Index 319. G Gaussian noise, Ground-loop current, 61

Transcription

1 Index A Absolute accuracy, 72 ADCs. See Analog-to-digital converters (ADCs) AFDD-T. See Automated frequency domain decomposition-tracking (AFDD-T) Aliasing, 81, 90 Analog-to-digital converters (ADCs) conversion time, 72 cross talk, 72 dynamic range, nonlinearity, 73 offset, 73 resolution, 71 sampling rate, 72 Applications, OMA damping estimation aerodynamic damping, 216 damping ratio, energy dissipation, 216 fracture mechanics, 217 frequency resolution, 217 frictional loss, 217 modal identification, 219 modal parameter estimates, 221 modal properties, 220 natural frequency, 218 noninvertible matrix, 219 physical criteria, 220 sensitivity analysis, 218 stabilization diagram, excitation system bending mode, 255 cable motion, 252 candidate modes, characteristics of sensors, 249, 251 dynamic responses, 248 dynamic tests, 247 environmental and operational loads, 243 frequency domain algorithm, 252 interaction effects, 248 kurtosis index, 245 mode classification, mode shape complexity and coherence, 255 narrowband excitation, 246 probability density function, 245 sensor layouts, short time Fourier transform, 244 source correlation, 256 spurious harmonics, 244 spurious peaks, 244 stair structure, 247 structural modes, 252, 254 transmissibility-based method, 246 vibration data, 249 mass normalized mode shapes approximated equations, 238 columns and beams, 240 frequency shifts, Gaussian white noise, 240 lightly damped structures, 238 mass-change methods, 237 mass-change strategy, 239 modal truncation effects, 239 natural frequency estimates, 242 r.c. frame, scaling factors, 237, squared modal displacements, stiffness-change methods, 238 measurement chain assessment AC cut-off frequency response, data acquisition modules, 214 dynamic response, 215 FE model, 213 low-amplitude vibrations, 214 masonry cell, 213 modal identification tests, 211 C. Rainieri and G. Fabbrocino, Operational Modal Analysis of Civil Engineering Structures: An Introduction and Guide for Applications, DOI / , # Springer Science+Business Media New York

2 316 Index Applications, OMA (cont.) output-only modal analysis technique, 213 reduction of modeling uncertainties, 211 sensor layouts, 215 star vault, numerical and experimental modal property estimates ambient vibration tests, 222 dynamic response, 237 finite element modeling, 228, frequency scatters, 236 geometric and structural survey, ground acceleration load, 233 ground motions, 222 mode shape correlation, 233, 236 natural frequencies, 232 output-only modal identification, PMRs, seismic assessment, 223 seismic capacity, 222 serviceability conditions, 223 shell elements, 236 structural response, 235 Tower of the Nations, predictive correlations accelerometers, 259 ambient vibration tests, 259 bell towers, bending mode, 262 elastic response spectrum, 258 Italian masonry towers, nondestructive structural investigation, 257 seismic performance, 257 seismic vulnerability, 258 torsional mode, 262 ARES. See Automated modal parameter extraction system (ARES) Auto-correlation functions, Automated frequency domain decompositiontracking (AFDD-T) algorithm, 296 architecture, 299, 300 environmental effects, 306 FDD-based algorithms, 301 histograms, 304, 306 integration, 303 L Aquila earthquake, 307, 308 principles and implementation, r.c. frame, 300 real-time estimation, 294 School of Engineering Main Building, statistical treatment, 294 Automated modal parameter extraction system (ARES) algorithm, 288 benchmark 4-DOF system, 289, 290 clustering techniques, 286 modal damping ratios, 285, 291 performance, SNR, 290 SSI method, 286, 287 stabilization diagram, 286, 287 state-machine architecture, 287 statistical analysis, 289 Automated modal parameter identification, 267 Automated modal tracking methods, 267 Automated OMA AFDD-T algorithm, 296 architecture, 299, 300 environmental effects, 306 FDD-based algorithms, 301 histograms, 304, 306 integration, 303 L Aquila earthquake, 307, 308 principles and implementation, r.c. frame, 300 real-time estimation, 294 School of Engineering Main Building, statistical treatment, 294 ARES algorithm, 288 benchmark 4-DOF system, 289, 290 clustering techniques, 286 modal damping ratios, 285, 291 performance, SNR, 290 SSI method, 286, 287 stabilization diagram, 286, 287 state-machine architecture, 287 statistical analysis, 289 LEONIDA algorithm, 275 applications, drawbacks, 271 FDD method, 270, integration, 303 LabVIEW environment, LSCF method, 269

