Some investigations on modal identification methods of ambient vibration structuresLe Thai HoaWind Engineering Research CenterTokyo Polytechnic UniversityDate: 2009/12/05

Contents1. Frequency-domain modal identification of ambient vibration structures using combined Frequency Domain Decomposition and Random Decrement Technique2. Time-domain modal identification of ambient vibration structures using Stochastic Subspace Identification3. Time-frequency-domain modal identification of ambient vibration structures using Wavelet Transform

Introduction Modal identification of ambient vibration structures has become a recent issue in structural health monitoring, assessment of engineering structures and structural control Modal parameters identification: natural frequencies, damping and mode shapes Some concepts on modal analysis Experimental/Operational Modal Analysis(EMA/OMA) Input-output/Output-only Modal Identification Deterministic/ Stochastic System Identification Ambient/ Forced/ Base Excitation Tests Time-domain/ Frequency-domain/ Time-scale planebased modal identification methods Nonparametric/ Parametric identification methods SDOF and MDOF system identifications .

Vibration tests/modal identificationAmbient Vibration TestsForcedVibration TestsRandom/StochasticDeterministic/StochasticExperimentalModal Analysis OperationalModal Analysis Output-only IdentificationAmbient loads &Micro tremor Shaker (Harmonic)Hummer (Impulse)Sine sweep (Harmonic) Base servo (White noise, Seismic loads)ExperimentalModal Analysis OperationalModal Analysis Input-output IdentificationOutput-only IdentificationFRF identificationTransfer Functions Removing harmonic & input effectsIndirect & direct identifications

Modal identification methodsTime domainFrequency domainTime-frequency planeIbrahim Time Domain (ITD)Random Decrement Technique (RDT)Stochastic Subspace Identification (SSI)Frequency Domain Decomposition (FDD)Wavelet Transform (WT)Hilbert-Huang Transform (HHT)[Time-scale Plane]Enhanced Frequency Domain Decomposition (EFDD)Eigensystem Realization Algorithm (ERA)Ambient vibration Output-only system identificationApplicable in conditions and combinedCommercial and industrial usesAcademic uses, under development

ARTeMIS Extractor 2009 FamilyThe State-of-the-Art software for Operational Modal AnalysisODS: Operational Deflection ShapesFDD: Frequency Domain Decomposition EFDD: Enhanced Frequency Domain DecompositionSSI: Stochastic Subspace Identification UPC: Unweighted Principal Component PC: Principal Component CVA: Canonical Variate AlgorithmCommercial Software for OMA

Commercial Frequency domainTime domainPackageODSFDDPeak Picking(No damping)EFDD(Damping)SSI(UPC)SSI(PC)SSI(CVA)ARTeMIS Light ARTeMIS HandyARTeMIS Pro

Uses of FDD, RDT and SSI[Time-scale Plane]For MDOF Systems

Response time seriesY(t)

Power SpectralDensity MatrixSYY(n)

Modal Parameters

FDD(POD, SVD)EFDD(POD, SVD)RDFunctionsDYY(t)RDTITDMRDTCovarianceMatrixRYY(t)Direct methodSSI-COV(POD, SVD)Direct methodData Matrix HY(t)SSI-DATA(POD, SVD)FDDRDTSSI

Comparison FDD, RDT and SSI Advantages: Dealing with cross spectral matrix, good for natural frequencies and mode shapes estimation Disadvantage: based on strict assumptions, leakage due to Fourier transform, damping ratios, effects of inputs and harmonics; closed frequenciesFDDRDT Advantages: Dealing with data correlation, removing noise and initial, good for damping estimation, SDOF systems Disadvantage: MDOF systems, short data record, natural frequencies and mode shapes combined with other methodsSSI Advantages: Dealing with data directly, no leakage and less random errors, direct estimation of frequencies, damping Disadvantage: Stabilization diagram, many parametersCurrent trends in modal identification: Combination between identification methods Refined techniques of identification methods Comparisons between identification methods

RDT to refine modal identificationTime-frequency DomainPossibilities of RDT combined with othermodal identification methodsMulti-mode RDT

Frequency Domain Decomposition (FDD)Random DecrementTechnique (RDT)

