Flutter stability and buffeting response have been the topics of most concern in the design state of longspan suspension bridges. Among approaches towards the aerodynamic stability, the aerodynamicbased control method which uses control surfaces to generate forces counteracting the unstable excitations has shown to be promising. This study focused on the mechanically controlled system using flaps; two flaps were attached on both sides of a bridge deck and were driven by the motions of the bridge deck. When the flaps moved, the overall cross section of the bridge deck containing these flaps was continuously changing. As a consequence, the aerodynamic forces also changed. The efficiency of the control was studied through the numerical simulation and experimental investigations. The values of quasisteady forces, together with the experimental aerodynamic force coefficients, were proposed in the simulation. The results showed that the passive flap control can, with appropriate motion of the flaps, solve the aerodynamic instability. The efficiency of the flap control on the full span of a simple suspension bridge was also carried out. The modebymode technique was applied for the investigation. The results revealed that the efficiency of the flap control relates to the mode number, the installed location of the flap, and the flap length.
1. Introduction
Flutter has been realized to be one of the most dangerous aeroelastics of structures in civil engineering, especially after the disaster of the Tacoma Narrow Bridge. Flutter relates to both wind loads and structural response, from the perspective of control structure interactions. The flutter vibration mode is caused by selfexcitation force from wind loads. When flutter occurs, the oscillatory motions regarding all degrees of freedom in the structure couple to create a single frequency called the
flutter frequency
. Also, the displacement is infinite by linear theory. Flutter could occur in both smooth and turbulent flows. The response of the bridge in turbulent flow in a wide range of wind speeds is called
buffeting response
. This response is normally used to determine the size of the structural members and requires careful evaluation at the design stage.
The question regarding the effective flutter and buffeting control is of growing interest as the span length of suspension bridges increases. This has resulted in a variety of techniques to reduce or suppress the aeroelastic response. The most common control methods can roughly be divided into the structurebased method and the aerodynamicsbased method. The former approach exploits stiffened girders and/ or eccentric mass to control the flutter phenomenon passively whereas the latter approach focuses on generating stabilizing aerodynamic forces from the flow to solve the flutter problem.
A high torsion rigidity trussstiffening girder was widely used in the design of a suspension bridge. For instance, the Akashi Kaikyo Bridge has a vertical stabilizer in the center span located along the centerline of the trusstype stiffening girder to improve aerodynamic stability. However, truss sections usually exhibit reasonably high wind forces (drag loading) which must be resisted by the bridge structure leading to the increase in the construction cost. The development of boxgirders with a nearly streamlined section could also achieve the wind stability of a long span bridge. A twindeck configuration with a center gap has been proven to be an effective means of improving the flutter characteristics (Fok et al., 2004a, 2004b; Kwok et al., 2012). However, the adequate stability cannot be assured beyond a certain limit in regard to length of spans (Brancaleoni, 1992; Ostenfeld and Larsen, 1992).
Research on aerodynamic control by using winglets and flaps (or being called control surfaces) was proposed and developed (Kobayashi et al., 1992, 1996, 1998, 2001 and 2005). The study was continued with experiments on passive flap control in turbulent and smooth flow (Phan and Kobayashi, 2011). However, the numerical simulation was not presented. Furthermore, the effects of flap control on the full span of the suspension bridge were not carried out. An extensive theoretical study on the active control of bridge flutter using a model similar to that proposed by Kobayashi was presented (Wilde et al., 1998; Preidikman and Mook, 1998; Nissen et al., 2004). The active model was then translated to a passive one (Wilde et al., 1999). Omenzetter et al. (2000, 2002) also proposed the passive flap control with springs and supplementary cables.
