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Analytical and experimental studies in the development of reduced-order computational models for nonlinear systems

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Analytical and experimental studies in the development of reduced-order computational models for nonlinear systems

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Content ANALYTICALANDEXPERIMENTALSTUDIESINTHEDEVELOPMENTOF REDUCED-ORDERCOMPUTATIONALMODELSFORNONLINEARSYSTEMS by FarzadTasbihgoo ADissertationPresentedtothe FACULTYOFTHEGRADUATESCHOOL UNIVERSITYOFSOUTHERNCALIFORNIA InPartialFulfillmentofthe RequirementsfortheDegree DOCTOROFPHILOSOPHY (CIVILENGINEERING) December2006 Copyright 2006 FarzadTasbihgoo Dedication tomywonderfulparentsandfabulouswife ii Acknowledgments I would first and foremost like to express my sincere appreciation for the guidance, assistance, support,andencouragementofmyadvisorProfessorSamiF.Masri,duringthecourseofmylong graduatestudies,andpreparationofthiswork. Hisbreadthknowledge, andexpertisehavebeen invaluableresourceformeinchoosing thedirectionofmy work. Heconstantlyencouragedme in exploring new technologies in other fields of engineering. Second, I would like to gratefully appreciate the constant support and guidance of Professor John P. Caffrey, in all phases of this investigation. Hisdeepknowledgeofexperimentalmechanicsandcomputer/electornicsystems, was a great source to obtain optimum and practical solutions to my problems during the course of my study. I would also like to thank the other members of my Ph.D committee, Professor L. CarterWellford,ProfessorJamesC.Anderson,andProfessorPetrosIoannou. I would particulary like to thank Professor Wellford, the former chairman of Civil Engi- neering department, for his great leadership, advisement, and support, from the very first day I enteredthegraduateprogramatUSC,andforhisexcellentteachingintheFiniteElementAnal- ysis course. I would also like to thank Professor Anderson, for a great experience in taking his graduate course of Seismic Design of structures. I want to acknowledge Professor Ioannou, for hisadvisementsandhisParameterIdentificationandAdaptiveControlcourse,whichhelpedme iii todevelopthetheoreticalframeworkofthisstudy. IwouldalsoliketothanksProfessorMihailo D. Trifunac, for his support and his excellent teaching in: (1) Earthquake Engineering and (2) EngineeringAnalysis. IwouldliketoacknowledgeProfessorAndrewW.Smyth,forhisassistanceduringtheinti- mal phase of this investigation in providing direction and guidance. I would like to thank Dr. Raymond W. Wolfe for his generosity in sharing the experimental data, which I used in the studyofmonitoringnonlinearviscousdamper. Iwouldalsoliketoacknowledgethesupportand guidanceofProfessorErikA.Johnson,ProfessorNajmedinMeshkati,andDr. FarzadNaeim. ImustmentionedthosethatcamebeforemystayatUSC.AtIranUniversityofScienceand Technology (IUST), where I completed my bachelor of civil engineering, I was also fortunate to have some of the very best in teaching and guidance. Professor Feridoon Amini, Professor MortezaZahedi,ProfessorHormozFamili,ProfessorAliKaveh,ProfessorJalilShahi,Professor Shamss Nobakht, and Professor Hamid Behbahani, for their constant support, encouragement, and guidance. Apart from all the faculty members, there are also others whom I like to thank, whohaveprovidedmeencouragement,support,andguidance,alongtheway,theyareMr. Alias- gharTaheriandMr. FeridoonKiaie. TherearelotmorepeopleIwouldliketoacknowledgeformakingmystayatUSCenjoyable. First of all, the staff of the Department, Irene Sollof, Vangie Reys, Tessie Jamanila, Lance Hill, AnnaNieto,LeonLemons,Mary-KayFernandes,andRobertBoeche. I would like to thank my colleagues in our research group, especially for hours of useful discussions. TheyareDr. MazenA.Wahbeh,Hae-Bum(Andrew)Yun,RezaD.Nayeri,Miguel G.Hernandez,ElenaKallinikidou,MohammadrezaJahanshahi,andRezaJafarkhani. iv Lastly,Imustthankmyfamilyforsupportingandinspiringmeincompletingthisinvestiga- tion. Myparents,MahnazandFaramarz,forconstantlyencouragingandinspiringmethroughout my studies from Razi high school, toward IUST university, and at USC. I must send my love to my sister, Farnaz, for being a great source of amusement and encouragement throughout this process. My in-laws, Mahdokht, Hassan, Noushin, Nasim, Niloofar, and Masoud Zarkesh, also deserve a major kudos. I would like to acknowledge my late grandfather, Mohsen Jourabchi, whohiswordsofwisdomwasagreatsourceofinspirationforme. Last but not least, there is my beloved wife, Nazanin, who has been just fabulous in putting up with Ph.D lifestyle. Without her and her endless love, I could never have gone through this processwithsuchanupbeatattitude. AsIhavealwaystoldeveryone, theverybestthingIhave doneinmylifewasmarryingmybelovedwife,Nazanin. v TableofContents Dedication ii Acknowledgments iii ListofTables ix ListofFigures x Abstract xvii Chapter1Introduction 1 1.1 BackgroundandMotivation 1 1.1.1 On-lineIdentificationofNonlinearHystereticSystems 2 1.1.2 Data-BasedModel-FreeRepresentationsofComplexNonlinearDynamic Systems 5 1.2 Approach 7 1.2.1 On-lineIdentificationofNonlinearSDOFSystem 7 1.2.2 IntegratedStudiesofComplexNonlinearDynamicalSystems 9 1.3 Scope 10 Chapter2ReconfigurableTestApparatusforNonlinear SDOF Systems 13 2.1 DescriptionofReconfigurableTestApparatus 14 2.2 SampleMeasurements 17 2.2.1 FreeVibrationofLinearSDOFSystem 17 2.2.2 NonlinearSystemwithaGapUnderRandomExcitation 18 2.2.3 HarmonicallyExcitedHystereticSDOFSystemwithTime-VaryingSlid- ingFrictionForce 20 2.3 IdentificationofReduced-OrderNonlinearModel 23 2.3.1 FormulationofOn-LineBouc-WenHystereticModelIdentificationPro- cedure 23 2.3.2 On-lineIdentificationAlgorithm 24 2.3.3 AnalysisofSampleTestResults 28 2.4 ApplicationtoMeasurementsfromTestApparatus 30 2.4.1 DerivationofaGeneralLinearParametricModel 30 2.4.2 IdentificationoftheHystereticRestoringForce 31 vi 2.5 ChapterSummary 34 Chapter3On-lineMonitoringofNonlinearViscousDampers 38 3.1 OverviewofOn-lineParametricIdentificationApproach 39 3.1.1 SimplifiedMathematicalModelofNonlinearViscousDampers 39 3.1.2 FormulationofthePolynomial-BasisModelforOn-lineIdentification 43 3.1.3 On-LineIdentificationBasedonAdaptiveLeast-Squares(ALS) 44 3.2 ExperimentalStudies 46 3.3 ApplicationoftheOn-lineIdentificationAlgorithm 47 3.3.1 On-lineIdentificationofHarmonicDatawiththeSimplifiedDesignModel 47 3.3.2 On-lineIdentificationofRandomDatawiththePolynomial-BasisModel 51 3.3.3 ValidationoftheOn-LineIdentificationforRandomDatawiththePoly- nomialBasisModel 55 3.4 Discussion 57 3.4.1 SimplifiedDesignModel 58 3.4.2 Polynomial-BasisModel 59 3.5 ChapterSummary 64 Chapter4Data-BasedModel-FreeRepresentationsofNonlinearDynamicSystems 65 4.1 FormulationofData-BasedModel-FreeRepresentationsofNonlinearElement 66 4.1.1 EquationofMotion 66 4.1.2 ModelingtheNonlinearNonconservativeRestoringForces 69 4.1.3 Polynomial-BasisModel 70 4.1.4 ArtificialNeuralNetwork(ANN)Model 72 4.2 Simulationofa2-DOFCoupledNonlinear“Joint”Component 76 4.2.1 MathematicalModel 76 4.2.2 Identification of Data-Based Model-Free Representation of the 2DOF CoupledNonlinearSystem 79 4.3 ExperimentalStudies 82 4.3.1 TestSetupofa2- DOF NonconservativeDissipative“Joint”System 82 4.4 Application 85 4.4.1 IdentificationDataSets 86 4.4.2 Polynomial-BasisModel 87 4.4.3 ArtificialNeuralNetwork 89 4.5 Discussion 93 4.5.1 ComparisonoftheTwoNonparametricMethods 97 4.6 ChapterSummary 97 Chapter5DevelopmentofReduced-OrderComputationalModelsforNonlinear“ JOINT” 99 5.1 Formulation 100 5.1.1 Data-BasedModel-FreeRepresentationofNonlinear“Joint”Element 101 5.1.2 Adding the Reduced-Order Model to the Equations-of-Motion of the System 104 5.1.3 SolvingtheNewUpdateNonlinearEquations-of-MotionoftheSystem 117 5.2 Simulation 117 vii 5.2.1 MathematicalModel 118 5.2.2 GenerationofSyntheticData 121 5.2.3 DerivationoftheData-BasedModel-FreeRepresentationoftheNonlin- ear“Joint”Element 121 5.2.4 MathematicalModeloftheSystemwithNonlinear“Joint”Element 124 5.2.5 ImplementationoftheReduced-OrderNeuralNetworkModelintoNas- tranFiniteElementCode 128 5.3 Application 132 5.3.1 Development of a Computational Model with the Reduced-Order Data- BasedRepresentationoftheNonlinear“Joint”Element 132 5.4 ChapterSummary 136 Chapter6Conclusion 138 Bibliography 141 AppendixAIntegrationofNormalizationProceduretoNeuralNetworkModel 149 A.1 FormulationofNormalizationProcedure 149 A.2 IntegrationofNormalizationProcedure 151 AppendixBNastranComputationalModelforNonlinear“JOINT”Element 154 B.1 OriginalModel 154 B.2 TheTrainedNeuralNetworkParameters 157 B.3 TheComputationalModelwithReduced-Order“Joint”Element 159 viii ListofTables 3.1 Samplesmeanvalues(μ)andstandarddeviations(σ)oftheunknownparameters (C andn)fromthesimplifieddesignmodel(Eq.(3.2)),obtainedthroughon-line identificationfor202harmonicdatasets. 49 3.2 Samples mean values (μ) and standard deviations (σ) of the unknown parame- ters (C and n) from the simplified design model (Eq. (3.2)), obtained through nonlinearoptimizationfor202harmonicdatasets. 49 ix ListofFigures 2.1 IdealizedmathematicalmodelofagenericnonlinearSDOFsystem. 14 2.2 Photographoftheassemblednonlineartestapparatus. 15 2.3 Photographoftime-varyingfrictionforceactuator. 15 2.4 Solidmodelofmainapparatuscomponents. 17 2.5 Systemarchitectureandwiringdiagramoftestsetup. 18 2.6 Experimental measurement corresponding to transient response of linear SDOF system. 19 2.7 TimehistoryplotsofnonlinearSDOFwithgapsubjectedtorandomexcitation. 20 2.8 PhasediagramsofexperimentalmeasurementsfromanonlinearSDOFwithgap subjectedtorandomexcitation. 21 2.9 ExperimentalmeasurementfromaSDOFsystemwithtime-varyingslidingfric- tiontoharmonicexcitation. 22 2.10 Phaseplotofexperimentalmeasurementscorrespondingtothehystereticsystem withslidingfrictionforcesunderharmonicexcitation. 23 2.11 Responseandphaseplotofsimulationdatacorrespondingtoahysteretic(Bouc- Wen)modelunderrandomexcitation. 29 2.12 Time-evolutionoftheidentificationparameters. 30 2.13 Contribution of the identified restoring force components based on simulation datafromahysteretic(Bouc-Wen)SDOFsystemunderrandomexcitation. 32 x 2.14 Response and phase plot of the experimental test corresponding to a hysteretic system. 33 2.15 Evolution of the identification parameters corresponding to the experimental datafromahystereticsystemunderrandomexcitation. 35 2.16 Comparison of the measured (solid line) and estimated (dashed line) restoring force in a hysteretic system subjected to stationary random excitation. (a) Time historyofrestoringforcer(t)overatimespantfrom0-120s;(b)plotofr(t)over a time span t from 55-65 s; (c) plot of r(t) over a time span t from 60-61 s; (d) phaseplotoftherestoringforceversusthecorrespondingsystemdisplacement. 36 2.17 Contribution of each parameter in the identification results based on the experi- mentalmeasurementsfromahystereticsystemunderrandomexcitation. 37 3.1 Simplifiedmodelandmathematicalrepresentationofnonlinearviscousdampers. 39 3.2 Overview of the experimental test setup for the 44.48 KN (10 kip) viscous damperatUniversityofSouthernCalifornia. 47 3.3 On-lineidentificationofadamperdatasetsubjectedtoharmonicoscillationwith frequency of 0.25 Hz and amplitude of ±0.051 m (±2 in). Parts (a) and (b) show the time-history of the identified parameters, C the damping coefficient, andntheexponent,respectively. Thedash-dotlineindicatestheoptimumvalue oftheparametersobtainedthroughstandardnonlinearleast-squarescurvefitting optimizationofthedesignmodel. Part(c)depictsthetime-domaincomparisonof the measured force (solid-line) and the identified force (dash-dot line). Parts(d) and (e) are the phase-domain comparison of the force-displacement and force- velocityofthemeasureddata(solidline)vs.theidentifieddata(dash-dotline). 48 3.4 ComparisonofthefrequencyhistogramandGaussianprobabilitydistributionof the unknown parameters, C and n, for 202 harmonic data sets, estimated with on-line identification algorithm and nonlinear optimization. Parts (a) and (b) show the frequency histogram and Gaussian probability distribution of the data (solid line) obtained through on-line identification, and parts (c) and (d) show the frequency histogram and Gaussian probability distribution of the data (solid line)obtainedthroughnonlinearoptimization. 50 3.5 Comparisonofthemeasuredforce(solidline)andtheidentifiedforce(dash-dot line). Part (a) shows a selected time-domain comparison of the measured force (solid-line)andtheidentifiedforce(dash-dotline),parts(b)and(c)aretheforce vs.displacementandforcevs.velocityphase-plotcomparisondepictingboththe measuredforce(solidline)andidentifiedforce(dash-dotline). 52 xi 3.6 Time-historyevolutionoftheunknownparametersofpolynomialmodelobtained from on-line identification algorithm (solid line) and the optimum values of the parametersderivedfromstandardleast-squaresestimation(dash-dotline). 53 3.7 Time-history evolution of the contribution of each of the identified parameters andtheircorrespondingbasis(θ i ×φ i ). Foreaseofcomparisonandillustration, all the plots are normalized to the maximum measured force and zoomed in to thelastten-secondsoftheidentificationprocess. 54 3.8 The validation results of the measured force (solid line) and estimated force (dash-dot line), where the identification and validation data are two completely differentsets. Part(a)isthetime-historyplotofforces,part(b)isthephase-plot offorcevs.displacement,andparts(c)isthephase-plotofforcevs.velocity. 56 3.9 The validation results of the measured force (solid line) and estimated force (dash-dot line) from the polynomial-basis model, where the identification and validation data are different segments of same data set. Part (a) is the time- historyplotofforces,part(b)isphase-plotofforcesvs.displacement,andparts (c)isthephase-plotofforcesvs.velocity. 57 3.10 On-line identification of data set subjected to broad-band random oscillation with amplitude of ±0.0254 m (±1 in). Parts(a) and (b) show the time-history of the identified parameters C, the damping coefficient, and n, the exponent, respectively, the dash-dot lines indicate the optimum value of the parameters obtainedthroughnonlinearleast-squarescurvefittingwiththesamemodel. Part(c) is the time-domain comparison of the measured force (solid-line) and the iden- tified force (dash-dot line). Parts(d) and (e) are the phase-domain comparison of the force-displacement and force-velocity of the measured data (solid line) vs.theidentifieddata(dash-dotline). 60 3.11 The relationship between the order of polynomial-basis model and the identifi- cation error. Part (a) shows a 3D surface of error as a function of polynomial orders, where the x-axis is the ˙ x order, the y-axis is the x order, and the z-axis istheidentificationerror. Parts(b)and(c)illustratethedependencyoftheiden- tification error to the orders of each terms while the order of the other term is fixed. 62 3.12 Correlationbetweentheasymptoticvaluesofselectedidentifiedparametersand theestimationerrorforthepolynomial-basismodelwithmodelorder,whenthe orders for x and ˙ x are equal in values. Parts (a) to (e) show the asymptotic values of identified parameters corresponding to ¨ x, x, ˙ x, x 3 , and ˙ x 3 terms with thepolynomialorder. Part(f)depictstherelationshipbetweentheidentification errorandthepolynomialorder. 63 xii 4.1 Freebodydiagramofagenericmasslessnonlinear“joint”element. 67 4.2 Mathematicalmodelofadiscrete4- DOF systemwithanonlinear“joint”element. 68 4.3 Schematic of data-based model-free representation of the nonlinear nonconser- vativerestoringforcesforthe2- DOF “joint”element. 70 4.4 Schematic of the neural network architecture for modeling the nonlinear restor- ingforcesofthecouplednonlinearsystemshowninFig.4.2. 74 4.5 Mathematical model of nonlinear coupled 2- DOF system used to generate the syntheticdatasets. 76 4.6 Syntheticdatasetsusedforidentificationofnonlinearcoupled“joint”. 78 4.7 Verification results of the polynomial-basis model identification with the syn- theticdatasetsusedinidentification. Parts(a)and(b)showthephase-plotcom- parison of the simulated forces (solid line) and the identified forces (dash-dot line)forrelativedisplacementvs. axialrestoringforceinthexdirection,andthe relativerotationvs. torsionalrestoringforceintheθ direction,respectively. 80 4.8 Validation results of the polynomial-basis model with the synthetic data sets, which were not used in the identification process. Parts (a) and (b) show the time-historycomparisonoftheidentifiedrestoringforces(dash-dotline)withthe simulatedrestoringforces(solidline)fortheaxialandtorsionalrestoringforces respectively, parts (c) and (d) show the phase-plot comparison of the relative displacement vs. axial restoring force, and the relative rotation vs. torsional restoringforceforthexdirectionandθ direction,respectively. 81 4.9 Verificationoftheneuralnetworkidentificationwiththesyntheticdatasetsused in identification. Parts (a) and (b) show the phase-plot comparison of the sim- ulated restoring forces (solid line) and the identified restoring forces (dash-dot line)forrelativedisplacementvs. axialrestoringforceinthexdirection,andthe relativerotationvs. torsionalrestoringforceintheθ direction,respectively. 82 4.10 Validationresultsoftheneuralnetworkidentificationwiththesyntheticdatasets notusedinidentification. Parts(a)and(b)showthetime-historycomparisonof theidentifiedrestoringforce(dash-dotline)withsimulatedrestoringforce(solid line)foraxialandtorsionalforcesrespectively,parts(c)and(d)showthephase- plot comparison of the relative displacement vs. axial restoring force, and the relativerotationvs. torsionalrestoringforce,forthexdirectionandθ direction, respectively. 83 xiii 4.11 Overviewofthe2- DOF nonlinear“joint”experimentaltestsetup. 84 4.12 Solid model and detailed view of the 2- DOF nonlinear experimental test setup. Part (a) is the solid model design of the test setup, and part (b) is the exploded viewofthenonlinear2- DOF “joint”. 85 4.13 Systemarchitectureandwiringdiagramoftestsetup. 86 4.14 Experimentaldatasetsusedintheidentificationofthenonlinearcoupled2- DOF “joint”. 88 4.15 Verification results of the polynomial-basis model identification with the exper- imental data sets used in identification. Parts (a) and (b) show the time-history comparisonofthemeasuredresortingforces(solidline)withtheidentifiedrestor- ing forces (dash-dot line) for the axial and torsional directions, respectively. Parts(c)and(d)showthephase-plotcomparisonofthemeasuredrestoringforce (solid line) and the identified restoring force (dash-dot line) for the relative dis- placementvs. theaxialrestoringforceinthexdirection,andtherelativerotation vs. thetorsionalrestoringforceintheθ direction,respectively. 90 4.16 Validation of the polynomial-basis model identification with the experimental datasetsnotusedinidentification. Parts(a)and(b)showthetime-historycom- parisonoftheidentifiedforce(dash-dotline)withthemeasuredforce(solidline) for the axial and torsional forces respectively. Parts (c) and (d) show the phase- plot comparison of the relative displacement vs. the axial restoring force, and the relative rotation vs. the torsional restoring force, for the x and θ directions, respectively. 91 4.17 Verification results for the neural network identification with the experimental datasetsusedinidentification. Parts(a)and(b)showthephase-plotcomparison of the measured force (solid line) and the identified force (dash-dot line) for relativedisplacementvs. axialrestoringforceinthexdirection,andtherelative rotationvs. torsionalrestoringforceintheθ direction,respectively. 92 4.18 Validationresultsfortheneuralnetworkidentificationwiththeexperimentaldata setsnotusedinidentification. Parts(a)and(b)showthetime-historycomparison oftheidentifiedforce(dash-dotline)withthemeasuredforce(solidline)forthe axial and torsional forces, respectively. Parts (c) and (d) show the phase-plot comparison of the relative displacement vs. the axial restoring force, and the relative rotation vs. the torsional restoring force, for the x and θ directions, respectively. 94 5.1 A nonlinear massless “joint” element is added to a linear model between nodes x i andx j ,eachwithn 1 - DOFs. 101 xiv 5.2 Schematicofthereduced-orderdata-basedmodel-freerepresentationofa 6- DOF nonlinear“joint”element. Thereduced-ordermodelhastherelativestatesofthe “joint” as the input and the nonlinear nonconservative restoring forces as the output. 102 5.3 Mathematicalmodelofalinear8- DOFssystem. 118 5.4 Mathematicalmodelofthe8- DOFssystemwithanonlinear“joint”element. 120 5.5 Mathematical models used in finite element simulation. Part (a) is the linear model,andpart(b)isthelinearmodelwithanonlinear“joint”elementconsist- ingofatwo-dimensionalgapandacouplingnonlinearitydefinedbythefunction g. 122 5.6 Theexternalforcesusedinthesimulationstudies. Part(a)isthetime-historyof theappliedaxialloadandpart(b)isthetime-historyoftheappliedtorqueforce. Notethatthetwoappliedforcesareuncorrelated. 122 5.7 Mathematicalmodelforthefiniteelementsimulationwherethenonlinear“joint” modelisrepresentedbyareduced-ordermodelobtainedthroughtheneuralnet- workmodelingtechnique. 124 5.8 Verificationtheresponsesofthesimulationmodelswiththenonlinearmathemat- icalelement(solidline)andwiththereduced-order“joint”element(dashed-dot line), subjected to the same random excitation used in the identification of the reduced-order “joint” element. Parts (a) and (b) shows the time-history com- parison of the axial and torsional restoring forces, and parts (c) and (d) are the phase-plotcomparisonoftherelativedisplacementvs. axialrestoring-forceand therelativerotationvs. torsionalrestoring-force,respectively. 130 5.9 Validationtheresponsesofthesimulationmodelswiththenonlinearmathemat- icalelement(solidline)andwiththereduced-order“joint”element(dashed-dot line), subjected to the different random excitation than those used in the identi- fication of the reduced-order “joint” element. Parts (a) and (b) shows the time- history comparison of the axial and torsional restoring forces, and parts (c) and (d)arethephase-plotcomparisonoftherelativedisplacementvs. axialrestoring- forceandtherelativerotationvs. torsionalrestoring-force,respectively. 131 5.10 Comparisonoftheexperimentalmeasurementsfromthe2- DOFnonlinear“joint” testsetup,withtheresponsesofthecomputationalmodelwiththereduced-order representation of the nonlinear “joint” element, when both systems were sub- jectedtoaxialloadonly. 134 xv 5.11 Comparisonoftheexperimentalmeasurementsfromthe2- DOFnonlinear“joint” testsetup,withtheresponsesofthecomputationalmodelwiththereduced-order representation of the nonlinear “joint” element, when both systems were sub- jectedtotorqueforceonly. 135 5.12 Comparisonoftheexperimentalmeasurementsfromthe2- DOFnonlinear“joint” testsetup,withtheresponsesofthecomputationalmodelwiththereduced-order representation of the nonlinear “joint” element, when both systems were sub- jectedsimultaneouslytouncorrelatedaxialloadandtorqueforce. 137 xvi Abstract Thisresearchworkreportsonanintegratedapproach,involvingcarefullyconductedexperimen- taltestsonreconfigurableapparatuses,thatallowsthestudyofabroadclassofgenericnonlinear phenomena (such as dead-space, dry friction, hysteresis, limited-slip, etc.), that generates high- quality experimental measurements, for the subsequent development of high-fidelity, nonlinear, reduced-order,mathematicalmodelsofdifferentformats,thatareusefulforthemonitoring,con- trol,andsimulationofrealisticnonlinearstructuralsystems. Progresses achieved in this study includes: (1) design, fabrication, assembly, instrumen- tation, calibration, and use of reconfigurable one- and two-dimensional test apparatuses, that allowtheconvenienttestingofmanyimportantclassesofnonlinearphenomena,bothstationary aswellasnonstationaryinnature;(2)evaluationandextensionofsomeusefulsystemidentifica- tion tools, involving parametric methods, such as the adaptive least-squares with the Bouc-Wen hysteresis model, as well as nonparametric methods, such as neural networks and polynomial- basis models, for dealing with realistic situations involving challenging hysteretic phenomena; (3) development of on-line monitoring schemes for large-scale viscous dampers, based on the adaptiveleast-squareswithaforgetting-factor,inconjunctionwithtwomodels: (a)thesimplified design model, and (b) a polynomial-basis model; and (4) development and implementation of a xvii methodologyforutilizingdata-basedmodelsofdiscretenonlinear“joint”elementstocreateeffi- cient system-level finite-element representations of multi-dimensional structures incorporating complexnonlinearelements. xviii Chapter1 Introduction 1.1 BackgroundandMotivation There is a strong interest in investigating nonlinear systems for analysis of experimental mea- surementsandfordevelopinghigh-fidelityreduced-ordernonlinearmodelsforavarietyofappli- cations, including control, simulation, and monitoring (Kerschen et al., 2006). One such exam- ple that is of considerable interest involves bolted joints whose deformations induces friction forces, hysteretic phenomena, etc. (Ibrahim and Pettit, 2003; Segalman, 2005, 2006). In order todevelopsoftwaresimulationpackagesforreliableemulationofthedynamicresponseofsuch complexnonlinearsystems,itisimportanttodevelopverifiable,physics-based,modelsthrough carefullydesignedandconductedexperimentaltests. However, there are practical challenges to performing comprehensive experimental inves- tigations. Among the hurdles encountered in performing experimental studies, is the time- consuming effort needed to design and deploy the test apparatus, particularly when there is a need to modify the physical parameters in order to allow comprehensive experimental studies 1 focused on understanding the influence and interaction of various system and excitation param- eters. Withtheabovediscussioninmind,thepurposeoftheexperimentalandanalyticalinvestiga- tionunderdiscussionistodesign,analyze,construct,calibrate,andevaluateareconfigurabletest apparatusforstudyinggenericnonlinearphenomena,widelyencounteredintheappliedmechan- ics field, such as elastic nonlinearities, nonlinear viscous forces, dry-friction, and time-varying hystereticphenomena. This work explores some of the fundamental interdisciplinary research areas in the field of structural control and health monitoring: (1) on-line identification of nonlinear hysteresis sys- tems,withtheapplicationtothestructuralhealthmonitoringoflarge-scaleviscousdampers,and (2) development of data-based model-free nonparametric representations of nonlinear “joint” elementsthatcanbeincorporatedintocomputationalmodels. 1.1.1 On-lineIdentificationofNonlinearHystereticSystems Problems involving the identification of structural systems exhibiting inelastic restoring forces with hereditary characteristics are widely encountered in the applied mechanics field. Repre- sentativeexamplesinvolvebuildingsunderstrongearthquakeexcitationsoraerospacestructures incorporating joints. Due to the hysteretic nature of the restoring force in such situations, the nonlinear force cannot be expressed in the form of an algebraic function involving the instanta- neous values of the state variables of the system. Consequently, much effort has been devoted, by numerous investigators, to develop models of hysteretic restoring forces and techniques to identifysuchsystems. Anoverviewofpublicationsaboutthissubjectisavailableintheworkof 2 Smyth et al. (2002); Vestroni and Noori (2002); Wen (1980); Yang et al. (2006); Yang and Lin (2004,2003). Also, one of the challenges in actively controlling the nonlinear dynamic response of struc- tural systems undergoing hysteretic deformations is the need for rapid identification of the non- linearrestoringforcesothattheinformationcanbeutilizedbyon-linecontrolalgorithms. Con- sequently,theavailabilityofamethodfortheon-lineidentificationofhystereticrestoringforces iscrucialforthepracticalimplementationofactivecontrolconcepts(Housneretal.,1997;Hous- nerandMasri,1990). Beforeinterestinadaptive(oron-line)identificationemerged,researchershadworkedonthe problem of the modeling and identification of nonlinear hysteretic systems in a “batch” mode, i.e.,wheretheentiretime-historyoftheexcitationandresponseisavailableattheoutset. Notable studies in this area include Andronikou and Bekey (1984); Benedettini et al. (1995); Capec- chi (1990); Caughey (1960, 1963); Chassiakos et al. (1995); Iwan (1966); Iwan and Cifuentes (1986);JayakumarandBeck(1987);Jennings(1964);Linetal.(2001);LohandChung(1993); Masrietal.(1991);Nietal.(1999);PengandIwan(1992);Roberts(1987);RobertsandSpanos (1990);SatoandQi(1998);Smythetal.(2001);Spanos(1981);Suesetal.(1988);Vinogradov and Pivovarov (1986); Wen and Ang (1987); Worden and Tomlinson (1988, 2001); Yar and Hammond(1987a,b). On-lineMonitoringofNonlinearViscousDampers The design of tall buildings and long-span bridges requires special attention to meet the large dynamic demands from wind and seismic events. Energy dissipative devices such as large- scale viscous dampers are often the preferred alternative of designers for their innate ability 3 to dramatically reduce member demands during large dynamic events. Viscous dampers have been used worldwide in a variety of applications and structures due to their effective energy dissipatingcapabilities(Aiken,1996;ChenandDuan,2000;Housneretal.,1997;Kitagawaand Midorikawa,1998;OuandLi,2004;ParkandKoh,2001;SpencerandNagarajaiah,2003;Wolfe etal.,2002). Therelianceonthesedevicestodissipateenergystemmingfromwindandseismic occurrencesmakesthemintegralcomponentstothesuccessofdesignstowithstandsuchevents. Failure of a viscous damper can portend potentially catastrophic localized or large-scale systemfailure,astheadjoiningmembersaresizedbasedontheenergyabsorbedbythedampers. Given the critical nature of the damper elements to the success of the design strategies being implemented on these large structures, a means of evaluating their in-situ health is imperative (Baker,1998;Caltrans,2003). In-situinspectionisacostlytraditionalmethodofinvestigatingandmonitoringthestructural health of viscous dampers. These inspections are limited in that very little knowledge can be gained of the continued performance capabilities of these devices from visual inspections. In fact,allthatisreadilyavailablefromsucheffortsrelatestoascertainingwhethertheconnection details remain sound and that leakage may have occurred if staining is evident. Beyond visual inspections, facility maintenance crews must physically remove dampers and ship them off-site forphysicalvalidationtesting. Theselimitationshavefueledintenseinterestinindustryandacademiccirclestowardsdevel- opment of methodologies to recover the state of the damper and possibly target deficiencies. However, on-line monitoring of viscous dampers can be an alternative method which has the potentialofsignificantlyreducinginspectioncosts. 4 The adaptive least-squares identification algorithm with a forgetting-factor is considered a promisingon-lineidentificationtoolfortrackingparametersofnonlinearsystems(Caffreyetal., 2004;Chassiakosetal.,1998;Smyth et al., 1999; Yang et al., 2006;Yang and Lin, 2004, 2003) In this study, two classes of models were used for modeling the nonlinear viscous damper: (1) the simplified design model (Miyamoto and Hanson, 2002; Soong and Dargush, 1997), and (2) anonparametricpolynomial-basismodel(Al-HadidandWright,1989,1990,1992;Masrietal., 2004, 1987a,b, 2006). The latter is a data-based model capable of capturing various types of nonlinear phenomena. The simplified design model, is used in this study due to its ability to identify the actual damping coefficient and exponent,C andn, respectively. A major limitation tothismodelisthatitonlyrepresentsthemaximuminduceddampingforce,anddoesnotcapture inherent damper hysteresis; therefore, the model is only valid for identification under harmonic excitationsforon-linemonitoring. Inordertoovercometheon-linemonitoringlimitationsofthe simplified design modelin the presence of wide-band random excitations, the polynomial-basis model was investigated. Previous studies have demonstrated the capabilities of the polynomial- basis model in representing various types of nonlinearities (Al-Hadid and Wright, 1989; Masri etal.,2004,2006). 1.1.2 Data-Based Model-Free Representations of Complex Nonlinear Dynamic Systems The modeling and identification of nonlinear nonconservative dissipative systems is a task that occurs frequently when dealing with problems that arise in the active control of structures, air- craft flutter, ships in motion, etc. This is a challenging problem that has been studied by many researchesinthepastseveraldecades. NotablestudiesinthisareaincludeBolotinetal.