Modeling and Estimation of Structural Damage

Jonathan M. Nichols received the B.Sc. degree from the University of Delaware in 1997 and the M. Sc. and Ph.D. degrees from Duke University in 1999 and 2002 respectively, all in Mechanical Engineering.?He is currently the Associate Superintendent for the Naval Research Laboratory Optical Sciences Division in Washington, D.C. His research interests include damage identification in structures, modelling and analysis of infrared imaging devices, signal and image processing, and parameter estimation. Kevin D. Murphy received the B.Sc. (Mechanical Engineering) and M. Sc. (Applied Mechanics) degrees from the University of Michigan in 1988 and 1990 respectively.?He received his Ph.D. from Duke University in 1994 in Mechanical Engineering.?He is currently a Professor and Mechanical Engineering Department Chair at the University of Louisville. His research focuses on the nonlinear mechanics, vibrations, and stability of structures for a broad variety of applications. Specific applications areas include: vibrations of damaged structures, adhesion/sticking contact in MEMS devices, and vibrations in manufacturing problems.

• Preface xi1 Introduction 11.1 Users' Guide 11.2 Modeling and Estimation Overview 21.3 Motivation 41.4 Structural Health Monitoring 71.4.1 Data-Driven Approaches 101.4.2 Physics-Based Approach 141.5 Organization and Scope 172 Probability 212.1 Probability Basics 232.2 Probability Distributions 252.3 Multivariate Distributions, Conditional Probability, and Independence 282.4 Functions of Random Variables 322.5 Expectations and Moments 392.6 Moment-Generating Functions and Cumulants 433 Random Processes 513.1 Properties of a Random Process 543.2 Stationarity 573.3 Spectral Analysis 613.3.1 Spectral Representation of Deterministic Signals 623.3.2 Spectral Representation of Stochastic Signals 653.3.3 Power Spectral Density 673.3.4 Relationship to Correlation Functions 713.3.5 Higher Order Spectra 743.4 Markov Models 813.5 Information Theoretics 823.5.1 Mutual Information 853.5.2 Transfer Entropy 873.6 Random Process Models for Structural Response Data 914 Modeling in Structural Dynamics 954.1 Why Build Mathematical Models? 964.2 Good Versus Bad Models – An Example 974.3 Elements of Modeling 994.3.1 Newton's Laws 1014.3.2 Background to Variational Methods 1014.3.3 Variational Mechanics 1034.3.4 Lagrange's Equations 1054.3.5 Hamilton's Principle 1084.4 Common Challenges 1144.4.1 Impact Problems 1144.4.2 Stress Singularities and Cracking 1174.5 Solution Techniques 1194.5.1 Analytical Techniques I – Ordinary Differential Equations 1194.5.2 Analytical Techniques II – Partial Differential Equations 1284.5.3 Local Discretizations 1314.5.4 Global Discretizations 1324.6 Volterra Series for Nonlinear Systems 1335 Physics-Based Model Examples 1435.1 Imperfection Modeling in Plates 1435.1.1 Cracks as Imperfections 1435.1.2 Boundary Imperfections: In-Plane Slippage 1455.2 Delamination in a Composite Beam 1515.3 Bolted Joint Degradation: Quasi-static Approach 1605.3.1 The Model 1615.3.2 Experimental System and Procedure 1645.3.3 Results and Discussion 1665.4 Bolted Joint Degradation: Dynamic Approach 1725.5 Corrosion Damage 1785.6 Beam on a Tensionless Foundation 1825.6.1 Equilibrium Equations and Their Solutions 1845.6.2 Boundary Conditions 1855.6.3 Results 1875.7 Cracked, Axially Moving Wires 1895.7.1 Some Useful Concepts from Fracture Mechanics 1915.7.2 The Effect of a Crack on the Local Stiffness 1935.7.3 Limitations 1945.7.4 Equations of Motion 1965.7.5 Natural Frequencies and Stability 1985.7.6 Results 1986 Estimating Statistical Properties of Structural Response Data 2036.1 Estimator Bias and Variance 2066.2 Method of Maximum Likelihood 2096.3 Ergodicity 2136.4 Power Spectral Density and Correlation Functions for LTI Systems 2186.4.1 Estimation of Power Spectral Density 2186.4.2 Estimation of Correlation Functions 2346.5 Estimating Higher Order Spectra 2406.5.1 