some investigations on modal identification methods of ambient vibration structures le thai hoa wind...
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Some investigations on modal identification methods of
ambient vibration structures
Le Thai HoaWind Engineering Research Center
Tokyo Polytechnic University
Date: 2009/12/05
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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
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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 ….
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Vibration tests/modal identification
Ambient Vibration Tests
ForcedVibration Tests
Random/StochasticRandom/Stochastic
Deterministic/Deterministic/StochasticStochastic
ExperimentalModal Analysis
OperationalModal Analysis
Output-only Identification
Ambient loads &Micro tremor
Shaker (Harmonic)Hummer (Impulse)Sine sweep (Harmonic) Base servo (White noise, Seismic loads)
ExperimentalModal Analysis
OperationalModal Analysis
Input-output Identification
Output-only Identification
FRF identificationTransfer Functions
Removing harmonic & input effects
Indirect & direct identifications
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Modal identification methods
Time domainTime domain Frequency domainFrequency domain Time-frequency planeTime-frequency plane
Ibrahim Time Domain (ITD)
Random Decrement Technique (RDT)
Stochastic Subspace Identification (SSI)
Frequency Domain Decomposition (FDD)
Wavelet Transform (WT)
Hilbert-Huang Transform (HHT)
[[Time-scale PlaneTime-scale Plane]]
Enhanced Frequency Domain Decomposition (EFDD)
Eigensystem Realization Algorithm (ERA)
Ambient vibration – Output-only system identificationAmbient vibration – Output-only system identification
Applicable in conditions and combined
Commercial and industrial uses
Academic uses, under development
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Commercial Frequency domain Time domain
Package ODS FDDPeak Picking(No damping)
EFDD(Damping)
SSI(UPC)
SSI(PC)
SSI(CVA)
ARTeMIS Light
ARTeMIS Handy
ARTeMIS Pro
ARTeMIS Extractor 2009 FamilyThe State-of-the-Art software for Operational Modal Analysis
ODS: Operational Deflection ShapesFDD: Frequency Domain Decomposition EFDD: Enhanced Frequency Domain DecompositionSSI: Stochastic Subspace Identification UPC: Unweighted Principal Component PC: Principal Component CVA: Canonical Variate Algorithm
Commercial Software for OMA
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Uses of FDD, RDT and SSI
[[Time-scale PlaneTime-scale Plane]]
For MDOF SystemsFor MDOF Systems
Response time series
Y(t)
Power SpectralDensity MatrixSYY(n)
Modal Parameters
FDD
(POD, SVD…)
EFDD
(POD, SVD…)
RDFunctions
DYY(t)
RDTITD
MRDT
CovarianceMatrixRYY(t)
Direct method SSI-COV
(POD, SVD…)
Direct method Data Matrix HY(t)
SSI-DATA
(POD, SVD…)
FD
DR
DT
SS
I
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Comparison FDD, RDT and SSIComparison 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
FDD
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
Current trends in modal identification:
Combination between identification methods
Refined techniques of identification methods
Comparisons between identification methods
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RDT to refine modal identification
OutputResponse
Time seriesY(t)
RDTRandom
Decrement Function
RDF
RDF-ITD & ERA
RDF-SSI-Covariance
Modal Parameters
RDF-BF PowerSpectralMatrix RDF-FDD
RDF-SSI-Data
Wavelet Transform (WT)
Hilbert-Huang Transform
Time Domain
Time-Frequency Plane
Frequency Domain
Time-frequency DomainPossibilities of RDT combined with othermodal identification methods
Multi-mode RDT
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Frequency Domain Decomposition (FDD)
Random DecrementTechnique (RDT)
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oFDD 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 DecompositionFrequency Domain Decomposition
o Relation between inputs excitation X(t) and output response Y(t) can be expressed via the complex FRF function matrix:
o Also FRF matrix written as normal pole/residue fraction form, we can obtain the output complex PSD matrix:
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o Output spectral matrix estimated from output data
o Output spectral matrix is decomposed (SVD, POD…)
Where: Spectral eigenvalues (Singular values) & Spectral eigenvectors (Singular vectors)
o ith modal shape identified at selected frequency
)()()()()()()(1
Tyi
N
iyiyi
TyyyyyS
rxll
k
ry
ry
ry
y
]:1[
]:1[
]:1[
2
1
cutllll
l
l
fxlxlyyyyyy
yyyyyy
yyyyyy
yy
SSS
SSS
SSS
S
)(...)()(
............
