STUDY OF PASSIVE DECK-FLAPS FLUTTER CONTROL SYSTEM ON
FULL BRIDGE MODEL. I: THEORY
By Piotr Omenzetter1, Krzysztof Wilde2 and Yozo Fujino3, Member, ASCE
ABSTRACT
A passive aerodynamic control method for suppression of the wind induced instabilities of a
very long span bridge is presented in this paper. The control system consists of additional
control flaps attached to the edges of the bridge deck. Control flap rotations are governed by
prestressed springs and additional cables spanned between the control flaps and an auxiliary
transverse beam supported by the main cables of the bridge. The rotational movement of the
flaps is used to modify the aerodynamic forces acting on the deck and provides aerodynamic
forces on the flaps used to stabilize the bridge. A time-domain formulation of self-excited
forces for the whole 3D suspension bridge model is obtained through a Rational Function
Approximation (RFA) of the generalized Theodorsen function and implemented in the FEM
formulation. This paper lays the theoretical groundwork for the one that follows.
Key Words: Aerodynamic control, long span bridges, FEM analysis, flutter, passive control,
rational function approximation, time-domain analysis
1 Ph.D., Postdoctoral Fellow, School of Civil and Structural Eng., Nanyang Technological University, 50 Nanyang Avenue,
Singapore 639798, E-mail: cpiotr@ntu.edu.sg, Phone: 65-790-4853, Fax: 65-791-0046; formerly Postdoctoral Fellow, Dept.
of Civil Eng., University of Tokyo, Japan
2 Ph.D., Assistant Professor, Dept. of Civil Eng., Technical University of Gdansk, Ul. Narutowicza 11/12, 80-952 Gdansk,
Poland, E-mail: wild@pg.gda.pl, Phone: 48-58-347-2051, Fax: 48-58-347-2044; formerly Associate Professor, Dept. of
Civil Eng., University of Tokyo, Japan
3 Ph.D., Professor, Dept. of Civil Eng., University of Tokyo, Hongo 7-3-1, Bunkyo-ku, Tokyo 113-8656, Japan, E-mail:
fujino@bridge.t.u-tokyo.ac.jp, Phone: 81-3-5841-6095, Fax: 81-3-5841-7454
1
INTRODUCTION
Rapid technological progress in bridge engineering has recently led to the construction
of the Akashi Kaikyo Bridge in Japan and the Great Belt Bridge in Denmark, with main spans
of 1990 m and 1624 m, respectively. Although new records in terms of size of bridge
structures have been set, structure-wind interaction phenomena, static as well as dynamic,
which are increasingly important as spans become longer and bridge girders more flexible,
may prohibit further development of span length. For super long bridges, flutter instability is
most often a governing design criterion since it may lead to the total collapse of a structure.
To control the flutter of long span bridges, application of various passive and active
devices such as the Tuned Mass Damper (Nobuto et al. 1988) and the eccentric mass method
(Branceleoni 1992) has been studied, but no satisfactory solution has yet been achieved.
The aerodynamic control of bridge flutter was first proposed by Ostenfeld and Larsen
(1992). According to their concept, active control uses additional control surfaces attached
beneath both edges of the deck through aerodynamically shaped pylons. The rotational
displacement of the control surfaces is actively adjusted by feedback control such that the
generated aerodynamic forces provide a stabilizing action on the deck. The first reported
research on this control system has been given by Kobayashi and Nagaoka (1992). They
conducted a wind tunnel experiment on the sectional model of a bridge and obtained an
increase of flutter wind speed by a factor of 2. Wilde and Fujino (1998) carried out a
theoretical analysis of such a system. They applied Rational Function Approximation (RFA)
to model unsteady aerodynamics and derived a time-domain equation of motion for the
control system. The suggested variable-gain output feedback law guaranteed the system’s
stability and allowed application of different control strategies at low and high wind speed.
2
The high cost of building and maintaining active control systems motivated Wilde et
al. (1998) to investigate, both analytically and experimentally, the concept of a passive system
utilizing control surfaces. In their study, control surface motion was governed by an additional
pendulum attached to the center of gravity of the deck. The analysis revealed a maximum
increase in the critical wind speed of 57%. The experimental results showed very good
agreement with the theoretical prediction for small motions of the control surfaces. However,
for larger motions, considerable discrepancy was noticed and the actual critical wind speed
was even higher than the theoretical prediction.
Aerodynamic control may also be achieved by additional flaps attached directly to the
edges of the bridge deck. In this system, the flow pattern around the deck is affected by the
motion of the flaps, and thus the stabilizing action comes not only from the aerodynamic
forces generated on control flaps but can also be achieved through modification of the
aerodynamic forces induced on the bridge deck. This control system will be referred to as the
deck-flaps system.
The proposed passive control system (Fig. 1) consists of auxiliary flaps attached
directly to the bridge deck. When the deck undergoes pitching motion or relative horizontal
motion with respect to the main cables, control flap rotation is governed by additional control
cables spanned between the control flaps and an auxiliary transverse beam supported by the
main cables of the bridge. Since the control cables can only pull the flaps but not push them,
additional prestressed springs are used to reverse the motion of the control surfaces. The
system shown in Fig. 1a, referred to as an asymmetric cable connection system, can work
properly for wind coming from only one direction and requires alteration of its configuration
as the wind direction changes. The stabilizing action of the system shown in Fig. 1b, referred
to as a symmetric cable connection system, is, on the contrary, independent of wind direction.
3
The investigation of the proposed control system on the sectional model of a
suspension bridge was carried out by Omenzetter et al. (2000a, 2000b). In their study,
horizontal motions of the deck and the main cables were ignored. They derived a nonlinear
equation of motion of the control system to take into account the lack of compressive stiffness
of the control cables and discussed conditions under which linearization of the equation of
motion was permissible. The study of effectiveness of the proposed passive control showed
that the asymmetric cable connection system can greatly improve the critical wind speed,
much beyond practical needs. The improvement in critical wind speed for the symmetric cable
connection system was, on the other hand, limited.
Analysis of the proposed control method on a simple sectional model of the bridge
deck-flaps system has brought a great deal of qualitative information. In this model, however,
the contribution of the whole multimodal bridge dynamics and aerodynamic interaction
cannot be studied. Moreover, the efficiency of the deck-flaps control system cannot be fully
estimated. Therefore, the aim of this paper is to extend the investigation into the efficiency of
the passive flutter control by the bridge deck-flaps system by studying a full three dimensional
model of a suspension bridge. This first paper, Part I, is devoted to derivation of an
aerodynamic model of the whole suspension bridge with the deck-flaps control system. First,
the derivation of the time-domain model of self-excited forces of the deck-flaps system
sectional model is briefly reported on, as an indispensable point for further analysis. This
derivation makes use of RFA of the frequency-domain formulation of the aerodynamic forces.
The formulation is then extended to the full bridge model by means of FEM. Part I provides
the mathematical formalism for the problem, while the following paper, Part II, presents
results of the numerical analysis.
4
SECTIONAL MODEL OF THE DECK-FLAPS SYSTEM
A theoretical description of the self-excited aerodynamic forces of an oscillating
airplane wing was derived from potential flow theory by Theodorsen (1935). Theodorsen and
Garrick (1943) extended this solution to characterize the nonstationary flow about a wingaileron-tab combination. Both solutions describe the unsteady aerodynamic forces due to
steady-state oscillations in terms of the frequency-dependent Theodorsen circulatory function.
An extension of Theodorsen’s theory to arbitrary motions was presented by Edwards (1977).
He introduced the generalized Theodorsen function to describe self-excited aerodynamic
forces caused by the arbitrary motion of a wing.
Roger (1977) proposed a modeling method which can transform the aeroelastic
equation of motion of an airplane into a frequency-independent time-domain equation. This
method approximates aerodynamic force coefficients by rational functions of the Laplace
variable. The problem in Roger’s formulation has a relatively large number of newly added
aerodynamic states, and modifications of his method were proposed by Dunn (1980) and
Karpel (1981). The application of RFA for flutter analysis of bridges of various cross-sections
was first reported by Wilde et al. (1996).
The model for self-excited aerodynamic forces for a sectional model of the bridge
deck-flaps system was proposed by Omenzetter et al. (2000a, 2000b). They used the
frequency-domain solution of Theodorsen and Garrick (1943) and developed, using RFA, its
time-domain counterpart.
The cross-section of a bridge deck with control flaps is shown in Fig. 2. The deck
together with flaps has a chord width of 2b and the distance between main cables axes is
denoted by 2bc . The length of the hanger cables is denoted by lh ( lh1 and lh2 at each end of the
finite element, respectively), and ah is the distance between the deck elastic center and hanger
5
ends. The leading and trailing flap hinges are positioned with respect to the deck elastic center
at r and r , respectively.
The displacement vector of the element cross-section is selected as

