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

2. 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

3. 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 ….

4. 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

5. 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

6. 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

7. 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…)

8. 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

9. 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

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

11. 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:

12. 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

13. 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)

14. 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

15. 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

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

17. 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

18. 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

19. 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

20. 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:

21. 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

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

23. 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

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

25. 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%

26. Spectral eigenvectors
FDD
99.9%
0.01%
0.07% 0%

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

28. Mode shapes estimation
FDD
Mode 4
Mode 5
MAC

29. Mode shapes comparison

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

31. 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

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

33. 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

34. 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

35. 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

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

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

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

39. 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)

40. 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

41. 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)

42. 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

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

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

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

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