Date: 2009/12/05 Some investigations on modal identification methods of ambient vibration structures Le Thai Hoa Wind Engineering Research Center Tokyo Polytechnic University

Contents 1. Frequency-domain modal identification of ambient vibration structures using combined Frequency Domain Decomposition and Random Decrement Technique 2. Time-domain modal identification of ambient vibration structures using Stochastic Subspace Identification 3. 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 plane–based modal identification methods Nonparametric/ Parametric identification methods  SDOF and MDOF system identifications ….

Vibration tests/modal identification Ambient loads & Micro tremor Indirect & direct identifications Experimental Modal Analysis Ambient Vibration Tests Output-only Identification Operational Modal Analysis Random/Stochastic Shaker (Harmonic) Hummer (Impulse) Sine sweep (Harmonic) Base servo (White noise, Seismic loads) FRF identification Transfer Functions Experimental Modal Analysis Input-output Identification Forced Vibration Tests Operational Modal Analysis Output-only Identification Deterministic/Stochastic Removing harmonic & input effects

Modal identification methods Ambient vibration – Output-only system identification Time domain Frequency domain Time-frequency plane Ibrahim Time Domain (ITD) Frequency Domain Decomposition (FDD) Wavelet Transform (WT) Eigensystem Realization Algorithm (ERA) Hilbert-Huang Transform (HHT) Enhanced Frequency Domain Decomposition (EFDD) Random Decrement Technique (RDT) Stochastic Subspace Identification (SSI) Applicable in conditions and combined [Time-scale Plane] Commercial and industrial uses Academic uses, under development

Commercial Software for OMA ARTeMIS Extractor 2009 Family The State-of-the-Art software for Operational Modal Analysis Commercial Frequency domain Time domain Package ODS FDD Peak Picking (No damping) EFDD (Damping) SSI (UPC) (PC) (CVA) ARTeMIS Light  ARTeMIS Handy ARTeMIS Pro ODS: Operational Deflection Shapes FDD: Frequency Domain Decomposition EFDD: Enhanced Frequency Domain Decomposition SSI: Stochastic Subspace Identification UPC: Unweighted Principal Component PC: Principal Component CVA: Canonical Variate Algorithm

Uses of FDD, RDT and SSI For MDOF Systems Power Spectral Density Response time series Y(t) Power Spectral Density Matrix SYY(n) FDD Modal Parameters (POD, SVD…) FDD EFDD (POD, SVD…) ITD RD Functions DYY(t) RDT MRDT RDT CovarianceMatrix RYY(t) Direct method SSI-COV (POD, SVD…) SSI [Time-scale Plane] Direct method Data Matrix HY(t) SSI-DATA (POD, SVD…)

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 frequencies Current trends in modal identification:  Combination between identification methods  Refined techniques of identification methods  Comparisons between identification methods RDT 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 methods SSI Advantages: Dealing with data directly, no leakage and less random errors, direct estimation of frequencies, damping Disadvantage: Stabilization diagram, many parameters

RDT to refine modal identification Output Response Time series Y(t) RDT Random Decrement Function RDF RDF-ITD & ERA RDF-SSI-Covariance Modal Parameters RDF-BF Power Spectral Matrix RDF-FDD RDF-SSI-Data Wavelet Transform (WT) Hilbert-Huang Transform Time Domain Time-Frequency Plane Frequency Domain Multi-mode RDT Possibilities of RDT combined with other modal identification methods Time-frequency Domain

Frequency Domain Decomposition (FDD) Random Decrement Technique (RDT)

Frequency Domain Decomposition 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 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:

Frequency Domain Decomposition Output spectral matrix estimated from output data Output response PSD matrix Output spectral matrix is decomposed (SVD, POD…) Frequencies & Damping Ratios Identification Where: Spectral eigenvalues (Singular values) & Spectral eigenvectors (Singular vectors) Mode shapes Identification ith modal shape identified at selected frequency

Random Decrement Techniques RDT extracts free decay data from ambient response of structures (as averaging and eliminating initial condition) & to Triggering condition Xo Xo RD function (Free decay)

Conditional correlation functions Random Decrement Techniques RD functions (RD signatures) are formed by averaging N segments of X(t) with conditional value Xo (Auto-RD signature) Conditional correlation functions (Cross-RD signature) N : Number of averaged time segments X0 : Triggering condition (crossing level) k : Length of segment

