اسم المؤلف
Rune Brincker , Carlos E. Ventura
التاريخ
3 مارس 2022
المشاهدات
534
التقييم
(لا توجد تقييمات)

Introduction to Operational Modal Analysis
Rune Brincker
Technical University of Denmark
Carlos E. Ventura
Contents
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 38vi Contents
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 121Contents vii
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 197viii Contents
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) 266Contents ix
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 324x Contents
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 Preface
Index
2DOF system, 29, 98, 119
Accelerometers, 4, 169, 179, 194, 214, 293, 294,
302
Actuator, 177
Sensitivity, 178
Aliasing, 62, 158, 182
Ambient vibration tests, 3, 293, 297
Analog-to-digital (AD) converter, 15, 153, 159
dynamic range of, 181
sigma-delta, 159, 181
Angular frequency, 59, 84, 90
dimensionless, 71
Antialiasing, 182
filter, 62, 158, 183, 208, 212, 298
Applications OMA,
Container ship, 284, 301
Tall building, 4, 292
Vasco da Gama Bridge, 297
AR model, 242, 251, 261, 276
free decay, 242
ARMA model, 75, 116, 120, 239, 244, 246, 275
covariance equivalent, 116, 120, 246
prediction error, 247
ARMA models in frequency domain, 275
unit circle, 275
ARTeMIS Extractor software, 254, 265, 302
Artificial excitation, 2, 164, 167
hammer input, 2, 165
random, 2, 165
shaker input, 2, 165
Auto correlation function, 25, 234, 244, 257
normalized, 25
time averaging, 25
symmetry relation, 25
Auto regressive (AR) matrices, 116
Automated identification, 291, 307, 315, 318,
319
OMA, 318
Band pass filter, 64, 209, 213, 217, 243, 250,
256, 319
Basic frequency domain (BFD) method, 265
Bernal equation, 222
Bias error, 153, 171, 196, 221
Bits, 180, 297
effective, 181
least significant, 181
Block Hankel matrix, 41, 220, 242, 248, 252, 255
Block matrix, 41, 230
Block Toeplitz matrix, 41, 220, 249, 255
Boltzmann constant, 185
Brownian motion, 177, 185, 187
Buoyant force, 172, 177, 187
Cable (wire), 7, 13, 150, 168, 175, 188, 197, 293
co-axial, 175
Calculus, 43
Canonical variate algorithm (CVA), 256
Capacitor, 174
Central limit theorem, 28, 30, 205
Characteristic equation, 95, 102
Characteristics of sensors, 169
Charge amplifier, 168, 175
Charge, 168, 173
density, 173354 Index
Clipping, 164, 196, 202
Closely spaced modes, 11, 82, 110, 112, 114,
141, 154, 164, 234, 237, 240, 266, 268,
271, 282, 284
Coherence, 8, 189, 191, 283, 317
function, 8
modal, 317
Companion matrix, 117, 119, 243, 246, 277
Complex mode indicator function
(CMIF), 266
Continuous time pole, 70, 78, 83, 117, 248,
317
Control current, 178
Controllability matrix, 252
Convolution, 28, 57, 59, 67, 73, 88, 90, 123, 127,
129, 131, 143, 221, 223, 323
matrix, 34
theorem, 57
Correlation function estimation, 218
direct, 219
unbiased, 222
random decrement, 224
tail and tapering, 233
Correlation function,
derivative of, 27
deterministic, 123
matrix, 41, 127, 129, 131, 141, 143, 219,
228, 234, 236, 245, 257
Covariance equivalent, 120, 246
Covariance matrix estimation
Covariance matrix, 47, 50, 131, 138, 157, 165
possible, 27
Cross correlation function, 26, 126, 128
Cross spectral density, 126, 138, 190
Crystal, 173, 175
Cut-off frequency, 203, 209, 216
Damping, 95, 244
Data acquisition, 151, 153, 158, 168, 179,
183, 188, 196, 293, 301
Data driven SSI, 254
Data quality, 173, 196, 198, 201