)(...)()(
)(...)()(
)(
21
22212
12111
)()()()( 111 TyyyyyS
)(),( yy
)(1 iyi
i
Frequencies & Frequencies & Damping Damping RatiosRatiosIdentificationIdentification
Mode shapesMode shapesIdentificationIdentification
Frequency Domain DecompositionFrequency Domain Decomposition
Output Output responseresponse
PSD matrixPSD matrix
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Random Decrement Techniques
Triggering condition Xo
Xo
RD function (Free decay)
RDT extracts free decay data from ambient response of structures (as averaging and eliminating initial condition)
&
to
0 0
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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
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Combined FDD-RDT diagram
Response Data Matrix
Y(t)
Cross Power Spectral Matrix
SYY(n)
1st Spectral Eigenvalue
1st Spectral Eigenvector
Natural Frequencies
Free Decay Fun. & Damping Ratios
Mode Shapes
POD, SVD, QR…
Data Matrix
Y(t)
Cross Power Spectral Matrix
SYY(n)
1st Spectral Eigenvalue
1st Spectral Eigenvector
Natural Frequencies
Free Decay Fun. & Damping Ratios
Mode Shapes
RDFun.
DYY(t)
POD, SVD, QR…RDT
FDD-RDT
FDD
Damping only
Natural Frequencies
…FDD Response Series at
Filtered FrequenciesFree Decay Fun. &
Damping Ratios
RDTBPFat fi
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Stochastic Subspace Identification(SSI)
o Covariance-driven SSI
o Data-driven SSI
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SSISSI 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
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State-space representationState-space representation Continuous stochastic state-space model
wk: process noise (disturbances, modeling, input)vk : sensor noise
:
Discrete stochastic state-space model
state-space model
Second-order equations First order equationsA: state matrix; C: output matrix X(t): state vector; Y(t): response vector
yk
vk
C
Stochastic system
A
wk
(
,
wk , vk : zero mean white noises with covariance matrix
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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 Block Toeplitz matrix
s: number of block rowsN-2s: number of block columns
shifted t
past
future
s: number of block rows
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SSI-COV and SSI-DATASSI-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 matrices
k: number of singular valuesk: system order
&
Modal parameters estimationMode shapes:
Poles:
Frequencies: Damping:
Hankel
Toeplitz
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Flow chart of SSI algorithmData Matrix
[Y(t)]
Data Rearrangement
Block Teoplitz MatrixRP [], RF[],
Block Hankel MatrixHP[], HF[]
SSI-COV
SSI-DATAOrthogonal Projection
Ps
Observability MatrixOs
System MatricesA, C
Modal ParametersStabilization Diagram
Covaria
nce
DataData order reduction
POD
POD
Hankel matrix
Data past/ future
Toeplitz matrix
Parameter k
Parameter s
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Modal identification of ambient vibration structures using combined Frequency Domain Decomposition and Random Decrement Technique
Numerical example
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Fullscale ambient measurement
Five-storey steel frame
GroundGround
Floor1Floor1
Floor2Floor2
Floor3Floor3
Floor4Floor4
Floor5Floor5
X
Y
Z
0 50 100 150 200 250 300-1.5
-1
-0.5
0
0.5
1
1.5x 10
-3
Time (s)
Dis
p.
(m
)
Floor5
0 50 100 150 200 250 300-1.5
-1
-0.5
0
0.5
1
1.5x 10
-3
Time (s)
Dis
p.
