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

공학석사학위논문

Time-domain Aeroelastic Analysis of Bridge using

a Truncated Fourier Series of

the Aerodynamic Transfer Function

공기동역학적 전달함수의 푸리에 급수 근사를 이용한

교량의 시간영역 공탄성 해석

2013년 2월

서울대학교 대학원

건설환경공학부

박 진 욱

ABSTRACT

This study presents the exact relation between the real and imaginary parts of aero-

dynamic transfer functions for deriving impulse response functions that satisfy the

causality condition. A truncated Fourier series is utilized to express the aerodynam-

ic transfer functions, and the causality condition is defined in terms of the coeffi-

cients of a Fourier cosine and sine series, which represent the real and imaginary

parts of the aerodynamic transfer functions, respectively. The impulse response

functions that satisfy the causality condition are obtained through the inverse Fourier

transform of the aerodynamic transfer functions that conform to the exact relation.

The coefficients of the Fourier series are determined by minimizing the error be-

tween the transfer functions formed by measured flutter derivatives and by the Fou-

rier series. Since the impulse response functions become a series of Dirac delta

functions in the truncated Fourier series approximation method, the aerodynamic

forces are easily evaluated as the sum of current and past displacements with the

same number of the terms in the Fourier series. This study proposes these the trun-

cated Fourier series approximation method. The validity of the truncated Fourier

series approximation method is demonstrated for two types of bluff sections and one

real bridge: a rectangular section with a width to depth ratio of 5, an H-type section

and 2nd

Jindo cable stayed bridge. Time-domain aeroelastic analyses are performed

for an elastically supported system with each section. The applicability of the trun-

cated Fourier series approximation method is also verified for a large-scale bridge,

2nd

Jindo cable stayed bridge. The truncated Fourier series approximation method

yields stable and accurate solutions for the examples efficiently.

KEY WORDS:

Impulse response function; Transfer function; Fourier Series; Causality condition;

Convolution integral; Aeroelastic analysis; Flutter derivative

Student Number: 2011-20978

Table of Contents

1. Introduction ................................................................................. 1

2. Causality Requirement in Aerodynamic Forces ......................... 5

3. Truncated Fourier Series Approximation Method ...................... 9

3.1. Fourier Series Representations of Aerodynamic Transfer functions .... 9

3.2. Minimization and Discretization ......................................................... 13

4. Applications and Verification ................................................... 15

4.1. Rectangular section of B/D=5 ............................................................. 20

4.2. H-type section ...................................................................................... 25

4.3. Large-scale Bridge: 2nd

Jindo Cable Stayed Bridge ............................ 30

5. Summary and Conclusions ....................................................... 38

REFERENCES ............................................................................. 40

List of Figures

Fig. 2.1. Aerodynamic forces and the corresponding displacements .............. 5

Fig. 4.1. Dimension of a cross-section considered: (a) rectangular section; (b)

H-type section; (c) section of 2nd

Jindo bridge ..................................... 15

Fig. 4.2. Transfer functions of the rectangular section for the lift force: (a)

imaginary part of hh component; (b) imaginary part of the h compo-

nent; (c) real part of the hh component; and (d) real part of the h com-

ponent ..................................................................................................... 22

Fig. 4.3. Transfer functions of the rectangular section for the moment: (a) im-

aginary part of h component; (b) imaginary part of the component;

(c) real part of the h component; and (d) real part of the component 23

Fig. 4.4. Free vibration responses at a wind velocity of 8.0m/s for the rectan-

gular section: (a) vertical displacement; and (b) rotational angle .......... 24

Fig. 4.5. Transfer functions of the H-type section for the lift force: (a) imagi-

nary part of the hh component; (b) imaginary part of the h component;

(c) real part of the hh component; and (d) real part of the h component 27

Fig. 4.6. Transfer functions of the H-type section for the moment: (a) imagi-

nary part of the h component; (b) imaginary part of the component;

(c) real part of the h component; and (d) real part of the component .28

Fig. 4.7. Forced vibration responses at a wind velocity of 6.0m/s for the H-

type section: (a) vertical displacement for s 2~0t ; (b) rotational an-

gle for s 2~0t ; (c) vertical displacement for s 20~18t ; and (d) ro-

tational angle for s 20~18t ................................................................. 29

Fig. 4.8. Mean velocity and velocity fluctuations of the wind flow .............. 31

Fig. 4.9. Transfer functions of the section of 2nd

Jindo bridge for the lift

force: (a) imaginary part of hh component; (b) imaginary part of the h

component; (c) real part of the hh component; and (d) real part of the h

component .............................................................................................. 35


Jindo bridge for the mo-

ment: (a) imaginary part of h component; (b) imaginary part of the

component; (c) real part of the h component; and (d) real part of the

component ....................................................................................... 36

Fig. 4.11. Responses at the middle of deck span at a wind velocity of 30.0m/s

and velocity fluctuations of the section of 2nd

Jindo bridge: (a) vertical

displacement for s 600~0t ; (b) vertical displacement for

s 100~0t ; and (c) vertical displacement for s 600~500t ; ............ 37

List of Tables

Table 4.1. Mechanical properties of the structural systems used in the exam-

ples ......................................................................................................... 18

Table 4.2. Time increment used for the time-domain analysis ...................... 19

Table 4.3. First-order derivatives of the lift and moment coefficients .......... 19

Table 4.4. Optimal penalty numbers used for the PFA .................................. 19

Table 4.5. The aerodynamic force coefficients and the first-order derivatives

of the coefficients of 2nd Jindo cable stayed bridge .............................. 32

1. Introduction

The importance of time-domain aeroelastic analysis has been increasingly empha-

sized in recent decades to consider various nonlinearities of a structural system

and/or non-stationary effects of air flows (Chen and Kareem 2003; Salvatori and

Borri 2007; Diana et al. 2010; Wu and Kareem 2011; Zhang et al. 2011; Chen 2012;

Jung et al. 2012). The critical issue in the time-domain aeroelastic analysis is the

formulation of impulse response functions required to transform self-excited aero-

dynamic forces defined in the frequency domain to the time domain. The impulse

response functions are obtained by the inverse Fourier transform of the aerodynamic

transfer functions formed with the flutter derivatives identified in wind-tunnel tests.