(1998a,b, 5 2002); Kounadis (1992, 1994, 1997, 2006); Sugiyama et al. (1995). The conventional method- ology of modeling nonconservative dissipative systems, is by representing the underlying non- linear dynamics of the system with a parametric model whose unknown parameters reflect the characteristics of the physical system. In general, this is quite a challenging approach, and usu- allythederivedparametricmodelsareconstrainedbytheassumptionsembeddedintheselection of the model class to be identified (Kerschen et al., 2006; Ljung, 1999; Worden and Tomlinson, 2001). Inthisstudy,afairlyrecentmodelingtechnique,basedondata-basednonparametricsystem identification, is presented for modeling nonconservative systems with generic nonlinear dis- sipative “joint” elements. The “joint” elements, in this study, are representatives of a class of dynamical systems with nonlinear characteristics such as dissipative damping, hysteresis, gaps, etc.,whicharetypicalofproblemsthatariseinthedynamicsoflarge-scalecivilstructureswith energydissipatingdevices,aerospacestructuresincorporatingjoints,systemswithdamagedele- ments, etc. (Housner et al., 1997; Quinn and Segalman, 2005; Segalman, 2005, 2006; Spencer andNagarajaiah,2003;Wolfeetal.,2002). Two of the promising nonparametric techniques have been utilized here to model the non- conservative dissipative restoring forces of a nonlinear “joint” element. These techniques fit a “black-box” mathematical model to the input and output measurements of the system response (Sjberg et al., 1995). The first technique is based on a polynomial-basis model, and the second method is based on artificial neural networks. Both modeling techniques have been shown to be powerful tools for modeling nonlinear dynamical system (Masri and Caughey, 1979; Masri et al., 1992, 1993, 1987a,b, 1999). These models are useful in applications where the overall 6 fidelity of the model, in representing the system, are important, such as health monitoring or control applications (Al-Hadid and Wright, 1989; Masri et al., 1989, 1996, 2000; Worden and Tomlinson, 1994; Worden et al., 1994). Moreover, it is shown herein that these models can be efficiently used in computational engines, for an accurate prediction and estimation of the nonlinearresponseofthesystemtobroad-bandexcitations. The nonparametric modeling techniques presented in this study have been a subject of research by many investigators; however, there are few reported studies concerning the mod- elingofthenonlineardissipativebehaviorofmulti-degree-of-freedom( MDOF)systemsthrough experimental and analytical studies, which is an essential step in understanding the actual char- acteristics and behavior of nonlinear MDOF dynamical systems (Caffrey et al., 2004; Kerschen etal.,2006). 1.2 Approach 1.2.1 On-lineIdentificationofNonlinearSDOFSystem This study reports the results of an analytical and experimental study to design, analyze, con- struct,test,andevaluateareconfigurabletest-bedtoallowtheconvenientperformanceofsophis- ticated experiments in a laboratory setting for investigating a broad category of important non- linear,time-varyingphenomenathatarewidelyencounteredintheappliedmechanicsfield. The essential elements of the test apparatus include an electro-dynamic exciter that drives an oscillating mass whose restoring force is easily adjustable from one resembling a linear SDOF mass-spring-damper system with constant coefficients, to one that can represent systems 7 with nonlinear elastic forces, to one that models systems possessing hysteretic properties with precisely-controlledtime-varyingcharacteristics. The apparatus design is economical to fabricate, convenient to manipulate, and it provides results that can be accurately replicated under repeated test combinations of system parameters and dynamic loads. The utility of the apparatus to provide an investigational tool for studying thedynamicresponseofsystemswithtime-varyingdry-frictionforcesthatgiverisetohysteretic phenomena is demonstrated. Sophisticated on-line system-identification techniques are used to estimate the parameters of reduced-order models that capture the dominant features of the physical model. It is shown that the apparatus under discussion is a useful research tool for investigatorsconductingstudiesinphysics-basedmodelsofgenericnonlineardynamicsystems. On-lineMonitoringofNonlinearViscousDampers Viscous dampers are increasingly incorporated into retrofit and new design strategies of large civil structures to dissipate energy from strong dynamic loads resulting from wind and seis- mic activity. Extended service life requirements and harsh environmental conditions dictate the importanceofreliableandefficientproceduresfordamperconditionassessmentbasedonvibra- tionsignatureanalysis. Theadaptiveleast-squaresidentificationalgorithmwithaforgetting-factorisappliedtotwo models: (1) the simplified design model, and (2) a polynomial-basis model which is capable of representing various types of nonlinearities. Such models are useful for in-situ identifica- tion of viscous dampers. This study was motivated by the need to quantify the ability of the algorithm to effectively identify unknown nonlinear system characteristics for health monitor- ing applications. The effects of persistence of excitation and modeling errors defined by over- 8 and under-parameterization are considered. Experimental data was utilized in these studies to demonstrate that the recursive adaptive least-squares algorithm is viable for real-time on-line systemidentificationapplicationsinvolvingcomplexnonlinearphenomena. 1.2.2 IntegratedStudiesofComplexNonlinearDynamicalSystems Ageneralprocedureispresentedfordevelopingdata-based,nonparametricmodelsofnonlinear multi-degree-of-freedom, nonconservative, dissipative systems. Two broad classes of methods are discussed: one relying on the representation of the system restoring forces in a polynomial- basis format, and the other using artificial neural networks to map the complex transformations relating the system state variables to the needed system outputs. A nonlinear two-degree-of- freedom system is used to formulate the approach under discussion and to generate synthetic data for calibrating the efficiency of the two methods in capturing complex nonlinear phenom- ena (such as dry friction, hysteresis, dead-space nonlinearities, and polynomial-type nonlin- earities) that are widely encountered in the applied mechanics field. Subsequently, a reconfig- urabletestapparatuswasusedtogenerateexperimentalmeasurementsfromaphysicalnonlinear “joint”involvingtwo-dimensionalmotion(translationandrotation)andcomplicatedinteraction forces between the different motion axes, among its internal elements. Both the polynomial- basisapproachandtheneural-networkmethodwereusedtodevelophigh-fidelity,nonparametric modelsofthephysicaltestarticle. The ability of the identified models to accurately “generalize” the essential features of the nonlinear system was verified by comparing the predictions of the models with experimental measurements from data sets corresponding to different excitations than those used for identi- fication purposes. It is shown that the identification techniques under discussion can be useful 9 tools for developing accurate simulation models of complex multi-dimensional nonlinear sys- temsunderbroad-bandexcitation. 1.3 Scope This thesis proceeds along two fronts: (1) an experimental phase involving the design and fabrication of an adjustable test apparatus for conducting studies on a generic “joint” element whichincorporatesimportantnonlinearcharacteristicssuchasnonlinearelasticproperties,hys- teretic characteristics, and dead-space nonlinearities involving friction, and (2) an analytical phasefocusedonthedevelopmentofatheoreticalframeworkforprocessingexperimentalstruc- turalresponsemeasurementsfromuncertainsystems,todevelopandevaluatetheutilityofsome promisinganalyticaltools. The thesis is organized as follows: Chapter 2 investigates the analytical and experimental studies of parametric system identification of nonlinear SDOF systems. The details of a recon- figurable test apparatus and its main components, which is built to investigate different classes of nonlinear phenomena under dynamic environments, are discussed. Sample measurements corresponding to different nonlinear regimes of time-varying nonlinear phenomena that can be inducedbythetestapparatusarepresented. Furthermore, the nonlinear system identification results, corresponding to measurements obtained from the apparatus, when it is configured to investigate the development of a reduced- order, low-complexity mathematical models corresponding to systems incorporating time- varyinghystereticelements,arepresentedinthischapter. 10 Chapter3investigatestheapplicationsoftheon-lineidentificationmethoddevelopedprevi- ouslyfornonlinearhystereticsystemssubjectedtoharmonicandarbitrarydynamicexcitationsin Chapter 2. The recursive adaptive least-squares identification algorithm with a forgetting-factor is applied to data sets derived from experimental testing of a viscous damper similar to those being incorporated in recent design strategies to dissipate dynamic forces. Issues of parameter convergence, model parameterization and modeling errors are studied for a range of types and levelsofexcitation. Resultsforvarioustestdefinitionsarepresented,concludingwithanoverall assessmentoftheidentificationalgorithmanditsapplicabilitytoreal-timein-situsystemhealth monitoring. Chapter 4 presents the analytical and experimental studies of nonparametric system iden- tification of nonlinear “joint” elements. In this chapter, the mathematical background and for- mulation of data-based model-free representations of nonlinear dissipative systems based on a polynomial-basis model and neural networks are discussed. The simulation studies, and details oftheexperimentaltestsetupofanonlinear2- DOF“joint”elementarepresented. Theutilization ofthemethodisdemonstratedthroughtheapplicationofthemodelingtoolswiththeexperimen- taldatasets. Chapter 5 discusses the development of reduced-order computational model of a nonlinear “joint”element. Inthischapterthedetailsofthemethodologyofhowtoincorporatingthedata- based model-free representation of a nonlinear “joint” obtained from laboratory experimental measurements,intoacomputationalmodel,whichconsistsofsimilarnonlinear“joint”element, are discussed. The methodology is calibrated with simulation models, and it is utilized with 11 the experimental data obtained from the studies of nonlinear 2- DOF “joint” element, which is explainedinChapter4. Chapter 6 presents the concluding remarks and the direction of future studies in the contin- uationandadvancementsoftheresultsofthisstudy. 12 Chapter2 ReconfigurableTestApparatusfor Nonlinear SDOFSystems The main purpose of this chapter is to provide the details of a reconfigurable test apparatus and the corresponding analytical studies in the development of reduced-order models for nonlinear SDOF systems. The experimental test setup was built to simulate the nonlinear response of SDOF systems with generic type of nonlinearities in order to utilize and calibrate the on-line identificationalgorithm. The details of the reconfigurable test apparatus and its main components, which is built to investigatedifferentclassesofnonlinearphenomenaunderdynamicenvironment,arediscussed in Section 2.1, typical measurements of the test setup are shown in Section 2.2, the formulation of the on-line identification algorithm based on adaptive least-squares with forgetting factor is explainedinSection2.3,andtheapplicationoftheon-lineidentificationalgorithmwithexperi- mentaldatasetsisreportedinSection2.4. 13 2.1 DescriptionofReconfigurableTestApparatus Thetestapparatuswasdesignedtorepresentasingle-degree-of-freedom(SDOF)nonlinearsys- tem having a sliding mass m that is connected by a nonlinear resilient element to a fixed point, while the mass is subjected to an external disturbance F(t). A simplified mathematical model of such a system is depicted in Fig. 2.1. Its main components are the lumped mass m and the nonlinearrestoringforceelementr(x, ˙ x). Figure2.1: IdealizedmathematicalmodelofagenericnonlinearSDOFsystem. Aphotographofthecompletelyassembledelectro-mechanicalapparatusisshowninFig.2.2 and Fig. 2.3. The labels indicate the major subcomponents of the apparatus. The system mass consists of a rigid block that is constrained to slide uniaxially on ball bearings. A computer- controlled electro-mechanical device is used to apply, through a stepper motor, a time-varying normal force to a bar which slides between two edges that are used to induce dry friction (Coulomb-like)forceswhenthebarisinstalledinthetestassembly. 14 Linear Bearing Accelerometer Load Cell Stepper Motor Spring Slide Element (Friction) Shaker Element with Dead-Space Nonlinearity Figure2.2: Photographoftheassemblednonlineartestapparatus. Moving Edge LVDT Spring Slide Stepper Motor Figure2.3: Photographoftime-varyingfrictionforceactuator. 15 AsolidmodelofthetestsetupisshowninFig.2.4. Itsmajorelementsare: • A rigid mass consisting of a solid plate constrained to move uniaxially through attached linearbearings. • A long-stroke electromagnetic force generator used to furnish the desired system excita- tion. • An adjustable, passive nonlinear element (dead-space nonlinearity). This part consists of twospringswithinashaftseparatedbyanadjustablegap. • A computer-controlled stepper motor that can apply a controllable level of normal force, inthecaseofslidingfrictiontests(Fig.2.3). • A collection of sensors for the direct measurement of the induced system force, accelera- tion, velocity, and, displacement, associated with the different system components. Force gauges, accelerometers, linear velocity transducers (LVT), and linear variable displace- menttransducers(LVDT)areused. Duethepresenceofhystereticphenomena,itiscrucial fromthesystemidentificationpointofviewtohavedirectpositionmeasurements(Smyth andPei,2000;Worden,1990). • A data acquisition board for the measurement and control of the test apparatus functions. TheNationalInstrumentLabVIEWsoftwareisusedtocreatetheGUIforthedataacqui- sitionandgenerationofexcitationforcesforthevibrationgenerator. A pictorial diagram indicating the inter-connection of the main system components, includ- ing the mechanical assembly, excitation sources, instrumentation network, adaptive nonlinear components,andthemastercomputer,isprovidedinFig.2.5. 16 Spring Linear Bearing Variable Friction Stepper Motor Dead-Space Nonlinearity Element Shaker Figure2.4: Solidmodelofmainapparatuscomponents. 2.2 SampleMeasurements To illustrate the wide range of test configurations and nonlinear phenomena that can be inves- tigated through the use of the apparatus under discussion, the following sections provide some samplemeasurementsfromavarietyofdynamictests. 2.2.1 FreeVibrationofLinearSDOFSystem The free vibrations, from successive impulse excitations, of the apparatus when configured as a linear SDOF system is shown in Fig. 2.6, in which part (a) shows a time history plot of the system’sdisplacement,part(b)showsthecorrespondingsystemvelocity,andpart(c)showsthe 17 Load Cell Kistler 9212 Multifunction DAQ Board AT-MIO-16XE-10 National Instruments Low Pass Butterworth Filter NI SCXI 1143 NI SCXI 1000 Chassis National Instruments Stepper Motor Parker – S DC Amplifier Model 136, Endevco Accelerometer Model 7290A-10 Endevco Stepper Drive Parker – SX series Charge Amplifier Kistler 5010B Electro-Seis Shaker, Model 113 APS Dynamics, INC Analog Digital Dual Mode Amplifier, Model 114 APS Dynamics, INC Figure2.5: Systemarchitectureandwiringdiagramoftestsetup. corresponding acceleration. A phase plot of the normalized velocity plotted versus the normal- izeddisplacementisshowninFig.2.6(d). 2.2.2 NonlinearSystemwithaGapUnderRandomExcitation Another illustration of the broad categories of nonlinear phenomena that can be represented in the test apparatus under discussion is demonstrated in the time-history plots shown in Fig. 2.7. These plots show the time history of the force and corresponding displacement of the system whenconfiguredtostudymemory-lessphenomena: inthiscaseadead-spacenonlinearity(gap) in the system’s stiffness characteristics. Notice that there are no obvious nonlinear response featuresintheindicatedtimehistoryrecords. 18 12 12.5 13 13.5 14 14.5 15 15.5 16 -0.8 -0.4 0 0.4 0.8 Time (sec) Displacement (in) (a) 12 12.5 13 13.5 14 14.5 15 15.5 16 -15 -7.5 0 7.5 15 Time (sec) Velocity (in/s) (b) 12 12.5 13 13.5 14 14.5 15 15.5 16 -250 -200 -150 -100 -50 Time (sec) Acceleration (in/s 2 ) (c) -1 -0.5 0 0.5 1 -1 -0.5 0 0.5 1 Normalized Displacement Normalized Velocity (d) Figure2.6: ExperimentalmeasurementcorrespondingtotransientresponseoflinearSDOFsys- tem. Thephaseplotofthesystemrestoringforceversusthecorrespondingdisplacementisshown in Fig. 2.8(a), a plot of the normalized velocity versus the normalized displacement is shown in Fig.2.8(b),andaphaseplotofthenormalizedrestoringforceversusthecorrespondingnormal- izedvelocityisshowninFig.2.8-(c). The fact that the nonlinearity in this case involves displacement-related terms, instead of velocity-relatedterms,isobviousfromanalysisoftheplotsinFig.2.8-(a)and-(c). Furthermore, theextentofthedead-spacenonlinearitycanbeeasilydiscernedfromFig.2.8-(a). AcompoundplotcombiningtheabovementionedthreephaseplotsisshowninFig.2.8-(d) in the form of a surface plot in which the vertical axis represents the nonlinear restoring force, while the two horizontal axes correspond to the system’s displacement and velocity. Nonpara- metricidentificationtechniquessuchastheRestoringForceMethod(MasriandCaughey,1979) 19 20 25 30 35 40 45 50 -0.8 -0.4 0 0.4 0.8 (a) Displacement (in) 20 25 30 35 40 45 50 -20 -10 0 10 20 Velocity (in/s) (b) 20 25 30 35 40 45 50 -20 -10 0 10 20 Restoring Force (lbf) Time (sec) (c) Figure2.7: TimehistoryplotsofnonlinearSDOFwithgapsubjectedtorandomexcitation. that seek to develop an approximating function for the indicated restoring force, are ideal for applicationinsuchsituations. 2.2.3 HarmonicallyExcitedHystereticSDOFSystemwithTime-VaryingSliding FrictionForce Theresponseoftheapparatuswhenconfiguredtoinvestigatethebehaviorofanonlinearsystem with a time-varying sliding friction force is shown in Fig. 2.9. Part (a) shows the displacement timehistory;part(b)thevelocitytimehistory;part(c)theaccelerationtimehistory;part(d)the 20 Figure 2.8: Phase diagrams of experimental measurements from a nonlinear SDOF with gap subjectedtorandomexcitation. force time history; and part (e) shows the time-evolution of the relative position of the stepper motorwhichcontrolsthemagnitudeofthenormalforcethatinducesfrictionalbehavior. About twofullperiodsoftime-variationofthenormalforceisshown. Notice that the output measured by the force gauge increases dramatically when the magni- tude of the normal force that modulates the dry friction force reaches its positive peak. Notice also that the increase in the friction component of the nonlinear restoring force lowers the sys- tem response, as clearly indicated in the significant reduction in the level of the corresponding system’sstatevariables. 21 0 5 10 15 20 25 30 -1 -0.5 0 0.5 1 (a) Displacement (in) 0 5 10 15 20 25 30 -20 -10 0 10 20 (b) Velocity (in/s) 0 5 10 15 20 25 30 -400 -200 0 200 400 (c) Acceleration (in/s 2 ) 0 5 10 15 20 25 30 -15 -7.5 0 7.5 15 (d) Force (lbf) 0 5 10 15 20 25 30 -6 -3 0 3 6 Time (sec) (e) Proportional Normal Force Figure 2.9: Experimental measurement from a SDOF system with time-varying sliding friction toharmonicexcitation. To further clarify the physics of the underlying phenomena observed during this test, the plot in Fig. 2.10-(a) shows a phase diagram of the force versus the system displacement, when the system is operating with a constant level of sliding friction, under steady-state harmonic excitation. The fact that the plotted trajectories do not fall exactly on top of each other is a clear indication of the variability (uncertainty) in the physical system parameters, even under thiscarefullycontrolledtest. Similarly, the phase diagram exhibited in Fig. 2.10-(b) corresponds to the time history seg- ment shown in Fig. 2.9. The significant change in the level of the friction forces during the 22 −1 −0.8 −0.6 −0.4 −0.2 0 0.2 0.4 0.6 0.8 1 −0.8 −0.6 −0.4 −0.2 0 0.2 0.4 0.6 0.8 1 Normalized Displacement Normalized Restoring Force −1 −0.8 −0.6 −0.4 −0.2 0 0.2 0.4 0.6 0.8 1 −1 −0.8 −0.6 −0.4 −0.2 0 0.2 0.4 0.6 0.8 1 Normalized Displacement Normalized Restoring Force (a)Constantfriction. (b)Time-varyingfriction. Figure 2.10: Phase plot of experimental measurements corresponding to the hysteretic system withslidingfrictionforcesunderharmonicexcitation. periodicvariationinthenormalforce,thatcontrolsthefrictionlevel,isclearlyevidentfromthis diagram,andthetransitionregionbetweenlow-levelandhigh-levelfrictionforcesiscapturedin theindicatedplot. 2.3 IdentificationofReduced-OrderNonlinearModel 2.3.1 Formulation of On-Line Bouc-Wen Hysteretic Model Identification Proce- dure Themotionofthesingle-degree-of-freedom(SDOF)systemtobeidentifiedisgovernedby m¨ x(t)+r(x(t), ˙ x(t)) = F(t) (2.1) where x(t) is the system displacement, r(x(t), ˙ x(t)) is the restoring force, and F(t) is the sys- tem’sexternalexcitation. Themassmofthesystemisassumedtobeknownoreasilyestimated, 23 andmeasurementsofF(t), ¨ x(t)areassumedtobeavailableateverytimestept k .Thevaluesof ˙ x(t) and x(t) are available either by direct measurements at times t k , or by integration of the signal ¨ x(t). If the restoring force r(x, ˙ x) has hysteretic characteristics, a model for such a force can be givenbythefollowingnonlineardifferentialequation(Bouc,1967;Wen,1976): r = r(x, ˙ x) (2.2) ˙ r = ( 1 η )[A˙ x−ν(β|˙ x||r| n−1 r−γ˙ x|r| n )] (2.3) Differentcombinationsoftheparametersη, A, ν, β, γandnwillproducesmoothhysteretic loopsofvarioushardeningorsofteningcharacteristics,withdifferentamplitudesandshapes. Let F(k) = F(t k ) ; x(k) = x(t k ) ; ˙ x(k) = ˙ x(t k ) ; ¨ x(k) = ¨ x(t k ) and r(k) = r(t k ). The systemequationofmotion(Eq. (2.1))isrewrittenas: r(k) = F(k)−m¨ x(k) (2.4) Hence,thevaluesofzattimet k areavailable,andtheidentificationproblemcanbestatedas: giventhemassm,andusingtheon-linemeasurementsof x, ˙ x,¨ x,andF,makeon-lineestimates oftheunknownparametersofthehystereticmodelexpressedbyEq. (2.3). 2.3.2 On-lineIdentificationAlgorithm Thehystereticmodel describedin Section 2.3.1, obeys the nonlinear differential equation (2.3). The model is parameterized linearly with respect to the coefficients (1/η)A, (1/η)νβ, and 24 (1/η)νγ, but nonlinearly with respect to the power n. It is, however, desirable to use a linearly parameterized estimator for the on-line estimation of hysteretic behavior, hence the following modificationofthemodelexpressedbyEq. (2.3)willbeused: ˙ r = ( 1 η ) " A˙ x− n=N X n=1 a n ν(β|˙ x||r| n−1 r−γ˙ x|r| n ) # (2.5) where the value of coefficient a n determines the contribution of power n to the hysteresis, and N is a large enough integer. For example, if the value of power n in the model (Eq. (2.3)) is n = 3,thenthecoefficientsa i inEq. (2.5)willbe: a 1 = 0,a 2 = 0,a 3 = 1. Forapplyingtheon-lineleast-squaresidentificationalgorithms,thedifferentialequationrep- resentingasystemshouldbeexpressedintheformofalinearStaticParametricModel(Ioannou andSun,1996): z = θ ∗T φ (SPM) (2.6) where z is the measurement vector, φ is the signal (or regressor) vector, and θ is the parameter (unknowns)vector. Note,thisisalinearcombinationofnonlinearresponsevariables. In SPM models, there is a linear relation between the parameters and the response, transfer function, and other inputs/outputs of the system (the parametric models for adaptive identifi- cation algorithms are not limited to SPM models (Ioannou and Sun, 1996)). The parameters of a linear SPM can be identified by using the adaptive least-squares algorithms (Chassiakos 25 et al., 1998; Ioannou and Datta, 1991; Smyth et al., 1999). Following the work of Ioannou and Sun(1996),themodifiedLeast-Squaresalgorithmwithforgettingfactorforidentifying θ(t),the estimate of θ ∗ , is obtained by solving ∇J(θ) = 0. The cost function J(θ) for the Recursive AdaptiveLeast-SquaresalgorithmwithForgetting-Factorisdefinedas: J(θ) = 1 2 Z t 0 exp −β(t−τ) [z(τ)−θ T φ(τ)] 2 m 2 s (τ) dτ + 1 2 exp −βt (θ−θ 0 ) T Q 0 (θ−θ 0 ) (2.7) where, Q 0 = Q T 0 > 0, β ≥ 0, θ 0 = θ(0). The recursive Least-Squares Algorithm for continuous-timeis: ˙ θ = Pφ (2.8) ˙ P =                βP −P φφ T m 2 s P, ifkP(t)k≤ R 0 0 otherwise (2.9) where, P(0) = P 0 = P T 0 ,||P 0 || ≤ R 0 , R 0 is the upper bound for||P||, m s is the normalizing signal,andβ ≥ 0istheforgettingfactor(IoannouandSun,1996). For the nonlinear hysteretic system defined by the Bouc-Wen model (Eq. (2.3)), the SPM can be derived by differentiating equation 2.4 with respect to time and redefining the ˙ z in sz notation, where s is the Laplace variable with zero initial condition (Ioannou and Sun, 1996). TheparametricmodelforEq. (2.1)thusbecomes: 26 F = m¨ x+r(x, ˙ x) (2.10) r = r(x, ˙ x) = F −m¨ x (2.11) ˙ r = sr = s(F −m¨ x) (2.12) ˙ r = ( 1 η ) " A˙ x− n=N X n=1 a n ν(β|˙ x||r| n−1 r−γ˙ x|r| n ) # (2.13) s(F −m¨ x) = ( 1 η ) " A˙ x− n=N X n=1 a n ν(β|˙ x||r| n−1 r−γ˙ x|r| n ) # (2.14) For avoiding the noise caused by differentiator block s in the identification process, both sides of the parametric model in Eq. (2.14) are filtered by a stable filter 1 Λ(s) , where Λ(s) is a Hurwitzpolynomial,hereΛ(s)ischosenass+λandλ > 0. s s+λ (F −m¨ x) = ( 1 s+λ ){ 1 η [A˙ x− n=N X n=1 a n ν(β|˙ x||r| n−1 r−γ˙ x|r| n )]} (2.15) s s+λ (F −m¨ x) | {z } z =                      A η −νβa 1 η νγa 1 η . . . −νβa N η νγa N η                      T | {z } θ ∗T                      1 s+λ ˙ x 1 s+λ |˙ x||r| 1−1 1 s+λ ˙ x|r| 1 . . . 1 s+λ |˙ x|r| N−1 1 s+λ ˙ x|r| N                      | {z } φ (2.16) 27 Equation(2.16)istheSPMforanonlinearhystereticsystemdefinedbytheBouc-Wennon- lineardifferentialequation. 2.3.3 AnalysisofSampleTestResults In this section, simulation results from a single-degree-of-freedom hysteretic system are pre- sentedtoillustratetheapplicationoftheon-lineidentificationalgorithmdevelopedinSection3. AhystereticsystemobeyingEq. (2.3)waschosenwiththefollowingparametervalues: η = 1, A = 5, ν = 1, β = 0.1, γ =−1, n = 2. (2.17) additionally,thesystemmassmhadavalueofunity. Assume that the experimental measurements for F(t) and ¨ x(t) are available over a time span much longer than the system characteristic period of interest. The corresponding system displacement x(t) and velocity ˙ x(t) of m can be found by direct measurement or through inte- gration of ¨ x(t). By discretizing the measurements, the values of r(k) can be obtained from Eq. (2.4). The method under consideration imposes no restrictions on the nature of the excitation sourcetobeusedasaprobingsignal. However,inordertoensurethattheparametersultimately convergetotheexactvalues,onemusthaveasufficientlyrichsignal,sothateachresponsecom- ponentinφ j (t)isrelativelylinearlyindependentfromtheotherφ k (t)’s(k6= j),thatis,thatone haspersistentexcitation(IoannouandSun,1996). Thewide-bandrandomexcitation F(t)usedinthissimulationisshowninFig.2.11(a),and theresponsedisplacementx(t)isshowninFig.2.11(b). Thecorrespondingrestoringforcetime history,obtainedbyremovingthecontributionoftheinertiaforcefromtheforcemeasurements, 28 0 5 10 15 20 25 30 35 40 45 50 −1.5 −1 −0.5 0 0.5 1 1.5 Time (sec) (b) Displacement (in) 0 5 10 15 20 25 30 35 40 45 50 −10 −8 −6 −4 −2 0 2 4 6 8 10 Time (sec) Excitation (lbf) (a) 0 5 10 15 20 25 30 35 40 45 50 −2.5 −2 −1.5 −1 −0.5 0 0.5 1 1.5 2 2.5 (c) Restoring Force (lbf) Time (sec) −1.5 −1 −0.5 0 0.5 1 1.5 −2.5 −2 −1.5 −1 −0.5 0 0.5 1 1.5 2 2.5 (d) Displacement (in) Restoring Force (lbf) Figure 2.11: Response and phase plot of simulation data corresponding to a hysteretic (Bouc- Wen)modelunderrandomexcitation. is shown in Fig. 2.11(c). The phase plane plot of the restoring force r versus x in Fig. 2.11(d) showsclearlythehystereticnatureofthesystem. By applying the above mentioned algorithm, the parameters converge to their asymptotic valueswithinaboutonesystemperiod(Fig.2.12). The asymptotic values for the parameters were ˆ θ = [5.000,0.002,0.002,−0.100,−1.001] whiletheactualparametersusedforthesimulationwereθ ∗ = [5,0,0,−0.1,−1]. Comparisonoftheresultsshowsthattheasymptoticvaluesoftheparameterswereveryclose to the actual values, including the parameters that should have vanishingly small values. The time-evolution of the identified restoring force components corresponding to the contribution 29 0 20 40 60 80 100 120 −2 −1 0 1 2 3 4 5 6 (a) Time θ 5 θ 1 θ 4 θ 2 θ 3 Figure2.12: Time-evolutionoftheidentificationparameters. of five (θ i .φ i ) components, is plotted in Fig. 2.13 in parts ((b)-(f)), together with the residual error (ˆ z−z ∗ ) shown in Fig. 2.13(h), found by subtracting from the reference signalz shown in Fig.2.13(a),thetotalsumofidentifiedforcesshowninFig.2.13(g). ItisclearfromFig.2.13thatthedominantcontributiontotheoverallrestoringforceisfrom θ 1 .φ 1 andθ 5 .φ 5 . Itisalsoclearthatthepeakvaluesoftheidentifiednonlinearforcecomponents coincidewiththelargeamplitudemotionrange. 2.4 ApplicationtoMeasurementsfromTestApparatus 2.4.1 DerivationofaGeneralLinearParametricModel Theparametricmodelderivedinsection2.3.1isforhystereticsystemsdefinedbyonlytheBouc- Wen differential equation (Eq. (2.3)). To derive a more general linear static parametric model 30 that represents a variety of SDOF models from linear to nonlinear, the damping, stiffness, and cubic nonlinearity polynomial terms were added to Eq. (2.16) in order to define a more general parametricmodel. Thefirstterm( A η ˙ x)intheBouc-Wenmodelisequivalenttothestiffnessterm. Forthegeneral parametricmodel,thistermwasreplacedwiththeactualstiffnessterm. s s+λ F −m¨ x | {z } z =                              k c d −νβa 1 η νγa 1 η . . . −νβa N η νγa N η                              T | {z } (θ ∗ ) T ×                              s s+λ x s s+λ ˙ x s s+λ x 3 1 s+λ |˙ x||r| 1−1 r 1 s+λ ˙ x|r| 1 . . . 1 s+λ |˙ x|r| N−1 r 1 s+λ ˙ x|r| N                              | {z } φ 2.4.2 IdentificationoftheHystereticRestoringForce A sample response record obtained through the test apparatus described in Section 2.1 was selected for application of the on-line identification technique under discussion. The system was excited with a wide band random signal. The response of the appratus (x, ˙ x and ¨ x) and the appliedforceF(t)weremeasuredateverydiscretetimestep(Fig.2.14). Thehystereticbehavior ofthesystemisevidentfromthephaseplotofresponse(Fig.2.14(f)). 31 0 20 40 60 80 100 120 −6 −3 0 3 6 Contribution of φ Vectors in Identification of z Vector z 0 20 40 60 80 100 120 −6 3 0 3 6 θ 1 . φ 1 0 20 40 60 80 100 120 −6 −3 0 3 6 θ 2 . φ 2 0 20 40 60 80 100 120 −6 −3 0 3 6 θ 3 . φ 3 0 20 40 60 80 100 120 −6 −3 0 3 6 θ 4 . φ 4 0 20 40 60 80 100 120 −6 −3 0 3 6 θ 5 . φ 5 0 20 40 60 80 100 120 −6 −3 0 3 6 Σ n i θ i .φ i 0 20 40 60 80 100 120 −6 −3 0 3 6 error Time sec (a) (b) (c) (d) (e) (f) (g) (h) Figure2.13: Contributionoftheidentifiedrestoringforcecomponentsbasedonsimulationdata fromahysteretic(Bouc-Wen)SDOFsystemunderrandomexcitation. 32 20 22 24 26 28 30 -15 -10 -5 0 5 10 15 Excitation (lbf) Time (sec) 20 22 24 26 28 30 -0.4 -0.2 0 0.2 0.4 0.6 Displacement (in) Time (sec) 20 22 24 26 28 30 -10 -5 0 5 10 Velocity (in/sec) Time (sec) 20 22 24 26 28 30 -300 -200 -100 0 100 200 300 Acceleration (in/sec 2 ) Time (sec) -0.4 -0.2 0 0.2 0.4 0.6 -10 -5 0 5 10 Velocity (in/sec) Displacement (in) -0.4 -0.2 0 0.2 0.4 0.6 -20 -15 -10 -5 0 5 10 15 20 Restoring Force (lbf) Displacement (in) Figure 2.14: Response and phase plot of the experimental test corresponding to a hysteretic system. 33 The equivalent mass of the system (the oscillating mass) was measured prior to the test. The number of the Bouc-Wen terms, N, was selected to be 2; therefore the total num- ber of the unknown parameters was(3 + 2N) = 7. At every time-step, the regressor vec- tor (φ) and the measurement vector (z) were constructed. The initial value for P(0) was selected to be 1000. The identification algorithm was applied to the parametric model, and the unknown parameters were identified for every time step (Fig. 2.15). A super position of the seven parameters is shown in Fig. 2.15(h). The asymptotic values for the parameters were ˆ θ = [48.48,0.18,−48.61,−0.12,−1.76,−0.00,0.12]. A comparison of the identified measurement vector (ˆ z) with the actual measurement vector (z) indicated an accurate identification (Fig. 2.16). The plot is Fig. 2.16(b) is a high-resolution plot of the time segment around t ≈ 55. Similarly, the plot in Fig. 2.16(c) is a further increase in resolution aroundt≈ 60. In each of the curves in Fig. 2.16, the solid line corresponds to the measurements,whilethedashedlineisfortheestimatedrestoringforce. The contribution of the various force components appearing in Eq. (2.16) to the overall restoringforceisshowninFig.2.17,inwhichplot(a)showsthemeasured(reference)resorting forcez,plots(b)-(i)showthecontributionoftheidentifiedtermsintheseriesexpansion,plot(j) is the sum of all the identified forces, and plot (h) shows the residual error corresponding to the difference(ˆ z−z). 