FDD for output-only identification based on strict points (1) Input uncorrelated white noises Input PSD matrix is diagonal and constant (2) Effective matrix decomposition of output PSD matrix Fast decay after 1st eigenvector or singular vectors for approximation of output PSD matrix (3) Light damping and full-separated frequencies Frequency Domain Decomposition Relation between inputs excitation X(t) and output response Y(t) can be expressed via the complex FRF function matrix: Also FRF matrix written as normal pole/residue fraction form, we can obtain the output complex PSD matrix:

Output spectral matrix estimated from output data Output spectral matrix is decomposed (SVD, POD) Where: Spectral eigenvalues (Singular values) & Spectral eigenvectors (Singular vectors) ith modal shape identified at selected frequency Frequencies & Damping RatiosIdentificationMode shapesIdentificationFrequency Domain DecompositionOutput responsePSD matrix

Random Decrement TechniquesTriggering condition XoXoRD function (Free decay) RDT extracts free decay data from ambient response of structures (as averaging and eliminating initial condition)&to00

Random Decrement Techniques RD functions (RD signatures) are formed by averaging N segments of X(t) with conditional value Xo(Auto-RD signature)(Cross-RD signature)N : Number of averaged time segmentsX0 : Triggering condition (crossing level)k : Length of segment Conditional correlation functions

Combined FDD-RDT diagram

Response Data Matrix Y(t)

Cross Power Spectral Matrix SYY(n)

1st Spectral Eigenvalue1st Spectral EigenvectorNatural FrequenciesFree Decay Fun. & Damping RatiosMode ShapesPOD, SVD, QR

Data Matrix Y(t)

Cross Power Spectral Matrix SYY(n)

1st Spectral Eigenvalue1st Spectral EigenvectorNatural FrequenciesFree Decay Fun. & Damping RatiosMode Shapes

RDFun.DYY(t)

POD, SVD, QRRDTFDD-RDTFDDDamping onlyNatural FrequenciesFDDResponse Series at Filtered FrequenciesFree Decay Fun. & Damping RatiosRDTBPFat fi

Stochastic Subspace Identification(SSI) Covariance-driven SSI Data-driven SSI

SSI SSI is parametric modal identification in the time domain. Some main characteristics are follows: Dealing directly with raw response time series Data order and deterministic input signal, noise are reduced by orthogonal projection and synthesis from decomposition SSI has firstly introduced by Van Overschee and De Moor (1996). Then, developed by several authors as Hermans and Van de Auweraer(1999); Peeters (2000); Reynder and Roeck (2008); and other. SSI has some major benefits as follows: Unbiased estimation no leakage Leakage due to Fourier transform; leakage results in unpredictable overestimation of damping No problem with deterministic inputs(harmonics, impulse) Less random errors: Noise removing by orthogonal projection

State-space representation Continuous stochastic state-space model wk: process noise (disturbances, modeling, input)vk : sensor noise: Discrete stochastic state-space model state-space modelSecond-order equationsFirst order equationsA: state matrix; C: output matrix X(t): state vector; Y(t): response vector yk vkCStochastic systemA wk(, wk , vk : zero mean white noises with covariance matrix

Data reorganizing Response time series as discrete data matrix N: number of samples M: number of measured points Reorganizing data matrix either in block Toeplitz matrix or block Hankel matrix as past (reference) and future blocksBlock Hankel matrix Block Toeplitz matrix s: number of block rowsN-2s: number of block columnsshifted tpastfutures: number of block rows

SSI-COV and SSI-DATA Projecting future block Hankel matrix on past one (as reference): conditional covariance Data order reduction via decomposing, approximating projection matrix Ps using first k values & vectors Observability matrix & system matricesk: number of singular valuesk: system order & Modal parameters estimationMode shapes:Poles:Frequencies:Damping:HankelToeplitz

Flow chart of SSI algorithmData Matrix [Y(t)]Data Rearrangement

Block Teoplitz MatrixRP [], RF[],Block Hankel MatrixHP[], HF[]SSI-COVSSI-DATAOrthogonal ProjectionPsObservability MatrixOsSystem MatricesA, CModal ParametersStabilization DiagramCovarianceDataData order reductionPODPODHankel matrixData past/ futureToeplitz matrixParameter kParameter s

Modal identification of ambient vibration structures using combined Frequency Domain Decomposition and Random Decrement TechniqueNumerical example

