofbasisfunctionsbecome Φ ={1, ˙ z, ˙ z 2 , ˙ z 3 ,z,z˙ z,z˙ z 2 ,z˙ z 3 ,z 2 ,z 2 ˙ z,z 2 ˙ z 2 ,z 2 ˙ z 3 ,z 3 ,z 3 ˙ z,z 3 ˙ z 2 ,z 3 ˙ z 3 } (4.6) Standardleast-squaresmethodscanthenbeusedtofindtheindividualcoefficientsassociated with each basis function as demonstrated in the work of Masri and Caughey (1979) and Masri etal.(1987a,b): G (i) = qmax X q=0 rmax X r=0 p (i) qr z q i ˙ z r i G (i) = Φ (i) p (i) p (i) = [Φ (i) ] † G (i) (4.7) whereG (i) is aN ×1 vector whose elements are the time history samples of the i th restoring function;p (i) is a (1 +q max )(1 +r max )× 1 vector of the unknown parameters (p (i) qr ), to be identifiedintheprocess;theΦ (i) isaN×(1+q max )(1+r max )matrixoftheknowntimehistories of the basis functions (see Eq. (4.5)); N is the number of time samples; and the superscript† denotesthepseudoinverse. 89 4.3.2 TheLinearCasewithUnknownForce As was pointed out earlier, the chain system identification approach requires the applied forces to be known. However, there are many practical applications in which determining the applied forces is not feasible or very costly. In such applications, the only available external force is commonly from ambient sources and cannot in general, be measured. In this subsection, the chain identification method is extended to handle these cases, by combining it with the idea of the NExT technique (James et al., 1993, 1996). To this end, we need to assume that the dominant behavior of the chain-like system under consideration is linear and that each of the externalforcesisastationarywhitenoise. Consider the i th element of a chain-like system shown in Fig. 4.1. For this element the mass-normalizedrestoringforcecanbewrittenas(seeEq. (4.4)) ¯ G (i) = ¯ F i (t)− ¨ x i + ¯ m i ¯ G (i+1) (4.8) where ¯ G (i) = G (i) m i , ¯ F i = F i m i , ¯ m i = m i+1 m i (4.9) Multiplying both sides of Eq. (4.8) by a shifted scalar reference responsex r (t−τ), taking the expectation,andconsideringthedefinitionofthecorrelationfunctionR(.)yields R xr ¯ G (i)(τ) =R xr ¯ F i (τ)−R xr¨ x i (τ)+ ¯ m i R xr ¯ G (i+1)(τ) (4.10) whereR xy (τ)≡E[x(t)y(t+τ)]. 90 BasedontheNExTtechnique,theR xr ¯ F i (τ)canbeeliminated,assumingtheunknownforce isastationarywhitenoiseprocess,hence R xr ¯ G (i)(τ) =−R xr¨ x i (τ)+ ¯ m i R xr ¯ G (i+1)(τ) (4.11) Sincetheacceleration ¨ x i isknownandtherestoringforceoftheupperlevel ¯ G (i+1) hasbeen already computed in the previous step, all the terms on the right hand side of the Eq. (4.11) are known. Consequently,thecross-correlationtime-historyofthei th restoringfunctionR xr ¯ G (i)(τ) canbecomputed. Onetheotherhand,assumingthedominantbehaviorofthesystemislinear,onecanusethe followinglinearexpansionofthemass-normalizedrestoringforce: ¯ G (i) (t) = k i m i z i (t)+ c i m i ˙ z i (t) = ¯ k i z i (t)+¯ c i ˙ z i (t) (4.12) where ¯ k i and ¯ c i arethemass-normalizedstiffnessanddampingofthei th element. Multiplying both sides of Eq. (4.12) by a shifted scalar reference responsex r (t−τ) and repeatingtheabovementionedprocessyields R xr ¯ G (i)(τ) = ¯ k i R xrz i (τ)+¯ c i R xr ˙ z i (τ) (4.13) ThelefthandsideofEq. (4.13)hasbeenalreadycomputedinthepreviousstep(seeEq. 4.11)), andthecross-correlationtime-historiesontherighthandside( R xrz i andR xr ˙ z i )canbecomputed directly. Hence,onecanusestandardleast-squaresmethodtofindtheunknownmass-normalized stiffnessanddampingcoefficientsforeachelementofthechain-likesystem. 91 The main advantages of the above mentioned formulation are; (a) the unknown force is eliminated from the chain identification procedure, and (b) the absolute value of the lumped mass of each element is not needed to be known. Instead, one needs to know the mass ratio of each element with respect to the lower-level element i.e., ¯ m i = m i+1 m i , which is much easier to determine, or estimate. For example, if the mass is uniformly distributed, ¯ m i is identically one forallelements. Itisworthnotingthat,forthetypeofsystemunderdiscussion,theglobalM −1 K andM −1 C matricescanbereconstructedusingtheidentifiedlocalparameters ¯ k i and¯ c i usingthefollowing formulations: M −1 K = ¯ k 1 + ¯ k 2 ¯ m 1 − ¯ k 2 ¯ m 1 − ¯ k 2 ¯ k 2 + ¯ k 3 ¯ m 2 − ¯ k 3 ¯ m 2 . . . . . . . . . − ¯ k n−1 ¯ k n−1 + ¯ k n ¯ m n−1 − ¯ k n ¯ m n−1 − ¯ k n ¯ k n (4.14) M −1 C = ¯ c 1 +¯ c 2 ¯ m 1 −¯ c 2 ¯ m 1 −¯ c 2 ¯ c 2 +¯ c 3 ¯ m 2 −¯ c 3 ¯ m 2 . . . . . . . . . −¯ c n−1 ¯ c n−1 +¯ c n ¯ m n−1 −¯ c n ¯ m n−1 −¯ c n ¯ c n (4.15) 92 4.4 IllustrativeExample 4.4.1 DescriptionoftheExampleandDamageScenario Toillustratetheapplicationofthemethodsunderdiscussionforthestructuraldamagedetection and localization, consider an n DOF hypothetical chain-like structure shown in Fig. 4.2. Let’s assumethatn = 100(100DOFs),andthatthesystemislinear,whichyieldsthelinearexpansion of the restoring forces (see Eq. (4.12)). Uncorrelated broadband stationary random forces are appliedateachDOF.Inthisexample,itisassumedthattheexcitationforcesareimmeasurable, resemblingambientexcitation. Theonlyavailablemeasurementsareassumedtobeacceleration time-histories at certain DOFs. The displacement and velocity time-histories are computed by integrating the acceleration records. To make the simulation case more realistic, acceleration recordsarefirstnoise-pollutedatalevelof5%oftheirrespectiveRMSvaluewithuncorrelated whitenoise,andthenthenoisecontaminatedaccelerationtime-historiesareintegratedonceand thentwicetocomputethecorrespondingvelocity,anddisplacementtime-histories,respectively. This integration sequence of noisy signals, introduces noise amplification effects which distort thevelocityanddisplacementestimates,asiscommonlythecaseinrealapplications. Anotherimportantissuetobeconsideredinpracticalapplications,istheissueofmodel-order reduction, which is directly related to the resolution of sensor placement. Moreover, the degree of sensor resolution influences the resolution of damage localization. To investigate the effect of model-order reduction in this example, the number of measurement stations are assumed to bemuchsmallerthanthenumberofstructuralDOFs. Toquantifythedegreeofthemodel-order reduction, one can use the ratio of ΔL L , in which ΔL is the distance between two consecutive sensor stations, andL is the total length of the chain-like structure. Similarly, for the discrete 93 example under discussion, one can use the ratio of Δn n to represent the degree of model order reduction,inwhichΔnisthenumberofstructuralDOFsbetweentwoconsecutivemeasurement stations,andnisthetotalnumberofstructuralDOFs. 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 Damage 2 Damage 1 Sensor Stations Measurement Station i Measurement Station i+1 eq k % 90 ) ( ) ( = −Damage No s Damage s k k s k Figure 4.2: Schematic plot of the 100 DOF linear chain-like example under discussion. There are 20 sensor stations, hence Δn n = 1 20 . Damage is introduced at two locations by reduction of stiffness of two distinct elements (out of 100) by 10%, which results in 2.17% reduction of equivalent stiffness between twoconsecutivemeasurementstations. In order to investigate the effectiveness of the global modal parameter approach (the cen- tralizedapproach)andthechainsystemidentificationapproach(thedecentralizedapproach)for change detection and localization, one needs to create a damage scenario for this example. To 94 this end, low levels of damage are introduced at two locations by introducing a 10% stiffness reductionattwodistinctelements(outofthe100existingelements)asshowninFig. 4.2. Based on the geometry shown in Fig. 4.2, one can find the relationship between the stiffness of each elementk s , and the equivalent stiffness between two consecutive measurement stationsk eq , as follows: k eq = 1 Δn k s (4.16) assuming the stiffness of the elements between two consecutive measurement stations are iden- tical. Consequently, an amount of α% reduction of stiffness at one element will result in the followingratiofortheequivalentstiffnessvalues: k s (Damage) k s (No−Damage) = 1− α 100 ⇒ k eq (Damage) k eq (No−Damage) = Δn(1−α/100) (Δn−1)(1−α/100)+1 (4.17) Forexample,a10%stiffnessreductionatoneelement,and Δn n = 5 100 resultsina2.17%reduc- tionofequivalentstiffnessbetweentwoconsecutivemeasurementstations. In order to quantify the uncertainty in the estimated parameters, the simulation was per- formedtogenerate100ensembles,eachincludingatleast2000periodsofthefirstfundamental modeofthesystem. 4.4.2 ModalIdentificationusingNExT/ERA Inthissubsection,theeffectivenessoftheglobalmodalparametersfordetectingandlocatingthe damage in the example under discussion is investigated. Figures 4.3-4.4 summarize the modal parameter identification results using the NExT/ERA method. The first column of these figures 95 shows the identified mode shapes for the reference (undamaged) and damage cases (solid and dashedlines,respectively). Thesecondandthirdcolumnsofthesefigures,showthehistograms andfittedGaussianprobabilitydensityfunctions(pdf)fortheidentifiednaturalfrequenciesand modal damping values, respectively. The Gaussian probability density functions are estimated based on matching the mean and standard deviation of the corresponding histograms. These statistical distributions are generated using the identification results of 100 time windows of 2000 seconds each (about 2000 periods of the first mode). Only the first five modes are plotted forcompactnessreasons. Ineachfigure,dashedandsolidlinesrepresenttheidentificationresults for reference and damage cases, respectively, and the short upward arrow represents the exact parameter value. μ and σ are the mean and standard deviation in each case, and CV denotes the Coefficient of Variation. Subscripts 1 and 2 indicate the modal parameters for the reference and damage cases, respectively. The ratio Δn n is 1 20 in this case. Table 4.1 shows definition of the symbols used in Fig. 4.3-4.4, and their corresponding values for the first mode. Please note thatinthistablewejustshowedthevaluesformodalfrequency. Similarterminologyisusedfor modaldamping. The first observation from Fig. 4.3-4.4 is that the statistical dispersion of the identified natural frequencies are extremely small which indicates very high confidence in the estimated modalparameters. CoefficientofVariation(CV)istheparametertypicallyusedasameasureof statistical dispersion. The values of the CV for the first four modal frequencies in this example range from 0.07% for the first mode to about 0.12% for the fourth mode. On the other hand however, one can clearly observe higher degrees of dispersion in the estimated modal damping values(formodaldamping,thevaluesoftheCVrangefrom4to14%). Thereasonisbehindthe 96 Table 4.1: Definition of the symbols used in Fig. 4.3-4.4, and their corresponding values for the first mode. Superscript(*)indicatestheexactparameter,andsubscripts1and2indicatethemodalparameters forreference(undamaged)anddamagecases,respectively. Symbol Valueforthe1 st mode Description MAC 100% ModalAssuranceCriterionbetweenthedamageand no-damagemodeshapes(seeEq. (4.18)) μ 1 0.955 E[f]: mean(stochastic)offrequency forreferencecase[Hz] σ 1 0.067 p E[f 2 ]−E[f] 2 : standarddeviationof frequencyforreferencecase μ 2 0.954 E[f]: mean(stochastic)offrequency fordamagecase[Hz] σ 2 0.067 p E[f 2 ]−E[f] 2 : standarddeviationof frequencyfordamagecase CV 1 0.07% 100σ 1 /μ 1 : coefficientofvariationof frequencyforreferencecase CV 2 0.07% 100σ 2 /μ 2 : coefficientofvariationof frequencyfordamagecase Δμ/μ 1 -0.19% 100(μ 2 −μ 1 )/μ 1 : percentagechangeinfrequency normalizedbymeanoffrequency Δμ/σ 1 -258.2% 100(μ 2 −μ 1 )/σ 1 : percentagechangeinfrequency normalizedbystandarddeviationoffrequency μ ∗ 1 0.9549 exact(deterministic)valueoffrequencyfor referencecase[Hz] μ ∗ 2 0.9539 exact(deterministic)valueoffrequencyfor damagecase[Hz] e μ1 0.01% 100(μ ∗ 1 −μ 1 )/μ ∗ 1 : normalizedidentificationerror inmeanoffrequencyforthereferencecase e μ2 0.01% 100(μ ∗ 2 −μ 2 )/μ ∗ 2 : normalizedidentificationerror inmeanoffrequencyforthedamagecase factthatestimationofthedamping-relatedtermsarecommonlylessaccuratethantheestimation ofthestiffness-relatedterms. Furthermore, it can be observed that despite using noisy acceleration records only (i.e., not using any force information, or exact velocity and displacement records), the estimated modal frequencies using the NExT/ERA approach show a very good match with their actual values. Forinstance,thethenormalizedidentificationerrorinmeanoffrequencyforthefirstfourmodes ranges from 0.01 to 0.04%, which indicates extremely high accuracy. As mentioned earlier, the 97 damping values are less accurate. The normalized identification error in the mean of damping valuesforthefirstfourmodesrangesfrom1to26%. The other important observation from these figures is that the damages did not make any noticeable changes in the mode shapes. The Modal Assurance Criterion (MAC) is commonly usedasanindicationofchangeinthemodeshapes(Pappaetal.,1992). TheMACvaluebetween twomodeshapesΦ i andΦ j isdefinedas MAC i,j = 100 |Φ H i Φ j | 2 (Φ H i Φ i )(Φ H j Φ j ) (4.18) where the superscriptH, denotes the Hermitian of a matrix. The MAC value ranges between 0 and 100; 0 for orthogonal and 100 for identical mode shapes. Figures 4.3-4.4 show that for the first four modes, the MAC values between the damage and no-damage mode shapes are 100%, whichindicatesnochangeatall. Consequently,nospatialinformationaboutthedamagelocation couldbeextractedfromthemodalparametersinthisexample. Presumablythereasonisthatthe modeshapesarenotsensitiveenoughtolowlevelsoflocaldamage. However, one can clearly observe the shift in the mean and pdf of the identified natural frequencies for the first four modes. It is worth noting that, except for the first mode for which the mean frequency shift is about 2.5σ, the frequency shifts for the other modes are all almost within a 1σ bound. On the other hand, no noticeable shift, was observed in the mean modal damping. 98 Table4.2: ValuescorrespondingtothestatisticalparametersshowninTable4.1forthefirstfivemodes. Parameter 1 st mode 2 nd mode 3 rd mode 4 th mode 5 th mode freq ζ freq ζ freq ζ freq ζ freq ζ μ 1 0.955 0.5 2.609 1.0 4.309 1.6 6.014 2.1 7.722 2.7 μ 2 0.954 0.5 2.606 1.0 4.305 1.6 6.007 2.1 7.720 2.7 CV 1 % 0.07 13.90 0.09 7.75 0.08 5.70 0.11 4.62 0.16 4.95 CV 2 % 0.07 14.14 0.09 7.76 0.08 5.80 0.12 4.53 0.16 5.02 Δμ/μ 1 % -0.19 -1.50 -0.08 0.03 -0.09 0.20 -0.12 -0.31 -0.03 0.12 Δμ/σ 1 % -258.2 -10.80 -94.82 0.43 -114.02 3.45 -103.43 -6.71 -15.61 2.36 e μ1 % -0.01 26.54 0.01 -1.91 0.04 -4.86 0.01 -6.00 0.02 -7.24 e μ2 % -0.01 24.82 0.01 -1.80 0.04 -4.58 0.01 -6.18 0.01 -7.11 4.4.3 DamageAssessmentandlocalizationusingChainSystemIdentification In this subsection, the effectiveness of the chain system identification approach in detecting and locating the damage in the example under discussion, is investigated. As was previously mentioned, in this example the excitation forces are assumed to be unknown, stationary, white noiseprocesses. Therefore,toimplementthechainidentificationapproach,oneneedstousethe cross-correlationtime-historiesinsteadoftheactualtime-histories(seeEq. (4.13)). Figure4.5showsacomparisontofurtherclarifytheideabehindtheuseofcross-correlation instead of the actual time-histories. From this figure, it can be seen that the contribution of the excitation force ( ¯ F i (t)) in the restoring force function of each link of the chain ( ¯ G (i) ), is fairly significantcomparedtotheotherterms(¯ m i ¯ G (i+1) −¨ x i ),whereasinR xr ¯ G (i),thecontributionof thecross-correlationoftheexcitationforceandthedisplacementofareferenceDOF( R xr ¯ F i ),is negligiblecomparedtotheotherterms(¯ m i R xr ¯ G (i+1)−R xr¨ x i ). Using the linear expansion of the restoring forces (Eq. (4.12)), one can identify the mass- normalized coefficients of displacement and velocity ( k i m i and c i m i , respectively) at each mea- surement station, individually. Figure 4.6 shows the histogram and fitted Gaussian pdf for the 99 mass-normalized stiffness and damping terms, respectively. These statistical distributions are generated using the identification results of 100 time windows of 2000 seconds each (about 2000periodsofthefirstmode). TheGaussianprobabilitydensityfunctionsareestimatedbased on matching the mean and standard deviation of the corresponding histograms. For the sake of compactness, the results at only three consecutive sensor stations close to the first damage location are presented. Similar results are observed for the second damage location. The corre- spondingstatisticalmean(μ),dispersion(CV),identificationerror(e μ ),anddetectabilityratios (Δμ/μ 1 andΔμ/σ 1 )areshowninTable4.3. It is important to note here that the mass-normalized damping term c i m i , is commonly much smaller than the mass-normalized stiffness term k i m i , and consequently, its estimation is much less accurate. From Fig. 4.6 and Table 4.3 one can see very large and very small coefficients of variation (CV) for the quantities c i m i and k i m i , respectively. Moreover, Table 4.3 shows that the leveloftheidentificationerror(e μ )for c i m i isextremelylargerthanitsforthe k i m i . Table 4.3: Statistical mean (μ), dispersion (CV), identification error (e μ ), and detectability ratios (Δμ/μ 1 and Δμ/σ 1 ) corresponding to the distributions of the mass-normalized coefficient of displace- ment (k i /m i ), andvelocity (c i /m i ) shown in Fig. 4.6. For definition of the symbols refer to Table 4.1. Notice that the first damage location is between sensor stations 2 and 3; consequently the correspond- ing damage effects should only be reflected in the properties of the restoring force functionG 3 (see Fig. 4.6(a 2 )andFig. 