x cs  hd b vd b 
 
hc1 b vc1 b hc 2 b vc 2 b  h1  h2 
(1)
where hd denotes horizontal displacement of the deck, v d vertical displacement of the deck,
 torsional rotation of the deck,  and  relative rotations of the flaps, hc1 and vc 1 horizontal
and vertical displacements of cable 1, hc2 and vc 2 horizontal and vertical displacements of
cable 2, and  h1 and  h2 inclination angles of the hanger cables. The apostrophe character (‘)
is used throughout the paper to denote transposition of a vector or matrix. The corresponding
vector of forces acting on the cross-section is denoted by Xcs .
In a simplified sectional model of the deck-flaps system the horizontal displacements
are neglected and the hanger cables are assumed inextensible, hence the vector of degrees of
freedom becomes

x s  vd b 
 
(2)
The corresponding vector of generalized forces is
s
X  Xv d b X

X
X

(3)
where Xv d is a lift force, and X , X  , and X are torsional moments acting on respective
degrees of freedom.
The equation of motion of the sectional model becomes
M ss x s  Css x s  K ss x s  X s
6
(4)
M ss , C ss , and K ss are system mass, damping, and stiffness matrices, respectively. The
dynamics of the control flaps is included and the matrix M ss becomes:
md  2mc b 2

s
M s  

 sym.
S b
S b
I  2m b
2
c c
r S  I
I
S b 
r S  I 
0 

I

(5)
The mass properties appearing in (5) are as follows: md is the mass of the deck together with
the flaps, mc is the mass of the main cable, S is the first order moment of inertia of the deck
together with the flaps about the deck elastic center, S and S are the first order moments of
inertia of the leading and trailing flaps about their hinges, respectively, I is the second order
moment of inertia of the deck together with the flaps about the deck elastic center, and I and
I are the second order moments of inertia of the leading and trailing flaps about their hinges,
respectively.


The damping and stiffness matrices are diagonal, i.e. Cs  diag cv d b , c , c , c ,
s

2

Kss  diag kvd b2 , k , k , k , with the entries corresponding to damping and stiffness
coefficients of respective degrees of freedom.
In this study, the only forces considered to be acting on the system are the self-excited
forces. The formula for the self-excited forces of the deck-flaps system can be written in
Laplace domain as (Omenzetter et al. 2000a)
2
2

 s
U ˜ s
U  ˜ s
U 
U 
ˆ
ˆ
L Xs   M sa s2   
C
s

K

C

s

RS
s

C

s

RS
L x 
a
a
2
1
b 
 b 
b 

b 


(6)
˜ s, K
˜ s , R , S , and S
U is the mean velocity of an oncoming wind. The matrices M sa , C
1
2
a
a
depend on system geometry, namely the location of the deck rotation center, flap size and
7
location of flap hinges. The procedure to obtain them is described in Omenzetter et al.
(2000a); expanded formulas are not shown here due to their length and complexity.
The function Csˆ appearing in (6) is the generalized Theodorsen function (Edwards
1977). The generalized Theodorsen function is approximated by rational functions of the
nondimensionalized Laplace variable as
n
l
a
C˜ sˆ  a0   i , sˆ  sb /U
ˆ  bi
i 1 s
(7)
where C˜ sˆ denotes approximation. The partial fractions, ai / sˆ  bi , are called lag terms,
because each represents a transfer function in which the output lags behind the input and
permits an approximation of the time delays inherent in unsteady aerodynamics. The
coefficients of the partial fractions, bi , are referred to as lag coefficients. Addition of each
partial fraction introduces into the resulting state-space realization new states referred to as
aerodynamic states. The number of partial fractions is denoted by nl , and is found as a
compromise between the precision of the approximation and the size of the state-space
realization.
The aerodynamic states are defined in Laplace domain as
1
 U  s  U  s
L x   sI 
R
E L x s 
 b  b 
 b  b
s
a
(8)
The matrices appearing in (8) have the forms
E sb  [b1 ,...,bnl ]S2  [1,...,1]S1
(9a)
nl times
and

Rsb  diag b1,...,bnl
8

(9b)
Introducing the approximation (7) and the aerodynamic states (8) into the formula for
the self-excited forces (6), and then taking the inverse Laplace transformation of (6) and (8),
yield the time-domain formula for self-excited forces
U
X s  M as x s  
b
2
2
 s s U  s s U  s s
 Ca x    K a x    Q x a

b
b
U
x as  
b
 s s U
 Rb xa  

b
 s s
 Eb x

(10a)
(10b)
where
˜ s  a RS 
˜ s  a RS , K s  K
C C
a
a
0
1
a
0
2
s
a
nl
 a RS ,
i
2
Q  R[a1,...,an l ]
s
(10c-e)
i1
The state-space equation of motion becomes


