Transcript Slide 1

Open Seminar
at Tokyo Polytechnic University
POD AND NEW INSIGHTS
IN WIND ENGINEERING
LE THAI HOA
Vietnam National University, Hanoi
PART 2
CONTENTS
 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
TOPIC 3
STOCHASTIC RESPONSE PREDICTION
OF WND-EXCITED STRUCTURES IN
FREQUENCY DOMAIN AND TIME
DOMAIN





Introduction
Response Prediction in Frequency Domain
Response Prediction in Time Domain
Numerical Examples
Remarks and Insights
Introduction (1)
Gust response prediction of structures due to turbulent
wind faces the difficulty in projecting the full-scale
buffeting forces on the structural generalized coordinates.
The joint acceptance function technique has been used for
this purpose the conventional approach.
Proper Orthogonal Decomposition (POD) and its Proper
Transformations decompose the full-scale buffeting forces
into so-called turbulent loading modes and projects onto
generalized coordinates and structural modes.
Introduction (2)
Structural Modal Transformation
Structures
Full-scale Gust
Forces
Double Modal Transformation
Generalized
Structural Modes
Turbulent Loading
Modes
Structural
Response
POD
Covariance Proper Transformation
in Time Domain
Surface Pressure Fields
Spectral Proper Transformation
Turbulent Wind Fields
in Frequency Domain
Fig. 22 Scheme on stochastic gust response prediction of structures
Introduction (3)
 Gust response of structures firstly proposed in the frequency
domain by Davenport 1962. Time domain gust response was
developed
Formulation
of the
stochastic gust response prediction
by Chen
1996
of structures applies both the POD-based Proper
 Double Modal Transformation (DMT) for gust response
Transformations
prediction
in the frequency domain proposed by Carassale and
Solari
Effects
of number
of and
low-order
1999,
application
for simple
frameturbulent
structuresloading
and
modes on
andSolari
global
responses
of for that of
buildings
by generalized
Carassale 1999,
2000;
Chen 2005;
bridges
by Solari 2005 using Spectral Proper Transformation.
structures
Interaction
between
the
structuralinmodes
and thedomain,
Stochastic
gust
response
is predicted
the frequency
turbulent
loading modes
thus
the time-domain
formulation have been required as new
line of the POD. Therefore, problems of unsteady forces,
nonlinear aerodynamics can be solved as further development
Stochastic Gust Response in Time Domain
 1DOF motion equation in i-th generalized coordinate:
i  2 i i i  i2 i  iT Fb(i ) (t )
 Turbulent fields u(t), w(t) are approximated as the CPT:
u (t )   u ~
xu (t ) 
~
M

~
uj xuj (t );
w(t )   w ~
x w (t ) 
j 1
~
M

~
wj x wj (t )
j 1
 1DOF equation in the generalized coordinate is expressed:
~
~
M
 T M ~

1
2
T
~



i (t )  2 iii (t )  i i (t )  UBi Cu uj xuj (t )  i Cw  wj xwj (t )
2
j 1
j 1


~ ~
~ ~
 (t )  2   (t )   2 (t )  1 UB A
i
i i i
i i
u xuj (t )  Aw xwj (t )
2
~
~
~
~
M
M
M
M
~
~
~
~
Au   Auij   iT Cu  uj ; Aw   Awij   iT Cw  wj

j 1
j 1
j 1
~ ~
Au , Aw : Cross modal coefficients

j 1
 Time histories of generalized responses obtained by using direct
integration methods (here Newton-beta method used). Finally, the
global responses are determined
Stochastic Gust Response in Frequency Domain
 Spectra of generalized response:
Mˆ