In order to study the effects of wind on structures, the time domain analysis may be the most effective approach. This approach can be done if forces acting on structures are expressed in the time domain. The famous selfexcited forces suggested by Scanlan and Tomko (1971) contain frequencydependent coefficients. These forces are functions of flutter derivatives and reduced frequency. These flutter derivatives are found by experimental testing. To use these forces in the time domain, they have to be transformed into time dependency forces. This can be achieved by using indicial functions (Costa and Borri, 2006; Scanlan et al., 1974) or rational functions (Chen et al., 2000; Chen and Kareem, 2001; Cao and Sarkar, 2012). The simpler approach is to use quasisteady theory. It is valid in both the time and frequency domain. These forces are the functions of the aerodynamic force coefficients: lift, drag, and moment coefficients.
In this study, the efficiency of mechanically driven flaps for controlling the aerodynamic stability of a bridge deck was investigated with both numerical simulation and experiment approaches whereby, a flap was mechanically controlled by the pitching motion of the deck and the aerodynamic damping or suppressing aerodynamic force was then produced. As shown in
Figure 1
, the flaps were installed on both sides of the bridge deck and allowed the pitching motion. Providing the appropriate pitching motion of, the aerodynamic suppressing forces produced by them overcome the aerodynamic excitation, thus the flutter and buffeting were suppressed. Sectional model tests were also conducted to investigate the effect of the mechanically driving flap on the aerodynamic control. To explore the effects of flap control on both twodimensional and threedimensional cases, the quasisteady theory has been selected with the related force coefficients obtained from the experiment in the wind tunnel. In addition, the numerical study on flutter and buffeting passive control for a superlong span suspension bridge (3,000 m central span) has been taken for the analysis. A threedimensional model in conjunction with modebymode analysis has been considered in an attempt to predict the flutter speed and buffeting response for full span with and without flaps. The method of analysis in essence is based on the quasisteady forces method as used in the twodimensional analysis.
2. Twodimensional analysis by quasisteady forces
The bridge deck section in this paper was a shallow box girder and was aerodynamically treated as a flat plate. The controlling flaps at both edges of the bridge deck were also assumed to be flat plates. Thus, the numerical model of the bridge deck with the controlling flaps is described as shown in
Figure 1
.
 2.1 Control method
Figure 2
presents the bridge deck system having flaps which were mechanically driven by the rotational motion of
A bridge deck with flaps installed at leading and trailing edges
the deck. The flap was fixed with a hinge at the longitudinal edge of the bridge deck. Two hanger cables from a main cable were fixed to an anchor beam at its ends. A cross beam of the deck system was fixed to the anchor beam with a hinge. The gears
G_{s}
and
G_{f}
were fastened to the anchor beam and flap, respectively. Both gears were connected by a driving belt. When a bridge deck caused a torsion motion in its natural oscillation mode, hanger ropes nearly kept their vertical figure. The relative rotation angle was transmitted to the rotation of the flap through the driving belt with a given amplifying factor
G
=
R_{s}/R_{f}
where
G_{s}
and
G_{f}
are the radiuses of the gear
G_{s}
and
G_{f}
, respectively. Thus, the flap was driven just following the pitching motion of the bridge deck mechanically.
The same mechanism of the flap driving system was installed at the leeward side of the bridge deck. When the flaps were driven by the above mentioned manner, if we set the headup motion of the bridge deck as
α
, the turning down of the leading edge flap was
γ
=
Gα
and the turning up the trailing edge flap was
β
=
Gα
The same flap motions were seen when the wind direction was reversed.
 2.2 Numerical analysis
The governing equations of motion with respect to the static equilibrium position of a twodimensional bridge deck excited by aerodynamic forces are given by
In which m is the mass per unit length;
I
is the moment of inertial per unit length;
ξ_{h}
,
ξ_{α}
are the damping ratios of heaving motion
h
and pitching motion
α
, respectively;
ω_{h}
,
ω_{α}
are the natural circular frequencies of heaving and pitching motion, respectively;
L
is the total aerodynamic lift force (positive when upward);
M
is the total aerodynamic
Bridge deck with the mechanically driven flaps.
moment (positive when clockwise).
L
and
M
are defined in Eqs. (2).