4.7) Parameter sensorstation2 sensorstation3 sensorstation4 k i /m i c i /m i k i /m i c i /m i k i /m i c i /m i μ 1 5503.0 0.5 5450.8 1.4 5391.9 -1.6 μ 2 5501.5 1.0 5330.7 1.3 5393.7 -2.2 CV 1 % 0.60 1023.3 0.7 429.3 0.7 339.3 CV 2 % 0.70 550.8 0.7 420.2 0.8 250.8 Δμ/μ 1 % -0.0 90.2 -2.2 -4.3 0.0 38.2 Δμ/σ 1 % -4.2 8.8 -330.8 1.0 4.6 -11.2 e μ1 % 4.4 -91.7 4.4 -78.6 4.3 -125.9 e μ2 % 4.4 -84.2 4.3 -79.0 4.3 -135.7 100 BasedonFig. 4.7,itisclearthatthefirstdamagelocation(betweenthesensorstations2and 3) must only affect the chain identified stiffness at sensor station 3, and that is precisely what happened. Although sensor stations 2 and 4 are also very close to the first damage location, no noticeable shift was observed in their local stiffness distributions. This means that the chain identification method can localize the damage by isolating its effect at its corresponding chain element. Consequently, for a larger number of chain elements (or sensor stations), one can expect a higher spatial resolution of damage localization. Figure 4.6(a 2 ) shows that the shift in the mean of k 3 m 3 is about 2.2%, and that is very close to the expected percentage of reduction of equivalentstiffnessatthefirstandseconddamagelocations(basedonEq. (4.17),for Δn n = 1 20 , k ∗ eq 2 −k ∗ eq 1 k ∗ eq 1 = 2.17%). Itmustbenotedthat μ 2 −μ 1 σ 1 isanimportantratiowhichindicatesthedegree of detectability of a change. In fact, in order to reliably interpret a change as damage, it has to be out of 3σ bound (99% confidence interval). From Fig. 4.6(a 2 ) it can be seen that the mean valueof k 3 m 3 forthedamagecasehasshiftedbeyondthe3σ bound. It can be concluded that, even for a very small level of damage, the chain system identifica- tionmethodwascapableofnotonlyitsdetection,butalsoofpinpointingitslocation. Resolution ofthedamagelocalizationdependsontheresolutionofthesensorplacement. Ontheotherhand, the sensor resolution plays a significant role in damage detectability. In fact, sensor resolution definesthedegreeofthemodel-orderreduction,andthemodel-orderreductionaffectsthecapa- bilityofdamagedetectability. Toclarifythisstatement,let’sinvestigatetheeffectofmodel-orderreductiononthedamage detectability in this example. To this end, three levels of model-order reduction are considered. 101 As was pointed out earlier, the ratio of Δn n is used to quantify the degree of model-order reduc- tion in this example. Table 4.4 summarizes the effects of the sensor resolution (or model-order reduction)ontheabsoluteerrorandtherelativechangeoftheestimatedparametersbythechain identificationmethod. Fromthistable,itisclearthatreducingthesensorresolution(orincreas- ing the model order-reduction) decreases the damage detectability of the method. For instance, byincreasing Δn n from5/100to20/100,itcanbeseenthattherelativechangeofequivalentstiff- nessatthethirdmeasurementstation,whichistheclosesttothefirstdamagelocation,decreased from3.3σ to2.5σ,whichindicatesalessenedcapabilityfordetectability. Ontheotherhand,onecanobservethatbydecreasingthenumberofmeasurementstations, theabsoluteidentificationerrorincreases. Infact,oneofthemainsourcesoftheabsoluteerrorin theestimationofk i /m i isassociatedwiththeerrorinthemassratios. Inthechainmethod,ithas been assumed that the mass is concentrated at the sensor locations. In reality however, mass is distributedthroughoutthestructuralDOFs,whosenumberismuchlargerthanthemeasurement stations. Consequently, decreasing the number of measurement stations increases the error due to the mass ratio estimation. Nonetheless, the absolute error in k i /m i due to the mass ratio’s errorisexactlythesameforthedamageandno-damagecases(seeFig. 4.6( a 1 toa 3 )),sincethe damage usually affects the stiffness, and not the mass. Consequently, the error due to the mass ratioestimationhasvirtuallynoeffectonthedamagedetectability. Another important issue to be considered here is the noise level, and its influence on the damage detectability and the absolute error in the identification results. Table 4.5 summarizes the effect of noise on the absolute error and relative change of the estimated parameters by the chainidentificationmethod,forthreelevelsofnoise-to-signalratios;5%,10%,and20%. Itcan 102 be clearly seen that, by increasing the noise level from 5% to 20%, the ratio of μ 2 −μ 1 σ 1 , which represents the degree of detectability, decreased from 330% to less than 100%. On the other hand, the noise level had no significant influence on the mean absolute error (e μ ). This can be explained by the fact that noise does not change the mean but it increased the variance of the estimatedparameters. Table 4.4: Effect of the sensor resolution (model-order reduction) on the absolute error and relative changeoftheestimatedparametersbythechainidentificationmethod. Theseresultsarefortheestimated k eq /m at sensor station 3, which is the closest to the the first damage location. Parameters μ and σ are the mean and standard deviation of the estimatedk eq /m at each measurement station, respectively. Superscript (*) indicates the exact parameter, and subscripts 1 and 2 denote the parameters for the no- damageanddamagecases,respectively. Δn n k ∗ s 2 −k ∗ s 1 k ∗ s 1 k ∗ eq 2 −k ∗ eq 1 k ∗ eq 1 μ ∗ 1 −μ1 μ ∗ 1 μ2−μ1 μ1 μ2−μ1 σ1 model changeof changeof absolute estimatedchangeof relativechange order localstiffnessat equivalentstiffnessat identification equivalentstiffnessat (detectability) reduction structuralDOF,% measurementDOF,% error,% measurementDOF,% % 5/100 10 -2.17 4.4 -2.2 -330.8 10/100 10 -1.10 9.3 -1.1 -263.3 20/100 10 -0.55 19.7 -0.6 -248.8 Table 4.5: Effect of the noise on the absolute error and relative change of the estimated parameters by the chain identification method. These results are for the estimatedk eq /m at sensor station 3, which is theclosesttothethefirstdamagelocation. Themodel-orderreductionratioisassumedtobe Δn n = 1/20. Parametersμ andσ are the mean and standard deviation of the estimatedk eq /m at each measurement station, respectively. Superscript (*) indicates the exact parameter, and subscripts 1 and 2 denote the parametersfortheno-damageanddamagecases,respectively. Noise k ∗ s 2 −k ∗ s 1 k ∗ s 1 k ∗ eq 2 −k ∗ eq 1 k ∗ eq 1 μ ∗ 1 −μ1 μ ∗ 1 μ2−μ1 μ1 μ2−μ1 σ1 RMS changeof changeof absolute estimatedchangeof relativechange level localstiffnessat equivalentstiffnessat identification equivalentstiffnessat (detectability) % structuralDOF,% measurementDOF,% error,% measurementDOF,% % 5 10 -2.17 4.4 -2.2 -330.8 10 10 -2.17 4.8 -2.1 -199.9 20 10 -2.17 4.5 -2.0 -97.5 103 μ1 =0.955Hz,μ2 =0.954Hz μ1 =0.50%,μ2 =0.50% MAC=100% CV1 =0.07%,CV2 =0.07% CV1 =13.90%,CV2 =14.40% 0 0.2 0.4 0 5 10 15 20 normalized relative disp Measurment Stations Damage 1 Damage 2 0.952 0.954 0.956 0.958 0 100 200 300 400 500 600 freq [Hz] pdf 0.2 0.4 0.6 0.8 0 1 2 3 4 5 6 ζ % pdf μ1 =2.609Hz,μ2 =2.606Hz μ1 =1.0%,μ2 =1.0% MAC=100% CV1 =0.09%,CV2 =0.09% CV1 =7.75%,CV2 =7.76% −0.5 0 0.5 0 5 10 15 20 normalized relative disp Measurment Stations Damage 1 Damage 2 2.6 2.61 2.62 0 50 100 150 200 freq [Hz] pdf 0.6 0.8 1 1.2 1.4 0 1 2 3 4 5 6 ζ % pdf (a) (b) (c) Figure 4.3: Modal parameter identification results for the chain-like example under discussion, using the NExT/ERA method. In order to quantify the uncertainty in the estimated parameters, the statistical averagingwasconductedover100ensembles,eachincludesatleast2000periodsofthefirstfundamental modeofthesystem. Ineachfigure,dashedandsolidlinesrepresenttheidentificationresultsforreference (undamaged) and damage cases, respectively. The short upward arrow represents the exact parameter value. μ and σ are the mean and standard deviation in each case, and CV denotes the Coefficient of Variation. The ratio Δn n is 1 20 in this case. Table 4.1 summarizes the definition of the symbols used in this figure. (a) projection of the mode shape on the sensor stations for the reference and damage cases(solidanddashedlines,respectively),(b)Histogramandestimatedprobabilitydensityfunctionfor the identified modal frequency corresponding to the reference structure and the damaged one (solid and dashedlines,respectively),and(c)Histogramandestimatedprobabilitydensityfunctionfortheidentified modal damping corresponding to the reference structure and the damaged one (solid and dashed lines, respectively). Values corresponding to the statistical parameters defined in Table 4.1 can be found in Table4.2. 104 μ1 =4.309Hz,μ2 =4.305Hz μ1 =1.60%,μ2 =1.60% MAC=100% CV1 =0.08%,CV2 =0.08% CV1 =5.70%,CV2 =5.80% −0.5 0 0.5 0 5 10 15 20 normalized relative disp Measurment Stations Damage 1 Damage 2 4.29 4.3 4.31 4.32 0 20 40 60 80 100 120 freq [Hz] pdf 1.2 1.4 1.6 1.8 2 0 1 2 3 4 5 ζ % pdf μ1 =6.014Hz,μ2 =6.007Hz μ1 =2.10%,μ2 =2.10% MAC=100% CV1 =0.11%,CV2 =0.12% CV1 =4.62%,CV2 =4.53% −0.5 0 0.5 0 5 10 15 20 normalized relative disp Measurment Stations Damage 1 Damage 2 5.98 6 6.02 6.04 0 10 20 30 40 50 60 freq [Hz] pdf 1.6 1.8 2 2.2 2.4 2.6 0 1 2 3 4 5 ζ % pdf μ1 =7.722Hz,μ2 =7.720Hz μ1 =2.70%,μ2 =2.70% MAC=100% CV1 =0.16%,CV2 =0.16% CV1 =4.95%,CV2 =5.02% −0.5 0 0.5 0 5 10 15 20 normalized relative disp Measurment Stations Damage 1 Damage 2 7.65 7.7 7.75 0 5 10 15 20 25 30 35 freq [Hz] pdf 2 2.5 3 3.5 0 0.5 1 1.5 2 2.5 3 ζ % pdf Figure4.4: Modalparameteridentificationresultsforthechain-likeexampleunderdiscussion,usingthe NExT/ERAmethod. Cont. 105 1500 1550 1600 1650 1700 −1 0 1 time [sec] ¯ m 11 ¯ G (12) − ¨x 11 1500 1550 1600 1650 1700 −1 0 1 time [sec] ¯ F 11 0 20 40 60 80 100 −1 0 1 time [sec] ¯ m 11 R x r ¯ G (12)−R xr¨x11 0 20 40 60 80 100 −1 0 1 time [sec] R x r ¯ F 11 (a)Actualtime-histories (b)Cross-correlationtime-histories Figure 4.5: Comparison between the actual time-histories (Eq. (4.8)), and the cross-correlation time- histories (Eq. (4.10)), for measurement station 11 (Fig. 4.2). It can be seen that the contribution of the excitation force ( ¯ F i (t)) in the restoring force function ( ¯ G (i) ), is fairly significant compared to the other terms (¯ m i ¯ G (i+1) − ¨ x i ), whereas inR xr ¯ G (i), the contribution of the cross-correlation of the excita- tion force and the displacement of a reference DOF (R xr ¯ Fi ), is negligible compared to the other terms (¯ m i R xr ¯ G (i+1) −R xr¨ xi ). Measurement station 20 was used as the reference. For ease of comparison, identicalordinatescalesareusedforeachcase. 106 0.98 0.99 1 1.01 1.02 0 10 20 30 40 50 60 pdf Normalized Coef. of Dsp, k i /m i 0.96 0.98 1 1.02 0 10 20 30 40 50 60 pdf Normalized Coef. of Dsp, k i /m i 0.98 1 1.02 0 10 20 30 40 50 pdf Normalized Coef. of Dsp, k i /m i (a 1 )k i /m i atsensorstation2 (a 2 )k i /m i atsensorstation3 (a 3 )k i /m i atsensorstation4 −20 0 20 40 0 0.01 0.02 0.03 0.04 pdf Normalized Coef. of Vel, c i /m i −10 0 10 0 0.02 0.04 0.06 0.08 0.1 pdf Normalized Coef. of Vel, c i /m i −10 0 10 0 0.02 0.04 0.06 0.08 0.1 0.12 pdf Normalized Coef. of Vel, c i /m i (b 1 )c i /m i atsensorstation2 (b 2 )c i /m i atsensorstation3 (b 3 )c i /m i atsensorstation4 Figure 4.6: Chain system identification results for the chain-like example under discussion. In order to quantify the uncertainty in the estimated parameters, the statistical averaging was conducted over 100 ensembles,eachincludesatleast2000periodsofthefirstfundamentalmodeofthesystem. Ineachfigure, dashedandsolidlinesrepresenttheidentificationresultsfortheno-damage(reference)anddamagecases, respectively, and the short upward arrow represents the exact parameter value. The abscissa for all the graphs are normalized to the corresponding values for the reference (no-damage) case in order to have identical abscissa scale. The ratio Δn n is 1 20 in this case. (a 1 ), (a 2 ), and (a 3 ) show the mass-normalized coefficient of displacement (k i /m i ) for the sensor stations 2, 3 and 4, respectively. (b 1 ), (b 2 ), and (b 3 ) showthemass-normalizedcoefficientof velocity(c i /m i )forthesensorstations2,3and4,respectively. Notice that the first damage location is between sensor stations 2 and 3; consequently the corresponding damage effects should only be reflected in the properties of the restoring force function G 3 (see Fig. 4.6(a 2 ) and Fig. 4.7). The corresponding statistical mean (μ), dispersion (CV), identification error (e μ ), anddetectabilityratios(Δμ/μ 1 andΔμ/σ 1 )areshowninTable4.3. 107 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 Damage 2 Damage 1 Sensor Stations Figure4.7: Sensorstationsanddamagelocationsforthe100DOFchain-likeexampleunderdiscussion. 108 4.5 UCLAFactorBuilding: DescriptionandInstrumentation The Factor Building, built in the late 1970’s, is a 17-story steel frame structure on the UCLA campus (Los Angeles, California). The building houses the UCLA center for the health sci- ences, including the school of nursing, the school of medicine, and the Jonsson comprehensive cancer center. With 15 stories above grade, and the height of 216.5 feet (65.99 meters), Fac- tor is recognized as the tallest building on the UCLA campus. Following the 1994 Northridge earthquake, the building was instrumented by the U.S. Geological Survey Advanced National Seismic System (ANSS) and the NSF Center for Embedded Networked Sensing (CENS). The instrumentation network consists of a dense array of 72-channel uniaxial Kinemetrics force- balance accelerometers, which made the Factor Building one of the most densely instrumented buildingsinthenorthAmerica(Baeketal.,2006). Theembeddedaccelerometersarecharacter- ized by a frequency bandwidth from DC to 50 Hz, a large amplitude range of±4g, and a wide dynamic range of 135 dB. These sensors were utilized in order to measure the relatively low- amplitude, free and forced-vibration responses. The sensors’ signals are continuously recorded atthesamplingrateof100Hertzusinga24-bitdataacquisitionsystem. As shown in Fig. 4.8, there are four accelerometers at each floor. For all the floors above grade, the accelerometers are oriented horizontally; two in the north-south, and two in the east- west directions, per each floor. For the basement and the subbasement, however, there are two horizontallyandtwoverticallyorientedaccelerometers. In this study, we consider three degrees of freedom (DOF) for each story above grade; one intheeast-westdirection( xDOF),oneinthenorth-southdirection( y DOF),andonetorsion(θ DOF). However, one needs to find the relationship between the mentioned DOFs (x,y, andθ) 109 (a) (b) Figure 4.8: (a) Overview of the Factor Building, (b) its overall dimensions and sensor loca- tions/directions. andthefouruniaxialaccelerometersateachfloor(twointheeast-westandtwointhenorth-south directions). To this end, the following formulas were developed based on the sensor-location diagramshowninFig. 4.9: ¨ x = ¨ x N + ¨ x S 2 , ¨ y = ¨ y E + ¨ y W 2 , ¨ θ = ¨ θ x + ¨ θ y 2 (4.19) where ¨ θ x = ¨ x S − ¨ x N r y N −r y S , ¨ θ y = ¨ y E − ¨ y W r x E −r x W (4.20) Theaccelerometercoordinateswithrespecttoselectedreferencepoint(i.e.,r y N ,r y S ,r x E ,and r x W )canbefoundintheworkofSkolniketal.(2006). Formoreinformationaboutthebuilding, itsinstrumentationanddataacquisitionsystem,thereaderisreferredtotheUCLAFactorseismic 110 N x & & S x & & W y & & E y & & Y X θ x W r x E r y N r y S r Ref. Point Figure4.9: Schematicplotofthesensorslayoutforeachfloorabovegrade. arrayhomepage,http://factor.gps.caltech.edu/,andtheworkofKohleretal.(2005)andSkolnik etal.(2006). 