2
*T
2
T
*T
 H (n)Cu { uj (n)uj (n) uj (n) }K (n)  H (n)  
1
j 1

S  (n)  ( UB) 2 


Mˆ
2
2
*T
2
T
*T
 H (n)C w { wj (n) wj (n) wj (n) }K (n)  H (n) 


j 1
1
S (n)  ( UB) 2 H Aˆu  u K 2 Aˆu*T H *T  H Aˆ w  w K 2 Aˆ w*T H *T
2
Mˆ
Mˆ
Mˆ
Mˆ
T
Aˆu (n)   Aˆuij (n)   i Cu uj (n); Aˆ w (n)   Aˆ wij (n)   iT Cw wj (n)

j 1

j 1
j 1
j 1
H(n) Frequency response matrix; K(n): Admittance function matrix
Aˆ , Aˆ Cross modal coefficients
u
w
 Spectra and root mean square of global response:

SU (n)   T S  (n);  U2   SU (n)dn;  r 
0
Mr
2

 r ,i ; r  h, p, a
r 1
h: vertical; p: horizontal; a: rotational
Numerical Example
 A line-like structure is used for demonstration and investigation
 Kaimal spectrum and Bush&Panofsky spectrum are used auto
spectral densities of longitudinal and vertical turbulences,
respectively. Davenport’s empirical formula is used for
spanwise coherence. Liepmann’s empirical function is used for
the aerodynamic admittance.
200 fu*2
3.36 f u*2
S uu (n) 
;
5 / 3 S ww ( n) 
n 1  10 f 5 / 3
n1  50 f 
 c n | ym  yk | 

COH (n,  mk )  exp  
 0.5(U m  U k ) 