0
I
s
x  
 s  U 2 s 
1
 s 
s
s 1

x


M

M
 s a  K s   b  K a    M ss  M as  Css   Ub
 



 x as  
  
U  s

0
  Eb

b


0
  xs 
2


1

U
 s
 
s
s
s  s
 x 
C
M

M
Q
a
 a    s
  b
 s
 xa 
U  s

  Rb

b
(11)
The augmented state vector contains aerodynamic states, x sa . In this RFA formulation addition
of one lag term results in addition of only one new aerodynamic state.
FEM MODEL OF A SUSPENSION BRIDGE WITH THE DECK-FLAPS SYSTEM
FEM flutter analyses of long-span bridges have been conducted by numerous authors
(e.g. Agar 1988, Agar 1989, Astiz 1998, Namini 1991, Namini et al. 1992, Pfeil and Batista
1995). However, they all employed a frequency-domain formulation of unsteady aerodynamic
forces and as a result had to use various computationally burdensome techniques to solve for
9
the critical wind speed. The present study attempts to derive an FEM formulation of the
bridge deck flutter problem with time-domain formulation of aerodynamic forces.
Structural Modeling
A full bridge model, in which each structural element, i.e. the towers, the main cables,
the deck and the hangers, is modeled using separate finite elements, requires a large number
of degrees of freedom, and hence results in a very large size of system matrices as well as
significant computational burden. While such a detailed analysis is indispensable for the
structural design of a bridge, it results in a flutter analysis that is needlessly complicated.
Simplified structural models of a suspension bridge were proposed by Abdel-Ghaffar (1978,
1979, 1980), for horizontal, torsional and vertical vibrations. His models offer a significant
reduction of computational burden and at the same time retain features of the structure
relevant for flutter analysis. In the present study, the structural modeling of a suspension
bridge according to Abdel-Ghaffar (1978, 1979, 1980) is extended to model a bridge deckflaps system with time-domain formulation of self-excited aerodynamic forces.
The simplified suspension bridge model (Abdel-Ghaffar 1978, 1979, 1980) is derived
under the following assumptions:

all stresses in the bridge follow Hooke’s law

the dead load of the structure is carried only by the main cables

the main cables are of uniform cross-section and of a parabolic profile under dead load

the cables are perfectly flexible

the vibrational hanger forces are considered to be distributed loads, as if the distance
between the hangers were very small

the hangers are inextensible
10

vibrational displacements from static equilibrium are small

longitudinal displacements of the girder are ignored
In the present analysis additional assumptions are included as follows:

the mass of the towers as well as their bending and torsional stiffness are ignored

energy due to warping of the girder is neglected
Structural degrees of freedom of the bridge deck-flaps finite element are shown in Fig.
3. The vector of element structural degrees of freedom, q e , is of size 18. The corresponding
vector of element generalized nodal forces is denoted as Xe .
The entries of the vector of displacements of the cross-section, x cs , are continuous
functions defined over the element length. The relationship between the vector of
displacements of the cross-section, x cs , and the nodal displacement vector, q e , is through the
shape function matrix, N ,
x cs  Nqe
(12)
The shape functions are chosen as third order polynomials for bending of the girder and first
order polynomials for torsion of the girder, lateral displacements of the main cables and
relative rotations of the flaps. The shape function matrix is shown in Appendix I.
The kinetic energy of the whole suspension bridge, T , can be found as a summation of
e
contributions from each of all the finite elements of the model, T ,
T
T
e
(13)
e
e
The kinetic energy, T , of a finite element of length L may be evaluated as:
L
1 cs cs cs
T   x M s x dx
20
e
11
(14)
where M css is the mass matrix of the element cross-section:
md b2





M css  






0
md b 2
0
S b
I
0
Sb
r S  I
I
0
S b
r S  I
0
I
0
0
0
0
0
mc b2
0
0
0
0
0
0
mc b2
0
0
0
0
0
0
0
mcb 2
sym.
0
0
0
0
0
0
0
0
mcb 2
0
0
0
0
0
0
0
0
0
0
0 
0 
0 
0 

0

0  (15)
0 
0 
0 
0 
0 

Thus, the element structural mass matrix, M es , is evaluated as
L
M 
e
s
 NM Ndx
cs
s
(16)
0
and shown in Appendix I.
Unlike the kinetic energy (13), the potential energy of the whole structure, V , cannot
be simply evaluated as a sum of contributions from each finite element. Instead, it can be
expressed as two summands, V and V˜ ,
V  V  V˜
(17)
V is the potential energy due to elastic deformations of the girder and hanger cables, rotations
of the flaps, as well as potential energy arising from gravity. It can be decomposed into
e
additive contributions from each finite element, V ,
V
V
e
e
e
The potential energy V stored in a finite element may be evaluated as
12
(18)
L
1
V 
2
e

 Lx  K
cs
cs
s
Lx cs dx
(19)
0
where matrix K css is diagonal and given as


Kcss  diag EI yb 2, EI zb 2, GI , k , k , H, H, H, H, md g 2, md g 2
(20)
In Equation (20), EIy represents bending rigidity of the girder in the xz plane, EIz bending
rigidity of the girder in the xy plane, GI torsional rigidity for twisting with respect to the x
axis, k  and k stiffness of the springs at the deck-flap connections, H tension force in the
main cable, and g gravity acceleration, respectively. Note that the displacement vector, x cs ,
cs
enters Equation (19) in the form of derivatives of its entries, Lx , where L is the matrix
ordinary differential operator shown in Appendix I.
Thus, the element stiffness matrix, K es , associated with the potential energy V e , is
evaluated as
L
K 
e
s

LN  K css LNdx
(21)
0
and shown in Appendix I.
V˜ is the potential energy due to elastic deformations of the main cables. It needs to be
expressed for the whole system as
2 
2


m

m
g

1
E
A
b


d
c
c c

V˜ 

2 L˜ 
 2H
 


e

 
˜ q  
N
 
e
c1
e

e
 
˜ q  
N
 
e
c1
e

e

 
˜ q  
N
 
e
c2
e

e


˜ q  (22)
N


e
c2
e
In Equation (22), Ec Ac is the axial stiffness of the main cable. The virtual length of the cable,
L˜ , is defined as
13
Lc
L˜ 

0
dx
3
cos 
(23)
where  is the angle of the main cable inclination, and Lc is the total main cable length.
˜ e and N
˜ e are given as
Element matrices N
c1
c2
L
L