Coherence Functions 2466.5.2 Bispectral Density Estimation 2486.5.3 Analytical Bicoherence for Non-Gaussian Signals 2576.5.4 Trispectral Density Function 2646.6 Estimation of Information Theoretics 2756.7 Generating Random Processes 2846.7.1 Review of Basic Concepts 2856.7.2 Data with a Known Covariance and Gaussian Marginal PDF 2876.7.3 Data with a Known Covariance and Arbitrary Marginal PDF 2906.7.4 Examples 2956.8 Stationarity Testing 3026.8.1 Reverse Arrangement Test 3046.8.2 Evolutionary Spectral Testing 3066.9 Hypothesis Testing and Intervals of Confidence 3126.9.1 Detection Strategies 3136.9.2 Detector Performance 3196.9.3 Intervals of Confidence 3277 Parameter Estimation for Structural Systems 3337.1 Method of Maximum Likelihood 3367.1.1 Linear Least Squares 3387.1.2 Finite Element Model Updating 3417.1.3 Modified Differential Evolution for Obtaining MLEs 3447.1.4 Structural Damage MLE Example 3477.1.5 Estimating Time of Flight for Ultrasonic Applications 3527.2 Bayesian Estimation 3637.2.1 Conjugacy 3657.2.2 Using Conjugacy to Assess Algorithm Performance 3667.2.3 Markov Chain Monte Carlo (MCMC) Methods 3747.2.4 Gibbs Sampling 3797.2.5 Conditional Conjugacy: Sampling the Noise Variance 3807.2.6 Beam Example Revisited 3837.2.7 Population-Based MCMC 3867.3 Multimodel Inference 3927.3.1 Model Comparison via AIC 3927.3.2 Reversible Jump MCMC 3978 Detecting Damage-Induced Nonlinearity 4038.1 Capturing Nonlinearity 4078.1.1 Higher Order Cumulants 4088.1.2 Higher Order Spectral Coefficients 4108.1.3 Nonlinear Prediction Error 4128.1.4 Information Theoretics 4148.2 Bolted Joint Revisited 4158.2.1 Composite Joint Experiment 4158.2.2 Kurtosis Results 4178.2.3 Spectral Results 4198.3 Bispectral Detection: The Single Degree-of-Freedom (SDOF), Gaussian Case 4218.3.1 Bispectral Detection Statistic 4228.3.2 Test Statistic Distribution 4238.3.3 Detector Performance 4258.4 Bispectral Detection: the General Multi-Degree-of-Freedom (MDOF) Case 4298.4.1 Bicoherence Detection Statistic Distribution 4338.4.2 Which Bicoherence to Compute? 4348.4.3 Optimal Input Probability Distribution for Detection 4368.5 Application of the HOS to Delamination Detection 4388.6 Method of Surrogate Data 4448.6.1 Fourier Transform-Based Surrogates 4468.6.2 AAFT Surrogates 4488.6.3 IAFFT Surrogates 4498.6.4 DFT Surrogates 4508.7 Numerical Surrogate Examples 4518.7.1 Detection of Bilinear Stiffness 4518.7.2 Detecting Cubic Stiffness 4568.7.3 Surrogate Invariance to Ambient Variation 4618.8 Surrogate Experiments 4648.8.1 Detection of Rotor – Stator Rub 4658.8.2 Bolted Joint Degradation with Ocean Wave Excitation 4678.9 Surrogates for Nonstationary Data 4758.10 Chapter Summary 4769 Damage Identification 4819.1 Modeling and Identification of Imperfections in Shell Structures 4819.1.1 Modeling of Submerged Shell Structures 4829.1.2 Non-Contact Results Using Maximum Likelihood 4879.1.3 Bayesian Identification of Dents 4929.2 Modeling and Identification of Delamination 5019.3 Modeling and Identification of Cracked Structures 5089.3.1 Cracked Plate Model 5089.3.2 Crack Parameter Identification 5109.3.3 Optimization of Sensor Placement 5239.4 Modeling and Identification of Corrosion 5279.4.1 Experimental Setup 5309.4.2 Results and Discussion 5329.5 Chapter Summary 53810 Decision Making in Condition-Based Maintenance 54310.1 Structured Decision Making 54410.2 Example: Ship in Transit 54510.2.1 Loading Data 54710.2.2 Ship "Stringer" Model 55210.2.3 Cumulative Fatigue Model 55910.3 Optimal Transit 56210.3.1 Problem Statement 56210.3.2 Solutions via Dynamic Programming 56310.3.3 Transit Examples 56510.4 Summary 568Appendix A Useful Constants and Probability Distributions 571Appendix B Contour Integration of Spectral Density Functions 575Appendix C Derivation of Terms for the Trispectrum of an MDOF Nonlinear Structure 581C.1 Simplification of CVIII pijk (τ1, τ2, τ3) 582C.2 Submanifold Terms in the Trispectrum 583C.3 Complete Trispectrum Expression 585Index 587