Transcript Some investigations in modal parameters identification of

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
Ambient
Vibration Tests
Random/Stochastic
Experimental
Modal Analysis
Operational
Modal Analysis
Shaker (Harmonic)
Hummer (Impulse)
Sine sweep (Harmonic)
Base servo (White
noise, Seismic loads)
Forced
Vibration Tests
Deterministic/Stochastic
Indirect & direct
identifications
Output-only
Identification
FRF identification
Transfer Functions
Experimental
Modal Analysis
Input-output
Identification
Operational
Modal Analysis
Output-only
Identification
Removing harmonic
& input effects
Modal identification methods
Ambient vibration – Output-only system identification
Time domain
Ibrahim Time
Domain (ITD)
Eigensystem
Realization
Algorithm (ERA)
Random
Decrement
Technique (RDT)
Stochastic
Subspace
Identification (SSI)
Frequency domain
Frequency Domain
Decomposition
(FDD)
Enhanced
Frequency Domain
Decomposition
(EFDD)
Time-frequency plane
Wavelet
Transform (WT)
Hilbert-Huang
Transform (HHT)
Applicable in conditions and combined
Commercial and[Time-scale
industrial Plane]
uses
Academic uses, under development
Commercial Software for OMA
ARTeMIS Extractor 2009 Family
The State-of-the-Art software for Operational Modal Analysis
Commercial
Package
Frequency domain
ODS
FDD
Peak Picking
(No damping)
EFDD
(Damping)
ARTeMIS Light


ARTeMIS Handy



ARTeMIS Pro



Time domain
SSI
(UPC)
SSI
(PC)
SSI
(CVA)



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
RDT
Direct method
Direct method
RD
Functions
DYY(t)
Covariance
Matrix
RYY(t)
Data Matrix
HY(t)
FDD
(POD, SVD…)
EFDD
(POD, SVD…)
ITD
MRDT
Modal
Parameters
RDT
Response
time series
Y(t)
FDD
SSI-COV
(POD, SVD…)
SSI-DATA
(POD, SVD…)
SSI
Power
Spectral
Density
Matrix
SYY(n)
[Time-scale Plane]
Comparison FDD, RDT and SSI
FDD
 Advantages: Dealing with cross spectral matrix, good for
natural frequencies and mode shapes estimation
 Disadvantage: based on strict assumptions, leakage
Current
trends in modal identification:
due to Fourier transform, damping ratios, effects of inputs
and
closed
frequencies
 harmonics;
Combination
between
identification methods
RDT
 Refined techniques of identification methods
 Advantages: Dealing with data correlation, removing noise
 initial,
Comparisons
identification
methods
and
good for between
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
Time Domain
Multi-mode RDT
RDF-ITD & ERA
RDF-SSI-Covariance
RDF-SSI-Data
Output
Response
Time series
Y(t)
RDT
Random
Decrement
Function
RDF
Frequency Domain
Power
Spectral
Matrix
RDF-BF
RDF-FDD
Wavelet Transform (WT)
Hilbert-Huang Transform
Possibilities of RDT
combined with other
modal identification
methods
Time-Frequency
Time-frequency
Domain
Plane
Modal
Parameters
Frequency Domain
Decomposition (FDD)
Random Decrement
Technique (RDT)
Frequency Domain Decomposition
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
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:
Frequency Domain Decomposition
o Output spectral matrix estimated from output data
Output  y1[1 : r ] 
response  y [1 : r ]


yk   2




 yl [1 : r ] 
lxr
PSD matrix  S y y ( ) S y y ( ) ... S y y ( ) 
 S ( ) S ( ) ... S ( )
y 2 y1
y2 y2
y 2 yl

S yy ( )  
 ...
...
...
... 