(m
)
Floor4
0 50 100 150 200 250 300-1
-0.5
0
0.5
1x 10
-3
Time (s)
Dis
p.
(m
)
Floor3
0 50 100 150 200 250 300-1
-0.5
0
0.5
1x 10
-3
Time (s)
Dis
p.
(m
)
Floor2
0 50 100 150 200 250 300-5
0
5x 10
-4
Time (s)D
isp
. (m
)
Floor1
0 50 100 150 200 250 300-1
0
1x 10
-4
Time (s)
Dis
p. (m
)
Ground
Output displacement (X)
Floor 5
Floor 4
Floor 3
Floor 2
Floor 1
Ground
5 minutes record
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Random decrement functions
0 5 10 15 20 25 30 35 40 45 50-8
-6
-4
-2
0
2
4
6
8x 10
-4 Floor5
Time (s)
Dis
p.(m
) Floor 5
0 5 10 15 20 25 30 35 40 45 50-8
-6
-4
-2
0
2
4
6
8x 10
-4 Floor4
Time (s)D
isp.(m
) Floor 4
0 5 10 15 20 25 30 35 40 45 50-5
0
5x 10
-4 Floor3
Time (s)
Dis
p.(m
) Floor 3
0 5 10 15 20 25 30 35 40 45 50-4
-3
-2
-1
0
1
2
3
4x 10
-4 Floor2
Time (s)
Dis
p.(m
) Floor 2
level crossing: segment: 50sno. of sample: 30000no. of samples in segment: 5000
Parameters
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Spectral eigenvalues
0 5 10 15 20 25 30
10-10
10-5
100
Frequency (Hz)
No
rma
lize
d e
ige
nva
lue
s
Eigenvalue1Eigenvalue2Eigenvalue3Eigenvalue4Eigenvalue5Eigenvalue6
13.69Hz11.45Hz
8.84Hz10.16Hz
19.75Hz
18.12Hz
1.73Hz
5.35Hz
Mode 1 Mode 2Mode 3
Mode 4Mode 5
0 5 10 15 20 25 3010
-40
10-30
10-20
10-10
100
Frequency (Hz)
Nor
mal
ized
PS
D
Singular value 1Singular value 2Singular value 3Singular value 4Singular value 5Singular value 6
1.73Hz 5.34Hz
8.82Hz10.16Hz
13.67Hz 18.02Hz
Mode 1 Mode 2Mode 3 Mode 4Mode 5
FDD
FDD-RDT
Eigenvalue1: 99.9% Eigenvalue2: 0.07%Eigenvalue3: 0.01%Eigenvalue4: 0%
Eigenvalue1: 100% Eigenvalue2: 0%Eigenvalue3: 0%Eigenvalue4: 0%
Natural frequencies (Hz) FDD FDD-RDTmode 1 1.73 1.73mode 2 5.35 5.34mode 3 8.84 8.82mode 4 13.69 13.67mode 5 18.12 18.02
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Spectral eigenvectors
99.9%
0.07%
0.01%
0%
FDD
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FDD-RDT
Spectral eigenvectors
100%
0%
0%
0%
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Mode shapes estimation
0 0.25 0.5 0.75 1Ground
Floor1
Floor2
Floor3
Floor4
Floor5Mode 1
FEMIdentified
-1 -0.5 0 0.5 1Ground
Floor1
Floor2
Floor3
Floor4
Floor5Mode 2
FEMIdentified
-1 -0.5 0 0.5 1Ground
Floor1
Floor2
Floor3
Floor4
Floor5Mode 3
FEMIdentified
-1 -0.5 0 0.5 1Ground
Floor1
Floor2
Floor3
Floor4
Floor5Mode 4
FEMIdentified
-1 -0.5 0 0.5 1Ground
Floor1
Floor2
Floor3
Floor4
Floor5Mode 5
FEMIdentified
Mode 1Mode 1
Mode 4Mode 4 Mode 5Mode 5
}}{{
||),(
2
ATAE
TE
ATE
AEMAC
Mode 2Mode 2 Mode 3Mode 3FDD