˜ e  T Ndx , N
˜e 
N
c1
v c1
c2
0
T
vc2
(24a, b)
Ndx
0
˜ e and N
˜ e , as well as T and T are shown in Appendix I.
Matrices N
vc 1
vc 2
c1
c2
Aerodynamic Modeling
In the process of extension of the sectional model of aerodynamic forces into a 3D
case, the only forces that are considered are those derived previously for the sectional model.
In other words, the self-excited drag force acting on the deck, as well as forces induced by
horizontal motions of the deck and the main cables are all ignored. Also, static wind forces
due to mean wind velocity component, as well as buffeting forces are neglected in this study.
The wind induced forces acting on an element cross-section can be written as
U
X  M x 
b
cs
cs
a
cs
2
2
 cs cs  U  cs cs  U  cs cs
 Ca x    K a x    Q x a

b
b
(25)
where x csa are aerodynamic states defined by the equation
U
x csa  
b
 s cs  U
 Rb xa  

b
 cs cs
 Eb x

The matrices appearing in (25) and (26) are as follows
14
(26)
011
M  041
0
 61
cs
a
01 6 
0 46 ,
0 66 

01 4
Msa
06 4
011
C  0 41
0
 61
cs
a
01 6 
0 46 ,
06 6 

014
Csa
06 4


cs
Q cs  01n l  Q s 06 n l  , E b  0nl 1



011 01 4
K  04 1 K sa
0
 61 06 4
01 6 
0 4 6 (27a-c)
0 66 

cs
a
Esb
0 nl 6

(27d, e)
Matrix R sb is given by (9b).
Due to the RFA of aerodynamic forces, the degrees of freedom of an element are
augmented by the element aerodynamic states
e
qa  qa,11

qa,1 2
qa, n l 1 qa, n l 2

(28)
where q ea is related to x csa (26) through the shape function matrix, N a ,
x csa  Naqea
(29)
The shape functions for aerodynamic states were selected as first order polynomials, and
matrix N a is shown in Appendix I.
The formula for the self-excited element generalized nodal forces can be stated as
U
Xe  M ea q e  
b
2
 e e U  e e
e e
 Ca q    K a q  Q q a

b
U
Eea q ea  
b
 e e U
 R bq a  

b
(30a)
 e e
 Eb q

(30b)
where the element aerodynamic matrices are given as
L
M 
e
a
 NM Ndx ,
cs
a
0
L
C 
e
a
 NC Ndx ,
cs
a
L
K 
e
a
0
 NK Ndx ,
cs
a
0
15
L
Q 
e
 NQ N dx (31a-d)
cs
a
0
L
E 
e
a

N a Na dx , Reb 
0
L

Na  Rsb Na dx , E eb 
0
L
N
a
Ecs Ndx
b
(31e-g)
0
The element aerodynamic matrices (31) are shown in Appendix I.
After the assemblage procedure and application of appropriate boundary conditions,
the global state-space equation of motion becomes


0
I

r  
2
 r     M  M 1 K   U  K   M  M 1 C   U
 s
a 
s
a  s
  a

    s
b
b





ra  
 U  1

0
  E a Eb

b


0
 r
2
 
1
  U 
 
 Ca     M s  M a  Q   r 
  b

 ra 
 U  1

  Ea R b

b
(32)
where r is the vector of global displacements and ra the vector of global aerodynamic states.
Matrices M s , C s and K s are global structural mass, damping and stiffness matrices,
respectively, whereas M a , Ca , K a , E a , E b and Rb are global aerodynamic matrices.
Equation (32) represents an extension of sectional state-space equation of motion (11) to the
full 3D aeroelastic bridge model.
PASSIVE AERODYNAMIC CONTROL
A study of the proposed passive control system on the sectional model was conducted
by Omenzetter et al. (2000a, 2000b). They derived a nonlinear equation of motion to account
for the lack of compressive stiffness of the control cables, and assumed finite bending
stiffness of the supporting beam. Through a study of the system response under wind
excitation, they showed that for sufficiently large prestressing moments at the deck-flap
connections, the control cable may always remain taut. Moreover, if the axial stiffness of the
control cables as well as bending stiffness of the supporting beam are large enough, flexural
16
deformations of these members can be ignored. Sufficient values of the prestressing moments
and members’ stiffness, to satisfy the aforementioned assumptions, were assessed and found
feasible.
A cross-section of the passive bridge deck-flaps system is shown in Fig. 4. The basic
geometric dimensions are the same as in Fig. 2. Additionally, the support points of the leading
and trailing flap by control cables are positioned at rc and rc , and the support points of the
additional cables on the transverse beam at xc  and xc . The only difference regarding
assumed displacement of the cross-section is such that, due to the presence of a transverse
beam of large axial stiffness, the independent horizontal motions of the cables, hc1 and hc2 ,
are now replaced by a common one, hc .
Assuming that the control cables are always taut and inextensible, and the bending
stiffness of the transverse beam is infinite, the formulas for flap rotations are:
  t  
t
lh  ah
hd  hc  ,
  t  
t
lh  ah
hd  hc 
(33a, b)
where control gains t  and t are determined from the geometry of the control system as
t 
rc  xc
rc  r
, t  
rc  xc
rc  r
(34a, b)
Sectional Model of Passive Control System
In the sectional study, the horizontal displacements are ignored. In such a case, the
rotations of the flaps are proportional to the rotation of the deck:
  t  ,   t 
where t  and t are the control gains given in (34).
17
(35a, b)
Thus, in this constrained coordinate system, the vector of degrees of freedom is

x s, c  v d b 
(36)
and the following relationship between the constrained coordinates (36) and unconstrained
ones (2) holds
x s  Ts xs, c
(37)
 1 0 0 0 
T s  
0 1 t  t  

(38)
Matrix T s is given as
The matrices entering state-space equation of motion for the constrained system can be
evaluated as
s, c
s
s s
s, c
s s s
s, c
s s s
M s  T M s T , Cs  T Cs T , K s  T K s T
(39a-c)
s, c
s
s s
s, c
s s s
s, c
s s s
M a  T M a T , Ca  T Ca T , K a  T K a T
(39d-f)

Q s, c  T s Q s , E s,b c  E sbT s
(39g, h)
s,c
and Rb is the same as given by (9b).
FEM Model of Passive Control System
It is assumed that the sectional constraints (33) result in the following constraints for the
element degrees of freedom:
qe  Te qe , c
18
(40)
where qe , c is the vector of degrees of freedom of passive control finite element. This vector is
of size 12 and can be obtained from the vector of element degrees of freedom of the
unconstrained element, q e (Fig. 3), by neglecting the entries corresponding to flap rotations
and substituting the independent horizontal degrees of freedom of the main cables with a
common one. Matrix Te is shown in Appendix I.
The passive control element matrices are given by the following expressions
e, c
e
e e
e, c
e
e e
M s  T M sT , K a  T K a T
(41a, b)
˜ e, c  N
˜ e Te , N
˜ e, c  N
˜ e Te
N
c1
c1
c2
c2
(41c, d)
e, c
e
e e
e, c
e e e
e, c
e
e e
M a  T M a T , Ca  T Ca T , K a  T K a T
(41e-g)