Fullscale ambient measurement Five-storey steel frameGroundFloor1Floor2Floor3Floor4Floor5Output displacement(X)Floor 5Floor 4Floor 3Floor 2Floor 1Ground5 minutes record

Random decrement functionsFloor 5Floor 4Floor 3Floor 2level crossing: segment: 50sno. of sample: 30000no. of samples in segment: 5000Parameters

Spectral eigenvaluesMode 1Mode 2Mode 3Mode 4Mode 5Mode 1Mode 2Mode 3Mode 4Mode 5FDDFDD-RDTEigenvalue1: 99.9% Eigenvalue2: 0.07%Eigenvalue3: 0.01%Eigenvalue4: 0% Eigenvalue1: 100% Eigenvalue2: 0%Eigenvalue3: 0%Eigenvalue4: 0%

Natural frequencies (Hz)FDDFDD-RDTmode 11.731.73mode 25.355.34mode 38.848.82mode 413.6913.67mode 518.1218.02

Spectral eigenvectors99.9%0.07%0.01%0%FDD

FDD-RDTSpectral eigenvectors100%0%0%0%

Mode shapes estimationMode 1Mode 4Mode 5Mode 2Mode 3FDDMAC

Mode shapes comparisonMode 1Mode 4Mode 5Mode 2Mode 3

Identified auto PSD functionsMAC=98%MAC=95%FDDMode 1Mode 2Mode 3Mode 4Mode 5

Identified free decay functionsMode 1Mode 4Mode 2Mode 3FDDMode 5Uncertainty in damping ratios estimation from free decay functions of modes 3 & 4Unclear with modes 2 & 5

Identified free decay functionsMode 1Mode 4Mode 2Mode 3Mode 5BetterFDD

FDD - Band-pass filteringX5(t)f1=1.73Hzf2=5.34Hzf3=8.82Hzf4=13.67Hzf5=18.02HzResponse time series at Floor 5 has been filtered on spectral bandwidth around each modal frequency Floor 5

Damping ratio via FDD-BPFMode 1Mode 2Mode 3Mode 4Mode 5Uncertainty in damping ratios estimated from free decay functions at modes 4 & 5Free decay functionsFloor 5

FDD - Band-pass filteringResponse time series at Floor 1 has been filtered on spectral bandwidth around each modal frequency Floor 1f1=1.73Hzf2=5.34Hzf3=8.82Hzf4=13.67Hzf5=18.02HzX1(t)

Damping ratio via FDD-BPFMode 1Mode 2Mode 3Mode 4Mode 5Free decay functionsFloor 1

Damping ratio via FDD-BPFSelected free decay functions for damping estimationMode 1Mode 2Mode 3Mode 4Mode 5

Time-domain modal identification of ambient vibration structures using Stochastic Subspace Identification Numerical example

Parameters formulated Number of measured points: M=6 Number of data samples: N=30000 Dimension of data matrix: MxN=6x30000 Number of block row: s=20:10:120 (11 cases) Number of block columns: N-2s Dimension of Hankel matrix: 2sMx(N-2s) Data parametersHankel matrix parametersSystem order parameters Number of system order: k=5:5:60 (12 cases) (Number of singular values used)

Projection functionss=50s=100s=150Data after orthogonal projection look like time-shifted sine functions

Effects of s on energy contributionsystem orders(k)(k)(k)(s)(s)(s)k=10 90-96% Energyk=15 92-97% Energyk=20 93-98% Energy

Frequency diagrammode 1mode 2mode 3mode 4mode 51.74Hz5.34HzPSD of response time series8.82Hz13.67Hz18.044Hzs=50k=5:5:60

Natural frequencies (Hz)FDDSSImode 11.731.74mode 25.355.34mode 38.848.82mode 413.6913.67mode 518.1218.04

Frequency diagrammode 1mode 2mode 3mode 4mode 5s=20:10:120k=60

Frequency diagramPSD of response time seriesmode 1mode 2mode 3mode 4mode 5s=20:10:120k=60

Damping diagrams=50k=5:5:60mode 10.18%mode 20.22%mode 30.46%

Damping diagrams=20:10:120k=60mode 10.18%mode 20.22%mode 30.47%

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