4.6 FactorBuildingIdentificationResults This section reports the results of the application of the proposed algorithms for the structural identification of the Factor Building. As previously mentioned, the Factor Building was instru- mented with 72 accelerometers at 17 stories. In this study, use is made only of the acceleration recordsof allthe15stories abovegrade, fora totalnumberof64 accelerometers(fourper each floorplusfourontheroof). Aspointedoutearlier,threeDOFsareconsideredforeachstory;one intheeast-westdirection( xDOF),oneinthenorth-southdirection( y DOF),andonetorsion(θ DOF).Consequently,thereare48DOFsforthe16floorsabovegrade(16(floors)×3(x,y,and θ DOFs)=48). The acceleration data were recorded at the sampling rate of 100 Hz. To implement the chain identification method, the velocity and displacement time-histories are also needed to be 111 knownforallthemeasurementstations. Tothisend,theaccelerationtime-historiesareintegrated numerically. Windowing, detrending, and band-pass filtering is required for that process. In order to reduce the computational time and storage requirements, the time-histories (after the integration process) were down-sampled to 50 Hz, since the frequency range of interest is less than 10 Hz. Figure 4.10 depicts a typical ambient vibration record of the acceleration time- historiesmeasuredatthe14thfloor(nearthetop)oftheFactorBuilding,andtheircorresponding velocityanddisplacementtime-histories,computedthroughdigitalsignalprocessing. 4.6.1 ModalParameterIdentificationResults TheERAmethodinconjunctionwiththeNExTtechnique(thatwasdiscussedindetailinchap- ter 2) was used to extract the modal parameters of the Factor Building, based on the measured accelerationrecordsonly. Toimplementtheidentificationmethodology,thefirststepistocom- pute the Cross-Correlation Function (CCF) between the response of the preselected reference DOF(orDOFs)andtheresponseofall48availableDOFs. It is important to note that one cannot rely on just one single reference DOF to reliably identify all modes. One single reference that is a proper selection for some modes, may not be proper for other modes. For this reason, it is recommended to use multiple reference DOFs (Nayerietal.,2006). In this study the sensor stations on the 5th, 8th, and 15th floor in x, y, and θ directions (for a total number of 9 DOFs) were selected as the reference DOFs. The CCFs can then be estimated using the inverse Fourier transform of the Cross-Power-Spectral Densities (CPSDs). The CPSDs are computed directly from the data. Random errors associated with the CPSD can beminimizedbywindowingandaveraging. Inthisstudy,aHanningwindowof163.84seconds, 112 with50%overlapwasusedforthecomputationoftheCPSDs. Sincethefirstfundamentalperiod of the building is around 2 seconds, a frame with the length of 163.84 seconds which contains about80cyclesofthefirstfundamentalmode,isareasonablechoice(Nayerietal.,2006). Figure 4.11showstypicalCPSDsandtheirassociatedCCFsfortheFactorBuilding. OncetheCCFsarecomputed, onecanformtheHankelmatrix(Eq. (2.8))toimplementthe ERA method. Based on the guidelines presented in the work of Cooper and Wright (1992) and Nayeri et al. (2006), the number of block rows in the Hankel matrix ((r + 1) in Eq. (2.8)) is selected to be 30. Once the number of block rows is selected, the number of block columns is determined based on the available data points in the cross-correlation time-histories. An autonomousalgorithmwasusedtodefinethefinalmodel-orderandtodistinguishspuriousnoise modes from the genuine modes based on some accuracy indicators. For more detail, the reader isreferredtotheworkofPappaandElliott(1993),Pappaetal.(1998),andNayerietal.(2006). Figures 4.12-4.17 show the identified modal parameters for the Factor Building. These results are extracted from a time-window of two hours of the recorded acceleration data. In the next subsectionthestatisticalvariabilityofthemodalparametersisdiscussed. 113 1900 1950 2000 −1 −0.5 0 0.5 1 x 10 −4 time [sec] X ACC [g] 1900 1950 2000 −1 −0.5 0 0.5 1 x 10 −4 time [sec] Y ACC [g] 1900 1950 2000 −4 −2 0 2 4 x 10 −3 time [sec] ¨θ Angular ACC [deg/sec 2 ] 1900 1950 2000 −0.06 −0.04 −0.02 0 0.02 0.04 0.06 time [sec] X VEL [mm/sec] 1900 1950 2000 −0.1 −0.05 0 0.05 0.1 time [sec] Y VEL [mm/sec] 1900 1950 2000 −1.5 −1 −0.5 0 0.5 1 1.5 x 10 −4 time [sec] ˙ θ Angular VEL[deg/sec] 1900 1950 2000 −0.015 −0.01 −0.005 0 0.005 0.01 0.015 time [sec] X DSP [mm] 1900 1950 2000 −0.015 −0.01 −0.005 0 0.005 0.01 0.015 time [sec] Y DSP [mm] 1900 1950 2000 −3 −2 −1 0 1 2 x 10 −5 time [sec] θ Angle [deg] (a)Acceleration (b)Velocity (c)Displacement Figure4.10: Atypicalambientvibrationrecordsoftheaccelerationtime-historiesmeasuredatthe14th floor of the Factor Building, and their corresponding velocity and displacement time-histories computed by digital signal processing. Column (a) corresponds to the measured accelerations; column (b) to the processedvelocities;andcolumn(c)totheprocesseddisplacements. 114 0 2 4 6 8 10 10 −5 10 0 Freq [Hz] CPSD ¨X 5 ¨X 15 (f)[dB] 0 20 40 60 80 −5 0 5 x 10 −4 time [sec] R ¨X 5 ¨X 15 (τ) 0 2 4 6 8 10 10 −5 10 0 Freq [Hz] CPSD ¨X 10 ¨X 15 (f)[dB] 0 20 40 60 80 −5 0 5 x 10 −4 time [sec] R ¨X 10 ¨X 15 (τ) (a)5thfloor,xDOF (b)10thfloor,xDOF Figure 4.11: Typical semi-logarithmic plots of the CPSDs and their associated CCFs for the Factor Building. Forthiscase,thereferenceDOFisselectedtobethexaxisofthesensoronthe15thfloor,the window size is 163.84 seconds, and the overlap is 50%. For ease of comparison, identical abscissa and ordinate scales are used for all displayed plots. (a)CPSD andR between acceleration of floors 5 and 15;(b)CPSD andRbetweenaccelerationoffloors10and15. 115 0 0.2 0.4 0 2 4 6 8 10 12 14 16 normalized relative disp X DOF 0 0.2 0.4 0 2 4 6 8 10 12 14 16 normalized relative disp Y DOF 0 0.2 0.4 0 2 4 6 8 10 12 14 16 normalized relative θ θ DOF (a)1stbendinginX,f=0.55Hz 0 0.2 0.4 0 2 4 6 8 10 12 14 16 normalized relative disp X DOF 0 0.2 0.4 0 2 4 6 8 10 12 14 16 normalized relative disp Y DOF 0 0.2 0.4 0 2 4 6 8 10 12 14 16 normalized relative θ θ DOF (b)1stbendinginY,f=0.59Hz Figure 4.12: Identified modal parameters of the Factor Building using the NExT/ERA method. These resultsareextractedfromatime-windowoftwohoursoftherecordedaccelerationdata. Figure(a)shows the projection of the first mode in the X, Y, and θ directions; Figure (b) shows the projection of the second mode in the X, Y, andθ directions. It is clear from the displayed mode shapes that the first mode correspondstobendinginprimarilytheX-direction,andthatthesecondmodecorrespondstobendingin primarilytheY-direction. 116 0 0.2 0.4 0 2 4 6 8 10 12 14 16 normalized relative disp X DOF 0 0.2 0.4 0 2 4 6 8 10 12 14 16 normalized relative disp Y DOF 0 0.2 0.4 0 2 4 6 8 10 12 14 16 normalized relative θ θ DOF (c)1sttorsion,f=0.82Hz −0.2 0 0.2 0.4 0 2 4 6 8 10 12 14 16 normalized relative disp X DOF −0.2 0 0.2 0.4 0 2 4 6 8 10 12 14 16 normalized relative disp Y DOF −0.2 0 0.2 0.4 0 2 4 6 8 10 12 14 16 normalized relative θ θ DOF (d)2ndbendinginX,f=1.65Hz Figure4.13: Identifiedmodalparameters. Cont. 117 −0.2 0 0.2 0.4 0 2 4 6 8 10 12 14 16 normalized relative disp X DOF −0.2 0 0.2 0.4 0 2 4 6 8 10 12 14 16 normalized relative disp Y DOF −0.2 0 0.2 0.4 0 2 4 6 8 10 12 14 16 normalized relative θ θ DOF (e)2ndbendinginY,f=1.85Hz −0.2 0 0.2 0.4 0 2 4 6 8 10 12 14 16 normalized relative disp X DOF −0.2 0 0.2 0.4 0 2 4 6 8 10 12 14 16 normalized relative disp Y DOF −0.2 0 0.2 0.4 0 2 4 6 8 10 12 14 16 normalized relative θ θ DOF (f)2ndtorsion,f=2.62Hz Figure4.14: Identifiedmodalparameters. Cont. 118 −0.2 0 0.2 0.4 0 2 4 6 8 10 12 14 16 normalized relative disp X DOF −0.2 0 0.2 0.4 0 2 4 6 8 10 12 14 16 normalized relative disp Y DOF −0.2 0 0.2 0.4 0 2 4 6 8 10 12 14 16 normalized relative θ θ DOF (g)3rdbendinginX,f=2.87Hz −0.2 0 0.2 0.4 0 2 4 6 8 10 12 14 16 normalized relative disp X DOF −0.2 0 0.2 0.4 0 2 4 6 8 10 12 14 16 normalized relative disp Y DOF −0.2 0 0.2 0.4 0 2 4 6 8 10 12 14 16 normalized relative θ θ DOF (h)3rdbendinginY,f=3.15Hz Figure4.15: Identifiedmodalparameters,Cont. 119 −0.2 0 0.2 0.4 0 2 4 6 8 10 12 14 16 normalized relative disp X DOF −0.2 0 0.2 0.4 0 2 4 6 8 10 12 14 16 normalized relative disp Y DOF −0.2 0 0.2 0.4 0 2 4 6 8 10 12 14 16 normalized relative θ θ DOF (i)3rdtorsion,f=4.19Hz −0.2 0 0.2 0 2 4 6 8 10 12 14 16 normalized relative disp X DOF −0.2 0 0.2 0 2 4 6 8 10 12 14 16 normalized relative disp Y DOF −0.2 0 0.2 0 2 4 6 8 10 12 14 16 normalized relative θ θ DOF (j)4thbendinginX&Y,f=4.24Hz Figure4.16: Identifiedmodalparameters,Cont. 120 −0.2 0 0.2 0.4 0 2 4 6 8 10 12 14 16 normalized relative disp X DOF −0.2 0 0.2 0.4 0 2 4 6 8 10 12 14 16 normalized relative disp Y DOF −0.2 0 0.2 0.4 0 2 4 6 8 10 12 14 16 normalized relative θ θ DOF (k)4thbendinginY,f=4.51Hz −0.2 0 0.2 0.4 0 2 4 6 8 10 12 14 16 normalized relative disp X DOF −0.2 0 0.2 0.4 0 2 4 6 8 10 12 14 16 normalized relative disp Y DOF −0.2 0 0.2 0.4 0 2 4 6 8 10 12 14 16 normalized relative θ θ DOF (l)5thbendinginX,f=5.21Hz Figure4.17: Identifiedmodalparameters,Cont. StatisticalAnalysisoftheModalParameters Beforeusingtheidentifiedmodalparametersfor reliablestructuralhealthmonitoringpurposes, theirstatisticalvariabilitymustbeinvestigatedandquantified. Therearemanysourcesotherthan damage that can exert variations, of some degree, in the modal parameters: nonlinearity in the system,temperaturevariations,moistureabsorption,soilstructureinteractions,nonstationarities, 121 and measurement noise and hysteresis. Hence, it is worthwhile to use statistical averaging to extract a distribution of the identified parameters, and then to use those distributions to assess thevariabilityoftheidentifiedparameters. Inordertohavestatisticallymeaningfulresults(whilesimultaneouslydealingwithmanage- able sizes of needed data files), 50 days of data (24 hours each) were considered in this study. Themodalparameteridentificationwasperformedovertime-windowsof2hourseach,andwith 50% overlap. This choice results in a total number of 1200 statistical ensembles (50 (days)× 24(time-windowsperday)). Figures4.18-4.19and4.20-4.21showthehistogramsandthefitted Gaussiandistributionsoftheestimatedmodalfrequenciesanddampingvalues,respectively. It is seen that, for most of the identified modes, the distribution of the estimated parameters closely fit a Gaussian distribution. One important observation from these plots is that the varia- tion in the frequency estimation is very small; A Coefficient of Variation (CV) is about 1∼2 % for most of the modal frequencies. On the other hand, the damping estimation variance is seen tobemuchhigher;theCVis20∼70%formostofthemodaldampingvalues. Thisobservation isconsistentwiththewell-knownfactthattypicallytheestimatedmodaldampingvaluesarenot asaccurateasthemodalfrequencies. However,itisworthnotingthattheestimatedmeanmodal dampingvaluesareallpositiveandwithinareasonablerange. 122 0.53 0.54 0.55 0.56 0.57 0 10 20 30 40 50 60 70 80 μ = 0.551 [Hz], CV =σ/μ =0.9% freq [Hz] pdf 0.57 0.58 0.59 0.6 0.61 0.62 0 10 20 30 40 50 60 70 μ = 0.595 [Hz], CV =σ/μ =1.0% freq [Hz] pdf (a)1stbendinginX (b)1stbendinginY 0.75 0.8 0.85 0.9 0 5 10 15 20 μ = 0.810 [Hz], CV =σ/μ =2.5% freq [Hz] pdf 1.6 1.65 1.7 1.75 0 5 10 15 20 25 μ = 1.665 [Hz], CV =σ/μ =1.0% freq [Hz] pdf (c)1sttorsion (d)2ndbendinginX 1.75 1.8 1.85 1.9 1.95 0 5 10 15 20 μ = 1.849 [Hz], CV =σ/μ =1.3% freq [Hz] pdf 2.2 2.4 2.6 2.8 3 0 1 2 3 4 5 μ = 2.606 [Hz], CV =σ/μ =3.5% freq [Hz] pdf (e)2ndbendinginY (f)2ndtorsion Figure 4.18: Probability densities of the estimated modal frequencies for the Factor Building. A total of 50 days of data (each 24 hours) were considered in this study. The modal parameter identification was conducted over time-windows of 2 hours each, and with 50% overlap, for a total number of 1200 statistical ensembles. The μ and σ are the mean and standard deviation in each case, and CV denotes the Coefficient of Variation. In each plot panel, a thin line indicates the outline of the histogram of the indicatedparameter,andthesolidlineindicatestheestimatedGaussianpdfhavingamatchingmeanand standarddeviationasthecorrespondinghistogram. 123 2.7 2.8 2.9 3 0 2 4 6 8 10 12 μ = 2.867 [Hz], CV =σ/μ =1.2% freq [Hz] pdf 2.8 3 3.2 3.4 0 1 2 3 4 5 6 7 μ = 3.108 [Hz], CV =σ/μ =2.1% freq [Hz] pdf (g)3rdbendinginX (h)3rdbendinginY 4 4.2 4.4 0 1 2 3 4 5 6 μ = 4.188 [Hz], CV =σ/μ =1.6% freq [Hz] pdf 4.1 4.2 4.3 4.4 0 2 4 6 8 10 μ = 4.212 [Hz], CV =σ/μ =1.0% freq [Hz] pdf (i)3rdtorsion (j)4thbendinginX&Y 4 4.2 4.4 4.6 4.8 5 0 0.5 1 1.5 2 2.5 3 3.5 μ = 4.529 [Hz], CV =σ/μ =2.6% freq [Hz] pdf 5 5.2 5.4 5.6 0 1 2 3 4 5 μ = 5.269 [Hz], CV =σ/μ =1.7% freq [Hz] pdf (k)4thbendinginY (l)5thbendinginX Figure4.19: ProbabilitydensitiesoftheestimatedmodalfrequenciesfortheFactorBuilding. Cont. 124 −5 0 5 10 0 0.05 0.1 0.15 0.2 0.25 μ = 2.9 %, CV =σ/μ =67.9% ζ % pdf −5 0 5 10 0 0.05 0.1 0.15 0.2 μ = 3.0 %, CV =σ/μ =68.8% ζ % pdf (a)1stbendinginX (b)1stbendinginY −5 0 5 10 0 0.05 0.1 0.15 0.2 0.25 μ = 3.5 %, CV =σ/μ =49.2% ζ % pdf 1 2 3 0 0.2 0.4 0.6 0.8 1 1.2 1.4 μ = 2.0 %, CV =σ/μ =16.2% ζ % pdf (c)1sttorsion (d)2ndbendinginX 1 2 3 0 0.2 0.4 0.6 0.8 1 1.2 1.4 μ = 1.9 %, CV =σ/μ =17.6% ζ % pdf 0 2 4 6 8 0 0.1 0.2 0.3 0.4 0.5 μ = 3.6 %, CV =σ/μ =27.0% ζ % pdf (e)2ndbendinginY (f)2ndtorsion Figure 4.20: Probability densities of the estimated modal damping for the Factor Building. A total of 50 days of data (each 24 hours) were considered in this study. The modal parameter identification was conductedovertime-windowintervalsof2hourseach,andwith50%overlap,foratotalnumberof1200 statistical ensembles. Note that (as seen from the plotted histograms) some of the estimated (identified) dampingvalueshadanegativemagnitude. 125 1 2 3 4 0 0.2 0.4 0.6 0.8 1 μ = 2.5 %, CV =σ/μ =17.1% ζ % pdf 0 2 4 0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 μ = 2.4 %, CV =σ/μ =28.7% ζ % pdf (g)3rdbendinginX (h)3rdbendinginY −4 −2 0 2 4 6 0 0.05 0.1 0.15 0.2 0.25 0.3 0.35 μ = 1.2 %, CV =σ/μ =102.5% ζ % pdf −4 −2 0 2 4 6 0 0.05 0.1 0.15 0.2 0.25 0.3 0.35 0.4 μ = 1.1 %, CV =σ/μ =102.3% ζ % pdf (i)3rdtorsion (j)4thbendinginX&Y 0 2 4 6 0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 μ = 3.1 %, CV =σ/μ =24.4% ζ % pdf −2 0 2 4 6 0 0.1 0.2 0.3 0.4 0.5 μ = 2.3%, CV =σ/μ =42.3% ζ % pdf (k)4thbendinginY (l)5thbendinginX Figure4.21: ProbabilitydensitiesoftheestimatedmodaldampingfortheFactorBuilding. Cont. 126 4.6.2 ChainSystemIdentificationResults This section presents the results of the chain system identification approach for identifying the interstory restoring functions of the Factor Building. As was mentioned earlier, the excitation forcesappliedtothebuildingareallfromtheambientsourcesandhenceimmeasurable. There- fore,inordertoimplementthechainidentificationapproach,oneneedstoassumethattheexci- tation forces are unknown stationary white noise processes, and then use the cross-correlation time-histories and the linear expansion of the restoring forces (Eq. (4.12)) to identify the mass- normalized coefficients of displacement and velocity (k i /m i and c i /m i , respectively) at each measurementstation,individually. Figure4.22showsrepresentativephaseandtime-historyplotsoftherestoringfunctionsasso- ciated with the 13th floor of the Factor Building, in the x, y, and θ directions. In this figure, the first column corresponds to the actual time-histories of the restoring functions by ignor- ing the force term (see Eq. (4.8)), whereas the second and third columns correspond to the cross-correlation time-histories of the restoring functions (see Eq. (4.12)). In Fig. 4.22, cross- correlationphasediagramsimplythattherestoringforcefunctionsareessentiallylinear. Hence, thelinearexpansionoftherestoringforcesinaseriesformcancapturethedominantbehaviorof the system. Furthermore, by comparing the solid and dashed lines in the Fig. 4.22, one can see a very good agreement between the actual and estimated (identified) cross-correlation restoring forces. As previously mentioned, the identification process was performed over a time-window intervalof2hours,andwith50%overlap,foratotalnumberof50days. Theresultingestimated parameters provide us with the opportunity to study the statistical variability of the identified 127 parameters. Figures 4.23-4.24 and 4.25-4.26 show the distribution of the estimated coefficient of displacement and velocity terms in the interstory restoring functions, respectively. For the sakeofcompactness,onlytheresultsforthe2nd,6th,9th,and16thfloorsarepresented. It is important to note here that the mass-normalized stiffness term ( k i /m i ) is the dominant termintherestoringfunction. Consequently,theestimationofthedampingterm(c i /m i )whose value is commonly much smaller than the stiffness term, is relatively much less accurate. From Fig. 4.25-4.26, onecanseeverylargecoefficientsofvariation(CV)andinsomecasesnegative values, forc i /m i . On the other hand, the stiffness term, which is much more important from damage detection point of view, is estimated with relatively low coefficients of variation, and its values for all the floors are positive, as it should be. It is worth noting that, in general, it was found that the coefficient of variation of the identification results tends to be higher (more scattered) for building locations where the strain energy is relatively small. This observation has implication regarding the advantage of using known excitation force where magnitude is substantiallylargerthantheambient(“noise”)level. 