# Introduction to Operational Modal Analysis

ISBN: 978-1-118-53514-1

Jun 2015

376 pages

Select type: O-Book

## Description

Comprehensively covers the basic principles and practice of Operational Modal Analysis (OMA).

• Covers all important aspects that are needed to understand why OMA is a practical tool for modal testing
• Covers advanced topics, including closely spaced modes, mode shape scaling, mode shape expansion and estimation of stress and strain in operational responses
• Discusses practical applications of Operational Modal Analysis
• Includes examples supported by MATLAB® applications
• Accompanied by a website hosting a MATLAB® toolbox for Operational Modal Analysis

## Related Resources

### Student

View Student Companion Site

Preface xi

1 Introduction 1

1.1 Why Conduct Vibration Test of Structures? 3

1.2 Techniques Available for Vibration Testing of Structures 3

1.3 Forced Vibration Testing Methods 4

1.4 Vibration Testing of Civil Engineering Structures 5

1.5 Parameter Estimation Techniques 5

1.6 Brief History of OMA 6

1.7 Modal Parameter Estimation Techniques 6

1.8 Perceived Limitations of OMA 10

1.9 Operating Deflection Shapes 10

1.10 Practical Considerations of OMA 11

1.11 About the Book Structure 13

References 15

2 Random Variables and Signals 17

2.1 Probability 17

2.1.1 Density Function and Expectation 17

2.1.2 Estimation by Time Averaging 19

2.1.3 Joint Distributions 21

2.2 Correlation 23

2.2.1 Concept of Correlation 23

2.2.2 Autocorrelation 24

2.2.3 Cross Correlation 25

2.2.4 Properties of Correlation Functions 27

2.3 The Gaussian Distribution 28

2.3.1 Density Function 28

2.3.2 The Central Limit Theorem 28

2.3.3 Conditional Mean and Correlation 30

References 31

3 Matrices and Regression 33

3.1 Vector and Matrix Notation 33

3.2 Vector and Matrix Algebra 35

3.2.1 Vectors and Inner Products 35

3.2.2 Matrices and Outer Products 36

3.2.3 Eigenvalue Decomposition 38

3.2.4 Singular Value Decomposition 40

3.2.5 Block Matrices 40

3.2.6 Scalar Matrix Measures 41

3.2.7 Vector and Matrix Calculus 43

3.3 Least Squares Regression 44

3.3.1 Linear Least Squares 44

3.3.2 Bias, Weighting and Covariance 47

References 52

4 Transforms 53

4.1 Continuous Time Fourier Transforms 53

4.1.1 Real Fourier Series 54

4.1.2 Complex Fourier Series 55

4.1.3 The Fourier Integral 58

4.2 Discrete Time Fourier Transforms 59

4.2.1 Discrete Time Representation 59

4.2.2 The Sampling Theorem 62

4.3 The Laplace Transform 66

4.3.1 The Laplace Transform as a generalization of the Fourier Transform 66

4.3.2 Laplace Transform Properties 67

4.3.3 Some Laplace Transforms 68

4.4 The Z-Transform 71

4.4.1 The Z-Transform as a generalization of the Fourier Series 71

4.4.2 Z-Transform Properties 73

4.4.3 Some Z-Transforms 73

4.4.4 Difference Equations and Transfer Function 75

4.4.5 Poles and Zeros 76

References 79

5 Classical Dynamics 81

5.1 Single Degree of Freedom System 82

5.1.1 Basic Equation 82

5.1.2 Free Decays 83

5.1.3 Impulse Response Function 87

5.1.4 Transfer Function 89

5.1.5 Frequency Response Function 90

5.2 Multiple Degree of Freedom Systems 92

5.2.1 Free Responses for Undamped Systems 93

5.2.2 Free Responses for Proportional Damping 95

5.2.3 General Solutions for Proportional Damping 95

