Estimating Multiple Aerodynamic Estimating Multiple Aerodynamic Admittance Functions using System Admittance Functions using System

Identification TechniquesIdentification Techniques

Le, Thai Le, Thai HoaHoaWind Engineering Research CenterWind Engineering Research Center

Tokyo Polytechnic UniversityTokyo Polytechnic University

Buffeting response theory of bridges under turbulence flow Buffeting response theory of bridges under turbulence flow has been basing on two major theories:has been basing on two major theories:QuasiQuasi--steady Theorysteady Theory

Relationship between: instantaneous wind fluctuations Relationship between: instantaneous wind fluctuations and instantaneous buffeting forces and instantaneous buffeting forces Aerodynamic Admittance FunctionsAerodynamic Admittance Functions

Strip TheoryStrip TheoryRelationship between point buffeting forces and spatial Relationship between point buffeting forces and spatial buffeting forces buffeting forces Spatial Coherence FunctionsSpatial Coherence Functions

• These theories rooted from aeronautical field for airfoils and elongated shapes. Boundary layer, turbulence flows and bluff bodies must be featured when they are used for civil structures

Introduction Introduction

Aerodynamic admittance functions are considered as the Aerodynamic admittance functions are considered as the transfer functions between input wind fluctuations and transfer functions between input wind fluctuations and output induced forces for a aim of correcting the quasioutput induced forces for a aim of correcting the quasi--steady theorysteady theory

Admittance functions can be determined by either Admittance functions can be determined by either empirical equations or physical measurements empirical equations or physical measurements

It is usually assumed that It is usually assumed that that equal contribution of uthat equal contribution of u--wind fluctuation and wwind fluctuation and w--wind fluctuation on the wind fluctuation on the aerodynamic admittance of buffeting forcesaerodynamic admittance of buffeting forces

Some shortcomings are follows: Some shortcomings are follows: 1/ Contribution of each wind fluctuating components 1/ Contribution of each wind fluctuating components

u(t), w(t) on aerodynamic admittanceu(t), w(t) on aerodynamic admittance2/ Effects of second2/ Effects of second--order squared terms, cross terms order squared terms, cross terms

of wind fluctuations uof wind fluctuations u22(t), w(t), w22(t), wu(t)(t), wu(t)3/ Effect of measurement noises 3/ Effect of measurement noises

Introduction Introduction

Recent literature reviewsRecent literature reviews• Sear’s (1941) and Lieppman (1953): Practical formulae

for admittance of lift force• Davenport (1962): Quasi-steady admittance function of

buffeting forces• Holscher (1992), Kawatani (1992), Sankaran (1992),

Larose (1999), Scanlan (2001), Hatanaka (2002,2008) and so on: Measured aerodynamic admittance functions of lift, drag, moment in isotropic turbulence flows

• Diana (2002), Cigada (2002): Measured admittance functions in active turbulence generator

• Caracoglia (2005), Tubino (2005): Revised relationship between admittance and flutter derivatives

• Sterling (2009), Baker (2010): Determined aerodynamic weighting functions in the time domain

Single-variate admittance functions

ObjectivesObjectives

1. Investigate contribution of wind fluctuation components on aerodynamic admittance functions, effects of squared terms and cross terms of wind fluctuations, measurement noise as well

2. Propose new approach of Multiple Aerodynamic Admittance Functions using system identification techniques in the frequency domain and the time domain

U+u+dx/dt

L(t)

D(t)

M(t)

(t)

z(t)

x(t)

Vw+dz/dt(t)

))(()(21)( 2 tBCtVtL L

))(()(21)( 2 tBCtVtD D

))(()(21)( 22 tCBtVtM M

• Relationship between instantaneous wind fluctuations and buffeting forces

)(2)( 22 tUuUtV

'02

221

0|00 )()())(( FF

FFFF CC

dCd

ddCCtC

UuwuUtV 2)( 2222 Or

UtwCC

UtuCBUtL DLL

)()()(221)( '2

• Quasi-steady buffeting forces (only lift for a sake of brevity)

• Corrected quasi-steady buffeting forces

UtwfCC

UtufCBUtL LwDLLuL

)()()()(2)(21)( '2

Shortcomings in the quasi-steady theory:

1/ Linearization of relative velocity

2/ First-order approximation of relative angle of attack

3/ Impossible for taking frequency components from wind-

structure interaction

4/ Impossible for dealing with unsteady buffeting forces and

‘memory effect’ in fluid

QuasiQuasi--steady buffeting forces & correctionsteady buffeting forces & correction

Correction

SingleSingle--variatevariate quasiquasi--steady admittancesteady admittance

)]()()()(4[)21()( 22'222 fSfCfSfCUBfS wwLwLuuLuLLL

• Quasi-steady buffeting forces in the frequency domain

)()(4)()( 2'2

0

22

fSLfSLfSUf

wwuu

LL

• Assumed)]()()( 222 fff LLwLu

• Quasi-steady single-variate admittance functions

LBCUL 20 2

1 '2'