S
(

)
S
(

)
...
S
(

)
y
y
y
y
 yl y1
 l x l x f cut
l 2
l l
1 1
1 2
1 l
o Output spectral matrix is decomposed (SVD, POD…)
N
S yy ( )   y ( ) y ( ) y ( )    yi ( ) yi ( ) yiT ( )
Frequencies &
i 1
Damping Ratios
T
S yy ( )  y1 ( ) y1 ( ) y1 ( )
Identification
Where:  y ( ),  y ( ) Spectral eigenvalues (Singular values) &
Spectral eigenvectors (Singular vectors)
Mode shapes
Identification
th
T
o i modal shape identified at selected frequency 
i   y1 (i )
i
Random Decrement Techniques
 RDT extracts free decay data from ambient response of
structures (as averaging and eliminating initial condition)
0
0
&
to
Triggering condition Xo
Xo
RD function (Free decay)
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
FDD
POD, SVD, QR…
1st Spectral
Eigenvalue
Response
Data Matrix
Y(t)
Free Decay Fun. &
Damping Ratios
Cross Power
Spectral Matrix
SYY(n)
1st Spectral
Eigenvector
FDD-RDT
POD, SVD, QR…
RDT
Data
Matrix
Y(t)
1st Spectral
Eigenvalue
RD
Fun.
DYY(t)
Natural
Frequencies
Cross Power
Spectral Matrix
SYY(n)
Mode Shapes
Natural
Frequencies
Free Decay Fun. &
Damping Ratios
1st Spectral
Eigenvector
Mode Shapes
Damping only
…
FDD
Natural
Frequencies
BPF
at fi
Response Series at
Filtered Frequencies
RDT
Free Decay Fun. &
Damping Ratios
Stochastic Subspace
Identification(SSI)
o Covariance-driven SSI
o 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
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 : zero mean white noises
with covariance matrix
vk
wk

C
A
Stochastic system
yk
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
Toeplitz
k: system order
 Observability matrix & system matrices