MACMAC
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Mode shapes comparisonMode 1Mode 1
Mode 4Mode 4 Mode 5Mode 5
Mode 2Mode 2 Mode 3Mode 3
0 0.2 0.4 0.6 0.8 1Ground
Floor 1
Floor 2
Floor 3
Floor 4
Floor 5
Normalized amplitude
Mode 1
FEMFDDFDD-RDT
-1 -0.5 0 0.5 1Ground
Floor 1
Floor 2
Floor 3
Floor 4
Floor 5
Normalized amplitude
Mode 2
FEMFDDFDD-RDT
-1 -0.5 0 0.5 1Ground
Floor 1
Floor 2
Floor 3
Floor 4
Floor 5
Normalized amplitude
Mode 3
FEMFDDFDD-RDT
-1 -0.5 0 0.5 1Ground
Floor 1
Floor 2
Floor 3
Floor 4
Floor 5
Normalized amplitude
Mode 4
FEMFDDFDD-RDT
-1 -0.5 0 0.5 1Ground
Floor 1
Floor 2
Floor 3
Floor 4
floor 5
Normalized amplitude
Mode 5
FEMFDDFDD-RDT
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Identified auto PSD functions
MAC=98%
MAC=95%
FDD
Mode 1Mode 2
Mode 3Mode 4
Mode 5
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Identified free decay functions
0 0.5 1 1.5 2 2.5 3 3.5 4 4.5 5-1.5
-1
-0.5
0
0.5
1
1.5
Time (s)
Nor
mal
ized
am
plitu
de
Free decay function of mode 1
0 0.5 1 1.5 2 2.5 3 3.5 4 4.5 5-1.5
-1
-0.5
0
0.5
1
1.5
Time (s)
Nor
mal
ized
am
plitu
de
Free decay function of mode 2
0 0.5 1 1.5 2 2.5 3 3.5 4 4.5 5-1.5
-1
-0.5
0
0.5
1
1.5
Time (s)
Nor
mal
ized
am
plitu
de
Free decay function of mode 3
0 0.5 1 1.5 2 2.5 3 3.5 4 4.5 5-1.5
-1
-0.5
0
0.5
1
1.5
Time (s)
Nor
mal
ized
am
plitu
de
Free decay function of mode 5
Mod
e 1
Mod
e 1
Mod
e 4
Mod
e 4
Mod
e 2
Mod
e 2
Mod
e 3
Mod
e 3
FDD
Mod
e 5
Mod
e 5
0 0.5 1 1.5 2 2.5 3 3.5 4 4.5 5-1.5
-1
-0.5
0
0.5
1
1.5
Time (s)N
orm
aliz
ed a
mpl
itude
Free decay function of mode 4
Uncertainty in damping ratios estimation from free decay functions of modes 3 & 4
Unclear with modes 2 & 5
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Identified free decay functionsM
ode
1M
ode
1
Mod
e 4
Mod
e 4
Mod
e 2
Mod
e 2
Mod
e 3
Mod
e 3
Mod
e 5
Mod
e 5
Better
0 0.5 1 1.5 2 2.5 3 3.5 4 4.5 5-1.5
-1
-0.5
0
0.5
1
1.5
Time (s)
No
rma
lize
d a
mp
litu
de
Free decay function for mode 1
0 0.5 1 1.5 2 2.5 3 3.5 4 4.5 5-1.5
-1
-0.5
0
0.5
1
1.5
Time (s)
No
rma
lize
d a
mp
litu
de
Free decay function for mode 2 FDD
0 0.5 1 1.5 2 2.5 3 3.5 4 4.5 5-1.5
-1
-0.5
0
0.5
1
1.5
Time (s)
No
rma
lize
d a
mp
litu
de
Free decay function for mode 3
0 0.5 1 1.5 2 2.5 3 3.5 4 4.5 5-1.5
-1
-0.5
0
0.5
1
1.5
Time (s)N
orm
aliz
ed
am
plit
ud
e
Free decay function for mode 4
0 0.5 1 1.5 2 2.5 3 3.5 4 4.5 5-1.5
-1
-0.5
0
0.5
1
1.5
Time (s)
No
rma
lize
d a
mp
litu
de
Free decay function for mode 5
