time-domain aeroelastic analysis of bridge using a truncated...

Jinwook Park, Seoul National University, Korea

Kilje Jung, University of Notre Dame, USA

Ho-Kyung Kim, Seoul National University, Korea

Hae Sung Lee, Seoul National University, Korea

Time-domain Aeroelastic Analysis of Bridge using a Truncated Fourier Series

of the Aerodynamic Transfer Function

The 8th China-Japan-Korea International Workshop on Wind Engineering May 11th, 2013

2 Seoul National University

1. Introduction

2. Causality Requirement

3. Truncated Fourier Series Approximation

4. Application and Verification - H-type section

5. Summary

Contents


Aerodynamic Forces induced by Motion of a Bridge

- Self-exited forces

No analytical solutions of the aerodynamic forces

Usually assumed as a linear system in frequency domain

• Determined through a Wind tunnel test or CFD simulation

For the time-domain analysis, the aerodynamic system should satisfy the

Causality Requirement.

Lae

Mae Aerodynamic

System

Strip Theory

Governing Equation : Navier-Stokes Eq.

Integration of the Pressure Distribution

h

a

U

B

1. Introduction


Linearized Aerodynamic System in Frequency Domain

- Adopting the Flutter Derivatives proposed by Scanlan

Mixed frequency/time domain

• Small-amplitude sinusoidal motion with a single frequency

where, : Reduced frequency

Basically based on the Aerodynamic Transfer Functions

where, F : Fourier transform

a

a

)(

)(

)()(

)()(

)(

)(

)()(

)()(

)(

)(

14

34

21

21

t

th

KAKA

KHKH

t

th

KAKA

KHKH

U

B

tM

tL

ae

ae

U

BK

a

a

aaa

a

)(

)(

)(

)(

)()()()(

)()()()(

)(

)(

3241

3241

F

hF

F

hF

KAKiKAKAKiKA

KHKiKHKHKiKH

MF

LF

h

hhh

ae

ae



Linearized Aerodynamic System in Time-domain

- Using a Convolution Integral Theorem

By taking the inverse Fourier transform

Aerodynamic Impulse Response Functions

a

,,for)(2

1)( hlkdeKt ti

klkl

-15

-10

-5

0

0 1 2 3 4

Imaginary partReal part

K=B/U

Mea

sure

d T

ran

sfer

Fu

nct

ion

Inverse FT

-4

0

4

-5 0 5 10

Evaluated IRF

Ut/B

Norm

aliz

ed m

agnit

ude

kl (t)=0 : Causality condition


a

a

aa

a

a

00

00

)()()()(

)()()()(

)(

)(

dtdht

dtdht

tM

tL

h

hhh

ae

ae


Causality Condition

- Definition

- Mathematical meaning

One-sided convolution Integral

Relationship between the real and imaginary part of TF

- Physical meaning

No aerodynamic force is generated before the deck moves

0for0)( tt

-4

0

4

-5 0 5 10

Evaluated IRFModified IRF

Ut/B

Norm

aliz

ed m

agnit

ude

0for0)sin)(cos)(()(0

tdtKtKt I

kl

R

klkl


a

a

aaa

a

tt

h

t

h

t

hh

ae

ae

dtdht

dtdht

tM

tL

00

00

)()()()(

)()()()(

)(

)(


Previous works - RFA (Rational Function Approximation)

: RFA is based on the solution to the ideally thin section.

: RFA has a limitation to its applicability to bluff sections.

- PFA (Penalty Function Approach), Jung et al. 2012

: PFA is applicable to the bluff sections

: Causality condition is weakly imposed as a penalty function through the optimization.

-Limitation of RFA, Jung et al. 2012

-20

-10

0

10

20

0 1 2 3 4 5

Measured transfer functionPFA

-400

-200

0

200

400

RFA (nr=1)RFA (nr=2)RFA (nr=3)

Rea

l par

t of

ha c

ompo

nent

B/U

- Aerodynamic transfer function for H-type section

- RFA is drawn against the right vertical axis.

- RFA yields unreliable results

Propose the efficient method that exactly satisfy the causality condition !!

