le thai hoa vietnam national university, hanoi open seminar at tokyo polytechnic university pod and...
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LE THAI HOALE THAI HOAVietnam National University, Hanoi Vietnam National University, Hanoi
Open Seminar at Tokyo Polytechnic University
POD AND NEW INSIGHTSPOD AND NEW INSIGHTSIN WIND ENGINEERING IN WIND ENGINEERING
PART 2
Introduction POD and its Proper Transformations in Time Domain and Frequency Domain New Insights in Wind Engineering Topic 1: POD and Pressure Fields Topic 2: POD and Wind Fields, Wind Simulation Topic 3: POD and Response Prediction Topic 4: POD and System Identification
Further Perspectives and Development
IntroductionIntroduction Response Prediction in Frequency Domain Response Prediction in Frequency Domain Response Prediction in Time Domain Response Prediction in Time Domain Numerical ExamplesNumerical Examples Remarks and Insights Remarks and Insights
STOCHASTIC RESPONSE PREDICTION STOCHASTIC RESPONSE PREDICTION OF WND-EXCITED STRUCTURES IN OF WND-EXCITED STRUCTURES IN FREQUENCY DOMAIN AND TIME FREQUENCY DOMAIN AND TIME
DOMAIN DOMAIN
TOPIC 3 TOPIC 3
Gust response prediction of structures due to turbulent Gust response prediction of structures due to turbulent wind faces the wind faces the difficulty in projecting the full-scale difficulty in projecting the full-scale buffeting forces on the structural generalized coordinatesbuffeting forces on the structural generalized coordinates. . The joint acceptance function technique has been used for The joint acceptance function technique has been used for this purpose the conventional approach. this purpose the conventional approach.
Proper Orthogonal Decomposition (POD) and its Proper Proper Orthogonal Decomposition (POD) and its Proper Transformations decompose the full-scale buffeting forces Transformations decompose the full-scale buffeting forces into so-called into so-called turbulent loading modesturbulent loading modes and projects onto and projects onto generalized coordinates and structural modes. generalized coordinates and structural modes.
Introduction (1)Introduction (1)
Structures
Full-scale Gust Forces
Generalized Structural Modes
Turbulent Loading Modes
Structural Response
Structural Modal Transformation
POD
Fig. 22 Scheme on stochastic gust response prediction of structures
Double Modal Transformation
Introduction (2)Introduction (2)
Covariance Proper Transformation in Time Domain Spectral Proper Transformation in Frequency Domain
Surface Pressure FieldsTurbulent Wind Fields
Gust response of structures firstly proposed in the frequency Gust response of structures firstly proposed in the frequency domain by domain by Davenport 1962Davenport 1962. Time domain gust response was . Time domain gust response was developed by developed by Chen 1996Chen 1996
Double Modal Transformation (DMT) for gust response Double Modal Transformation (DMT) for gust response prediction in the frequency domain proposed by prediction in the frequency domain proposed by Carassale and Carassale and Solari 1999, Solari 1999, application for simple frame structures and application for simple frame structures and buildings by buildings by Carassale 1999, Solari 2000; Chen 2005;Carassale 1999, Solari 2000; Chen 2005; for that of for that of bridges by bridges by Solari 2005 Solari 2005 using Spectral Proper Transformation.using Spectral Proper Transformation.