Q e, c  T e Qe , E e,b c  E ebTe
(41h, i)
The aerodynamic states are not subjected to any constraints, and therefore matrices E ea (31e)
e
and R b (31f) are the same for the controlled system as well as the uncontrolled one.
CONCLUSIONS
The derivation of an aerodynamic model of the whole suspension bridge with the
deck-flaps control system is presented in this paper. First, the derivation of the time-domain
model of self-excited forces of the sectional model of the deck-flaps system is briefly
explained. This derivation makes use of RFA of the frequency-domain formulation of the
aerodynamic forces. The formulation is then extended to the full bridge model by means of
FEM. In Part II of this paper, numerical simulations of the passive deck-flaps control systems
are conducted and discussion of the results is provided.
19
APPENDIX I. FEM MATRICES
The assumed polynomial shape functions can be expressed as combinations of the
following two linear functions
1 x   1 
x
x
, 2 x  
L
L
(42a, b)
and the shape function matrix, N ,is as follows (refer to (12))
 2
 3  21 
0
 1 

2
0
 L1 2

0
12 3  21 

0
L12 2

0
0


0
0

0
0


0
0


0
0
N 
 2
 3  22 
0
 2

 L122
0


0
22 3  22 

0
 L122

0
0








0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
1
0
0
0
0
0
1
0
0
1
0
0
0
0
0
1
0
0
0
0
0
0
12 3  21  0
L12 2
0
0
0
bc

2b 1
0
0
0
0
0
0
0
1
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
22 3  22 
L122
2
0
0

bc

2b 1
0
0
12 3  21 
L122
0
0
0
0
0
0
2
0
0 2
0 2 3  22 
0
 L122
b
0
 c 2
2b
0
0
0
0
0
0
0
0
0
2
0
0
0
0
0
0
2
0
0
0
bc

2b 2
0
0
0
0
0
0
2
0
0



 
 
b 1  2 12 3  21  b 1  2 12 3  21 

lh1 lh2 
lh1 lh2 

1 2  2
1 2  2
bL  1 2
bL  1 2 
lh1 lh2 
lh1 lh2 

0
0

0
0



  
  
ah  1  2 1
ah  1  2 1

l h1 lh2 
lh1 lh2 

0
0

0
0


 

b 1  2 1
0
lh1 lh2 


 

0
b 1  2 1
l h1 lh2 

1 2  2
1 2  2

 b  2 3  22  b   2 3  22 
lh1 lh2 
lh1 l h2 

1 2  2
1 2  2

bL  12
bL  12

lh1 lh2 
lh1 lh2 

0
0

0
0

1 2 
1 2 
 ah   2
 ah   2

lh1 lh2 
lh1 lh2 

0
0

0
0


 
b 1  2 2
0

lh1 lh2 


 

0
b 1  2 2
lh1 lh2 

(43)
The matrix ordinary differential operator, L , is diagonal and given as follows (refer to (19))
 d 2 d 2 d
d d d d

L  diag 2 , 2 , , 1,1, , , , ,1, 1
dx dx dx

dx dx dx dx
The shape function matrix for aerodynamic states, N a , is as follows (refer to (29))
20
(44)
1
0
N a  
0

2
0
0
0
0
1 2
0
0
0
1
0
0 1
0 2

2 
nl
(45)
The element structural matrices are as follows (non-listed entries are 0 ):
symmetric mass matrix M es (refer to (16))
Mse 1,1  Mse 10,10 
Mse 1,10 
13md b 2 L
11md b2 L2
, Mse 1,2   Mse 10,11 
35
210
9md b2 L
13md b2 L2
m b2 L3
, Mse 1,11   Mse 2,10  
, Mse 2,2  d
70
420
105
Mse 2,11  
13md  2mc b 2 L
md b 2 L3
, Mse 3,3  Mse 12,12  
140
35
11md  2mc b2 L2
7S bL
, Mse 3,5  Mse 12,14   
M 3, 4   M 12,13 
210
20
e
s
e
s
Mse 3,6   Mse 12,15 
Mse 3,12  
7S bL
20
, Mse 3, 7  Mse 12,16 
7S bL
20
9md  2mc b2 L
13md  2mc b 2 L2
, Mse 3,13  Mse 4,12  
70
420
Mse 3,14   Mse 5,12  
Mse 3,16   Mes 7,12 
3S bL
3S bL
, Mse 3,15  Mse 6,12   
20
20
3S bL
m  2mc b 2L3
, Mse 4,4   Mse 13,13  d
20
105
S  bL2
S bL2
e
e
, M s  4, 6    M s 13,15  
M 4,5  M 13,14 
20
20
e
s
e
s
21
S bL2
md  2mc b 2 L3

e
,
M 4,7  M 13,16 
Ms 4,13  
20
140
e
s
e
s
S bL2
S bL2
e
e
,
Ms 4,15  Ms 6,13 
M 4,14    M 5,13 
30
30
e
s
e
s
2I  mc bc2 L
S bL2

e
e
, Ms 5,5  Ms 14,14 
M 4,16   M 7,13 
30
6
e
s
e
s
Mse 5,6  Mse 14,15 
Mse 5,14 
r S
 

 I L
3
2I  m b L ,
Mse 5,16  Mse 7,14 
2
c c
12
r S



 I L
6
, Mse 5,7  Mse 14,16 
Mse 5,15  Mse 6,14 
, Mse 6,6  Mse 15,15 
Mse 7,7  Mse 16,16 
 



 I L
3

 I L
6
I L
3
, Mse 6,15 
I L
6
I L
I L
, Mse 7,16  
3
6
Mse 8,8  Mse 9,9  Mse 17,17  Mse 18,18 
Mse 8,17  Mse 9,18 
r S
r S
mcb 2 L
3
mc b2 L
m b2 L3
, Mse 11,11  d
6
105
symmetric stiffness matrix K es (refer to (21))
K 1,1 
e
s
12EIy b 2
L3
10 3 md gb 2 L
6EI y b2 15 7 md gb2 L2
e




   
, Ks 1,2  
lh1 lh2  35
L2
lh1 lh2  420
16 5 m gbah L
16 5 m gb 2 L
e
e
Kse 1,5     d
, Ks 1,8  Ks 1,9     d
lh1 lh2  60
lh1 lh2  120
22
(46)
Kse 1,10   
12EI yb 2
L3
 1
6EIy b2  7 6 md gb 2 L2
1 9m gb 2 L
, Kse 1,11 
    d
   
lh1 lh2  140
L2
lh1 lh2  420
 5
 5
4 m gbah L
4  m gb2 L
, Kse 1,17  Kse 1,18     d
Kse 1,14      d
lh1 lh2  60
lh1 lh2  120
Kse 2,2 
4EIy b 2
L
 5
 2
3  m gb2 L3
1 m gbah L2
, Kse 2,5     d
    d
lh1 lh2  840
lh1 lh2  60
 2 1 m gb 2 L2
6EIy b 2  6
7 
md gb2 L2
e