2.5 ChapterSummary The main purpose of this chapter was to illustrate the design, construction, and evaluation of theperformanceofaneasilyreconfigurabletest apparatusforconducting basicresearchstudies 34 0 20 40 60 80 100 120 -60 -30 0 30 60 θ 1 (a) 0 20 40 60 80 100 120 -1 -0.5 0 0.5 1 θ 2 (b) 0 20 40 60 80 100 120 -100 -50 0 50 100 θ 3 (c) 0 20 40 60 80 100 120 -1 -0.5 0 0.5 1 θ 4 (d) 0 20 40 60 80 100 120 -50 -25 0 25 50 θ 5 (e) 0 20 40 60 80 100 120 -1 -0.5 0 0.5 1 θ 6 (f) 0 20 40 60 80 100 120 -1 -0.5 0 0.5 1 θ 7 (g) Time sec 0 20 40 60 80 100 120 -50 0 50 -50 0 (h) Time sec Super Posing θs Figure 2.15: Evolution of the identification parameters corresponding to the experimental data fromahystereticsystemunderrandomexcitation. 35 0 15 30 45 55 60 65 75 90 105 120 −30 −20 −10 0 10 20 30 (a) Restoring Force (lbf) 55 56 57 58 59 60 61 62 63 64 65 −30 −20 −10 0 10 20 30 (b) Restoring Force (lbf) 60 60.2 60.4 60.6 60.8 61 −20 −15 −10 −5 0 5 10 15 20 Time (sec) (c) Restoring Force (lbf) −0.6 −0.4 −0.2 0 0.2 0.4 0.6 −18 −12 −6 0 6 12 18 Displacement (in) Restoring Force (lbf) (d) Figure2.16: Comparisonofthemeasured(solidline)andestimated(dashedline)restoringforce in a hysteretic system subjected to stationary random excitation. (a) Time history of restoring force r(t) over a time span t from 0-120 s; (b) plot of r(t) over a time span t from 55-65 s; (c) plot of r(t) over a time span t from 60-61 s; (d) phase plot of the restoring force versus the correspondingsystemdisplacement. in nonlinear phenomena. The main features of the apparatus are described and its capability to model the behavior of a system with wide range of selectable nonlinear characteristics that encompass those usually encountered in the applied mechanics field. Using adaptive estima- tionapproaches, amethodispresentedfortheon-lineidentificationofhystereticsystemsunder arbitrary dynamic environments. It is shown through the use of simulation studies that the pro- posed approach can yield reliable estimates of the hysteretic restoring force under a wide range ofexcitationlevelsandresponseranges. 36 0 20 40 60 80 100 120 −30 −15 0 15 30 Contribution of φ Vectors in Identification of z Vector z 0 20 40 60 80 100 120 −30 −15 0 15 30 θ 1 . φ 1 0 20 40 60 80 100 120 −30 −15 0 15 30 θ 2 . φ 2 0 20 40 60 80 100 120 −30 −15 0 15 30 θ 3 . φ 3 0 20 40 60 80 100 120 −30 −15 0 15 30 θ 4 . φ 4 0 20 40 60 80 100 120 −30 −15 0 15 30 θ 5 . φ 5 0 20 40 60 80 100 120 −30 −15 0 15 30 θ 6 . φ 6 0 20 40 60 80 100 120 −30 −15 0 15 30 θ 7 . φ 7 0 20 40 60 80 100 120 −30 −15 0 15 30 Σ n i θ i .φ i 0 20 40 60 80 100 120 −30 −15 0 15 30 error Time sec (a) (b) (c) (e) (f) (g) (h) (i) (j) (k) Figure2.17: Contributionofeachparameterintheidentificationresultsbasedontheexperimen- talmeasurementsfromahystereticsystemunderrandomexcitation. 37 Chapter3 On-lineMonitoringofNonlinear ViscousDampers This chapter investigates the applications of the on-line identification method developed previ- ouslyfornonlinearhystereticsystemssubjectedtoharmonicandarbitrarydynamicexcitationsin Chapter 2. The recursive adaptive least-squares identification algorithm with a forgetting-factor is applied to data sets derived from experimental testing of a viscous damper similar to those being incorporated in recent design strategies to dissipate dynamic forces. Issues of parameter convergence, model parameterization and modeling errors are studied for a range of types and levelsofexcitation. Resultsforvarioustestdefinitionsarepresented,concludingwithanoverall assessmentoftheidentificationalgorithmanditsapplicabilitytoreal-timein-situsystemhealth monitoring. The chapter is organized as follows: Section 3.1 presents the mathematical background of on-lineidentificationandtheformulationofthemodels,Section3.2detailstheexperimentaltest 38 setup,Section3.3illustratesapplicationoftheon-linemonitoringofthesimplifieddesignmodel andthepolynomial-basismodel,andSection3.4discussestheresults. 3.1 OverviewofOn-lineParametricIdentificationApproach 3.1.1 SimplifiedMathematicalModelofNonlinearViscousDampers An assembled nonlinear viscous damper in any structure can be modeled as a nonlinear SDOF (Hart and Wong, 1999; Soong and Dargush, 1997). Fig. 3.1 illustrates a schematic of a viscous dampertestassemblyandtheequivalentsimplifiedmathematicalmodelofthenonlineardamper. (a)Schematicofnonlinearviscousdampertestassembly (b)Simplifiedmathematicalrepresentationofdamper Figure3.1: Simplifiedmodelandmathematicalrepresentationofnonlinearviscousdampers. Theequationofmotionofageneralnonlinear SDOF systemisgovernedby: m¨ x+r(x, ˙ x) = F(t) (3.1) where m is the effective moving mass, x, ˙ x, and ¨ x are responses to the applied force F(t), and r(x, ˙ x)isthenonlinearrestoringforce. 39 The nonlinear restoring force of viscous dampers can be characterized by a variety of para- metric ornonparametric mathematical models (Wolfe et al., 2002; Yun et al., 2006). In orderto derive the on-line identification algorithm for viscous dampers, two models have been selected representingthedampernonlinearrestoringforce. These models are: (1) the simplified design model; and (2) a nonparametric polynomial- basis model. The simplified design model is a physics-based model applicable to harmonically excited systems such as vibrating machinery. It is also relevant for damper parameter valida- tion purposes, wherein low amplitude sinusoidal excitations are applied. A major drawback to thismathematicalsystemrepresentationisthatitonlyreplicatesthemaximuminduceddamping force for harmonic input. Thus, the nonparametric polynomial-basis model is also investigated, as it has been shown capable of detecting system response to the broad-band excitations antici- patedinstructuralfieldapplications. TheSimplifiedDesignModel Thesimplifieddesignmodelisusedextensivelybydampermanufacturersandstructuraldesign engineerstodeterminethemaximuminduceddampingforce(HartandWong,1999;Miyamoto and Hanson, 2002; Soong and Dargush, 1997). The mathematical formulation of the simplified designmodelisdescribedby: F d = Csgn(˙ x)|˙ x| n (3.2) where F d is the damping force, ˙ x is the damper velocity, C is the damping coefficient, sgn(.) isthesignumfunctionwhichtakesthesignofitsargument,|.|istheabsolutefunction,andnis theexponentdefiningthedegreeofdampernonlinearity. 40 FormulationoftheSimplifiedDesignModelforOn-LineIdentification ThesimplifieddesignmodelisahighlynonlinearmodelwithrespecttounknownparametersC andn. Inordertoderiveastaticparametricmodelforon-lineidentificationofviscousdampers utilizingthesimplifieddesignmodel,wheretheunknownparametersandmeasurementsexhibit a linear relationship (Ioannou and Sun, 1996), several simplifying assumptions are necessary. These assumptions are: (1) the inertia force m¨ x is negligible in comparison to the nonlinear restoringforce,(2)thenonlinearrestoringforceismodeledbythesimplifieddesignmodel,and (3) the damping force F d and the velocity ˙ x are in-phase. The first assumption results from theauthors’previousstudies(Yunetal.,2006), whichdemonstratedthatthecontributionofthe inertiaforce isinsignificantincomparison to the nonlinear restoring force. The last assumption isvalidbasedonthesimplifieddesignmodelmathematicalformulation(Eq.(3.2)). It will be shown herein that the formulation of the simplified design model conforms to the assumption that if the value of the damping coefficient C is positive, then the damping force F d and the velocity ˙ x are in-phase. Recent studies (Wolfe et al., 2002; Yun et al., 2006) reveal that the damping force and velocity are in-phase when the viscous dampers are subjected to low frequency harmonic excitations, but occasionally are out-of-phase when the dampers are subjected to random excitations. This is one of the major limitations to on-line monitoring by usingthesimplifieddesignmodelforbroad-bandinput. In order to derive the static parametric model of the simplified design model (Eq. (3.2)), to facilitatenumericalimplementation,thesimplifieddesignmodelisreformulatedinanequivalent form: 41 sgn(F d )|F d | = Csgn(˙ x)|˙ x| n (3.3) Eqs. (3.2) and (3.3) are mathematically identical in real number space (F d ∧ ˙ x ∈ R). The derivation of the static parametric model form of the simplified design model is obtained from Eq.(3.3)throughthestepsexpressedbyEqs.(3.4)through 3.7: Log(sgn(F d )|F d |) = Log(Csgn(˙ x)|˙ x| n ) (3.4) Log(sgn(F d ))+Log(|F d |) = Log(C)+Log(sgn(˙ x)) (3.5) +nLog(|˙ x|) Log(|F d |)+Log(sgn(F d ))−Log(sgn(˙ x)) = Log(C)+nLog(|˙ x|) (3.6) Log(|F d |)+Log( sgn(F d ) sgn(˙ x) ) = Log(C)+nLog(|˙ x|) (3.7) Consideringthethirdassumptioninmodelingtheviscousdamperswiththesimplifieddesign model, the damping force F d and damper velocity ˙ x have the same direction and sign. Conse- quently, it follows that, Log(sgn(F d )/sgn(˙ x)) = Log(1) = 0. Hence, the static parametric modelforthesimplifieddesignmodel(Eq.(3.2))issimply: Log(|F d |) | {z } z = Log(C) n | {z } θ ∗ T ×     1 Log(|˙ x|)     | {z } φ (3.8) 42 whichisoftheformz = θ ∗ T φ,wherez isthemeasurementvector,θ ∗ isthevectorofunknown parameters,andφisthesignal(orregressor)vector. 3.1.2 FormulationofthePolynomial-BasisModelforOn-lineIdentification Thepolynomial-basismodelisanonparametricdata-basedmodelwhichismathematicallycapa- ble of capturing the response of various types of nonlinear phenomenon based on input/output data(Masrietal.,2004,2005,2006). Themostefficientformofthepolynomial-basismodeloccurswhenthebasisareorthogonal andcomplete(Beckmann,1973;WestwickandKearney,2003),suchasChebychevpolynomials (MasriandCaughey,1979;WordenandTomlinson,2001). Inordertoderiveanorthogonalbasis andthecorrespondingcoefficientsfromagivendataset,theentiresubspaceofdataisnecessary forintegrationandnormalizationpurposes. Derivation of an orthogonal basis in real-time is not possible for on-line monitoring, due to the absence of future data points at every instance. Hence, the polynomial-basis must be appliedinanonorthogonalform,suchasapower-seriespolynomialmodel. Moreover,byusing the nonorthogonal polynomial-basis, the optimum order of the polynomial would not be the minimum required order for capturing the behavior of the data, due to dependency between the parameters(Beckmann,1973;WestwickandKearney,2003). However, researchhasshownthe polynomial-basismodeltobeusefulwhenappliedinthiscontext (Al-HadidandWright,1989, 1990,1992;Masrietal.,2004,2006;Wolfeetal.,2002). The nonlinear restoring force of viscous dampers can be modeled by a polynomial-basis in theformofapower-seriesexpansionofstates: 43 m¨ x+r(x, ˙ x) = F(t) (3.9) r(x, ˙ x)≈ ˆ r(x, ˙ x) = p X i q X j a ij x i ˙ x j (3.10) where m is the effective moving mass, ¨ x is the acceleration of the effective mass, x and ˙ x are thestatevariables,r(x, ˙ x)isthenonlinearrestoringforce,F(t)istheappliedforce,pandq are thepolynomialorders,andthe(a ij )saretheunknownpolynomialcoefficients. Inordertoderivethestaticparametricmodelforthepolynomial-basisformulation,thepoly- nomialorders,pandq,shouldbespecified. Byassigningintegervalues,k andl,fortheorders, andexpandingtheseriesterms,thestaticparametricmodelformisderivedas: F(t) = m¨ x+ k X i=0 l X j=0 a ij x i ˙ x j (3.11) ˆ F(t) |{z} z = m a 00 a 10 ... a kl | {z } θ ∗ T × ¨ x x 0 ˙ x 0 x 1 ˙ x 0 ... x k ˙ x l T | {z } φ (3.12) where z is the measurement vector, θ ∗ is the vector of unknown parameters, and φ is the signal (orregressor)vector. 3.1.3 On-LineIdentificationBasedonAdaptiveLeast-Squares(ALS) Theon-lineparametricsystemidentificationmethodisbasedontheadaptiveleast-squaresalgo- rithm with a forgetting-factor. This method is capable of identifying unknown parameters of nonlinear, nonstationary models in the form of a linear static parametric model in real-time 44 (tracking the parameters’ values at every step). The word linear here means that the parame- ters of the system (θ ∗ ) and the signal vectors of the system (φ) have a linear relationship, even thoughthesystemhasanonlinearresponse. Thegeneralformofastaticparametricmodelmay bedescribedby(IoannouandDatta,1991;IoannouandSun,1996): z = θ ∗ T φ (3.13) wherevectorsz,φ,andθ ∗ areasdefinedforEq.(3.12)previously. The adaptive least-squares algorithm with a forgetting-factor for identifying θ(t), which is theestimateofθ ∗ (whereθ ∗ isthe“exact”valueofθ)inthestaticparametricmodel(Eq.(3.13)), is obtained by solving ∇J(θ) = 0. The cost function J(θ) for the recursive adaptive least- squaresalgorithmwithforgetting-factorisdefinedas: J(θ) = 1 2 Z t 0 exp −β(t−τ) [z(τ)−θ T φ(τ)] 2 m 2 s (τ) dτ + 1 2 exp −βt (θ−θ 0 ) T Q 0 (θ−θ 0 )(3.14) where,Q 0 = Q T 0 > 0,β ≥ 0,θ 0 = θ(0). Therecursiveleast-squaresalgorithmforcontinuous-timemonitoringis: ˙ θ = Pφ and ˙ P =                βP −P φφ T m 2 s P, ifkP(t)k≤ R 0 0 otherwise (3.15) 45 where, P(0) = P 0 = P T 0 ,||P 0 || ≤ R 0 , R 0 is the upper bound for||P||, m s is the normalizing signal,andβ ≥ 0istheforgetting-factor(IoannouandSun,1996). Theconditionforparameterconvergenceoftheadaptiveleast-squaresalgorithm(Eq.(3.15)) is that the signal vector φ of the static parametric model (Eq. (3.13)) be persistently excited (IoannouandSun,1996). 3.2 ExperimentalStudies Testdatautilizedinthisresearchwasobtainedfroma44.48KN(10kip)viscousdamperwitha velocityratingof1.78m/s(70in/sec)anda0.305m(12in)stroke. Theexperimentaltestswere performed on the campus of the University of Southern California using a specially designed test apparatus. The laboratory facility incorporates an 48.93 KN (11 kip) MTS Systems actua- tor fitted with a 0.0057 m 3 /s (90 gpm) servo valve to meet the velocity requirements of the test damper. Frequencies up to 2.5 Hz at 0.15 m (6 in) displacement were possible with this setup. Test displacement reductions to a 0.013 m (0.5 in) level increased the attainable frequency to 30 Hz. The damper and actuator were mounted on a specially designed reactive mass table to isolate the test from external influences. The damper/actuator interface was connected with lin- ear bearings to preclude out-of-plane bending relative to the axis of the damper. Special care wastakentoensurethealignmentofthedamper,actuatorandmountingbracketcenterlineswith high fidelity laser accuracy to reduce out-of-plane effects stemming from the facility construc- tion. Photographs of the test set-up are provided in Fig. 3.2. Further information regarding the experimentaltestsetup,instrumentations,anddatacollectionsarereportedbyWolfe(2002). 46 (a) (b) Figure 3.2: Overview of the experimental test setup for the 44.48 KN (10 kip) viscous damper atUniversityofSouthernCalifornia. 3.3 ApplicationoftheOn-lineIdentificationAlgorithm 3.3.1 On-lineIdentificationofHarmonicDatawiththeSimplifiedDesignModel Duetothemathematicallimitationsinthederivationofthestaticparametricmodelforthesim- plified design model, only harmonic data sets were used for associated on-line identification. Fig. 3.3 illustrates the on-line identification results of a harmonic data set subjected to a sinu- soidal excitation with frequency of 0.25 Hz and amplitude of±0.051 m (±2 in). Figs. 3.3-(a) and 3.3-(b) show the time-history evolution of the identified parameters, C and n, respectively. Fig. 3.3-(c) shows the time-domain comparison of the measured force (solid line) and the iden- tified force (dash-dot line). Figs. 3.3-(d) and 3.3-(e) are the phase-domain comparisons of the force-displacementandforce-velocityofthemeasureddata(solidline)versustheidentifieddata (dash-dotline). Inordertoverifytheon-lineidentificationresults,theunknownparametersofthesimplified designmodel(Eq.(3.2)),thedampingcoefficientC andtheexponentn,wereestimatedthrough standard nonlinear least-squares curve fitting optimization (Dennis and Schnabel, 1996). The 47 0 10 20 30 40 50 60 0 50 100 150 200 C (KN(s/m) n ) t (sec) (a) 0 10 20 30 40 50 60 −2 −1 0 1 2 n t (sec) (b) 15 16 17 18 19 20 21 22 23 24 25 −10 0 10 Force (KN) t (sec) (c) −0.06 −0.04 −0.02 0 0.02 0.04 0.06 −10 0 10 Force (KN) Displacement (m) (d) −0.1 −0.05 0 0.05 0.1 −10 0 10 Force (KN) Velocity (m/s) (e) Figure 3.3: On-line identification of a damper data set subjected to harmonic oscillation with frequencyof0.25Hzandamplitudeof±0.051m(±2in). Parts(a)and(b)showthetime-history of the identified parameters, C the damping coefficient, and n the exponent, respectively. The dash-dotlineindicatestheoptimumvalueoftheparametersobtainedthroughstandardnonlinear least-squares curve fitting optimization of the design model. Part(c) depicts the time-domain comparison of the measured force (solid-line) and the identified force (dash-dot line). Parts(d) and (e) are the phase-domain comparison of the force-displacement and force-velocity of the measureddata(solidline)vs.theidentifieddata(dash-dotline). dash-dot line, in Figs. 3.3-(a) and 3.3-(b), indicates the optimum value of the unknown param- eters, C and n, obtained through standard nonlinear least-squares curve fitting optimization of thesimplifieddesignmodel,superposedoverthetime-historyevolutionofthesameparameters obtainedthroughon-lineidentification(solidline). As shown in Figs. 3.3-(a) and 3.3-(b), the on-line identification of the unknown parameters convergestotheiroptimumvalueswithinthefirstperiodofthedata. Astatisticalcomparisonwasmadebetweentheoptimumvaluesoftheunknownparameters (C and n) obtained through nonlinear optimization, with the average value of the unknown 48 Table 3.1: Samples mean values (μ) and standard deviations (σ) of the unknown parameters (C and n) from the simplified design model (Eq. (3.2)), obtained through on-line identification for 202harmonicdatasets. On-lineIdentification NonlinearDamping Exponent Coefficient(C) (KNs n /m n ) (n) Sample Standard Sample Standard Mean(μ) Deviation(σ) Mean(μ) Deviation(σ) 48.27 27.52 0.64 0.20 Table 3.2: Samples mean values (μ) and standard deviations (σ) of the unknown parameters (C andn)fromthesimplifieddesignmodel(Eq.(3.2)),obtainedthroughnonlinearoptimizationfor 202harmonicdatasets. NonlinearOptimization NonlinearDamping Exponent Coefficient(C) (KNs n /m n ) (n) Sample Standard Sample Standard Mean(μ) Deviation(σ) Mean(μ) Deviation(σ) 59.00 31.61 0.71 0.20 parameters (averaging values within the last two cycles of the identification) obtained from the on-line identification, to verify the on-line identification results of all available harmonic data sets (202 data sets). The statistical parameters of fitted normal distributions over all of the data aresummarizedinTables3.1and3.2andillustratedinFig.3.4. Figs. 3.4-(a) and 3.4-(b) show the frequency histogram and Gaussian probability distribu- tionofthedata(solidline)fortheaveragevaluesoftheunknownparameters,C andn,obtained throughon-lineidentification. Figs.3.4-(c)and3.4-(d)depictthefrequencyhistogramandGaus- sianprobabilitydistributionofthedata(solidline)forthevaluesoftheunknownparameters,C 49 On-lineIdentification 0 50 100 150 0 0.005 0.01 0.015 0.02 0.025 0.03 Data Density 0 0.5 1 1.5 0 0.5 1 1.5 2 2.5 3 Data Density C idn n idn (a) (b) NonlinearOptimization 0 50 100 150 0 0.005 0.01 0.015 0.02 0.025 0.03 Data Density 0 0.5 1 1.5 0 0.5 1 1.5 2 2.5 3 Data Density C opt n opt (c) (d) Figure3.4: ComparisonofthefrequencyhistogramandGaussianprobabilitydistributionofthe unknownparameters,C andn,for202harmonicdatasets,estimatedwithon-lineidentification algorithmandnonlinearoptimization. Parts(a)and(b)showthefrequencyhistogramandGaus- sian probability distribution of the data (solid line) obtained through on-line identification, and parts(c)and(d)showthefrequencyhistogramandGaussianprobabilitydistributionofthedata (solidline)obtainedthroughnonlinearoptimization. and n, obtained through nonlinear optimization. The superposed Gaussian distribution has the samemeanandvarianceasthecorrespondinghistograms. It is shown only for comparison purposes, even though it covers the physically infeasible range of parameters. It is seen that both approaches yield nearly the same measures of the samplemeanandstandarddeviationofthetwodamperparametersC andn. 50 3.3.2 On-lineIdentificationofRandomDatawiththePolynomial-BasisModel In order to demonstrate the utility of the on-line identification results of the polynomial-basis model (Eq. (3.12)) for random data sets, the model order should be selected prior to the com- putation. Foron-lineidentificationofabroad-bandrandomdatasetwithamplitudeof ±0.0254 m (±1 in), the model order was set to k = l = 5, yielding a total of thirty-seven ( 37) unknown parameters,withthestaticparametricmodelexpressedas: F(t) = m¨ x+ 5 X i=0 5 X j=0 a ij x i ˙ x j (3.16) ˆ F(t) = [m a 00 a 01 a 02 ... a 33 ... a 54 a 55 ] × (3.17) [¨ x x 0 ˙ x 0 x 0 ˙ x 1 x 0 ˙ x 2 ... x 3 ˙ x 3 ... x 5 ˙ x 4 x 5 ˙ x 5 ] T Theon-lineidentificationresultsfortherandomdatasetwithamplitudeof ±0.013m(±0.5 in) in are illustrated in Fig. 3.5. Fig. 3.5-(a) shows the time-domain comparison between the measured force (solid line) and identified force (dash-dot line), and Figs. 3.5-(b) and 3.5-(c) illustrate the force-displacement and force-velocity phase-domain comparison using both the measuredforce(solidline)andidentifiedforce(dash-dotline). The normalized mean-square error between the measured data and identified data is 4.65%, indicating a good agreement between the measured and identified forces (Worden and Tomlin- son,2001). In order to verify the on-line identification results, the unknown parameters of the polynomial-basis model (Eq. (3.17)) were identified through standard least-squares estimation 51 20 21 22 23 24 25 26 27 28 −15 −10 −5 0 5 10 15 Force (KN) Time (sec) (a) −0.02 −0.01 0 0.01 0.02 −15 −10 −5 0 5 10 15 Force (KN) Displacement (m) (b) −0.08 −0.06 −0.04 −0.02 0 0.02 0.04 0.06 −15 −10 −5 0 5 10 15 Force (KN) Velocity (m/s) (c) Figure 3.5: Comparison of the measured force (solid line) and the identified force (dash-dot line). Part (a) shows a selected time-domain comparison of the measured force (solid-line) and the identified force (dash-dot line), parts (b) and (c) are the force vs. displacement and force vs. velocity phase-plot comparison depicting both the measured force (solid line) and identified force(dash-dotline). techniques(Masrietal.,2006;Mendel,1995). Fig.3.6showsthecomparisonofthetime-history evolutionoftwelve(12)selectedunknownparameters(solidline)andtheoptimumvalueofthe unknown parameters obtained by least-squares estimation (dash-dot line). The corresponding basistermofeachparameterisdisplayedatthetopofeachplot. Clearly, most of the parameters, with few exceptions, converged to their optimum values withinafewcyclesoftheidentificationcommencing. Thoseparameterswhichdidnotconverge to their optimum values, are the parameters which are insignificant in contributing to the total (combinedforcecomponents)identifiedforce. 52 0 20 40 60 −50 0 50 100 θ0 φ(¨x) 0 20 40 60 −500 0 500 1000 1500 θ2 φ(x 0 ˙ x 1 ) 0 20 40 60 −1000 −500 0 500 1000 θ3 φ(x 0 ˙ x 2 ) 0 20 40 60 −500 0 500 1000 θ4 φ(x 0 ˙ x 3 ) 0 20 40 60 −600 −400 −200 0 200 400 θ5 φ(x 0 ˙ x 4 ) 0 20 40 60 −500 0 500 θ6 φ(x 0 ˙ x 5 ) 0 20 40 60 −1500 −1000 −500 0 500 θ7 φ(x 1 ˙ x 0 ) 0 20 40 60 −2000 −1000 0 1000 2000 θ8 φ(x 1 ˙ x 1 ) 0 20 40 60 −3000 −2000 −1000 0 1000 θ9 φ(x 1 ˙ x 2 ) 0 20 40 60 −2000 −1000 0 1000 2000 3000 θ10 φ(x 1 ˙ x 3 ) 0 20 40 60 −1000 −500 0 500 1000 θ11 φ(x 1 ˙ x 4 ) 0 20 40 60 −1000 −500 0 500 1000 1500 θ12 φ(x 1 ˙ x 5 ) 0 20 40 60 −1500 −1000 −500 0 500 1000 θ13 φ(x 2 ˙ x 0 ) 0 20 40 60 −4000 −2000 0 2000 θ14 φ(x 2 ˙ x 1 ) 0 20 40 60 −3000 −2000 −1000 0 1000 2000 θ15 φ(x 2 ˙ x 2 ) 0 20 40 60 −2000 −1000 0 1000 2000 3000 θ16 φ(x 2 ˙ x 3 ) 0 20 40 60 −2000 −1000 0 1000 θ17 φ(x 2 ˙ x 4 ) 0 20 40 60 −1000 0 1000 2000 θ18 φ(x 2 ˙ x 5 ) 0 20 40 60 −2000 0 2000 4000 6000 θ19 φ(x 3 ˙ x 0 ) 0 20 40 60 −8000 −6000 −4000 −2000 0 2000 θ20 φ(x 3 ˙ x 1 ) 0 20 40 60 −4000 −2000 0 2000 4000 θ21 φ(x 3 ˙ x 2 ) 0 20 40 60 −6000 −4000 −2000 0 2000 θ22 φ(x 3 ˙ x 3 ) 0 20 40 60 −2000 0 2000 4000 6000 θ23 φ(x 3 ˙ x 4 ) 0 20 40 60 −4000 −2000 0 2000 θ24 φ(x 3 ˙ x 5 ) 0 20 40 60 −2000 0 2000 4000 θ25 φ(x 4 ˙ x 0 ) 0 20 40 60 −4000 −2000 0 2000 4000 θ26 φ(x 4 ˙ x 1 ) 0 20 40 60 −4000 −2000 0 2000 4000 θ27 φ(x 4 ˙ x 2 ) 0 20 40 60 −4000 −2000 0 2000 4000 θ28 φ(x 4 ˙ x 3 ) 0 20 40 60 −2000 0 2000 4000 θ29 φ(x 4 ˙ x 4 ) 0 20 40 60 −6000 −4000 −2000 0 2000 θ30 φ(x 4 ˙ x 5 ) 0 20 40 60 −3000 −2000 −1000 0 1000 2000 θ31 φ(x 5 ˙ x 0 ) t (sec) 0 20 40 60 −2000 0 2000 4000 θ32 φ(x 5 ˙ x 1 ) t (sec) 0 20 40 60 −3000 −2000 −1000 0 1000 2000 θ33 φ(x 5 ˙ x 2 ) t (sec) 0 20 40 60 −2000 −1000 0 1000 2000 3000 θ34 φ(x 5 ˙ x 3 ) t (sec) 0 20 40 60 −4000 −2000 0 2000 θ35 φ(x 5 ˙ x 4 ) t (sec) 0 20 40 60 −2000 0 2000 4000 θ36 φ(x 5 ˙ x 5 ) t (sec) Figure 3.6: Time-history evolution of the unknown parameters of polynomial model obtained from on-line identification algorithm (solid line) and the optimum values of the parameters derived from standard least-squares estimation (dash-dot line). The corresponding basis term ofeachparameterisdisplayedinthetitleeachplotplot. ThecontributionofeachoftheidentifiedparametersisillustratedinFig.3.7throughatime- history evolution of the identified parameters and their corresponding basis (θ T i ×φ i ). Fig. 3.7 revealsthatthetime-historyevolutionofthecontributionofidentificationterms( θ T i ×φ i ). For ease of comparison and illustration, all the plots are normalized with respect to the maximum measured force and zoomed in to the last ten-seconds of the identification process. 53 50 55 60 −1 −0.5 0 0.5 1 θ0 φ(¨x) 50 55 60 −1 −0.5 0 0.5 1 θ2× φ(x 0 ˙ x 1 ) 50 55 60 −1 −0.5 0 0.5 1 θ3× φ(x 0 ˙ x 2 ) 50 55 60 −1 −0.5 0 0.5 1 θ4× φ(x 0 ˙ x 3 ) 50 55 60 −1 −0.5 0 0.5 1 θ5× φ(x 0 ˙ x 4 ) 50 55 60 −1 −0.5 0 0.5 1 θ6× φ(x 0 ˙ x 5 ) 50 55 60 −1 −0.5 0 0.5 1 θ7× φ(x 1 ˙ x 0 ) 50 55 60 −1 −0.5 0 0.5 1 θ8× φ(x 1 ˙ x 1 ) 50 55 60 −1 −0.5 0 0.5 1 θ9× φ(x 1 ˙ x 2 ) 50 55 60 −1 −0.5 0 0.5 1 θ10× φ(x 1 ˙ x 3 ) 50 55 60 −1 −0.5 0 0.5 1 θ11× φ(x 1 ˙ x 4 ) 50 55 60 −1 −0.5 0 0.5 1 θ12× φ(x 1 ˙ x 5 ) 50 55 60 −1 −0.5 0 0.5 1 θ13× φ(x 2 ˙ x 0 ) 50 55 60 −1 −0.5 0 0.5 1 θ14× φ(x 2 ˙ x 1 ) 50 55 60 −1 −0.5 0 0.5 1 θ15× φ(x 2 ˙ x 2 ) 50 55 60 −1 −0.5 0 0.5 1 θ16× φ(x 2 ˙ x 3 ) 50 55 60 −1 −0.5 0 0.5 1 θ17× φ(x 2 ˙ x 4 ) 50 55 60 −1 −0.5 0 0.5 1 θ18× φ(x 2 ˙ x 5 ) 50 55 60 −1 −0.5 0 0.5 1 θ19× φ(x 3 ˙ x 0 ) 50 55 60 −1 −0.5 0 0.5 1 θ20× φ(x 3 ˙ x 1 ) 50 55 60 −1 −0.5 0 0.5 1 θ21× φ(x 3 ˙ x 2 ) 50 55 60 −1 −0.5 0 0.5 1 θ22× φ(x 3 ˙ x 3 ) 50 55 60 −1 −0.5 0 0.5 1 θ23× φ(x 3 ˙ x 4 ) 50 55 60 −1 −0.5 0 0.5 1 θ24× φ(x 3 ˙ x 5 ) 50 55 60 −1 −0.5 0 0.5 1 θ25× φ(x 4 ˙ x 0 ) 50 55 60 −1 −0.5 0 0.5 1 θ26× φ(x 4 ˙ x 1 ) 50 55 60 −1 −0.5 0 0.5 1 θ27× φ(x 4 ˙ x 2 ) 50 55 60 −1 −0.5 0 0.5 1 θ28× φ(x 4 ˙ x 3 ) 50 55 60 −1 −0.5 0 0.5 1 θ29× φ(x 4 ˙ x 4 ) 50 55 60 −1 −0.5 0 0.5 1 θ30× φ(x 4 ˙ x 5 ) 50 55 60 −1 −0.5 0 0.5 1 θ31× φ(x 5 ˙ x 0 ) t (sec) 50 55 60 −1 −0.5 0 0.5 1 θ32× φ(x 5 ˙ x 1 ) t (sec) 50 55 60 −1 −0.5 0 0.5 1 θ33× φ(x 5 ˙ x 2 ) t (sec) 50 55 60 −1 −0.5 0 0.5 1 θ34× φ(x 5 ˙ x 3 ) t (sec) 50 55 60 −1 −0.5 0 0.5 1 θ35× φ(x 5 ˙ x 4 ) t (sec) 50 55 60 −1 −0.5 0 0.5 1 θ36× φ(x 5 ˙ x 5 ) t (sec) Figure 3.7: Time-history evolution of the contribution of each of the identified parameters and their corresponding basis (θ i × φ i ). For ease of comparison and illustration, all the plots are normalizedtothemaximummeasuredforceandzoomedintothelastten-secondsoftheidenti- ficationprocess. It is obvious from this representation of individual parameters over time that only a few are significant in their contribution to the total restoring force, while some terms cancel each other (e.g. θ 4 ×φ(x 0 ˙ x 3 )andθ 6 ×φ(x 0 ˙ x 5 )). 54 3.3.3 ValidationoftheOn-LineIdentificationforRandomDatawiththePolyno- mialBasisModel To demonstrate the utility of the on-line identification process of the polynomial-basis model to predict system response, two data sets were selected with identical frequency content, but differentamplitudes. Thedatasetwithahigherpeakamplitude(±0.0254mor±1in)wasused to initially identify the unknown parameters (train the system model); thereafter, the identified parameterswereusedtopredicttheforcevalueoftheseconddatasethavingaloweramplitude inputsignal(±0.013mor±0.5in). Forthispurpose,theestimatedforceiscomputedby, ˆ z est = θ T ×φ(t), (3.18) where ˆ z est istheestimateofthemeasuredforce,θ T isavectorcontainingtheasymptoticvalues of the identified unknown parameters from the identification data set, and φ(t) is the measure- ment vector (measured data) from the validation data set. Fig. 3.8 shows the validation results of the measured force (solid line) and estimated force (dash-dot line) for the polynomial-basis model, where Fig. 3.8-(a) represents the time-history plot of forces, Fig. 3.8-(b) the phase-plot offorcesvs.displacement,andFig.3.8-(c)thephase-plotofforcevs.velocity. The normalized mean-square error between the estimated force from the validation process and the measured force is about 11.40%, which is significantly larger than the corresponding error for the “training” phase. If the identification and validation process were computed using the same data set, basically employing a segment of the the data for identification, and another segmentwhichisnotusedinidentification,forvalidation(IoannouandSun,1996;Ljung,1999), 55 20 21 22 23 24 25 26 27 28 −6 −4 −2 0 2 4 6 Force (KN) Time (sec) (a) −0.01 −0.005 0 0.005 0.01 −6 −4 −2 0 2 4 6 Force (KN) Displacement (m) (b) −0.06 −0.04 −0.02 0 0.02 0.04 0.06 −6 −4 −2 0 2 4 6 Force (KN) Velocity (m/s) (c) Figure 3.8: The validation results of the measured force (solid line) and estimated force (dash- dot line), where the identification and validation data are two completely different sets. Part (a) is the time-history plot of forces, part (b) is the phase-plot of force vs. displacement, and parts (c)isthephase-plotofforcevs.velocity. the results indicate a more accurate estimation, with a mean-square error of 1.40% (Fig. 3.9). Thisresultisasexpecteddueagaintosatisfyingthepersistenceofexcitationconcerns(Ioannou andSun,1996). Ingeneral,forhighlynonlinearsystems,ifthevalidationdatasetcontainsdifferent(usually greater) dynamic characteristics (different amplitude or frequency range) than the identification dataset,theidentifiedmodelwillnotreplicatethebehaviorofthevalidationdatasetcompletely (IoannouandSun,1996). 56 20 21 22 23 24 25 26 27 28 −6 −4 −2 0 2 4 6 Force (KN) Time (sec) (a) −0.01 −0.005 0 0.005 0.01 −6 −4 −2 0 2 4 6 Force (KN) Displacement (m) (b) −0.06 −0.04 −0.02 0 0.02 0.04 0.06 −6 −4 −2 0 2 4 6 Force (KN) Velocity (m/s) (c) Figure3.9: Thevalidationresultsofthemeasuredforce(solidline)andestimatedforce(dash-dot line)fromthepolynomial-basismodel,wheretheidentificationandvalidationdataaredifferent segments of same data set. Part (a) is the time-history plot of forces, part (b) is phase-plot of forcesvs.displacement,andparts(c)isthephase-plotofforcesvs.velocity. 3.4 Discussion The present study demonstrates the capabilities and limitations of the on-line monitoring of the simplified design model and a nonparametric polynomial-basis model. The on-line monitoring of the simplified design model is a useful tool for monitoring the design parameters of vis- cous dampers subjected to harmonic excitations, which are typical test cases used for testing and developing damper specifications (HITEC, 1996, 1998a,b, 1999), or when designing struc- tures (Hart and Wong, 1999; Miyamoto and Hanson, 2002; Soong and Dargush, 1997). The polynomial-basis model was shown to be capable of identifying system response to broad-band 57 excitations. Also, the authors have demonstrated in previous studies (Wolfe et al., 2002; Yun etal.,2006),thatidentificationwithorthonormalbasismodels(Chebyshevpolynomial)orblack box models (Artificial Neural Networks) provides efficient and robust models for identification and change detection, in batch-mode identification, when all the data is available during the identificationprocess. 3.4.1 SimplifiedDesignModel Comparisonoftheon-linemonitoringandnonlinearoptimizationofthesimplifieddesignmodel (Fig. 3.4), indicates that the estimated values of the design parameters (Eq. (3.9)) converge to theiroptimumvalueswithinthefirstcycleofdata,forharmonicexcitation. Thisenablesmanu- facturersanddesignengineerstoverifythe“actual”designparametersbymonitoringthedesign model, during structural testing, in order to accurately estimate the design parameters for dif- ferent harmonic load cases. However, due to the mathematical limitations imposed through the formulation of the simplified design model (Eq. (3.7)), tracking the design parameters under randomexcitations,typicalinmostpracticalapplications,isnotfeasible. Atissueisthefactthat harmonic excitations do not produce “trained” data identification models that contain enough frequencyperturbationstoaccuratelyreplicateexcitingsignalscontainingdifferentfrequencies, whetherharmonicinnatureorbroad-band. Itwasassumedthatthedampingforceandthedampervelocityarein-phasewhenthestatic parametric model was derived for the simplified design model in Section 3.1.1. As noted previ- ously,theauthors(Wolfeetal.,2002;Yunetal.,2006)havepreviouslyshownthatthedamping force and velocity are generally in-phase when the viscous dampers are subjected to low fre- quency harmonic excitations (except in some narrow regions of the phase-plane), and not when 58 subjected to random excitations. In order to demonstrate this limitation of the simplified design model, a random data set with amplitude of ±0.0254 m (±1 in) was used for identification. Fig.3.10illustratestheon-lineidentificationresultsofthesimplifieddesignmodelwiththeran- dom data set. Figs.3.10-(a) and 3.10-(b) show the time-history of the identified parameters C and n, respectively (the dash-dot lines indicate the optimum value of the parameters obtained through nonlinear least-squares curve fitting with the same model). Fig. 3.10-(c) presents the time-domain comparison of the measured force (solid-line) and the identified force (dash-dot line),andFigs.3.10-(d)and3.10-(e)arethephase-domaincomparisonoftheforce-displacement andforce-velocityofthemeasureddata(solidline)vs.theidentifieddata(dash-dotline). Clearly, the simplified design model is incapable of accurately tracking the unknown parameters, C and n,whenthedamperissubjectedtorandomexcitations. 