Modeling and Estimation of Structural Damage
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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.
Content
Preface xi
1 Introduction 1
1.1 Users' Guide 1
1.2 Modeling and Estimation Overview 2
1.3 Motivation 4
1.4 Structural Health Monitoring 7
1.4.1 Data-Driven Approaches 10
1.4.2 Physics-Based Approach 14
1.5 Organization and Scope 17
2 Probability 21
2.1 Probability Basics 23
2.2 Probability Distributions 25
2.3 Multivariate Distributions, Conditional Probability, and Independence 28
2.4 Functions of Random Variables 32
2.5 Expectations and Moments 39
2.6 Moment-Generating Functions and Cumulants 43
3 Random Processes 51
3.1 Properties of a Random Process 54
3.2 Stationarity 57
3.3 Spectral Analysis 61
3.3.1 Spectral Representation of Deterministic Signals 62
3.3.2 Spectral Representation of Stochastic Signals 65
3.3.3 Power Spectral Density 67
3.3.4 Relationship to Correlation Functions 71
3.3.5 Higher Order Spectra 74
3.4 Markov Models 81
3.5 Information Theoretics 82
3.5.1 Mutual Information 85
3.5.2 Transfer Entropy 87
3.6 Random Process Models for Structural Response Data 91
4 Modeling in Structural Dynamics 95
4.1 Why Build Mathematical Models? 96
4.2 Good Versus Bad Models - An Example 97
4.3 Elements of Modeling 99
4.3.1 Newton's Laws 101
4.3.2 Background to Variational Methods 101
4.3.3 Variational Mechanics 103
4.3.4 Lagrange's Equations 105
4.3.5 Hamilton's Principle 108
4.4 Common Challenges 114
4.4.1 Impact Problems 114
4.4.2 Stress Singularities and Cracking 117
4.5 Solution Techniques 119
4.5.1 Analytical Techniques I - Ordinary Differential Equations 119
4.5.2 Analytical Techniques II - Partial Differential Equations 128
4.5.3 Local Discretizations 131
4.5.4 Global Discretizations 132
4.6 Volterra Series for Nonlinear Systems 133
5 Physics-Based Model Examples 143
5.1 Imperfection Modeling in Plates 143
5.1.1 Cracks as Imperfections 143
5.1.2 Boundary Imperfections: In-Plane Slippage 145
5.2 Delamination in a Composite Beam 151
5.3 Bolted Joint Degradation: Quasi-static Approach 160
5.3.1 The Model 161
5.3.2 Experimental System and Procedure 164
5.3.3 Results and Discussion 166
5.4 Bolted Joint Degradation: Dynamic Approach 172
5.5 Corrosion Damage 178
5.6 Beam on a Tensionless Foundation 182
5.6.1 Equilibrium Equations and Their Solutions 184
5.6.2 Boundary Conditions 185
5.6.3 Results 187
5.7 Cracked, Axially Moving Wires 189
5.7.1 Some Useful Concepts from Fracture Mechanics 191
5.7.2 The Effect of a Crack on the Local Stiffness 193
5.7.3 Limitations 194
5.7.4 Equations of Motion 196
5.7.5 Natural Frequencies and Stability 198
5.7.6 Results 198
6 Estimating Statistical Properties of Structural Response Data 203
6.1 Estimator Bias and Variance 206
6.2 Method of Maximum Likelihood 209
6.3 Ergodicity 213
6.4 Power Spectral Density and Correlation Functions for LTI Systems 218
6.4.1 Estimation of Power Spectral Density 218
6.4.2 Estimation of Correlation Functions 234
6.5 Estimating Higher Order Spectra 240
6.5.1 Coherence Functions 246
6.5.2 Bispectral Density Estimation 248
6.5.3 Analytical Bicoherence for Non-Gaussian Signals 257
6.5.4 Trispectral Density Function 264
6.6 Estimation of Information Theoretics 275
6.7 Generating Random Processes 284
6.7.1 Review of Basic Concepts 285
6.7.2 Data with a Known Covariance and Gaussian Marginal PDF 287
6.7.3 Data with a Known Covariance and Arbitrary Marginal PDF 290
6.7.4 Examples 295
6.8 Stationarity Testing 302
6.8.1 Reverse Arrangement Test 304
6.8.2 Evolutionary Spectral Testing 306