Kse 2,8  Kse 2,9     d


, Ks 2,10  
lh1 lh2  120
L2
l h1 lh2  420
Kse 2,11 
2EIy b 2
L
 1 1 m gb2 L3
 1 1 m gbah L2
e
    d
, Ks 2,14     d
lh1 l h2  280
lh1 l h2  60
 1 1 m gb 2 L2
Kse 2,17  Kse 2,18  K se 8,11   Kse 9,11     d
lh1 lh2  120
Kse 3,3  Kse 12,12    Kse 3,12 
12EI zb2 12Hb2

L3
5L
Kse 3, 4  Kse 3,13  K se 4,12  K se 12,13 
Kse 4, 4  Kse 13,13 
Kse 5,5 
6EIz b2 Hb2

L2
5
4EIz b2 4Hb2 L
2EI zb 2 Hb2 L
, Kse 4,13 


L
15
L
15
 3 1 m gbah L
GI Hbc2 
3
1 m ga2 L

    d h , Kse 5,8  Kse 5,9     d
L
2L lh1 lh2  12
lh1 lh 2  24
 4
 1
5 m gbah L
1 m gbah L2
e
Kse 5,10     d
, Ks 5,11     d
lh1 lh2  60
lh1 lh2  60
23
Kse 5,14  
GI Hbc2 
1 1 m ga2 L

    d h
L
2L lh1 l h2  12
 1
1 m gbah L
Kse 5,17  K se 5,18  Kse 8,14  Kse 9,14      d
lh1 lh2  24
Kse 6,6  Kse 15,15 
Kse 8,8  Kse 9,9 
k L
k L
kL
kL
, Kse 6,15   , Kse 7,7  Kse 16,16    , Kse 7,16  
3
6
3
6
 4 5 m gb 2 L
Hb2 
3 1 m gb 2 L
e
e
    d
, Ks 8,10  Ks 9,10     d
L
lh1 lh2  24
lh1 lh1  120
Kse 8,17  K es 9,18  
K 10,10  
e
s
Hb2 
1 1  m gb2 L
    d
L
lh1 lh1  24
 3 10 md gb 2 L
6EI yb 2  7 15 md gb2 L2
e




   
, Ks 10,11  
lh1 lh1  35
L2
l h1 lh1  420
12EIy b 2
L3
 5 16 m gbah L
 5 16 m gb 2 L
e
e
Kse 10,14     d
, Ks 10,17  Ks 10,18     d
lh1 lh1  60
l h1 lh1  120
K 11,11 
e
s
4EI yb 2
L
 3 5  m gb2 L3
 1 2  m gbah L2
e
    d
, Ks 11,14      d
lh1 lh1  840
lh1 lh1  60
 1 2 m gb 2 L2
GI
Hbc2 
1 3  m ga2 L
e
Kse 11,17  Kse 11,18     d
    d h
, Ks 14,14   
lh1 l h1  120
L
2L lh1 lh1  12
 1 3 m gbah L
Kse 14,17  Kse 14,18     d
lh1 lh1  24
Kse 17,17  Kse 18,18 
Hb2 
1 3  m gb2 L
    d
L
lh1 lh1  24
˜e , N
˜ e , T and T are as follows (refer to (24))
Matrices N
vc 1
vc 2
c1
c2
24
(47)
˜ e  0 0
N
c1


˜ e  0 0
N
c2


L2
bL
 c
12
4b
L
2
L2
12
L
2
bc L
4b
L2
12
0 0 0 0 0 0
L
2

0 0 0 0 0 0
L
2
L2

12

bc L
4b
bc L
4b

0 0 0 0 (48a)


0 0 0 0  (48b)