Once the impulse response functions are known for a given problem, the aerody-

namic forces can be evaluated in the time domain through a one-sided convolution

integral.

To perform a one-sided convolution integral, the impulse response functions

should satisfy the causality condition (Jung et al. 2012), which states that the im-

pulse response functions vanish for the negative time domain from the physical

point of view. However, the flutter derivatives identified without consideration of

the causality condition cannot lead to the impulse response functions satisfying the

causality condition. Therefore, the aerodynamic transfer functions need to be modi-

fied so as to satisfy the causality condition. It seems that the rational function ap-

proximation (RFA) has been the only method used to modify the aerodynamic trans-

fer functions (Chen et al. 2000; Caracoglia and Jones 2003; Salvatori and Borri

2007; Zhang et al. 2011), until Jung et al. (2012) proposed a new approach based on

the finite element method (FEM). Despite its popularity, however, Caracoglia and

Jones (2003) reported on the potential limitations of the RFA related to its applica-

bility to bluff sections. Jung et al. (2012) also demonstrated that the RFA produces

erroneous steady-state responses for a slab-on-stringer type deck section. The limi-

tations of the RFA stems from the fact that the rational functions cannot reasonably

approximate intricate aerodynamic transfer functions that are frequently observed

for bluff sections. Furthermore, Zhang et al. (2011) pointed out that the minimiza-

tion process to determine the coefficients of rational functions may yield unreasona-

ble solutions unless proper constraints are imposed on the coefficients.

To overcome the drawbacks of the RFA, Jung et al. (2012) proposed a FEM-

based approach, which is referred to hereafter as the penalty function approach

(PFA). In their approach, the causality condition is weakly imposed as a penalty

function in the minimization to modify aerodynamic transfer functions using the

cubic spline interpolation. Although their approach yields accurate and stable re-

sults even for a bluff section, a rather complicated FEM-based formulation is re-

quired, and the penalty number should be determined iteratively. Moreover, the

convolution integrals should be evaluated through numerical integration from 0 to

the current time, which requires a huge computational effort for large-scale struc-

tures.

In this study, the exact relation between the real and imaginary parts of aerody-

namic transfer functions to satisfy the causality condition is derived using a truncat-

ed Fourier series. Since the real and imaginary parts of the aerodynamic transfer

function are even and odd functions in the frequency domain, respectively, the Fou-

rier cosine and sine series are separately applied to represent the individual part. A

linear term is added in the Fourier sine series to avoid oscillations of the truncated

Fourier series caused by a possible discontinuity at the maximum frequency. The

coefficients of the Fourier series are determined using a minimization scheme simi-

lar to that adopted in the RFA and PFA. However, the minimization procedure in

the truncated Fourier series approximation method becomes much simpler than that

in the PFA by the virtue of the exact expression for the causality condition. As the

aerodynamic transfer function is expressed with a Fourier series in the frequency

domain, the corresponding impulse response function becomes a series of Dirac del-

ta functions with the same number of terms as are used in the Fourier series. As a

result, the convolution integrals are evaluated by simple summations of a few past

displacements, which reduces computational effort considerably compared to the

PFA.

The truncated Fourier series approximation method (TFA) and the PFA pro-

posed by Jung et al. (2012) are conceptually equivalent to each other. The differ-

ences between two methods lie in strategies to impose the causality condition and

trial functions to modify the aerodynamic transfer functions. The PFA employs a

FEM-based formulation using piecewise cubic spline interpolation, while the Ray-

leigh-Ritz type representation of the aerodynamic transfer function is adopted with a

Fourier series in the truncated Fourier series approximation method. Concerning the

causality condition, however, the TFA gives exact solutions, but the PFA yields only

approximated solutions because of the weak enforcement of the causality condition.

The accuracy and effectiveness of the TFA are demonstrated through numerical

examples for two typical bluff sections and one real bridge: a rectangular section

with a width to depth (B/D) ratio of 5, an H-type slab-on-stringer section and 2nd

Jindo cable stayed bridge. The TFA successfully yields the aerodynamic transfer

functions satisfying the causality condition exactly for the sections. For the time-

domain analysis, elastically supported systems with the two sections are considered.

The TFA is also applied the large-scale bridge, 2nd

Jindo cable stayed bridge, with

buffeting forces by velocity fluctuations using modal analysis. The responses ob-

tained using the TFA are in good agreement with those obtained by the PFA, while

the computational time is dramatically reduced.

2. Causality Requirement in Aerodynamic Forces

Fig. 2.1. Aerodynamic forces and the corresponding displacements

The aerodynamic forces induced by motions of an object, shown in Fig. 2.1, in a

stationary wind flow are expressed by the convolution integrals in the time-domain

(Chen et al. 2000; Caracoglia and Jones 2003; Salvatori and Borri 2007; Zhang et al.

2011; Jung et al. 2012).

))()()(

)((2

1)(

00

2

t

h

t

hhaedtd

B

htBUtL

))()()(

)((2

1)(

00

22

tt

haedtd

B

htBUtM

(2.1)

where aeL and aeM = the aerodynamic lift force and moment, respectively; h and

= the vertical and rotational displacement, respectively; = air density; U = mean

cross wind velocity; and B = width of the section. The real function, kl for

h, Lae

,Mae

U

B

,, hlk is the kl-component of the impulse response function representing the

aerodynamic force in the k direction at time t induced by the unit impulse motion of

an object in the l direction at 0t . The one-sided convolution integrals in Eq. (2.1)

are valid if and only if every component of the impulse response function vanishes

identically for the negative time domain (Jung et al. 2012), that is, 0kl for 0t ,

which is referred to as the causality condition. The causality condition represents

the physical fact that aerodynamic forces are induced only after an object moves.

Each component of the impulse response function is defined as the inverse Fou-

rier transform of the corresponding component of the transfer function of the aero-

dynamic forces in the frequency domain:

deit tiR

kl

I

klkl)(

2

1)( (2.2)

Here, kl = the kl-component of the aerodynamic transfer function; i = the imagi-

nary unit; and superscript I and R indicate the imaginary and real part of a complex

variable, respectively. The aerodynamic transfer function in Eq. (2.2) is expressed

in terms of flutter derivatives identified in wind tunnel tests (Scanlan and Tomko

1971).