3.4.2 Polynomial-BasisModel This study has shown that with some caution in application, the polynomial-basis model is a promising tool that is capable of tracking the measured response of nonlinear viscous dampers (Fig. 3.5). Practical implementation of the polynomial-basis model requires that the user define theorderofthemodelpriortoimplementationofthetrackingalgorithm. Foreaseofdiscussion, the order in this section refers to the order of both the x and ˙ x terms. The simplest criteria for definingtheorderisbasedontheestimationerror,yieldingtheorderlevelthatproducesanerror value less than a predefined threshold. For example, if the required error threshold is specified as5%,theordershouldbeincreaseduntiltheerrorfallswithinthisbound. Theacceptablevalue oftheerrordependsontheapplication,butingeneralanormalizedmean-squareerrorbelow 5% isconsideredagoodestimation(WordenandTomlinson,2001). 59 0 10 20 30 40 50 60 0 50 100 150 200 C (KN(s/m) n ) t (sec) (a) 0 10 20 30 40 50 60 0 0.5 1 n t (sec) (b) 15 16 17 18 19 20 21 22 23 24 25 −20 −10 0 10 20 Force (KN) t (sec) (c) −0.01 −0.005 0 0.005 0.01 −10 −5 0 5 10 Force (KN) Displacement (m) (d) −0.06 −0.04 −0.02 0 0.02 0.04 0.06 −10 −5 0 5 10 Force (KN) Velocity (m/s) (e) Figure 3.10: On-line identification of data set subjected to broad-band random oscillation with amplitudeof±0.0254m(±1in). Parts(a)and(b)showthetime-historyoftheidentifiedparam- eters C, the damping coefficient, and n, the exponent, respectively, the dash-dot lines indicate theoptimumvalueoftheparametersobtainedthroughnonlinearleast-squarescurvefittingwith the same model. Part(c) is the time-domain comparison of the measured force (solid-line) and the identified force (dash-dot line). Parts(d) and (e) are the phase-domain comparison of the force-displacement and force-velocity of the measured data (solid line) vs. the identified data (dash-dotline). SensitivityofPolynomialBasisModeltoBasisOrder There are some challenges in applying the polynomial-basis model with the adaptive least- squares tracking algorithm as a structural health monitoring tool. One of the main issues is under- and over-parameterization of the model. Fig. 3.11 illustrates the model-order and error relationship. Fig.3.11-(a)showsathree-dimensionalsurfaceofthecomputederrorasafunction of polynomial order, where the x-axis represents the ˙ x order, the y-axis corresponds to the x 60 order, and the z-axis is the identification mean-square error. Fig. 3.11-(b) illustrates the depen- dency of the identification mean-square error on the x order of each term, while the ˙ x order of theothertermsisfixed. Fig.3.11-(c)illustratesthedependencyoftheidentificationmean-square error on the ˙ x order of each term, while the x order of the other terms is fixed (in these figures the fixed-term order is set to 5). There are two major observations evident from this figure: (i) the model is largely invariant in regard to the order of x, whereas the order of ˙ x controls the identification error, and (ii) the identification mean-square error will not always decrease with increasingmodelorder. Another important limitation of the polynomial-basis model is the variation of the param- eters’ asymptotic values with the model order. Figs. 3.12-(a) to 3.12-(e) show the asymptotic valuesforsomeoftheselectedidentifiedparameters,correspondingto ¨ x,x, ˙ x,x 3 ,and ˙ x 3 ,with respect to the polynomial order. Fig. 3.12-(f) displays the variation of the identification error withrespecttothemodelorder. There are several important points that can be discussed from Figs. 3.11 and 3.12: (i) the identification error decreases with the model order until the number of basis increases such that the over-parameterization effect occurs, (ii) the values of the parameters change with the order, where this effect is attributed to the lack of orthogonality in the polynomial-basis model(Beckmann, 1973), and (iii) the mass value is the only parameter that remains largely invariant with respect to the model order until the system becomes over-parameterized. This latter fact allows the mass parameter (in the present application) to be used as a measure or indicatorofmodelover-parameterization. 61 0 5 10 15 20 25 30 35 0 10 20 30 0 10 20 30 40 50 60 70 80 90 100 Order ofx Order of ˙ x Error % (a) 0 5 10 15 20 25 30 35 0 10 20 30 40 50 60 70 80 90 100 Order of x (˙ x order is fixed at 5) Polynomial order Error % 0 5 10 15 20 25 30 35 0 10 20 30 40 50 60 70 80 90 100 Order of ˙ x (x order is fixed at 5) Polynomial order Error % (b) (c) Figure3.11: Therelationshipbetweentheorderofpolynomial-basismodelandtheidentification error. Part (a) shows a 3D surface of error as a function of polynomial orders, where the x-axis isthe ˙ xorder,they-axis isthexorder,andthez-axis istheidentificationerror. Parts(b)and(c) illustratethedependencyoftheidentificationerrortotheordersofeachtermswhiletheorderof theothertermisfixed. 62 2 8 14 20 26 32 38 −1 −0.5 0 0.5 1 θ i (¨x) Polynomial order (a) Nominal value 2 8 14 20 26 32 38 −1 −0.5 0 0.5 1 θ i (x) Polynomial order (b) Nominal value 2 8 14 20 26 32 38 −1 −0.5 0 0.5 1 θ i (˙ x 3 ) Polynomial order (c) Nominal value 2 8 14 20 26 32 38 −1 −0.5 0 0.5 1 θ i (x) Polynomial order (d) Nominal value 2 8 14 20 26 32 38 −1 −0.5 0 0.5 1 θ i (x 3 ) Polynomial order (e) Nominal value 2 8 14 20 26 32 38 0 5 10 15 20 MSE % Polynomial order (f) Error % Figure 3.12: Correlation between the asymptotic values of selected identified parameters and the estimation error for the polynomial-basis model with model order, when the orders for x and ˙ x are equal in values. Parts (a) to (e) show the asymptotic values of identified parameters corresponding to ¨ x, x, ˙ x, x 3 , and ˙ x 3 terms with the polynomial order. Part (f) depicts the relationshipbetweentheidentificationerrorandthepolynomialorder. InfluenceofExcitationCharacteristicsonIdentificationResults Furthermore, due to the nature of nonlinear systems, where the system response is a nonlinear function of the amplitude and spectral content of the excitation (Worden and Tomlinson, 2001), the identified parameters of the nonlinear viscous dampers with the on-line monitoring of the polynomial-basis model are limited to predicting the response of the damper subjected to an excitation with similar dynamic characteristics (in amplitude and spectral content). This limita- tionisillustratedinFigs.3.8and 3.9. Theestimationoftheforcewiththeparametersidentified with another data set (Fig. 3.8), produces a greater error for the random data set with lower amplitude (≈ 11.4%), whereas the identification of the force using segments of the measured 63 data from the same exciting signal (Fig. 3.9), results in substantially reduced error (≈ 1.4%). This is a general limitation of all nonlinear system identification procedures (Kerschen et al., 2006). 3.5 ChapterSummary In this Chapter, the on-line monitoring application of two mathematical model representations of nonlinear viscous dampers, one based on the simplified design model, and the second pred- icated on the polynomial-basis model, were investigated through experimental studies. It was demonstrated that monitoring with the simplified design model would provide a useful tool for design engineers and manufacturers to verify the design parameters under harmonic testing, as well as for in-situ parameter identification for systems subjected to harmonic input. However, it was also shown that the simplified design model is not applicable for identification purposes when the system is excited by random signals. To circumvent this limitation, the polynomial- basis model was shown to be capable of identifying the behavior of nonlinear viscous dampers subjectedtorandomexcitations. In order to implement a practical on-line structural health monitoring methodology, the model for detecting changes in nonlinear viscous dampers should be capable of capturing the anticipated breadth of response which the device may experience during its service life, and should employ a semi-on-line monitoring strategy. This latter condition ensures that each data set considered incorporates adequate data (from the identifiability point of view) to produce a robustmodelthatisinsensitivetonoiseandmodelingerrors. 64 Chapter4 Data-BasedModel-Free RepresentationsofNonlinearDynamic Systems This chapter proceeds along two fronts: (1) an analytical phase focused on the development of a theoretical framework for processing experimental structural response measurements from uncertain systems, to develop and evaluate the utility of some promising analytical tools, based onpolynomialbasismodelsandneuralnetworks,fordevelopingreduced-order,data-basedcom- putational representations of nonlinear dynamic systems, and (2) an experimental phase involv- ingthedesignandfabricationofanadjustabletestapparatusforconductingstudiesonageneric two-dimensional“joint”elementwhichincorporatesimportantnonlinearcharacteristicssuchas nonlinear elastic properties, hysteretic characteristics, and dead-space nonlinearities involving friction. 65 The chapter is organized as follows: Section 4.1 presents the mathematical background and formulation of data-based model-free representations of a nonlinear dissipative 2- DOF system based on a polynomial basis model and neural networks, Section 4.2 presents the simulation studies,Section4.3detailstheexperimentaltestsetup,Section4.4demonstratestheapplication ofthemethodologythroughexperimentaldatasets,andtheresultsarediscussedinSection4.5. 4.1 Formulation of Data-Based Model-Free Representations of NonlinearElement 4.1.1 EquationofMotion Consider a discrete n-degrees-of-freedom ( n- DOF) system incorporating nonlinear nonconser- vative dissipative elements, which is subjected to directly applied excitation forces F(t). The motionofthisnonlinearsystemisassumedtobegovernedbythesetofequations, M¨ x(t)+R(x, ˙ x,p) = F(t) (4.1) where x(t) is the displacement vector of order n,M is a constant matrix that characterizes the inertia forces,R(x, ˙ x,p) is the restoring force vector of nonlinear nonconservative forces, p is the vector of system-specific parameters, and F(t) is the n-column vector of directly applied forces. The nonlinear nonconservative components, in this study, are presented as massless nonlin- ear “joint” elements, that are located between two lumped masses of the system. Figure 4.1 66 i total F ) , x r(x, p & i M i total T i i x , x & j j x , x & j total T j M i j i j where x x x x x x : & & & − = − = j total F Figure4.1: Freebodydiagramofagenericmasslessnonlinear“joint”element. shows the free body diagram of a massless nonlinear “joint” element that is located between DOF- i and - j, with constant mass matrices ofM i andM j that characterizes the inertia forces; x i , ˙ x i ,x j ,and ˙ x j whicharethestateofthe DOFs;r(x, ˙ x,p)whichrepresentstherestoringforce vector of nonlinear nonconservative forces of the “joint” element; and F total i , F total j , T total i , andT total j whicharethesumoftheresultingforces(externalandinternal)appliedtothe DOFs- i and- j ofthesystem. The nonlinear restoring force of the “joint” element shown in Fig. 4.1, can be obtained by subtracting the corresponding inertia forces of each DOFs from the sum of the resulting forces appliedtothe“joint”elementofthesystem. For illustration of the methodology, consider a 4- DOFs system shown in Fig. 4.2, where the systemhastwolumpedmasses,m 1 andm 2 ,withrotationalmassmomentsofinertias,I 1 andI 2 , and it is subjected to external forces F 1 (t) and F 2 (t), and torques T 1 (t) and T 2 (t). The masses have axial motion and rotation about their axes, which are uncoupled. The system consists of linear and torsional elastic springs k x and k θ , linear and torsional viscous dampers c x and c θ , andanonlinearelementg(x, ˙ x,θ, ˙ θ,p),whererestoringforcecharacteristicsisafunctionofthe 67 2 2 I m 2 θ 2 x 1 1 I m 1 θ 1 x ) , , , , ( p x x g θ θ & & 2 2 2 2 θ θ c k c k x x , , 2 2 2 2 θ θ c k c k x x , , 3 3 3 3 θ θ c k c k x x , , 2 2 F T ) , , , ( θ θ & & x x r 1 1 F T 1 1 1 1 θ θ c k c k x x , , 1 2 1 2 1 2 1 2 x x x x x x x x x x where θ θ θ θ θ θ & & & & & & − = − = − = − = , , : Figure4.2: Mathematicalmodelofadiscrete4- DOF systemwithanonlinear“joint”element. system states and set of element-specific parameters p. The nonlinear element is connected to thesystemthroughsetofelasticsprings. Theequations-of-motionforthe 4- DOFssystemshowninFig.4.2aregovernedby: m 1 ¨ x 1 +k x 1 x 1 +c x 1 ˙ x 1 +r 1 (x, ˙ x,θ, ˙ θ) = F 1 (t) (4.2) m 2 ¨ x 2 +k x 2 x 2 +c x 2 ˙ x 2 −r 1 (x, ˙ x,θ, ˙ θ) = F 2 (t) (4.3) I 1 ¨ θ 1 +k θ 1 θ 1 +c θ 1 ˙ θ 1 +r 2 (x, ˙ x,θ, ˙ θ) = T 1 (t) (4.4) I 2 ¨ θ 2 +k θ 2 θ 2 +c θ 2 ˙ θ 2 −r 2 (x, ˙ x,θ, ˙ θ) = T 2 (t) (4.5) where x 1 and x 2 are the displacement of the masses, θ 1 and θ 2 are the angle of rotation of the masses,andr 1 andr 2 arethenonlinearrestoringforces. In general, the nonlinear restoring forces are functions of the system states, where in this example,thenonlinearrestoringforcesofthe2- DOF “joint”elementare: 68 r 1 (x, ˙ x,θ, ˙ θ) = g(x, ˙ x,θ, ˙ θ,p)+v 1 (x)+u 1 (˙ x) (4.6) r 2 (x, ˙ x,θ, ˙ θ) = g(x, ˙ x,θ, ˙ θ,p)+v 2 (θ)+u 2 ( ˙ θ) (4.7) where x (= x 2 − x 1 ) is the relative axial displacement of the two masses, ˙ x (= ˙ x 2 − ˙ x 1 ) is the relative linear velocity, θ (= θ 2 −θ 1 ) is the angle of the relative rotation, ˙ θ (= ˙ θ 2 − ˙ θ 1 ) is therelativeangularvelocity,v 1 andv 2 arefunctionsdescribingtheforcesoflinearandtorsional elastic springs, u 1 u 2 are functions describing the linear and torsional damping forces, and g is afunctiondescribingthenonlinearforcesofthe“joint”element. The nonlinear restoring forces of the 2- DOF “joint” element in the 4- DOF system can be obtained by subtracting the inertia forces from the sum of the resulting forces applied to the massesofthe“joint”element,whichare: r 1 (x, ˙ x,θ, ˙ θ) = (F 1 (t)−k x 1 x 1 −c x 1 ˙ x 1 )−m 1 ¨ x 1 (4.8) r 2 (x, ˙ x,θ, ˙ θ) = (T 1 (t)−k θ 1 θ 1 −c θ 1 ˙ θ 1 )−I 1 ¨ θ 1 (4.9) 4.1.2 ModelingtheNonlinearNonconservativeRestoringForces Rather than attempting to derive analytical expression for each components of the nonlinear restoringforces(g,u i ,andv i inEqs.(4.6)and(4.7)),experiencehasshown(Masrietal.,2006), it is a more efficient approach to represent the nonlinear restoring forces with nonparametric data-based models. Figure 4.3 shows inputs/outputs schematic of a generic reduced-order data- based model representation of a nonlinear nonconservative restoring forces for a 2- DOF “joint” 69 Data-Based Model-Free 2DOF Element ) (t x ) , , , ( 1 θ θ & & x x r ) (t x & ) (t θ ) , , , ( 2 θ θ & & x x r ) (t θ & Figure4.3: Schematicofdata-basedmodel-freerepresentationofthenonlinearnonconservative restoringforcesforthe2- DOF “joint”element. element, where the nonlinear nonconservative restoring forces are modeled as a mathematical functionofthe“joint”relativestates. Two classes of nonparametric data-based models have been investigated throughout of this study for modeling the nonlinear restoring forces of the 2- DOF nonlinear “joint” components, whichare: • Polynomial-basis model, where the nonlinear restoring forces are represented by a linear combinationofapowerseriesexpansionofthesystemstates, • Artificial neural network, where a mathematical black-box model is used to represent the nonlinearrestoringforces. 4.1.3 Polynomial-BasisModel The polynomial-basis model is a nonparametric model, were the restoring force is defined by a linearcombinationofthepower-seriesexpansionofthesystemstates. Forthenonlinear“joint”elementmodelshowninFig.4.2anditsrestoringforcesdescribed by Eqs. (4.6) and (4.7), the number of DOFs representing the “joint” is 2 and the number of 70 required states for the models is 4. The polynomial-basis model representing the nonlinear restoringforces,foreach DOF wouldbe: r 1 (x, ˙ x,θ, ˙ θ,p)≈ ˆ r 1 (x, ˙ x,θ, ˙ θ) = m X i=0 n X j=0 o X k=0 p X l=0 a ijkl (x i ˙ x j θ k ˙ θ l ) (4.10) r 2 (x, ˙ x,θ, ˙ θ,p)≈ ˆ r 2 (x, ˙ x,θ, ˙ θ) = m X i=0 n X j=0 o X k=0 p X l=0 b ijkl (x i ˙ x j θ k ˙ θ l ) (4.11) wherem,n,o,andp,areintegernumbersindicatingthemaximumorderofeachstate,a ijkl and b ijkl are the unknown coefficients of the basis, and ˆ r i is the estimate of the nonlinear restoring force,for DOF- i. Thenumberofunknowncoefficientsforthepolynomial-basismodeldependsonthelargest powerofeachstateandnumberofstates. Forthe2- DOFpolynomial-basismodel(Eqs.(4.10)and (4.11)),thenumberoftheunknowncoefficientsforeach DOFis(m+1)(n+1)(o+1)(p+1). For example, if the powers m, n, o, and p are set to 3, then ˆ θ(x, ˙ x) would consists of 512 unknown terms(ˆ q 1 and ˆ q 2 eachwouldhavetotalof256coefficients)intheformof: ˆ r 1 (x, ˙ x,θ, ˙ θ) = a 0000 (x 0 ˙ x 0 θ 0 ˙ θ 0 )+a 0001 (x 0 ˙ x 0 θ 0 ˙ θ 1 )+...+ (4.12) a 1012 (x 1 ˙ x 0 θ 1 ˙ θ 2 )+...+a 3333 (x 3 ˙ x 3 θ 3 ˙ θ 3 ) ˆ r 2 (x, ˙ x,θ, ˙ θ) = b 0000 (x 0 ˙ x 0 θ 0 ˙ θ 0 )+b 0001 (x 0 ˙ x 0 θ 0 ˙ θ 1 )+...+ (4.13) b 2313 (x 2 ˙ x 3 θ 1 ˙ θ 3 )+...+b 3333 (x 3 ˙ x 3 θ 3 ˙ θ 3 ) The unknown coefficients, a ijkl and b ijkl , can be identified by applying standard nonlinear least-squaresestimationmethods(Ljung,1999;Mendel,1995)totheaboveequations. 71 4.1.4 ArtificialNeuralNetwork(ANN)Model Artificial neural networks are a powerful nonparametric modeling technique, inspired by the human brain’s biological neural network architecture, that one capable of modeling and identi- fication variety of complex nonlinear systems (Dayhoff, 1990). The application of neural net- workmodelingapproachesinnonlinearsystemidentificationandstructuraldynamicshavebeen demonstrated by several researchers (Bani-Hani and Ghaboussi, 1998; Kosmatopoulos et al., 2001; Masri et al., 1992). The neural network modeling approaches are based on “black-box” mathematical modeling, without incorporating any physical characteristics of the system, and by only providing the inputs and outputs measurements of the system. The main limitation of ANNistheabsenceofthephysicalcharacteristicsofthesystemintheidentificationmodel. For particular applications, where the physics of the system is not as important as the performance ofthemodel,theANNwouldbetheidealmodelingtechnique. Thearchitectureandtrainingalgorithmsaretwoimportantmeasuresinmodelinganysystem withneuralnetworks,whicharecontrolledbytheapplicationtypeandtheproblemcharacteris- tics(Haganetal.,1995). NetworkArchitecture In the data-based modeling approach, the restoring forces of the nonlinear “joint” element (Fig. 4.1) are modeled as a nonlinear function of the system states. Therefore, the neural net- workarchitectureselectedformodelingthenonlinearrestoringforcesofthe2- DOF“joint”com- ponent (Fig. 4.2), is a multilayer feedforward network, which is the recommended architecture, 72 for function approximation applications, using the neural networks (Demuth and Beale, 2005; Haganetal.,1995). Amultilayerfeedforwardneuralnetworkiscapableofapproximatinganyfunction,provided there are enough neurons in hidden layers (Hornik et al., 1989). Moreover, it has been mathe- maticallyproventhatatwo-layer(aninputlayerandahiddenlayer)feedforwardnetworkwitha sigmoidfunctioninthefirstlayer,andalinearfunctioninthesecondlayer,providedasufficient number of neurons is used in the hidden layer, would approximate any function (Hagan et al., 1995). The linear function of the output layer enables the network of generalization beyond the inputrangeoftheinputsignals,whereasthesigmoidtransferfunctioninthehiddenlayerallows thenetworktolearnthenonlinearaswellaslinearrelationbetweentheinputandoutputvectors (DemuthandBeale,2005). Theschematicofthenetworkarchitectureformodelingthenonlinearrestoringforcesofthe nonlinear 2- DOF “joint” system, is shown in Fig. 4.4. The number of neurons in the input layer dependsonthenumberofrelativestatesinthemodel,whereasthenumberoftheneuronsinthe outputlayerdependsonthenumberofthe DOFsofthe“joint”. Themathematicalmodelofthedesignedtwo-layerfeedforwardneuralnetworkis: {a 1 } l×1 = tanh([W 1 ] l×2n 1 ×{P} 2n 1 ×1 | {z } inputs +{b 1 } l×1 ) (4.14) {T} n 1 ×1 | {z } outputs = [W 2 ] n 1 ×l ×{a 1 } l×1 +{b 2 } n 1 ×1 (4.15) 73 Figure 4.4: Schematic of the neural network architecture for modeling the nonlinear restoring forcesofthecouplednonlinearsystemshowninFig.4.2. where{a 1 } is the hidden layer output,{P} = [x, ˙ x,θ, ˙ θ] T is the input vector constructed from therelativestatesofthe“joint”,the{T} = [r 1 ,r 2 ] T istheoutputvectorcontainingthenonlinear restoringforcesofthesystem,n 1 isthenumberof DOFsofthenonlinear“joint”element(which in this study is 2), l is the number of neurons in the hidden layer,W 1 andW 2 are the network weights,andb 1 andb 2 arethenetworkbiases. NetworkTraining The backpropagation training algorithm was used to train the neural network model (Eq. 4.15), where the weights and biases values are updated based on Levenberg-Marquardt optimization (Bayesianregularization)method,whichallowsthetrainednetworktogeneralizewell(Demuth andBeale,2005;Haganetal.,1995). 74 Forimprovingthetrainingperformance,thetrainingwasperformedinseveralstages(Masri et al., 1992). First, the weights and biases of the network were initialized with random values, then part of the data were fed to the network and trained the network to reach a relatively high errortolerancelevel(thenetworkperformanceindicatorforerrortolerancelevelwasnormalized mean-square error), thereafter, the values of the weights and biases which were obtained by the first training were used as an initial values for the second stage of training. In the second stage, the number of training data points was increased to twice as much as the first training data set, with the error tolerance level set to half of the first stage. This incremental training procedure was repeated till all the data points were fed to the network in the last stage. The optimum valuesoftheweightsandbiasesattheendofthelaststageofthetrainingwasconsideredasthe optimumvaluesofthetrainednetwork. However, there is a possibility that the obtained weights and biases were not the global optimum values for the trained network with the given data sets. In order to be certain that the global optimum values of the network parameters have been obtained, the incremental training procedure was repeated several times (about 5 repetitions) with different initial random values forthenetworkparameters. Ifthevaluesoftheparametersafterallthereparationsremainsclose, thenthosevaluesareconsideredastheoptimumparametersvaluesforthetrainednetwork. It is worth noting that another data manipulation procedure that could improve the network training performance and generalization capability, is to normalize the input and output data sets,priortothenetworktrainingprocedure,tohavearangeof±1,andzeromean(Demuthand Beale,2005;Haganetal.,1995). Thenormalizationprocedurecanbeintegratedintotheneural network architecture, by modifying the weights and biases, without changing the mathematical 75 Figure 4.5: Mathematical model of nonlinear coupled 2- DOF system used to generate the syn- theticdatasets. formulation of the model. The integration of the normalization process, into the mathematical modeloftheneuralnetworkarchitecture,isdiscussedinAppendixA. 4.2 Simulationofa 2-DOFCoupledNonlinear“Joint”Component 4.2.1 MathematicalModel In order to illustrate and calibrate the methods under discussion, consider the example of finite elementmodelshowninFig.4.5. Thistwo-dimensionalmodel(linearmotionin xaxisandrota- tionθ about thex axis) consists of 4 massesm i , with moment of inertiaI m i , i = 1,2,3,4, that areinterconnectedwithelasticelementsbetweenmassesm 1 - m 2 andm 3 - m 4 ,andwithanonlin- ear element between m 2 - m 3 consisting of a gap element in both directions with a clearance of d x andd θ ,andacouplingelementdefinedmathematicallyasafunctionofsystemrelativestates (g(x, ˙ x,θ, ˙ θ))informofEq.(5.54). The coupling element, g(x, ˙ x,θ, ˙ θ), adds a nonlinear force in either direction when the rel- ative displacement of the other direction, exceeds the gap size, in the gap element. The combi- nation of the coupling element and the gap element, emulates a nonlinear 2- DOF “joint” com- ponent, where the friction force is a function of the relative velocity and position. Note that the 76 nonlinearcouplingforceterm,isaddedtothesystemwhenboth DOFsareinmotion. Figure4.6 shows phase-plots of the nonlinear “joint” model responses, when the system was subjected to broadbandrandomexcitations. Thefunctionaldependenceofg onitsstatevariableswaschosentobe: g(x, ˙ x,θ, ˙ θ) =                                −k 0x ˙ xθ, if|x| ≥dx &|θ|<d θ ; −k 0θ ˙ θx, if|x|<dx &|θ| ≥d θ ; −k 0x ˙ xθ−k 0θ ˙ θx, if|x| ≥dx &|θ| ≥d θ ; 0, if|x|<dx &|θ|<d θ . (4.16) TrainingDataSets For an accurate identification of nonlinear dynamical systems, it is essential that the identifica- tion data sets provide a complete representation of the nonlinear characteristics of the system (IoannouandSun,1996;Kerschenetal.,2006;Masrietal.,2006;SmythandPei,2000;Tasbih- gooetal.,2006). Therefore,thedatasetsselectedfortheidentificationofthesimulationmodel, weredatasetsofthesystemresponseswhensubjectedtobroadbandrandomexcitations,inorder tocaptureallthemodesofthesystemwithintheexcitationfrequencyrange. However, for capturing the coupling effects of the nonlinear component through the identi- ficationprocess,itwasnecessarythatthedatasetsincorporatethecouplingeffectsofthe DOFs, because the identified model depends on the input and output data. Therefore, three data sets wererequired tocapturethecorrectbehaviorofthesystem. 77 5 10 15 20 25 −1000 0 1000 Time (sec) (a) Axial RF (lbf) 5 10 15 20 25 −1000 0 1000 Time (sec) (b) Torsional RF (lbf−in) −0.5 0 0.5 −1000 −500 0 500 1000 Relative Displacement (in) (c) Axial Restoring Force (lbf) −0.5 0 0.5 −1000 −500 0 500 1000 Relative Rotation (rad) (d) Torsional Restoring Force (lbf−in) (a)Dataset1: Externalforceappliedtoxaxisonly. 5 10 15 20 25 −1000 0 1000 Time (sec) (a) Axial RF (lbf) 5 10 15 20 25 −1000 0 1000 Time (sec) (b) Torsional RF (lbf−in) −0.5 0 0.5 −1000 −500 0 500 1000 Relative Displacement (in) (c) Axial Restoring Force (lbf) −0.5 0 0.5 −1000 −500 0 500 1000 Relative Rotation (rad) (d) Torsional Restoring Force (lbf−in) (b)Dataset2: Externalforceappliedtoθ axisonly. 5 10 15 20 25 −1000 0 1000 Time (sec) (a) Axial RF (lbf) 5 10 15 20 25 −1000 0 1000 Time (sec) (b) Torsional RF (lbf−in) −0.5 0 0.5 −1000 −500 0 500 1000 Relative Displacement (in) (c) Axial Restoring Force (lbf) −0.5 0 0.5 −1000 −500 0 500 1000 Relative Rotation (rad) (d) Torsional Restoring Force (lbf−in) (c)Dataset3: Externalforceappliedtobothxandθ axes. Figure4.6: Syntheticdatasetsusedforidentificationofnonlinearcoupled“joint”. These data sets, which are shown in Fig. 4.6, represent the system response when there was excitation on each of the axes individually, and when the excitation forces were applied to both axes simultaneously. Moreover, the data were normalized across all the sets prior to the identificationprocessinordertohaveazeromeanwithamplitudeof±1. 78 4.2.2 IdentificationofData-BasedModel-FreeRepresentationofthe2DOFCou- pledNonlinearSystem Polynomial-BasisModel For identification of the synthetic data with the polynomial-basis model (Eq. (4.10)), the order of the model was set to 3 (m=n=o=p=3), thus resulting in total number of 512 (256 × 2) unknown parameters. Each set of parameters (256 coefficients) for q 1 and q 2 representations (Eqs.(4.10)and(4.11))whereidentifiedusingstandardnonlinearleast-squaresmethods(Ljung, 1999;Mendel,1995). Figure 4.7 shows the verification results of the polynomial-basis model identification with thedatasetsusedinidentification,whereFigs.4.7-(a)and-(b)showthephase-plotcomparison ofthesimulatedforce(solidline)andtheidentifiedforce(dash-dotline)fortherelativedisplace- mentvs. axialrestoringforceinthexdirection, andtherelativerotationandtorsionalrestoring forceintheθ direction,respectively. Inordertoutilizetheapplicationofthemodel-freerepresentationthroughapolynomial-basis model, new sets of data were generated with completely different broadband random excitation levels, whichhadthesamespectralcontentsas theidentificationdatasets. Thereafter, theiden- tifiedmodelwasusedtopredicttheresponseofthesystemandtocompareitwiththegenerated data sets, for validation purposes. Figure 4.8 shows the validation results, where parts (a) and (b) show the time-history comparison of the identified restoring forces (dash-dot line) with the simulated restoring forces (solid line), for the axial and torsional restoring forces respectively, parts(c)and(d)showthephase-plotcomparisonoftherelativedisplacementvs. axialrestoring 79 −0.4 −0.2 0 0.2 0.4 0.6 −800 −600 −400 −200 0 200 400 600 800 Relative Displacement (in) (a) Axial Restoring Force (lbf) −0.4 −0.2 0 0.2 0.4 −600 −400 −200 0 200 400 600 Relative Rotation (rad) (b) Torsional Restoring Force (lbf−in) Figure 4.7: Verification results of the polynomial-basis model identification with the synthetic datasetsusedinidentification. Parts(a)and(b)showthephase-plotcomparisonofthesimulated forces (solid line) and the identified forces (dash-dot line) for relative displacement vs. axial restoring force in the x direction, and the relative rotation vs. torsional restoring force in the θ direction,respectively. force, and the relative rotation vs. torsional restoring force for the x direction and θ direction, respectively. ArtificialNeuralNetwork The neural network architecture used for modeling the nonlinear restoring forces of the 2- DOF model (Fig. 4.2), was a two-layer feedforward network, with 40 neurons in the hidden layer. The number 40 was selected after several trial and error tests with lower and higher number of neurons,tilltheperformanceandconvergenceratesofthenetworkwereacceptable. Figure 4.9 shows the verification results of the neural network model identification with the data sets used in identification, where Figs. 4.9-(a) and -(b) show the phase-plot comparison of thesimulatedrestoringforce(solidline)andtheidentifiedrestoringforce(dash-dotline)forthe 80 10 11 12 13 14 15 16 17 18 19 20 −1000 −500 0 500 1000 Time (sec) (a) Axial Restoring Force (lbf) 15 16 17 18 19 20 21 22 23 24 25 −1000 −500 0 500 1000 Time (sec) (b) Torsional Restoring Force (lbf−in) −0.4 −0.2 0 0.2 0.4 0.6 −600 −400 −200 0 200 400 600 Relative Displacement (in) (c) Axial Restoring Force (lbf) −0.4 −0.2 0 0.2 0.4 0.6 −1000 −800 −600 −400 −200 0 200 400 600 800 1000 Relative Rotation (rad) (d) Torsional Restoring Force (lbf−in) Figure 4.8: Validation results of the polynomial-basis model with the synthetic data sets, which were not used in the identification process. Parts (a) and (b) show the time-history comparison of the identified restoring forces (dash-dot line) with the simulated restoring forces (solid line) for the axial and torsional restoring forces respectively, parts (c) and (d) show the phase-plot comparison of the relative displacement vs. axial restoring force, and the relative rotation vs. torsionalrestoringforceforthexdirectionandθ direction,respectively. relative displacement vs. axial restoring force in the x direction, and the relative rotation and torsionalrestoringforceintheθ directionrespectively. The validation data sets were used to test the generalization capability of the neural net- work model. Figure 4.10 shows the validation results, where parts (a) and (b) show the time- history comparison of the identified restoring forces (dash-dot line) with simulated restoring 81 −0.4 −0.2 0 0.2 0.4 0.6 −1000 −500 0 500 1000 Relative Displacement (in) (a) Axial Restoring Force (lbf) −0.4 −0.2 0 0.2 0.4 0.6 −1000 −500 0 500 1000 Relative Rotation (rad) (b) Torsional Restoring Force (lbf−in) Figure 4.9: Verification of the neural network identification with the synthetic data sets used in identification. Parts(a)and(b)showthephase-plotcomparisonofthesimulatedrestoringforces (solid line) and the identified restoring forces (dash-dot line) for relative displacement vs. axial restoring force in the x direction, and the relative rotation vs. torsional restoring force in the θ direction,respectively. forces (solid line) for axial and torsional forces respectively, parts (c) and (d) show the phase- plot comparison of the relative displacement vs. axial restoring force, and the relative rotation vs. torsionalrestoringforceforthexdirectionandθ direction,respectively. Theverificationandvalidationresultsillustratedabove,clearlyindicatethecapabilityofthe neural network models in accurately approximating data-based model-free representations of complexnonlinearnonconservativesystems. 