Tvc 1  0 0 0 0 0 0 1 0 0 0 0 
(48c)
Tvc 2  0 0 0 0 0 0 0 0 1 0 0 
(48d)
The element aerodynamic matrices are as follows (refer to (31)); (non-listed entries are
0 ):
symmetric matrix M ea
13Macs22 L
11Macs22 L2
e
e
, Ma 3, 4   Ma 12,13  
M 3,3  M 12,12  
35
210
e
a
e
a
7Macs23 L
7Macs24 L
e
e
, Ma 3,6   Ma 12,15 
M 3,5  M 12,14  
20
20
e
a
e
a
7Macs25 L
9Macs22 L
e
, Ma 3,12  
M 3, 7  M 12,16 
20
70
e
a
e
a
Mae 3,13  Mae 4,12  
13Macs22 L2
3Macs23 L
, Mae 3,14   Mae 5,12  
420
20
3Macs24 L
3Macs25 L
e
e
, Ma 3,16   Ma 7,12 
M 3,15  M 6,12  
20
20
e
a
e
a
Macs22 L3
Macs23 L2
e
e
M 4,4   M 13,13 
, Ma 4,5  Ma 13,14 
105
20
e
a
e
a
Macs24 L2
Macs25 L2
e
e
, Ma 4,7  M a 13,16 
M 4,6   M 13,15 
20
20
e
a
e
a
25
Macs22 L3
Macs23 L2
Macs24 L2
e
e
e
e
, Ma 4,14    Ma 5,13 
, Ma 4,15  Ma 6,13 
M 4,13  
140
30
30
e
a
7M csa 25 L2
Macs33 L
Macs34 L
e
e
e
e
, Ma 5,5  Ma 14,14 
, Ma 5,6  Ma 14,15 
M 4,16 
20
3
3
e
a
Macs35 L
Macs33 L
Macs34 L
e
e
e
, Ma 5,14 
, Ma 5,15  Ma 6,14 
M 5,7  M 14,16 
3
6
6
e
a
e
a
Mae 5,16  Mae 7,14 
Macs35 L
M cs L
, Mae 6,6  Mae 15,15  a 44
6
3
Macs45 L
Macs44 L
Macs45 L
e
e
e
, Ma 6,15 
, Ma 6,16  Ma 7,15 
M 6,7  M 15,16 
3
6
6
e
a
e
a
Macs55 L
Macs25 L2
Macs55 L
e
e
, Ma 7,13  
, Ma 7,16 
M 7,7  M 16,16 
3
30
6
e
a
e
a
(49)
matrix C ea
Cae 3,3  Cae 12,12 
13Cacs22 L
11Cacs22 L2
, Cae 3, 4  Cae 4,3  Cae 12,13  Cae 13,12 
35
210
7Cacs23 L
7Ccsa 24 L
e
e
, Ca 3,6  Ca 12,15 
C 3,5  C 12,14  
20
20
e
a
e
a
7Cacs25L
9Cacs22 L
e
e
, Ca 3,12   Ca 12,3 
C 3,7  C 4,16  C 12,16 
20
70
e
a
e
a
e
a
13Cacs22 L2
3Cacs23 L
e
e
, Ca 3,14   Ca 12,5 
C 3,13  C 4,12  C 12, 4  C 13,3  
420
20
e
a
e
a
e
a
Cae 3,15  Cae 12,6  
e
a
3Cacs24 L
3Ccs L
, Cae 3,16   Cae 12,7  a 25
20
20
Cacs22 L3
Cacs23 L2
e
e
, Ca 4,5  Ca 13,14 
C 4,4   C 13,13 
105
20
e
a
e
a
26
Cacs24 L2
Cacs25 L2
e
e
, Ca 4,7  Ca 13,16 
C 4,6  C 13,15 
20
20
e
a
e
a
Cacs22 L3
Cacs23 L2
e
e
, Ca 4,14   Ca 13,5 
C 4,13  C 13,4  
140
30
e
a
e
a
Cacs24 L2
7Cacs32 L
e
e
, Ca 5,3  Ca 14,12  
C 4,15  C 13,6 
30
20
e
a
e
a
Cae 5,4  Cae 14,13 
Cacs32 L2
C cs L
, Cae 5,5  Cae 14,14  a 33
20
3
Cacs34 L
Cacs35 L
e
e
, Ca 5,7  Ca 14,16 
C 5,6  C 14,15 
3
3
e
a
e
a
3Cacs32 L
Cacs32 L2
e
e
, Ca 5,13  Ca 14,4  
C 5,12  C 14,3 
20
30
e
a
e
a
Cacs33 L
Cacs34 L
e
e
, Ca 5,15  Ca 14,6 
C 5,14  C 14,5 
6
6
e
a
e
a
Cae 5,16  Cae 14,7 
Cacs35 L
7Ccsa 42 L
, Cae 6,3  Cae 15,12  
6
20
Cacs42 L2
Cacs43 L
e
e
, Ca 6,5  Ca 15,14  
C 6,4  C 15,13 
20
3
e
a
e
a
Ccsa 44 L
Cacs45 L
e
e
, Ca 6,6  Ca 15,16 
C 6,6  C 15,15 
3
3
e
a
e
a
3Cacs42 L
Cacs42 L2
e
e
, Ca 6,13  Ca 15,4  
C 6,12  C 15,3 
20
30
e
a
e
a
Cae 6,14  Cae 15,5 
Cacs43 L
Ccs L
, Cae 6,15  Cae 15,6   a 44
6
6
27
Cacs45 L
7Cacs52 L2
e
e
, Ca 7,3  Ca 16, 4 
C 6,16  C 15,7 
6
20
e
a
e
a
Cacs52 L2
Cacs53 L
e
e
, Ca 7,5  Ca 16,14 
C 7,4  C 16,13 
20
3
e
a
e
a
Cacs54 L
Cacs55 L
e
e
, Ca 7,7  Ca 16,16 
C 7,6  C 16,15 
3
3
e
a
e
a
Cae 7,12  Cae 16,3 
3Cacs52 L
C cs L2
C cs L
, Cae 7,13   a 52 , Cae 7,14  Cae 16,5  a 53
20
30
6
Cacs54 L
Cacs55 L
e
e
, Ca 7,16  Ca 16,7 
C 7,15  C 16,6 
6
6
e
a
e
a
Cacs25 L2
7Cacs52 L
e
, Ca 16,12 
C 13, 7  
30
20
e
a
(50)
matrix K ea : its entries can be obtained from (50) by replacing Cacsij with Kacsij .
L
3



E ea 






L
6
L
3
0
0
0
0
0
0
0
0
L
3
sym.
28
0 

0 


0

0
L 
6 
L 

3 
(51)
Rb 11L
 3



Reb  





Rb11 L
6
Rb11 L
 0
 0
cs
 7Q21 L
 20
cs 2
 Q21 L
 20
cs
 Q31 L
 3cs
Q L
 41
3
 cs
Q51 L

3
 0
 0
Q e  
0

0
 3Q cs L
21

20
 Q cs L2
21
 30
cs
 Q31
L
 6
cs
 Q41
L
 6
cs
 Q51
L
 6
 0
 0
3
0
0
0
0
0
0
0
0
Rb n l n l L
3
sym.
0
0
cs
3Q21
L
20
cs 2
Q21
L
30
cs
Q31
L
6
Qcs41L
6
cs
Q51
L
6
0
0
0
0
cs
7Q21
L
20
cs 2
Q21
L

20
cs
Q31
L
3
Qcs41L
3
cs
Q51
L
3
0
0
0
0
cs
7Q2n
L
l
20
cs 2
Q2n
L
l
20
cs
Q3n l L
3
cs
Q4n
L
l
3
Q5ncsl L
3
0
0
0
0
cs
3Q2n
L
l
20
cs 2
Q2n
L
l

30
Q3ncsl L
6
cs
Q4n
L
l
6
Q5ncsl L
6
0
0
29


0 


0

0 
Rb n l n l L

6 
Rb n l n l L

3 
0





30

cs
Q3nl L

6 
cs
Q4n
L
l

6
cs
Q5n
L 
l

6
0 
0 

0

0
cs
7Q2n l L 

20
cs 2 
Q2n l L

20 
cs
Q3nl L 
3 
cs
Q4n l L 
3 
cs
Q5n
L 
l
3 
0 
0 
(52)
0
0
cs
3Q2n
L
l
20
cs 2
Q2n
L
l
(53)
 0
 0
cs
7E12 L
 20
cs 2
 E12 L
 20
cs
 E13 L
 3cs
E L
 14
3
 cs
E15 L

3
 0
 0
e
E b  
0

0
3E cs L
12

20
 E cs L2
12
 30
 E13cs L
 6
 E14cs L
 6
 E15cs L
 6
 0
 0
0
0
cs
3E12
L
20
E12cs L2
30
E13cs L
6
E14cs L
6
E15cs L
6
0
0
0
0
7E12cs L
20
E12cs L2

20
E13cs L
3
E14cs L
3
E15cs L
3
0
0
0
0
7E ncsl 2 L
20
Encsl 2 L2
20
cs
Enl 3 L
3
Encsl 4 L
3
Encsl 5 L
3
0
0
0
0
3Encsl 2 L
20
Encsl 2 L2

30
Encsl 3 L
6
Encsl 4 L
6
Encsl 5 L
6
0
0
Matrix Te is given as follows (refer to (41))
30





30

cs
Enl 3 L

6 
E ncsl 4 L

6
Encsl 5 L 

6
0 
0 

0

0
7Encsl 2 L 

20
cs 2 
En 2 L
 l 
20
Encsl 3 L 
3 
cs
E nl 4 L 
3 
cs
Enl 5 L 
3 
0 
0 
0
0
3Encsl 2 L
20
Encsl 2 L2
(54)
 1
 0
 0
 0