 Static aerodynamic coefficients and its first-derivatives as
C L  0.158, C D  0.041, C M  0.174, CL'  3.73, CD'  0, CM'  2.06
Modal Analysis and Structural Modes
Torsional component
Vertical component
0.2
0.61Hz
mode 1
0
-0.2
1
2E-5
1
mode 2
0.80Hz
5
9
0.85Hz
13 17 21 25 29
mode 3
9
1.29Hz
13 17 21 25 29
-0.2
1
2E-4
5
9
13 17 21 25 29
mode 4
1.19Hz
-2E-4
1
mode 5
5
9
0.85Hz
13 17 21 25 29
mode 3
5
9
13 17 21 25 29
mode 6
1.45Hz
-0.02
1
2E-5
5
9
1.58Hz
13 17 21 25 29
mode 7
0.1
1
5
9
13
17
21
25
5
9
1.29Hz
29
1.63Hz
0
5
9
1.68Hz
13 17 21 25 29
mode 9
-0.1
1
5E-4
13 17 21 25 29
mode 5
0
0
-5E-4
1
-5E-4
1
5
9
13 17 21 25 29
Structural nodes
mode 8
9
13 17 21 25 29
1.85Hz
-2E-5
1
0.02
5
9
1.58Hz
13 17 21 25 29
mode 7
5
9
13 17 21 25 29
Structural nodes
1
Fig. 23Mode
Normalized
structural modes
5
9
1.19Hz
13 17 21 25 29
mode 4
-0.02
1
2E-5
5
9
1.45Hz
13 17 21 25 29
mode 5
-0.02
1
0.02
-0.02
1
-2E-5
1
5E-5
5
9
1.63Hz
13 17 21 25 29
mode 8
0
5
9
1.68Hz
13 17 21 25 29
mode 9
0
mode 10
-1E-5
1
0.02
0
0
5
mode 2
0
0
0
-2E-4
1
5E-4
-1
1
0.02
0.80Hz
0
0
0
-0.2
1
2E-4
mode 1
0
0
5
1E-5
0.61Hz
0
0
-2E-5
1
0.2
0.2
-5E-5
1
0.02
5
9
1.85Hz
13 17 21 25 29
mode 10
0
5
9
13 17 21 25 29
Structural nodes
Mode 3
-0.02
1
5
9
13 17 21 25 29
Structural nodes
Simulated Wind Time-series for Time-domain Analysis
Simulated u-turbulence
10
node 6
node 1
10
0
-10
0
10
20
30
40
50
60
70
80
90
0
-10
100
0
-10
0
10
20
30
40
50
60
70
80
90
-10
100
node 3
node 8
0
10
20
30
40
50
60
70
80
90
100
node 9
node 4
40
50
60
70
80
90
100
0
10
20
30
40
50
60
70
80
90
100
0
10
20
30
40
50
60
70
80
90
100
0
10
20
30
40
50
60
70
80
90
100
0
10
20
30
40
50
60
70
80
90
100
10
0
0
10
20
30
40
50
60
70
80
90
0
-10
100
10
node 10
10
node 5
30
0
-10
10
0
-10
20
10
0
-10
10
0
10
-10
0
10
node 7
node 2
10
0
10
20
30
40
50
60
70
80
90
100
0
-10
Time (sec.)
Time (sec.)
Simulated w-turbulence
5
node 6
node 1
5
0
-5
0
10
20
30
40
50
60
70
80
90
100
0
0
10
20
30
40
50
60
70
80
90
100
node 3
node 8
0
10
20
30
40
50
60
70
80
90
100
node 9
node 4
40
50
60
70
80
90
100
0
10
20
30
40
50
60
70
80
90
100
0
10
20
30
40
50
60
70
80
90
100
0
10
20
30
40
50
60
70
80
90
100
0
10
20
30
40
50
60
70
80
90
100
5
0
0
10
20
30
40
50
60
70
80
90
0
-5
100
5
5
node 10
node 5
30
0
-5
5
0
-5
20
5
0
-5
10
0
-5
5
-5
0
5
node 7
node 2
5
-5
0
-5
0
10
20
30
40
50
60
70
80
90
100
0
-5
Time (sec.)
Fig. 24 Simulated time series at structural nodes
Time (sec.)
Time Histories of Global Gust Forces
Node 5
10
7.5
2
1.5
5
1
Moment (tf.m)
Lift (tf)
Node 5
2.5
0
-2.5
-5
-7.5
-10
0.5
0
-0.5
-1
-1.5
0
-2
10 20 30 40 50 60 70 80 90 100
0
Node 15
Moment (tf.m)
Lift (tf)
Node 15
10
7.5
5
2.5
0
-2.5
-5
-7.5
-10
-12.5
0
10 20 30 40 50 60 70 80 90 100
Time (sec.)
10 20 30 40 50 60 70 80 90 100
2
1.5
1
0.5
0
-0.5
-1
-1.5
-2
-2.5
0
10 20 30 40 50 60 70 80 90 100
Time (sec.)
Fig. 25 Time histories of global gust responses in nodes 5&15 atU=20m/s
Time Histories of Global Responses
Node 5
2 x Max. amplitudeNode 5
0.04
max
min
Amplitude
0.01
0
-0.01
-0.02
-0.03
0.0120.04
0.15
0.1
0.05
0.02
0.009
-0.02
-0.04
-0.06
0.003
-0.08
x 10
10
20
30
-7
0 40
50
60
100
0 5
10 15
2070 25 8030 90
35 40
45
Time (sec.)
Node 5
0.4
max
min
0
-1
10
20
50
10 15
2030 25 4030Time35
4060 45 70
(sec.)
80
90
100
80
90
100
Node 15
Node 15
Node 15
0.040.01
2 x Max. Amplitude
0.3
0.2
0.1
max
min
0.008
Rot. dips. (deg.)
Vert. dips. (m)
1
0-0.10
0 5
0.006
0.03
0.004
0.002
0.02
Amplitude
0
2
Amplitude
Rotational disp. (deg.)
3
0
0.006
-0.04
-0.05
max
min
0.06
Rot. dips. (deg.)
Vert. dips. (m)
0.02
Node 5
0.0150.08
0.2
0.03
Node 15
0.1
Amplitude
0.05
Vertical disp. (m)
Node 15
Node 5
0
-0.002
0.01
-0.004
-2
-0.006
0
0
-3
-4
0
10
20
30
40
5
10 15 20 25 30 35 40 45
50
60
Time (sec.)
70
80
Mean velocity
(m/s)90