Combined FDD-RDT diagram POD, SVD, QR… Natural Frequencies 1st Spectral Eigenvalue Response Data Matrix Y(t) Cross Power Spectral Matrix SYY(n) Free Decay Fun. & Damping Ratios 1st Spectral Eigenvector Mode Shapes FDD-RDT POD, SVD, QR… Natural Frequencies RDT 1st Spectral Eigenvalue Data Matrix Y(t) RD Fun. DYY(t) Cross Power Spectral Matrix SYY(n) Free Decay Fun. & Damping Ratios 1st Spectral Eigenvector Mode Shapes Damping only FDD BPF RDT Natural Frequencies Response Series at Filtered Frequencies Free Decay Fun. & Damping Ratios … at 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 ( State-space representation : , Continuous stochastic state-space model state-space model Second-order equations First order equations A: state matrix; C: output matrix X(t): state vector; Y(t): response vector Discrete stochastic state-space model wk: process noise (disturbances, modeling, input) vk : sensor noise wk vk wk , vk : zero mean white noises with covariance matrix  C yk A Stochastic system

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 blocks Block Hankel matrix shifted t Block Toeplitz matrix past future s: number of block rows N-2s: number of block columns s: 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 Hankel k: number of singular values k: system order Toeplitz Observability matrix & system matrices  & Modal parameters estimation Mode shapes: Poles: Frequencies: Damping:

Flow chart of SSI algorithm Data Matrix [Y(t)] SSI-COV Covariance Block Teoplitz Matrix RP [], RF[], Data past/ future Data Rearrangement Parameter s Block Hankel Matrix HP[], HF[] Data order reduction Data Orthogonal Projection Ps SSI-DATA Hankel matrix POD Parameter k Observability Matrix Os System Matrices A, C Toeplitz matrix POD Modal Parameters Stabilization Diagram

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

Fullscale ambient measurement 5 minutes record Floor 5 Floor5 Floor 4 Floor4 Floor 3 Floor3 Floor 2 Floor2 X Y Z Floor 1 Floor1 Ground Ground (X) Five-storey steel frame Output displacement

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

Spectral eigenvalues FDD FDD-RDT Natural frequencies (Hz) FDD FDD-RDT Mode 1 Mode 2 Mode 3 Mode 4 Mode 5 FDD Eigenvalue1: 99.9% Eigenvalue2: 0.07% Eigenvalue3: 0.01% Eigenvalue4: 0% Natural frequencies (Hz)   FDD FDD-RDT mode 1 1.73 mode 2 5.35 5.34 mode 3 8.84 8.82 mode 4 13.69 13.67 mode 5 18.12 18.02 Mode 1 Mode 2 Mode 3 Mode 4 Mode 5 FDD-RDT Eigenvalue1: 100% Eigenvalue2: 0% Eigenvalue3: 0% Eigenvalue4: 0%

Spectral eigenvectors FDD 99.9% 0.01% 0.07% 0%

Spectral eigenvectors FDD-RDT 100% 0% 0% 0%

Mode shapes estimation FDD Mode 4 Mode 5 MAC

Mode shapes comparison

Identified auto PSD functions FDD MAC=95% Mode 1 Mode 2 Mode 3 Mode 4 Mode 5 MAC=98%

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

Identified free decay functions FDD Mode 1 Mode 2 Mode 3 Mode 4 Mode 5 Better

FDD - Band-pass filtering Floor 5 X5(t) f1=1.73Hz f2=5.34Hz f3=8.82Hz f4=13.67Hz f5=18.02Hz Response time series at Floor 5 has been filtered on spectral bandwidth around each modal frequency

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

FDD - Band-pass filtering Floor 1 X1(t) f1=1.73Hz f2=5.34Hz f3=8.82Hz f4=13.67Hz f5=18.02Hz Response time series at Floor 1 has been filtered on spectral bandwidth around each modal frequency

Damping ratio via FDD-BPF Free decay functions Floor 1 Mode 2 Mode 1 Mode 3 Mode 4 Mode 5

Damping ratio via FDD-BPF Selected free decay functions for damping estimation Mode 2 Mode 1 Mode 3 Mode 4 Mode 5

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

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

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

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

Frequency diagram Natural frequencies (Hz) FDD SSI mode 1 1.73 1.74 k=5:5:60 Natural frequencies (Hz)   FDD SSI mode 1 1.73 1.74 mode 2 5.35 5.34 mode 3 8.84 8.82 mode 4 13.69 13.67 mode 5 18.12 18.04 PSD of response time series

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

Frequency diagram mode 5 mode 4 mode 3 mode 1 mode 2 s=20:10:120 k=60 PSD of response time series

Damping diagram mode 1 0.18% mode 3 mode 2 0.46% 0.22% s=50 k=5:5:60

Damping diagram mode 1 0.18% mode 2 mode 3 0.22% 0.47% s=20:10:120 k=60