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| موضوع: كتاب Introduction to Operational Modal Analysis الخميس 3 مارس 2022 - 14:53 | |
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أخواني في الله أحضرت لكم كتاب Introduction to Operational Modal Analysis Rune Brincker Technical University of Denmark Carlos E. Ventura University of British Columbia, Canada
و المحتوى كما يلي :
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 7.2.7 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 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 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 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 instrument, 153 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 Broadband excitation, 1 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 Faraday’s law, 175, 177 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 distribution matrix, 347 Load, traffic, 165, 205 Load, wind, 12, 165, 205, 287 Loading conditions, 2, 13, 205 Loading source correlation, 164 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 Moving stochastic loads, 165 Multiple degree of freedom (MDOF) system, 81, 92, 98 Multiple input loading, 8, 11, 124, 167, 243 Noise floor, 153, 183, 186 ADC, 159, 181, 196 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 overlap-add filtering, 204 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 ADC, 180, 181, 196 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 Traffic load , 165, 205 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 loading, 2, 165 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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