3 Index 317 MAC sequences, SSI technique, 269 time domain filtering method, 270 problem statement, vibration-based monitoring, Auto-power spectral density function, 56 Autoregressive moving average (ARMA) models ARMAV, 117 companion matrix, 119 IV method, linear time-invariant system, 117 minimal realization, 118 observability canonical state-space realization, 118 prediction error, Auto-regressive moving average vector (ARMAV) model, 117 B Basic frequency domain (BFD) method, , Blind source separation (BSS) techniques applicability of, 166 classification, sources, 166 static mixtures, 167 use of, C Canonical variate analysis (CVA), 157, Common-mode rejection ratio (CMRR), 61 Complex mode indicator function (CMIF), 130 Complex numbers, 23 27, 53 Coordinate modal assurance criterion (COMAC), Correlation functions, Covariance-driven stochastic subspace identification (Cov-SSI) method BR variant, 157 canonical angles, 157 CVA variant of, 157 Laplace variable, 158 noise sources, 158 observability/controllability matrix, 154, 155 state-space model, 153 Toeplitz matrix, Cross-correlation functions, 55 D Damping estimation aerodynamic damping, 216 damping ratio, energy dissipation, 216 fracture mechanics, 217 frequency resolution, 217 frictional loss, 217 modal identification, 219 modal parameter estimates, 221 modal properties, 220 natural frequency, 218 noninvertible matrix, 219 physical criteria, 220 sensitivity analysis, 218 stabilization diagram, DAQ Assistant, Data acquisition ADCs, aliasing, 81, 90 anti-aliasing filter, 82 chi-square goodness-of-fit test, 85 clipping, 85 CMRR, 61 common-mode voltage range, 61 data pretreatment, data storage, differential measurement system, 60 file storage, filtering and decimation, 91 FIR filters, 83 floating signal sources, 60, 61 grounded signal sources, 60 ground-loop current, 61 hardware selection, IIR filters, 83 intermittent noise spikes, modal analysis test, 59 mode shape merging, MySQL database, offsets, 88 power line pickup, 88 pseudodifferential systems, 62 reference sensors, 79 roving sensors, 78 sampling frequency, 81 sensor installation, signal dropouts, 88, 89 spurious trends, steep learning curve, 70 transducers dynamic range, 67 force-balance accelerometers, 65

4 318 Index Data acquisition (cont.) offset error, 67 Peterson noise curves, piezoelectric sensors, sensor resolution, 67 sensor self noise, 67 settling time, 67 wired vs. wireless, Data-driven stochastic subspace identification (DD-SSI) method, 117 CVA, 166 Hankel matrix, Kalman filter state, 159, 162 modal parameter, 165 orthogonal and oblique projections, 159 principal component algorithm, 165 state-space matrices, UPC algorithm, 165 Data processing BFD method, , FDD method CMIF, 130 frequency-spatial domain decomposition, 133 k-th mode, 131 MAC rejection level, 132 modal coordinates, PSD matrix, 131 natural frequency estimates, 200 output-only modal identification, SDOF system, 132 singular value plots, , 201 singular vectors vs. mode shapes, 131 spatial filtering, 133 frequency domain parametric methods common-denominator model, 134 cost function, 134 LSCF method (see Least squares complex frequency (LSCF) method) LSFD method, 134 p-lscf method (see Poly-reference least squares complex frequency (p-lscf) method) influence of sensor layout, 206 mode shape estimates, OMA (see Operational modal analysis (OMA)) quality checks and comparisons AutoMAC matrix, COMAC, 190 CrossMAC matrix, 191 ECOMAC, 190 MAC, model updating, 185 mode shapes, MSF, 188 natural frequencies, NMD, 188 verified model, 185 SOBI, SSI, structural dynamics models (see Structural dynamics models) time domain methods (see Time domain methods) Discrete Fourier transform (DFT), 27, 42, 53 E Eigensystem realization algorithm (ERA), 147 Eigenvalue decomposition (EVD), 48 Enhanced coordinate modal assurance criterion (ECOMAC), 190 Euler s identities, Excitation system bending mode, 255 cable motion, 252 candidate modes, characteristics of sensors, 249, 251 dynamic responses, 248 dynamic tests, 247 environmental and operational loads, 243 frequency domain algorithm, 252 interaction effects, 248 kurtosis index, 245 mode classification, mode shape complexity and coherence, 255 narrowband excitation, 246 probability density function, 245 sensor layouts, short time Fourier Transform, 244 source correlation, 256 spurious harmonics, 244 spurious peaks, 244 stair structure, 247 structural modes, 252, 254 transmissibility-based method, 246 vibration data, 249 Experimental modal analysis (EMA), 1 3 F Fast Fourier transform (FFT), 27 Finite element method (FEM), 2, 228, Finite impulse response (FIR) filters, 83 Forward innovation model, 116 Fourier transform