5.2.4 Transfer Function and FRF Matrix for Proportional Damping 96

5.2.5 General Damping 99

5.3 Special Topics 107

5.3.1 Structural Modification Theory 107

5.3.2 Sensitivity Equations 109

5.3.3 Closely Spaced Modes 110

5.3.4 Model Reduction (SEREP) 114

5.3.5 Discrete Time Representations 116

5.3.6 Simulation of OMA Responses 119

References 121

6 Random Vibrations 123

6.1 General Inputs 123

6.1.1 Linear Systems 123

6.1.2 Spectral Density 125

6.1.3 SISO Fundamental Theorem 128

6.1.4 MIMO Fundamental Theorem 129

6.2 White Noise Inputs 130

6.2.1 Concept of White Noise 130

6.2.2 Decomposition in Time Domain 131

6.2.3 Decomposition in Frequency Domain 134

6.2.4 Zeroes of the Spectral Density Matrix 137

6.2.5 Residue Form 139

6.2.6 Approximate Residue Form 140

6.3 Uncorrelated Modal Coordinates 143

6.3.1 Concept of Uncorrelated Modal Coordinates 143

6.3.2 Decomposition in Time Domain 144

6.3.3 Decomposition in Frequency Domain 145

References 147

7 Measurement Technology 149

7.1 Test Planning 149

7.1.1 Test Objectives 149

7.1.2 Field Visit and Site Inspection 150

7.1.3 Field Work Preparation 150

7.1.4 Field Work 151

7.2 Specifying Dynamic Measurements 152

7.2.1 General Considerations 152

7.2.2 Number and Locations of Sensors 154

7.2.3 Sampling Rate 158

7.2.4 Length of Time Series 159

7.2.5 Data Sets and References 160

7.2.6 Expected Vibration Level 162

7.3 Sensors and Data Acquisition 168

7.3.1 Sensor Principles 168

7.3.2 Sensor Characteristics 169

7.3.3 The Piezoelectric Accelerometer 173

7.3.4 Sensors Used in Civil Engineering Testing 175

7.3.5 Data Acquisition 179

7.3.6 Antialiasing 182

7.3.7 System Measurement Range 182

7.3.8 Noise Sources 183

7.3.9 Cabled or Wireless Sensors? 187

7.3.10 Calibration 188

7.3.11 Noise Floor Estimation 191

7.3.12 Very Low Frequencies and Influence of Tilt 194

7.4 Data Quality Assessment 196

7.4.1 Data Acquisition Settings 196

7.4.2 Excessive Noise from External Equipment 197

7.4.3 Checking the Signal-to-Noise Ratio 197

7.4.4 Outliers 197

7.5 Chapter Summary – Good Testing Practice 198

References 199

8 Signal Processing 201

8.1 Basic Preprocessing 201

8.1.1 Data Quality 202

8.1.2 Calibration 202

8.1.3 Detrending and Segmenting 203

8.2 Signal Classification 204

8.2.1 Operating Condition Sorting 204

8.2.2 Stationarity 205

8.2.3 Harmonics 206

8.3 Filtering 208

8.3.1 Digital Filter Main Types 209

8.3.2 Two Averaging Filter Examples 210

8.3.3 Down-Sampling and Up-Sampling 212

8.3.4 Filter Banks 213

8.3.5 FFT Filtering 213

8.3.6 Integration and Differentiation 214

8.3.7 The OMA Filtering Principles 216

8.4 Correlation Function Estimation 218

8.4.1 Direct Estimation 219

8.4.2 Biased Welch Estimate 221

8.4.3 Unbiased Welch Estimate (Zero Padding) 222

8.4.4 Random Decrement 224

8.5 Spectral Density Estimation 229

8.5.1 Direct Estimation 229

8.5.2 Welch Estimation and Leakage 229

8.5.3 Random Decrement Estimation 232

8.5.4 Half Spectra 233

8.5.5 Correlation Tail and Tapering 233

References 237

9 Time Domain Identification 239

9.1 Common Challenges in Time Domain Identification 240

9.1.1 Fitting the Correlation Functions (Modal Participation) 240

9.1.2 Seeking the Best Conditions (Stabilization Diagrams) 242

9.2 AR Models and Poly Reference (PR) 242

9.3 ARMA Models 244