Data segment, 28, 64, 121, 151, 159
overlap, 159
Data set, 7, 11, 19, 41, 63, 205, 217, 234, 236,
243, 245, 250, 253, 257
reference, 160
Decimation (down sampling), 159, 181, 190,
209, 212, 298, 342
Decomposition, 131, 145
eigenvalue, 38, 93, 101, 119, 243, 246, 277,
348, 351
frequency domain (FDD), 8, 11, 154, 208,
261, 266, 272, 294, 299, 304
modal, 96, 115, 117, 132, 134, 138, 140, 143,
145, 155, 268, 270, 273
singular value, 12, 27, 40, 155, 191, 197, 232
Delta function, 68, 70, 73, 77, 87, 89, 97, 130,
133, 211
Derivative, 24, 45, 58, 63, 65, 68, 83, 169, 215,
226, 228
of correlation function, 27, 127, 228
matrix, 43
vector, 44
Design matrix, 45, 49, 52, 310
Detrending, 203
Diagonal matrix, 27, 34, 36, 38, 40, 47, 93, 97,
101, 105, 109, 118, 143, 241, 248, 252,
256, 262, 267, 348, 351
Dielectric material, 173
Difference equation, 71, 75, 77, 116, 246, 275
Differentiation theorem, 58, 63, 68, 70, 73, 90,
352
Dimensionless frequency, 100, 172, 275, 318
Direct Current (DC), 61, 64, 92, 130, 138, 168,
170, 172, 175, 179, 189, 203, 209, 211,
215, 268, 272, 275
Discrete Fourier transform (DFT), 53, 61, 211,
229, 262, 264, 272
Discrete impulse response, 76
Discrete time, 15, 55, 57, 59, 61, 63, 65, 71, 73,
77, 118, 206, 212, 218, 231, 235, 241,
243, 245, 247, 250, 263
pole, 117, 248, 250
representation, 59, 61, 11
system matrix, 118, 251, 256
Displacement sensor, 177, 179
Distribution function, 18, 20
DOF, 10, 29, 70, 76, 78, 81, 87, 89, 95, 98, 102,
105, 107, 114, 119, 124, 138, 168, 175,
177, 179, 190, 229, 246, 272, 274, 283,
310, 313, 319
active, 115
experimental, 115, 324
FE model, 324
fitting, 314
reference, 283, 323
Down-sampling, 212
up-sampling, 212
Duhamel’s integral, 88
Dynamic matrix, 93Index 355
Dynamic range, 153, 169, 171, 175, 179, 181,
183, 207, 302
Dynamic response, 3, 8, 29, 152, 155, 168, 287
Effective bits, 181
Eigenvalue, 27, 38, 42, 93, 101, 103, 108, 114,
117, 136, 240, 243, 248, 250, 252, 262,
277, 348, 350
decomposition, 38, 93, 101, 119, 243, 250,
277, 348
Eigenvector, 38, 93, 96, 98, 101, 103, 106, 108,
110, 113, 117, 250, 252, 269, 271, 348,
350, 352
Eigensystem realization algorithm (ERA), 251
Eigenvalue decomposition, (EVD) 38, 93, 101,
119, 243, 246, 250, 277, 348, 351
Eigenvalue problem, 33, 38, 93, 101, 106, 108,
110, 113, 117, 157
Energy dissipation, 81, 98
Energy, 62, 81, 83, 97, 105, 123, 182, 185, 188,
209, 212, 221, 297
distribution of, 1, 125
Enhanced frequency domain decomposition
(EFDD), 271, 302, 306
Environmental conditions, 3, 204, 289, 319
Ergodic, 21
Errors,
clipping, 164, 196, 202
drop-outs, 202
spikes, 12, 193, 197, 202
Estimation,
force estimation, 307, 323
mode shape expansion, 307, 311, 320, 325
stress and strain, 307, 324
Excitation, 2, 4, 11, 87, 143, 147, 153, 166, 175,
177, 179, 186, 189, 205, 284, 287, 290,
298, 302, 303, 338
micro tremor, 162
Expectation, 17, 22, 24, 30, 47, 133, 222, 225,
255
of function, 19
Experimental modal analysis (EMA), 1, 301
Fast Fourier transform (FFT), 53
FDD, 8, 154, 165, 191, 197, 202, 261, 266, 268,
342, 344, 346
approximations, 267
dominating singular value, 270
harmonics, 208
noise separation, 191, 268
orthogonality, 270
pole estimation, 271
FE model, 151, 157, 285, 287, 289, 306, 311,
313, 324
Feedback signal, 178
FEMtools, 287
FFT filtering, 121, 213, 215
Field test, 151, 284
Filter bank, 213
Filter,
antialiasing, 62, 158, 182, 208, 212, 298
cut-off frequency, 182