4.3 ExperimentalStudies 4.3.1 TestSetupofa 2- DOFNonconservativeDissipative“Joint”System A test apparatus was designed to simulate the behavior of a nonlinear nonconservative dissipa- tive 2- DOF “joint” element, in order to utilize the application of the discussed algorithms for derivationofdata-basedmodel-freerepresentationsofsuchsystems. 82 12 13 14 15 16 17 18 19 −1000 −500 0 500 1000 Time (sec) (a) Axial Restoring Force (lbf) 16 17 18 19 20 21 22 23 24 −500 0 500 Time (sec) (b) Torsional Restoring Force (lbf−in) −0.8 −0.6 −0.4 −0.2 0 0.2 0.4 0.6 0.8 −1000 −500 0 500 1000 Relative Displacement (in) (c) Axial Restoring Force (lbf) −0.4 −0.2 0 0.2 0.4 0.6 −800 −600 −400 −200 0 200 400 600 800 Relative Rotation (rad) (d) Torsional Restoring Force (lbf−in) Figure 4.10: Validation results of the neural network identification with the synthetic data sets not used in identification. Parts (a) and (b) show the time-history comparison of the identified restoring force (dash-dot line) with simulated restoring force (solid line) for axial and torsional forcesrespectively,parts(c)and(d)showthephase-plotcomparisonoftherelativedisplacement vs. axialrestoringforce,andtherelativerotationvs. torsionalrestoringforce,forthexdirection andθ direction,respectively. Photographs of the completely assembled test and the nonlinear “joint” element are shown in Fig. 4.11. The labels indicates the major subcomponents of the test apparatus and directions of the relative motion within the “joint” element. The solid model design and the details of the nonlinear“joint”designareshowninFig.4.12. 83 Thetestsetupconsistsoftwocomputer-controlledelector-mechanicalservodrivesthatgen- erate external excitations in two independent directions. The motion of the drives is transferred throughashaft,withuniversaljointsateachendtoprovidedecoupledmotions,tothenonlinear “joint”element. Theappliedforcestothesystemaremeasuredthroughtwosetsofstraingauges mountedontheshaftinaxialandtorsionalconfigurations. Therelativemotions,intheaxialand rotational directions, of the “joint” are measured with four sets of optical encoders. Two linear andangularaccelerometersareusedtomeasuretheabsoluteaccelerationofthe“joint”. The data acquisition system included a DAQ-board, three counter-boards, controller, and a chassis, in order to have synchronized measurements. A pictorial diagram indicating the inter-connection of the main system components, including the mechanical assembly, excita- tionsources,instrumentationnetwork,andsensors,isprovidedinFig.4.13. Three different programs were developed for the discussed experimental test setup. These programs were: (a) a data acquisition program developed in LabVIEW programming environ- ment for real-time recording and analysis of the data; (b) a low level program written in “C” AFSOR Contractors Meeting 2005 – Analytical and Experimental Studies in Nonlinear System Modeling 2DOF Experimental Setup F T 2DOF Joint F=Applied Force M=Applied Torque Rotary Shaker Linear Shaker The 2DOF nonlinear joint test setup. The system consists of a linear and rotary shaker. The forces are applied to the joint through a set of universal joints at each end of the driving shaft. AFSOR Contractors Meeting 2005 – Analytical and Experimental Studies in Nonlinear System Modeling Nonlinear 2DOF Joint The 2DOF nonlinear joint consists of four rotary bearings and four linear bearings in order to provide uncoupled motion in the linear (x) and rotary (ө) directions. 1 2 1 2 Rotation Relative nt Dispalceme Relative θ θ θ − = = − = = x x x x1 ө1 x2 ө2 (a)Overview (b)Nonlinear2- DOF “joint” Figure4.11: Overviewofthe2- DOF nonlinear“joint”experimentaltestsetup. 84 (a) (b) Figure4.12: Solidmodelanddetailedviewofthe2- DOF nonlinearexperimentaltestsetup. Part (a)isthesolid modeldesignofthe test setup, and part (b) is the exploded view of the nonlinear 2- DOF “joint”. languageforsimulatingthebroadbandrandomexcitationfortheservomotors;and(c)acontrol programwritteninVisualBasicfortheservocontroller. There were some programming and practical challenges involved in the discussed pro- grammes, few can be pointed here which were: synchronization of the clocks on the counter- boardsandtheDAQ-board;mixsensors/signaldatacollection(digitalsignalsfromopticalcoun- ters and continuous signals from accelerometers and strain gages); over-writing the servo con- trollersinordertofollowrandomsignals;theprocessautomation,etc. 4.4 Application In this section the identification results of the experimental data sets for the 2- DOF nonlinear “joint”testsetupispresentedforthepolynomial-basisandneuralnetworkmodel. Inthissection description of the identification (training) data set is discussed, followed by the identification results for both methods. The identification methods are performed in the verification and val- idation stages. In the verification stage, a set of data is used in the identification task, and then 85 Endevco DC Amplifier Model 136 National Instruments NI SCXI 1000 Chassis NI SCXI 1143 Low Pass NI SCXI 1520 Strain Gage National Instruments PXI 1042 Chassis PXI 6602 Counter Module PXI 6052E DAQ board 2 Set Strain Gages measuring - Torque force - Axial force Parker 406 series Linear servo motor 2 Endevco Accelerometer Nonlinear 2DOF “Joint” s - Linear: Model 7290A-10 Analog Digital - Rotary: Model 7302BM4 Parker 6k6 motion controller Parker Gemini Servo Drives National Instruments NI PCI 6509 Digital I/O User Interface in LabVIEW Parker BE series Rotary servo Two DOF Motion - Linear -Rotation NI Terminal Block 2 Optical Encdoers (USDigital) Linear & Rotray + Figure4.13: Systemarchitectureandwiringdiagramoftestsetup. anothersetofdata,withsimilardynamiccharacteristicstotheidentificationdataset,isusedfor thevalidationoftheidentifiedmodelfromthefirststage. 4.4.1 IdentificationDataSets Theexperimentaldatasetsusedforidentification,correspondedtomeasurementsobtainedfrom the system’s responses when it was subjected to broadband random excitations. Because of the couplingeffectsbetweenthe DOFsonthe“joint”componentintheexperimentaltestsetup,three different data sets were used simultaneously for identification process, in order to capture the correctphysicsofthesystem. ThedatasetsusedforidentificationareshowninFig.4.14,where part(a)istheresponseofthe“joint”elementwhenitwassubjectedtobroadbandexcitationonly 86 on thex axis, part (b) iswhen there was broadband excitation only on theθ axis, and part (c)is the response of the system when there were two uncorrelated broadband random excitations on bothxandθ axesatthesametime. 4.4.2 Polynomial-BasisModel IdentificationandVerification For modeling the 2- DOF nonlinear “joint” with the polynomial-basis model, the order of the model(Eqs.(4.10)and(4.11))wassetto3(m=n=o=p=3),inwhichthetotalnumberofunknown parameters would be 512 (2×256). The values of the unknown parameters were obtained by usingstandardnonlinearleast-squaresidentificationmethods(Ljung,1999;Mendel,1995). Figure 4.15 shows the verification results of the identified polynomial-bases model, where parts (a) and (b) show the time-history comparison of the measured resorting forces (solid line) with the identified restoring forces (dash-dot line) for the axial and torsional directions, respec- tively. Parts (c) and (d) show the phase-plot comparison of the measured restoring force (solid line) and the identified restoring force (dash-dot line) for the relative displacement vs. the axial restoringforceinthexdirection,andtherelativerotationvs. thetorsionalrestoringforceinthe θ direction,respectively. Validation For utilizing the data-based model-free representation of the nonlinear “joint” element with the polynomial-basis model approach, another broadband random data set was used to validate the 87 5 10 15 20 25 −50 0 50 Time (sec) (a) Axial RF (lbf) 5 10 15 20 25 −50 0 50 Time (sec) (b) Torsional RF (lbf−in) −0.2 −0.1 0 0.1 0.2 −60 −40 −20 0 20 40 60 Relative Displacement (in) (c) Axial Restoring Force (lbf) −0.2 −0.1 0 0.1 0.2 −60 −40 −20 0 20 40 60 Relative Rotation (rad) (d) Torsional Restoring Force (lbf−in) (a)Dataset1: Externalforceappliedtothexaxisonly. 5 10 15 20 25 −50 0 50 Time (sec) (a) Axial RF (lbf) 5 10 15 20 25 −50 0 50 Time (sec) (b) Torsional RF (lbf−in) −0.2 −0.1 0 0.1 0.2 −60 −40 −20 0 20 40 60 Relative Displacement (in) (c) Axial Restoring Force (lbf) −0.2 −0.1 0 0.1 0.2 −60 −40 −20 0 20 40 60 Relative Rotation (rad) (d) Torsional Restoring Force (lbf−in) (b)Dataset2: Externalforceappliedtotheθ axisonly. 5 10 15 20 25 −50 0 50 Time (sec) (a) Axial RF (lbf) 5 10 15 20 25 −50 0 50 Time (sec) (b) Torsional RF (lbf−in) −0.2 −0.1 0 0.1 0.2 −60 −40 −20 0 20 40 60 Relative Displacement (in) (c) Axial Restoring Force (lbf) −0.2 −0.1 0 0.1 0.2 −60 −40 −20 0 20 40 60 Relative Rotation (rad) (d) Torsional Restoring Force (lbf−in) (c)Dataset3: Externalforceappliedtothebothxandθ axes. Figure 4.14: Experimental data sets used in the identification of the nonlinear coupled 2- DOF “joint”. 88 identifiedmodel. Inordertoobtainthevalidationdatasets,anothersetofexperimentalmeasure- ments was conducted, with uncorrelated broadband random excitations, with a different excita- tion from what was used in the identification stage, but with similar dynamic characteristics. Figure 4.16 shows the validation results of the polynomial-basis model with the experimental data sets not previously used in identification. Parts (a) and (b) show the time-history compari- son of the identified force (dash-dot line) with the measured force (solid line) for the axial and torsional forces respectively. Parts (c) and (d) show the phase-plot comparison of the relative displacement vs. the axial restoring force, and the relative rotation vs. the torsional restoring force,forthexandθ directions,respectively. 4.4.3 ArtificialNeuralNetwork IdentificationandVerification The neural network model used to represent the experimental “joint” component, had the same network architecture discussed in Section 4.1.4 and shown in Fig. 4.4, with 40 neurons in the hidden layer. The optimum weights and biases of the network were obtained through the back- propagationalgorithm,asdiscussedinSection4.1.4. Figure4.17showstheverificationresultsfortheneuralnetworkidentificationwiththeexper- imentaldatasets,whereparts(a)and(b)showthetime-historycomparisonoftheidentifiedforce (dash-dotline)withthemeasuredforce(solidline)fortheaxialandtorsionalforcesrespectively. Parts(c)and(d)showthephase-plotcomparisonoftherelativedisplacementvs. theaxialrestor- ing force, and the relative rotation vs. the torsional restoring force, for the x and θ directions, respectively. 89 10 12 14 16 18 20 22 24 26 28 30 −60 −40 −20 0 20 40 60 Time (sec) (a) Axial Restoring Force (lbf) 10 12 14 16 18 20 22 24 26 28 30 −60 −40 −20 0 20 40 60 Time (sec) (b) Torsional Restoring Force (lbf−in) −0.2 −0.15 −0.1 −0.05 0 0.05 0.1 0.15 0.2 −60 −40 −20 0 20 40 60 Relative Displacement (in) (c) Axial Restoring Force (lbf) −0.1 −0.05 0 0.05 0.1 −60 −40 −20 0 20 40 60 Relative Rotation (rad) (d) Torsional Restoring Force (lbf−in) Figure 4.15: Verification results of the polynomial-basis model identification with the experi- mentaldatasetsusedinidentification. Parts(a)and(b)showthetime-historycomparisonofthe measured resorting forces (solid line) with the identified restoring forces (dash-dot line) for the axial and torsional directions, respectively. Parts (c) and (d) show the phase-plot comparison of themeasuredrestoringforce(solidline)andtheidentifiedrestoringforce(dash-dotline)forthe relativedisplacementvs. theaxialrestoringforceinthexdirection,andtherelativerotationvs. thetorsionalrestoringforceintheθ direction,respectively. 90 10 12 14 16 18 20 22 24 26 28 30 −60 −40 −20 0 20 40 60 Time (sec) (a) Axial Force (lbf) 10 12 14 16 18 20 22 24 26 28 30 −60 −40 −20 0 20 40 60 Time (sec) (b) Torsional Force (lbf) −0.2 −0.15 −0.1 −0.05 0 0.05 0.1 0.15 0.2 −60 −40 −20 0 20 40 60 Relative Displacement (in) (b) Axial Force (lbf) −0.1 −0.05 0 0.05 0.1 −60 −40 −20 0 20 40 60 Relative Rotation (rad) (d) Torsional Force (lbf) Figure4.16: Validationofthepolynomial-basis modelidentification withthe experimentaldata sets not used in identification. Parts (a) and (b) show the time-history comparison of the identi- fied force (dash-dot line) with the measured force (solid line) for the axial and torsional forces respectively. Parts (c) and (d) show the phase-plot comparison of the relative displacement vs. the axial restoring force, and the relative rotation vs. the torsional restoring force, for the x and θ directions,respectively. 91 2 4 6 8 10 12 14 16 18 20 −60 −40 −20 0 20 40 60 Time (sec) (a) Axial Restoring Force (lbf) 2 4 6 8 10 12 14 16 18 20 −60 −40 −20 0 20 40 60 Time (sec) (b) Torsional Restoring Force (lbf−in) −0.2 −0.15 −0.1 −0.05 0 0.05 0.1 0.15 0.2 −60 −40 −20 0 20 40 60 Relative Displacement (in) (c) Axial Restoring Force (lbf) −0.2 −0.15 −0.1 −0.05 0 0.05 0.1 0.15 0.2 −60 −40 −20 0 20 40 60 Relative Rotation (rad) (d) Torsional Restoring Force (lbf−in) Figure4.17: Verificationresultsfortheneuralnetworkidentificationwiththeexperimentaldata sets used in identification. Parts (a) and (b) show the phase-plot comparison of the measured force (solid line) and the identified force (dash-dot line) for relative displacement vs. axial restoring force in the x direction, and the relative rotation vs. torsional restoring force in the θ direction,respectively. 92 Validation For demonstrating the generalization capability of the neural network model representation of the nonlinear 2- DOF “joint” element, the identified model was used with the validation data set (as described in Section 4.4.2) to predict the nonlinear nonconservative forces. Figure 4.18 shows the validation results for the neural network identification with the experimental data set not used in the identification, where parts (a) and (b) show the time-history comparison of the identified force (dash-dot line) with the measured force (solid line) for the axial and torsional forces, respectively. Parts (c) and (d) show the phase-plot comparison of the relative displace- ment vs. the axial restoring force, and the relative rotation, vs. the torsional restoring force for thexandθ directions,respectively. 4.5 Discussion Derivation and evaluation of the data-based model-free representations of a generic nonlinear nonconservative 2- DOF “joint” system were discussed throughout this chapter. The modeling techniqueswerefirstcalibratedwithsyntheticdatasetsandthenvalidatedthroughexperimental data sets. The results in Sections 4.2 and 4.3 demonstrated the capabilities of these modeling tools in capturing the behavior of highly nonlinear systems. However, in utilizing these tools, there are some adjustable parameters and implementation issues, that the user should take into consideration prior to using these tools. Some of these important points, which may influence theresultsandtheimplementationofthetools,arediscussedbelow. 93 2 4 6 8 10 12 14 16 18 20 −60 −40 −20 0 20 40 60 Time (sec) (a) Axial Restoring Force (lbf) 2 4 6 8 10 12 14 16 18 20 −60 −40 −20 0 20 40 60 Time (sec) (b) Torsional Restoring Force (lbf−in) −0.2 −0.15 −0.1 −0.05 0 0.05 0.1 0.15 0.2 −60 −40 −20 0 20 40 60 Relative Displacement (in) (c) Axial Restoring Force (lbf) −0.2 −0.15 −0.1 −0.05 0 0.05 0.1 0.15 0.2 −60 −40 −20 0 20 40 60 Relative Rotation (rad) (d) Torsional Restoring Force (lbf−in) Figure 4.18: Validation results for the neural network identification with the experimental data sets not used in identification. Parts (a) and (b) show the time-history comparison of the identi- fied force (dash-dot line) with the measured force (solid line) for the axial and torsional forces, respectively. Parts (c) and (d) show the phase-plot comparison of the relative displacement vs. the axial restoring force, and the relative rotation vs. the torsional restoring force, for the x and θ directions,respectively. 94 The order of the polynomial-basis model and the architecture of the neural network are two important factors in the utilizations of the tools. These factors define the number of unknown parameters in the models, which would cause over- and under-parametrization effects in the identification process (Demuth and Beale, 2005; Hagan et al., 1995; Smyth, 1998; Tasbihgoo et al., 2006). In order to have a robust model, it is necessary to select the optimum values for theseproblem-dependentfactors. Inthisstudy,thevalueoftheoptimumorderofthepolynomial-basismodel,andthenumber of neurons, in the hidden layer of the neural network model, were obtained through trial and error of the identification process with the synthetic data, and then through the repetition of the process with the experimental data sets. Because of the mathematical and physical similarities between the simulation model and the experimental test setup, the order and neuron numbers, thatwereobtainedwiththesyntheticdatasets,remaindthesameintheidentificationprocessof theexperimentaldatasets. Another important issue that users should be aware of, is the selection of the identification training data sets. Due to the data-based modeling nature of these tools, the training data sets areverycriticalinthederivationofanaccuratemodel. Itisnecessarythatthetrainingdatasets provide sufficient information in regard to the characteristics of the system, with respect to the spectralcontentandfrequenciesofinterest,inordertocapturethedesiredmodesofthesystem. For instance, the data sets used for modeling the 2- DOF nonlinear coupled “joint” system (Fig.4.11),consistedofmeasurementsoftheresponseofthesystemtobroadbandrandomexci- tations, in order to excite all the nonlinear modes in the frequency range of interest. Moreover, three different data sets were used simultaneously in the identification and the training process 95 in order to capture the coupling effects of the “joint” system; however, if only one set of the data, for example the one that both DOFs were excited (Fig. 4.6-(c)), was only used in the iden- tification stage, then the models would have failed in the generalization tests, due to the lack of informationaboutthecouplingofthe DOFsinthetrainingdataset. Theknowledgeofthesystemmassmatrixisnotcrucialforthediscussedmodelingandiden- tification techniques. If the value of the mass matrix is not available, the absolute acceleration data will be added to the list of the inputs, and the output data will be the sum of the applied forces from the system, instead of only the nonlinear restoring forces (Fig. 4.3), and the mass and moment of inertia terms will be added to the list of unknown parameters in the polynomial basismodel (Eqs.(4.10)and (4.11). However, the two-layer feedforward architecture (Fig. 4.4) remainsthesamefortheneuralnetworkmodel. Despite the excellent capability of these models for the identification and modeling non- linear systems, it is important to point out that these models rely heavily on the nature of the input/outputdatasets. Theidentifiedmodelsarevalidwithintherangeofthetrainingdatasets. The data-based models may result in significant errors, if the models are subjected to inputs incorporating signals beyond the training data sets characteristics, with respect to spectral con- tentortheamplituderange. The user should be aware that, although the modeling tools under discussion demonstrated an adequate capability in capturing and modeling the response of a generic nonlinear, noncon- servative“joint”system,theprocedureswillremainthesameforsystemswithdifferenttypesof nonlinearities,buttheparameters(suchastheorderinthepolynomial-basismodelorthenumber of neurons in the neural network model), as well as the training data sets may be significantly 96 different. It will be cost-effective and time-efficient, if the identification tools were tuned with syntheticdatasetsfirst,andthenappliedtoexperimentaldatasets. 4.5.1 ComparisonoftheTwoNonparametricMethods The modeling techniques discussed in this study are two completely different approaches from themathematicalformulationandimplementationaspects; however, bothmethodswouldresult in data-based model-free representations of nonlinear systems. The selection of which method to use is based on the identification application and objectives. For example, for a fast model identification, the polynomial-basis model would be a better choice, or for a high-fidelity rep- resentation the neural network approach would be the ideal method. In the simulation phase of this study, the neural network model took a longer duration in the training stage in comparison to the identification stage of the polynomial-basis model (about 180 times longer), although the performance(accuracy)ofthetrainedneuralnetworkmodelwasmuchhigherthantheidentified polynomial-basismodel(about 16timeslowerpercentageerrorinthevalidationtestcases). 4.6 ChapterSummary In this Chapter, the application of two nonparametric modeling approaches, one using a polynomial-basis model and the other based on artificial neural networks, were discussed and evaluated for modeling nonlinear, nonconservative, dissipative systems. The tools use input/output data of the system to obtain data-based, model-free computational models of com- plexnonlinearsystems,whichsubsequentlycanbeusedincomputationalenginesforsimulation 97 studies, condition assessment, and damage detection applications. The methods were first cali- bratedthroughsimulationstudiesandthereafterappliedtoexperimentaldatasetscorresponding toa2- DOF couplednonlinear“joint”system. Thegeneralizationcapabilityofthemethodswere demonstratedthroughvalidationtestsusingsyntheticandexperimentaldatasets. 98 Chapter5 DevelopmentofReduced-Order ComputationalModelsforNonlinear “JOINT” The main purpose of this chapter is to formulate the implementation of a reduced-order com- putationalmodelofamulti-degrees-of-freedom( MDOF)system, consistingofnonlinear“joint” elements, with a data-based model-free nonparametric representation of the nonlinear “joint” elements,whichisobtainedfromexperimentalmeasurements,asdiscussedinChapter4. Theimplementationofthemethodologyisdemonstratedbasedonneuralnetworkmodeling techniques, as a representative of nonparametric identification methods. The procedure may vary in details, if another nonparametric identification technique is used; however, the general conceptofthemethodologywillremainthesame. 99 It should be noted that, due to the complexity in the formulation, and for simplification in theillustrationofthemethodology,itisassumedthatthe MDOFsystemconsistsofonlyonenon- linear “joint” element. However, the formulation, and the implementation of the methodology, wouldremainessentiallythesame,ifthesystemconsistsofmultiplenonlinear“joint”elements. Inordertoaddthereduced-ordermodel-freerepresentationofthenonlinear“joint”element totheequations-of-motionofasystem,itisrequiredtofirstderivethedata-basedmodel-freerep- resentation of the nonlinear “joint” element through either synthetic or experimental data sets, and thereafter incorporate the reduced-order model into the linear set of equations-of-motion. After deriving the new set of nonlinear equations-of-motion of the system with the nonlinear “joint”element,thetime-historyresponseofthesystemcanbeobtainedthroughstandardnumer- icaltime-marchingtechniques. Thechapterisorganizedasfollows: Section5.1presentsthedetailsoftheimplementationof adata-basedmodel-freerepresentationofanonlinear“joint”element,inacomputationalmodel ofa MDOFsystem,Section5.2presentsthesimulationstudies,andSection5.3demonstratesthe applicationofthemethodologyinconjunctionwiththeexperimentaldatasets. 5.1 Formulation Thegoverningequations-of-motionforalinear n-degrees-of-freedom(n- DOF)systemis: [M] n×n {¨ x} n×1 +[C] n×n { ˙ x} n×1 +[K] n×n {x} n×1 ={F} n×1 (5.1) 100 where, [M], [C], and [K] are the system matrices representing the inertia, damping, and stiffness terms, respectively,{¨ x},{ ˙ x}, and{x} are the system state vectors of acceleration, velocity,anddisplacement,and{F}isthevectorofexternalforces. Assume that a massless nonlinear “joint” element is to be incorporated between two of the nodes of the system, for instance {x i } and {x j }, each with n 1 - DOFs, as shown in Fig. 5.1, where the maximum value of the n 1 is 6, which represents a node with 6- DOFs. Therefore, the linear assumption of the system is no longer valid, and the equations-of-motion of the system (Eq.(5.1))shouldbemodifiedtorepresentthenewnonlinearsystem. i i x , x & i M j j x , x & j M Figure 5.1: A nonlinear massless “joint” element is added to a linear model between nodes x i andx j ,eachwithn 1 - DOFs. 5.1.1 Data-BasedModel-FreeRepresentationofNonlinear“Joint”Element Inordertodemonstratethemethodology,theneuralnetworkarchitecture,asdiscussedinChap- ter4,isusedasarepresentativemodelingtechnique;howeverthederivationofthemethodology for other data-base model-free methods, such as polynomial-basis model, would be similar. In general, data-based model-free techniques, represent the nonlinear restoring forces of a “joint” element,throughtherelativestatesofthe“joint”. Theschematicofthedata-basedreduced-order 101 modelofanonlinear“joint”elementisshowninFig.5.2,wheretherelativestatesofthenonlin- ear “joint” are the input to the model and the nonlinear nonconservative restoring forces are the outputofthemodel. ) x , R(x p p & T M M M z y x z y x rf rf rf rf rf rf ] , , , , , [ } RF { p = T p p p pz py px z y x x x x x x x ] , , , , [ } x { p θ θ θ & & & & & & & = Non-linear Reduced-Order 6DOF Element T p p p pz py px z y x x x x x x x ] , , , , [ } x { p θ θ θ = Figure 5.2: Schematic of the reduced-order data-based model-free representation of a 6- DOF nonlinear “joint” element. The reduced-order model has the relative states of the “joint” as the inputandthenonlinearnonconservativerestoringforcesastheoutput. Assume that, the nonlinear “joint” element was placed between the nodes {x i } and {x j }, eachwithn 1 - DOFs,andstatevectorsdescribedby: {x i } = [x ix ,x iy ,x iz ,x iθx ,x iθy ,x iθz ] T (5.2) {x j } = [x jx ,x jy ,x jz ,x jθx ,x jθy ,x jθz ] T (5.3) The relative states of the nonlinear “joint” element in the system level model, can be repre- sentedbyanewvirtualnode{x p }as: {x p } n 1 ×1 = {x i } n 1 ×1 −{x j } n 1 ×1 (5.4) {˙ x p } n 1 ×1 = {˙ x i } n 1 ×1 −{˙ x j } n 1 ×1 (5.5) 102 Thenonlinearnonconservativerestoringforcesinducedbythenonlinear“joint”elementcan beexpressedby: {RF p } = [rf x ,rf y ,rf z ,rf Mx ,rf My ,rf Mz ] T (5.6) ThedetailsofderivingthenonlinearnonconservativerestoringforcesarediscussedinChapter4. For modeling the “joint” with the neural network architecture, a two-layer feedforward model,asdiscussedinSection4.1.4,withsigmoidfunctionsinthefirstlayerandlinearfunctions inthesecondlayer,isusedtoderivethedata-basedmodelofthenonlinear“joint”element. The inputandoutputoftheneuralnetworkmodelare: {P} =        {x p } n 1 ×1 {˙ x p } n 1 ×1        (2n 1 )×1 , {T} ={RF} n 1 ×1 (5.7) andthemathematicalmodelrepresentingthetwo-layerfeedforwardneuralnetworkwouldbe: {a 1 } l×1 = G 1 ([W 1 ] l×2n 1 {P} 2n 1 ×1 | {z } inputs +{b 1 } l×1 ) (5.8) {T} n 1 ×1 | {z } outputs = [W 2 ] n 1 ×l {a 1 } l×1 +{b 2 } n 1 ×1 (5.9) where [W 1 ] , [W 2 ], {b 1 }, and {b 2 } are the weights and biases of the network, G 1 (.) is the hyperbolic-tangent sigmoid function ( tanh),{P} is the network input,{a 1 } is the output of the first(hidden)layer,and{T}isthenetworkoutput. 103 Inordertoincorporatetheneuralnetworkmodelofthenonlinear“joint”element(Eqs.(5.8) and (5.9)) in to the equations-of-motion of n- DOF system (Eq. (5.1)), it is required to partition the weight matrix [W 1 ], and the input vector{P}, into submatrices and subvectors associated withonlythedisplacementandthevelocitystatesas: [W 1 ] l×2n 1 ×{P} 2n 1 ×1 = [w 1x ] l×n 1 [w 1˙ x ] l×n 1 ×        {x p } n1×1 {˙ x p } n1×1        (5.10) therefore,Eq.(5.8)canberewritteninpartitionedformas: {a 1 } l×1 = G 1     [w 1x ] l×n 1 [w 1˙ x ] l×n 1 ×        {x p } n1×1 {˙ x p } n1×1        +{b 1 } l×1     (5.11) Thepartitionedneuralnetworkmodelwouldbe: {a 1 } l×1 = G 1 [w 1x ] l×n 1 ×{x p } n1×1 +[w 1˙ x ] l×n 1 ×{˙ x p } n1×1 +{b 1 } l×1 (5.12) {T} n 1 ×1 = [W 2 ] n 1 ×l ×{a 1 } l×1 +{b 2 } n 1 ×1 (5.13) 5.1.2 Adding the Reduced-Order Model to the Equations-of-Motion of the Sys- tem The next step involves incorporating the reduced-order partitioned neural network model of the nonlinear “joint” element (Eqs. (5.12) and (5.13)) into the system equations-of-motions (Eq.(5.1)),inaformthat,whentheresponseofthesystemiscomputedinaniterativesolution, at every step of the solution, the values of the nonlinear restoring forces of the “joint” element 104 canbeobtainedbasedonthesystemcurrentstateatthatstep,andbeaddedtothesystemexternal forcesforthenextstep. Inordertoillustratethemethodologyofhowtoaddthereduced-ordermodeltothesystem, the partitioned neural network model is added to the system in several stages. Equations (5.14) and (5.15) show the stage numbers, along with their corresponding mathematical part of the neuralnetworkmodel. {a 1 } l×1 = G 1        [w 1x ] l×n 1 ×{x p } n1×1 | {z } 1 +[w 1˙ x ] l×n 1 ×{˙ x p } n1×1 | {z } 1 +{b 1 } l×1 | {z } 2        | {z } 3 (5.14) {T} n 1 ×1 =   [W 2 ] n 1 ×l ×{a 1 } l×1 +{b 2 } n 1 ×1 | {z } 4    | {z } 5 (5.15) where stage 1 is adding the relative states as a set of new virtual points to the system, stage 2 is the addition of the partitioned weights, [w 1x ] and [w 1˙ x ], and the bias,{b 1 }, of the first layer to the system matrices and computation of their product with the relative states, stage 3 is the evaluation of the functional value G 1 (.) for the results of the stage 2, stage 4 is the addition of the weight, [W 2 ], and the bias, {b 2 }, of the second layer to the system, and computation of theirproductwiththestage3results,andstage5istheadditionofthenonlinearrestoringforces generatedthroughstage4totheexternalforcesvectorofthesystem. The mathematical details of adding the neural network model to the system equations, at eachofthestagesdiscussedabove,areexplainedinthefollowingsections. Notethateverystage is organized as following: (1) description of the new states of the system resulting from that 105 stage; (2)introductionofnewsetsofvirtualpoints,ifneededatthestage; (3)descriptionofthe expanded mathematical portion of the neural network model to be incorporated to the system equations; and (4) the final form of the updated system matrices, the updated states, and the updatedforcevectorareshown. Stage1 Thefirststageistheadditionoftherelativestatesvector,{P},describedbyEqs.(5.4)and(5.5), to the equations-of-motion of the system (Eq. (5.1)). For adding the relative states equations to thesystem,bothequationsarerewritteninanequivalentmatrix-formas: [0] l×l | {z } [M (1) ] ×{0} l×1 | {z } { ¨ x (1) } + −[I] n 1 ×n 1 [I] n 1 ×n 1 [I] n 1 ×n 1 | {z } [C (1) ] ×                {˙ x i } n l ×1 {˙ x j } n 1 ×1 {˙ x p } n 1 ×1                | {z } { ˙ x (1) } (5.16) + −[I] n 1 ×n 1 [I] n 1 ×n 1 [I] n 1 ×n 1 | {z } [K (1) ] ×                {x i } n l ×1 {x j } n 1 ×1 {x p } n 1 ×1                | {z } { x (1) } ={0} ×1 | {z } { F (1) } The Eq. (5.16) can be incorporated into the system equations by simply adding the corre- spondingmatricesandvectorstothesystemcharacteristicmatricesandstatevectors. 106 Theupdatedsystemstatevectorswouldbe: {¨ x 1 } =        {¨ x} n×1 {0} n 1 ×1        s 1 ×1 , { ˙ x 1 } =        {˙ x} n×1 {˙ x p } n 1 ×1        s 1 ×1 , {x 1 } =        {x} n×1 {x p } n 1 ×1        s 1 ×1 and the updated system characteristic matrices with their corresponding updated state vectors wouldbe: M 1 ×{¨ x 1 } =     [M] n×n [0] n 1 ×n 1     s 1 ×s 1        {¨ x} n×1 {0} n 1 ×1        s 1 ×1 (5.17) C 1 ×{ ˙ x 1 } =     [C] n×n [e] n×n 1 [I] n 1 ×n 1     s 1 ×s 1        {˙ x} n×1 {˙ x p } n 1 ×1        s 1 ×1 (5.18) K 1 ×{x 1 } =     [K] n×n [e] n×n 1 [I] n 1 ×n 1     s 1 ×s 1        {x} n×1 {x p } n 1 ×1        s 1 ×1 (5.19) where s 1 (= n+n 1 ) is the order of the new system matrices, [ I ] r×r is the identity matrix of size ofr, [e] n×n 1 is a transition matrix correlating the DOFsx i andx j to the virtual DOF x p in formof: 107 [e] n×n 1 =                 [0] l 1 ×n 1 −[I] n 1 ×n 1 [0] l 2 ×n 1 [I] n 1 ×n 1 [0] l 3 ×n 1                 n×n 1 (5.20) where the values of l 1 , l 2 , and l 3 depends on the location of the DOFs x i and x j in the original stiffnessmatrix([K] n×n )assembly. Theupdatedforcevectorwouldbe: {F 1 } =        {F} n×1 {0} n 1 ×1        s 1 ×1 (5.21) Notethattheemptypartitionsinallofthematricesandvectorsrepresentzerotermsinthose partitions. Stage2 The second stage is the addition of the reduced-order model derived by the partitioned neural networkmodel(Eqs.