100
0
0
-0.008
-0.01
5
0
10 15 20 25 30 35 40 45
Mean
velocity
(m/s)
10
20
30
40
50
60
70
Time (sec.)
Fig. 26 Time histories of global responses in nodes 5&15 at U=20m/s
Generalized Response Spectra
Vertical displacement
10
0
10
mode 10
mode 7
2
mode 4
10
Rotational displacement
4
mode 3
mode 8
mode 5
2
10
0
Sa(n)
Sh(n) (m2.s)
10
Vertical displacement
4
mode 1
mode 2
10
Rotational displacement
10
10
10
-2
10
target
10modes
5 modes
First modes
-4
10
-6
0
10
0.2
0.4 0.6
0.8
1 1.2 1.4 1.6
Frequency n(Hz)
1.8
2
2.2
2.4
-2
target
10modes
5 modes
First modes
-4
-6
0
0.2
0.4 0.6
0.8
1 1.2 1.4 1.6
Frequency n(Hz)
1.8
2
2.2
2.4
Fig. 27 Effect of turbulent modes on spectra of generalized responses
in node 15 (at U=20m/s)
Global Response Spectra
10
10
10
10
-1
10
-2
30 modes (target)
10 modes
5 modes
First mode
mode 10
10
-1
mode 7
mode 9
10
0
mode 4
0
30 modes (target)
10 modes
5 modes
First mode
Rotational displacement
mode 3
10
SA(n) (deg2.s)
SH(n) (m2.s)
10
1
Rotational displacement
Vertical displacement
mode 5
mode 6
mode 8
10
2
mode 2
10
mode 1
Vertical displacement
-3
-2
10
-3
-4
0
0.2 0.4 0.6 0.8
1
1.2 1.4 1.6 1.8
Frequency n(Hz)
2
2.2 2.4
10
-4
-5
0
0.2 0.4 0.6 0.8
1
1.2 1.4 1.6 1.8
2
2.2 2.4
Frequency n(Hz)
Fig. 28 Effect of turbulent modes on global responses spectra in node 15
(at U=20m/s)
Cross Modal Coefficients
Interaction between u-turbulent spectral modes and structural modes
Auu1111
A
Lift
0.4
0.3
0.2
Au11
13 12
11 10
tur bulent modes (u)
0.1
cr oss modal coefficient
0.5
| AuLij |
15 14
2 1 1 2
9
7 8
6
5
3 4 str uctur al
modes
1
0.8
0.6
0.4
Au11
10
6 5
4 3
1.2
| AuMij |
15 14
0
9 8
7
Au 31
Au 33
13 12
11 10
tur bulent modes (u)
0.2
Moment
Au13
cr oss modal coefficient
Au13
0
9 8
7
6 5
4 3
2 1 10 9
6 5
8 7
2
4 3
1
str uctur al modes
Between w-turbulent spectral modes and structural modes
4
Lift
Au11
3
2
1
15 14
13 12
11 10
tur bulent modes (w )
0
9 8
7
6 5
4 3
2 1 1 2
5 6
3 4
9
7 8
10
str uctur al modes
7
| AwMij |
6
5
4
3
2
1
15 14
13 12
11 10
tur bulent modes (w )
0
9 8
7
6 5
4 3
2 1 10 9
Fig. 29 Influence of spectral mode on structural modes
6 5
8 7
2
4 3
1
str uctur al modes
Moment
5
cr oss modal coefficient
6
cr oss modal coefficient
| AwLij |
Maximum Amplitude of Structure due to CPT
Vertical displacement (m)Vertical displacement (m)
0.1
N.Modes
Node 5
%
Node15
%
30
0.040
100
0.093
100
0.04
20
0.037
93
0.080
86
0.02
10
0.028
70
0.069
74
520
25
0.023
5830
0.053
57
N.modes
Node 5
%
Node15
%
30
.0027
100
.0078
100
20
.0026
96
.0075
96
10
.0021
78
.0071
91
5
.0018
67
.0049
63
0.08
Max amplitude(m)
30 modes
20 modes
10 modes
5 modes
0.06
0
5
Max amplitude(deg.)
0.01
0.008
0.006
30 modes
20 modes
10 modes
5 modes
10
15
Deck nodes
Rotational displacement (degree)
Rotational displacment (degree)
0.004
0.002
0
5
10
15
Deck nodes
20
25
30
Fig. 30 Effect of covariance modes on global responses at all deck nodes
Maximum Amplitude of Structure due to SPT
Max amplitude (deg.)
Max amplitude (m)
Vertical displacement Vertical displacement (m)
0.1
30 modes
10 modes
5 modes
First mode
N.modes
Node 5
%
Node15
%
30
0.067
100
0.147
100
10
0.066
99
0.147
99
5
0.064
95
0.144
97
0.05
0
5
10
15
Deck nodes
20
25
30
1
0.058
86
0.131
88
N.modes
Node 5
%
Node15
%
30
.0069
100
0.015
100
10
.0068
98
0.015
99
5
.0065
93
0.014
95
1
.0059
84
0.012
80
Rotational displacement (degree)
Tosional displacement (deg.)
30 modes
0.01
10 modes
5 modes
First mode
0.005
0
5
10
15
Deck nodes
20
25
30
Fig. 31 Effect of spectral modes on global responses at all deck nodes
Remarks and Insights
 Framework on the gust response of bridges is formulated in
both the time domain and frequency domain using both PODbased Proper Transformations with comprehensive approach
of spatially-correlated turbulent field.