128 −2 −1 0 1 2 x 10 −3 −3 −2 −1 0 1 2 3 Z [mm] ¯ G (13) X DOF −2 0 2 x 10 −4 −6 −4 −2 0 2 4 6 x 10 −4 R xrZ R xr ¯ G (13) X DOF MSE=3.89 % actual estimated 5 10 15 20 −6 −4 −2 0 2 4 6 8 x 10 −4 time [sec] R x r ¯ G (13) X DOF MSE=3.89 % actual estimated −1 0 1 x 10 −3 −2.5 −2 −1.5 −1 −0.5 0 0.5 1 1.5 2 Z [mm] ¯ G (13) Y DOF −2 0 2 x 10 −4 −8 −6 −4 −2 0 2 4 6 x 10 −4 R xrZ R xr ¯ G (13) Y DOF MSE=3.35 % actual estimated 5 10 15 20 −6 −4 −2 0 2 4 6 8 x 10 −4 time [sec] R x r ¯ G (13) Y DOF MSE=3.35 % actual estimated −4 −2 0 2 x 10 −6 −8 −6 −4 −2 0 2 4 6 8 x 10 −3 θ [deg] ¯ G (13) θ DOF −2 −1 0 1 2 x 10 −4 −1.5 −1 −0.5 0 0.5 1 1.5 x 10 −6 R xrθ R xr ¯ G (13) θDOF MSE=11.50 % actual estimated 5 10 15 20 −1 −0.5 0 0.5 1 1.5 x 10 −6 time [sec] R x r ¯ G (13) θ DOF MSE=11.50 % actual estimated Figure 4.22: Representative phase and time-history plots of the restoring force functions associated with the 13th floor of the factor building, in x, y, and θ directions. The first column corresponds to the actual time-histories of the restoring forces by ignoring the force term (see Eq. (4.8)), whereas the second and third columns correspond to the cross-correlation time-histories of the restoring forces (see Eq. (4.12)). Inthesecondandthirdcolumns,thesolidanddashedlinesrepresenttheactualandestimated (identified) restoring force functions, respectively. MSE is the Mean-Squared-Error of the estimation. Lackofsmoothnessofsomeofthedisplayedcasesisduetodatapointdecimation,andnotatruereflection ofthephysicsoftheunderlyingrestoringforces. SymbolR xy standsforE[x(t)y(t+τ)];x r corresponds tothereferenceDOFs(X,Y,andθ directionsofthe10thfloor);z =x 13 −x 12 . 129 1400 1600 1800 2000 2200 2400 2600 2800 0 0.5 1 1.5 2 2.5 x 10 −3 μ = 2132.3, CV =σ/μ =8.1% pdf Coef. of Dsp, k i /m i 1400 1500 1600 1700 1800 1900 2000 0 1 2 3 4 5 6 x 10 −3 μ = 1686.7, CV =σ/μ =4.5% pdf Coef. of Dsp, k i /m i (a)2ndfloor,x (d)6thfloor,x 4500 5000 5500 6000 6500 7000 7500 8000 0 0.2 0.4 0.6 0.8 1 1.2 x 10 −3 μ = 6219.9, CV =σ/μ =6.0% pdf Coef. of Dsp, k i /m i 1200 1300 1400 1500 1600 0 1 2 3 4 5 6 7 8 x 10 −3 μ = 1432.4, CV =σ/μ =3.6% pdf Coef. of Dsp, k i /m i (b)2ndfloor,y (e)6thfloor,y 0 2000 4000 6000 8000 0 0.5 1 1.5 2 2.5 3 3.5 4 4.5 x 10 −4 μ = 4034.3, CV =σ/μ =24.6% pdf Coef. of Dsp, k i /m i 2400 2600 2800 3000 3200 3400 0 0.5 1 1.5 2 2.5 3 3.5 4 x 10 −3 μ = 2879.1, CV =σ/μ =3.8% pdf Coef. of Dsp, k i /m i (c)2ndfloor,θ (f)6thfloor,θ Figure 4.23: Sample distributions of the estimated coefficient of displacement term in the interstory restoring functions. Coefficient of displacement is the mass-normalized stiffness term ( k i /m i ). For the sakeofcompactness,onlytheresultsforthe2nd,6th,9th,and16thfloorsareshown. Otherfloorlocations havesimilarresults. 130 300 400 500 600 0 0.002 0.004 0.006 0.008 0.01 0.012 μ = 492.9, CV =σ/μ =8.0% pdf Coef. of Dsp, k i /m i 0 500 1000 1500 0 0.2 0.4 0.6 0.8 1 1.2 1.4 1.6 1.8 2 x 10 −3 μ = 783.2, CV =σ/μ =26.9% pdf Coef. of Dsp, k i /m i (g)9thfloor,x (j)16thfloor,x 400 600 800 1000 1200 0 0.5 1 1.5 2 2.5 3 3.5 4 4.5 x 10 −3 μ = 830.7, CV =σ/μ =10.9% pdf Coef. of Dsp, k i /m i 0 200 400 600 800 1000 0 0.5 1 1.5 2 2.5 3 3.5 4 x 10 −3 μ = 500.6, CV =σ/μ =21.9% pdf Coef. of Dsp, k i /m i (h)9thfloor,y (k)16thfloor,y 600 800 1000 1200 1400 0 0.5 1 1.5 2 2.5 3 3.5 4 x 10 −3 μ = 960.8, CV =σ/μ =11.0% pdf Coef. of Dsp, k i /m i −200 0 200 400 600 800 1000 1200 0 0.5 1 1.5 2 2.5 x 10 −3 μ = 485.0, CV =σ/μ =36.5% pdf Coef. of Dsp, k i /m i (i)9thfloor,θ (l)16thfloor,θ Figure 4.24: Sample distributions of the estimated coefficient of displacement term in the interstory restoringfunctions. Cont. 131 −150 −100 −50 0 50 100 150 0 0.002 0.004 0.006 0.008 0.01 0.012 0.014 μ = 7.9, CV =σ/μ =403.8% pdf Coef. of Vel, c i /m i −100 −50 0 50 100 150 0 0.002 0.004 0.006 0.008 0.01 0.012 0.014 μ = 8.3, CV =σ/μ =376.9% pdf Coef. of Vel, c i /m i (a)2ndfloor,x (d)6thfloor,x −100 0 100 200 300 0 1 2 3 4 5 6 7 8 x 10 −3 μ = 73.5, CV =σ/μ =74.4% pdf Coef. of Vel, c i /m i −40 −20 0 20 40 60 0 0.005 0.01 0.015 0.02 0.025 0.03 0.035 μ = 9.5, CV =σ/μ =138.6% pdf Coef. of Vel, c i /m i (b)2ndfloor,y (e)6thfloor,y −200 −100 0 100 200 0 1 2 3 4 5 6 7 8 9 x 10 −3 μ = 29.6, CV =σ/μ =157.2% pdf Coef. of Vel, c i /m i −150 −100 −50 0 50 100 150 0 0.002 0.004 0.006 0.008 0.01 0.012 μ = 1.3, CV =σ/μ =2571.8% pdf Coef. of Vel, c i /m i (c)2ndfloor,θ (f)6thfloor,θ Figure 4.25: Sample distributions of the estimated coefficient of velocity term in the interstory restor- ing functions. Coefficient of velocity is the mass-normalized damping term ( c i /m i ). For that sake of compactness,onlytheresultsforthe2nd,6th,9th,and16thfloorsarepresented. 132 It should be noted that in this study, it is assumed that the mass is uniformly distributed throughout the floors of the building, which results in all the mass ratios to be identically one. That assumption increases the absolute estimation error. Nonetheless, the estimation error in k i /m i associatedwiththeerrorinthemassratioswouldbeexactlythesameforthedamageand no-damage cases, since the damage commonly affects the stiffness, and not the mass. Conse- quently, the error due to mass-ratio estimation has no effect on the damage detectability of this approach. In fact, the absolute error is not important from the damage detection point of view, buttherelativechangeis. It would be interesting to check the consistency of the identified local parameters with the identified global modal parameters. To this end, one can reconstruct the global M −1 K and M −1 C matrices using the identified chain system local parameters (k i /m i and c i /m i ) based on the formulations presented in the Eq. (4.14) and (4.15). Figures 4.27 and 4.28 show some representative modal parameters which were estimated using the reconstructed global matrices, M −1 K andM −1 C. By simple Comparison between some representative global modes recon- structedfromchainidentificationresultsandthemodesidentifiedusingtheNExT/ERAmethod shown in Fig. 4.29, one can clearly see the good agreement between the identified local and globalmodelsofthebuilding. 133 −40 −20 0 20 40 0 0.005 0.01 0.015 0.02 0.025 0.03 0.035 0.04 μ = −2.9, CV =σ/μ =365.3% pdf Coef. of Vel, c i /m i −50 0 50 0 0.005 0.01 0.015 0.02 0.025 μ = 4.6, CV =σ/μ =398.3% pdf Coef. of Vel, c i /m i (g)9thfloor,x (j)16thfloor,x −100 −50 0 50 100 0 0.002 0.004 0.006 0.008 0.01 0.012 0.014 0.016 0.018 0.02 μ = 6.2, CV =σ/μ =353.6% pdf Coef. of Vel, c i /m i −50 0 50 0 0.005 0.01 0.015 0.02 0.025 0.03 0.035 0.04 μ = 1.7, CV =σ/μ =620.1% pdf Coef. of Vel, c i /m i (h)9thfloor,y (k)16thfloor,y −100 −50 0 50 100 0 0.002 0.004 0.006 0.008 0.01 0.012 0.014 0.016 0.018 μ = −0.9, CV =σ/μ =2492.6% pdf Coef. of Vel, c i /m i −50 0 50 0 0.005 0.01 0.015 0.02 0.025 μ = 1.6, CV =σ/μ =1082.3% pdf Coef. of Vel, c i /m i (i)9thfloor,θ (l)16thfloor,θ Figure4.26: Sampledistributionsoftheestimatedcoefficientofvelocitytermintheinterstoryrestoring functions. Cont. 134 0 0.2 0.4 0 2 4 6 8 10 12 14 16 normalized relative disp X DOF 0 0.2 0.4 0 2 4 6 8 10 12 14 16 normalized relative disp Y DOF 0 0.2 0.4 0 2 4 6 8 10 12 14 16 normalized relative θ θ DOF (a)1stbendinginX,f=0.54Hz (b)1stbendinginY,f=0.58Hz (c)1sttorsion,f=0.78Hz −0.2 0 0.2 0.4 0 2 4 6 8 10 12 14 16 normalized relative disp X DOF −0.4 −0.2 0 0.2 0.4 0 2 4 6 8 10 12 14 16 normalized relative disp Y DOF −0.4 −0.2 0 0.2 0.4 0 2 4 6 8 10 12 14 16 normalized relative θ θ DOF (d)2ndbendinginX,f=1.59Hz (e)2ndbendinginY,f=1.81Hz (f)2ndtorsion,f=2.59Hz Figure 4.27: Some representative modal parameters of the Factor Building which were estimated using thereconstructedglobalmatrices,M −1 K andM −1 C. ElementsoftheM −1 K andM −1 C areestimated usingtheidentifiedchainsystemlocalparameters(k i /m i andc i /m i )basedontheformulationspresented inEq. (4.14)and(4.15). 135 −0.4 −0.2 0 0.2 0.4 0 2 4 6 8 10 12 14 16 normalized relative disp X DOF −0.4 −0.2 0 0.2 0.4 0 2 4 6 8 10 12 14 16 normalized relative disp Y DOF −0.4 −0.2 0 0.2 0.4 0 2 4 6 8 10 12 14 16 normalized relative disp Y DOF (g)3rdbendinginX,f=2.81Hz (h)3rdbendinginY,f=3.11Hz (i)2ndbendinginY,f=1.81Hz −0.4 −0.2 0 0.2 0.4 0 2 4 6 8 10 12 14 16 normalized relative θ θ DOF −0.4 −0.2 0 0.2 0.4 0 2 4 6 8 10 12 14 16 normalized relative disp X DOF −0.4 −0.2 0 0.2 0.4 0 2 4 6 8 10 12 14 16 normalized relative disp Y DOF (j)2ndtorsion,f=2.59Hz (k)3rdbendinginX,f=2.81Hz (l)3rdbendinginY,f=3.11Hz Figure 4.28: Some representative modal parameters of the Factor Building which were estimated using the reconstructed global matrices(M −1 K and M −1 C) based on the local chain identification results. Cont. 136 0 0.2 0.4 0 2 4 6 8 10 12 14 16 normalized relative disp X DOF 0 0.2 0.4 0 2 4 6 8 10 12 14 16 normalized relative disp Y DOF 0 0.2 0.4 0 2 4 6 8 10 12 14 16 normalized relative disp θ DOF 1stbendinginX 1stbendinginY 1sttorsion MAC=99.9% MAC=99.9% MAC=99.8% Δf/f ERA =1.8% Δf/f ERA =1.6% Δf/f ERA =4.8% Figure 4.29: Comparison between some representative global modes reconstructed from chain identi- fication results (solid lines with circles) and the modes identified using the NExT/ERA method (dashed lines with squares). MAC (Modal Assurance Criterion) is the measure of comparison between the mode shapesidentifiedusingtheERAandtheChainmethod,andΔf/f ERA = (f ERA −f chain )/f ERA isthe percentageofdifferencebetweentheidentifiednaturalfrequencies. 4.7 Discussion VariabilityoftheEstimatedParametersDuetoEnvironmentalEffects As was previously mentioned, there are many sources other than damage that can cause notice- able variations in the estimated (identified) dynamic properties of a structure. These sources of variation can be divided into three main categories: (1) environmental conditions such as temperature variation, soil condition, and humidity, (2) operational condition, such as traffic 137 conditionsandexcitationsources,and(3)measurementandprocessing,errorsincludingnonsta- tionarity,measurementnoiseandhysteresis,anderrorsassociatedwithdigitalsignalprocessing. In this subsection the focus is on the variability of the estimated parameters due to temperature variations. Previous studies by Peeters and Roeck (2001a), Farrar et al. (1997), Sohn et al. (1999), and Cornwell et al. (1999), have shown that temperature effects seem to be the most significant cause of variability in the structural modal parameters. Temperature variations not only change the material properties by changing the modulus of elasticity, but can also change the bound- ary conditions as well (Peeters and Roeck, 2001a). In some cases, the variability of structural parameters caused by temperature effects is so significant that is can mask the changes due to damage. Hence, damage detection would be virtually impossible in such cases. Consequently, understanding and detecting (quantifying) the effect of temperature conditions on the estimated localandglobalstructuralparametersisofgreatinterest. As mentioned earlier, in this study the identification procedures were performed every two hourswith1houroverlap, foratotalnumberof50days. Thatresultsin24datapointsforeach single day (one for each hour of a day). Consequently, there are 50 ensembles (corresponding to 50 days) for each hour of a day. Figures 4.30 to 4.32 show the time-history of modal fre- quencies for a 24-hour period, starting and ending at midnight. Note that the last hour values differs by two hours from the starting value, hence, the start and end data points do not coin- cide. The second and third rows of these figures show the comparison between the estimated frequencydistributionsfornight(10PM-2AM)andday(10AM-2PM).Table4.6summarizes thestatisticalvaluescorrespondingtothedistributionsshowninFig. 4.30-4.34. 138 In the mentioned figures, the mean frequency at each hour was found by averaging over 50 daysatthesamehour. Fromthesefiguresitcanbeseenthattherearenoticeablechangesinthe modalfrequenciesduringa24-hourperiod. Onecanclearlyseethatthemaximumandminimum daily values of the modal frequencies happen at midnight and midday, respectively. However, the degree of variation differs mode to mode, ranging from 1% for the first two modes, up to 6% for the 4th bending mode. On the other hand, Fig. 4.35 shows the temperature time-history profilefora24-hourperiodinLosAngelesarea,wherethebuildingunderdiscussionislocated. Thisprofilewascomputedbyaveragingthehourlytemperaturedatafor50consecutivedays,in whichthedaysareexactlythesameastheaccelerationrecordingdatesinthisstudy. Figure4.36 compares the frequency variation profile (Fig. 4.30-4.32) with the temperature variation profile (Fig. 4.35). Based on this figure one can clearly observe the correlation between temperature andfrequencyvariationsina24-hourperiod. Inaddition,fromFig. 4.36,itisobservedthatthere is a time delay between frequency and temperature variations. In fact, the frequency variations lagged behind the temperature variation. This can be explained by heat diffusion phenomenon. The displayed temperature profile is found using the air temperature records and not using the actual structural materials’ temperature. In reality, it takes a while for materials to warm up or cool down. It is interesting to note here that the delay time for warm-up period (the first half of day) is around 6 hours, but the delay time for the cool-down period (the second half of day) is around2hours. ThethirdrowofFig. 4.30-4.32showsthat,forsomemodes,themeanvalueofthefrequency atmiddayisshiftedoutof95%confidenceintervalsofitscorrespondingdistributionatmidnight. This indicates that the frequency variations due to temperature variations could be significant; 139 however, this change must not be regarded as structural damage. On the other hand, low levels of structural damage may not change the modal frequencies as much as the temperature vari- ations do. Therefore, in order to use changes in modal frequencies as damage indicators, one needs to exclude the temperature effects by taking its variations into account. To this end, the frequency distributions should be computed separately for different temperature ranges. Such a need requires many years of data acquisition for different weather and seasonal conditions. In otherwords, ifoneidentifiesthefrequencydistributionsforvarioustemperatureconditions, the confidenceintervalswouldbenarrower(smallerCVs),andhencethepotentialstructuraldamage wouldbemoredetectable. The above mentioned statement can be confirmed by comparing the distributions of modal frequencies in Fig. 4.30-4.34 with those in Fig. 4.18-4.19. For instance, for the 4th bending mode in Y, based on Fig. 4.19(k), in which the distributions are plotted for the whole day, the valueofCVforthe4thbendingmodeinYis2.6%,whereasFig. 4.33,inwhichthedistributions areplottedfordifferenttemperatureranges,showssmallervaluesofCVforthesamemode(the CVs are 1.5% and 1.1% for midday and midnight, respectively). Furthermore, based on the above discussion, the presence of double peaks in the frequency histogram of some modes in Fig. 4.18-4.19 can be explained. This is due to the fact that there are actually two (or more) distinctfrequencydistributionsforduringthedayandnightperiods. Similarly, the same argument is valid for the chain identification results. Figure 4.37 shows thedistributionsofthemass-normalizedstiffnesstermforsomerepresentativestoriesoftheFac- tor Building. Again, the solid and dashed lines represent the estimated frequency distributions fornight(10PM-2AM)andday(10AM-2PM)periods,respectively. Fromthisfigure,itcanbe 140 observedthatthemeanstiffnessvaluesatmiddayaresmallerthantheircorrespondingvaluesat midnight,whichcanbetiedtothetemperaturevariationsinoneday. Itshouldbereemphasized that much larger number of daily ensembles, in different weather and seasonal conditions, are requiredtohavemorereliable(accurate)distributions. 