(5.12)and(5.13))totheupdatedsystem,derivedinstage1,whichrequires toadditionoftwosetsofnewvirtualpoints,{S 1 } l×1 and{ ˙ S 1 } l×1 ,definedas: {S 1 } l×1 +{ ˙ S 1 } l×1 = [w 1x ] l×n 1 ×{x p } n1×1 +[w 1˙ x ] l×n 1 ×{˙ x p } n1×1 +{b 1 } l×1 (5.22) 108 Equation(5.22)isrewrittenandrearrangedinanequivalentmatrix-form,inordertodemon- stratehowtheequationwillbeintegratedintotheupdatedsystemequationsas: −[w 1x ] l×n 1 ×{x p } n1×1 +{S 1 } l×1 −[w 1˙ x ] l×n 1 ×{˙ x p } n1×1 +{ ˙ S 1 } l×1 ={b 1 } l×1 (5.23) [0] l×l | {z } [M (2) ] ×{0} l×1 | {z } { ¨ x (2) } + −[w 1˙ x ] l×n 1 [I] l×l | {z } [C (2) ] ×        {˙ x p } n1×1 { ˙ S 1 } l×1        | {z } { ˙ x (2) } + (5.24) −[w 1x ] l×n 1 [I] l×l | {z } [K (2) ] ×        {x p } n1×1 {S 1 } l×1        | {z } { x (2) } ={b 1 } l×1 | {z } { F (1) } The addition of the Eq. (5.24) to the updated system equations, from stage 1, would results innewsystemstatesas: {¨ x 2 } =                {¨ x} n×1 {0} n 1 ×1 {0} l×1                s 2 ×1 , { ˙ x 2 } =                {˙ x} n×1 {˙ x p } n 1 ×1 { ˙ S 1 } l×1                s 2 ×1 , (5.25) {x 2 } =                {x} n×1 {x p } n 1 ×1 {S 1 } l×1                s 2 ×1 109 The matrices [M (2) ], [C (2) ], and [K (2) ] can be added to the updated mass ( M 1 ), stiffness ( K 1 ), and damping ( C 1 ) matrices, obtained from stage 1, that would results in newsystemmatricesas: M 2 ×{¨ x 2 } =         [M] n×n [0] n 1 ×n 1 [0] l×l         s 2 ×s 2 ×                {¨ x} n×1 {0} n 1 ×1 {0} l×1                s 2 ×1 (5.26) C 2 ×{ ˙ x 2 } =         [C] n×n [e] n×n 1 [I] n 1 ×n 1 [−w 1˙ x ] l×n 1 [I] l×l         s 2 ×s 2 ×                {˙ x} n×1 {˙ x p } n 1 ×1 { ˙ S 1 } l×1                s 2 ×1 (5.27) K 2 ×{x 2 } =         [K] n×n [e] n×n 1 [I] n 1 ×n 1 [−w 1x ] l×n 1 [I] l×l         s 2 ×s 2 ×                {x} n×1 {x p } n 1 ×1 {S 1 } l×1                s 2 ×1 (5.28) andtheupdatedforcevectorwouldbe: {F 2 } =                {F} n×1 {0} n 1 ×1 {b 1 } l×1                s 2 ×1 (5.29) 110 where s 2 (= n+n 1 +l) is the order of the new system matrices, [ I ] r×r is the identity matrix of size of r, [e ] n×n 1 is a transition matrix (Eq. (5.20)). Note that the empty partitions in all of thematricesandvectorsrepresentzerotermsinthosepartitions. Stage3 The next stage is the computation of the output of the first-layer of the neural network model, {a 1 },describedbytheEq.(5.12), throughtheG 1 functionevaluationofthevirtualstates,{S 1 } and{ ˙ S 1 }, which wereobtained in the pervious stage. In order to implement the function evalu- ationinthesystemequations,anothersetofvirtualpoints,{S 2 },isdefinedas: {S 2 } l×1 = G 1 {S 1 } l×1 +{ ˙ S 1 } l×1 (5.30) RewritingEq.(5.30)inamatrix-formas: [0] l×l | {z } [M (3) ] ×{0} l×1 | {z } { ¨ x (3) } +[0] l×l | {z } [C (3) ] ×{0} l×1 | {z } { ˙ x (3) } +[I] l×l | {z } [K (3) ] ×{S 2 } l×1 | {z } { x (3) } = G 1 {S 1 } l×1 +{ ˙ S 1 } l×1 | {z } { F (3) } (5.31) andaddingtheEq.(5.31)totheupdatedsystemwouldresultinnewstatevectorsas: (5.32) {¨ x} =                        {¨ x 3 } n×1 {0} n 1 ×1 {0} l×1 {0} l×1                        , s 3 ×1 { ˙ x 3 } =                        {˙ x} n×1 {˙ x p } n 1 ×1 { ˙ S 1 } l×1 {0} l×1                        , s 3 ×1 {x 3 } =                        {x} n×1 {x p } n 1 ×1 {S 1 } l×1 {S 2 } l×1                        s 3 ×1 111 theupdatedsystemmatricesandtheircorrespondingstatesas: M 3 ×{¨ x 3 } = (5.33)             [M] n×n [0] n 1 ×n 1 [0] l×l [0] l×l             s 3 ×s 3 ×                        {¨ x 3 } n×1 {0} n 1 ×1 {0} l×1 {0} l×1                        s 3 ×1 C 3 ×{ ˙ x 3 } = (5.34)             [C] n×n [e] n×n 1 [I] n 1 ×n 1 [−w 1˙ x ] l×n 1 [I] l×l [0] l×l             s 3 ×s 3 ×                        {˙ x} n×1 {˙ x p } n 1 ×1 { ˙ S 1 } l×1 {0} l×1                        s 3 ×1 K 3 ×{x 3 } = (5.35)             [K] n×n [e] n×n 1 [I] n 1 ×n 1 [−w 1x ] l×n 1 [I] l×l [I] l×l             s 3 ×s 3 ×                        {x} n×1 {x p } n 1 ×1 {S 1 } l×1 {S 2 } l×1                        s 3 ×1 112 andtheupdateforcevectoras: {F 3 } =                        {F} n×1 {0} n 1 ×1 {b 1 } l×1 {G 1 ({S 1 } l×1 +{ ˙ S 1 } l×1 )} l×1                        s 3 ×1 (5.36) wheres 3 (= n+n 1 +l+l)istheorderofthenewsystemmatrices,[I] r×r istheidentitymatrix of size of r, [e ] n×n 1 is a transition matrix (Eq. (5.20)). Note that the empty partitions in all of thematricesandvectorsrepresentzerotermsinthosepartitions. Stage4 The next stage is the computation of the second equation of the neural network model (Eq.(5.13)),whichrequirestheintroductionofthelastsetofvirtualpoints,{S 3 },definedas: {S 3 } n 1 ×1 = [W 2 ] n 1 ×l ×{S 2 } l×1 +{b 2 } n 1 ×1 (5.37) where the values of the virtual points{S 2 } are obtained from the pervious stage. Rewriting the aboveequationinmatrix-formas: [0] l×l | {z } [M (4) ] ×{0} l×1 | {z } { ¨ x (4) } +[0] l×l | {z } [C (4) ] ×{0} l×1 | {z } { ˙ x (4) } + (5.38) −[W 2 ] n 1 ×l [I] n 1 ×n 1 | {z } [K (4) ] ×        {S 2 } l×1 {S 3 } n 1 ×1        | {z } { x (4) } ={b 2 } l×1 | {z } { F (4) } 113 andaddingtheaboveequationtotheupdatedstatesfromtheperviousstage,wouldresultinnew statesas: {¨ x 4 } =                                {¨ x} n×1 {0} n 1 ×1 {0} l×1 {0} l×1 {0} n 1 ×1                                , s 4 ×1 { ˙ x 4 } =                                {˙ x} n×1 {˙ x p } n 1 ×1 { ˙ S 1 } l×1 {0} l×1 {0} n 1 ×1                                , s 4 ×1 (5.39) {x 4 } =                                {x} n×1 {x p } n 1 ×1 {S 1 } l×1 {S 2 } l×1 {S 3 } n 1 ×1                                s 4 ×1 theupdatesystemmatricesandtheircorrespondingstateswouldbe: M 4 ×{¨ x 4 } = (5.40)                 [M] n×n [0] n 1 ×n 1 [0] l×l [0] l×l [0] n 1 ×n 1                 s 4 ×s 4 ×                                {¨ x} n×1 {0} n 1 ×1 {0} l×1 {0} l×1 {0} n 1 ×1                                s 4 ×1 114 C 4 ×{ ˙ x 4 } = (5.41)                 [C] n×n [e] n×n 1 [I] n 1 ×n 1 [−w 1˙ x ] l×n 1 [I] l×l [0] l×l [0] n 1 ×n 1                 s 4 ×s 4 ×                                {˙ x} n×1 {˙ x p } n 1 ×1 { ˙ S 1 } l×1 {0} l×1 {0} n 1 ×1                                s 4 ×1 K 4 ×{x 4 } = (5.42)                 [K] n×n [e] n×n 1 [I] n 1 ×n 1 [−w 1x ] l×n 1 [I] l×l [I] l×l [−W 2 ] n 1 ×l [I] n 1 ×n 1                 s 4 ×s 4 ×                                {x} n×1 {x p } n 1 ×1 {S 1 } l×1 {S 2 } l×1 {S 3 } n 1 ×1                                s 4 ×1 4 andtheupdatedforcevectorwouldbe: {F 4 } =                                {F} n×1 {0} n 1 ×1 {b 1 } l×1 {G 1 ({S 1 } l×1 +{ ˙ S 1 } l×1 )} l×1 {b 2 } n 1 ×1                                s 4 ×1 (5.43) 115 wheres 4 (= n+n 1 +l+l+n 1 )istheorderofthenewsystemmatrices,[I] r×r istheidentity matrixofsizeofr,[e] n×n 1 isatransitionmatrix(Eq.(5.20)). Notethattheemptypartitionsin allofthematricesandvectorsrepresentzerotermsinthosepartitions. Stage5 The last step is the addition of the resulting nonlinear restoring forces, from the neural network implementation, to the system external force vector, for the DOFs that the nonlinear “joint” elementhasbeenaddedto(DOFsiandj inhere). Theresultingnonlinearrestoringforcecanbe derivedbasedonthevaluesobtainedfromperviousstageforthevirtualpoints,{S 3 },as: {T} n 1 ×1 = ({S 3 } n 1 ×1 ) (5.44) Theupdatedforcevectorwouldbe: {F 5 } =                                                                  {F} n×1 +                                {0} l 1 ×1 {S 3 } n 1 ×1 {0} l 2 ×1 −{S 3 } n 1 ×1 {0} l 3 ×1                                n×1 {0} n 1 ×1 {b 1 } l×1 {G 1 ({S 1 } l×1 +{ ˙ S 1 } l×1 )} l×1 {b 2 } n 1 ×1                                                                  s 4 ×1 (5.45) 116 where the values of l 1 , l 2 , and l 3 depend on the location of the DOFs x i and x j in the original stiffness matrix ([K] n×n ) assembly. Note that, the added nonlinear restoring forces are equal in value and opposite in direction. The rest of the system matrices and states would remain the sameasstage4(Eqs.(5.40)-(5.42)). 5.1.3 SolvingtheNewUpdateNonlinearEquations-of-MotionoftheSystem Thenewnonlinearequationsofmotionforthesystemwouldbe: M 4 s 4 ×s 4 ¨ x 4 s 4 ×1 + C 4 s 4 ×s 4 ˙ x 4 s 4 ×1 + K 4 s 4 ×s 4 {x 4 } s 4 ×1 = F 5 s 4 ×1 (5.46) By using any time-marching techniques, suitable for nonlinear systems (such as Runge- Kutta),theaboveequationscanbesolved(throughcalculatingthetime-historyresponse). An important note that should be pointed out is that, after incorporating the reduced-order modelofthenonlinear“joint”elementintothelinearsetofequations-of-motion,theupdatedsets ofequations-of-motionwouldbea nonlinearset,althoughtheformulationmayappearasalinear set of equations. The updated force vector (Eq. (5.45)) is a state-dependent (time-dependent) vector,whosevaluewillbeupdatedbasedonthetime-historyresponseofthesystem; therefore theupdatedsystemshouldbesolvedwithnumericalcomputationalmethodsthataresuitablefor useincomputingthetime-historyresponseofnonlinearsystems. 5.2 Simulation Inordertodemonstratetheapplicationofthemethodologyunderdiscussion,considertheexam- pleof8- DOFssystemshowninFig.5.3,werethesystemconsistsoffouruncoupledmassesm i , 117 4 x θ 4 x 1 1 T F 2 2 x x θ , 2 2 2 2 θ θ c k c k x x , , 1 1 1 1 θ θ c k c k x x , , 4 4 4 4 θ θ c k c k x x , , 5 5 5 5 θ θ c k c k x x , , 1 1 x x θ , 3 3 x x θ , 3 3 3 3 θ θ c k c k x x , , 1 1 I m , 4 4 I m , 3 3 I m , 2 2 I m , Figure5.3: Mathematicalmodelofalinear8- DOFssystem. with rotational mass moment of inertia I i , with linear and rotational stiffness k x i and k θ i , and linear and torsional viscous damping c x i and c θ i , where i = 1,2,3,4. The system is subjected toanexternalaxialloadandatorqueforceappliedtothemassm 1 ,simultaneously. 5.2.1 MathematicalModel Themathematicalmodelrepresentingthemotionofthe8- DOF systemisgivenby: [M] 8×8 {¨ x} 8×1 +[C] 8×8 { ˙ x} 8×1 +[K] 8×8 {x} 8×1 ={F} 8×1 (5.47) wherethestatevectorsare: {x} = [x 1 , θ x 1 , x 2 , θ x 2 , x 3 , θ x 3 , x 4 , θ x 4 ] T (5.48) {˙ x} = [˙ x 1 , ˙ θ x 1 , ˙ x 2 , ˙ θ x 2 , ˙ x 3 , ˙ θ x 3 , ˙ x 4 , ˙ θ x 4 ] T (5.49) {¨ x} = [¨ x 1 , ¨ θ x 1 , ¨ x 2 , ¨ θ x 2 , ¨ x 3 , ¨ θ x 3 , ¨ x 4 , ¨ θ x 4 ] T (5.50) 118 themassmatrixis: [M] =                              m 1 0 0 0 0 0 0 0 0 I m 1 0 0 0 0 0 0 0 0 m 2 0 0 0 0 0 0 0 0 I m 2 0 0 0 0 0 0 0 0 m 3 0 0 0 0 0 0 0 0 I m 3 0 0 0 0 0 0 0 0 m 4 0 0 0 0 0 0 0 0 I m 4                              (5.51) thestiffnessmatrixis: [K] = (5.52)                              k x 1 +k x 2 0 −k x 2 0 0 0 0 0 0 k θ 1 +k θ 2 0 −k θ 2 0 0 0 0 −k x 2 0 k x 1 +k x 2 0 −k x 3 0 0 0 0 −k θ 2 0 k θ 2 +k θ 3 0 −k θ 3 0 0 0 0 −k x 3 0 k x3 +k x 4 0 −k x 4 0 0 0 0 −k θ 3 0 k θ 3 +k θ 4 0 −k θ 4 0 0 0 0 −k x 4 0 k x 4 +k x 5 0 0 0 0 0 0 −k θ 4 0 k θ 4 +k θ 5                              119 thedampingmatrixis: [C] = (5.53)                              c x 1 +c x 2 0 −c x 2 0 0 0 0 0 0 c θ 1 +c θ 2 0 −c θ 2 0 0 0 0 −c x 2 0 c x 1 +c x 2 0 −c x3 0 0 0 0 −c θ 2 0 c θ 2 +c θ 3 0 −c θ 3 0 0 0 0 −c x 3 0 c x 3 +c x 4 0 −c x 4 0 0 0 0 −c θ 3 0 c θ 3 +c θ 4 0 −c θ 4 0 0 0 0 −c x 4 0 c x 4 +c x 5 0 0 0 0 0 0 −c θ 4 0 c θ 4 +c θ 5                              andtheexternalforcevectoris:{F} = [F 1 (t), T 1 (t), 0, 0, 0, 0, 0, 0] T . Assume that a massless nonlinear “joint” element has been added to the system between massesm 2 andm 3 ,asshowninFig.5.4. 1 1 I m , 4 x θ 4 x ) , , , ( θ θ & & x x Q 1 1 T F 4 4 I m , 2 2 x x θ , 3 3 I m , 2 2 2 2 θ θ c k c k x x , , 1 1 1 1 θ θ c k c k x x , , 4 4 4 4 θ θ c k c k x x , , 5 5 5 5 θ θ c k c k x x , , 2 3 2 3 2 3 2 3 x x x x x x x x x x where θ θ θ θ θ θ & & & & & & − = − = − = − = , , : 1 1 x x θ , 2 2 I m , 3 3 x x θ , Figure5.4: Mathematicalmodelofthe8- DOFssystemwithanonlinear“joint”element. 120 5.2.2 GenerationofSyntheticData For obtaining accurate simulation data sets, the finite element models of the two cases, shown by Figs. 5.3 and 5.4, were developed with the MSC.Nastran finite element analysis software (MSC.Software Corporation, 2000). Figure 5.5 shows the schematic of the simulation models, wherepart(a)showsthelinearmodelandpart(b)showsthemodelwithanonlinear“joint”ele- mentconsistingofatwo-dimensionalgapandacouplingnonlinearitydefinedbyamathematical functiong. Thetime-historyoftheappliedloadsareshowninFig.5.6. Thefunctionaldependenceofg onitsstatevariableswaschosentobe: g(x, ˙ x,θ, ˙ θ) =                                −k 0x ˙ xθ, if|x| ≥dx &|θ|<d θ ; −k 0θ ˙ θx, if|x|<dx &|θ| ≥d θ ; −k 0x ˙ xθ−k 0θ ˙ θx, if|x| ≥dx &|θ| ≥d θ ; 0, if|x|<dx &|θ|<d θ . (5.54) 5.2.3 Derivation of the Data-Based Model-Free Representation of the Nonlinear “Joint”Element The synthetic data obtained from the simulation model shown in Fig. 5.5-(a), were used to trainandidentifythedata-basedmodel-freerepresentationofthenonlinear“joint”elementwith the neural network. The identification procedures were discussed in details in Chapter 4, Sec- tion4.1.4. ThevaluesofthetrainedweightsandbiasesareshownbyEq.5.55. 121 (a) (b) Figure 5.5: Mathematical models used in finite element simulation. Part (a) is the linear model, andpart(b)isthelinearmodelwithanonlinear“joint”elementconsistingofatwo-dimensional gapandacouplingnonlinearitydefinedbythefunctiong. 0 5 10 15 20 25 30 −1000 −500 0 500 1000 Time (sec) (a) Axial Force (lbf) 0 5 10 15 20 25 30 −1000 −500 0 500 1000 Time (sec) (b) Torque Force (lbf−in) Figure5.6: Theexternalforcesused inthesimulationstudies. Part(a)is thetime-historyofthe applied axial load and part (b) is the time-history of the applied torque force. Note that the two appliedforcesareuncorrelated. 122 [W 1 ] =                                                               3.030 0.000 0.000 −0.009 0.527 −0.080 0.085 −0.109 −0.795 0.202 −0.081 −0.028 0.125 −0.389 −0.231 −0.411 4.544 −0.004 −0.006 −0.001 4.482 0.022 0.001 0.017 0.076 −0.467 −0.224 −0.255 0.078 1.357 0.032 0.020 0.060 −0.687 0.180 −0.078 . . . 0.189 0.496 0.065 −0.009 0.169 −0.125 −0.140 −0.015 −0.463 0.035 0.041 0.087 −0.481 −0.039 −0.033 −0.053 −0.764 0.025 −0.053 −0.110 −0.485 −0.161 −0.082 −0.089                                                               30×4 , {b 1 } =                                                                                                                            −0.080 0.311 0.914 0.823 2.057 −2.267 −0.849 1.012 −0.604 . . . −0.583 0.108 0.175 0.176 −0.771 0.019                                                                                                                            30×1 (5.55) [W 2 ] =     1.392 0.816 −0.983 −0.156 ... −0.802 −1.152 −0.797 −0.011 −0.057 0.088 −0.085 ... −0.340 0.002 0.301     2×30 {b 2 } =        0.012 −0.284        2×1 123 5.2.4 MathematicalModeloftheSystemwithNonlinear“Joint”Element After obtaining the reduced-order data-based representation of the nonlinear “joint” element, a new mathematical model was constructed to incorporate the reduced-order model based on the methodology under discussion, and analyzed through the finite element analysis. Figure 5.7 showstheschematicofthemathematicalmodelwiththereduced-orderelement. Figure 5.7: Mathematical model for the finite element simulation where the nonlinear “joint” model is represented by a reduced-order model obtained through the neural network modeling technique. Inordertoimplementthereduced-ordermodelinthefiniteelementmodel,anew DOF,x 100 wasdefinedtorepresenttherelativestatesofthe“joint”elementas: x 100 = x 3 −x 2 , θ x100 = θ x3 −θ x2 ˙ x 100 = ˙ x 3 − ˙ x 2 , ˙ θ x100 = ˙ θ x3 − ˙ θ x2 (5.56) theneuralnetworkmodeltobeaddedtothefiniteelementmodelwouldbe:        RF x 100 RF θ x100        2×1 = [W 2 ] 2×1 ×{a 1 } 30×1 +{b 2 } 2×1 (5.57) 124 {a 1 } 30×1 = tanh              [W 1 ] 30×4 1 ×                        x 100 θ x 100 ˙ x 100 ˙ θ x 100                        4×1 +{b 1 } 30×1              (5.58) In order to add the neural network model to the system, first the [W 1 ] matrix should be partitionedas: [W 1 ] =                                                                                                                3.030 0.000 0.527 −0.080 −0.795 0.202 0.125 −0.389 4.544 −0.004 4.482 0.022 0.076 −0.467 . . . 0.189 0.496 0.169 −0.125 −0.463 0.035 −0.481 −0.039 −0.764 0.025 −0.485 −0.161                                                       30×2 | {z } [w 1x ]                                                       0.000 −0.009 0.085 −0.109 −0.081 −0.028 −0.231 −0.411 −0.006 −0.001 0.001 0.017 −0.224 −0.255 . . . 0.065 −0.009 −0.140 −0.015 0.041 0.087 −0.033 −0.053 −0.053 −0.110 −0.082 −0.089                                                       30×2 | {z } [w 2x ]                                                          30×4 (5.59) 125 The transition matrix (Eq. (5.20)) that relates the DOF x 100 to the system DOFs, x 3 and x 4 wouldbe: [e] =                              0 0 0 0 1 0 0 1 −1 0 0 −1 0 0 0 0                              8×2 (5.60) where n = 8, n 1 = 2, l = 30, l 1 = 2, l 2 = 0, and l 3 = 2, therefore s 4 , the order of the new updatedsystemmatricesiss 4 = 8+2+30+30+2 = 72. Thenewupdatedstateswouldbe: {¨ x 4 } =                                {¨ x} 8×1 {0} 2×1 {0} 30×1 {0} 30×1 {0} 2×1                                72×1 , { ˙ x 4 } =                                {˙ x} 8×1 {˙ x p } 2×1 { ˙ S 1 } 30×1 {0} 30×1 {0} 2×1                                72×1 , {x 4 } =                                {x} 8×1 {x p } 2×1 {S 1 } 30×1 {S 2 } 30×1 {S 3 } 2×1                                72×1 (5.61) 126 theupdatedsystemmatriceswouldbe: K =                 [K] 8×8 [e] 8×2 [I] 2×2 [−w 1x ] 30×2 [I] 30×30 [I] 30×30 [−W 2 ] 2×30 [I] 2×2                 72×72 (5.62) C =                 [C] 8×8 [e] 8×2 [I] 2×2 [−w 1˙ x ] 30×2 [I] 30×30 [0] 30×30 [0] 2×2                 72×72 (5.63) M =                 [M] 8×8 [0] 2×2 [0] 30×30 [0] 30×30 [0] 2×2                 72×72 (5.64) 127 andtheupdatedforcevectorwouldbe: {F} =                                                                                                                                                     F 1 (t) T 1 (t) 0 0 0 0 0 0                                                          8×1 +                                                          0 0 −RF x100 −RF θ x100 RF x100 RF θ x100 0 0                                                          8×1 {0} 2×1 {b 1 } 30×1 tanh({S 1 } 30×1 +{ ˙ S 1 } 30×1 ) {b 2 } 2×1                                                                                            72×1 (5.65) Note that, the nonlinear restoring forces are added in equal and opposite amounts, to the DOFs that the nonlinear “joint” element was associated with. The empty partitions, in all of the matricesandvectors,representzerotermsinthosepartitions. 5.2.5 ImplementationoftheReduced-OrderNeuralNetworkModelintoNastran FiniteElementCode TheMSC.Nastranfiniteelementanalysissoftware(MSC.SoftwareCorporation,2000)hasbeen usedtovalidatetheimplementationoftheneuralnetworkmodel. 128 The details of the Nastran input files (deck cards) for the model with the nonlinear “joint” element (Fig. 5.5-(b)) and the model with reduced order element (Fig. 5.7) are provided in AppendixB. Verification Inordertoverifytheimplementationofthereduced-ordermodelintothecomputationalengine, Nastran,theresponseofthetwomodels,themodelwiththemathematicalnonlinearityfunctions (Fig, 5.5-(a)) and the model with the reduced-order “joint” element (Fig. 5.7), which were sub- jectedtothesamerandomexcitationthatwereusedfortrainingofthereduced-order“joint”,are comparedinFig.5.8. Figures5.8-(a)and-(b)showthetime-historycomparisonoftheaxialandtorsionalrestoring forces of the original model (solid line) and the reduced-order model (dashed-dot line), respec- tively. Figures5.8-(c)and-(d)depictthephase-plotcomparisonoftherelativedisplacementvs. axialrestoring-forceandtherelativerotationvs. torsionalrestoring-force,oftheoriginalmodel (solidline)andthereduced-ordermodel(dashed-dotline),respectively. Validation For utilizing the methodology, the response of the two models, the model with the mathemat- ical nonlinearity functions (Fig, 5.5-(a)) and the model with the reduced-order “joint” element (Fig.5.7), whichweresubjectedtodifferentrandomexcitationsthanthoseusedforthetraining ofthereduced-order“joint”element,arecompared,andshowninFig.5.9. 129 10 11 12 13 14 15 16 17 18 19 20 −500 0 500 Time (sec) (a) Axial Restoring Force (lbf) 10 11 12 13 14 15 16 17 18 19 20 −500 0 500 Time (sec) (b) Torsional Restoring Force (lbf−in) −0.4 −0.2 0 0.2 0.4 0.6 −800 −600 −400 −200 0 200 400 600 800 Relative Displacement (in) (c) Axial Restoring Force (lbf) −0.4 −0.2 0 0.2 0.4 0.6 −800 −600 −400 −200 0 200 400 600 800 Relative Rotation (rad) (d) Torsional Restoring Force (lbf−in) Figure5.8: Verificationtheresponsesofthesimulationmodelswiththenonlinearmathematical element (solid line) and with the reduced-order “joint” element (dashed-dot line), subjected to thesamerandomexcitationusedintheidentificationofthereduced-order“joint”element. Parts (a) and (b) shows the time-history comparison of the axial and torsional restoring forces, and parts (c) and (d) are the phase-plot comparison of the relative displacement vs. axial restoring- forceandtherelativerotationvs. torsionalrestoring-force,respectively. 130 15 16 17 18 19 20 21 22 23 24 25 −1000 −500 0 500 1000 Time (sec) (a) Axial Restoring Force (lbf) 15 16 17 18 19 20 21 22 23 24 25 −1000 −500 0 500 1000 Time (sec) (b) Torsional Restoring Force (lbf−in) −0.8 −0.6 −0.4 −0.2 0 0.2 0.4 0.6 0.8 −1000 −500 0 500 1000 Relative Displacement (in) (c) Axial Restoring Force (lbf) −0.8 −0.6 −0.4 −0.2 0 0.2 0.4 0.6 0.8 −1000 −800 −600 −400 −200 0 200 400 600 800 1000 Relative Rotation (rad) (d) Torsional Restoring Force (lbf−in) Figure 5.9: Validation the responses of the simulation models with the nonlinear mathematical element (solid line) and with the reduced-order “joint” element (dashed-dot line), subjected to thedifferentrandomexcitationthanthoseusedintheidentificationofthereduced-order“joint” element. Parts(a)and(b)showsthetime-historycomparisonoftheaxialandtorsionalrestoring forces,andparts(c)and(d)arethephase-plotcomparisonoftherelativedisplacementvs. axial restoring-forceandtherelativerotationvs. torsionalrestoring-force,respectively. 131 Figures5.9-(a)and-(b)showthetime-historycomparisonoftheaxialandtorsionalrestoring forces of the original model (solid line) and the reduced-order model (dashed-dot line), respec- tively. Figures5.9-(c)and-(d)depictthephase-plotcomparisonoftherelativedisplacementvs. axialrestoring-forceandtherelativerotationvs. torsionalrestoring-force,oftheoriginalmodel (solidline)andthereduced-ordermodel(dashed-dotline),respectively. Theresultsindicatethatthereduced-orderdata-basedmodelofthenonlinear“joint”element, isanaccuraterepresentationoftheoriginalnonlinearmathematicalmodel. 5.3 Application In order to demonstrate the application of the reduced-order computational models in practical problems,theexperimentaltestsetupofthe2- DOFnonlinear“joint”model,whichwasdiscussed indetailsinChapter4,isusedasanexample. The development of the data-based model-free representation of the experimental results with the artificial neural network are discussed in detail in Chapter 4. The reduced-order model implementation, in the Nastran computational code, is similar to the simulation model imple- mentationdiscussedinSection5.2. 5.3.1 Development of a Computational Model with the Reduced-Order Data- BasedRepresentationoftheNonlinear“Joint”Element The data-based model-free representation of the nonlinear 2- DOF “joint” experiment, is used to develop a computational finite element model of a system that contains a nonlinear “joint” element,basedonthemethodologyunderdiscussioninthissection. 132 Figures5.10, 5.11,and 5.12illustratethephase-plotresponsesofthecomputationalmodel with the data-based model-free representation of the 2- DOF experimental “joint” element along withthephase-plotresponseoftheexperimentalmeasurementsfromthe 2- DOF“joint”element. Figure 5.10 shows the comparison results, in which both systems were subjected to axial load only, where parts (a) and (b) are the phase-plot response of the computational model with thereduced-order“joint”elementfortherelativedisplacementvs.axialrestoringforce,andthe relative rotation vs. torsional restoring force, respectively. Parts (c) and (d) show the phase- plot response of the 2- DOF nonlinear “joint” element experimental test results for the relative displacement vs. axial restoring force, and the relative rotation vs. torsional restoring force, respectively. Figure 5.11 shows the comparison results, in which both systems were subjected to torque force only, where parts (a) and (b) are the phase-plot response of the computational model with thereduced-order“joint”elementfortherelativedisplacementvs.axialrestoringforce,andthe relative rotation vs. torsional restoring force, respectively. Parts (c) and (d) show the phase- plot response of the 2- DOF nonlinear “joint” element experimental test results for the relative displacement vs. axial restoring force, and the relative rotation vs. torsional restoring force, respectively. Figure 5.12 shows the comparison results, in which both systems were subjected simulta- neously to uncorrelated axial load and torque force, where parts (a) and (b) are the phase-plot responseofthecomputationalmodelwiththereduced-order“joint”elementfortherelativedis- placementvs.axialrestoringforce,andtherelativerotationvs.torsionalrestoringforce,respec- tively. Parts (c) and (d) show the phase-plot response of the 2- DOF nonlinear “joint” element 133 −0.2 −0.15 −0.1 −0.05 0 0.05 0.1 0.15 0.2 −60 −40 −20 0 20 40 60 Relative Displacement (in) Axial Restoring Force (lbf) −0.2 −0.15 −0.1 −0.05 0 0.05 0.1 0.15 0.2 −60 −40 −20 0 20 40 60 Relative Rotation (rad) Torsional Restoring Force (lbf−in) (a) (b) −0.2 −0.15 −0.1 −0.05 0 0.05 0.1 0.15 0.2 −60 −40 −20 0 20 40 60 Relative Displacement (in) Axial Restoring Force (lbf) −0.2 −0.15 −0.1 −0.05 0 0.05 0.1 0.15 0.2 −60 −40 −20 0 20 40 60 Relative Rotation (rad) Torsional Restoring Force (lbf−in) (c) (d) Figure 5.10: Comparison of the experimental measurements from the 2- DOF nonlinear “joint” testsetup,withtheresponsesofthecomputationalmodelwiththereduced-orderrepresentation of the nonlinear “joint” element, when both systems were subjected to axial load only. Parts (a) and (b) are the phase-plots of the computational model with the reduced-order “joint” ele- ment,fortherelativedisplacementvs.axialrestoringforceandtherelativerotationvs.torsional restoring force, respectively, whereas parts (c) and (d) are the phase-plots of the 2- DOF “joint” experimentfortherelativedisplacementvs.axialrestoringforceandtherelativerotationvs.tor- sionalrestoringforce,respectively. 134 −0.2 −0.15 −0.1 −0.05 0 0.05 0.1 0.15 0.2 −60 −40 −20 0 20 40 60 Relative Displacement (in) Axial Restoring Force (lbf) −0.2 −0.15 −0.1 −0.05 0 0.05 0.1 0.15 0.2 −60 −40 −20 0 20 40 60 Relative Rotation (rad) Torsional Restoring Force (lbf−in) (a) (b) −0.2 −0.15 −0.1 −0.05 0 0.05 0.1 0.15 0.2 −60 −40 −20 0 20 40 60 Relative Displacement (in) Axial Restoring Force (lbf) −0.2 −0.15 −0.1 −0.05 0 0.05 0.1 0.15 0.2 −60 −40 −20 0 20 40 60 Relative Rotation (rad) Torsional Restoring Force (lbf−in) (c) (d) Figure 5.11: Comparison of the experimental measurements from the 2- DOF nonlinear “joint” test setup, with the responses of the computational model with the reduced-order representa- tion of the nonlinear “joint” element, when both systems were subjected to torque force only. Parts (a) and (b) are the phase-plots of the computational model with the reduced-order “joint” element, for the relative displacement vs. axial restoring force and the relative rotation vs. tor- sional restoring force, respectively, whereas parts (c) and (d) are the phase-plots of the 2- DOF “joint”experimentfortherelativedisplacementvs.axialrestoringforceandtherelativerotation vs.torsionalrestoringforce,respectively. 135 experimental test results for the relative displacement vs. axial restoring force, and the relative rotationvs.torsionalrestoringforce,respectively. Clearly the results shownin Figs. 5.10, 5.11, and 5.12, indicate that the reduced-order data- based model-free representation of the nonlinear 2- DOF “joint” element, which was obtained throughidentificationofthemodelwiththeexperimentalmeasurements,hascapturedthedom- inantbehaviorofthenonlinear“joint”element,andcanbeusedincomputationalmodelsincor- porating multiple nonlinear “joint” elements, in order to simulate and predict the system-level response. 5.4 ChapterSummary In this chapter the development of the reduced-order computational model of a generic com- plex nonlinear “joint” element was discussed, and its application was demonstrated through a data-based model-free representation of the nonlinear “joint” element, with the artificial neural network model. A series of simulation studies were performed to calibrate the methodology first, and thereafter, the utilization of the method was demonstrated through comparison to the experimental measurements of the 2- DOF nonlinear “joint” test setup, which was explained in detailinChapter4. Theresultsindicatethepowerofthemethodologyinemployingadata-basedmodel-freerep- resentation of a nonlinear “joint” element through laboratory testings, and subsequently using a computational system-level model based on the reduced-order nonlinear “joint” element to pre- dictthenonlinearsystemresponsetoexcitationdatasetsthatdifferfromthoseusedinobtaining theidentifiedmodel. 136 −0.2 −0.15 −0.1 −0.05 0 0.05 0.1 0.15 0.2 −60 −40 −20 0 20 40 60 Relative Displacement (in) Axial Restoring Force (lbf) −0.2 −0.15 −0.1 −0.05 0 0.05 0.1 0.15 0.2 −60 −40 −20 0 20 40 60 Relative Rotation (rad) Torsional Restoring Force (lbf−in) (a) (b) −0.2 −0.15 −0.1 −0.05 0 0.05 0.1 0.15 0.2 −60 −40 −20 0 20 40 60 Relative Displacement (in) Axial Restoring Force (lbf) −0.2 −0.15 −0.1 −0.05 0 0.05 0.1 0.15 0.2 −60 −40 −20 0 20 40 60 Relative Rotation (rad) Torsional Restoring Force (lbf−in) (c) (d) Figure 5.12: Comparison of the experimental measurements from the 2- DOF nonlinear “joint” testsetup,withtheresponsesofthecomputationalmodelwiththereduced-orderrepresentation of the nonlinear “joint” element, when both systems were subjected simultaneously to uncor- related axial load and torque force. Parts (a) and (b) are the phase-plots of the computational model with the reduced-order “joint” element, for the relative displacement vs. axial restoring force and the relative rotation vs. torsional restoring force, respectively, whereas parts (c) and (d) are the phase-plots of the 2- DOF “joint” experiment for the relative displacement vs. axial restoringforceandtherelativerotationvs.torsionalrestoringforce,respectively. 137 Chapter6 Conclusion This study developed and evaluated some powerful approaches for analyzing, modeling, moni- toring and control of complex nonlinear MDOF systems, regardless of their scale and type (i.e., spanningtherangefromMEMS,toaerospacesystems,tolargecivilstructures,tophysiological systems). The main purpose of this study was to design, construct and evaluate the performance of an easily reconfigurable test apparatus for conducting basic research studies in nonlinear phenom- ena. Themainfeaturesoftheapparatusaredescribedanditscapabilitytomodelthebehaviorof physicalsystemswithawiderangeof selectablenonlinear characteristicsthat encompassthose usually encountered in the applied mechanics field. Using adaptive estimation approaches, a method is presented for the on-line identification of hysteretic systems under arbitrary dynamic environments. Itisshown,throughtheuseofsimulationstudies,thattheproposedapproachcan yield reliable estimates of the hysteretic restoring force under a wide range of excitation levels andresponseranges. 138 Theon-linemonitoringapplicationoftwomathematicalmodelrepresentationsofnonlinear viscous dampers, one based on the simplified design model, and the second predicated on the polynomial-basis model, were investigated through experimental studies. It was demonstrated that monitoring with the simplified design model would provide a useful tool for design engi- neers and manufacturers to verify the design parameters under harmonic testing, as well as for in-situ parameter identification for systems subjected to harmonic input. However, it was also shown that the simplified design model is not applicable for identification purposes when the systemisexcitedbyrandomsignals. Tocircumventthislimitation,thepolynomial-basismodel was shown to be capable of identifying the behavior of nonlinear viscous dampers subjected to randomexcitations. In order to implement a practical on-line structural health monitoring methodology, the model for detecting changes in nonlinear viscous dampers should be capable of capturing the anticipated breadth of response which the device may experience during its service life, and should employ an on-line monitoring strategy. This latter condition ensures that each data set consideredincorporatesadequatedata(fromtheidentifiabilitypointofview)toproducearobust modelthatisinsensitivetonoiseandmodelingerrors. Results of this study show that adaptive least-squares identification approaches applied in conjunctionwithnonparametric,nonlinearmodelshavethepotentialtobeusefultoolforimple- mentingstructuralhealthmonitoringapproachesthatarebasedonvibrationsignatureanalysis. The application of two nonparametric modeling approaches, one using a polynomial-basis model and the other based on artificial neural networks, were discussed and evaluated for mod- eling nonlinear, nonconservative, dissipative systems. The tools use input/output data of the 139 system to obtain data-based, model-free computational models of complex nonlinear systems, which subsequently can be used in computational engines for simulation studies, condition assessment, and damage detection applications. The methods were first calibrated through sim- ulation studies and thereafter applied to experimental data sets corresponding to a 2DOF cou- pled nonlinear “joint” system. The generalization capability of the methods were demonstrated throughvalidationtestsusingsyntheticandexperimentaldatasets. Futureinvestigationsareplannedtoproceedalongtwofronts,(1)theimplementationofthe on-line identification technique for real world structural systems, such as monitoring of large- scale viscous dampers assembled on long-span suspended bridges, and (2) the comparison of thesystem-levelresponseofa MDOF systemwithnonlinear“joint”elementswithitsequivalent modelwithreduced-orderdata-basedrepresentationofthe“joint”elementthroughexperimental measurements, such as structural frames with energy dissipating devices (e.g., viscous dampers or base isolators). 