*

3

2*

2

2*

4

2*

1

2 , HKHiKiHKHiKi R

h

I

h

R

hh

I

hh

*

3

2*

2

2*

4

2*

1

2 , AKAiKiAKAiKi RIR

h

I

h

(2.3)

where UBK / = the non-dimensional reduced frequency where = the angular

frequency of oscillation; and *

mH and

*

mA ( 4,3,2,1m ) = the flutter derivatives.

The aerodynamic transfer function in Eq. (2.3) is hereafter referred to as the meas-

ured transfer function.

As the impulse response function is a real function, the following relations

should hold:

)(cos0

R

klkltdt , )(sin

0

I

klkltdt (2.4)

Eq. (4) implies that )(R

kl and )( I

kl are an even and odd function, respectively, in

the frequency domain. The impulse response function for 0t and the causality

condition for 0t becomes as follows:

0

)sin)(cos)((1

)( dttt I

kl

R

klkl for 0t (2.5)

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

dttt I

kl

R

klkl for 0t (2.6)

The causality condition in Eq. (2.6) implies that a certain relationship exists be-

tween the real and imaginary part of the aerodynamic transfer function. Since, how-

ever, such a relation is generally not considered in the identification of flutter deriva-

tives, the measured transfer functions should be modified so as to satisfy the causali-

ty condition. The RFA is the most widely adopted approach for imposing the cau-

sality condition. However, several researchers have reported that the RFA yields

erroneous results for bluff objects due to the limitations of rational functions in rep-

resenting non-monotonic, intricate measured transfer functions (Caracoglia and

Jones 2003; Zhang et al. 2011; Jung et al. 2012). The PFA proposed by Jung et al.

(2012) overcomes the limitations of RFA by interpolating the aerodynamic transfer

function with the piecewise cubic spline and imposing the causality condition as a

penalty function. Their approach has been successfully applied to a bluff section.

However, they did not present the exact relation between the real and imaginary part

of the aerodynamic transfer function to satisfy the causality condition. Moreover,

the PFA requires a considerable computational effort to determine the optimal penal-

ty number and to perform the convolution integrals in Eq. (2.1). A rigorous and ef-

ficient scheme for exactly imposing the causality condition is formulated in the fol-

lowing.

3. Truncated Fourier Series Approximation Method

3.1. Fourier Series Representations of Aerodynamic Transfer

Functions

A simple and straightforward approach is formulated to exactly impose the cau-

sality condition on the aerodynamic transfer function using a truncated Fourier series.

The aerodynamic transfer function is generally expressed in terms of the reduced

frequency, and is defined up to the maximum reduced frequency, m axK , adopted in

actual wind-tunnel tests. Therefore, each component of the modified transfer func-

tion is expressed as a truncated Fourier series with the period of m ax2K . Since the

real and imaginary part of the aerodynamic transfer function are an even and odd

function, respectively, the Fourier cosine series and the Fourier sine series are sepa-

rately adopted for the individual part as follows:

N

n

n

klkl

I

kl

N

n

n

klkl

R

kl

KK

nbKbK

KK

naaK

1 m ax

0

1 m ax

0

sin)(

cos)(

(3.1)

where kl = the kl component of the modified transfer function;

n

kla and

n

klb = un-

known coefficients of the Fourier series; and N = the number of terms in the Fourier

series. The linear term is introduced in the imaginary part of Eq. (3.1) to prevent

oscillations of the Fourier sine series caused by a discontinuity between the Fourier

sine series and the measured transfer function at m axKK . The discontinuity arises

because the Fourier sine series in Eq. (3.1) becomes zero at m ax

KK while the

measured transfer function usually has a non-zero value. A very large number of

terms should be included in the Fourier sine series to model the discontinuity with-

out the linear term.

Substituting Eq. (3.1) into Eq. (2.5) yields the expression for the modified im-

pulse response function.

N

n

n

kl

n

klN

n

n

kl

n

kl

klkl

N

n

n

kl

N

n

n

kl

klkl

I

kl

R

kl

I

kl

R

klkl

ba

K

n

U

Bt

ba

K

n

U

Bttb

U

Bta

dKtB

UKK

K

nb

B

U

dKtB

UKK

K

na

B

U

dKtB

UKKb

B

UdKt

B

UKa

B

U

dKtB

UKKt

B

UKK

B

U

dtKtKt

1 m ax1 m ax

00

1 0 m ax

1 0 m ax

0

0

0

0

0

0

2)(

2)()()(

)sin()sin(1

)cos()cos(1

)sin(1

)cos(1

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

)sin)(cos)((1

)(

(3.2)

where kl = the kl-component of the modified impulse response function. For the

derivation of Eq. (3.2), the following relations are utilized:

0

)cos(1

)( dKtB

UK

B

Ut ,

0

2

2

)sin(1

)( dKtB

UKK

B

Ut (3.3)

The first three terms in the last equation of Eq. (3.2) vanish for 0t , while the last

term does not unless n

kl

n

klab . Therefore, the causality condition for the modified

transfer function in Eq. (3.1) is defined as:

n

kl

n

klab for Nn ,,1 (3.4)

Enforcement of Eq. (3.4) on Eq. (3.1) and Eq. (3.2) leads to the final expressions for

the modified transfer function and the corresponding impulse response function that

exactly satisfy the causality requirement.

N

n

n

klkl

I

kl

N

n

n

klkl

R

kl

KK

naKb

KK

naa

1 m ax

0

1 m ax

0

sin

cos

(3.5)

N

n

n

klklklklK

n

U

Btat

U

Bbtat

1 m ax

00 )()()()( (3.6)

The unknown coefficients in Eq. (3.5) are easily determined via minimizing the

errors between the measured and modified transfer functions. The minimization

process proposed by Jung et al. (2012) is adopted in this study, and is described

briefly in the next section. Once the coefficients are obtained, the aerodynamic

forces in Eq. (2.1) are evaluated without numerical integration.