6.9 Hypothesis Testing and Intervals of Confidence 312
6.9.1 Detection Strategies 313
6.9.2 Detector Performance 319
6.9.3 Intervals of Confidence 327
7 Parameter Estimation for Structural Systems 333
7.1 Method of Maximum Likelihood 336
7.1.1 Linear Least Squares 338
7.1.2 Finite Element Model Updating 341
7.1.3 Modified Differential Evolution for Obtaining MLEs 344
7.1.4 Structural Damage MLE Example 347
7.1.5 Estimating Time of Flight for Ultrasonic Applications 352
7.2 Bayesian Estimation 363
7.2.1 Conjugacy 365
7.2.2 Using Conjugacy to Assess Algorithm Performance 366
7.2.3 Markov Chain Monte Carlo (MCMC) Methods 374
7.2.4 Gibbs Sampling 379
7.2.5 Conditional Conjugacy: Sampling the Noise Variance 380
7.2.6 Beam Example Revisited 383
7.2.7 Population-Based MCMC 386
7.3 Multimodel Inference 392
7.3.1 Model Comparison via AIC 392
7.3.2 Reversible Jump MCMC 397
8 Detecting Damage-Induced Nonlinearity 403
8.1 Capturing Nonlinearity 407
8.1.1 Higher Order Cumulants 408
8.1.2 Higher Order Spectral Coefficients 410
8.1.3 Nonlinear Prediction Error 412
8.1.4 Information Theoretics 414
8.2 Bolted Joint Revisited 415
8.2.1 Composite Joint Experiment 415
8.2.2 Kurtosis Results 417
8.2.3 Spectral Results 419
8.3 Bispectral Detection: The Single Degree-of-Freedom (SDOF), Gaussian Case 421
8.3.1 Bispectral Detection Statistic 422
8.3.2 Test Statistic Distribution 423
8.3.3 Detector Performance 425
8.4 Bispectral Detection: the General Multi-Degree-of-Freedom (MDOF) Case 429
8.4.1 Bicoherence Detection Statistic Distribution 433
8.4.2 Which Bicoherence to Compute? 434
8.4.3 Optimal Input Probability Distribution for Detection 436
8.5 Application of the HOS to Delamination Detection 438
8.6 Method of Surrogate Data 444
8.6.1 Fourier Transform-Based Surrogates 446
8.6.2 AAFT Surrogates 448
8.6.3 IAFFT Surrogates 449
8.6.4 DFT Surrogates 450
8.7 Numerical Surrogate Examples 451
8.7.1 Detection of Bilinear Stiffness 451
8.7.2 Detecting Cubic Stiffness 456
8.7.3 Surrogate Invariance to Ambient Variation 461
8.8 Surrogate Experiments 464
8.8.1 Detection of Rotor - Stator Rub 465
8.8.2 Bolted Joint Degradation with Ocean Wave Excitation 467
8.9 Surrogates for Nonstationary Data 475
8.10 Chapter Summary 476
9 Damage Identification 481
9.1 Modeling and Identification of Imperfections in Shell Structures 481
9.1.1 Modeling of Submerged Shell Structures 482
9.1.2 Non-Contact Results Using Maximum Likelihood 487
9.1.3 Bayesian Identification of Dents 492
9.2 Modeling and Identification of Delamination 501
9.3 Modeling and Identification of Cracked Structures 508
9.3.1 Cracked Plate Model 508
9.3.2 Crack Parameter Identification 510
9.3.3 Optimization of Sensor Placement 523
9.4 Modeling and Identification of Corrosion 527
9.4.1 Experimental Setup 530
9.4.2 Results and Discussion 532
9.5 Chapter Summary 538
10 Decision Making in Condition-Based Maintenance 543
10.1 Structured Decision Making 544
10.2 Example: Ship in Transit 545
10.2.1 Loading Data 547
10.2.2 Ship "Stringer" Model 552
10.2.3 Cumulative Fatigue Model 559
10.3 Optimal Transit 562
10.3.1 Problem Statement 562
10.3.2 Solutions via Dynamic Programming 563
10.3.3 Transit Examples 565
10.4 Summary 568
Appendix A Useful Constants and Probability Distributions 571
Appendix B Contour Integration of Spectral Density Functions 575
Appendix C Derivation of Terms for the Trispectrum of an MDOF Nonlinear Structure 581
C.1 Simplification of CVIII pijk (t1, t2, t3) 582
C.2 Submanifold Terms in the Trispectrum 583
C.3 Complete Trispectrum Expression 585
Index 587