21

LBCUL

)()()( 2'

22

fSLfSUf

ww

LLL

In case of CL=0

SingleSingle--variatevariate nonnon--linear admittancelinear admittance• Comprehensive form of non-linear buffeting forces

in the frequency domain

])(

)()(

)(

)()()()(4[)21()(

422'

422

222'

22222

22

2

22

2 UfS

fCU

fSfC

UfSfC

UfSfCBUfS

wwLwL

uuLuL

wwLwL

uuLuLLL

• Assumed)()()()()( 22222

22 fffff LLwLuLwLu

• Non-linear single-variate admittance functions

)()()()(4)()(

222220

20

22'220

42

fSLfSLfSULfSULfSUf

wwuuwwuu

LLL

UuwuUtV 2)( 2222

MISO system identification model MISO system identification model in the frequency domainin the frequency domain

HLu(f)

HLu2(f)

HLw(f)

HLw2(n)

Su(f)

Su(f)

Su2(f)

SLu(f)

SLu2(f)

Sw(f)

Sw2(f)

SLw(f)

SLw2(f)

Sw(f)

SL(f)

SN(f)

Output

Input

Su(f), Sw(f), Su2(f), Sw2(f): multiple inputs

SL(f), SD(f), SM(f): outputs

H(f): transfer functions

Sn(t): measurement noise

• Multiple-input and Single-output (MISO) system model has been used for estimating Multiple Admittance Functions

)()(|)(| 2

fSfSfH

uu

LuLu

)()(

|)(|22

2

22

fSfS

fHuu

LuLu

)()(|)(| 2

fSfSfH

ww

LwLw

)()(

|)(|22

2

22

fSfS

fHww

LwLw

• Multiple (local) transfer functions

Multiple Multiple admittance functionsadmittance functions

;4

|)(|)( 20

222

LfHUf Lu

Lu

;|)(|)( 2'

222

LfHUf Lw

Lw

;|)(|

)( 20

242 2

2 LfHU

f LuLu

20

242 |)(|

)( 2

2 LfHU

f LwLw

• Estimating multiple aerodynamic admittance functions

• Comprehensive relationship between multiple inputs xi (i=1..N) multiple outputs yj (j=1..M) can be expressed in matrix form

22

22

|)(||)(|

|)(||)(|

)()(

)()(

)()(

)()(

1

111

1

111

1

111

fHfH

fHfH

fSfS

fSfS

fSfS

fSfS

NMM

M

NNN

N

MNN

M

xyxy

xyxy

xxxx

xxxx

yxyx

yxyx

)(|)(|)(1

2 fSfHfSkikjji xx

N

kxyyx

)()()()()()( 22 fSfSfSfSfSfS NLwLwLuLuLL

[Bendat and Piersol, 1993]

)()(|)(|)(|)(|)(|)(|)(|)(|)( 222222222 fSfSfHfSfHfSfHfSfHfS NwwLwwwLwuuLuuuLuLL

Multiple admittance function

Framework for multiple admittance functionsMeasured wind fluctuations and measured buffeting forces

Force coefficients and first-order derivative coefficients

Multiple transfer functions

Multiple admittance functions

Contribution of each wind fluctuations, effects of squared fluctuations and measurement noise

Unsteady buffeting forces in time domainUnsteady buffeting forces in time domain• Unsteady buffeting forces can be expressed via Impulse

Response Functions and convolution integrals

t

Lw

t

Lu dU

wtIdU

utIUtL0 0

2 ])()()()()[21()( [Lin et el. 1986][ Scanlan 1993]

• With influence of squared terms of wind fluctuations, we have complete form of unsteady buffeting lift as follows

t

Lw

t

Lu dU

wtIdU

utIUtL0 0

2 )()()()()[21()(

])()()()(0 2

2

02

2

22 t

Lw

t

Lud

UwtId

UutI

ILu(t), ILw(t): Aerodynamic Weighting Functions(Impulse Response Functions)

)]*(1)*(1)[21()( 2 wIU

uIU

UtL LwLu (*): convolution operator

Weighting Functions (Impulse Response Functions)

1/ Considered as transfer functions in the time domain

In special case, they were known as Aerodynamic Weighting

Functions in the time domain

2/ Dealing with past, present states of buffeting forces or

‘Memory Effect’ in unsteady fluid via convolution operation

3/ Used for predicting unsteady buffeting response prediction

in the time domain, but determined via admittance functions

Relationship between multiple admittance Relationship between multiple admittance functions and weighting functionsfunctions and weighting functions

• Mathematical relationship between Admittance Functions and Weighting Functions (Impulse Response Functions)can be established as follows

)(2)( fBCfI LuLLu )()(2)( ' fCCBfI LwDLLw

2/)()( 22 UfBCfI LuLLu 2/)()( 22 UfBCfI LwLLw

)(),(),(),( 22 fIfIfIfI LwLuLwLu : Fourier transform of aerodynamic weighting functions