))()()(

)(1)()(

(2

1)(

1 m ax

00

1 m ax

002

N

n

n

hhh

N

n

n

hhhhhhae

K

n

U

Btat

U

Bbta

K

n

U

Btha

BU

thb

B

thaBUtL

))()()(

)(1)()(

(2

1)(

1 m ax

00

1 m ax

0022

N

n

n

N

n

n

hhhae

K

n

U

Btat

U

Bbta

K

n

U

Btha

BU

thb

B

thaBUtM

(3.7)

Evaluating the aerodynamic forces in Eq. (3.7) requires N past displacements from

the present time at every time interval of )/()/(m ax

KUB . The time increment

used in a time-marching algorithm for the equations of motion of a given structure

should be a division of )/()/(m ax

KUB by an integer so that all past displacements

in Eq. (3.7) are available for calculation. Eq. (3.7) contains only N past displace-

ments, while the convolution integral with the impulse response functions obtained

by the PFA requires the complete time histories of the displacements evaluated at all

time steps in a time-marching algorithm for the equations of motion. The TFA

greatly improves computational efficiency in the evaluation of the aerodynamic

forces, especially, for a large-scale structure.

3.2. Minimization and Discretization

The unknown coefficients in Eq. (3.5) are determined by minimizing the errors

between the measured and modified transfer functions. The minimization procedure

proposed by Jung et al. (2012) is given as:

max

2

max

2

0

2

2

0

2

2))((

)1(

2

1))((

2

1)(Min

K

I

klkl

I

kl

L

I

kl

kl

K

R

klkl

R

kl

L

R

kl

kl

klkldK

wdK

w

kl

aaaa

(3.8)

where klw = a prescribed weighting factor ranging from 0 to 1, which adjusts the

relative weight between the real and imaginary part of the aerodynamic transfer

function in the minimization. Each term in the object function is normalized with

respect to its own L2-norm denoted as 2L

to level the magnitude of each term. With

this normalization, an equal weighting of 2/1kl

w for all kl can be adopted.

A cubic spline, which is a piecewise third-order polynomial, is utilized to form

the continuous transfer functions with discretely identified flutter derivatives (Jung

et al. 2012). The detailed procedures for the interpolation of measured transfer func-

tions using cubic spline were proposed in Jung et al. (2012), and are not presented

here. The modified transfer function in Eq. (3.5) is written in a matrix form.

kl

RR

klK aN )( , kl

II

klK aN )( (3.9)

where

)coscos01(m axm ax

KK

NK

K

R N

)sinsin0(m axm ax

KK

NK

KKI

N

TN

klklklklklaaba )( 100 a

(3.10)

As the minimization problem of Eq. (3.8) is a quadratic form with respect to the un-

known coefficients, the first-order optimality condition yields the following linear

algebraic equation:

maxmax

maxmax

00

00

)()()()(

))()((

K

sp

I

kl

TII

kl

K

sp

R

kl

TRR

kl

kl

K

ITII

kl

K

RTRR

kl

dKWdKW

dKWdKW

NN

aNNNN

(3.11)

Here, 2

2

)(/Lsp

R

klkl

R

klwW ;

2

2

)(/)1(Lsp

I

klkl

I

klwW ; and

sp

i

kl)( = the spline-

interpolated measured transfer function for RIi , formed with the flutter deriva-

tives. The integrals in the left-hand side of Eq. (3.11) are performed analytically,

whereas the trapezoidal rule is employed for integrating the right-hand side terms.

The orthogonal property of the trigonometric function is utilized for the analytical

integration of Eq. (3.11).

4. Applications and Verification

(a)

(b)

(c)

Fig. 4.1. Dimension of a cross-section considered: (a) rectangular section; (b) H-type

section; (c) section of 2nd

Jindo bridge

3.0 cm

Barrier: 0.5 cm 1.8 cm, 0.8 cm from the edge

46.0 cm

30.0 cm

6.0 cm

12.55 m

The TFA is applied to modify the measured transfer functions of the two bluff sec-

tions and one real bridge shown in Fig. 4.1 and to perform aeroelastic analyses in the

time domain. The first section, a rectangular section with a B/D ratio of 5, is fre-

quently employed in wind tunnel tests as a representative bluff section. The second

section is an H-type section with a B/D ratio of 9.6 including the barrier, and simu-

lates a slab-on-girder type section. The third section is the section of 2nd Jindo cable

stayed bridge which is not bluffer than H-type section. No case study for a stream-

lined section is presented in this study because measured transfer functions of a

streamlined section vary monotonically with the reduced frequency, and a conven-

tional method such as the RFA yields accurate solutions for streamlined sections.

Although the B/D ratio of the H-type section is larger than that of the rectangu-

lar section, the H-type section is subject to more intricate aerodynamic forces than

the rectangular section. This is because strong turbulence is generated around the H-

type section due to the empty spaces between the two girders and two barriers. Con-

sequently, the measured transfer functions of the H-type section exhibit wigglier

variations than those of the rectangular section.

For aeroelastic analyses of the rectangular and H-type section in the time do-

main, an elastically supported system, in which each section is supported with a ver-

tical and rotational spring, is considered. The equations of motion per unit length

are defined as follows:

)()( tLtLhkhchm exaehhh

)()( tMtMkcmexae

(4.1)

where j

m , j

c and j

k are the mass, damping and stiffness in the direction of

,hj , respectively, and are summarized in Table 4.1 for each section. exL and

exM are the external excitation forces in the h and direction, respectively. The

mechanical properties of the rectangular section are used in free-vibration tests at the

wind tunnel laboratory of Seoul National University, Korea, and those of the H-type

section are taken from data reported by Kim and King (2007). The Newmark-

method with 4/1 and 2/1 is adopted for the time integration of Eq. (4.1).

The time increment for the Newmark- method is determined by dividing the time

interval between two Dirac delta functions by the smallest integer that yields a time

increment of less than 0.01s for each section. Table 4.2 summarizes the procedures

used in determining the time increment for the wind speeds adopted in the

aeroelastic analyses. An air density of 1.25 3kg/m is used. The flutter onset veloci-

ty is estimated to be 9.4 m/s for the rectangular section and 8.6 m/s for the H-type

section through the frequency-domain flutter analysis based on a complex eigenval-

ue analysis (Matsumoto et al. 2008).