141 0 4 8 12 16 20 24 −3 −2.5 −2 −1.5 −1 −0.5 0 frequency percentage change time of day [Hour] 0 4 8 12 16 20 24 −3 −2.5 −2 −1.5 −1 −0.5 0 frequency percentage change time of day [Hour] 0.53 0.54 0.55 0.56 0.57 0.58 0 20 40 60 80 100 freq [Hz] pdf Night: 10PM−2AM Day: 10AM−2PM 0.57 0.58 0.59 0.6 0.61 0.62 0.63 0 10 20 30 40 50 60 70 80 freq [Hz] pdf Night: 10PM−2AM Day: 10AM−2PM −4 −2 0 2 4 0 20 40 60 80 100 (ν−μ 1 )/σ 1 pdf Night: 10PM−2AM Day: 10AM−2PM −5 0 5 0 10 20 30 40 50 60 70 80 (ν−μ 1 )/σ 1 pdf Night: 10PM−2AM Day: 10AM−2PM 1stbendinginX,CV1=1.2%,CV2=0.9% 1stbendinginY,CV1=1.0%,CV2=1.2% Δμ/μ1=0.1%,Δμ/σ1=4.5% Δμ/μ1=-0.01%, Δμ/σ1=-2.0% Figure 4.30: The first row shows the time-histories of Some representative modal frequencies of the Factor Building in a 24-hour period (one day), in which the corresponding frequencies at each hour are computed by averaging over 50 days. The second and third rows show the comparison between the estimated frequency distributions for night (10 PM-2 AM) and day (10 AM-2 PM). In the captions, subscripts 1 and 2 correspond to the distributions for night: 10 PM-2 AM (solid lines) and day: 10 AM-2 PM (dashed lines), respectively. The parameters μ 1 andμ 2 represent the mean of frequency, for night(10 PM-2 AM) and day (10 AM-2 PM), respectively; Δμ/μ 1 is 100(μ 2 −μ 1 )/μ 1 , and represent the percentage change of the mean frequency normalized by mean frequency for the night; Δμ/μ 1 is 100(μ 2 −μ 1 )/μ 1 ,andrepresentthepercentagechangeofthemeanfrequencynormalizedbythestandard deviationoffrequency;andtheCVdenotestheCoefficientofVariation. 142 0 4 8 12 16 20 24 −5 −4 −3 −2 −1 0 frequency percentage change time of day [Hour] 0 4 8 12 16 20 24 −3 −2.5 −2 −1.5 −1 −0.5 0 0.5 1 frequency percentage change time of day [Hour] 0.7 0.75 0.8 0.85 0.9 0 5 10 15 20 25 30 freq [Hz] pdf Night: 10PM−2AM Day: 10AM−2PM 1.65 1.7 1.75 0 10 20 30 40 50 60 70 freq [Hz] pdf Night: 10PM−2AM Day: 10AM−2PM −6 −4 −2 0 2 4 0 5 10 15 20 25 30 (ν−μ 1 )/σ 1 pdf Night: 10PM−2AM Day: 10AM−2PM −5 0 5 0 10 20 30 40 50 60 70 (ν−μ 1 )/σ 1 pdf Night: 10PM−2AM Day: 10AM−2PM 1sttorsion,CV1=1.9%,CV2=2.5% 2ndbendinginX,CV1=1.0%,CV2=0.4% Δμ/μ1=-2.9%, Δμ/σ1=-152.2% Δμ/μ1=-1.8%, Δμ/σ1=-179.2% Figure4.31: Modalfrequencytime-histories,Cont. 143 0 4 8 12 16 20 24 −3 −2.5 −2 −1.5 −1 −0.5 0 0.5 1 frequency percentage change time of day [Hour] 0 4 8 12 16 20 24 −3 −2.5 −2 −1.5 −1 −0.5 0 0.5 1 frequency percentage change time of day [Hour] 1.8 1.85 1.9 1.95 0 10 20 30 40 50 60 70 freq [Hz] pdf Night: 10PM−2AM Day: 10AM−2PM 2.8 2.85 2.9 2.95 3 0 5 10 15 20 25 30 35 freq [Hz] pdf Night: 10PM−2AM Day: 10AM−2PM −5 0 5 0 10 20 30 40 50 60 70 (ν−μ 1 )/σ 1 pdf Night: 10PM−2AM Day: 10AM−2PM −5 0 5 0 5 10 15 20 25 30 35 (ν−μ 1 )/σ 1 pdf Night: 10PM−2AM Day: 10AM−2PM 2ndbendinginY,CV1=1.0%,CV2=0.4% 3rdbendinginX,CV1=1.0%,CV2=0.5% Δμ/μ1=-2.5%, Δμ/σ1=-242.1% Δμ/μ1=-2.1%, Δμ/σ1=-246.1% Figure4.32: Modalfrequencytime-histories,Cont. 144 0 4 8 12 16 20 24 −5 −4 −3 −2 −1 0 1 frequency percentage change time of day [Hour] 0 4 8 12 16 20 24 −7 −6 −5 −4 −3 −2 −1 0 1 frequency percentage change time of day [Hour] 2.9 3 3.1 3.2 3.3 3.4 0 2 4 6 8 10 12 14 freq [Hz] pdf Night: 10PM−2AM Day: 10AM−2PM 3.5 4 4.5 0 1 2 3 4 5 6 7 8 9 freq [Hz] pdf Night: 10PM−2AM Day: 10AM−2PM −5 0 5 0 2 4 6 8 10 12 14 (ν−μ 1 )/σ 1 pdf Night: 10PM−2AM Day: 10AM−2PM −10 −5 0 5 0 2 4 6 8 (ν−μ 1 )/σ 1 pdf Night: 10PM−2AM Day: 10AM−2PM 3rdbendinginY,CV1=1.9%,CV2=1.2% 3rdtorsion,CV1=1.2%,CV2=2.6% Δμ/μ1=-3.6%, Δμ/σ1=-195.7% Δμ/μ1=-5.1%, Δμ/σ1=-409.9% Figure4.33: Modalfrequencytime-histories,Cont. 145 Table 4.6: Comparison between the estimated frequency for night (10 PM-2 AM) and day (10 AM-2 PM), based on the distributions shown in Fig. 4.30-4.34. In this table, subscripts 1 and 2 correspond to the distributions for night: 10 PM-2 AM and day: 10 AM-2 PM, respectively. The parameters μ 1 andμ 2 represent the mean of frequency, for night(10 PM-2 AM) and day (10 AM-2 PM), respectively; Δμ/μ 1 is 100(μ 2 −μ 1 )/μ 1 , and represent the percentage change of the mean frequency normalized by mean frequency for the night; Δμ/μ 1 is 100(μ 2 −μ 1 )/μ 1 , and represent the percentage change of the mean frequency normalized by the standard deviation of frequency; and the CV denotes the Coefficient ofVariation. mode MaxΔf % CV 1 % CV 2 % Δμ/μ 1 % Δμ/σ 1 % 1stbendinginX -1.1 1.2 0.9 0.1 4.5 1stbendinginY -1.0 1.0 1.2 -0.01 -2.0 1sttorsion -4.6 1.9 2.5 -2.9 -152.2 2ndbendinginX -2.5 1.0 0.4 -1.8 -179.2 2ndbendinginY -2.8 1.0 0.4 -2.5 -242.1 3rdbendinginX -2.4 1.0 0.5 -2.1 -246.1 3rdbendinginY -4.2 1.9 1.2 -3.6 -195.7 3rdtorsion -6.3 1.2 2.6 -5.1 -409.9 4thbendinginX&Y -1.4 1.1 0.7 -0.9 -84.4 3rdbendinginY -6.0 1.5 1.1 -4.7 -315.6 146 0 4 8 12 16 20 24 −3 −2.5 −2 −1.5 −1 −0.5 0 frequency percentage change time of day [Hour] 0 4 8 12 16 20 24 −6 −5 −4 −3 −2 −1 0 frequency percentage change time of day [Hour] 4.1 4.2 4.3 4.4 0 2 4 6 8 10 12 14 16 freq [Hz] pdf Night: 10PM−2AM Day: 10AM−2PM 4.2 4.4 4.6 4.8 5 0 2 4 6 8 10 freq [Hz] pdf Night: 10PM−2AM Day: 10AM−2PM −5 0 5 0 2 4 6 8 10 12 14 16 (ν−μ 1 )/σ 1 pdf Night: 10PM−2AM Day: 10AM−2PM −6 −4 −2 0 2 4 0 2 4 6 8 10 (ν−μ 1 )/σ 1 pdf Night: 10PM−2AM Day: 10AM−2PM 4thbendinginX&Y,CV1=1.1%,CV2=0.7% 4thbendinginY,CV1=1.5%,CV2=1.1% Δμ/μ1=-0.9%, Δμ/σ1=-84.4% Δμ/μ1=-4.7%, Δμ/σ1=-315.6% Figure4.34: Modalfrequencytime-histories,Cont. 147 4 8 12 16 20 24 19 20 21 22 23 24 25 26 27 28 29 time of day [Hr] Mean Temp [C] Figure 4.35: Temperature time-history in a 24-hour period in Los Angeles area. This time-history was computedbyaveragingthehourlytemperaturedataover50dayscorrespondingtotheaccelerationrecod- ingdates. HourlytemperaturedatawasobtainedfromNationalOceanicandAtmosphericAdministration, U.S.departmentofCommerce(http: //cdo.ncdc. noaa.gov /ulcd /ULCD). 148 0 4 8 12 16 20 24 −1 −0.8 −0.6 −0.4 −0.2 0 0.2 normalized percentage change time of day [Hour] frequency temperature 0 4 8 12 16 20 24 −1 −0.8 −0.6 −0.4 −0.2 0 0.2 normalized percentage change time of day [Hour] frequency temperature 2ndbendinginY 3rdbendinginX 0 4 8 12 16 20 24 −1 −0.8 −0.6 −0.4 −0.2 0 0.2 normalized percentage change time of day [Hour] frequency temperature 0 4 8 12 16 20 24 −1 −0.8 −0.6 −0.4 −0.2 0 0.2 normalized percentage change time of day [Hour] frequency temperature 3rdbendinginY 4thbendinginY Figure 4.36: Comparison between the temperature and frequency variations in a 24-hour period. For compactnessreasons,theresultsareonlyplottedforfourrepresentativemodes. PleasenotethattheYaxis of these figures shows the normalized percentage of change, which is the percentage change divided by the maximum percentage change. For temperature variation, the profile is multiplied by-1 for improved clarity. It is observed that there is a time delay between frequency and temperature variations. In fact, thefrequencyvariationslaggedbehindthetemperaturevariation. Thiscanbeexplainedbyheatdiffusion phenomenon. Thedisplayedtemperatureprofileisfoundusingtheairtemperaturerecordsandnotusing the actual structural materials’ temperature. In reality, it takes a while for materials to warm up or cool down. Itisinterestingtonoteherethatthedelaytimeforwarm-upperiod(thefirsthalfofday)isaround 6hours,butthedelaytimeforthecool-downperiod(thesecondhalfofday)isaround2hours. 149 1000 1500 2000 2500 3000 0 0.5 1 1.5 2 2.5 x 10 −3 pdf Coef. of Dsp, k i/m i (a)2ndfloor,x,CV1=8.30%, CV2=8.40%,Δμ/μ1=-3.9%, Δμ/σ1=-47.3% 4500 5000 5500 6000 6500 7000 7500 8000 0 0.2 0.4 0.6 0.8 1 1.2 1.4 1.6 1.8 x 10 −3 pdf Coef. of Dsp, k i/m i (b)2ndfloor,y,CV1=5.4%, CV2=4.0%,Δμ/μ1=-0.8%, Δμ/σ1=-14.2% −2000 0 2000 4000 6000 8000 0 1 2 3 4 5 6 x 10 −4 pdf Coef. of Dsp, k i/m i (c)2ndfloor,θ,CV1=17.7%, CV2=26.3%,Δμ/μ1=-29.9%, Δμ/σ1=-169.5% 200 300 400 500 600 700 0 0.002 0.004 0.006 0.008 0.01 0.012 pdf Coef. of Dsp, k i/m i (d)9thfloor,x,CV1=7.1%, CV2=11.7%,Δμ/μ1=-7.5%, Δμ/σ1=-105.5% 200 400 600 800 1000 1200 1400 0 1 2 3 4 5 6 x 10 −3 pdf Coef. of Dsp, k i/m i (e)9thfloor,y,CV1=8.8%, CV2=15.5%,Δμ/μ1=-5.0%, Δμ/σ1=-56.8% 200 400 600 800 1000 1200 1400 0 1 2 3 4 5 6 x 10 −3 pdf Coef. of Dsp, k i/m i (f)9thfloor,θ,CV1=7.0%, CV2=14.6%,Δμ/μ1=-12.4%, Δμ/σ1=-176.9% −500 0 500 1000 1500 2000 0 0.5 1 1.5 2 2.5 3 3.5 x 10 −3 pdf Coef. of Dsp, k i/m i (g)16thfloor,x,CV1=15.7%, CV2=37.8%,Δμ/μ1=-14.8%, Δμ/σ1=-94.2% −200 0 200 400 600 800 1000 0 1 2 3 4 5 6 x 10 −3 pdf Coef. of Dsp, k i/m i (h)16thfloor,y,CV1=14.0%, CV2=30.7%,Δμ/μ1=-17.2%, Δμ/σ1=-122.8% −500 0 500 1000 1500 0 0.5 1 1.5 2 2.5 x 10 −3 pdf Coef. of Dsp, k i/m i (i)16thfloor,θ,CV1=40.3%, CV2=37.3%,Δμ/μ1=20.4%, Δμ/σ1=50.5% Figure4.37: Distributionsoftheestimatedcoefficientsofthedisplacementtermintheinterstoryrestor- ing functions. Coefficient of displacement is the mass-normalized stiffness term ( k i /m i ). For the sake of compactness, only the results for the 2nd, 9th, and 16th floors are shown. The solid and dashed lines represent the estimated distributions for night (10 PM-2 AM) and day (10 AM-2 PM), respectively. In the captions, subscripts 1 and 2 correspond to the distributions for night: 10 PM-2 AM (solid lines) and day: 10 AM-2 PM (dashed lines), respectively. The parameters μ 1 and μ 2 represent the mean param- eter value for night(10 PM-2 AM) and day (10 AM-2 PM), respectively; Δμ/μ 1 is 100(μ 2 −μ 1 )/μ 1 , and represent the percentage change of the mean value normalized by the mean for the night; Δμ/μ 1 is 100(μ 2 −μ 1 )/μ 1 , and represent the percentage change of the mean value normalized by the standard deviation;andtheCVdenotestheCoefficientofVariation. 150 4.8 SummaryandConclusions This study presents the results of two time-domain identification techniques applied to a full- scale17-storybuildingbasedonambientvibrationmeasurements. TheFactorBuildingisasteel frame structure located on the UCLA campus (Los Angeles, California). Following the 1994 Northridge earthquake, this building was instrumented permanently with a dense array of 72- channel accelerometers, and since then the acceleration data has been continuously recorded. This unique collection of data provided the opportunity to evaluate the effectiveness of various identificationmethodologiesonafull-scalestructure,andunderrealisticfieldconditions. As mentioned, two types of time-domain identification methods are implemented. The first methodistheeigensystemrealizationalgorithminconjunctionwiththenaturalexcitationtech- nique, which is used to extract the modal parameters (Natural frequencies, mode shapes, and modaldamping)ofthebuilding,basedonambientvibrationrecords. Thistypeofidentification methodology is regarded as a global (or centralized) approach, since it deals with the global dynamicpropertiesofthestructure. Thesecondmethodisatime-domainidentificationtechniqueforchain-likeMDOFsystems that are not necessarily linear. Such a class of problems encompasses many practical applica- tions, including tall buildings like the one under discussion. The method uses the information about the state variables of each element of the chain-like system to express the system charac- teristics in terms of some polynomial basis functions. Since in this method the identification of eachlinkofthechainisperformedindependently,itisregardedaslocal(ordecentralized)iden- tification methodology. For the same reason, this method can be easily adopted for large sensor 151 network architectures in which the centralized approaches are not feasible due to massive stor- age, power, bandwidth, and computational requirements. Generally this method can be applied to linear or nonlinear chain-like MDOF systems. Nonetheless, it requires the applied forces to be known. However, for full scale infrastructures like the one under discussion, providing mea- surable excitation isvery costly, andone needsto justrelyon theavailable ambientexcitations. In this study the chain identification procedure was generalized to handle the linear cases with unmeasurable external forces, by combining the chain identification approach with the idea of theNExTtechnique. Anexampleofa100-DOFlinearchain-likestructurewaspresentedtoillustratetheapplica- tionoftheabovementionedapproachesforstructuraldamagedetectionandlocalization. Based on the results of this example, it is concluded that, even for very small levels of damage, the chain system identification approach is capable of not only detecting the damage but also of pinpointing its location. By contrast, the NExT/ERA approach appeared to be less sensitive to low levels of local damage, and it could not provide any spatial information about the damage location. It was shown that for the example under discussion, increasing the noise level and/or reducingthesensorresolution(increasingthemodel-orderreduction),bothhavenegativeeffects onthedamagedetectabilityofthesesmethods. the last section reported the results of the application of the proposed local and global tech- niques for structural identification of the Factor Building. To have a statistically meaningful results, 50 days of the recorded acceleration data were considered in this study. The modal parameter and chain identification procedures were performed over time windows of 2 hours each and with 50% overlap. Using the NExT/ERA method, 12 dominant modes of the building 152 wereidentified;5bendinginE-W,4bendinginN-S,and3torsionalmodes. Itwasobservedthat variationsinthefrequencyestimationarerelativelysmall;CVsareabout1∼2%formostofthe estimated modal frequencies. On the other hand, the damping estimation variance was seen to bemuchhigher;CVsareabout20∼70%formostoftheestimatedmodaldampingvalues. This observation can be explained by the fact that the estimated modal damping values are typically muchlessaccuratethanthemodalfrequencies. Chain system identification was successfully implemented using the output-only data acquired form the Factor Building. Distributions of the estimated coefficients of displacement intheinterstoryrestoringfunctions(whicharethemass-normalizedlocalstiffnessvalues)show higherdegreesofvariability(largerCVs)comparedtotheglobalmodalparameters. Itisbelieved that the reason behind this is the fact that the chain system identification uses the interstory rel- ative motion which is extremely small. In fact, the response of the building due to the ambient excitation is so tiny (in the order of micro-meter), that the relative motion between the floors is in the noise level. That increases the identification error and consequently the variability of the estimated local parameters. In the future it is recommended to use a shaker (or shakers) to gen- erate measurable and dominant excitation forces to adequately excite the building. That would increasetheinterstories’relativemotionandenhancethesignal-to-noiseratiosbyseveralorders of magnitude. In that case, since the dominant excitation force is measurable and known, one would be able to use the nonlinear expansion of the restoring forces, and identify the nonlinear restoringforces. Inordertochecktheconsistencybetweenthelocallyandgloballyidentifiedparameters,the localmass-normalizedstiffnessanddampingtermsidentifiedusingthechainsystem,wereused 153 to reconstruct the global modal parameters. The reconstructed modal parameters, show a very goodagreementwiththemodalparametersidentifiedusingtheNExT/ERAmethod. Variability of the estimated parameters due to temperature variations was also investigated. It is shown that there is a strong correlation between the modal frequency variations and the temperature variations, in a 24-hour period. The maximum and minimum values of the modal frequencies happen at midnights, and middays respectively, which corresponded to the lowest and highest temperature values in a day. However, the degree of variation differs from mode to mode,rangingfrom1%forthefirsttwomodes,to6%for4thbendingmodeoftheFactorBuild- ing. Consequently, in order to use the changes in the modal frequencies as damage indicators, one needs to exclude the temperature effects by taking its variations into account. It is worth notingthatmuchlargernumberofdailyensemblesindifferentweatherandseasonalconditions arerequiredtohavemorereliable(accurate)statisticaldistributions. 