Only
few basicGust
turbulent
modes
contributeofdominantly
Thus,
Unsteady
Response
Prediction
structures
and effectively
onto
the
global
of bridges.
formulated
thanks
the
SPT gust
and response
CPT in both
frequency
Concretely,
thetime
firstdomain
spectral using
turbulent
modeResponse
contributes
domain
and the
Impulse
significantly on the gust response, whereas more basic
Functions
Model will be next development
covariance modes are required for the gust response.
 Effective turbulent field and cross modal coefficients can be
refined for simulating the turbulent field and estimating the
gust response by using few turbulent modes and effective
spectral band.
TOPIC 4
SYSTEM IDENTIFICATION OF WNDEXCITED STRUCTURES IN FREQUENCY
DOMAIN AND TIME DOMAIN





Introduction
Frequency Domain Decomposition (FDD)
Stochastic Subspace Identification (SSI)
Numerical Examples
Remarks and Insights
Introduction (1)
 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.
Input-output identification
Small-scale system: Forced exciters
Due to impulse or forced shaker…
Known
Input
X(t)
System
Noise u(t)
Known
Output
Y(t)
Output-only identification
Large-scale system: Ambient exciters
Due to traffic, wind, wave, sound…
Unknown
Input
X(t)
White Noise
Process
System
Noise u(t)
Known
Output
Y(t)
Introduction (2)
Output-only System
Identification
Stochastic Subspace
Identification (SSI)
Nonparametric Methods
Parametric Methods
 Projection
and
decomposition
of
either
or covariance
in Frequency Domain
inspectral
Time Domain
P. Overschee
matrices of output response time series, even dealing with
B.D Moor,1996
Peak
Picking
SSI with AR, ARMA
directly output
data
must
be
needed
Technique (PPT)
J.H. Weng et.
al., 2008
 Some robustness
numerical methods are used such as QR
Frequency Domain
SSI with Covariance
Decomposition
(FDD)
Decomposition, Least Squares, Singular Value Decomposition…
Enhanced Frequency
J.S. Bandat
 Advantage
of
POD
will
be
exploited for thisSSIpurpose
with Data of
Domain Decomposition
et al., 1993
R. Brincker
decomposition
Covariance Matrix
et al., 2001
Spectral Matrix of
Output Data
A. Yoshida
Y. Tamura,2004
L. Carassale,
F.Percivale,2007
Fig. 32 Only-output system identification methods
of Output Data
Hankel Matrix of
Output Data
Comparison between FDD and SSI
FDD
SSI
 Formulated in the frequency
domain
 Based on spectral matrix of
measured output data
 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
 Less accurate identification
in cases of high noises
 High applicable for closed
modal identification
 Easier in identification
 Prior knowledge of modal
frequencies is required
Frequency Domain Decomposition (FFD)
 Firstly, cross spectral matrix estimated from measured output data
 S y1 y1 ( ) S y1 y2 ( )
 S ( ) S ( )
y 2 y1
y2 y2
S yy ( )  
 ...
...