slope, 209
FIR, 203, 209, 211, 215, 218
IIR, 209, 215, 218
Filtering,
band-pass, 125, 209, 213, 218, 243, 250, 253,
256, 306
high-pass, 168, 175, 203, 210, 213, 215, 217,
298
low-pass, 182, 209, 212, 215, 298
Finite impulse response (FIR) filters, 209
Flat triangular window, 233, 236, 265
Force identification, 307, 324
Force-balance accelerometer (FBA), 169
Fourier integral, 58, 66, 213
Fourier series, 54, 61, 71, 230
complex, 55, 71, 126
real, 54
Fourier transform, 53, 56, 58, 64, 66, 68, 71, 73,
90, 121, 123, 125, 127, 129, 134, 143,
145, 172, 201, 203, 211, 213, 215, 221,
224, 229, 232, 236, 266, 275, 320
discrete, 59, 61, 211, 262, 264, 272
Free decay, 69, 83, 85, 87, 95, 97, 101, 105, 120,
348
Frequency bandwidth, 60, 185
Frequency domain identification, 190, 208, 261,
317
Frequency domain poly reference (FDPR), 261
Frequency range, 1, 4, 9, 149, 153, 175, 215,
290, 297, 303, 305, 341, 343
Frequency resolution, 55, 60, 271, 298
Frequency response function (FRF), 4, 7, 9, 64,
75, 89, 91, 99, 123, 138, 172, 190, 204,
209, 211, 213, 215, 266, 275, 307
matrix, 35, 96, 104, 120, 124, 130, 136, 164,
324, 351
Frequency shift, 113
Fundamental equations, 128356 Index
Fundamental theorem, MIMO, 129, 165
Fundamental theorem, SIMO, 128
Gaussian distribution, 28, 207
Gaussian noise, 214
white noise, 230
Generalized angle, 36, 282, 309
Generalized stabilization , 315
Geophone, 168, 173, 175, 177, 215
generator constant , 177
Global positioning system (GPS), 150, 187
Good testing practice, 12, 198
Ground loop, 186
Ground motion, 4, 165, 167, 338
Half spectrum, 215, 237, 262, , 264, 276
Hammer input, 2, 4, 13, 165
Hankel matrix, 41, 220, 241, 243, 248, 250, 252,
255, 263
Hanning window, 50, 204, 214, 222, 250, 256,
298
Harmonics, 54, 206, 208, 217, 222
FDD plot, 208
kurtosis, 207
Heaviside partial fraction expansion theorem, 98,
135
Heritage Court Tower, 63, 206, 234, 236, 239,
261, 268, 287, 335
Hermitian matrix, 14, 39
Hooke’s law, 173
Ibrahim time domain (ITD), 8, 41, 142, 224, 240,
248, 315, 317
Identity matrix, 34, 110, 114, 117, 143, 155, 323
Impulse response function (IRF), 9, 29, 70, 76,
87, 123, 132, 143, 212, 239, 242, 348
matrix, 124, 348
SDOF system, 87
Independent inputs, 13, 164, 166
Independent white noise, 164
Infinite impulse response (IIR), 209
Influence of tilt, 194, 196
Initial conditions, 84, 86, 95, 118, 209, 227, 251,
255, 275, 349
Inner product, 35, 37, 39, 44, 48, 94, 102, 109,
112, 114, 124, 141, 268, 272, 317, 323
Integrated circuit piezoelectric (ICP), 169, 175
Integrated electronics piezoelectric (IEPE), 175
Integration theorem, 59, 63, 67, 73
Integration, 27, 43, 58, 64, 128, 214, 223, 232
time domain, 215
frequency domain, 216
Joint distribution, 21
Kronecker delta, 56
Kurtosis, 202, 207
Laplace transform table, 69, 98
Laplace transform, 53, 66, 68, 71, 73, 83, 89, 96,
103, 105, 123, 130, 348, 350
Laplace variable, 123, 351
Law of elasticity, 82
Leakage, 120, 159, 203, 214, 217, 221, 223, 229,
232, 298, 302
Least significant bit (LSB), 180
Least squares, 44, 50, 243, 256, 276, 309
Lightly damped structures, 29, 83, 265, 268
Linear operation, 35, 57
Linear regression, 48, 272
Linear system, 107, 123, 172
Linear time invariant system, 87
Load, wind, 12, 165, 205, 287
Local correspondence (LC) principle, 109, 313
Logarithmic decrement, 65, 272
MAC, matrix, 282, 284, 287, 291, 299, 301, 308,
313, 317, 344, 346