154 Chapter5 OptimumStrategiesforDeployingPassiveand Semi-activeMultiple-UnitImpactDampers underStochasticExcitation 5.1 Introduction Background Amongthenumeroustypesofdampingdevicesthathavebeendevelopedandappliedforatten- uating undesirable oscillations, the class of dampers that exploit “impact damping” phenomena for vibration reduction provides some useful features that are ideal for certain situations where ruggedness,reliability,andinsensitivitytotemperatureextremesarearequirementforhandling the encountered operating conditions. The most common example of such environments are combustion chambers and turbine blades where the structure is under severe temperature, pres- sure,andcentrifugalloads. Duetoruggednessandinsensitivitytotemperature,impactdampers arecapabletobeutilizedinsuchenvironmentstoattenuatestructuralvibrationandincreasethe their fatigue life cycle significantly. For example, Panossian (1992) investigated the potential application of impact dampers to reduce vibration in a space shuttle main engine liquid oxygen inlet tee. Duffy et al. (2000) used an impact damper to reduce the vibration in rotor blades. 155 Variety of other applications of impact dampers was investigated, for example vibration reduc- tionofantennae(Simonian,1995),tennisracket(S.,1995),boringtools(EmaandMarui,2000; Oledzkietal.,1999),androcketengineturbopumps(Mooreetal.,1997). Members of this class of dampers include the single-particle impact damper, multi- unit/single-particle impact dampers, multi-particle impact dampers, arrays of particle dampers, and hybrid impact dampers that utilize a combination of momentum transfer devices with fea- tures characteristic of other classes of linear or nonlinear dampers (e.g. dynamic vibration neu- tralizers with motion-limiting stops). The numerous publications cited in the References sec- tion, include analytical, computational, and experimental investigations of the class of damping devicesthatutilizefeaturesoftheimpactdampingmechanism. Thelonglistofpublicationscitedabove,atteststoboththebroadinterdisciplinarynatureof thevariousissuesinvolvedinthemodeling,analysis,simulation,design,anddeploymentofthis familyofdampers,aswellasdemonstratesthecontinuingeffortbymanyinvestigatorstoaddress and resolve some of the many open questions that still await solution in regard to this highly nonlinearclassofdampingdevices,whenoperatingunderarbitrarydynamicenvironments. Thefactthatthemotionofeventhesimplestmanifestationofthisdampersystem(asingle- particle/single-unit)understeady-stateharmonicexcitationcangiverisetoverycomplexchaotic motion,isoneindicationofthechallengesencounteredintryingtofullyunderstandandanalyze the physics involved in the three-dimensional operation of this family of dampers (even when a singleparticleisinvolved)underbroad-band,nonstationary,multi-componentexcitation. While there are some appealing vibration-control features of the family of impact dampers as discussed above, there are also some accompanying undesirable characteristics: impulsive 156 loads transmitted during the momentum exchange phase of the coupled system motion, and the attendant noise and potential local deformations accompanying the plastic collisions among the system components. Furthermore, since the sensitivity of the primary system’s response to the “tuning” of the optimum gap size in a single unit damper is quite dependent on the magnitude of the coefficient of restitution and other levels of inherent (dry friction) damping encountered during the damper operation, optimum design strategies have been investigated over the years. However, there are still many unresolved issues needing study in regard to investigating the performanceofthisclassofdevicesunderbroad-bandexcitation,stationaryornot. Motivation Ithasbeenpreviouslyshown,bothanalyticallyandexperimentally(Masri,1967a,b,1970b)that, foragiventotallevelofimpactdampermassratio,itismoreadvantageoustodistributethetotal particle mass in several units operating in parallel. The use of such an array of impact dampers leads to a reduction in the peak impulsive damping force, a considerable lessening of noise pollution,andareducedsensitivitytothedamper(s)gapsize. Furthermore, with the dramatic advancements in the field of material science focusing on material microstructure design (Christodoulou and Venables, 2003), new classes of structural materialsarebeingdevelopedforaspecificsetoffunctionalproperties. Suchdevelopmentsare paving the way for the production of composite materials which can have their micro structure tailoredtoachievespecificfunctionalities(Vecchio,2005). It,thus,doesnotrequireagreatleap of imagination to see the possibility, in the not too distant future, of composite materials with embeddedmicro-channelsthatprovidethefunctionalityofarraysofmulti-unitparticledampers. 157 Such composite materials can provide, in a distributed manner, the essential functionality of a granularmaterialdamper(Maley,2001;Panossian,1990;WangandYang,2000). Withtheaboveinmind,thegoalofthepresentinvestigationwastoperformfurtheranalyti- cal and experimental studies to develop optimum strategies for dealing with situations in which vibration attenuation devices that are based on impact damping phenomena can be designed to provideeffectiveandrobustdampingperformance,whilesimultaneouslybeingrelativelyinsen- sitive to variations in the spectral characteristics of wide-band excitations, both the stationary and nonstationary type. It will also be shown that, depending on the level of sophistication of the design, a significant improvement in performance can be achieved if an application allows theincorporationofadaptivestiffnesscharacteristicsofthedampers. Scope The contents of this paper are arranged as follows: Section 2 presents the governing equations of motion for a single-degree-of-freedom (SDOF) linear system that is provided with an arbi- trary number of non-similar single-particle dampers operating under an arbitrary dynamic load. Thisrepresentsamulti-unitimpactdamper(MUID).InSection3,theresponseofsuchaMUID under broad-band stationary random excitation is simulated with a very broad range of system designparameters,withthegoalofascertainingtheoptimumdesignspecificationsinachieving a level of vibration control with a specific level of total damper mass ratio. In Section 4, the performance of a MUID under non-stationary random excitation is investigated, and the domi- nantfeaturesoftheresponsearecomparedandcontrastedwiththeoptimumdesignstrategyfor the stationary excitation case. Section 5 presents simulation results for an adaptive nonlinear device corresponding to a semi-active impact damper (SAID) in which the damper gap size is 158 actively adjusted to implement the concept of “parameter control”, which is shown to lead to a significant enhancement in the operating efficiency of a MUID when subjected to stationary or non-stationaryexcitation. 5.2 MathematicalModel The model shown in Fig. 5.1 represents a primary systemM provided with a nonlinear aux- iliary multi-unit impact damper, in which each unit consists of a mass m k that is coupled to M by a piecewise linear dashpot c 2 and spring k 2 , with both c 2 and k 2 having “dead space” characteristicsrepresentedbyclearanced k . M X F(t) K C 2 2 ,C K 2 k d k m . . . k Z Figure5.1: Modelofthemultipleunitimpactdamper In this study it was assumed that the units in the multi-unit impact damper are parallel, so there is no internal interaction between the particles. The equation of motion for the N-unit impactdamperis: 159 ¨ x =−2ζω n ˙ x−ω 2 n x+ f(t) M + N X k=1 μ k ω 2 2 G(z k )+2ζ 2 ω 2 H(z k , ˙ z k )+μ s gsgn(˙ z k ) (5.1) ¨ z k = ¨ x− ω 2 2 G(z k )+2ζ 2 ω 2 H(z k , ˙ z k )+μ s gsgn(˙ z k ) , k = 1,2,...,N (5.2) Where x: primarysystemdisplacement. z k : relativedisplacementofthek’thparticlewithrespecttotheprimarysystem. f(t): externalexcitationforce. ζ: primarysystemfractionofcriticaldamping. ω n : primarysystemnaturalfrequency. ζ 2 : fractionofcriticaldampingoftheimpactdamper“stops”. ω 2 : naturalfrequencyoftheimpactdamper“stops”. M: massoftheprimarysystem. μ k : massratioofthek’thparticlem k /M. μ s : frictioncoefficientbetweenparticlesandtheprimarysystem. g: accelerationduetogravity. G(z k ) and H(z k , ˙ z k ) are nonlinear functions shown in Fig. 5.2, representing the dead space characteristicsofthemodel. Byproperchoiceofω 2 ,thenonlinearspringsG(z k )cansimulatearigidbarriertoanydegree of accuracy. Based on previous studies (Masri and Ibrahim, 1973a,b), the ratio ofω 2 /ω n = 20 160 1 1 2 d 2 d − Z ) (z G ) , ( z z H z z & & Z 2 d 2 d − 1 1 (a) (b) Figure5.2: Nonlinearfunction(a)G(z k )and(b)H(z k , ˙ z k ) is appropriate. The parameterζ 2 in conjunction withH(z k , ˙ z k ) provides means for simulating inelastic impacts, ranging from the completely plastic up to the elastic ones, so the value of any desired coefficient of restitution (e) can be adjusted by setting the appropriate value forζ 2 according to Fig. 5.3. The curves shown in Fig. 5.3 are based on the results of Masri and Ibrahim(1973a,b). 5.3 StationaryRandomExcitationofMulti-UnitImpactDampers Inthissection,theeffectivenessofmulti-unitimpactdampers(MUID)understationaryrandom excitationsisinvestigated. TheRoot-Mean-Square(RMS)level σ x ofthedisplacementxisused to quantify the effectiveness of MUID as a function of different system parameters such as gap clearance, mass ratio, and primary system damping. Normalized (non-dimensional) version of the RMS can be obtained by dividingσ x byσ x 0 which is the RMS level of the primary system displacement in the absence of the MUID. In these simulations, the primary system natural frequencywaschosentobe1Hz(ω n = 2π). 161 0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1 ζ 2 e ζ=0.01, µ=0.10 ω 2 /ω n =5 ω 2 /ω n =20 Experiment Figure5.3: Dependenceofcoefficientofrestitutioneondampingparameterζ 2 InordertoachievestationarityintheRMSlevel,thesimulationsweredoneforatleast2000 periodsoftheprimarysystem. Figures 5.4to 5.7showtheRMSratioofthedisplacementasa functionoftheclearanceratio(d/σ x 0 )fordifferentvaluesofμ,e,ζ,andthenumberofparticles. Particle units are assumed to be in parallel such that there is no interaction between them. The case where the particle units are all together has been investigated experimentally and analyti- callyinRef. (PapalouandMasri,1996a,b,1998). Inthosecases,thereisaninteractionbetween particles due to friction and internal collisions, which reduces the effectiveness of MUID. The gap clearance of all units is assumed to be equal, however the initial position of particles are distributedrandomlysoastoaccountforrealisticsituationswherethereisalwaysacertainlevel ofuncertaintyinevennominallyidenticalphysicalparameters. Inthisway,theimpacttimingof eachunitisdifferentthantheotherunits. 162 0 5 10 15 0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1 d / σ x0 σ x / σ x0 e=0.75, μ=0.10, ζ=0.01, effect of number of particles 1 particle 2 particles 10 particles 100 particles Figure 5.4: RMS response levels for the pri- mary system with e = 0.75, μ = 0.10, and ζ = 0.01. (effectofnumberofparticles) 0 5 10 15 0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1 d / σ x0 σ x / σ x0 100 particles, μ=0.10, ζ=0.01, effect of e e=0.75 e=0.25 Figure 5.5: RMS response levels for the pri- marysystemwithμ = 0.10,ζ = 0.01and100 particle units. (effect of coefficient of restitu- tione) ItisclearthattheRMSleveloftheresponsefortheimpactdampedsystemexhibitsadefinite minimumforcertainclearanceratios. Thisoptimumsituationfortheimpacttimingcorresponds to two impacts per ‘cycle’ of the response (Masri and Caughey, 1966). If the clearance ratio is too small, the particle will collide with the ends of its container more than two times, on aver- age, per response cycle. Therefore, the small clearance does not allow for substantial momen- tum exchange between the colliding masses, so that only slight attenuation of the response is achieved. On the other hand, for large gap clearances, too few impacts occur since the particles do not acquire enough momentum to travel from one end of the container to the opposite end, on the average twice per response cycle. In the limit, as d → 0 or d → ∞ the response of the MUID will revert back to that of the linear SDOF primary system. Therefore, there is an optimum value for the gap clearance ratio, but as one can conclude from Fig. 5.4 to 5.7, the mentionedoptimumclearancedepends,inanonlinearfashion,onsomesystemparameterssuch asμ,e,ζ,andthetotalnumberofparticles. 163 0 5 10 15 0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1 d / σ x0 σ x / σ x0 100 particles, e=0.75, ζ=0.01, effect of μ μ=0.01 μ=0.05 μ=0.10 μ=0.20 Figure 5.6: RMS response levels for the pri- marysystemwithe = 0.75,ζ = 0.01,and100 particleunits. (effectofmassratioμ) 0 5 10 15 0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1 d / σ x0 σ x / σ x0 100 particles, e=0.75, µ=0.10, effect of ζ ζ=0.01 ζ=0.05 ζ=0.10 Figure 5.7: RMS response levels for the pri- marysystemwithμ = 0.10,e = 0.75,and100 particle units. (effect of primary system damp- ingζ) Figure 5.4showstheeffectofthenumberofparticleunitsontheperformanceoftheMUID. It is observed that a MUID is more effective in reducing the RMS level of the response as opposed to a single unit one with the sameμ ,e, andζ. However, there is not much difference in the RMS response level between the results of a 10- and 100-unit damper with the other parameters being the same. Hence, a further increase of the number of units above a certain number would not result in further response reduction. Another important conclusion from the mentionedfigureisthat,asthenumberofunitsincreases,thesensitivityofvibrationattenuation tochangesind,decreases. ThisresultsinawideroptimummarginofdfortheMUIDcompared tothesingleunitone. Figure 5.5showstheeffectofthecoefficientofrestitutioneontheperformanceofMUID.It isobservedfromthediagramthatforsmalld’slowere’sleadtoincreasedmechanicalenergydis- sipation,andhence,amoreeffectivedamper. Sincehighe’simplytheavailabilityofmorekinetic energyfortheparticlestooscillatewith,theoptimumclearanceisincreasedaseincreases. This 164 50 55 60 65 70 75 80 85 90 −100 −50 0 50 100 time impact force (a) single unit impact damper 50 55 60 65 70 75 80 85 90 −10 0 10 time impact force (b) multiple unit impact damper with 100 particles Figure 5.8: Comparing the impact force level for (a) single unit impact damper and (b) MUID, where everyotherparameter(μ,e,ζ,andd)remainsthesame. factisclearinFig. 5.5. AlthoughtheoptimumRMSreductionisslightlymoreforthee = 0.25 as opposed toe = 0.75, it can be seen that the optimum RMS ratio is essentially the same for e’sintherangeof0.2<e< 0.8. AnothersignificantobservationthatcanbegleanedfromFig. 5.5isthatthesensitivityofMUIDtochangesindincreasesasedecreases,whichresultsinthe narrower optimum clearance for smaller e’s. Consequently, MUID designed with a relatively highvalueofecantolerateabroaderrangeofexcitationlevels,whilestillperformingneartheir optimumlevel. Figure 5.6 shows that for a given set of design parameters (e = 0.75, ζ = 0.01, and 100 units)theincreaseinμwillincreasethereductionintheresponselevel, butthereductionisnot directly proportional to the increase inμ. In fact, Fig. 5.6 shows that the effectiveness per unit μwilldecreaseinanonlinearmannerasthemassratioμincreases. 165 0 0.05 0.1 0.15 0.2 0.25 0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1 minimum( σ x / σ x0 ) µ MUID with 100 particles, e=0.75 ζ=0.01 ζ=0.05 ζ=0.10 Figure 5.9: Effect of viscous damping and mass ratio on the performance of MUID with 100 particles ande = 0.75 Figure 5.7 shows that for a given set of parameters (e = 0.75, μ = 0.10, and 100 units) theeffectivenessofthedamperincreasesastheprimarysystemdampingζ decreases,sothatthe maximumeffectofMUID wouldbeachievedforaprimary systemwithanegligibleamountof inherentdamping. Another significant advantage of a MUID as opposed to a single unit one is the substantial reduction of the peak level of impulsive forces exchanged during momentum transfer. This results in (1) the attendant reduction in the accompanying plastic deformations induced in the impact process, and (2) the appreciable attenuation of the noise level surrounding the operation of the damping device. Figure 5.8 compares the impact forces for a single and a multiple unit (100 particles) impact damper where all other parameters (μ ,e, ζ, and d) are the same. It is obviousthatthepeaklevelofimpactforceforaMUIDismuchlowerforasingleunitone. 