0
 t b
lh1  ah
 t b
lh1  ah
 0
 0
e
T  
0
 0
 0

0

0

 0

 0
 0

 0
APPENDIX II.
0 0 0
1 0 0
0 1 0
0
0
0
0
0
0
0
0
0
0 0 0
0 0 0
0 0 0
0
0
0
0 0 1
0 0 0
0
1
0
0
t b
0
0
0 0 0
0 0 0
0
0
0
0 0 0
0
0
0 0 0
0
0
0
1
0 0 0
0 0 0
0 0 0
0
0
0
0
0
0
0
t b
lh2  ah
t b
1
0
0
0
0
0
1
0
0
0
0
1
0 0 0
t
0 0 0
t
0 0 0
0 0 0
0
0
0 0 0 t

0 0 0
0 0 0
0 0 0
0
0
0
lh1  ah
tb
 
lh1  ah
1
1
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0 0 0
0
0
0 0 0
0
0
0 0 0
0 0 0
0
0
0
0
0 0 0 t
0
0
0
0
lh2  ah
0
0
0
1
0
0




0

0

0


0

0

0


0

0

0

0

0
t b 

lh2  ah 
t b 
 

lh2  ah

1

1

0
0
0
(55)
REFERENCES
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Abdel-Ghaffar, A. M. (1980). “Vertical vibration analysis of suspension bridges.” Journal of
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Agar, T. J. A. (1988). “The analysis of aerodynamic flutter of suspension bridges.” Journal of
Computers and Structures, 30(3), 593-600.
31
Agar, T. J. A. (1989). “Aerodynamic analysis of suspension bridges by a modal technique.”
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Washington, D.C.
Edwards, J. W. (1977). “Unsteady aerodynamic modeling for arbitrary motions.” AIAA
Journal, 15(1), 593-595.
Karpel, M. (1981). "Design for active and passive flutter suppression and gust alleviation."
NASA CR-3482, National Aeronautics and Space Administration, Washington, D.C.
Kobayashi, H., and Nagaoka, H. (1992). "Active control of flutter of a suspension bridge."
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Namini, A. H. (1991). “Analytical modeling of flutter derivatives as finite elements.” Journal
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cable suspended bridges.” Journal of Structural Engineering, ASCE, 118(6), 15091526.
Nobuto, J., Fujino, Y., and Ito, M. (1988). “ A study on the effectiveness of TMD to suppress
a coupled flutter of bridge deck.” Journal of Structural Mechanics and Earthquake
Engineering, JSCE, 398(I-10), 413-416 (in Japanese).
32
Omenzetter, P., Wilde, K., and Fujino, Y. (2000a). “Suppression of wind induced instabilities
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Wind Engineering and Industrial Aerodynamics, 87(1), 61-79.
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33
Wilde, K., Fujino, Y., and Masukawa, J. (1996). “Time domain modeling of bridge deck
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APPENDIX III.
NOTATION
The following symbols are used in this paper:
Ac = cross section area of the main cable
ah = vertical distance between hanger cable end and deck rotation center
ai = coefficients of rational function approximation
b = half chord width of deck together with flaps
bi = lag coefficients
C = the Theodorsen function
C˜ = approximation of the Theodorsen function
cs
e
s
˜ s , Cs, c = aerodynamic matrix
Ca , Ca , Ca , Ca , C
a
a
Ccss , C s , C ss , Cs,s c = structural damping matrix
c = damping coefficient associated with 
 degree of freedom
E = Young modulus of the girder
s
cs
e
e
s, c
E a , E b , E b , E a , E b , E b = matrix defining aerodynamic states
Ec = Young modulus of the main cable
G = shear modulus of the girder
g = gravity acceleration
H = tension force in the main cable
hc , hc1 , hc2 = horizontal displacement of the main cables
hd = horizontal displacement of the deck
34
 axis
I = second order moment of inertia of the deck with respect to 
I = second order mass moment of inertia of the deck
I , I = second order mass moment of inertia of the flap
ec
cs
e
s
˜ s = aerodynamic matrix
Ka , Ka , Ka , Ka , Ka , K
a
e, c
cs
e
s
s, c
K s , K s , K s , K s , K s , K s = structural stiffness matrix
 degree of freedom
k = stiffness coefficient associated with 
L 
 = Laplace transform of 

L = finite element length
L˜ = virtual length of the cable defined by Eq. (23)
Lc = total length of the main cable
lh , lh1 , lh2 = length of hanger cables
e, c
cs
e
s
s, c
M a , M a , M a , M a , M a , M a = aerodynamic matrix
e
e, c
cs
s
s, c
M s , M s , M s , M s , M s , M s = structural mass matrix
mc = mass of the main cable
md = mass of the deck
N = shape function matrix for structural degrees of freedom
N a = shape function matrix for aerodynamic degrees of freedom
˜ e, c , N
˜ e, c , N
˜e , N
˜ e = matrices describing the elastic stiffness of the main cable finite
N
c1
c2
c1
c2
element
nl = number of lag terms
Q , Q cs , Q e, c , Q e , Q s , Q s, c = aerodynamic matrix
q e , qe , c = vector of element structural degrees of freedom
35
q ea = vector of aerodynamic degrees of freedom
R = matrix defined in Eq. (6)
R cb , R eb , R sb , Rsb , c = matrix defining aerodynamic states
r = vector of global structural degrees of freedom
ra = vector of global aerodynamic degrees of freedom
rc , rc = location of support points of the flaps by control cables
r , r = location of flap hinge
S1 , S 2 = matrix defined in Eq. (6)
S = first order mass moment of inertia of the deck
S , S = first order mass moment of inertia of a flap about its hinge
s , sˆ = ordinary and nondimensionalized Laplace variable
T , T e = kinetic energy
Te = transformation matrix defined by Eq. (40)
T s = transformation matrix defined by Eq. (38)
Tvc 1 , Tvc 2 = matrices defined by Eq. (24)
t  , t = gearing factors for flap rotation
U = mean wind velocity
V , V , V˜ , V e = potential energy
vc 1 , vc 2 = vertical displacement of the main cables
v d = vertical displacement of the deck
Xcs , Xe , Xs = vector of generalized forces
Xv d = aerodynamic lift force
 coordinate
X = aerodynamic moment corresponding to 
36
x cs , x s , x s, c = vector of sectional structural degrees of freedom
x csa , x sa = aerodynamic states
xc , xc = location of support points of control cables by the transverse beam
 = torsional displacement of the deck
 = torsional displacement of the leading flap
 = torsional displacement of the trailing flap
 h1 , h2 = inclination angels of hanger cables
Superscripts
c = corresponding to controlled system
cs = corresponding to cross section
e = corresponding to finite element
s = corresponding to sectional model
  = transposition
Subscripts
a = corresponding to aerodynamic properties
s = corresponding to structural properties
 = corresponding to the leading flap or its rotation
 = corresponding to the trailing flap or its rotation
37
38
39
40
41