 S yl y1 ( ) S yl y2 ( )
 y1[1 : r ] 
 y [1 : r ]


yk   2




 yl [1 : r ] 
lxr
... S y1 yl ( ) 
... S y2 yl ( )
...
... 

... S yl yl ( ) 
 l x l x f cut
 Secondly, cross spectral matrix is decomposed using POD
S yy ( ) y ( )   y ( ) y ( )
then
S yy ( )   y () y () y ()T
 ( ),  ( ) Eigenvalue and eigenvector matrices
In the (FDD,
knowledge of modal frequencies is
rior
 )  diagp
( )  ( )   ( )
Where:
y
y
 (n)  (nto
 ( ) required
) 
 ( ) the modal parameters
generally
identify
y
y1
y
y1
y2
y2
yl
yl
l x l x f cut
l x l x f cut
 ith modal identification, we decompose at selected frequency
S yy (i ) y (i )   y (i ) y (i ) then S yy (i )   y (i ) y (i ) y (i )T

 y (i )  diag  yi
 yi   yi l x l
Thus, the ith mode:
and

 y (i )   y1  y 2   yl
i   y1

l xl
i
State-space Representation in SSI (1)
My(t )  Cy (t )  Ky(t )  F (t )
 Continuous state-space representation:
 x (t )  Aˆ x(t )  Bˆ u (t )

ˆ
ˆ
 y (t )  Cx(t )  Du (t )
Where: x(t )  { y(t ), y (t )}T F (t )  Uu(t )
Stochastic
identification
requires
finding system parameters: A, C,
0
I
0




ˆ   LM 1K  LM 1C 
ˆ  LM 1U mxr
Aˆ  
Bˆ   1 
C
D


1

1
mx
2
n
modal parameters,
damping ratios from ambient
 M K  M C frequencies
 M U and
2 nx 2 n
2 nxr
output
x(t): measurements
state vector; Aˆ ,yBˆk,(t)
Cˆ : state matrix; input matrix; output matrix
 Discrete state-space representation at interval time
tk  kt
Inputs
 xk 1  Axk  Bu k

 yk  Cxk  Du k
ˆ
Where: xk  x(kt ) A  AAt , B  ( A  I ) Aˆ 1Bˆ , C  Cˆ , D  Dˆ
k
k
Input-output relationship: yk  CA x0  Du k   CA Bu k i  CA x0   (CAi 1B)uk i
i 1
i 0
If inputs are white noises (during free vibration), model reduce:
k
 xk 1  Axk  sk

 yk  Cxk  vk
Broad-band white noises
i 1
k
SSI-DATA (2)
 Reconstructing output data to Hankel matrices (past, futurestates)
 y1[1 : r ] 
 y [1 : r ]


yk   2




 yl [1 : r ] 
lxr
 y0
y
1
Yp  
 ...

 yi 1
y j 1 
y2 ...
y j 
... ...
... 

y j ... yi  j  2 
y1 ...
lixj
 yi
y
i 1
Yf  
 ...

 y2i 1
yi  j 1 
... yi  j 
...
... 