The measured transfer functions are formed with discretely identified flutter de-

rivatives using the cubic spline interpolation. The first-order derivatives of the lift

coefficient, LC , and moment coefficients, MC , with respect to the attack angle are

given in Table 4.3. These parameters are utilized to impose the boundary conditions

for the cubic spline interpolation. The attack angle is assumed to be positive for the

nose-down direction. The lift and moment coefficients of the rectangular section

were measured at the wind tunnel laboratory of Mokpo National University in Mok-

po, Korea, while those of the bluff H-type section are taken from the reference of

Kim and King (2007).

Several researchers have already reported limitations of the RFA in applications

to bluff sections (Caracoglia and Jones 2003; Zhang et al. 2011; Jung et al. 2012).

Especially, Jung et al. (2012) showed the RFA leads to erroneous steady-state solu-

tions for the H-type section. It appears that a comparison of the TFA with the RFA

would be meaningless, and thus the results obtained by the TFA are compared with

those by the PFA proposed by Jung et al. (2012). The optimal penalty numbers used

for the PFA are given in Table 4.4. The optimal penalty numbers for the H-type sec-

tions are taken from the published data by Jung et. al (2012), and those for the rec-

tangular section are estimated in this study in accordance with the method proposed

in their work.

Rectangular section H-type section

Masses hm ( kg/m ) 5.902 3.640

m ( /mmkg 2 ) 0.229 0.102

Dampings hc ( kg/s/m) 1.626 1.003

c ( /s/mmkg 2 ) 0.064 0.022

Stiffnesses hk ( N/m/m ) 2160.8 1332.6

k ( m/mN ) 403.3 106.2

Frequencies hf ( Hz ) 3.05 3.05

f ( Hz ) 6.68 5.13

Time increment used for the time-domain analysis

Type of section m axK

U

B(s)

m axKU

B (s) Integer divisor

Time increment

(s)

Rectangular section 4.55 0.0375 0.0259 3 0.00863

H-type section 4.19 0.0767 0.0575 6 0.00958

Section of

2nd

Jindo bridge 12.56 0.4183 0.1046 8 0.05229

Type of section 0

L

C

0

M

C

Rectangular section -7.65 -0.77

H-type section -9.85 -0.72

penalty numbers used for the PFA

Type of section hh h h

Rectangular section 0.03 0.56 0.06 0.18

H-type section 0.19 0.09 0.40 0.80

Section of

2nd

Jindo bridge 0.019 0.029 0.095 1.1

4.1. Rectangular section of B/D = 5

Each component of the measured transfer function of this section is formed us-

ing the flutter derivatives extracted at the wind tunnel laboratory of Seoul National

University in Seoul, Korea, and is modified using the TFA and PFA. Figs. 4.2 and

4.3 show the transfer functions in the lift and rotational directions, respectively. The

transfer functions obtained by the TFA for ,2N 5 and 10 are drawn together with

the measured transfer functions and the modified transfer functions by the PFA. The

differences between the measured and modified transfer functions indicate the de-

grees of violation of the causality condition in the measured transfer functions. It is

clearly seen that the TFA quickly converges as N increases. Although some minor

differences are found in the lift components of the transfer function modified by the

TFA and PFA, both methods yield nearly identical results.

Time-domain aeroelastic analyses are performed on the elastically supported

section using Eq. (3.7) for 10N and the convolution integral based on the PFA,

respectively. A free vibration is induced by initial displacements of 1 cm in the ver-

tical direction and 0.01 rad in the rotational direction. Fig. 4.4 shows the time histo-

ries of the displacements at a wind velocity of 8 m/s. Although the applied wind

velocity is close to the flutter onset velocity of 9.4 m/s, the performance of both

methods is stable in the time-marching algorithm for the equations of motion and

practically identical results are obtained.

Although the TFA and the PFA yield almost identical results, the TFA requires

much less computational effort to perform convolution integral in Eq. (2.1) because

only the N past displacements appear in Eq. (3.7). To demonstrate the computation-

al efficiency of the TFA, a time-domain analysis is performed up to 200 s on a desk-

top computer with a single core running at 3.2 GHz. The computation time required

to complete the aeroelastic analysis using a program developed with MATLAB

R2011b (MathWorks 2011) is 1.59 s for the TFA and 33.5 s for the PFA. The TFA

reduces the computational time by 95 % compared to PFA. Therefore, the TFA can

be used to perform a time-domain aeroelastic analysis even for a large-scale struc-

ture efficiently without any loss of accuracy.

(a) (b)

(c) (d)

Fig. 4.2. Transfer functions of the rectangular section for the lift force: (a) imaginary

part of the hh component; (b) imaginary part of the h component; (c) real part of

the hh component; and (d) real part of the h component

-30

-20

-10

0

10

20

0 1 2 3 4 5

Measured transfer functionPFATFA (N=2)TFA (N=5)TFA (N=10)

Rea

l p

art

of

hh

co

mp

onen

t

K=B/U

-30

-20

-10

0

10

20

0 1 2 3 4 5


Imag

inar

y p

art

of

h

co

mp

onen

t

K=B/U

-30

-20

-10

0

10

20

0 1 2 3 4 5


Rea

l par

t o

f h

co

mp

onen

t

K=B/U

-30

-20

-10

0

10

20

0 1 2 3 4 5


Imag

inar

y p

art

of

hh

co

mp

onen

t

K=B/U

(a) (b)

(c) (d)

Fig. 4.3. Transfer functions of the rectangular section for the moment: (a) imaginary

part of the h component; (b) imaginary part of the component; (c) real part of

the h component; and (d) real part of the component

-5

0

5

10

0 1 2 3 4 5

Measured transfer functionPFATFA (N=2)TFA (N=5)TFA (N=10)Im

agin

ary p

art

of

h c

om

ponen

t

K=B/U

-5

0

5

10

0 1 2 3 4 5


Rea

l par

t of

h c

om

ponen

t

K=B/U

-5

0

5

10

0 1 2 3 4 5


Imag

inar

y p

art

of

com

ponen

t

K=B/U

-5

0

5

10

0 1 2 3 4 5


Rea

l par

t of

com

ponen

t

K=B/U

(a)

(b)

Fig. 4.4. Free vibration responses at a wind velocity of 8.0m/s for the rectangular

section: (a) vertical displacement; and (b) rotational angle

-2

0

2

0 1 2 3

TFA (N=10) PFA

Pa

rtic

ula

r so

l.