Stochastic gust response is predicted in the frequency domain, Stochastic gust response is predicted in the frequency domain, thus the time-domain formulation have been required as new thus the time-domain formulation have been required as new line of the PODline of the POD. Therefore, problems of unsteady forces, . Therefore, problems of unsteady forces, nonlinear aerodynamics can be solved as further development nonlinear aerodynamics can be solved as further development
Formulation of the stochastic gust response prediction Formulation of the stochastic gust response prediction of structures applies both the POD-based Proper of structures applies both the POD-based Proper TransformationsTransformations Effects of number of and low-order turbulent loading Effects of number of and low-order turbulent loading modes on generalized and global responses of modes on generalized and global responses of structures structures Interaction between the structural modes and the Interaction between the structural modes and the turbulent loading modesturbulent loading modes
Introduction (3)Introduction (3)
1DOF motion equation in i-th generalized coordinate:1DOF motion equation in i-th generalized coordinate:
Turbulent fields u(t), w(t) are approximated as the CPT:Turbulent fields u(t), w(t) are approximated as the CPT:
1DOF equation in the generalized coordinate is expressed: 1DOF equation in the generalized coordinate is expressed:
: : Cross modal coefficientsCross modal coefficients
Time histories of generalized responses obtained by using direct Time histories of generalized responses obtained by using direct integration methods (here Newton-beta method used). Finally, the integration methods (here Newton-beta method used). Finally, the global responses are determinedglobal responses are determined
M
j
M
jwjwjw
Tiujuju
Tiiiiiii txCtxCUBttt
~
1
~
1
2 )(~)(~2
1)()(2)(
wu AA~
,~
)(~~)(~~
2
1)()(2)( 2 txAtxAUBttt wjwujuiiiiii
M
j
M
jwjw
Tiwijw
M
j
M
juju
Tiuiju CAACAA
~
1
~
1
~
1
~
1
~~;
~~
M
jwjwjww
M
jujujuu txtxtwtxtxtu
~
1
~
1
)(~)(~)(;)(~)(~)(
)(2 )(2 tF ib
Tiiiiiii
Stochastic Gust Response in Time DomainStochastic Gust Response in Time Domain
Spectra of generalized response: Spectra of generalized response:
H(n) Frequency response matrix; K(n): Admittance function matrixH(n) Frequency response matrix; K(n): Admittance function matrix Cross modal coefficients Cross modal coefficients Spectra and root mean square of global response:Spectra and root mean square of global response:
TTM
j
Twjwjwjw
M
j
TTTujujuju
nHnKnnnCnH
nHnKnnnCnH
UBnS*2
ˆ
1
*2
ˆ
1
*2*2
2
)()(})()()({)(
)()(})()()({)(
)2
1()(
wu AA ˆ,ˆ
TTwww
TTuuu HAKAHHAKAHUBnS **2**22 ˆˆˆˆ)
2
1()(
M
j
M
jwjw
Tiwijw
M
j
M
juju
Tiuiju nCnAnAnCnAnA
ˆ
1
ˆ
1
ˆ
1
ˆ
1
)()(ˆ)(ˆ;)()(ˆ)(ˆ
aphrdnnSnSnSrM
rirrUU
TU ,,;;)(;)()(
1
2,
0
2
h: vertical; p: horizontal; a: rotationalh: vertical; p: horizontal; a: rotational
Stochastic Gust Response in Frequency DomainStochastic Gust Response in Frequency Domain
A line-like structure is used for demonstration and investigationA line-like structure is used for demonstration and investigation Kaimal spectrum and Bush&Panofsky spectrum are used auto Kaimal spectrum and Bush&Panofsky spectrum are used auto
spectral densities of longitudinal and vertical turbulences, spectral densities of longitudinal and vertical turbulences, respectively. Davenport’s empirical formula is used for respectively. Davenport’s empirical formula is used for spanwise coherence. Liepmann’s empirical function is used for spanwise coherence. Liepmann’s empirical function is used for the aerodynamic admittance.the aerodynamic admittance.