&
 Modal parameters estimation
Mode shapes:
Poles:
Frequencies:
Damping:
Flow chart of SSI algorithm
Data Matrix
[Y(t)]
Data past/ future
Data Rearrangement
Data order reduction
Orthogonal Projection
Ps
SSI-COV
Block Teoplitz Matrix
RP [], RF[],
Parameter s
Block Hankel Matrix
HP[], HF[]
POD Parameter k
SSI-DATA
Hankel matrix
Observability Matrix
Os
System Matrices
A, C
POD
Modal Parameters
Stabilization Diagram
Toeplitz matrix
Numerical example
Modal identification of ambient
vibration structures using combined
Frequency Domain Decomposition
and Random Decrement Technique
Fullscale ambient measurement
5 minutes record
-3
1.5
Floor 5
x 10
Floor5
1
Floor5
Disp. (m)
0.5
0
-0.5
-1
-3
x 10
-1.5
1.50
50
100
150
Time (s)
50
100
150
Time (s)
50
100
150
Time (s)
50
100
150
Time (s)
50
100
150
Time (s)
50
100
150
Time (s)
1
Floor4
Disp. (m)
0.5
Floor 4
200
250
300
Floor4
0
-0.5
-1
-3
x 10
-1.5
10
0.5
Disp. (m)
Floor3
300
Floor3
-0.5
-3
Disp. (m)
0.5
Floor2
Floor 2
200
250
300
Floor2
0
-0.5
Z
Disp. (m)
x 10
-1
50
X
Disp. (m)
Y
Floor 1
200
250
300
Floor1
-4
Ground
200
250
300
Ground
200
250
300
0
-1
0
Five-storey steel frame
-4
0
x 10
-5
10
Ground
250
0
x 10
-1
10
Floor1
Floor 3
200
Output displacement (X)
Random decrement functions
Floor5
-4
8
x 10
Floor 5
6
2
0
-2
-4
-6
-8
0
5
10
15
20
25
Time (s)
30
x 10
40
45
50
45
50
45
50
45
50
Floor 4
6
4
Disp.(m)
35
Floor4
-4
8
2
0
-2
-4
-6
-8
0
5
10
15
20
25
Time (s)
30
35
40
Floor3
-4
5
x 10
Disp.(m)
Floor 3
0
-5
0
5
10
15
20
25
Time (s)
30
35
40
Floor2
-4
4
x 10
Floor 2
3
2
Disp.(m)
Parameters
level crossing:
segment: 50s
no. of sample: 30000
no. of samples in
segment:
5000
Disp.(m)
4
1
0
-1
-2
-3
-4
0
5
10
15
20
25
Time (s)
30
35
40
Spectral eigenvalues
Mode 1
Mode 2
0
10
Mode 3
1.73Hz
FDD
Eigenvalue1: 99.9%
Eigenvalue2: 0.07%
Eigenvalue3: 0.01%
Eigenvalue4: 0%
Normalized eigenvalues
5.35Hz
Mode 4
8.84Hz
10.16Hz
11.45Hz13.69Hz
-5
10
Eigenvalue1
Eigenvalue2
Eigenvalue3
Eigenvalue4
Eigenvalue5
Eigenvalue6
Mode 5
19.75Hz
18.12Hz
Natural frequencies (Hz)
FDD
FDD-RDT
0
10
15
20 1.73 25
30
mode5 1
1.73
Frequency (Hz)
2Mode 2
5.35
5.34
Modemode
1
Mode8.84
3 Mode 4 8.82
10
mode
3
1.73Hz
5.34Hz
10.16Hz
Mode 5Singular value 1
value 2
mode 4 8.82Hz 13.67Hz
13.69 18.02Hz 13.67 Singular
Singular value 3
10
value 4
mode 5
18.12
18.02 Singular
Singular value 5
-10
10
0
FDD-RDT
Eigenvalue1: 100%
Eigenvalue2: 0%
Eigenvalue3: 0%
Eigenvalue4: 0%
Normalized PSD
-10
Singular value 6
-20
10
-30
10
-40
10
0
5
10
15
Frequency (Hz)
20
25
30
Spectral eigenvectors
FDD
99.9%
0.01%
0.07%
0%
Spectral eigenvectors
FDD-RDT
100%
0%
0%
0%
Mode shapes estimation
Mode 1
Mode 3
Mode 2
Mode 1
Floor5
Floor4
Mode 2
Mode 3
Floor5
FDD
Floor5
FEM
Identified
FEM
Identified
Floor4
Floor4
Floor3
Floor3
Floor3
Floor2
Floor2
Floor2
Floor1
Floor1
Floor1
FEM
Identified
Ground
0
0.25
0.5
0.75
Ground
-1
1
Mode 4
0.5
1
Floor5
Floor4
Floor4
Floor3
Floor3
Floor2
Floor2
Floor1
Floor1
0
0.5
1
Ground
-1
-0.5
0
0.5
1
| ET  A |2
MAC(E ,  A )  T
{E E }{ AT  A}
FEM