Ver

tica

l d

isp

lace

men

t (c

m)

Time (s)

-2

0

2

0 1 2 3

TFA (N=10) PFA

Pa

rtic

ula

r so

l.

Time (s)

Ro

tati

on

al a

ng

le (

1

0-2

rad

)

4.2. H-type section

Jung et al. (2012) demonstrated the limitation of the RFA for this section. This

cross section is examined again in this study, with the same setups, to demonstrate

the validity of the TFA. The flutter derivatives extracted by Kim and King (2007) at

the Boundary Layer Wind Tunnel Laboratory of the University of Western Ontario

in Ontario, Canada, are adopted.

Figs. 4.5 and 4.6 show the modified transfer functions evaluated using the TFA

and PFA, respectively, along with the measured ones. As in the previous example,

the TFA and the PFA yield almost the same results, even though some differences

are observed in the real parts of the hh and h components in a region of high re-

duced frequency. Since the TFA yields a closer solution to the measured transfer

function, it is believed that the TFA represents actual physical phenomena better

than the PFA. To ensure the convergence of the TFA, the number of series terms is

varied as 2, 5 and 10. As shown in the figures, the modified transfer functions with

5 terms are closely convergent to those with 10 terms.

The accuracy of the TFA in a time-domain simulation is examined for the sec-

tion mounted on springs and subjected to harmonic excitations. The applied excita-

tion forces are as follows:

tM

L

M

Lex

ex

ex

sin

0

0 (4.2)

where N/m 100L ; m/mN 1

0M ; and rad/s 8

ex . The excitation frequency

is set to around the average of the two mechanical frequencies of the structure. Fig.

4.7 shows the vertical displacement and rotational angle of the section calculated

using the TFA with 10 terms, the PFA and the particular solution presented in Jung

et al. (2012). The time histories of the displacements are calculated for 20s. The

transient responses induced by the suddenly applied excitation force for the first 2s

are shown in Figs. 7(a) and 7(b), and the steady-state responses for the last 2s are

given in Figs. 7(c) and 7(d). Negligible differences are observed in the vertical tran-

sient responses calculated by the TFA and PFA due to the difference in the real part

of the hh component. No noticeable difference is found among the steady-state re-

sponses by the TFA, the PFA and the particular solution of Eq. (4.1) under the con-

dition of Eq. (4.2).

(a) (b)

(c) (d)

Fig. 4.5. Transfer functions of the H-type section for the lift force: (a) imaginary part

of the hh component; (b) imaginary part of the h component; (c) real part of the hh

component; and (d) real part of the h component

-20

-10

0

10

20

0 1 2 3 4 5


Imag

inat

y p

art

of

hh c

om

ponen

t

K=B/U

-20

-10

0

10

20

0 1 2 3 4 5


Rea

l par

t of

hh c

om

ponen

t

K=B/U

-20

-10

0

10

20

0 1 2 3 4 5


Rea

l par

t of

h

com

ponen

t

K=B/U

-20

-10

0

10

20

0 1 2 3 4 5


Imag

inar

y p

art

of

h

com

ponen

t

K=B/U

(a) (b)

(c) (d)

Fig. 4.6. Transfer functions of the H-type section for the moment: (a) imaginary part

of the h component; (b) imaginary part of the component; (c) real part of the

h component; and (d) real part of the component

-4

-2

0

2

4

0 1 2 3 4 5


agin

ary p

art

of

h c

om

ponen

t

K=B/U

-4

-2

0

2

4

0 1 2 3 4 5


Imag

inar

y p

art

of

com

ponen

t

K=B/U

-4

-2

0

2

4

0 1 2 3 4 5


Rea

l par

t of

h c

om

ponen

t

K=B/U

-4

-2

0

2

4

0 1 2 3 4 5


Rea

l par

t of

com

ponen

t

K=B/U

(a)

(b)

(c)

(d)

Fig. 4.7. Forced vibration responses at a wind velocity of 6.0m/s for the H-type sec-

tion: (a) vertical displacement for s 2~0t ; (b) rotational angle for s 2~0t ; (c)

vertical displacement for s 20~18t ; and (d) rotational angle for s 20~18t

-5

0

5

0 1 2

TFA (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

TFA (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. TFA (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. TFA (N=10) PFA

Pa

rtic

ula

r so

l.

Time (s)

Ro

tati

on

al a

ng

le (

1

0-2

rad

)

4.3 Large-scale Bridge: 2nd

Jindo Cable Stayed Bridge

Time-domain aeroelastic analysis is performed for a large-scale bridge. 2nd

Jindo

Cable Stayed Bridge is adopted to verify the applicability of TFA for a large-scale

bridge. 2nd

Jindo Cable Stayed Bridge has main span length of 344 m and is located

between Jindo and Haenam, Korea. The modal analysis is used for a real bridge with

multi-degree of freedom unlike a section with two-degree of freedom. In a modal

analysis, the responses are expressed in terms of the generalized coordinates and

dimensionless modal values as follows (Katsuchi et al 1999):

i

iitBqxhtxh )()(),(

i

iitqxtx )()(),(

(4.3)

where x = the coordinates along the deck span; ih and i = the dimensionless

modal values of i-th mode for the vertical and rotational displacement, respectively;

iq = the generalized coordinates of i-th mode. Then, the equation of motion for the

generalized coordinates iq is:

iiiiiiiiItfqqq /)(2 2 (4.4)

where i and i = the damping ratio and circular natural frequency of the i-th mode,

respectively; if and iI = generalized force and inertia of the i-th mode, respectively.

Here, the generalized force of i-th mode if is defined as follows:

l

aeiaeii dxMBLhtf0

)()( (4.5)

where l = deck span length. The generalized force of i-th mode if is evaluated by

substituting Eq. (3.7) into Eq. (4.5).