Static aerodynamic coefficients and its first-derivatives asStatic aerodynamic coefficients and its first-derivatives as
;
501
200)(
3/5
2*
fn
funSuu
3/5
2*
101
36.3)(
fn
ufnSww
,174.0,041.0,158.0 MDL CCC 06.2,0,73.3 ''' MDL CCC
)(5.0
||exp),(
km
kmmk UU
yyncnCOH
Numerical ExampleNumerical Example
Fig. 23 Normalized structural modesFig. 23 Normalized structural modesMode 1Mode 1 Mode 3Mode 3
1 5 9 13 17 21 25 29-0.2
0
0.2
1 5 9 13 17 21 25 29-0.2
0
0.2
1 5 9 13 17 21 25 29-2E-5
0
2E-5
1 5 9 13 17 21 25 29-2E-4
0
2E-4
1 5 9 13 17 21 25 29-0.2
0
0.2
1 5 9 13 17 21 25 29
1 5 9 13 17 21 25 29-2E-4
0
2E-4
1 5 9 13 17 21 25 29-0.1
0
0.1
1 5 9 13 17 21 25 29-5E-4
0
5E-4
Structural nodes1 5 9 13 17 21 25 29
-5E-4
0
5E-4
Structural nodes
mode 1 mode 2
mode 3 mode 4
mode 5 mode 6
mode 7
mode 8
mode 10
mode 9
1 5 9 13 17 21 25 29-1
0
1
1 5 9 13 17 21 25 29-1E-5
0
1E-5
1 5 9 13 17 21 25 29-0.02
0
0.02
1 5 9 13 17 21 25 29-0.02
0
0.02
1 5 9 13 17 21 25 29-2E-5
0
2E-5
1 5 9 13 17 21 25 29-2E-5
0
2E-5
1 5 9 13 17 21 25 29-0.02
0
0.02
1 5 9 13 17 21 25 29-5E-5
0
5E-5
1 5 9 13 17 21 25 29-0.02
0
0.02
Structural nodes1 5 9 13 17 21 25 29
-0.02
0
0.02
Structural nodes
mode 1 mode 2
mode 3 mode 4
mode 5 mode 5
mode 7 mode 8
mode 9 mode 10
0.61Hz 0.80Hz
1.19Hz0.85Hz
1.29Hz 1.45Hz
1.58Hz 1.63Hz
1.68Hz 1.85Hz
0.61Hz
0.85Hz
1.29Hz
1.58Hz
1.68Hz
0.80Hz
1.19Hz
1.45Hz
1.63Hz
1.85Hz
Vertical component Torsional component
Modal Analysis and Structural ModesModal Analysis and Structural Modes
Simulated Wind Time-series for Time-domain AnalysisSimulated Wind Time-series for Time-domain Analysis
0 10 20 30 40 50 60 70 80 90 100-10
0
10
no
de
1
0 10 20 30 40 50 60 70 80 90 100-10
0
10
no
de
2
0 10 20 30 40 50 60 70 80 90 100-10
0
10
no
de
3
0 10 20 30 40 50 60 70 80 90 100-10
0
10
no
de
4
0 10 20 30 40 50 60 70 80 90 100-10
0
10
no
de
5
Time (sec.)
0 10 20 30 40 50 60 70 80 90 100-10
0
10
no
de
6
0 10 20 30 40 50 60 70 80 90 100-10
0
10
no
de
7
0 10 20 30 40 50 60 70 80 90 100-10
0
10
no
de
8
0 10 20 30 40 50 60 70 80 90 100-10
0
10
no
de
9
0 10 20 30 40 50 60 70 80 90 100-10
0
10
no
de
10
Time (sec.)
0 10 20 30 40 50 60 70 80 90 100-5
0
5
no
de
60 10 20 30 40 50 60 70 80 90 100
-5
0
5
no
de
7
0 10 20 30 40 50 60 70 80 90 100-5
0
5n
od
e 8
0 10 20 30 40 50 60 70 80 90 100-5
0
5
no
de
9
0 10 20 30 40 50 60 70 80 90 100-5
0
5
no
de
10
Time (sec.)
0 10 20 30 40 50 60 70 80 90 100-5
0
5
no
de
1
0 10 20 30 40 50 60 70 80 90 100-5
0
5
no
de
2
0 10 20 30 40 50 60 70 80 90 100-5
0
5
no
de
3
0 10 20 30 40 50 60 70 80 90 100-5
0
5
no
de
4
0 10 20 30 40 50 60 70 80 90 100-5
0
5
no
de
5
Time (sec.)
Fig. 24 Simulated time series at structural nodesFig. 24 Simulated time series at structural nodes
Simulated u-turbulence
Simulated w-turbulence
0 10 20 30 40 50 60 70 80 90 100-10
-7.5
-5
-2.5
0
2.5
5
7.5
10
Lift
(tf
)
Node 5
0 10 20 30 40 50 60 70 80 90 100-2
-1.5
-1
-0.5
0
0.5
1
1.5
2
Mom
ent
(tf.
m)
Node 5
0 10 20 30 40 50 60 70 80 90 100-12.5
-10-7.5
-5-2.5
02.5
57.510
Lift
(tf
)
Node 15
Time (sec.)
0 10 20 30 40 50 60 70 80 90 100-2.5
-2-1.5
-1-0.5
00.5
11.5
2
Mom
ent
(tf.
m)
Node 15
Time (sec.)
Time Histories of Global Gust Forces Time Histories of Global Gust Forces
Fig. 25 Time histories of global gust responses in nodes 5&15 atU=20m/s
0 10 20 30 40 50 60 70 80 90 100- 0.05
- 0.04
- 0.03
- 0.02
- 0.01
0
0.01
0.02
0.03
0.04
0.05
Time (sec.)
