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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