5 Index 319 complex numbers, Euler s identities, DFT, 27, 42, 53 FFT, 27 properties, 27 Fraction polynomial models circular correlation function, 122 common-denominator model, 120 cross-power spectra, 121 MFD, 120 Frequency domain decomposition (FDD) method, 103, automation of, 270 CMIF, 130 frequency-spatial domain decomposition, 133 k-th mode, 131 LEONIDA, MAC rejection level, 132 modal coordinates, PSD matrix, 131 SDOF system, 132 singular value plots, singular vectors vs. mode shapes, 131 spatial filtering, 133 structural modes, 277 threshold-based peak detection, 278 time-consuming sensitivity tests, 277 Frequency domain parametric methods, Frequency response function (FRF), 6, 106, 107 G Gaussian noise, Ground-loop current, 61 I Ibrahim time domain (ITD), 150 Impulse response function (IRF), 6, 108 Independent component analysis (ICA), Infinite impulse response (IIR) filters, 83 Instrumental variable (IV) method, Integrated electronics piezoelectric (IEPE) accelerometers, 63 J Joint approximate diagonalization (JAD) technique, 171 L LabVIEW event structure, vs. other programming languages, 14 programming in, 12 software development in, usage, 14 Least squares complex exponential (LSCE) method, 147 Least squares complex frequency (LSCF) method common-denominator model, generalized transform variable, 135 Jacobian matrix, 138 modal identification technique, 134 natural frequency estimates, 203 numerator coefficients, 139 output-only modal identification, 201 residue matrices, r-th mode, damping ratio, 139 scalar matrix fraction, 136 software development, 202 Toeplitz matrices, 138 z-domain formulation, 136 Least squares frequency domain (LSFD) method, 134 LEONIDA algorithm, 275 applications, drawbacks, 271 FDD method, 270, integration, 303 LabVIEW environment, LSCF method, 269 MAC sequences, SSI technique, 269 time domain filtering method, 270 M Matrix fraction description (MFD), 120 Mean phase deviation (MPD), 180, 182 Modal assurance criterion (MAC) sequences, Modal overlap factor (MOF), 180 Modal phase collinearity (MPC), 180, 181 Modal scale factor (MSF), 188 Mode shape estimates, complexity plot, 180 MPC, 180, 181 MPD, 180, 182 Multi degree of freedom (MDOF) systems, 7 MySQL database,

6 320 Index N Natural excitation techniques (NExT), 103 ERA, 147 Hankel matrix, ITD, 150 LSCE, 147 Prony s equation, 147 system matrix, 150 Normalized modal difference (NMD), 188 Nyquist frequency, 81 O Operational deflection shapes (ODSs), 103 Operational modal analysis (OMA) advantages and disadvantages, ambient vibration modal identification, 2 applications of data processing, 8 input-output testing, 9 modal parameters, 9 PSD matrix, 8 vibration-based damage detection, 9 10 broadband excitation, 105 classification of, combined system, data acquisition, 4 definition, 2 diagonal matrices, 6 EMA, 1 3 FDD, 103 forward problems, 4 frequency response function, 6 generalities block diagram, front panel, 13 high order model, 126 impulse response function, 6 inverse problems, 4 LabVIEW (see LabVIEW) linearity, 104 linear time-invariant system, 4 5 low order model, 126 MDOF systems, 7 modal mass, 6 mode shapes, 5 natural frequencies, 5 NExT, 103 noise, 4 observability, 104 ODS, 103 organization of, physically realizable, 7 random decrement technique, SDOF system, 7 signal-to-noise ratio, 4 software development design patterns, 16 event structure, producer/consumer architecture, 18, 20 state machine, VI hierarchy, spatial filter, 12 SSI, 103 stable system, 7 stationarity, 104 structural health monitoring, 3 testing types, 7 transmissibility function, virtual instruments, 13, Output-only modal identification, P Parametric OMA methods mathematical modes, 192 noise modes, 191 stabilization diagrams for, Peak picking method, , Poly-reference least squares complex frequency (p-lscf) method closely spaced modes, 142 companion matrix, cost function, 143 error formulation, 143 Jacobian matrix, single complex-valued matrix, Principal component analysis (PCA), 167 Probability density function, 29, 34, Probability theory, R Random data analysis auto-and cross-correlation functions, 55 auto-power spectral density function, 56 complex numbers, 23 27, 53 condition number κ, 51 error norms, Euler s identities, Fourier transform, DFT, 27, 42, 53 FFT, 27 properties, 27 Gaussian noise, auto-correlation of, least squares method, 52 matrix algebra decomposition method, 46 eigenvector, 48 EVD, 48