Mass change matrix, 112, 321, 323
Mass matrix, 93, 107, 110, 113, 269, 314, 321,
323, 352
Mass normalized mode shapes, 10, 91, 94
MATLAB, 15, 50, 61, 64, 218, 229, 309
Matrix measure, 41
determinant, 28, 41, 157
Frobenius norm, 42
p-norm, 42
trace, 42
Maxwell-Boltzman, 185
MDOF system, 81, 92, 98
Mean value, 17, 19, 23, 29, 31, 49, 54, 61, 171,
202, 211
Measurement chain, 151, 183, 198
Measurement duration, 153
Measurement range, 182, 184
Measurement system, 13, 149, 153, 162, 179,
182, 184, 186, 196, 207, 338Index 357
Memory time of system, 29, 159, 197
Micro electro-mechanical systems (MEMS),
169, 171, 179
Micro seismic excitation, 17, 162
Microtremor, 6, 162, 205
Modal Assurance Criterion (MAC), 282, 284,
287, 291, 299, 301, 308, 313, 317, 344,
346
Modal coherence, 317
Modal confidence factor (MCF), 317
Modal coordinate, 96, 103, 105, 115, 123, 132,
143, 145, 155, 159, 190, 265, 267, 269,
271, 273, 319, 324, 350
Modal damping, 8, 81, 95, 102, 105, 153, 222,
271, 298, 300, 351
Modal filtering, 196, 271, 273, 307, 319, 324
Modal force, 96, 143, 324
Modal indicators, 307, 315, 317
Modal mass, 81, 94, 97, 99, 102, 105, 107, 111,
113, 124, 147, 268, 320, 323, 351
Modal participation, 139, 141, 240, 244
Modal transfer function, 97, 103
Mode shape bias, 141
Mode shape cluster, 109, 314
Mode shape complexity, 345
Mode shape expansion, 307, 311, 320, 325
Mode shape matrix, 93, 97, 101, 105, 108, 115,
132, 143, 155, 241, 266, 272, 311, 313,
319, 324, 349
Mode shape merging, 283
Mode shape scaling, 158, 307, 316, 320, 323,
351
Mode shape subspaces
Model order, 242, 277, 282, 298, 315
Model reduction (SEREP), 82, 114, 312
Model updating, 285, 287, 342
Moving average, 75, 116, 302, 209, 211, 218
Moving standard deviation, 202
Multiple degree of freedom (MDOF) system, 81,
92, 98
Noise floor, 153, 183, 186
FDD, 193, 272
inherent, 177
sensor, 164, 170, 172, 177, 179
Noise modes, 3, 240, 242, 244, 262, 264, 272,
274, 277, 281, 298, 315
Noise sources,
electrical, 10, 153, 183, 186, 188
self-noise, 184, 191
Noise spectrum, 159, 171, 191, 193
Noise tail, 233, 234, 235
N-order realization, 252
Normal equations, 46, 276
Normal modes, 93, 95, 99, 101, 105, 110, 131,
139, 145, 266, 319
Notation, 23, 33, 37, 50, 67, 72, 131, 337, 347
Nyquist,
band, 62, 64, 72, 182, 203, 209, 213, 216, 342
frequency, 60, 62, 64, 71, 153, 158, 182, 212,
232, 275, 298, 342
Observability matrix, 252, 255
Observation equation, 347
Observation matrix, 347
OMA filtering, 208, 216
Operating condition, 1, 5, 11, 97, 164, 205
sorting, 204
Operating deflection shapes (ODS), 10, 320
generalized, 320
Orthogonality, 38, 54, 56, 93, 96, 101, 103, 109,
143, 352
closely spaced modes, 110, 114
geometrically, 269, 270, 272
mass matrix, 94, 108
stiffness matrix, 94, 108
damping matrix, 95
Outer product, 36, 39, 97, 109, 124, 241, 262,
270, 350
Outlier, 12, 196, 197, 202
Oversampling, 159
Oversized models, 277, 315
Parseval theorem. 125, 127, 130, 159, 170, 181,
182, 212, 221, 230
Partial fraction expansion, 98, 135
Participation, factor, 139, 141, 240, 244
Particular solution, 118, 247, 350
Peak picking, 7, 8
FDD, 7, 202, 271, 299, 304, 342
Periodic function, 54, 57, 61, 71
Permanent magnet, 175, 177, 178
Permittivity, 174
Perturbed eigenfrequency, 323
Perturbed system, 311, 314
Peterson low noise model (LNM), 162, 163358 Index
Physical mode, 2, 242, 244, 250, 253, 256, 264,
315, 317, 349
Physical rank, 154, 192