Not only self-exited forces but also buffeting forces are considered in the time-

domain analysis of this real bridge. While the self-exited forces are caused by mean

cross wind velocity U, the buffeting forces are caused by velocity fluctuations of the

wind u, v, w in Fig. 4.8.

Fig. 4.8. Mean velocity and velocity fluctuations of the wind flow

The velocity fluctuations are generated by ARMA (Auto-regression moving-

average) technique through von-Karman spectrum without considering the admit-

U

v w

u

tance function. The correlations of space are reflected through the coherence func-

tion at 28 points on the girder. The time interval of generated velocity fluctuations is

0.05 sec and the velocity fluctuations are generated for a wind velocity of 30 m/s

until 600 sec. The buffeting forces are defined by the velocity fluctuations under the

assumption of slowly varying gust action (Katsuchi et al 1999):

U

wC

U

uCBUM

U

wC

U

uCBUD

U

wC

C

U

uCBUL

M

Mb

DDb

D

L

Lb

0

22

0

2

0

2

22

1

22

1

22

1

(4.6)

where bL , bD and b

M = the buffeting lift, drag force and moment, respectively;

DC = drag coefficients. The aerodynamic force coefficients and the first-order de-

rivatives of that with respect to the attack angle are given in Table 4.5. The time in-

crement that is closest to 0.05 sec is determined and the generated velocity fluctua-

tions are linearly interpolated by interval to the corresponding time increment.

. The aerodynamic force coefficients and the first-order derivatives of the

coefficients of 2nd

Jindo cable stayed bridge

Type of

section DC LC M

C 0

D

C

0

L

C

0

M

C

Section of

2nd

Jindo

bridge

0.197 0.022 0.047 0.182 1.997 0.489

The 20 modes which have a high level of contribution in the direction of verti-

cal and rotational displacement for the section are used for the modal analysis. Actu-

ally the 20 modes occupy most of the level of contribution for time-domain

aeroelastic analysis in this example. The flutter derivatives are extracted at the wind

tunnel laboratory of Seoul National University in Seoul, Korea.

Figs. 4.9 and 4.10 show the measured transfer function and modified transfer

function obtained by PFA and TFA for N=2, 5, and 10. The TFA and PFA also

yields almost the same transfer functions for the section of 2nd

Jindo cable stayed

bridge. Even, the transfer function modified by TFA is closer to the measured trans-

fer function than the transfer function modified by PFA at the real part of hh and h

components of the transfer function. The large differences between measured trans-

fer functions and modified transfer functions by TFA and PFA at real part of h and

components of the transfer function means that the components of measured

transfer function violate the causality condition as that. The convergence of the TFA

for the section of 2nd

Jindo cable stayed bridge is better than the H-type bluff section,

but 10 terms of truncated Fourier series basis are adopted for a time domain analysis.

Fig. 4.11 shows the vertical displacement at the middle of deck span for the

self-exited forces and buffeting forces. As Fig. 4.11(a), which shows the vertical

displacement for total analysis time, is not clear for the comparison between PFA

and TFA, Figs. 4.11(b) and 4.11(c) for time s 100~0t and s 600~500t re-

spectively are additionally attached. The responses by TFA are almost the same with

those by PFA.

To check the applicability of the TFA for a real bridge, the computation times

between PFA and TFA during convolution integration are compared and a devel-

oped program for a time-domain analysis is Fortran 77. The computation time is

25.7 sec for PFA and 7.9 sec for the TFA. This difference of computation time can

be considered as a meaningless difference. However, if the nonlinearities of struc-

tural system are considered, time interval is decreased and the number of degree of

freedom, the number of considered mode for modal analysis, is increased, the differ-

ence of computation time between PFA and TFA can be huge and important.

(a) (b)

(c) (d)


Jindo bridge for the lift force: (a)

imaginary part of the hh component; (b) imaginary part of the h component; (c)

real part of the hh component; and (d) real part of the h component

-20

-10

0

10

20

0 2 4 6 8 10 12 14


agin

ary p

art

of

hh c

om

ponen

t

K=B/U

-20

-10

0

10

20

0 2 4 6 8 10 12 14


Rea

l par

t of

hh c

om

ponen

t

K=B/U

-20

-10

0

10

20

0 2 4 6 8 10 12 14


Imag

inar

y p

art

of

h

com

ponen

t

K=B/U

-20

-10

0

10

20

0 2 4 6 8 10 12 14


Rea

l par

t of

h

com

ponen

t

K=B/U

(a) (b)

(c) (d)


Jindo bridge for the moment: (a)

imaginary part of the h component; (b) imaginary part of the component; (c)

real part of the h component; and (d) real part of the component

-8

-4

0

4

8

0 2 4 6 8 10 12 14


Imag

inar

y p

art

of

h c

om

ponen

t

K=B/U

-8

-4

0

4

8

0 2 4 6 8 10 12 14


Rea

l par

t of

h c

om

ponen

t

K=B/U

-1

-0.5

0

0.5

1

0 2 4 6 8 10 12 14


Imag

inar

y p

art

of

com

ponen

t

K=B/U

-1

-0.5

0

0.5

1

0 2 4 6 8 10 12 14


Rea

l par

t of

com

ponen

t

K=B/U

(a)

(b)

(c)

Fig. 4.11. Responses at the middle of deck span at a wind velocity of 30.0m/s and

velocity fluctuations of the section of 2nd

Jindo bridge: (a) vertical displacement for

s 600~0t ; (b) vertical displacement for s 100~0t ; and (c) vertical displace-

ment for s 600~500t ;

-0.1

0.0

0.1

0 200 400 600

TFA (N=10) PFA

Ver

tica

l d

ispla

cem

ent

(cm

)

Time (s)

-0.1

0.0

0.1

0 50 100

TFA (N=10) PFA

Ver

tica

l d

ispla

cem

ent

(cm

)

Time (s)

-0.1

0.0

0.1

500 550 600

TFA (N=10) PFA

Ver

tica

l d

ispla

cem

ent

(cm

)

Time (s)

5. Summary and Conclusions

A rigorous method is proposed to strongly enforce the causality condition in the im-

pulse response function required to perform one-sided convolution integrals for a

time-domain aeroelastic analysis. The exact relation between the real and imaginary

parts of the aerodynamic transfer function is derived by expressing each part with a

truncated Fourier series. The coefficients of the Fourier series are determined

through the minimization of errors between the measured transfer function and the

Fourier series. The impulse response function corresponding to the modified trans-

fer function turns out to be a series of the Dirac delta functions with the same num-

ber of terms as are used in the truncated Fourier series. Consequently, the convolu-

tion integrals contain only a few terms on the current and past displacements, and

the computational efficiency is greatly improved compared with the PFA.