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FDD - Band-pass filtering
-2
0
2x 10
-3 Bandpass Filtering
-2
0
2x 10
-3
-2
0
2x 10
-4
Am
p.(
m)
-5
0
5x 10
-5
-5
0
5x 10
-5
0 50 100 150 200 250 300-2
0
2x 10
-5
Time (s)
Filtered at mode 1
Filtered at mode 2
Filtered at mode 3
Filtered at mode 4
Filtered at mode 5
Original output
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
0 2 4 6 8 10 12 14 16 18 2010
-30
10-25
10-20
10-15
10-10
10-5
Frequency (Hz)
PSD
Bandpass Filtering
Filtered at mode 1
Filtered at mode 2 Filtered
at mode 3Filtered at mode 5
Filtered at mode 4
Floor 5
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Damping ratio via FDD-BPF
0 5 10 15 20 25 30 35 40 45 50-5
0
5x 10
-4
Time (s)
Am
p. (
m)
RD function
Filtered at mode 1
Mod
e 1
Mod
e 1
0 5 10 15-4
-2
0
2
4x 10
-5
Time (s)
Am
p.(m
)
RD function
Filtered at mode 2
Mod
e 2
Mod
e 2
0 1 2 3 4 5 6 7 8 9 10-4
-2
0
2
4x 10
-6
Time (s)
Am
p.(m
)
RD function
Filtered at mode 3
Mod
e 3
Mod
e 3
0 0.5 1 1.5 2 2.5 3 3.5 4 4.5 5-3
-2
-1
0
1
2
3x 10
-6
Time (s)A
mp.
(m)
RD function
Filtered at mode 4
Mod
e 4
Mod
e 4
0 0.5 1 1.5 2 2.5 3 3.5 4 4.5 5-3
-2
-1
0
1
2
3x 10
-6
Time (s)
Am
p.(m
)
RD function
Filtered at mode 5
Mod
e 5
Mod
e 5 Uncertainty in damping
ratios estimated from free decay functions at modes 4 & 5
Free decay functionsFloor 5
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FDD - Band-pass filtering
Response time series at Floor 1 has been filtered on spectral bandwidth around each modal frequency
Floor 1
-5
0
5x 10
-4 Bandpass Filtering
-5
0
5x 10
-4
-2
0
2x 10
-5
-1
0
1x 10
-5
0 50 100 150 200 250 300-1
0
1x 10
-5
Time (s)
-5
0
5x 10
-5
Am
p.(
m)
Original output
Filtered at mode 1
Filtered at mode 2
Filtered at mode 3
Filtered at mode 4
Filtered at mode 5
f1=1.73Hz
f2=5.34Hz
f3=8.82Hz
f4=13.67Hz
f5=18.02Hz
X1(t)
0 2 4 6 8 10 12 14 16 18 2010
-30
10-20
10-10
100
Frequency (Hz)
PSD
Bandpass Filtering
Filtered at mode 3
Filtered at mode 2
Filtered at mode 5
Filtered at mode 1
Filtered at mode 4
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Damping ratio via FDD-BPFM
ode
1M
ode
1
Mod
e 2
Mod
e 2
Mod
e 3
Mod
e 3
Mod
e 4
Mod
e 4
Mod
e 5
Mod
e 5
Free decay functions
0 5 10 15 20 25 30 35 40 45 50-1.5
-1
-0.5
0
0.5
1
1.5x 10
-4
Time (s)
Am
p.(m
)
RD function
Filtered at mode 1
0 5 10 15-3
-2
-1
0
1
2
3x 10
-5
Time (s)
Am
p.(m
)
RD function
Filtered at mode 2
0 1 2 3 4 5 6 7 8 9 10-5
0
5x 10
-6
Time (s)
Am
p.(m
)
RD function
Filtered at mode 3
0 0.5 1 1.5 2 2.5 3 3.5 4 4.5 5-2
-1
0
1
2x 10
-6
Time (s)A
mp.