Piezoelectric accelerometer, 173, 175, 191, 193
Piezoelectric sensor, 168, 173, 175, 188
Piezoelectric coefficient, 174
Poles and zeros, 76
Poles, 70, 74, 89, 119, 137, 240, 256, 262, 267,
271, 274, 300, 310
complex conjugate, 84, 91, 132, 179
continuous time, 70, 78, 83, 85, 117, 248, 317
discrete time, 76, 78, 117, 248, 250
filter, 209, 215, 218
repeated, 110, 114, 271
stable, 282
Poly reference (PR), 240, 242, 261, 267, 276,
311, 315
PolyMax, 9, 261, 276
Posteriori indicator, 317
Potential modal ratio (PMR), 8, 283
Power spectral density, 126, 163, 185, 302
Prediction error, 202, 247
Preprocessing, 201
principal components (PC), 256, 344
Principle of local correspondence (LC), 109
Projection matrix, 255, 323
Probability density function, 17, 19, 21, 25, 28,
207
Proportional damping, 81, 95, 97, 104, 106
Pseudo inverse, 46, 115, 155, 252, 272, 311, 313,
319, 324
Quality assessment, 196
Quality checks and comparison, 196, 202
Random decrement (RD) technique, 31, 224,
236, 239, 248, 255
signature, 225, 255
estimation, 232
Random error, 10, 221, 229, 233, 345
Random signal (process), 17, 25, 29, 143, 153,
207, 218, 229, 255
Rank of problem, 154
Rational fractional form, 137
Realization, 8, 17, 19, 23, 157, 251, 253
Reduced mass matrix, 323
Reference sensor, 7, 11, 150, 152, 160, 189, 283,
293, 302, 306, 328, 341
recommended number of, 161
Regression, 14, 33, 48, 75, 240, 249, 262, 272,
281, 310
Repeated identification, 309
Residue matrix, 98, 141
Resolution, 169, 212, 338
frequency, 271, 298
sensor, 170
spatial, 160, 306
Response covariance matrix, 139
Response data filtering, 316
Reversal property, 57, 59, 67, 73, 127, 129, 144,
145, 221
Rotation angle, 113, 115
Roving sensors, 150, 152, 160, 161, 293, 338
Sampling frequency, 60, 151, 158, 181, 212, 218,
298
Sampling interval (time step), 15, 60, 76, 158,
219, 234, 236, 251, 298
Sampling rate, 63, 158, 181, 294, 298, 338, 341
Sampling theorem, 62, 212
Scaled mode shape, 97, 108, 113, 283, 321, 323
Scaling factor, 13, 58, 158, 170, 228, 284, 320,
322, 351
SD matrix, 127, 133, 136, 138, 140, 145, 154,
221, 233, 262, 266, 268, 270, 272
SDOF system, 70, 76, 78, 82, 84, 87, 89, 95,
102, 177, 272
Second Law of Motion, 82
Segmenting, 203, 213, 221, 223, 230, 236
Seismic mass (proof mas), 168, 177, 178, 186,
187, 194
Seismic noise power spectral density, 163
Seismograph, 168
Sensitivity analysis, 285, 287, 314
Sensitivity equations, 82, 109, 311, 321
Sensor characteristics, 169
Sensor element, 169, 176, 184, 186, 194
Sensor frequency response function (FRF), 188
Sensor noise, 164, 193
1/f noise, 185, 193
electromagnetic, 184, 185, 197
Sensor principles, 168, 169
electromagnetic, 168, 169, 170, 177
piezoelectric, 168, 169, 170, 173, 191
force-balance , 168, 169, 170, 177, 188, 294,
338
Sensor temperature sensitivity, 171
bias sensitivity, 171, 179
scaling factor sensitivity, 171
Sensor, calibration factor, 170, 198, 202Index 359
sensitivity, 169, 170
Sensor dynamic range, 153, 170, 175, 179, 302
Sensor location, 8, 152, 155, 157, 294, 338, 341
Servo accelerometer, 179
Servo amplifier, 178
Shaker input, 165
Shannon’s theorem, 62, 212
Side lobe, 233, 234, 236
Signal classification, 204
Signal dropouts, 12, 197, 201
Signal-to-noise ratio, 12, 153, 159, 164, 166,
184, 189, 197
Simulation, 12, 29, 49, 51, 119, 138
FE model, 152, 156, 158
OMA response, 119, 158
Single input, 12, 124, 128, 131, 166, 240, 249,
269
