le thai hoa vietnam national university, hanoi open seminar at tokyo polytechnic university pod and...

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


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


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

)


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


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


Rotational displacement (degree)

MaximumMaximum Amplitude of Structure due Amplitude of Structure due to SPTto SPT


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


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


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


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.


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

}}{{

||),(

2

ATAE

TE

ATE

AEMAC

THANK YOU VERY MUCH THANK YOU VERY MUCH FOR YOUR KIND ATTENTIONFOR YOUR KIND ATTENTION

どもありがとうございます。

le thai hoa vietnam national university, hanoi open seminar at tokyo polytechnic university pod and...

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