7 Index 321 identity matrix, 46 Moore Penrose pseudoinverse, 49 nonsingular matrix, 47 null space, 48 orthogonal, 47 real-valued/complex valued, 46 singular matrix, 47 square matrix, SVD, 49, 51 symmetric, 47 trace, 46 unitary matrix, 47 sampling interval, 26 Shannon s theorem, 26 SRP (see Stationary random processes (SRP)) statistics, 54 Random decrement (RD) technique, S Second order blind identification (SOBI) auto-correlation, 172 blind modal identification, 168 BSS techniques applicability of, 166 classification, sources, 166 static mixtures, 167 use of, complexity plot, drawback of, 168 JAD technique, 171 natural frequencies and damping ratios, 172 random response, sample software, 206 sensor layout, 197, 205 whitening matrix, Signal-to-noise ratio (SNR), 4, 290 Single degree of freedom (SDOF) system, 7 Singular value decomposition (SVD), 49, Spectral density functions coherence function, 41 coincident, 40 cross, 39 Fourier transforms, 38 Hanning window, 43 Hermitian matrix, 43 and OMA, quadrature, 40 random error, rectangular window, 42 two-sided, 39 Welch procedure, 41 Wiener Khinchin relations, 40 Star vault, State-space models continuous-time, 111 covariance equivalent model, 113 DD-SSI, 117 direct transmission matrix, 111 forward innovation model, 116 Kalman filter, measurement noise, 113 non-steady state Kalman gain, 115 observation equation, optimal predictor, 114 process noise, 113 realization, 111 Ricatti equation, 115 state equation, state prediction error, 114 state vector, 109 Stationary and ergodic random processes (SERP), 28 Stationary random processes (SRP) bias error, 44 correlation coefficient, 31 correlation functions auto-and cross-, 36 DFT, 27 FFT, 27 weakly ergodic, 36 covariance function, 30 damping ratio, 45, 46 Gaussian probability, 32 joint probability, 29 mean square error, 44 mean square value, 30 normalized rms error, probability density function, 29, 34 probability distribution function, 29 random/stochastic process, 28 sample function, 28 sample record, 28 SERP, 28 spectral density functions auto-spectral density functions, 40, 41 Blackman Tukey procedure, 41 coherence function, 41 coincident spectral density function, 40 cross-spectral density functions, 39, 40 Gaussian probability density function, 32 Hanning window, 43 quadrature spectral density function, 40

8 322 Index Stationary random processes (SRP) (cont.) rectangular window, DFT, 42 Welch procedure, 41 Wiener Khinchin relations, 40 standard deviation, statistically independent, 30 weakly stationary random processes, 28 Stochastic subspace identification (SSI), 103, 269 Cov-SSI method (see Covariance-driven Stochastic Subspace Identification (Cov-SSI) method) DD-SSI method (see Data-driven stochastic subspace identification (DD-SSI) method) Structural damping, 216 Structural dynamics models ARMA models ARMAV, 117 companion matrix, 119 linear time-invariant system, 117 minimal realization, 118 observability canonical state-space realization, 118 fraction polynomial models circular correlation function, 122 common-denominator model, 120 cross-power spectra, 121 MFD, 120 positive power spectra, 123 frequency response, FRF, 106, 107 impulse response, IRF, 108 MDOF system, 105 positive power spectra, 108 residue matrix, 107 spatial model, 106 state-space models (see State-space models) time domain modal, 108 UMPA, un-scaled mode shapes, 108 T Time domain methods, 127 ARMA model, Cov-SSI method BR variant, 157 canonical angles, 157 CVA variant of, 157 Laplace variable, 158 noise sources, 158 observability/controllability matrix, 154, 155 state-space model, 153 Toeplitz matrix, DD-SSI method CVA, Hankel matrix, Kalman filter state, 159, 162 modal parameter, 165 orthogonal and oblique projections, 159 principal component algorithm, 165 state-space matrices, UPC algorithm, 165 NExT-type procedures ERA, 147 Hankel matrix, ITD, 150 LSCE, 147 Prony s equation, 147 system matrix, 150 SOBI algorithm (see Second order blind identification (SOBI)) Transducers dynamic range, 67 force-balance accelerometers, 65 offset error, 67 Peterson noise curves, piezoelectric sensors, sensor resolution, 67 sensor self noise, 67 settling time, 67 U Unified matrix polynomial approach (UMPA), Unweighted principal component (UPC) algorithm, 165 V Virtual instruments (VIs ), 13, 15 16