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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
7.2.7 Loading Source Correlation and Artificial Excitation 164
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 Advanced Subjects 307
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
- Gregg Easterbrook
The engineering field that studies the modal properties of systems under ambient vibrations or normal operating conditions is called Operational Modal Analysis (OMA) and provides useful methods for modal analysis of many areas of structural engineering. Identification of modal properties of a structural system is the process of correlating the dynamic characteristics of a mathematical model with the physical properties of the system derived from experimental measurements.
It is fair to say that processing of data in OMA is challenging; one can even say that this is close to torturing the data, and it is also fair to say that fiddling around long enough with the data might lead to some strange or erroneous results that might look like reasonable results. One of the aims of this book is to help people who use OMA techniques avoid ending up in this situation, and instead obtain results that are valid and reasonable.
In OMA, measurement data obtained from the operational responses are used to estimate the parameters of models that describe the system behavior. To fully understand this process, one should have knowledge of classical structural mechanics, matrix analysis, random vibration concepts, application-specific simplifying assumptions, and practical aspects related to vibration measurement, data acquisition, and signal processing.
OMA testing techniques have now become quite attractive, due to their relatively low cost and speed of implementation and the recent improvements in recording equipment and computational methods. Table 1.1 provides a quick summary of the typical applications of OMA and how these compare with classical modal testing, also denoted experimental modal analysis (EMA), which is based on controlled input that is measured and used in the identification process.
Table 1.1 General characteristics of structural response
Source: Adapted from American National Standard: "Vibration of Buildings - guidelines for the measurement of vibrations and their effects on buildings," ANSI S2.47-1990 (ASA 95-1990).
The fundamental idea of OMA testing techniques is that the structure to be tested is being excited by some type of excitation that has approximately white noise characteristics, that is, it has energy distributed over a wide frequency range that covers the frequency range of the modal characteristics of the structure. However, it does not matter much if the actual loads do not have exact white noise characteristics, since what is really important is that all the modes of interest are adequately excited so that their contributions can be captured by the measurements.
Referring to Figure 1.1, the concept of nonwhite, but broadband loading can be explained as follows. The loading is colored, thus does not necessarily have an ideal flat spectrum, but the colored loads can be considered as the output from an imaginary (loading) filter that is loaded by white noise.
Figure 1.1 Illustration of the concept of OMA. The nonwhite noise loads are modeled as the output from a filter loaded by a white noise load
It can be proved that the concept of including an additional filter describing the coloring of the loads does not change the physical modes of the system, see Ibrahim et al. [1] and Sections 7.2.7 and 8.3.7. The coloring filter concept shows that in general what we are estimating in OMA is the modal model for "the whole system" including both the structural system and the loading filter.
When interpreting the modal results, this has to be kept in mind, because, some modes might be present due to the loading conditions and some might come from the structural system. We should also note that in practice we often estimate a much larger number of modes than the expected physical number of modes of the considered system.
This means that we need to find ways to justify which modes belong to the structural system, which modes might describe the coloring of the loading, and finally which modes are just noise modes that might not have any physical meaning. These kinds of considerations are important in OMA, and will be further illustrated later in this book.
We can conclude these first remarks by saying that OMA is the process of characterizing the dynamic properties of an elastic structure by identifying its natural modes of vibration from the operating responses. Each mode is associated with a specific natural frequency and damping factor, and these two parameters can be identified from vibration data from practically any point on the structure. In addition, each mode has a characteristic "mode shape," which defines the spatial distribution of movement over the entire structure.
Vibration measurements are made for a variety of reasons. They could be used to determine the natural frequencies of a structure, to verify analytical models of the structure, to determine its dynamic response under various environmental conditions, or to monitor the condition of a structure under various loading conditions. As structural analysis techniques continually evolve and become increasingly sophisticated, awareness grows of potential shortcomings in their representation of the structural behavior. This is prevalent in the field of structural dynamics. The justification and technology exists for vibration testing and analysis of structures.
Large civil engineering structures are usually too complex for accurate dynamic analysis by hand. It is typical to use matrix algebra based solution methods, using the finite element method of structural modeling and analysis, on digital computers. All linear models have dynamic properties, which can be compared with testing and analysis techniques such as OMA.
Let us discuss in some detail the two main types of modal testing: the EMA that uses controlled input forces and the OMA that uses the operational forces.
Both forced vibration and in-operation methods have been used in the past and are capable of determining the dynamic characteristics of structures. Forced vibration methods can be significantly more complex than in-operation vibration tests, and are generally more expensive than in-operation vibration tests, especially for large and massive structures. The main advantage of forced vibration over in-operation vibration is that in the former the level of excitation and induced vibration can be carefully controlled, while for the latter one has to rely on the forces of nature and uncontrolled artificial forces (i.e., vehicle traffic in bridges) to excite the structure, sometimes at very low levels of vibration. The sensitivity of sensors used for in-operation vibration measurements is generally much higher than those required for forced vibration tests.
By definition, any source of controlled excitation being applied to any structure in order to induce vibrations constitutes a forced vibration test. In-operation tests that rely on ambient excitation are used to test structures such as bridges, nuclear power plants, offshore platforms, and buildings. While ambient tests do not require traffic shutdowns or interruptions of normal operations, the amount of data collected is significant and it can be a complex task to analyze this data thoroughly.
The techniques for data analysis are different. The theory for forced vibration tests of large structures is well developed and is almost a natural extension of the techniques used in forced vibration tests of mechanical systems. In contrast, the theory for ambient vibration tests still requires further development.
Forced vibration tests or EMA methods are generally used to determine the dynamic characteristics of small and medium size structures. In rare occasions, these methods are used on very large structures because of the complexity associated with providing significant levels of excitation to a large, massive structure. In these tests, controlled forces are applied to a structure to induce vibrations. By measuring the structure's response to these known forces, one can determine the structure's dynamic properties. The measured excitation and acceleration response time histories are used to compute frequency response functions (FRFs) between a measured point and the point of input. These FRFs can be used to determine the natural frequencies, mode shapes, and damping values of the structure using well-established...
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