* Jung, K., Kim, H. K., and Lee, H. S. (2012). “Evaluation of impulse response functions for convolution integrals of aerodynamic forces by optimization with a penalty function.” J. Eng. Mech., 138(5), 519-529.



Comparison between Previous Work and Proposed Method

RFA

Rayleigh-Ritz type

approximation

Rational function

Strongly enforcement

Limitation of application

to the bluff section

PFA

Finite element method

Cubic spline

Weakly enforcement

Applicable to the bluff

section

Proposed

Rayleigh-Ritz type

approximation

Sine, Cosine function

Strongly enforcement

Applicable to the bluff

section

Approximation method

Basis function

Causality condition

Applicability



Approximation using Truncated Fourier Series

- Fourier series representations of the transfer functions

- Need of the linear term( ) in the imaginary part

N

n

n

klk l

I

k l

N

n

n

klk l

R

kl

KK

nbKbK

KK

naaK

1 max

0

1 max

0

sin)(

cos)(: modified transfer function

: coefficients of the Fourier series

: the number of terms in the Fourier series

: maximum reduced frequency

n

kl

n

kl ba ,

N

maxK

Kbkl

0

Without the linear term, the discontinuity arises at Kmax . The discontinuity causes Gibbs phenomenon. A large number of terms should be included.

I

klI

kl

I

kl

maxKK



Enforcement of the Causality Condition

- IRF corresponding to the modified transfer function

: IRF becomes a series of Dirac-delta functions.

- Causality condition for the modified transfer function

Nnab n

kl

n

kl ,,1for 0for0)( ttk l

N

n

n

kl

n

kl

N

n

n

kl

n

klk lk l

I

k l

R

kl

I

k l

R

klk l

ba

K

n

U

Bt

ba

K

n

U

Bttb

U

Bta

dKtB

UKKt

B

UKK

B

U

dtKtKt

1 max1 max

00

0

0

2)(

2)()()(

)]sin()()cos()([1

)sin)(cos)((1

)(

0t 0t



Final Aerodynamic Forces

))()()(

)(1)()(

(2

1)(

))()()(

)(1)()(

(2

1)(

1 max

00

1 max

0022

1 max

00

1 max

002

aaaaaa

aaa

aaa

aaa

aaa

N

n

n

N

n

n

hhhae

N

n

n

hhh

N

n

n

hhhhhhae

K

n

U

Btat

U

Bbta

K

n

U

Btha

BU

thb

B

thaBUtM

K

n

U

Btat

U

Bbta

K

n

U

Btha

BU

thb

B

thaBUtL

a

a

aaa

a

tt

hae

t

h

t

hhae

dtdB

htBUtM

dtdB

htBUtL

00

22

00

2

)()()(

)(2

1)(

)()()(

)(2

1)(

To perform the convolution integrals, only N past displacements are required.



Determination of Coefficients of the Fourier Series

- The unknown coefficients are determined by minimizing the errors between the measured and modified transfer functions.

max

2

max

2

0

2

2

0

2

2))((

)1(

2

1))((

2

1)(Min

K

I

klkl

I

k l

L

I

kl

kl

K

R

klkl

R

kl

L

R

kl

klklk l dK

wdK

w

kl

aaaa

TN

klklklklkl aaba )( 100 a : (N+2) unknowns

0

kl

kl

a: (N+2) equations

k lw : weighting factor, : L2-norm,

: measured transfer functions interpolated by cubic spline IR

kl

,

2L



Condition of the Problem

- Jung el al.(2012) demonstrated the limitation of RFA for this H-type section

- Governing equation

- Dimension of the H-type bluff section, Kim and King (2007)

- External forces

: To compare the steady-state responses with those obtained by measured transfer function.