Identified
FEM
Identified
Ground
-1
MAC
Mode 5
Mode 4
-0.5
0
Mode 5
Floor5
Ground
-1
-0.5
-0.5
0
0.5
1
Mode shapes comparison
Mode 1
Mode 2
Mode 3
Mode 2
Mode 1
Mode 3
Floor 5
Floor 5
Floor 5
FEM
FDD
FDD-RDT
FEM
FDD
FDD-RDT
FEM
FDD
FDD-RDT
Floor 4
Floor 4
Floor 4
Floor 3
Floor 3
Floor 3
Floor 2
Floor 2
Floor 2
Floor 1
Floor 1
Floor 1
Ground
0
0.2
0.4
0.6
0.8
Normalized amplitude
Ground
-1
1
Mode 4
1
Mode 5
Mode 4
Mode 5
Floor 5
floor 5
FEM
FDD
FDD-RDT
FEM
FDD
FDD-RDT
Floor 4
Floor 4
Floor 3
Floor 3
Floor 2
Floor 2
Floor 1
Floor 1
Ground
-1
-0.5
0
0.5
Normalized amplitude
-0.5
0
0.5
Normalized amplitude
1
Ground
-1
-0.5
0
0.5
Normalized amplitude
1
Ground
-1
-0.5
0
0.5
Normalized amplitude
1
Identified auto PSD functions
FDD
MAC=95%
Mode 1
Mode 2
Mode 3
Mode 4
MAC=98%
Mode 5
Identified free decay functions
1.5
1
1
0.5
0
-0.5
-1
0.5
1
1.5
2
2.5
Time (s)
3
3.5
4
4.5
0.5
0
-0.5
-1
-1.5
0
5
Mode 2
1.5
-1.5
0
0.5
1
1
0.5
0
-0.5
-1
1
1.5
2
2.5
Time (s)
3
3.5
4
4.5
5
Normalized amplitude
2.5
Time (s)
3
3.5
4
4.5
5
4
4.5
5
0.5
0
-0.5
-1
-1.5
0
0.5
1
1.5
2
2.5
Time (s)
3
3.5
Uncertainty in damping
ratios estimation from
free decay functions of
modes 3 & 4
Free decay function of mode 5
1.5
Mode 5
2
Mode 4
1
0.5
1.5
Free decay function of mode 4
1.5
Normalized amplitude
Normalized amplitude
Mode 3
Free decay function of mode 3
1.5
-1.5
0
FDD
Free decay function of mode 2
Normalized amplitude
Normalized amplitude
Mode 1
Free decay function of mode 1
1
0.5
0
-0.5
Unclear with modes 2 & 5
-1
-1.5
0
0.5
1
1.5
2
2.5
Time (s)
3
3.5
4
4.5
5
Identified free decay functions
1
0.5
0
-0.5
-1
Mode 2
1
-1.5
0
0.5
0
-0.5
-1
0.5
1
1.5
2
2.5
Time (s)
3
3.5
4
4.5
-1.5
0
5
0.5
1
1
0.5
0
-0.5
-1
3
3.5
4
4.5
5
4
4.5
5
0
-0.5
-1
0.5
1
1.5
2
2.5
Time (s)
3
3.5
4
4.5
5
-1.5
0
0.5
1
Free decay function for mode 5
1
Normalized amplitude
2.5
Time (s)
0.5
1.5
Mode 5
2
Mode 4
1
0.5
0
-0.5
Better
-1
-1.5
0
1.5
Free decay function for mode 4
1.5
Normalized amplitude
Normalized amplitude
Mode 3
Free decay function for mode 3
1.5
-1.5
0
FDD
Free decay function for mode 2
1.5
Normalized amplitude
Normalized amplitude
Mode 1
Free decay function for mode 1
1.5
0.5
1
1.5
2
2.5
Time (s)
3
3.5
4
4.5
5
1.5
2
2.5
Time (s)
3
3.5
FDD - Band-pass filtering
Floor 5
10
-5
2
Bandpass Filtering
x 10
Bandpass Filtering
-3
Filtered
at mode 1
0
-2
10
-10 x 10-3
Filtered
at mode 2
2
f1=1.73Hz
0
X5(t)
Original output
Filtered
at mode 3
Filtered
at mode 4
PSD
Amp.(m)
-2
10
10
-152
x 10
-4
Filtered at mode 2
-2
x 10
-20
-5
f3=8.82Hz
0
Filtered at mode 3
-5
5
Filtered at mode 1
f2=5.34Hz
0
5
Filtered
at mode 5
x 10
-5
-250 f4=13.67Hz
10 -5
2
Filtered at mode 4
x 10
0
-30
10 -20
0
-5
f5=18.02Hz
Filtered at mode 5
2
50
4
6
100
150
8 Time
10(s) 12
Frequency (Hz)
200
14
16
250
18
300
20
Response time series at Floor 5 has been filtered on
spectral bandwidth around each modal frequency
Damping ratio via FDD-BPF
Free decay functions
RD function
-4
5
x 10
Floor 5
RD function
-5
4
x 10
Mode 2
Filtered at mode 2
Amp.(m)