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Yar, M.andHammond, J.K.(1987a). “Modellingand response of bilinear hystereticsystems.” ASCE,JournalofEngineeringMechanics,113,1000–1013. Yar,M.andHammond,J.K.(1987b). “Parameterestimationforhystereticsystems.”Journalof SoundandVibration,117(1),161172. Yun,H.,Tasbihgoo,F.,Masri,S.F.,Caffrey,J.P.,Wolfe,R.W.,Makris,N.,andBlack,C.(2006). “Comparison of modeling approaches for full-scale nonlinear viscous dampers.” Journal of VibrationandControl,(Acceptedforpublication). 148 AppendixA IntegrationofNormalization ProceduretoNeuralNetworkModel A.1 FormulationofNormalizationProcedure The two-layer feedforward neural network model used in this study, were trained with normal- ized input and outputs, therefore, for generalization and validation test cases, the input data should be normalized and the output data should be de-normalized with the normalization fac- tors used in the training stages. The normalization and de-normalization can be integrated into thenetworkmathematicalmodel bymodifyingtheweightsandbiasesof thenetwork. Thepro- cedureofintegrationisexplainedherefortheneuralnetworkmodelusedinmodelingthe2- DOF “joint”modelinChapter4,wherethenumberofinputsare4,numberofoutputsis2,numberof neuronsinthehiddenlayerisl. 149 Thenetworkarchitecture,whichisshowninFig.4.4,hasthemathematicalmodeldescribed byEqs.A.1andA.2. {a 1 } l×1 = G 1 ([W 1 ] l×4 × {P} 4×1 | {z } normalizedinputs +{b 1 } l×1 ) (A.1) T 2×1 | {z } normalizedoutputs = [W 2 ] 2×l {a 1 } l×1 +{b 2 } 2×1 (A.2) where [W 1 ] , [W 2 ], {b 1 }, and {b 2 } are the weights and biases of the network, G 1 (.) is the tangent-hyperbolic function, P = [x i ,θ i , ˙ x i , ˙ θ i ] T is the network normalized input, T = [RF x i ,RF θ i ] T isthenetworknormalizedoutput,and{a 1 }istheoutputofthefirstlayer. Assume the network were trained using r sets of data, with the input sets {P} i = [x i ,θ i , ˙ x i , ˙ θ i ] T , and the output sets {T} i = [RF x i ,RF θ i ] T , where i = 1,2,...,r. The train- ing data sets should be normalized across all the r sets (Demuth and Beale, 2005), by using the maximum and minimum of all the network inputs and outputs data sets. For example, lets assumethatthemaximumandminimumofallthex i dataisx min andx max ,thenallthex i data shouldbenormalizedas: x i = (2×(x i −x min )/(x max −x min ))−1,therefore,thenormalization vectorofthetrainingdatawouldbe: {P} i =                        x i θ i ˙ x i ˙ θ i                        =                        (2×(x i −x min )/(x max −x min ))−1 (2×(θ i −θ min )/(θ max −θ min ))−1 (2×(˙ x i − ˙ x min )/(˙ x max − ˙ x min ))−1 (2×( ˙ θ i − ˙ θ min )/( ˙ θ max − ˙ θ min ))−1                        (A.3) 150 andthenormalizedoutputis: {T} i =        RF x i RF θ i        =        (2×(RF x i −RF x min )/(RF xmax −RF x min ))−1 (2×(RF θ i −RF θ min )/(RF θmax −RF θ min ))−1        (A.4) A.2 IntegrationofNormalizationProcedure In order to integrate the normalization process into the neural network model, the normalized inputvector(Eq.(A.3)),isrewritteninamatrix-formas:                        x i θ i ˙ x i ˙ θ i                        | {z } P =             2/(x max −x min ) 0 0 0 0 2/(θ max −θ min ) 0 0 0 0 2/(˙ x max − ˙ x min ) 0 0 0 0 2/( ˙ θ max − ˙ θ min )             | {z } a 1n (A.5) ×                        x i θ i ˙ x i ˙ θ i                        | {z } P +                        −(x max +x min )/(x max −x min ) −(θ max +θ min )/(θ max −θ min ) −(˙ x max + ˙ x min )/(˙ x max − ˙ x min ) −( ˙ θ max + ˙ θ min )/( ˙ θ max − ˙ θ min )                        | {z } b 1n andinasimplifiedformis: {P} 4×1 = [a 1n ] 4×4 ×{P} 4×1 +{b 1n } 4×1 (A.6) 151 Thede-normalizedoutputvectorinamatrix-formis:        RF x i RF θ i        | {z } T =     (RF xmax −RF x min )/2 0 0 (RF θmax −RF θ min )/2     | {z } a 2n (A.7) ×        RF x i RF θ i        | {z } T +        (RF xmax +RF x min )/2 (RF θmax +RF θ min )/2        | {z } b 2n andinasimplifiedformis: {T} 2×1 = [a 2n ] 2×2 ×{T} 2×1 +{b 2n } 2×1 (A.8) Ifvalueofthe{P}intheEq.(A.1)isreplacedwithitsequivalentvaluederivedinEq.(A.6) theformulationoftheneuralnetworkmodelwouldchangeas: {a 1 } l×1 = G 1 [W 1 ] l×4 ×([a 1n ] 4×4 ×{P} 4×1 +{b 1n } 4×1 )+{b 1 } l×1 (A.9) and by regrouping the terms in the original form of the network model, a new model would be achievedas: {a 1 } l×1 = G 1 ( [W 1 ] l×4 ×[a 1n ] 4×4 | {z } W 1 ×{P} 4×1 + (A.10) [W 1 ] l×4 ×{b 1n } 4×1 +{b 1 } l×1 | {z } b 1 whichisinthesameformastheEq.(A.1)intheoriginalnetworkmodel. 152 Ifthevalueofthe{T}intheEq.(A.8)isreplacedwithitsoriginalneuralnetworkformula- tion(Eq.(A.2)),itwouldresultinanewformforthemodeas: {T} 2×1 = [a 2n ] 2×2 ×{ [W 2 ] 2×l {a 1 } l×1 +{b 2 } 2×1 } 2×1 +{b 2n } 2×1 (A.11) and by regrouping the terms, the new form of the neural network model, which is in the same formoftheoriginalmodel,wouldbe: {T} 2×1 = [a 2n ] 2×2 ×[W 2 ] 2×l | {z } W 2 {a 1 } l×1 + [a 1n ] 2×2 ×{b 2 } 2×1 +{b 2n } 2×1 | {z } b 2 (A.12) Therefore, by integration the normalization process into the neural network model, an updated form of the neural network model is achieved, where the weights and biases of the updatedmodelcontainthenormalizationinformation. Theupdateformofthemodelis: {a 1 } l×1 = G 1 ( W 1 l×4 {P} 4×1 | {z } inputs + b 1 l×1 ) (A.13) {T} 2×1 | {z } outputs = W 2 2×l {a 1 } l×1 + b 2 2×1 (A.14) where the W 1 and W 2 arethe normalized weight matrices,{b 1 } and{b 2 } are the normal- ized biases vectors,{P} is the input to the network,{T} is the output of the network, and{a 1 } isthefirstlayeroutput. 153 AppendixB NastranComputationalModelfor Nonlinear“JOINT”Element B.1 OriginalModel The Nastran input file for the model with the nonlinear mathematical “joint” element, as dis- cussedinChapter5andshownbyFig.5.5-(a),is: NASTRAN NLINES=99999999 ID AF2 2DOF GAP IN RX & X COUPLED FORCE IN RX & X DIFFERENT SOL SEDTRAN TIME 10000 DIAG=10 CEND SET 1 = 3,4,100 SET 2 = 2 SET 3 = 6,7 SET 4 = 100 ECHO = SORT(EXCEPT TABLED1) DISPLACEMENT = 1 VELOCITY = 1 ACCELERATION = 1 OLOAD = 2 FORCE = 3 SPC = 1 DLOAD = 300 TSTEP = 1 NONLINEAR=5 NLLOAD = 4 MPC=1 BEGIN BULK $=========================================================== 154 $ BULk DATA $=========================================================== $ PARAMETERS $=========================================================== PARAM,AUTOSPC,NO PARAM,GRDPNT,0 $=========================================================== $ MPC EQUATIONS $=========================================================== MPC,1,3,1,1.,100,1,1. ,,4,1,-1. MPC,1,3,4,1.,100,4,1. ,,4,4,-1. $=========================================================== $ NONLINEAR ELEMENT $=========================================================== NOLIN1,5,100,1,-1800.,100,1,102 NOLIN1,5,100,4,-1800.,100,4,102 NOLIN2,5,100,1,-500.,100,11,200,4 NOLIN2,5,100,4,-500.,100,14,200,1 NOLIN1,5,200,1,1.,100,1,104 NOLIN1,5,200,4,1.,100,4,105 $=========================================================== $ LINEAR TABLE $=========================================================== TABLED1,101 ,-1.,-1.,1.,1.,ENDT $=========================================================== $ PIECE WISE NONLINEAR TABLE $=========================================================== TABLED1,102 ,-1.2,-1.,-.2,0.,.2,0.,1.2,1. ,ENDT TABLED1,104 ,-1.2,1.,-.2,0.,.2,0.,1.2,1. ,ENDT TABLED1,105 ,-1.2,1.,-.2,0.,.2,0.,1.2,1. ,ENDT $=========================================================== $ DYNAMIC LOAD $=========================================================== TLOAD1 101 201 2 TLOAD1 102 202 3 DLOAD 300 1.0 1.0 101 1.0 102 DAREA 201 2 1 1000. DAREA 202 2 4 1000. $=========================================================== $ EXCITATION FILE INCLUDE ’RANDEXCIT2.DAT’ INCLUDE ’RANDEXCIT3.DAT’ $ LOAD STEP SIZE TSTEP 1 3000000 0.00001 50 $=========================================================== $ CONSTRAINT SET 1 : FIXED ENDS 155 $=========================================================== SPC 1 1 123456 0. SPC 1 6 123456 0. $=========================================================== $ PROPERTY 4 : INNER ROD PBAR 4 1 1. 1. 1. 1. $=========================================================== $ MATERIAL 1 : AISI 4340 STEEL MAT1 1 2.9E+7 0.32 $=========================================================== $ GRID POINTS $=========================================================== GRID 1 0 0. 0. 0. 0 2356 GRID 2 0 0. 0. 0. 0 2356 GRID 3 0 .9 0. 0. 0 2356 GRID 4 0 1.1 0. 0. 0 2356 GRID 5 0 2. 0. 0. 0 2356 GRID 6 0 2. 0. 0. 0 2356 GRID 100 0 0. 0. 0. 0 2356 GRID 200 0 0. 0. 0. 0 2356 $=========================================================== $ LINEAR SPRINGS $=========================================================== CELAS2 1 1000. 1 1 2 1 0. CELAS2 2 1000. 5 1 6 1 0. CELAS2 3 200. 3 1 4 1 0. CELAS2 4 1. 200 1 $=========================================================== $ TORSIONAL SPRINGS $=========================================================== CELAS2 11 1000. 1 4 2 4 0. CELAS2 12 1000. 5 4 6 4 0. CELAS2 13 200. 3 4 4 4 0. $=========================================================== $ BARS $=========================================================== CBAR 6 4 2 3 0. 1. 0. CBAR 7 4 4 5 0. 1. 0. CBAR 8 4 3 4 0. 1. 0. 14 14 $=========================================================== $ LINEAR DAMPING $=========================================================== CDAMP2 1 10.0 1 1 2 1 CDAMP2 2 10.0 5 1 6 1 CDAMP2 3 6.32 3 1 4 1 $=========================================================== $ TORSIONAL DAMPING $=========================================================== CDAMP2 11 10.0 1 4 2 4 CDAMP2 12 10.0 5 4 6 4 CDAMP2 13 6.32 3 4 4 4 $=========================================================== $ MASS TERMS $=========================================================== 156 CONM2 1 2 0 1. 0. 0. 0. +EL A +EL A 1. 0. 0. 0. 0. 0. CONM2 2 5 0 1. 0. 0. 0. +EL B +EL B 1. 0. 0. 0. 0. 0. CONM2 3 3 0 1. 0. 0. 0. +EL C +EL C 1. 0. 0. 0. 0. 0. CONM2 4 4 0 1. 0. 0. 0. +EL D +EL D 1. 0. 0. 0. 0. 0. ENDDATA B.2 TheTrainedNeuralNetworkParameters The trained (identified) parameters of the neural network model representing the nonlinear “joint” element, in the simulation model (Fig. 5.5-(a)), which its mathematical formulation is givenbyEqs.(5.8)and(5.9),arepresentedinthissection. Theweightmatrix,[W 1 ] 30×4 ,forthe firstlayeris: -2.7765469e+001 -1.1208999e-001 1.2899544e+001 2.4757063e-001 -1.6986926e+001 -3.5573070e+000 1.1863059e+001 -1.3878458e-001 9.1016274e+000 -1.5373637e-001 -1.7083566e+001 -7.2844768e+000 -2.5919399e+001 -4.0898740e-001 -1.2913346e+001 5.5120588e-001 -8.9687597e+000 3.9235692e+000 5.8696331e+000 2.1639940e+000 8.9972224e+000 -1.8552882e-001 -1.7270160e+001 -6.9348635e+000 -1.9232475e+000 1.6456444e+001 -5.0252780e-001 5.9198562e+000 3.1244133e+002 1.8241467e+000 -8.2681299e+001 -1.4258228e-001 1.6646272e+001 -1.0498477e-001 -3.2606493e+001 -2.8985328e-001 1.8985855e+001 3.8322310e+000 -1.4661278e+001 1.3921516e-001 -3.2400832e+000 -8.2194965e+000 -1.2499127e+000 -3.4892869e+000 -9.3297143e+000 3.9153446e+000 8.0243204e+000 2.3053288e+000 3.1721018e+000 8.3339896e+000 1.3929550e+000 3.5724614e+000 -7.3285074e+001 4.1585888e-001 1.2285887e+002 -9.1138235e-001 9.2274769e-001 4.5587614e+000 -6.1705999e+000 -4.2890026e-002 -9.8534608e+000 4.0443346e+000 1.1198751e+001 2.6220462e+000 -2.1646986e+001 -4.2323422e+000 1.7831967e+001 -1.4803670e-001 2.1399645e+000 -1.5888369e+001 1.2743518e+000 -5.4915635e+000 -7.3306507e+001 4.1670057e-001 1.2320046e+002 -9.1375969e-001 -9.1477202e+000 2.0679148e-001 1.7846878e+001 6.7212178e+000 2.0674415e+000 -1.5987327e+001 2.1841807e+000 -5.2604892e+000 -1.0160402e+001 4.2302168e+000 1.3526231e+001 2.9339439e+000 -1.4131018e+000 -8.7532002e+000 -3.8792938e+000 -3.6527611e+000 8.3408078e-001 4.2658933e-002 -1.7169339e+000 2.6426602e+000 1.4461822e+001 2.4058702e-001 2.4916461e+001 -1.7027906e-002 3.5321257e+001 -6.4671165e-002 -1.2977998e+001 -2.9256128e-001 -1.4431838e+001 -9.9194132e-002 6.3413470e-001 8.7705473e-002 1.5576832e+000 2.9658873e-003 1.8948943e+001 1.5551581e-001 -8.6364170e-001 -4.2334363e-002 1.9605071e+000 -2.6298579e+000 -1.3333208e+000 -4.5817238e+000 5.3584729e+000 3.1227295e-002 thebiasvector,{b 1 } 30×2 ,forthefirstlayeris: 5.6752051e+000 -3.4277607e-002 2.1789902e+000 7.3139264e+000 157 8.3102207e-002 2.0698633e+000 -1.9556338e-002 -4.5604666e+001 5.3098087e+000 6.1737608e-003 3.3997154e-001 7.8304061e-002 -3.9342307e-001 1.1670605e+000 1.3446260e-002 7.6828404e-002 2.8641473e-002 2.6716995e-002 1.1696484e+000 -1.9994171e+000 3.1059293e-002 8.0264438e-002 -3.6057098e-001 3.4744404e-001 -6.3776025e+000 -5.9868545e+000 -4.4721782e+000 4.0812983e+000 -3.3685573e-001 -1.6594120e-002 theweightmatrix,[W 2 ] 2×30 ,forthesecondlayeris: columns1to5 -1.6179094e+003 5.4533097e+001 1.2708431e+002 7.2556194e+002 6.0957494e+002 -7.6703383e+001 1.4937295e+002 2.8550702e+002 8.8665996e+002 3.6488573e+002 columns6to10 -2.5734158e+002 2.9102863e+002 4.3256741e+000 -2.2380137e+001 9.4061315e+001 -5.9536159e+002 2.0083464e+001 -1.1587846e-001 -1.7074467e+002 2.6374156e+002 columns11to15 6.1763503e+002 -1.3576985e+003 5.8064018e+002 -1.3852945e+002 -9.0282588e+001 6.4290761e+001 -7.7874753e+002 6.0128862e+001 -2.3037429e+002 -8.9605415e+001 columns16to20 1.2706278e+003 4.1591476e+001 5.9873002e+002 1.3833574e+002 -1.2813846e+002 7.0344159e+002 1.1613180e+002 -4.9643264e+000 2.2979730e+002 -3.0668011e+002 columns21to25 -3.0595185e+002 -5.2626853e+002 -4.1678394e+001 2.1309709e+002 3.8611167e+002 2.4674944e+001 -2.8766002e+002 -2.1541040e+000 3.3638558e+002 4.0101112e+002 columns26to30 -3.2342584e+002 -7.1184577e+002 9.2676496e+001 2.1462033e+002 -1.0014841e+002 -1.2150567e+001 2.0178634e+001 2.2455059e+002 3.3655901e+002 -9.0800532e+001 andthebiasvector,{b 2 } 2×1 ,forthesecondlayeris: 1.7690229e+002 -4.5479208e+002 158 B.3 The Computational Model with Reduced-Order “Joint” Ele- ment TheNastraninputfile,forthesimulationmodelwiththereduced-order“joint”element(Fig.5.7), ispresentedinthissection. Theinputfile,isthemodifiedversionoftheoriginalinputfilebased onthemethodologydiscussedinChapter5. NASTRAN NLINES=99999999 ID AF2 2DOF GAP IN RX & X FORCE F & T with Neural Network Model SOL SEDTRAN TIME 10000 DIAG=10 CEND SET 1 = 3,4,100,3001,3002 SET 2 = 2 SET 3 = 6,7 SET 4 = 100 ECHO = SORT(EXCEPT TABLED1) DISPLACEMENT = 1 VELOCITY = 1 ACCELERATION = 1 OLOAD = 2 FORCE = 3 SPC = 1 DLOAD = 300 TSTEP = 1 NONLINEAR=5 NLLOAD = 4 MPC=1 K2PP=STIF B2PP=DAMP LOADSET=601 BEGIN BULK $=========================================================== $ Add the neural network model $=========================================================== INCLUDE ’NEURALCP.DAT’ $=========================================================== $ Add a table representing the tanh function $=========================================================== INCLUDE ’TANH.DAT’ $=========================================================== $ BULk DATA $=========================================================== $ PARAMETERS $=========================================================== PARAM,AUTOSPC,NO PARAM,GRDPNT,0 $=========================================================== $ MPC EQUATIONS $=========================================================== MPC,1,3,1,1.,100,1,1. ,,4,1,-1. MPC,1,3,4,1.,100,4,1. 159 ,,4,4,-1. $=========================================================== $ NONLINEAR ELEMENT $=========================================================== NOLIN1,5,100,1,-1.,3001,0,101 NOLIN1,5,100,4,-1.,3002,0,101 $=========================================================== $ LINEAR TABLE $=========================================================== TABLED1,101 ,-1.,-1.,1.,1.,ENDT TABLED1,103 ,-1.,1.,1.,1.,ENDT $=========================================================== $ DYNAMIC LOAD $=========================================================== DLOAD 300 1.0 1.0 101 1.0 102 1.0 103 TLOAD1 101 201 4 TLOAD1 102 202 5 TLOAD1 103 203 103 DAREA 201 2 1 1000. DAREA 202 2 4 1000. $=========================================================== $ EXCITATION FILE INCLUDE ’RANDEXCIT4.DAT’ INCLUDE ’RANDEXCIT5.DAT’ $ LOAD STEP SIZE TSTEP 1 3000000 0.00001 50 $=========================================================== $ CONSTRAINT SET 1 : FIXED ENDS $=========================================================== SPC 1 1 123456 0. SPC 1 6 123456 0. $=========================================================== $ PROPERTY 4 : INNER ROD PBAR 4 1 1. 1. 1. 1. $=========================================================== $ MATERIAL 1 : AISI 4340 STEEL MAT1 1 2.9E+7 0.32 $=========================================================== $ GRID POINTS $=========================================================== GRID 1 0 0. 0. 0. 0 2356 GRID 2 0 0. 0. 0. 0 2356 GRID 3 0 .9 0. 0. 0 2356 GRID 4 0 1.1 0. 0. 0 2356 GRID 5 0 2. 0. 0. 0 2356 GRID 6 0 2. 0. 0. 0 2356 GRID 100 0 0. 0. 0. 0 2356 $=========================================================== $ LINEAR SPRINGS $=========================================================== CELAS2 1 1000. 1 1 2 1 0. CELAS2 2 1000. 5 1 6 1 0. $=========================================================== 160 $ TORSIONAL SPRINGS $=========================================================== CELAS2 11 1000. 1 4 2 4 0. CELAS2 12 1000. 5 4 6 4 0. $=========================================================== $ BARS $=========================================================== CBAR 6 4 2 3 0. 1. 0. CBAR 7 4 4 5 0. 1. 0. CBAR 8 4 3 4 0. 1. 0. 14 14 $=========================================================== $ LINEAR DAMPING $=========================================================== CDAMP2 1 10.0 1 1 2 1 CDAMP2 2 10.0 5 1 6 1 $=========================================================== $ TORSIONAL DAMPING $=========================================================== CDAMP2 11 10.0 1 4 2 4 CDAMP2 12 10.0 5 4 6 4 $=========================================================== $ MASS TERMS $=========================================================== CONM2 1 2 0 1. 0. 0. 0. +EL A +EL A 1. 0. 0. 0. 0. 0. CONM2 2 5 0 1. 0. 0. 0. +EL B +EL B 1. 0. 0. 0. 0. 0. CONM2 3 3 0 1. 0. 0. 0. +EL C +EL C 1. 0. 0. 0. 0. 0. CONM2 4 4 0 1. 0. 0. 0. +EL D +EL D 1. 0. 0. 0. 0. 0. ENDDATA TheNEURALCP.DATfile: $=========================================================== $ Virtual Points S1 $=========================================================== SPOINT 1001 THRU 1030 $=========================================================== $ Virtual Points S2 $=========================================================== SPOINT 2001 THRU 2030 $=========================================================== $ Virtual Points S3 $=========================================================== SPOINT 3001 THRU 3002 $=========================================================== $ Nonlinear load sequence for adding the restoring forces $=========================================================== LSEQ 601 203 501 $=========================================================== $ b1 biases values $=========================================================== 161 SLOAD * 501 1001 -0.4304609E+00 SLOAD * 501 1002 -0.4907857E-01 SLOAD * 501 1003 -0.3852082E-01 SLOAD * 501 1004 0.2209561E+01 SLOAD * 501 1005 -0.1083731E-02 SLOAD * 501 1006 0.2666481E+01 SLOAD * 501 1007 -0.2016870E+01 SLOAD * 501 1008 -0.1345522E+01 SLOAD * 501 1009 0.1744954E+01 SLOAD * 501 1010 0.2882075E+00 SLOAD * 501 1011 0.5173473E+00 SLOAD * 501 1012 0.2242066E+01 SLOAD * 501 1013 -0.6291215E+00 SLOAD * 501 1014 -0.2768647E+01 SLOAD * 501 1015 0.4950535E+00 SLOAD * 501 1016 -0.3912226E-01 SLOAD * 501 1017 0.3355688E+00 SLOAD * 501 1018 0.3363718E+00 SLOAD * 501 1019 -0.2048855E+01 SLOAD * 501 1020 -0.5567257E-01 SLOAD * 501 1021 -0.1215441E+01 SLOAD * 501 1022 -0.2763053E+01 SLOAD * 501 1023 -0.4491771E+01 SLOAD * 501 1024 0.9204103E-01 SLOAD * 501 1025 -0.1325088E+00 SLOAD * 501 1026 -0.2757728E+01 SLOAD * 501 1027 -0.2360687E+01 SLOAD * 501 1028 0.2097802E+01 SLOAD * 501 1029 -0.2823106E+01 SLOAD * 501 1030 -0.1167039E-01 $=========================================================== $ b2 biases values $=========================================================== SLOAD * 501 3001 0.6894368E+03 SLOAD * 501 3002 -0.9732772E+02 $=========================================================== $ w1x wrights values $=========================================================== DMIG * STIF 0 1 1 DMIG * STIF 1001 0 * 1001 1 0.1000000E+01 DMIG * STIF 100 1 * 1001 0 0.3146320E+01 DMIG * STIF 100 4 * 1001 0 0.9123461E-02 DMIG * STIF 1002 0 * 1002 1 0.1000000E+01 DMIG * STIF 100 1 * 1002 0 -0.6155591E-01 DMIG * STIF 100 4 * 1002 0 -0.1301218E+01 DMIG * STIF 1003 0 * 1003 1 0.1000000E+01 DMIG * STIF 100 1 * 1003 0 -0.7801713E-02 162 DMIG * STIF 100 4 * 1003 0 -0.1321751E+01 DMIG * STIF 1004 0 * 1004 1 0.1000000E+01 DMIG * STIF 100 1 * 1004 0 0.3220967E+00 DMIG * STIF 100 4 * 1004 0 -0.2813771E+01 DMIG * STIF 1005 0 * 1005 1 0.1000000E+01 DMIG * STIF 100 1 * 1005 0 -0.9087936E-02 DMIG * STIF 100 4 * 1005 0 -0.1827865E+00 DMIG * STIF 1006 0 * 1006 1 0.1000000E+01 DMIG * STIF 100 1 * 1006 0 -0.3566934E-01 DMIG * STIF 100 4 * 1006 0 0.1331358E+02 DMIG * STIF 1007 0 * 1007 1 0.1000000E+01 DMIG * STIF 100 1 * 1007 0 -0.8548578E+01 DMIG * STIF 100 4 * 1007 0 -0.3283610E-01 DMIG * STIF 1008 0 * 1008 1 0.1000000E+01 DMIG * STIF 100 1 * 1008 0 -0.6409224E+01 DMIG * STIF 100 4 * 1008 0 -0.1577215E-01 DMIG * STIF 1009 0 * 1009 1 0.1000000E+01 DMIG * STIF 100 1 * 1009 0 0.2110013E+01 DMIG * STIF 100 4 * 1009 0 0.6028383E-02 DMIG * STIF 1010 0 * 1010 1 0.1000000E+01 DMIG * STIF 100 1 * 1010 0 -0.1772850E+01 DMIG * STIF 100 4 * 1010 0 -0.4731330E-02 DMIG * STIF 1011 0 * 1011 1 0.1000000E+01 DMIG * STIF 100 1 * 1011 0 0.5315289E+01 DMIG * STIF 100 4 * 1011 0 0.8436223E-02 DMIG * STIF 1012 0 * 1012 1 0.1000000E+01 DMIG * STIF 100 1 * 1012 0 -0.3221340E+01 DMIG * STIF 100 4 163 * 1012 0 -0.7264226E-02 DMIG * STIF 1013 0 * 1013 1 0.1000000E+01 DMIG * STIF 100 1 * 1013 0 -0.5120529E+01 DMIG * STIF 100 4 * 1013 0 -0.9309812E-02 DMIG * STIF 1014 0 * 1014 1 0.1000000E+01 DMIG * STIF 100 1 * 1014 0 0.4322081E-01 DMIG * STIF 100 4 * 1014 0 -0.1483722E+02 DMIG * STIF 1015 0 * 1015 1 0.1000000E+01 DMIG * STIF 100 1 * 1015 0 -0.3751210E+01 DMIG * STIF 100 4 * 1015 0 -0.1421752E-01 DMIG * STIF 1016 0 * 1016 1 0.1000000E+01 DMIG * STIF 100 1 * 1016 0 -0.5539321E+00 DMIG * STIF 100 4 * 1016 0 -0.5306135E-02 DMIG * STIF 1017 0 * 1017 1 0.1000000E+01 DMIG * STIF 100 1 * 1017 0 -0.1293461E+00 DMIG * STIF 100 4 * 1017 0 -0.2318377E+00 DMIG * STIF 1018 0 * 1018 1 0.1000000E+01 DMIG * STIF 100 1 * 1018 0 -0.1141949E+00 DMIG * STIF 100 4 * 1018 0 -0.3856836E+00 DMIG * STIF 1019 0 * 1019 1 0.1000000E+01 DMIG * STIF 100 1 * 1019 0 0.5432705E-01 DMIG * STIF 100 4 * 1019 0 0.9986582E+01 DMIG * STIF 1020 0 * 1020 1 0.1000000E+01 DMIG * STIF 100 1 * 1020 0 -0.3060077E-01 DMIG * STIF 100 4 * 1020 0 -0.2068620E+01 DMIG * STIF 1021 0 * 1021 1 0.1000000E+01 DMIG * STIF 100 1 * 1021 0 0.5473984E+01 DMIG * STIF 100 4 * 1021 0 0.2535700E-01 164 DMIG * STIF 1022 0 * 1022 1 0.1000000E+01 DMIG * STIF 100 1 * 1022 0 0.1387382E+02 DMIG * STIF 100 4 * 1022 0 -0.1309931E-01 DMIG * STIF 1023 0 * 1023 1 0.1000000E+01 DMIG * STIF 100 1 * 1023 0 0.9936054E-02 DMIG * STIF 100 4 * 1023 0 0.1905389E+02 DMIG * STIF 1024 0 * 1024 1 0.1000000E+01 DMIG * STIF 100 1 * 1024 0 0.9932293E-01 DMIG * STIF 100 4 * 1024 0 0.1844699E+00 DMIG * STIF 1025 0 * 1025 1 0.1000000E+01 DMIG * STIF 100 1 * 1025 0 -0.3654851E-01 DMIG * STIF 100 4 * 1025 0 -0.3175081E+01 DMIG * STIF 1026 0 * 1026 1 0.1000000E+01 DMIG * STIF 100 1 * 1026 0 0.1394258E+02 DMIG * STIF 100 4 * 1026 0 -0.1300648E-01 DMIG * STIF 1027 0 * 1027 1 0.1000000E+01 DMIG * STIF 100 1 * 1027 0 0.3487464E+00 DMIG * STIF 100 4 * 1027 0 -0.3676153E+01 DMIG * STIF 1028 0 * 1028 1 0.1000000E+01 DMIG * STIF 100 1 * 1028 0 -0.4325827E-01 DMIG * STIF 100 4 * 1028 0 -0.1022680E+02 DMIG * STIF 1029 0 * 1029 1 0.1000000E+01 DMIG * STIF 100 1 * 1029 0 0.4245251E-01 DMIG * STIF 100 4 * 1029 0 -0.1533989E+02 DMIG * STIF 1030 0 * 1030 1 0.1000000E+01 DMIG * STIF 100 1 * 1030 0 -0.3728285E+01 DMIG * STIF 100 4 * 1030 0 -0.1297742E-01 $ 165 $=========================================================== $ W2 wrights values $=========================================================== DMIG * STIF 2001 0 * 2001 0 0.1000000E+01 DMIG * STIF 2002 0 * 2002 0 0.1000000E+01 DMIG * STIF 2003 0 * 2003 0 0.1000000E+01 DMIG * STIF 2004 0 * 2004 0 0.1000000E+01 DMIG * STIF 2005 0 * 2005 0 0.1000000E+01 DMIG * STIF 2006 0 * 2006 0 0.1000000E+01 DMIG * STIF 2007 0 * 2007 0 0.1000000E+01 DMIG * STIF 2008 0 * 2008 0 0.1000000E+01 DMIG * STIF 2009 0 * 2009 0 0.1000000E+01 DMIG * STIF 2010 0 * 2010 0 0.1000000E+01 DMIG * STIF 2011 0 * 2011 0 0.1000000E+01 DMIG * STIF 2012 0 * 2012 0 0.1000000E+01 DMIG * STIF 2013 0 * 2013 0 0.1000000E+01 DMIG * STIF 2014 0 * 2014 0 0.1000000E+01 DMIG * STIF 2015 0 * 2015 0 0.1000000E+01 DMIG * STIF 2016 0 * 2016 0 0.1000000E+01 DMIG * STIF 2017 0 * 2017 0 0.1000000E+01 DMIG * STIF 2018 0 * 2018 0 0.1000000E+01 DMIG * STIF 2019 0 * 2019 0 0.1000000E+01 DMIG * STIF 2020 0 * 2020 0 0.1000000E+01 DMIG * STIF 2021 0 * 2021 0 0.1000000E+01 DMIG * STIF 2022 0 * 2022 0 0.1000000E+01 DMIG * STIF 2023 0 * 2023 0 0.1000000E+01 DMIG * STIF 2024 0 * 2024 0 0.1000000E+01 DMIG * STIF 2025 0 * 2025 0 0.1000000E+01 DMIG * STIF 2026 0 * 2026 0 0.1000000E+01 166 DMIG * STIF 2027 0 * 2027 0 0.1000000E+01 DMIG * STIF 2028 0 * 2028 0 0.1000000E+01 DMIG * STIF 2029 0 * 2029 0 0.1000000E+01 DMIG * STIF 2030 0 * 2030 0 0.1000000E+01 $ $=========================================================== $ W2 weight values $=========================================================== DMIG * STIF 3001 0 * 3001 0 0.1000000E+01 DMIG * STIF 2001 0 * 3001 0 -0.5623356E+04 DMIG * STIF 2002 0 * 3001 0 -0.8052010E+04 DMIG * STIF 2003 0 * 3001 0 0.4381072E+04 DMIG * STIF 2004 0 * 3001 0 0.1028240E+04 DMIG * STIF 2005 0 * 3001 0 -0.8285400E+04 DMIG * STIF 2006 0 * 3001 0 0.4394805E+03 DMIG * STIF 2007 0 * 3001 0 -0.3435309E+03 DMIG * STIF 2008 0 * 3001 0 -0.3303654E+03 DMIG * STIF 2009 0 * 3001 0 0.1084735E+03 DMIG * STIF 2010 0 * 3001 0 -0.4745345E+04 DMIG * STIF 2011 0 * 3001 0 0.4386562E+04 DMIG * STIF 2012 0 * 3001 0 -0.3822198E+02 DMIG * STIF 2013 0 * 3001 0 0.5756126E+04 DMIG * STIF 2014 0 * 3001 0 0.2129749E+04 DMIG * STIF 2015 0 * 3001 0 0.8979717E+03 DMIG * STIF 2016 0 * 3001 0 0.6355622E+03 DMIG * STIF 2017 0 * 3001 0 0.1577562E+04 DMIG * STIF 2018 0 * 3001 0 -0.1561197E+04 DMIG * STIF 2019 0 * 3001 0 0.1317185E+04 DMIG * STIF 2020 0 * 3001 0 0.6233176E+04 DMIG * STIF 2021 0 167 * 3001 0 0.1516456E+04 DMIG * STIF 2022 0 * 3001 0 0.9831821E+03 DMIG * STIF 2023 0 * 3001 0 0.4430615E+00 DMIG * STIF 2024 0 * 3001 0 0.5197559E+04 DMIG * STIF 2025 0 * 3001 0 -0.1805284E+04 DMIG * STIF 2026 0 * 3001 0 -0.1026726E+04 DMIG * STIF 2027 0 * 3001 0 0.5556146E+03 DMIG * STIF 2028 0 * 3001 0 0.1232628E+04 DMIG * STIF 2029 0 * 3001 0 -0.1695170E+04 DMIG * STIF 2030 0 * 3001 0 -0.2278541E+04 $ DMIG * STIF 3002 0 * 3002 0 0.1000000E+01 DMIG * STIF 2001 0 * 3002 0 -0.3513779E+04 DMIG * STIF 2002 0 * 3002 0 -0.1763890E+04 DMIG * STIF 2003 0 * 3002 0 0.1007716E+04 DMIG * STIF 2004 0 * 3002 0 -0.5185487E+02 DMIG * STIF 2005 0 * 3002 0 -0.1272976E+04 DMIG * STIF 2006 0 * 3002 0 0.1658115E+02 DMIG * STIF 2007 0 * 3002 0 -0.7883356E+03 DMIG * STIF 2008 0 * 3002 0 0.2803558E+04 DMIG * STIF 2009 0 * 3002 0 -0.1888595E+04 DMIG * STIF 2010 0 * 3002 0 -0.2121365E+04 DMIG * STIF 2011 0 * 3002 0 -0.4358868E+04 DMIG * STIF 2012 0 * 3002 0 0.1991842E+04 DMIG * STIF 2013 0 * 3002 0 -0.7755273E+04 DMIG * STIF 2014 0 * 3002 0 -0.1277164E+04 DMIG * STIF 2015 0 * 3002 0 -0.6906018E+04 DMIG * STIF 2016 0 * 3002 0 -0.4900844E+04 DMIG * STIF 2017 0 168 * 3002 0 0.3351924E+02 DMIG * STIF 2018 0 * 3002 0 -0.2325785E+02 DMIG * STIF 2019 0 * 3002 0 -0.1509501E+03 DMIG * STIF 2020 0 * 3002 0 -0.2662023E+04 DMIG * STIF 2021 0 * 3002 0 -0.1762120E+04 DMIG * STIF 2022 0 * 3002 0 0.2147362E+04 DMIG * STIF 2023 0 * 3002 0 0.1084607E+03 DMIG * STIF 2024 0 * 3002 0 -0.7073681E+03 DMIG * STIF 2025 0 * 3002 0 0.2019467E+04 DMIG * STIF 2026 0 * 3002 0 -0.2085812E+04 DMIG * STIF 2027 0 * 3002 0 -0.1654733E+02 DMIG * STIF 2028 0 * 3002 0 0.5317734E+02 DMIG * STIF 2029 0 * 3002 0 0.1260782E+04 DMIG * STIF 2030 0 * 3002 0 0.5246555E+04 $ $=========================================================== $ w2x wrights values $=========================================================== DMIG * DAMP 0 1 1 DMIG * DAMP 100 1 * 1001 0 0.3309367E-03 DMIG * DAMP 100 4 * 1001 0 0.1113029E-01 DMIG * DAMP 100 1 * 1002 0 0.1253914E-01 DMIG * DAMP 100 4 * 1002 0 -0.5961781E-03 DMIG * DAMP 100 1 * 1003 0 -0.2745703E-01 DMIG * DAMP 100 4 * 1003 0 -0.5223356E-03 DMIG * DAMP 100 1 * 1004 0 0.4582218E-01 DMIG * DAMP 100 4 * 1004 0 0.2375875E-02 DMIG * DAMP 100 1 * 1005 0 -0.2484306E-01 DMIG * DAMP 100 4 * 1005 0 -0.7732314E-03 DMIG * DAMP 100 1 * 1006 0 -0.2277587E-01 DMIG * DAMP 100 4 169 * 1006 0 -0.1086023E-03 DMIG * DAMP 100 1 * 1007 0 -0.7025771E-03 DMIG * DAMP 100 4 * 1007 0 0.1758298E-01 DMIG * DAMP 100 1 * 1008 0 -0.5382765E-03 DMIG * DAMP 100 4 * 1008 0 -0.2068620E-01 DMIG * DAMP 100 1 * 1009 0 0.9036889E-05 DMIG * DAMP 100 4 * 1009 0 -0.4143512E-01 DMIG * DAMP 100 1 * 1010 0 -0.5023304E-03 DMIG * DAMP 100 4 * 1010 0 -0.4753906E-03 DMIG * DAMP 100 1 * 1011 0 0.4047427E-03 DMIG * DAMP 100 4 * 1011 0 -0.4141244E-03 DMIG * DAMP 100 1 * 1012 0 -0.3443037E-03 DMIG * DAMP 100 4 * 1012 0 0.4010641E-01 DMIG * DAMP 100 1 * 1013 0 -0.3871721E-03 DMIG * DAMP 100 4 * 1013 0 -0.1839331E-02 DMIG * DAMP 100 1 * 1014 0 -0.2393643E-01 DMIG * DAMP 100 4 * 1014 0 0.1352290E-04 DMIG * DAMP 100 1 * 1015 0 -0.2214878E-03 DMIG * DAMP 100 4 * 1015 0 -0.6269668E-02 DMIG * DAMP 100 1 * 1016 0 -0.4479391E-03 DMIG * DAMP 100 4 * 1016 0 -0.2383377E-01 DMIG * DAMP 100 1 * 1017 0 0.7984621E-01 DMIG * DAMP 100 4 * 1017 0 0.5092525E-03 DMIG * DAMP 100 1 * 1018 0 0.8099600E-01 DMIG * DAMP 100 4 * 1018 0 0.1100415E-02 DMIG * DAMP 100 1 * 1019 0 -0.8845699E-02 DMIG * DAMP 100 4 * 1019 0 -0.6802283E-04 DMIG * DAMP 100 1 * 1020 0 -0.1088751E-01 170 DMIG * DAMP 100 4 * 1020 0 -0.8044046E-04 DMIG * DAMP 100 1 * 1021 0 0.3285729E-03 DMIG * DAMP 100 4 * 1021 0 0.2718512E-01 DMIG * DAMP 100 1 * 1022 0 -0.2462129E-03 DMIG * DAMP 100 4 * 1022 0 -0.5042539E-02 DMIG * DAMP 100 1 * 1023 0 0.1544007E-02 DMIG * DAMP 100 4 * 1023 0 0.4316818E-03 DMIG * DAMP 100 1 * 1024 0 0.1650431E-01 DMIG * DAMP 100 4 * 1024 0 0.2179786E-02 DMIG * DAMP 100 1 * 1025 0 -0.1086126E-02 DMIG * DAMP 100 4 * 1025 0 -0.3976801E-04 DMIG * DAMP 100 1 * 1026 0 -0.2445685E-03 DMIG * DAMP 100 4 * 1026 0 0.2281926E-02 DMIG * DAMP 100 1 * 1027 0 0.7446733E-01 DMIG * DAMP 100 4 * 1027 0 0.1791563E-02 DMIG * DAMP 100 1 * 1028 0 -0.1486444E-01 DMIG * DAMP 100 4 * 1028 0 0.1130291E-03 DMIG * DAMP 100 1 * 1029 0 -0.2317654E-01 DMIG * DAMP 100 4 * 1029 0 0.1249174E-04 DMIG * DAMP 100 1 * 1030 0 -0.2593187E-03 DMIG * DAMP 100 4 * 1030 0 -0.2045866E-01 $ $=========================================================== $ G(.) load values of 1st layer $=========================================================== NOLIN1 5 2001 0 1.000 1001 0 112 NOLIN1 5 2002 0 1.000 1002 0 112 NOLIN1 5 2003 0 1.000 1003 0 112 NOLIN1 5 2004 0 1.000 1004 0 112 NOLIN1 5 2005 0 1.000 1005 0 112 NOLIN1 5 2006 0 1.000 1006 0 112 NOLIN1 5 2007 0 1.000 1007 0 112 NOLIN1 5 2008 0 1.000 1008 0 112 NOLIN1 5 2009 0 1.000 1009 0 112 171 NOLIN1 5 2010 0 1.000 1010 0 112 NOLIN1 5 2011 0 1.000 1011 0 112 NOLIN1 5 2012 0 1.000 1012 0 112 NOLIN1 5 2013 0 1.000 1013 0 112 NOLIN1 5 2014 0 1.000 1014 0 112 NOLIN1 5 2015 0 1.000 1015 0 112 NOLIN1 5 2016 0 1.000 1016 0 112 NOLIN1 5 2017 0 1.000 1017 0 112 NOLIN1 5 2018 0 1.000 1018 0 112 NOLIN1 5 2019 0 1.000 1019 0 112 NOLIN1 5 2020 0 1.000 1020 0 112 NOLIN1 5 2021 0 1.000 1021 0 112 NOLIN1 5 2022 0 1.000 1022 0 112 NOLIN1 5 2023 0 1.000 1023 0 112 NOLIN1 5 2024 0 1.000 1024 0 112 NOLIN1 5 2025 0 1.000 1025 0 112 NOLIN1 5 2026 0 1.000 1026 0 112 NOLIN1 5 2027 0 1.000 1027 0 112 NOLIN1 5 2028 0 1.000 1028 0 112 NOLIN1 5 2029 0 1.000 1029 0 112 NOLIN1 5 2030 0 1.000 1030 0 112 $ 172