166 Figure 5.9 summarizes the effects ofμ andζ on the optimum performance of MUID with 100 units ande = 0.75. Again it is clear that for a givenζ, the optimum response reduction is not a linear function of the mass ratio. Also, one can conclude that even with very small mass ratios, a properly-designed MUID is capable of substantial attenuation of the RMS response level(about50%withμ = 5%)inlightly-dampedsystems. Up to now, it is assumed that the gap clearances are identical for all units. It would be usefultoinvestigatethebehaviorofaMUIDwithdifferentclearancesforeachunit. Tothisend, the performance of a MUID with randomly distributed (uniform distribution) gap clearances is investigated. AMUIDwith50particles,e = 0.75,ζ = 0.01,andatotalμ = 0.10isconsidered. The 50 gap clearances are uniformly distributed betweend min andd max . The resulting RMS ratios(σ x /σ x 0 )areplottedagainst(d min /σ x 0 )and(d max /σ x 0 )inFig. 5.10. 5 10 15 0 5 10 0.45 0.5 0.55 0.6 0.65 0.7 0.75 0.8 d min /σ x0 d max /σ x0 σ x /σ x0 0.5 0.5 0.5 0.55 0.55 0.55 0.55 0.55 0.6 0.6 0.6 0.65 d max / σ x0 d min / σ x0 2 4 6 8 10 12 14 2 4 6 8 10 12 14 each contour line shows the value of (σ x / σ x0 ) Optimum Region (a) (b) Figure 5.10: The resulting RMS ratios ( σx σx 0 ) for the uniform distribution of gap clearances between d min andd max ,for50particleunits,e = 0.75,ζ = 0.01,andμ = 0.10,(a)3Dplot,(b)contourplot The reason that the above diagonal elements are empty is that d max must be greater than d min . Allthevaluesalongthediagonalcorrespondtod min =d max ,soifoneplotsthediagonal values of (σ x /σ x 0 ) versus d = d min = d max , exactly the same results, obtained previously 167 for the equal gap clearances, would be achieved. The most important conclusion from this figure is that, by uniform distribution of d k ’s in the optimum region shown in the contour plot (approximately 2 < d min σx 0 < 6, 4 < dmax σx 0 < 8, andd max > d min ), MUID results in more than 50%reductioninRMSlevel. Ontheotherhand,thisfigureshowsthatwithrandomlydistributed d k ’s, the optimum RMS reduction is much less sensitive, so that even outside of the mentioned optimumregion,MUIDcanresultinanappreciableRMSreduction. 5.4 Nonstationary Random Excitation of Multi-Unit Impact Dampers InthissectiontheperformanceoftheMUIDundernonstationaryexcitations(t)isinvestigated. Oneconvenientmeansofgeneratingasyntheticnonstationaryrandomexcitationistomodulate astationaryrandomsignaln(t)throughmultiplicationbyadeterministicenvelopefunctiong(t), asfollows(Masri,1978): s(t) =g(t)n(t) (5.3) with g(t) =a 1 exp(a 2 t)+a 3 exp(a 4 t) (5.4) wheren(t)isthestationaryrandomexcitation,ands(t)istheresultingnonstationarypart. Bya properchoiceofa 1 ,a 2 ,a 3 ,anda 4 ,onecangenerateavarietyofnonstationaryexcitations,such asearthquake-likeexcitations. Simulationsweredoneforthesamesystemdiscussedintheprevioussection. TheRMSfor the resulting non-ergodic process was computed by averaging over a large ensemble of records 168 (over200ensembles). Forthecasestudiedinthispaper,thesimulationshowsthattheresulting RMS will not change after the ensemble size approaches 200 records. Three different envelope functions(g(t))wereconsideredinthisstudy,correspondingtoa“fast”,“medium”,and“slow” rateofenvelopedecay. 0 5 10 15 20 25 −1 −0.8 −0.6 −0.4 −0.2 0 0.2 0.4 0.6 0.8 1 excitation, g=g 1 t / T a 1 =1, a 2 =−1, a 3 =−1, a 4 =−1.5 excitation envelope function g 1 (t) Figure 5.11: Exponential envelope function g 1 (t) = exp(−t)−exp(−1.5t)andtheresult- ingnonstationaryrandomexcitation. 0 5 10 15 20 25 0 0.02 0.04 0.06 0.08 0.1 0.12 0.14 0.16 0.18 0.2 t / T σ x g=g 1 without MUID With MUID, e=0.75, μ=0.10, ζ=0.01, 100 particles Figure 5.12: Transient RMS response of the primary system by averaging over 200 ensem- bles. System parameters: μ = 0.10,ζ = 0.01, e = 0.75, d = d opt , and 100 particle units. Excitationenvelope:g(t) =g 1 (t). Figures 5.11 to 5.16 show the effectiveness of the MUID in reducing the RMS response (μ = 0.10,ζ = 0.01,e = 0.75,d = d opt , and 100 particle units) for three different envelope functions. Forthesethreecases,theRMSwascalculatedbyaveragingover200records,andthe optimumgapclearance(d opt )waschosenbasedonthemaximumreductioninthepeakvalueof theRMS. Table 5.1showstheratioofthepeakRMS(σ max /σ 0max )andtheratiooftheareaunderthe RMStimehistory R σ x dt/ R σ x 0 dt forthethreementionedcases. Onecanconcludefromthe displayed results that a MUID is significantly effective reducing the area under the RMS time historycurve(whichisanindicationoftheresponse“intensity”). However,theeffectivenessof 169 0 10 20 30 40 50 60 −1 −0.8 −0.6 −0.4 −0.2 0 0.2 0.4 0.6 0.8 1 excitation, g=g 2 t / T a 1 =1, a 2 =−0.2, a 3 =−1, a 4 =−1.5 excitation envelope function g 2 (t) Figure 5.13: Exponential envelope function g 2 (t) = exp(−0.2t)− exp(−1.5t) and the resultingnonstationaryrandomexcitation. 0 10 20 30 40 50 60 0 0.2 0.4 0.6 0.8 1 1.2 1.4 t / T σ x g=g 2 without MUID With MUID, e=0.75, μ=0.10, ζ=0.01, 100 particles Figure 5.14: Transient RMS response of the primary system by averaging over 200 ensem- bles. System parameters: μ = 0.10,ζ = 0.01, e = 0.75, d = d opt , and 100 particle units. Excitationenvelope:g(t) =g 2 (t). Table5.1: Summaryofthenonstationarysimulationresults. Envelopefunction PeakRMSratio AreaundertheRMStimehistory g(t) σmax σ0max R σx dt R σx 0 dt g 1 (t) 0.82 0.32 g 2 (t) 0.73 0.35 g 3 (t) 0.62 0.50 a MUID in reducing the peak RMS of the nonstationary response is not significant, especially when the envelope function duration is short. The reason for this behavior is due to the MUID nature: it takes a while for the particles to acquire enough momentum for effective vibration attenuation. By increasing the envelope duration, the behavior of the MUID improves, and gets closertothestationaryone. Forexample,forg(t) =g 3 (t),thenonstationaryexcitationduration isabout100naturalperiodsofthesystem,andtheresultingreductioninthepeakRMSisabout 38%,whichisclosertowhatwasachievedforthesamesituationinthestationaryexcitationcase (Fig. 5.9showsabout55%reductioninthestationaryRMSwithsimilarsystemparameters). 170 0 50 100 150 −1 −0.8 −0.6 −0.4 −0.2 0 0.2 0.4 0.6 0.8 1 excitation, g=g 3 t / T a 1 =1, a 2 =−0.03, a 3 =−1, a 4 =−0.4 excitation envelope function g 3 (t) Figure 5.15: Exponential envelope function g 3 (t) = exp(−0.03t)− exp(−0.4t) and the resultingnonstationaryrandomexcitation. 0 50 100 150 0 0.5 1 1.5 2 2.5 t / T σ x g=g 3 without MUID With MUID, e=0.75, μ=0.10, ζ=0.01, 100 particles Figure 5.16: Transient RMS response of the primary system by averaging over 200 ensem- bles. System parameters: μ = 0.10,ζ = 0.01, e = 0.75, d = d opt , and 100 particle units. Excitationenvelope:g(t) =g 3 (t). Figure 5.17summarizestheperformanceofaMUIDinreducingtheratioofthepeakRMS, andtheareaundertheRMStimehistorycurve,fordifferentmassratios. 5.5 Stationary and Nonstationary Random Excitation of Semi- ActiveImpactDampers Intheprevioussections,itwasshownthatMUIDsarerelativelyefficientvibrationneutralizersin attenuating the response of oscillating systems subjected to stationary, as well as nonstationary, randomexcitations. However,asinanypassivedevice,evenwhenthecharacteristicsofaMUID have been optimized for a given operating condition, its efficiency is limited in handling wide- band random excitations due to its inability of continuously adapting its characteristics to the evolving environment. This limitation is particularly significant in applications where not only theRMSleveloftheresponse,butthepeaklevelaswellisofconcern. 171 0 0.05 0.1 0.15 0.2 0.25 0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1 min( σ max / σ 0,max ) and min (∫ σ dt / ∫ σ 0 dt) μ MUID with 100 particles, e=0.75, and ζ=0.01, Excitation envelop g(t)=g 2 (t) min ( σ max / σ 0,max ) min (∫ σ dt / ∫ σ 0 dt ) Figure5.17: EffectofmassratioontheperformanceofMUIDinnonstationaryrandomexcitationwith envelopefunctiong(t) =g 2 (t) Masri et al. (1989) introduced and implemented a semi-active algorithm which renders the impact damper fixed gap sized an adjustable parameterd(t) that requires a minuscule amount of energy to control. The resulting adaptive nonlinear system has significantly superior per- formance in comparison to an optimized passive system. The essential feature of this adaptive control strategy is similar to the approach of Nudehi et al. (2006) who used adjustable stiffness approachesforefficientactivecontrolstrategies. The aforementioned algorithm to produce a Semi-Active Impact Damper (SAID) is quite effective in greatly reducing the RMS response, as well as the peak response, in both stationary and nonstationary cases. The control algorithm exploits the fact that the maximum momentum transferinvolvedintheimpactprocesscanbeachievediftheimpactsoccurattheinstantoftime correspondingtopeakvelocityoftheprimarysystem. Consequently,tomaximizetheefficiency 172 of an impact damper between two consecutive impacts, the gap clearances should be adjusted onlinesothatthefollowingtwoconditionsaresatisfied: • Foreachdamperunit,animpactismadetooccurwhenthevelocityofthecorresponding primary system has reached its peak value. This instant corresponds to the zero-crossing oftheprimarysystemdisplacement. • The velocities of the various set(s) of the two colliding masses must be opposite to each other at the time of impact. This condition insures that the impact process will stabilize themotionoftheprimarysystem(KaryeaclisandCaughey,1989a,b). To illustrate the application of this approach, a representative segment of the motion of a linear SDOF system being controlled by such an algorithm is shown in Fig. 5.18. As seen from this figure, the optimum control impacts are applied twice per fundamental period of the primarysystem. Theefficiencyofthiscontrolstrategyisdemonstratedbypresentingnumerical simulation results for both stationary and nonstationary excitations. Figure 5.19 summarizes the efficiency of a SAID in terms of RMS level reduction of a SDOF system subjected to a stationary random excitation. Figure 5.20 compares the RMS ratios for SAID and MUID with identical system parameters (e = 0.75,ζ = 0.01), and identical stationary random excitations. The optimum value of the gap clearances was chosen for MUID. It can be clearly seen from this figure that, for any given mass ratios, the response RMS ratio using the SAID is much lowerthanthecorrespondingonewhenusingtheMUIDatitsoptimumcondition. Forexample, withμ = 0.10 the RMS ratios for MUID and SAID with similar parameters are 43% and 24% respectively,whichcorrespondsto≈45%improvementusingtheSAID. 173 0 Disp (a) 0 Vel (b) 0 Relative Disp & Vel (c) 10 10.5 11 11.5 12 12.5 13 13.5 14 0 t / T impact force (d) x 2 ˙ x 1 ˙ x 2 x 1 z ˙ z Figure 5.18: Time history of a representative segment of a linear SDOF system, that is harmonically excited and provided with SAID having μ = 0.10 and e = 0.75. For clarity, the amplitude of all the plotted quantities have been normalized. (a) Absolute displacement of the primary (x 1 ) and the secondary (x 2 ) system. (b) Absolute velocity of the primary ( ˙ x 1 ) and the secondary ( ˙ x 2 ) system. (c) Relative displacement (z) and velocity (˙ z) between the primary and secondary systems. (d) Total impact force Theoretically (i.e., neglecting parameter uncertainty effects), the SAID with multiple units will result in essentially the same performance achieved by a single-unit version. This is due to the fact that, when the SAID controller detects the peak velocity, it will command all the units for the impact, exactly at the same time. So, if it is one unit or 100 units, the resulting (cumulative) impact force would be the same. However, in practice, there is a slight difference betweentheactualtimeofimpacts,andthelevelofmomentumexchange,duetovariationsinthe evolving particle(s) motion. This “re-distribution” of the total momentum exchange across the 174 0 0.05 0.1 0.15 0.2 0.25 0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1 σ x / σ x0 µ SAID with one unit, e=0.75 ζ=0.01 ζ=0.05 ζ=0.10 Figure 5.19: Effect of viscous damping and mass ratio on the efficiency of SAID (one unit,e = 0.75) subjectedtostationaryrandomexcitation. many particles involved in semi-active, and multi-unit impact dampers results in a more robust andefficientvibrationcontroldevice. 175 0 0.05 0.1 0.15 0.2 0.25 0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1 μ ζ=0.01, e=0.75 σ x / σ x0 MUID SAID Figure 5.20: RMS ratio comparison between SAID (one unit, e = 0.75, ζ = 0.01) and MUID (100 units,e = 0.75,ζ = 0.01)subjectedtostationaryrandomexcitation. 176 Figure 5.21 compares the efficiency of SAID and MUID in reducing the transient RMS response of a SDOF system subjected to nonstationary random excitations, for three different excitation envelope functions. It is clear from the results that a significant improvement can be achievedusingSAID,asopposedtoMUID.TheimprovementdoesnotapplyonlyontheRMS levelreduction,butalsoonthepeakresponsereduction. Figures 5.22 and 5.23 compare the ratio of the area under the RMS time history, and the peakRMSratioforthenonstationarycasewithenvelopefunctiong(t) =g 2 (t),respectively. The optimum value of the gap clearance was chosen for the MUID. One can clearly see that using theSAIDgreatlyimprovestheattenuationoftheRMSresponse,aswellasthepeakresponse. In addition to the improvement in vibration attenuation, there are other advantages in using theSAIDthatcanbesummarizedasfollows: • Virtually no prior information regarding the system model and the global dynamic char- acteristics is needed. This is an important advantage, especially for vibration control of unknown and/or time varying systems. Even for designing a passive damping device, such as a Tuned Mass Damper (TMD) or MUID, the designer needs to know the system parameters (e,ζ,μ, and level of excitation for a MUID, andμ,ω n , and frequency range ofexcitationforaTMD)whichmightnotbeavailableinsomepracticalapplications. • The behavior of passive MUID is highly amplitude dependent. The reason is that the optimumgapclearancewillchangewiththechangeinexcitationlevel(indicatedbyσ x 0 ). Unlike passiveone, SAID can adapt itselffor change in theexcitation level, such thatthe optimumconditionismaintained,regardlessofamplitudelevel. • Whethertheprimarysystemislinearornonlinearhasnobearingonthealgorithm. 177 • The algorithm is very simple to apply, and therefore the online computation is extremely fast. • Sincethecontrollerusestheavailablesystems’smomentumtoattenuateitsvibration, the controlenergyneededisverysmall(justsomeelectricpulsestoadjustthed). 178 0 5 10 15 20 25 0 0.05 0.1 0.15 0.2 0.25 t / T σ x g=g 1 without MUID or SAID With MUID, e=0.75, μ=0.10, ζ=0.01, 100 particles With SAID, e=0.75, μ=0.10, ζ=0.01, one unit (a) 0 10 20 30 40 50 60 0 0.5 1 1.5 t / T σ x g=g 2 without MUID or SAID With MUID, e=0.75, μ=0.10, ζ=0.01, 100 particles With SAID, e=0.75, μ=0.10, ζ=0.01, one unit (b) 0 50 100 150 0 0.5 1 1.5 2 2.5 t / T σ x g=g 3 without MUID or SAID With MUID, e=0.75, μ=0.10, ζ=0.01, 100 particles With SAID, e=0.75, μ=0.10, ζ=0.01, one unit (c) Figure5.21: ComparisonofSAIDandMUIDinreducingthetransientRMSresponseofaSDOFsystem subjected to nonstationary random excitations. System parameters: μ = 0.10, ζ = 0.01, e = 0.75. Excitationenvelope: (a)g(t) =g 1 (t),(b)g(t) =g 2 (t),and(c)g(t) =g 3 (t). 179 0 0.05 0.1 0.15 0.2 0.25 0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1 min (∫ σ dt / ∫ σ 0 dt ) μ Excitation envelope g(t)=g 2 (t) MUID with 100 particles, e=0.75, μ=0.10, ζ=0.01 SAID with one unit, e=0.75, μ=0.10, ζ=0.01 Figure5.22: ComparisonofSAIDandMUID in reducing the area under the RMS time his- tory in nonstationary random excitation with envelopefunctiong(t) =g 2 (t). 0 0.05 0.1 0.15 0.2 0.25 0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1 min( σ max / σ 0,max ) μ Excitation envelope g(t)=g 2 (t) MUID with 100 particles, e=0.75, μ=0.10, ζ=0.01 SAID with one unit, e=0.75, μ=0.10, ζ=0.01 Figure5.23: ComparisonofSAIDandMUID in reducing the peak RMS ratio in nonstation- ary random excitation with envelope function g(t) =g 2 (t). 