2
Amp. (m)
Mode 1
Filtered at mode 1
0
0
-2
-5
0
5
10
15
20
30
35
40
45
-4
0
50
x 10
5
10
RD function
-6
3
x 10
Filtered at mode 3
Filtered at mode 4
Mode 4
2
Amp.(m)
2
Amp.(m)
Mode 3
15
Time (s)
RD function
-6
4
25
Time (s)
0
1
0
-1
-2
-2
-4
0
1
2
3
4
6
7
8
9
10
-3
0
0.5
1
1.5
2
2.5
Time (s)
3
3.5
4
4.5
5
RD function
-6
3
5
Time (s)
x 10
Amp.(m)
Mode 5
Filtered at mode 5
2
1
0
-1
-2
-3
0
0.5
1
1.5
2
2.5
Time (s)
3
3.5
4
4.5
5
Uncertainty in damping
ratios estimated from
free decay functions at
modes 4 & 5
FDD - Band-pass filtering
Floor 1
0
10
5
x 10
Bandpass
Filtering
Bandpass Filtering
-4
0
Original output
-5
5
x 10
-5
-10
PSD
Amp.(m)
10
5
-4
Filtered
f1=1.73Hz
at mode 1
0
x 10
-5
Filtered
at mode 2
Filtered
at mode 3
Filtered
at mode 4
f2=5.34Hz
0
X1(t)
Filtered at mode 1
Filtered
at mode 5
Filtered at mode 2
-5
2
x 10
f3=8.82Hz
0
-20
-2
10
1
-5
x 10
Filtered at mode 3
-5
f4=13.67Hz
0
Filtered at mode 4
-1
1
x 10
-5
-300 f5=18.02Hz
10 -10
0
2
50
Filtered at mode 5
4
6
100
8 Time
10(s) 12
Frequency (Hz)
150
200
14
16
250
18
20
300
Response time series at Floor 1 has been filtered on
spectral bandwidth around each modal frequency
Damping ratio via FDD-BPF
Free decay functions
RD function
-4
1.5
x 10
Floor 1
RD function
-5
3
x 10
1
2
0.5
1
0
0
-0.5
-1
-1
-2
-1.5
0
5
10
15
20
30
35
40
45
-3
0
50
x 10
5
10
RD function
-6
2
x 10
Mode 4
Filtered at mode 4
Amp.(m)
1
Amp.(m)
Mode 3
Filtered at mode 3
0
0
-1
-5
0
1
2
3
4
5
Time (s)
6
7
8
9
10
RD function
-6
1
x 10
Filtered at mode 5
0.5
Amp.(m)
Mode 5
15
Time (s)
RD function
-6
5
25
Time (s)
Mode 2
Filtered at mode 2
Amp.(m)
Amp.(m)
Mode 1
Filtered at mode 1
0
-0.5
-1
0
0.5
1
1.5
Time (s)
2
2.5
3
-2
0
0.5
1
1.5
2
2.5
Time (s)
3
3.5
4
4.5
5
Damping ratio via FDD-BPF
Selected free decay functions for damping estimation
RD function
-4
1.5
x 10
RD function
-5
3
x 10
0.5
1
0
-1
-1
-2
10
15
20
25
Time (s)
30
35
40
45
-3
0
50
Amp.(m)
1
0
0
-1
-2
-4
0
1
2
3
4
5
Time (s)
6
7
8
9
10
RD function
-6
x 10
Filtered at mode 5
0.5
Amp.(m)
x 10
Filtered at mode 4
2
Amp.(m)
Mode 3
15
RD function
-6
2
Filtered at mode 3
1
10
Time (s)
RD function
x 10
5
Mode 4
5
-6
4
Mode 5
0
-0.5
-1.5
0
Mode 2
Filtered at mode 2
2
Amp.(m)
Amp.(m)
Mode 1
Filtered at mode 1
1
0
-0.5
-1
0
0.5
1
1.5
Time (s)
2
2.5
3
-2
0
0.5
1
1.5
2
2.5
Time (s)
3
3.5
4
4.5
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)
Projection functions
s=50
After orthogonal projection
-4
10
s=100
x 10
After orthogonal projection
-4
8
s=150
x 10
4
-2
-4
8
x 10
100
150
200
Time (ms)
After orthogonal projection
250
100
-4
8
s=50, column=50
4
4
Amplitude
6
2
0
x 10
200
300
400
Time (ms)
After orthogonal projection
500
-6
0
600
-2
6
50
-4
100
150
200
Time (ms)
After orthogonal projection
250
300
400
500
600
Time (ms)
After orthogonal projection
0
5
0
-4
100
x 10
200
300
400
Time (ms)
After orthogonal projection
500
-6
0
600
100
200
-4
5
x 10
300
400
500
600
Time (ms)
After orthogonal projection
Data after orthogonal projection look like time-shifted
sine functions