Aerodynamic weighting functions

Inverse FourierTransform

MISO system identification model MISO system identification model in the time domainin the time domain

hLu(t)

hLu2(t)

hLw(t)

hLw2(t)

u(t)

u(t)

u2(t)

LLu(t)

LLu2(t)

w(t)

w2(t)

LLw(t)

LLw2(t)

w(t)

L(t)

n(t)

OutputInput

• Multiple-Input, Single-Output(MISO) system model has beenused for estimating Multiple Impulse Response Functions

u(t), w(t), u2(t), w2(t):

multiple inputs

L(t), D(t), M(t): outputs

I(t): impulse response functions

n(t): measurement noise

h2Lu()

h2Lu

2()

h2Lw()

h2Lw

2()

Ru()

Ru()

Ru2()

RLu()

RLu2()

Rw()

Rw2()

RLw()

RLw2()

Rw()

RL()

Rn()

OutputInput

Mod

el 1

Mod

el 2

Multiple impulse response functionsMultiple impulse response functions• Comprehensive relationship between multiple inputs xi (i=1..N)

multiple outputs yj (j=1..M) can be expressed))(*)(()(

1txthty

k

N

k xyj kj (*): convolution operator

)(*)()(*)()(*)()(*)()( 2222 twthtwthtuthtuthtL LwLwLuLu

• Lift can be modeled as

)(\)()( tutLthLu )(\)()( 22 tutLthLu

)(\)()( twtLthLw )(\)()( 22 twtLthLw

(\): deconvolution operator

• Multiple impulse response functions (Multiple weighting functions) can be determined

)(21)( tUIth LuLu

Known impulse responsefunctions

)(21)( 22 tIth LuLu

)(21)( tUIth LwLw )(

21)( 22 tIth LwLw

Framework for multiple weighting functionsMeasured wind fluctuations and measured buffeting forces

Force coefficients and first-order derivative coefficients

Multiple impulse response functions

Multiple aerodynamic weighting functions

Contribution of each wind fluctuations, effects of squared fluctuations and measurement noise

Experimental dataExperimental data

-1

-0.8

-0.6

-0.4

-0.2

0

0.2

0.4

0.6

0.8

1

-10 -6 -2 2 6 10(deg)

CL

1

1.5

2

2.5

3

-10 -6 -2 2 6 10(deg)

CD

-0.2

-0.15

-0.1

-0.05

0

0.05

0.1

0.15

0.2

-10 -6 -2 2 6 10

CM

(deg)

●：U=8m/s

Δ：U=12m/s

Lift Coefficient　C L

-2

-1

0

1

2

-14 -10 -6 -2 2 6 10 14

C L

Drag Coefficient　C D

-5.0

-4.0

-3.0

-2.0

-1.0

0.0

1.0

2.0

3.0

4.0

5.0

-14 -10 -6 -2 2 6 10 14

C D

Moment Coefficient　C M

-0.3

-0.2

-0.1

0

0.1

0.2

0.3

-14 -10 -6 -2 2 6 10 14

C M

●：5m/s

△：10m/s

B/D=5

B/D=20

Model B/D=5 Model B/D=20

CL CD CM CL’ CD’ CM’0 1.09 0 6.41 0 0.600 1.42 0 7.06 0 0.70

B/D=5

B/D=5

Results and DiscussionsResults and DiscussionsSingle-variate quasi-steady admittance functions

Single-variate non-linear admittance functions

Effect of squared terms of wind fluctuations u2(t), w2(t) is

supposed to influence increasingly with respect to bluffer sections

and approaching flow’s higher turbulence intensity

Results and DiscussionsResults and DiscussionsB/

D=

5

Multiple transfer functions between lift and wind fluctuations

Multiple transfer functions between moment and wind fluctuations

Results and DiscussionsResults and DiscussionsMultiple admittance functions of lift, moment

Multiple admittance functions of lift and moment

B/D

=5B/

D=2

0

It is noting that effect of u-wind fluctuation on lift, moment

cannot be evaluated in this investigation due to zero balanced

force coefficient (CL=0) of the symmetrical girders

Results and DiscussionsResults and DiscussionsB/D=5

Effects of wind fluctuation, squared fluctuation and noise

Results and DiscussionsResults and DiscussionsB/D=5

Lift

Impulse response functions

Power spectra of Impulse response functions

Results and DiscussionsResults and DiscussionsImpulse response functions

Results and DiscussionsResults and DiscussionsAerodynamic weighting functions

Further worksFurther works

CL CD CM CL’ CD’ CM’0 1.09 0 6.41 0 0.600 1.42 0 7.06 0 0.70

B/D=5

B/D=5Sym

met

rical

Elon

gate

d Bl

uffe

rG

irder

s

Bluffer bridge girders: These values are no longer zeros

• Investigating multiple admittance functions and multipleimpulse response functions for practical girders

• Development of system identification technique of impulse response functions in the time domain