Abstract (if available)

Abstract This research work reports on an integrated approach, involving carefully conducted experimental tests on reconfigurable apparatuses, that allows the study of a broad class of generic nonlinear phenomena (such as dead-space, dry friction, hysteresis, limited-slip, etc.), that generates high-quality experimental measurements, for the subsequent development of high-fidelity, nonlinear, reduced-order, mathematical models of different formats, that are useful for the monitoring, control, and simulation of realistic nonlinear structural systems.

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Asset Metadata

Creator Tasbihgoo, Farzad (author)

Core Title Analytical and experimental studies in the development of reduced-order computational models for nonlinear systems

School Viterbi School of Engineering

Degree Doctor of Philosophy

Degree Program Civil Engineering

Publication Date 09/27/2006

Defense Date 07/28/2006

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

Tag computational models,Experimental and Molecular Pathology,nonlinear systems,OAI-PMH Harvest,reduced-order,system identification

Language English

Advisor Masri, Sami F. (committee chair), Anderson, James C. (committee member), Caffrey, John P. (committee member), Ioannou, Petros A. (committee member), Wellford, L. Carter (committee member)

Creator Email tasbihgo@usc.edu

Permanent Link (DOI) https://doi.org/10.25549/usctheses-m43

Unique identifier UC1109536

Identifier etd-Tasbihgoo-20060927 (filename),usctheses-m40 (legacy collection record id),usctheses-c127-3354 (legacy record id),usctheses-m43 (legacy record id)

Legacy Identifier etd-Tasbihgoo-20060927.pdf

Dmrecord 3354

Document Type Dissertation

Rights Tasbihgoo, Farzad

Type texts

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

Repository Name Libraries, University of Southern California

Repository Location Los Angeles, California

Repository Email cisadmin@lib.usc.edu