9.4 Ibrahim Time Domain (ITD) 248

9.5 The Eigensystem Realization Algorithm (ERA) 251

9.6 Stochastic Subspace Identification (SSI) 254

References 258

10 Frequency-Domain Identification 261

10.1 Common Challenges in Frequency-Domain Identification 262

10.1.1 Fitting the Spectral Functions (Modal Participation) 262

10.1.2 Seeking the Best Conditions (Stabilization Diagrams) 263

10.2 Classical Frequency-Domain Approach (Basic Frequency Domain) 265

10.3 Frequency-Domain Decomposition (FDD) 266

10.3.1 FDD Main Idea 266

10.3.2 FDD Approximations 267

10.3.3 Mode Shape Estimation 269

10.3.4 Pole Estimation 271

10.4 ARMA Models in Frequency Domain 275

References 278

11 Applications 281

11.1 Some Practical Issues 281

11.1.1 Modal Assurance Criterion (MAC) 282

11.1.2 Stabilization Diagrams 282

11.1.3 Mode Shape Merging 283

11.2 Main Areas of Application 284

11.2.1 OMA Results Validation 284

11.2.2 Model Validation 285

11.2.3 Model Updating 285

11.2.4 Structural Health Monitoring 288

11.3 Case Studies 291

11.3.1 Tall Building 292

11.3.2 Long Span Bridge 297

11.3.3 Container Ship 301

References 306

12.1 Closely Spaced Modes 307

12.1.1 Implications for the Identification 308

12.1.2 Implications for Modal Validation 308

12.2 Uncertainty Estimation 309

12.2.1 Repeated Identification 309

12.2.2 Covariance Matrix Estimation 310

12.3 Mode Shape Expansion 311

12.3.1 FE Mode Shape Subspaces 311

12.3.2 FE Mode Shape Subspaces Using SEREP 312

12.3.3 Optimizing the Number of FE Modes (LC Principle) 313

12.4 Modal Indicators and Automated Identification 315

12.4.1 Oversized Models and Noise Modes 315

12.4.2 Generalized Stabilization and Modal Indicators 315

12.4.3 Automated OMA 318

12.5 Modal Filtering 319

12.5.1 Modal Filtering in Time Domain 319

12.5.2 Modal Filtering in Frequency Domain 320

12.5.3 Generalized Operating Deflection Shapes (ODS) 320

12.6 Mode Shape Scaling 320

12.6.1 Mass Change Method 321

12.6.2 Mass-Stiffness Change Method 322

12.6.3 Using the FEM Mass Matrix 323

12.7 Force Estimation 323

12.7.1 Inverting the FRF Matrix 324

12.7.2 Modal Filtering 324

12.8 Estimation of Stress and Strain 324

12.8.1 Stress and Strain from Force Estimation 324

12.8.2 Stress and Strain from Mode Shape Expansion 325

References 325

Appendix A Nomenclature and Key Equations 327

Appendix B Operational Modal Testing of the Heritage Court Tower 335

B.1 Introduction 335

B.2 Description of the Building 335

B.3 Operational Modal Testing 336

B.3.1 Vibration Data Acquisition System 338

B.4 Vibration Measurements 338

B.4.1 Test Setups 341

B.4.2 Test Results 341

B.5 Analysis of the HCT Cases 342

B.5.1 FDD Modal Estimation 342

B.5.2 SSI Modal Estimation 343

B.5.3 Modal Validation 343

References 346

Appendix C Dynamics in Short 347

C.1 Basic Equations 347

C.2 Basic Form of the Transfer and Impulse Response Functions 348

C.3 Free Decays 348

C.4 Classical Form of the Transfer and Impulse Response Functions 349

C.5 Complete Analytical Solution 350

C.6 Eigenvector Scaling 351

C.7 Closing Remarks 351

References 352

Index 353

""This is an interesting book for anybody dealing with vibrations, density functions, and with data and signal processing.......I certainly recommend it as a textbook for graduate study in universities."" (Zentralblatt MATH 2016)