Am
plitu
de
Node 5
0 10 20 30 40 50 60 70 80 90 100- 0.1
- 0.08
- 0.06
- 0.04
- 0.02
0
0.02
0.04
0.06
0.08
0.1
Time (sec.)
Am
plitu
de
Node 15
0 10 20 30 40 50 60 70 80 90 100- 4
- 3
- 2
- 1
0
1
2
3x 10-7
Time (sec.)
Am
plitu
de
Node 5
0 10 20 30 40 50 60 70 80 90 100- 0.01
- 0.008
- 0.006
- 0.004
- 0.002
0
0.002
0.004
0.006
0.008
0.01
Time (sec.)
Am
plitu
de
Node 15
Time Histories of Global Responses Time Histories of Global Responses
Fig. 26 Time histories of global responses in nodes 5&15 at U=20m/s
Node 5 Node 15
Verti
cal d
isp.
(m)
Rota
tiona
l dis
p. (d
eg.)
2 x Max. amplitude
2 x Max. Amplitude
0 5 10 15 20 25 30 35 40 450
0.05
0.1
0.15
0.2V
ert.
dips
. (m
)
Node 5
0 5 10 15 20 25 30 35 40 450
0.003
0.006
0.009
0.012
0.015
Rot
. dip
s. (
deg.
)
Node 5
0 5 10 15 20 25 30 35 40 450
0.1
0.2
0.3
0.4
Ver
t. di
ps. (
m)
Node 15
Mean velocity (m/s)
0 5 10 15 20 25 30 35 40 450
0.01
0.02
0.03
0.04
Rot
. dip
s. (
deg.
)
Node 15
Mean velocity (m/s)
maxmin
maxmin
maxmin
maxmin
Fig. 27 Effect of turbulent modes on spectra of generalized responses Fig. 27 Effect of turbulent modes on spectra of generalized responses in node 15 (at U=20m/s)in node 15 (at U=20m/s)
0 0.2 0.4 0.6 0.8 1 1.2 1.4 1.6 1.8 2 2.2 2.410
-6
10-4
10-2
100
102
104
Frequency n(Hz)
Sh(n
) (m
2 .s)
Vertical displacement
mod
e 1
mod
e 2
mod
e 5
mod
e 8
target10modes
5 modesFirst modes
0 0.2 0.4 0.6 0.8 1 1.2 1.4 1.6 1.8 2 2.2 2.410
-6
10-4
10-2
100
102
104
Frequency n(Hz)
Sa(n
)
Rotational displacement
mod
e 3
mod
e 4
mod
e 10
mod
e 7
target10modes
5 modesFirst modes
Vertical displacement Rotational displacement
Generalized Response SpectraGeneralized Response Spectra
Fig. 28 Effect of turbulent modes on global responses spectra in node 15 Fig. 28 Effect of turbulent modes on global responses spectra in node 15 (at U=20m/s)(at U=20m/s)
0 0.2 0.4 0.6 0.8 1 1.2 1.4 1.6 1.8 2 2.2 2.410
-4
10-3
10-2
10-1
100
101
102
Frequency n(Hz)
SH
(n) (
m2 .s
)
Vertical displacement
30 modes (target)10 modes5 modesFirst mode
mod
e 1
mod
e 2
mod
e 5
mod
e 6
mod
e 8
0 0.2 0.4 0.6 0.8 1 1.2 1.4 1.6 1.8 2 2.2 2.410
-5
10-4
10-3
10-2
10-1
100
Frequency n(Hz)
SA
(n) (
deg
2 .s)
Rotational displacement
30 modes (target)10 modes5 modesFirst mode
mod
e 3
mod
e 7
mod
e 9
mod
e 10
mod
e 4
Vertical displacement Rotational displacement
Global Response SpectraGlobal Response Spectra
11uA 11uA
11uA
11uA
Fig. 29 Influence of spectral mode on structural modesFig. 29 Influence of spectral mode on structural modes
13uA
123456789101112131415
1 2 3 4 5 6 7 8 9 10
0
0.1
0.2
0.3
0.4
0.5
cross
modal co
effi
cien
t
turbulent modes (u) structural modes
|| LijuA
Lift
123456789101112131415
1 2 3 4 5 6 7 8 9 10
0
1
2
3
4
5
6
cross
modal co
effi
cien
t
turbulent modes (w)structural modes
|| LijwA
Lift
123456789101112131415
12345678910
0
0.2
0.4
0.6
0.8
1
1.2
cross
modal co
effi
cien
t
turbulent modes (u)structural modes
|| MijuA
Mom
ent
123456789101112131415
12345678910
0
1
2
3
4
5
6
7
cross
modal co
effi
cien
t
turbulent modes (w) structural modes
|| MijwA
Mom
ent
Between w-turbulent spectral modes and structural modes
Interaction between u-turbulent spectral modes and structural modes11uA13uA 31uA33uA
Cross Modal CoefficientsCross Modal Coefficients
5 10 15 20 25 300
0.02
0.04
0.06
0.08
0.1M
ax
am
plit
ud
e(m
)
Deck nodes
5 10 15 20 25 300
0.002
0.004
0.006
0.008
0.01
Ma
x a
mp
litu
de
(de
g.)