The applicability and effectiveness of the TFA are demonstrated through the

examples on two bluff sections and one real bridge: a rectangular section of B/D=5 ,

an H-type section and 2nd

Jindo cable stayed bridge. The time-domain aeroelastic

analysis on an elastically supported system with each section is performed using the

impulse response functions formed by the TFA and PFA. It is shown that the TFA

yields converging results for the intricately varying transfer functions of both sec-

tions with 10N , and stably performs the Newmark- method for a wind speed

near the flutter onset velocity. The time histories of the displacements obtained by

the TFA and PFA are practically identical. It is believed that a truncated Fourier

series with 5 terms may lead to sufficiently accurate results for practical purposes.

The TFA rigorously defines the causality condition of the transfer functions for

the aerodynamic forces, and the convergence to the exact solution is guaranteed by

the virtue of the Fourier series. A detailed time-domain aeroelastic analysis even for

a large-scale structure can be efficiently performed using the TFA.

REFERENCES

Caracoglia, L., and Jones, N. P. (2003). “Time domain vs. frequency domain charac-

terization of aeroelastic forces for bridge deck sections.” J. Wind. Eng. Ind.

Aerodyn., 91(3), 371–402.

Chen, X., Matsumoto, M., and Kareem, A. (2000). “Time domain flutter and buffet-

ing response analysis of bridges.” J. Eng. Mech., 126(1), 7–16.

Chen, X., and Kareem, A. (2003). “Aeroleastic analysis of bridges: Effects of turbu-

lence and aerodynamic nonlinearities.” J. Eng. Mech., 129(8), 885–895.

Chen, X. (2012). “Prediction of buffeting response of long span bridges to transient

nonstationary winds.” Proc., Colloq. on Bluff-Body Aerodynamics and Appli-

cations (BBAA VII), Shanghai, China.

Diana, G., Argentini, T., and Muggiasca, S. (2010). “Aerodynamic instability of a

bridge deck section model: Linear and nonlinear approach to force modeling.”

J. Wind. Eng. Ind. Aerodyn., 98(6-7), 363-374.

Katsuchi, H., Jones, N. P., and Scalan, R. H. (1999). “Multimode coupled flutter and

buffeting analysis of the Akashi-Kaikyo bridge.” J. Structural Eng., 125(1),

60-70.

Jung, K., Kim, H. K., and Lee, H. S. (2012). “Evaluation of impulse response func-

tions for convolution integrals of aerodynamic forces by optimization with a

penalty function.” J. Eng. Mech., 138(5), 519-529.

Kim, J. D., and King, J. P. C. (2007). “The development of wind tunnel test tech-

nique for an aeroelastic buffeting analysis of long-span bridges.” BLWTL-

SS19-2007-DRAFT report submitted to Korea Wind Engineering Research

Center, the Boundary Layer Wind Tunnel Laboratory of the University of

Western Ontario.

Matsumoto, M., Okubo, K., Ito, Y., Matsumiya, H., and Kim, G. (2008). “The com-

plex branch characteristics of coupled flutter.” J. Wind. Eng. Ind. Aerodyn., 96,

1843-1855.

MathWorks. (2011). Matlab reference guide (R2011b). MathWorks, Natick, Massa-

chusetts, USA.

Salvatori, L., and Borri, C. (2007). “Frequency- and time-domain methods for the

numerical modeling of full-bridge aeroelasticity.” Comput. Struct., 85(11-14),

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Scanlan, R. H., and Tomko, J. J. (1971). “Airfoil and bridge deck flutter deriva-

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초 록

이 연구는 causality 조건을 만족하는 임펄스 응답함수를 유도하기 위하여

공기동역학적 전달함수의 실수부와 허수부 사이의 명확한 관계를 제시한다. Truncat-

ed 푸리에 급수가 공기동역학적 전달함수를 나타내기 위해 이용되고 causality 조건은

푸리에 코사인, 사인 급수의 계수들에 대해 정의된다. 여기서 푸리에 코사인, 사인

급수는 각각 전달함수의 실수부와 허수부를 나타낸다. 푸리에 급수의 계수들로

정의된 명확한 관계를 가지는 전달함수를 푸리에 역변환 함으로써 causality 조건을

만족하는 임펄스 응답함수를 얻을 수 있다. 여기서 푸리에 급수의 계수들은 실험을

통해 추출되는 플러터 계수로 얻어지는 측정치 전달함수와 푸리에 급수로 표현되는

전달함수간의 최소제곱오차를 이용하여 결정된다. 이 푸리에 급수 근사법에서

임펄스 응답함수는 Dirac-delta 함수들의 시리즈가 되기 때문에 공기력을 푸리에

급수의 기저 개수만큼의 과거 응답과 현재 응답의 합으로 쉽게 구할 수 있다. 이

연구에서 제안하는 푸리에 급수 근사법의 타당성은 두 개의 bluff 단면인 폭-깊이 비

5의 직사각형 단면, H형 단면과 하나의 실교량인 제 2진도대교에 적용함으로써

검증된다. 두 개의 bluff단면에 대한 시간영역 공탄성 해석은 탄성지지 시스템에 대해

수행된다. 또한 규모가 큰 실교량에 대한 푸리에 급수 근사법의 적용성도 확인한다.

푸리에 급수 근사법은 이 예제들에 대해 효율적으로 안정적이고 정확한 결과를

보인다.

주요어: 임펄스 응답합수, 전달함수, 푸리에 급수, causality 조건, 중첩적분,

공탄성 해석, 플러터 계수

학번: 2011-20978

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

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