(m)
RD function
Filtered at mode 4
0 0.5 1 1.5 2 2.5 3-1
-0.5
0
0.5
1x 10
-6
Time (s)
Am
p.(m
)
RD function
Filtered at mode 5
Floor 1
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Damping ratio via FDD-BPFSelected free decay functions for damping estimation
Mod
e 1
Mod
e 1
Mod
e 2
Mod
e 2
Mod
e 3
Mod
e 3
Mod
e 4
Mod
e 4
Mod
e 5
Mod
e 5
0 0.5 1 1.5 2 2.5 3-1
-0.5
0
0.5
1x 10
-6
Time (s)
Am
p.(m
)
RD function
Filtered at mode 5
0 5 10 15-3
-2
-1
0
1
2
3x 10
-5
Time (s)
Am
p.(m
)
RD function
Filtered at mode 2
0 5 10 15 20 25 30 35 40 45 50-1.5
-1
-0.5
0
0.5
1
1.5x 10
-4
Time (s)
Am
p.(m
)
RD function
Filtered at mode 1
0 1 2 3 4 5 6 7 8 9 10-4
-2
0
2
4x 10
-6
Time (s)
Am
p.(m
)
RD function
Filtered at mode 3
0 0.5 1 1.5 2 2.5 3 3.5 4 4.5 5-2
-1
0
1
2x 10
-6
Time (s)A
mp.
(m)
RD function
Filtered at mode 4
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Time-domain modal identification of ambient vibration structures using Stochastic Subspace Identification
Numerical example
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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 parameters
Hankel matrix parameters
System order parameters
Number of system order: k=5:5:60 (12 cases) (Number of singular values used)
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Projection functions
0 50 100 150 200 250 300-4
-2
0
2
4
6
8
10x 10
-4
Time (ms)
Am
plit
ude
After orthogonal projection
s=50, column=10
0 50 100 150 200 250 300-6
-4
-2
0
2
4
6
8x 10
-4
Time (ms)
Am
plit
ud
e
After orthogonal projection
s=50, column=50
0 50 100 150 200 250 300-4
-2
0
2
4
6x 10
-4
Time (ms)
Am
plit
ude
After orthogonal projection
s=50, column=200
0 50 100 150 200 250 300-5
0
5x 10
-4
Time (ms)
Am
plit
ude
After orthogonal projection
s=50, column=500
0 100 200 300 400 500 600-6
-4
-2
0
2
4
6
8x 10
-4
Time (ms)
Am
plit
ude
After orthogonal projection
s=100, column=10
0 100 200 300 400 500 600-6
-4
-2
0
2
4
6
8x 10
-4
Time (ms)
Am
plit
ude
After orthogonal projection
s=100, column=50
0 100 200 300 400 500 600 700 800 900-6
-4
-2
0
2
4
6x 10
-4
Time (ms)
Am
plit
ud
e
After orthogonal projection
s=100, column=10
0 100 200 300 400 500 600 700 800 900-6
-4
-2
0
2
4
6x 10
-4
Time (ms)
Am
plit
ud
e
After orthogonal projection
s=150, column=50
0 100 200 300 400 500 600 700 800 900-5
0
5x 10
-4
Time (ms)
Am
plit
ud
e
After orthogonal projection
s=150, column=200
0 100 200 300 400 500 600 700 800 900-4
-2
0
2
4
6x 10
-4
Time (ms)
Am
plit
ud
e
After orthogonal projection
s=150, column=500
0 100 200 300 400 500 600-5
0
5x 10
-4
Time (ms)
Am
plit
ud
e
After orthogonal projection
s=100, column=200
0 100 200 300 400 500 600-5
0
5x 10
-4
Time (ms)
Am
plit
ud
e
After orthogonal projection
s=100, column=500
s=50s=50 s=100s=100 s=150s=150
Data after orthogonal projection look like time-shifted sine functions