Deck nodes
30 modes20 modes10 modes5 modes
30 modes20 modes10 modes5 modes
Fig. 30 Effect of covariance modes on global responses at all deck nodes
Vertical displacement (m)
Rotational displacement (degree)
Maximum Amplitude of Structure due to CPT Maximum Amplitude of Structure due to CPT
N.ModesN.Modes Node 5Node 5 %% Node15Node15 %%
3030 0.0400.040 100100 0.0930.093 100100
2020 0.0370.037 9393 0.0800.080 8686
1010 0.0280.028 7070 0.0690.069 7474
55 0.0230.023 5858 0.0530.053 5757
N.modesN.modes Node 5Node 5 %% Node15Node15 %%
3030 .0027.0027 100100 .0078.0078 100100
2020 .0026.0026 9696 .0075.0075 9696
1010 .0021.0021 7878 .0071.0071 9191
55 .0018.0018 6767 .0049.0049 6363
Vertical displacement (m)
Rotational displacment (degree)
5 10 15 20 25 300
0.05
0.1
Deck nodes
Max
am
plitud
e (m
)30 modes10 modes5 modesFirst mode
5 10 15 20 25 300
0.005
0.01
Deck nodes
Max
am
plitud
e (d
eg.)
30 modes10 modes5 modesFirst mode
Fig. 31 Effect of spectral modes on global responses at all deck nodes
Vertical displacement
Rotational displacement (degree)
MaximumMaximum Amplitude of Structure due Amplitude of Structure due to SPTto SPT
N.modesN.modes Node 5Node 5 %% Node15Node15 %%
3030 0.0670.067 100100 0.1470.147 100100
1010 0.0660.066 9999 0.1470.147 9999
55 0.0640.064 9595 0.1440.144 9797
11 0.0580.058 8686 0.1310.131 8888
N.modesN.modes Node 5Node 5 %% Node15Node15 %%
3030 .0069.0069 100100 0.0150.015 100100
1010 .0068.0068 9898 0.0150.015 9999
55 .0065.0065 9393 0.0140.014 9595
11 .0059.0059 8484 0.0120.012 8080
Vertical displacement (m)
Tosional displacement (deg.)
Framework on the gust response of bridges is formulated in Framework on the gust response of bridges is formulated in both the time domain and frequency domain using both POD-both the time domain and frequency domain using both POD-based Proper Transformations with based Proper Transformations with comprehensive approach of comprehensive approach of spatially-correlated turbulent fieldspatially-correlated turbulent field. .
Only few basic turbulent modes contribute dominantlyOnly few basic turbulent modes contribute dominantly and effectively on the global gust response of bridgesand effectively on the global gust response of bridges. .
Concretely, the first spectral turbulent mode contributes Concretely, the first spectral turbulent mode contributes significantly on the gust response, whereas more basic significantly on the gust response, whereas more basic covariance modes are required for the gust response. covariance modes are required for the gust response.
Effective turbulent field and cross modal coefficients can be Effective turbulent field and cross modal coefficients can be refined for simulating the turbulent field and estimating the refined for simulating the turbulent field and estimating the gust response by using few turbulent modes and effective gust response by using few turbulent modes and effective spectral band.spectral band.