180 5.6 SummaryandConclusions The performance of particle dampers whose behavior under broad-band excitations involves internal friction and momentum transfer is a highly complex nonlinear process that is not amenable to exact analytical solutions. While numerous analytical and experimental studies have been conducted over many years to develop strategies for modeling and controlling the behavior of this class of vibration dampers, no guidelines currently exist for determining opti- mumstrategiesformaximizingtheperformanceofparticledampers,whetherinasingleunitor inarraysofdampers,underrandomexcitation. This paper reports on a comprehensive study concerning the development and evaluation of practical design strategies for maximizing the damping efficiency of multiple-unit particle dampers under random excitation, both the stationary and nonstationary types. High-fidelity simulation studies are conducted with a variable number of multi-unit dampers ranging from one to a hundred, with the magnitude of the “dead-space” nonlinearity being a random variable with a prescribed distribution spanning a feasible range of parameters. Results of the com- putational studies are calibrated with carefully-conducted experiments with single-unit/single- particle,single-unit/multi-particle,andmultiple-unit/multi-particledampers. It is shown that a wide latitude exists in trade-offs between high vibration attenuation over a narrow range of damper gap size, versus slightly reduced attenuation over a much broader range. Theoptimumconfigurationcanbeachievedthroughtheuseofmultipleparticledampers designed in accordance with the procedure presented in the paper. A Semi-active algorithm is introducedtoimprovetheRMSlevelreduction,aswellasthepeakresponsereduction. Theutil- ity of the approach is demonstrated through numerical simulation studies involving broad-band 181 stationary random excitation, as well as highly non-stationary excitations resembling typical earthquakegroundmotions. 182 Chapter6 Conclusions Inthefirstpartofthisstudy,threetime-domaintechniqueswereconsideredforthemodalparam- eteridentificationofthenewCarquinezbridge,amodernlongsuspensionbridge,basedonambi- ent and forced vibration measurements collected before this new bridge was opened for traffic. These three methods are: the eigensystem realization algorithm (ERA), the ERA with data cor- relations(ERA/DC),andtheleastsquares(LS)algorithm. Inordertoimplementthesemethods using output-only information, the natural excitation technique (NExT) was first used to con- vertthenonhomogeneousequationofmotiontoahomogeneousone. Anautonomousalgorithm waspresentedtodistinguishthegenuinestructuralmodesfromspuriousnoiseorcomputational modes. OneimportantissueintheNExTtechniqueistheselectionofaproperreferenceDOF.Since theoptimumaccuracyfordifferentmodesoccursfordifferentchoicesofthereferenceDOFs,itis preferabletousemultiplereferencesasopposedtoasinglereference. Identificationformulations were modified to include many reference points simultaneously, or one at a time. The study shows that the ERA/DC method requires less initial model overspecification in the presence of noise, and it can be computationally faster than the ERA. Another issue in the implementation of the mentioned techniques in real experimental applications is choosing the right values for user-selectable parameters. Some useful guidelines for the selection of critical parameters were presented. Toverifythoseguidelines,someparameterstudieswereperformed. Itwasshownthat 183 these identification techniques are capable of being used in online structural health monitoring schemes,andincalibratingandvalidatingassociatedfiniteelementmodels. The second chapter of this thesis deals with the structural identification and monitoring of a full-scale 6-story building based on ambient vibration measurements. As a result of the 1994 Northridgeearthquake,acriticalsix-storybuildinginthemetropolitanLosAngelesregion(The LongBeachPublicSafetyBuilding)wasfoundtoneedsignificantseismicmitigationmeasures. Thebuildingwasinstrumentedwith14state-of-the-artstrong-motionaccelerometerstoacquire extensive ambient vibration data sets at regular intervals that covered the whole construction phase,duringwhichthebuildingevolvedfromitsoriginalconditiontotheretrofittedstatus. Two multi-input/multi-output state space methods that are suitable for experimental modal parame- ter identification of structures were considered. These two methods are the ERA and the ERA withdatacorrelations(ERA/DC).Theidentificationmethodologywasimplementedinconjunc- tion with the LBPSB data during its retrofit, in order to determine the evolution of the modal properties of the subject building during the various phases of its retrofit process, and correlate thechangesintheidentifiedstructuralfrequencieswiththetimethatspecificstructuralchanges were made. The main challenges with this unique experimental study were: (a) low resolution sensor placement, which results in high model-order reduction, and (b) the ambient excitation was so small that the higher modal displacements were in the noise level, hence not adequately excited. The third chapter of this study present the results of two time-domain identification tech- niques applied to a full-scale 17-story building based on ambient vibration measurements. The 184 Factor Building is a steel frame structure located on the UCLA campus (Los Angeles, Califor- nia). Following the 1994 Northridge earthquake, this building was instrumented permanently with a dense array of 72-channel accelerometers, and since then the acceleration data has been continuously recorded. This unique collection of data provided the opportunity to evaluate the effectivenessofvariousidentificationmethodologiesonafull-scalestructure,andunderrealistic fieldconditions. Two types of time-domain identification methods were implemented on the Factor Build- ing. The first method is the eigensystem realization algorithm in conjunction with the natural excitation technique, which is used to extract the modal parameters (natural frequencies, mode shapes, and modal damping) of the building, based on ambient vibration records. This type of identification methodology is regarded as a global (or centralized) approach, since it deals with the global dynamic properties of the structure. The second method is a time-domain identifi- cation technique for chain-like MDOF systems. Such a class of problems encompasses many practical applications, including tall buildings like the one under discussion. The method uses the information about the state variables of each element of the chain-like system to express the system characteristics in terms of some polynomial basis functions. Since in this method the identification of each link of the chain is performed independently, it is regarded as a local (or decentralized) identification methodology. For the same reason, this method can be easily adopted for large sensor network architectures in which the centralized approaches are not fea- sibleduetomassivestorage,power,bandwidth,andcomputationalrequirements. Generallythis method can be applied to linear or nonlinear chain-like MDOF systems. On the other hand, it requires the applied forces to be known. However, for full scale infrastructures like the one 185 underdiscussion,providingmeasurableexcitationisverycostly,andoneneedstojustrelyonthe availableambientexcitations. Thechainidentificationprocedurewasgeneralizedinthisstudyto handlethelinearcaseswithunmeasurableexternalforces,bycombiningthechainidentification approachwiththeideaoftheNExTtechnique. Anexampleofa100-DOFlinearchain-likestructurewaspresentedtoillustratetheapplica- tionoftheabovementionedapproachesforstructuraldamagedetectionandlocalization. Based on the results of this example, it is concluded that, even for very small levels of damage, the chainsystemidentificationapproachiscapableofnotonlydetectingthedamagebutalsoofpin- pointing its location. On the other hand, the NExT/ERA approach appeared to be less sensitive tolowlevelsoflocaldamage,anditcouldnotprovideanyspatialinformationaboutthedamage location. It was shown that, for the example under discussion, increasing the noise level and/or reducingthesensorresolution(increasingthemodel-orderreduction),bothhaveanegativeeffect ondamagedetectabilityofthesesmethods. The last section reported the results of the application of the proposed local and global techniques for structural identification of the Factor Building. To have statistically meaning- ful results, 50 days of the recorded acceleration data were considered in this study. The modal parameter and chain identification procedures were performed over time windows of 2 hours each and with 50% overlap. Using the NExT/ERA method, 12 dominant modes of the building were identified; 5 bending in E-W, 4 bending in N-S, and 3 torsional modes. It was observed thatvariationsinthefrequencyestimationarerelativelysmall;CVsareabout1∼2%formostof the estimated modal frequencies. On the other hand, the damping estimation variance was seen to be much higher; CVs were about 20∼70% for most of the estimated modal damping values. 186 That can be explained by the fact that the estimated modal damping values are typically much lessaccuratethanthemodalfrequencies. Chain system identification was successfully implemented using the output-only data acquired form the Factor Building. Distributions of the estimated coefficients of displacement intheinterstoryrestoringfunctions(whicharethemass-normalizedlocalstiffnessvalues)show higherdegreesofvariability(largerCVs)comparedtotheglobalmodalparameters. Itisbelieve that the reason behind this is the fact that the chain system identification uses the interstory rel- ative motion, which is extremely small. In fact, the response of the building due to the ambient excitation is so tiny, that the relative motion between the floors is within the data “noise level”. Thissituationincreasestheidentificationerror,andconsequentlythevariabilityoftheestimated local parameters. It is recommended that, a shaker (or shakers) be used to generate measurable anddominantexcitationforcestoadequatelyexcitethebuilding. Thatwouldincreasetheinter- stories’ relative motion and enhance the signal-to-noise ratios by several orders of magnitude. Inthatcase,sincethedominantexcitationforceismeasurableandknown,onewouldbeableto usethenonlinearexpansionoftherestoringforces,andidentifythenonlinearrestoringforces. Inordertochecktheconsistencybetweenthelocallyandgloballyidentifiedparameters,the local mass-normalized stiffness and damping terms identified using the chain system, are used to reconstruct the global modal parameters. The reconstructed modal parameters, show a very goodarrestmentwiththemodalparametersidentifiedusingtheglobalNExT/ERAmethod. The variability of the estimated parameters due to temperature variations was investigated. It was shown that there is a strong correlation between the modal frequency variations and the temperature variations in a 24-hour period. The maximum and minimum values of the modal 187 frequencies occur at midnights, and middays, respectively, which corresponded to the lowest and highest temperature values in a day. However, the degree of variation differs from mode to mode, ranging from 1% for the first two modes, to 6% for 4th bending mode of the Factor Building. Consequently,inordertousethechangesinthemodalfrequenciesasdamageindica- tors,oneneedstoexcludethetemperatureeffectsbytakingitsvariationsintoaccount. Itshould be mentioned that a much larger number of daily ensembles, in different weather and seasonal conditions,arerequiredtohavemorereliablestatisticaldistributions. The last part of this study dealt with the performance of particle dampers. The behavior of these kind of dampers under broad-band excitations is a highly complex nonlinear process that isnotamenabletoexactanalyticalsolutions. Whilenumerousanalyticalandexperimentalstud- ieshavebeenconductedovermanyyearstodevelopstrategiesformodelingandcontrollingthe behavior of this class of vibration dampers, no guidelines currently exist for determining opti- mumstrategiesformaximizingtheperformanceofparticledampers,whetherinasingleunitor in arrays of dampers, under random excitation. This part of thesis reports on a comprehensive study concerning the development and evaluation of practical design strategies for maximiz- ing the damping efficiency of multiple-unit particle dampers under random excitation, both the stationaryandnonstationarytypes. High-fidelitysimulationstudieswereconductedwithavari- able number of multi-unit dampers ranging from one to a hundred, with the magnitude of the “dead-space”nonlinearitybeingarandomvariablewithaprescribeddistributionspanningafea- sible range of parameters. Results of the computational studies were calibrated with carefully- conducted experiments with single-unit/single-particle, single-unit/multi-particle, and multiple- unit/multi-particle dampers. It was shown that a wide latitude exists in the trade-offs between 188 high vibration attenuation over a narrow range of damper gap size, versus a slightly reduced attenuation over a much broader range. The optimum configuration can be achieved through theuseofmultipleparticledampersdesignedinaccordancewiththeprocedurepresentedinthe paper. A Semi-active algorithm was introduced to improve the RMS level reduction, as well as the peak response reduction. 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Abstract (if available)
Abstract
One of the issues complicating the reliability assessment of structural health monitoring (SHM) methodologies slated for implementation under field conditions for damage detection in conjunction with typical infrastructure systems, is the paucity of experimental measurements from such structures. This study evaluates some promising SHM methodologies based on actual vibration measurements, obtained under realistic field conditions from three different cases of full-scale civil infrastructures. The first part of this study provides a comprehensive and comparative study of three time-domain identification algorithms applied to extract the modal parameters of the New Carquinez Bridge which is a newly-constructed long-span bridge that was monitored, in its virgin state, over a relatively long period of time, with a state-of-the-art dense sensor array. The second part of this study evaluates the usefulness of some identification techniques to determine the evolution of the modal properties of a full-scale 6-story building which has recently undertaken a 6-month seismic retrofit process, and to correlate the changes in the identified structural frequencies with the time that specific structural changes were made. The third part of this study presents the results of two time-domain identification techniques applied to a full-scale 17-story building, based on ambient vibration measurements. The UCLA Factor Building was instrumented permanently with a dense array of 72-channel accelerometers. The first identification method used in this case is the NExT/ERA, which is regarded as a global (or centralized) approach, and the second method is a chain system identification technique. Since in this method the identification of each link of the chain is performed independently, it is regarded as local (or decentralized) identification methodology. To have a statistically meaningful result, 50 days of the recorded data are considered in this study.
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Creator
Dehghan Nayeri, Mohammad Reza
(author)
Core Title
Analytical and experimental studies in system identification and modeling for structural control and health monitoring
School
Viterbi School of Engineering
Degree
Doctor of Philosophy
Degree Program
Mechanical Engineering
Publication Date
06/05/2007
Defense Date
05/14/2007
Publisher
University of Southern California
(original),
University of Southern California. Libraries
(digital)
Tag
OAI-PMH Harvest,Structural health monitoring,system identification and modelling
Language
English
Advisor
Masri, Sami F. (
committee chair
), Ghanem, Roger (
committee member
), Wellford, L. Carter (
committee member
), Yang, Bingen (
committee member
)
Creator Email
dehghann@usc.edu
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https://doi.org/10.25549/usctheses-m514
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UC1335270
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etd-DehghanNayeri-20070605 (filename),usctheses-m40 (legacy collection record id),usctheses-c127-504792 (legacy record id),usctheses-m514 (legacy record id)
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etd-DehghanNayeri-20070605.pdf
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504792
Document Type
Dissertation
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Dehghan Nayeri, Mohammad Reza
Type
texts
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University of Southern California
(contributing entity),
University of Southern California Dissertations and Theses
(collection)
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Libraries, University of Southern California
Repository Location
Los Angeles, California
Repository Email
cisadmin@lib.usc.edu
Tags
system identification and modelling