Amplitude
Amplitude
0
Amplitude
s=100, column=200
2
900
-2
-4
s=50, column=200
4
800
2
2
-6
0
300
700
s=150, column=50
-4
x 10
200
4
-2
-4
-6
0
100
-4
x 10
s=100, column=50
6
6
-4
-6
0
300
0
-2
Amplitude
-4
Amplitude
0
-2
50
2
2
0
-4
0
s=100, column=10
4
Amplitude
6
Amplitude
Amplitude
6
2
x 10
s=100, column=10
8
4
After orthogonal projection
-4
6
s=50, column=10
0
700
800
900
s=150, column=200
0
-2
-4
0
50
-4
5
x 10
100
150
200
Time (ms)
After orthogonal projection
250
300
-5
0
100
-4
5
x 10
200
300
400
Time (ms)
After orthogonal
projection
500
-5
0
600
100
200
-4
6
x 10
300
400
500
600
Time (ms)
After orthogonal
projection
s=100, column=500
s=50, column=500
700
800
900
s=150, column=500
0
Amplitude
Amplitude
Amplitude
4
0
2
0
-2
-5
0
50
100
150
Time (ms)
200
250
300
-5
0
100
200
300
Time (ms)
400
500
600
-4
0
100
200
300
400
500
Time (ms)
600
700
800
900
Effects of s on energy contribution
Singular values
Singular velues
0.5
0.5
s=20
s=40
s=60
s=80
s=100
s=120
0.4
0.3
0.4
0.2
0.1
0.1
system orders
2
3
4
5
6
7
Number of singular values
Energy contribution
(k)
8
9
0
20
10
20
10
1
2
3
4
5
100
Singular
Singular
Singular
Singular
Singular
value
value
value
value
value
1
2
3
4
5
120
30
20
10
2
3
(k)
4
5
6
7
Number of singular values
Energy cummulation
8
9
0
20
10
40
(s)
60
80
Number of block row s
Energy cummulation
100
100
120
100
80
70
k=10  90-96% Energy
k=15  92-97% Energy
k=20  93-98% Energy
60
50
40
2
3
4
5
6
7
Number of singular values)
(k)
8
9
Fisrt
First
First
First
First
90
Percentage (%)
s=20
s=40
s=60
s=80
s=100
s=120
90
Percentage (%)
(s)
40
Percentage (%)
Percentage (%)
30
30
1
60
80
Number of block row s
50
40
0
1
40
Energy contribution
50
s=20
s=40
s=60
s=80
s=100
s=120
value
value
value
value
value
0.3
0.2
0
1
Singular
Singular
Singular
Singular
Singular
80
70
60
50
40
10
30
20
40
60
80
Number of block row s
(s)
100
120
singular value
2 singular values
3 singular values
4 singular values
5 singular values
18.044Hz
mode 5
mode 3
8.82Hz
mode 2
5.34Hz
mode 1
1.74Hz
50
13.67Hz
Frequency diagram
60
mode 4
Frequency diagram
Number of poles
40
30
20
10
0
0
10
2
4
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
6
8
10
12
14
16
18
(Hz)
mode 5Natural frequency18.12
18.04
s=50
k=5:5:60
20
-5
X5
PSD
PSD of response time series
10
10
-10
-15
0
2
4
6
8
10
12
Natural frequency (Hz)
14
16
18
20
mode 5
mode 4
mode 3
50
mode 1
60
mode 2
Frequency diagram
s=20:10:120
k=60
System order k
40
30
20
10
0
0
2
4
6
8
10
12
Natural frequency (Hz)
14
16
18
20
Frequency diagram
120
110
Number of block rows s
100
90
s=20:10:120
k=60
80
70
60
50
40
30
20
0
10
2
4
6
8
10
12
Natural frequency (Hz)
14
16
18
20
-5
X5
PSD
PSD of response time series
10
10
-10
-15
0
2
4
6
8
10
12
Natural frequency (Hz)
14
16
18
20
Damping diagram
60
50
s=50
k=5:5:60
Number of poles
40
30
20
10
0
0.1
0.2
0.3
0.4
0.5
0.6
Damping ratio (%)
0.7
0.8
0.9
Damping diagram
80
70
s=20:10:120
k=60
System order k
60
50
40
30
20
10
0
0.1
0.2
0.3
0.4
0.5
0.6
Damping ratio (%)
0.7
0.8
0.9