... y2i  j  2 
lixj
yi 1 ...
yi  2
...
y2i
 Orthogonal projection of Hankel matrices, the decomposing the
projection using the POD
T
T
T
P

VDV
then
i
Pi  Y f Yp   Y f Yp YpYp Yp
 Identifying the system matrices: A, C
A   i  i and C  i *
Where:  i from i without last l row;  i fromi without first l row
 i first l rows of i
i  VD1/ 2 Extended observability matrix
 Identifying modal parameter: A   A ATA then   C A
Refining 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 Frame
Floor5
Floor 4
Floor 3
Floor 2
Floor 1
Ground
Fig. 33 5-storey steel frame at test site
Disaster Prevention Research Institute (DPRI), Kyoto University
Measured Velocities and Integrated Responses
5
x 10
Measured Velocity at Floor 1
-3
Measured Velocity at Floor 5
0.015
4
0.01
2
Amplitude (m/s)
Amplitude (m/s)
3
1
0
-1
0.005
0
-0.005
-2
-0.01
-3
-4
0
100
150
Time (s)
200
250
300
Displacement at Floor 1
-4
2
2
0
-2
-4
0
-0.015
0
Amplitude (m)
Amplitude (m)
4
x 10
50
50
100
150
Time (s)
200
250
300
x 10
50
100
150
Time (s)
200
250
300
250
300
Displacement at Floor 5
-3
1
0
-1
-2
0
50
100
150
Time (s)
Fig. 34 Measured velocities and integrated displacements
200
PSD of Output Response
10
Floor 1
-5
1.736Hz
5.341Hz
8.853Hz
11.43Hz
18.05Hz
19.76Hz
-10
PSD
10
13.66Hz
10
10
10
-15
-20
0
5
10
20
25
30
25
30
Floor 5
-5
1.736Hz
5.341Hz
8.853Hz
11.43Hz
13.66Hz
19.76Hz
-10
PSD
10
15
Frequency (Hz)
10
10
-15
-20
0
5
10
15
20
Fig. 35 High-resolved PSD of output response time series
Spectral eigenvalues
Mode 1
Frequencies and order of modes are
Mode 2 identified via combination with FEM
Mode 3
Ei ( f ) 
 ( f
k 1
i
100
)
  ( f
i 1
k 1
i
99.90%
Energy (%)
80
N f cut off
k
Mode 5
Energy contribution of eigenvalues and eigenvectors
Energy contribution of ith
eigenvalues & eigenmodes
f cut off
Mode 4
k
)
60
40
20
0.07%
0
1
2
0.01%
0.00%
3
4
Eigenvalues
0.00%
0.00%
5
6
Spectral eigenvectors
99.9%
0.07%
Spectral eigenvectors
0.01%
0%
Mode shapes estimation
Mode 1
Mode 1
Mode 3
Unscaled mode shapes
Mode 4
Mode shapes estimation
Floor4
Mode 2
Mode 1
Floor5
Mode 3
Floor5
Floor5
FEM
Identified
FEM
Identified
Floor4
Floor4
| ET  A |2
MAC(E , Floor3
A) 
{ET E }{ AT  A}
Floor3
Floor3
Floor2
Floor2
Floor2
Floor1
Floor1
Floor1
FEM
Identified
Ground
0
0.25
0.5
0.75
Ground
-1
1
-0.5
Floor5
Floor4
Floor4
Floor3
Floor3
Floor2
Floor2
Floor1
Floor1
1
0.5
1
FEM
Identified
FEM
Identified
-0.5
0.5
0
Ground
-1
-0.5
0
0.5
1
Mode 5
Mode 4
Floor5
Ground
-1
0
0.5
1
Ground
-1
-0.5
0
Mode shapes
THANK YOU VERY MUCH
FOR YOUR KIND ATTENTION
どもありがとう ございます。