Remarks and InsightsRemarks and Insights
Thus, Unsteady Gust Response Prediction of structures formulated thanks to the SPT and CPT in both frequency domain and the time domain using Impulse Response Functions Model will be next development
IntroductionIntroduction Frequency Domain Decomposition (FDD)Frequency Domain Decomposition (FDD) Stochastic Subspace Identification (SSI) Stochastic Subspace Identification (SSI) Numerical ExamplesNumerical Examples Remarks and Insights Remarks and Insights
SYSTEM IDENTIFICATION OF WND-SYSTEM IDENTIFICATION OF WND-EXCITED STRUCTURES IN FREQUENCY EXCITED STRUCTURES IN FREQUENCY
DOMAIN AND TIME DOMAIN DOMAIN AND TIME DOMAIN
TOPIC 4 TOPIC 4
Introduction (1)Introduction (1)
SystemKnownInput X(t)
KnownOutput Y(t)
Input-output identification
Noise u(t)
Small-scale system: Forced excitersDue to impulse or forced shaker…
UnknownInput X(t)
Output-only identification
Noise u(t)
Large-scale system: Ambient excitersDue to traffic, wind, wave, sound…
KnownOutput Y(t)
System
White Noise Process
System identification using ambient vibration measurements of wind-excited structures is recent challenging with new techniques in sensing and assessment System identification methods are mostly used in frequency domain, based on orthogonal decomposition of spectral matrix of measured output data Most recent techniques are developed in time domain, directly dealing with the measured output data.
Introduction (2)Introduction (2)
P. OverscheeB.D Moor,1996
Output-only System Identification
Parametric Methodsin Time Domain
Peak Picking Technique (PPT)
Frequency Domain Decomposition (FDD)
Enhanced Frequency Domain Decomposition
SSI with AR, ARMA
SSI with Covariance
SSI with DataStoc
hasti
c Su
bspa
ce
Iden
tifica
tion
(SSI
)
Fig. 32 Only-output system identification methods
Nonparametric Methods in Frequency Domain
R. Brincker et al., 2001
J.S. Bandat et al., 1993
A. Yoshida Y. Tamura,2004
J.H. Weng et. al., 2008
L. Carassale, F.Percivale,2007
Covariance Matrix of Output Data
Hankel Matrix of Output Data
Spectral Matrix of Output Data
Projection and decomposition of either spectral or covariance matrices of output response time series, even dealing with directly output data must be needed Some robustness numerical methods are used such as QR Decomposition, Least Squares, Singular Value Decomposition… Advantage of POD will be exploited for this purpose of decomposition
Comparison between FDD and SSIComparison between FDD and SSI
Formulated in the frequency domain Based on spectral matrix of measured output data
Less accurate identification in cases of high noises High applicable for closed modal identification Easier in identification Prior knowledge of modal frequencies is required
FDD SSI
Formulated in the time domain Directly deal with measured output data or covariance matrix High accurate identification in cases of high noises High applicable for closed modal identification More complicated Prior knowledge of modal frequencies not is required
Frequency Domain Decomposition (FFD)Frequency Domain Decomposition (FFD) Firstly, cross spectral matrix estimated from measured output data
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
Secondly, cross spectral matrix is decomposed using POD
)()()()( yyyyyS then TyyyyyS )()()()(
Where: Eigenvalue and eigenvector matrices )(),( yy
ith modal identification, we decompose at selected frequency i
)()()()( iyiyiyiyyS
cutfxlxlylyyy diag )()()()( 21
cutfxlxlylyyy nn )()()()( 21
then TiyiyiyiyyS )()()()(
lxlyiyiyiiy diag )( and
lxlylyyiy 21)(
Thus, the ith mode: 1yi
QRDecompositionLeast SquaresSingular Values Decomposition…
In the FDD, prior knowledge of modal frequencies is generally required to identify the modal parameters
State-space Representation in SSI (1)State-space Representation in SSI (1) Continuous state-space representation: )()()()( tFtKytyCtyM
Ttytytx )}(),({)(
)(ˆ)(ˆ)(
)(ˆ)(ˆ)(
tuDtxCty
tuBtxAtx
)()( tUutF Where:
nnxCMKM
IA
22
11
0ˆ
nxr
UMB
2
1
0ˆ
nmxCLMKLMC 2
11ˆ mxrULMD 1ˆ
x(t): state vector; : state matrix; input matrix; output matrix CBA ˆ,ˆ,ˆ
Discrete state-space representation at interval time tktk
kkk
kkk
DuCxy
BuAxx 1
Where: )( tkxxk DDCCBAIABAA tA ˆ,ˆ,ˆˆ)(, 1ˆ
Input-output relationship:
k
iik
ikk
iik
ik
kk uBCAxCABuCADuxCAy
0
10
1
10 )(
Inputs
If inputs are white noises (during free vibration), model reduce:
kkk
kkk
vCxy
sAxx 1
Broad-band white noises
Stochastic identification requires finding system parameters: A, C, modal parameters, frequencies and damping ratios from ambient output measurements yk(t)
SSI-DATA (2)SSI-DATA (2) Reconstructing output data to Hankel matrices (past, futurestates)
lixjjiji
j
j
p
yyy
yyy
yyy
Y
21
21
110
...