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Effects of s on energy contribution
1 2 3 4 5 6 7 8 9 100
0.1
0.2
0.3
0.4
0.5
Number of singular values
Singular values
s=20s=40s=60s=80s=100s=120
1 2 3 4 5 6 7 8 9 100
10
20
30
40
50
Number of singular values
Pe
rce
nta
ge
(%
)
Energy contribution
s=20s=40s=60s=80s=100s=120
1 2 3 4 5 6 7 8 9 1030
40
50
60
70
80
90
100
Number of singular values)
Pe
rce
nta
ge
(%
)
Energy cummulation
s=20s=40s=60s=80s=100s=120
20 40 60 80 100 1200
0.1
0.2
0.3
0.4
0.5
Number of block row s
Singular velues
Singular value 1Singular value 2Singular value 3Singular value 4Singular value 5
20 40 60 80 100 1200
10
20
30
40
50
Number of block row s
Pe
rce
nta
ge
(%
)
Energy contribution
Singular value 1Singular value 2Singular value 3Singular value 4Singular value 5
20 40 60 80 100 12030
40
50
60
70
80
90
100
Number of block row s
Pe
rce
nta
ge
(%
)Energy cummulation
Fisrt singular valueFirst 2 singular valuesFirst 3 singular valuesFirst 4 singular valuesFirst 5 singular values
system orders (k)
(k)
(k)
(s)
(s)
(s)
k=10 90-96% Energyk=15 92-97% Energyk=20 93-98% Energy
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0 2 4 6 8 10 12 14 16 18 200
10
20
30
40
50
60N
um
be
r o
f p
ole
s
Natural frequency (Hz)
Frequency diagram
Frequency diagram
0 2 4 6 8 10 12 14 16 18 2010
-15
10-10
10-5
Natural frequency (Hz)
PS
D
X5
mod
e 1
mod
e 2
mod
e 3
mod
e 4
mod
e 5
1.74
Hz
5.34
Hz
PSD of response time series
8.82
Hz
13.6
7Hz
18.0
44H
z
s=50k=5:5:60Natural frequencies (Hz)
FDD SSImode 1 1.73 1.74mode 2 5.35 5.34mode 3 8.84 8.82mode 4 13.69 13.67mode 5 18.12 18.04
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Frequency diagram
0 2 4 6 8 10 12 14 16 18 200
10
20
30
40
50
60S
yste
m o
rder
k
Natural frequency (Hz)
mod
e 1
mod
e 2
mod
e 3
mod
e 4
mod
e 5
s=20:10:120k=60
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Frequency diagram
0 2 4 6 8 10 12 14 16 18 2020
30
40
50
60
70
80
90
100
110
120N
um
be
r o
f b
lock r
ow
s s
Natural frequency (Hz)
0 2 4 6 8 10 12 14 16 18 2010
-15
10-10
10-5
Natural frequency (Hz)
PS
D
X5
PSD of response time series
mode 1
mod
e 2
mod
e 3
mod
e 4
mod
e 5
s=20:10:120k=60
![Page 45: Some investigations on modal identification methods of ambient vibration structures Le Thai Hoa Wind Engineering Research Center Tokyo Polytechnic University](https://reader035.vdocuments.mx/reader035/viewer/2022062422/56649ea05503460f94ba3a72/html5/thumbnails/45.jpg)
Damping diagram
0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.90
10
20
30
40
50
60
Num
ber
of p
oles
Damping ratio (%)
s=50k=5:5:60
mode 1
0.18%
mode 2
0.22% mod
e 3
0.46%
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0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.90
10
20
30
40
50
60
70
80
Sys
tem
ord
er k
Damping ratio (%)
Damping diagram
s=20:10:120k=60
mode 1
0.18%mod
e 2
0.22% mod
e 3
0.47%