Singular value decomposition (SVD), 12, 27, 40,
165, 191, 197, 256, 266, 302, 342
Singular value plot, 166, 294
Singular vector, 27, 40, 191, 252, 256, 267, 269,
271, 317
Site inspection, 150
Sorting, 206, 244
operating condition, 204
Spatial resolution, 160, 306
Spatial solution, 93
Specifications, 152, 287
Spectral density estimation, 229
Spectral density function, 7, 14, 138, 192, 201,
208, 222, 224, 229, 232, 234, 342
auto, 125
cross, 126
Spectral density matrix, 126, 127, 133, 136, 138,
140, 145, 154, 221, 233, 262, 266, 268,
270, 272
modal decomposition of, 134, 136, 138, 140,
146, 266, 268
zeroes of , 137, 138
Square error, 24, 45
SSI method, 8, 239, 254, 256, 258, 282, 298,
300, 302, 304, 306, 342, 344, 346
Stabilization diagram, 300, 304, 344
Stable, unstable mode, 316
Standard deviation, 19, 28, 47, 49, 51, 156, 181,
207, 229, 236, 298, 300, 317, 345
Moving, 202
State space, 104, 106, 124, 243, 252, , 300, 304,
347, 349
format, 100, 117
formulation, 81, 100, 116, 119, 246, 347, 349
Stationarity, 10, 19, 25, 28, 129, 166, 205, 207,
225
Stiffness matrix, 93, 95, 107, 114
Stochastic subspace identification (SSI), 8, 239,
254, 256, 258, 282, 298, 300, 302, 304,
306, 342, 344, 346
Strain gauges, 170, 173, 214
Structural health monitoring (SHM), Damage
detection, 4, 10, 13, 281, 284, 288, 290,
306, 318
Structural modification theory, 82, 107, 311
Synthesized signal, 264, 319
System equivalent reduction expansion process
(SEREP), 114, 312, 323
System matrix, 104, 117, 253, 256, 347, 351,
352
discrete time, 118, 249, 251, 253
Tapering, 203, 218, 222, 233, 234, 265
Window, 229, 232
Test objective, 149, 151
Test planning, 11, 149, 155
Thermal noise voltage, 184
Time averaging, 19, 22, 25, 207, 225
Time shift property, 57, 59, 67, 73, 75, 275
Time-frequency plot, 206
Toeplitz matrix, 41, 220
Trace, 42
Transducer, 4, 149, 151, 153, 179
Transducer Electronic Data Sheet (TEDS), 175
Transfer function, 8, 35, 75, 79, 106, 176, 179,
204, 209, 211, 348
matrix, modal, 96, 97, 103, 124, 130, 275
SDOF system, 70, 89, 96, 98
state space, 104, 348, 250
Transformation matrix, 36, 115, 311, 322
Transient response, 351, 118
Triggering condition, 225, 232, 237
level, positive point, positive velocity, 227
Troubleshooting, 320
Uncertainty estimation, 48, 50, 52, 307, 309
Uncertainty, 10, 13, 50, 155, 159,180, 182, 187,
228, 271, 284, 286, 300, 307, 309, 317,
322
close modes, 345
model, 157
Uncorrelated modal coordinates, 123, 143, 145,
268360 Index
Unit step function, 69, 73
derivative of, 69
Unitary matrix, 34
Unweighted principal component (UPC), 256,
244
Up-sampling, 63, 65, 212
Variable, random, 17, 19, 21, 23
Variance, 19, 21, 24, 26, 29, 31
Vibration data, 3, 5, 8, 338
Vibration level, 10, 13, 162, 198, 301
Weight matrix, 351
Welch estimate, 221, 224, 229, 332, 235
biased, 221, 223
unbiased, 222, 234
White noise, 2, 5, 7, 29, 123, 130, 135, 142, 146,
185, 229, 247, 267, 302
band limited, 131
concept of, 130
Wind loaded structures, 12, 165, 205, 287
Window,
basic lag, 224
exponential, 233, 235
flat-triangular, 233, 235, 237, 265
Hanning, 50, 204, 214, 218, 222, 230, 237
overlap, 159, 203, 214, 221, 230,
236, 298
triangular, 204, 224, 233
Wired (cabled) sensors, system, 150, 187
Wireless, 13, 150, 187
time synchronization, 187
Zero padding, 63, 204, 222, 224
Z-transform, 53, 71, 74
table, 75
unit circle, 72, 79

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