............
...
...
lixjjiii
jiii
jiii
f
yyy
yyy
yyy
Y
22212
21
11
...
............
...
...
Orthogonal projection of Hankel matrices, the decomposing the projection using the POD
pTpp
Tpfpfi YYYYYYYP
Ti VDVP then
Identifying the system matrices: A, CiiA *
iC and Where: from without last l row; from without first l rowi
i i ii first l rows of i
2/1VDi Extended observability matrix Identifying modal parameter: T
AAAA ACthen
lxrl
k
ry
ry
ry
y
]:1[
]:1[
]:1[
2
1
QRDecompositionLeast SquaresSingular Values Decomposition…
Refining FDD, SSI by Wavelet AnalysisRefining FDD, SSI by Wavelet Analysis System identification techniques of structures from ambient natural excitations usually has many difficulties associated with high noises, low and closed eigenvalues (frequencies), a lot of effects on measured output data Idea of the time-frequency analysis (wavelet analysis) can be applicable for to system identification or refinement of FDD, SSI, because of some following reasons:
Wavelet analysis reveals time information of sources of excitation and eigenvalues ocurrance Wavelet analysis eliminates and localizes the system noise Wavelet analysis decomposes and localizes at many frequency bands Especially, wavelet analysis does high resolution on low frequency bands that clearly separate low and closed eigenvalues
Field Measurements of 5-storey Steel FrameField Measurements of 5-storey Steel Frame
Fig. 33 5-storey steel frame at test site Disaster Prevention Research Institute (DPRI), Kyoto University
Ground
Floor 1
Floor 2
Floor 3
Floor 4
Floor5
Measured Velocities and Integrated ResponsesMeasured Velocities and Integrated Responses
Fig. 34 Measured velocities and integrated displacements
0 50 100 150 200 250 300-4
-3
-2
-1
0
1
2
3
4
5x 10
-3
Time (s)
Am
plit
ud
e (
m/s
)
Measured Velocity at Floor 1
0 50 100 150 200 250 300-0.015
-0.01
-0.005
0
0.005
0.01
0.015
Time (s)
Am
plit
ud
e (
m/s
)
Measured Velocity at Floor 5
0 50 100 150 200 250 300-4
-2
0
2
4x 10
-4
Time (s)
Am
plitu
de (m
)
Displacement at Floor 1
0 50 100 150 200 250 300-2
-1
0
1
2x 10
-3
Time (s)
Ampl
itude
(m)
Displacement at Floor 5
PSD of Output ResponsePSD of Output Response
Fig. 35 High-resolved PSD of output response time series
0 5 10 15 20 25 3010
-20
10-15
10-10
10-5
PS
D
Floor 5
0 5 10 15 20 25 3010
-20
10-15
10-10
10-5
Frequency (Hz)
PS
D
Floor 1
1.736Hz5.341Hz
8.853Hz11.43Hz
13.66Hz19.76Hz
18.05Hz
1.736Hz5.341Hz
8.853Hz11.43Hz
13.66Hz 19.76Hz
Spectral eigenvaluesMode 1
Mode 2
Mode 3Mode 4
Mode 5
Energy contribution of ith eigenvalues & eigenmodes
N
i
f
kki
f
kkif
offcutoffcut
iffE
1 11)( )()(
1 2 3 4 5 60
20
40
60
80
100
Eigenvalues
Ene
rgy
(%)
Energy contribution of eigenvalues and eigenvectors
99.90%
0.00% 0.00% 0.00%0.01%0.07%
Frequencies and order of modes are identified via combination with FEM
Spectral eigenvectors
Spectral eigenvectors
99.9%
0.07%
0.01%
0%
Mode shapes estimation
Unscaled mode shapesUnscaled mode shapes
Mode Mode 11
Mode Mode 11
Mode Mode 33
Mode Mode 44
Mode shapes estimation
Mode shapesMode shapes
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
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ATAE
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ATE
AEMAC
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