Hz15.5,Hz05.3

m/mN2.106,N/m/m6.1332

/s/mmkg022.0,kg/s/m003.1

/mmkg102.0,kg/m640.3

2

2

a

a

a

a

ff

kk

cc

mm

h

h

h

h

)()(

)()(

tMtMkcm

tLtLhkhchm

exae

exaehhh

aaa

aaa

3kg/m25.1smU /6

rad/s8,sinm/mN1

N/m10

exex

ex

ext

M

L



hh Components of the Transfer Functions for the Lift Force

-20

-10

0

10

20

0 1 2 3 4 5

Measured transfer functionPFAProposed method (N=2)Proposed method (N=5)Proposed method (N=10)

Imag

inat

y p

art

of

hh c

om

ponen

t

K=B/U

I

hh

-20

-10

0

10

20

0 1 2 3 4 5


Rea

l par

t of

hh c

om

ponen

tK=B/U

R

hh

- The PFA and Proposed method yields almost the same results. - Even, the Proposed method yields closer solution to the ‘Measured’ than the PFA. - For the proposed method, only 5 terms of Fourier series are enough convergent.



ha Components of the Transfer Functions for the Lift Force

-20

-10

0

10

20

0 1 2 3 4 5


Imag

inar

y p

art

of

ha

com

ponen

t

K=B/U

I

ha

-20

-10

0

10

20

0 1 2 3 4 5


Rea

l p

art

of

ha

co

mp

on

ent

K=B/U

R

ha

- The PFA and Proposed method yields almost the same results. - The differences with ‘Measured’ indicates the degrees of violation of the causality condition in the ‘Measured’.



ah Components of the Transfer Functions for the Moment

-4

-2

0

2

4

0 1 2 3 4 5

Measured transfer functionPFAProposed method (N=2)Proposed method (N=5)Proposed method (N=10)Im

agin

ary p

art

of a

h c

om

ponen

t

K=B/U

I

ha

-4

-2

0

2

4

0 1 2 3 4 5


Rea

l par

t of a

h c

om

ponen

tK=B/U

R

ha

-The results for the moment can be similarly interpreted with those for the lift force



aa Components of the Transfer Functions for the Moment

-4

-2

0

2

4

0 1 2 3 4 5


Imag

inar

y p

art

of aa

com

ponen

t

K=B/U

I

aa

-4

-2

0

2

4

0 1 2 3 4 5


Rea

l par

t of aa

com

ponen

tK=B/U

R

aa

-The results for the moment can be similarly interpreted with those for the lift force



Responses

- Using Newmark-beta method with time interval of 0.0096 sec

- The PFA, proposed method and particular solution have no noticeable difference.

-5

0

5

0 1 2

Proposed method (N=10) PFA

Pa

rtic

ula

r so

l.

Ver

tica

l d

isp

lace

men

t (c

m)

Time (s)

-5

0

5

0 1 2

Proposed method (N=10) PFA

Pa

rtic

ula

r so

l.

Time (s)

Ro

tati

on

al a

ng

le (

1

0-2

rad

)

-5

0

5

18 19 20

Particular sol. Proposed method (N=10) PFA

Pa

rtic

ula

r so

l.

Ver

tica

l d

isp

lace

men

t (c

m)

Time (s)

-5

0

5

18 19 20

Particular sol. Proposed method (N=10) PFA

Pa

rtic

ula

r so

l.

Time (s)

Ro

tati

on

al a

ng

le (

1

0-2

rad

)

)(th )(ta



Comparison of Computation Time

- Computer: 2.4GHz single core

- Program: MATLAB R2001b

- Analysis time: 200 sec

- Time interval: 0.00958 sec

PFA Proposed method (N=10)

Computation time 42.7 sec 2.6 sec

Reduction of 94 %


Proposed method can be used to perform a time-domain aeroelastic analysis even for a large-scale structure efficiently without any loss of accuracy.


5. Summary

Causality Requirement

Propose the Exact and Efficient Method using Truncated Fourier Series

- Causality condition is strongly imposed.

- IRF becomes Dirac-delta functions (Reduction of the computation)

Application and Verification through the example of H-type bluff section

ACKNOWLEDGEMENT This research was supported by the grant (09CCTI-

A052531-05-000000) from the Ministry of Land, Transport and Maritime Affairs of Korean government through the Core Research Institute at Seoul National University for Core Engineering Technology Development of Super Long Span Bridge R&D Center.

Thank you !

time-domain aeroelastic analysis of bridge using a truncated...

Documents