By using the classical thin airfoil theory (Anderson 1984), the static aerodynamic lift
L
and the aerodynamic moment
M
are computed by:
In which
ρ
is the air density,
B
= 2
b
is the bridge deck width (see
Figure 1
),
U
is the wind speed,
W
is the vertical wind,
α
is the angle of attack of the bridge deck,
β
is the angle of the trailing edge flap and
γ
is the angle of the leading edge flap.
The vertical wind is defined as follows:
where
is the mean vertical wind speed, which is assumed to be zero;
w
(
t
) is the vertical wind speed. The W was taken from a gust simulation of the wind tunnel. The coefficients
C'
are the slopes of the aerodynamic force coefficients. They are explained as follows:
where
θ
_{0}
=cos
^{1}
c
and c indicates the location of a hinge of the flap as shown in
Figure 1
.
The flaps are passively driven by the pitching motion:
With the known values of all parameters, Eqs. (1) were solved by using the RungeKutta method to obtain the critical speed and the buffeting responses.
 2.3 Numerical results based on the experimental data
The model dimensions and the dynamic properties of the investigated system are shown in
Table 1
. In still air, the damping ratios and the natural frequencies were measured when the flaps were fixed.
The vertical wind speed
W
was actively simulated in the Eiffel type wind tunnel based on the Karman’s spectrum with the turbulence intensity of 5% (Kobayashi, et al. 1992, 1994). The results shown in
Figure 3
were at the reduced wind speed
U
/(
ω_{α}b
)=4.3, where
ω_{α}
is the circular frequency of pitching motion (see
Table 1
).
The sensitivity numerical study in regard to the critical speed was performed with the combination of the coefficients
X
=
G
and
Y
=
G
where
G
varied from 5.5 to 5.5. The instability phenomenon of the bridge deck, which could be flutter, divergence, or large buffeting response, was defined when the pitching response exceeded 0.17 radians in root mean square (RMS). The wind speed at this phenomenon is called the critical speed
U_{cr}
. The equations of motion in the turbulent flow were solved by the Runge
Model dimensions and dynamic properties of the investigated system
Model dimensions and dynamic properties of the investigated system
The slopes of lift and moment coefficients
The slopes of lift and moment coefficients
Time serials of the vertical wind velocity w at U/(ω_{α}b)=4.3
Effect of G on the flutter control
Effect of G on buffeting control at U/(ω_{α}b) = 3.08
Kutta method. The slopes of the aerodynamic force coefficients for the numerical simulation were obtained from experiments done in smooth flow. These values are listed in
Table 2
.
Figure 4
, in which the vertical axis shows the critical speed
U_{cr}
compared with that of no control case
U
_{cr,0}
, shows the variation of the critical wind speed due to the gain of the control surfaces,
G
. Increase in positive gain resulted in the improvement of the critical wind speed. The maximum ratio
U_{cr}
/
U
_{cr,0}
was approximately 1.86 when
G
varied from 5.5 to 5.5. In the case of a further negative decrease of the gain
G
, the critical wind speed became smaller than that of the bridge deck without the control.
To check the effect of the amplifying factor
G
on buffeting, the percentage of reduction effect for the buffeting response (
PRE
) is introduced:
where
RMS_{cc}
and
RMS
_{c0}
are the root mean squares of the root mean squares buffeting response for control cases and noncontrol cases, respectively; the subscript
x
indicates the heaving
h
or pitching
α
.
Figure 5
shows the effect of the amplifying factor
G
on the buffeting at
U
/(
ω_{α}b
)=3.08. When
G
increased,
PRE_{x}
increased.
PRE_{x}
reduced and received a negative value when
G
< 0.
3. Wind tunnel test
 3.1 Wind tunnel model
The twodimensional wind tunnel tests were conducted to investigate the effects of the mechanical flaps on the controlling of the windinduced oscillations. The springmounting of the model and flaps are shown in
Figure 6
. Details of the cross sectional shapes are also included.
The wind tunnel model
The model was fixed with the hinge at the point H
_{α}
of the supporting frame. The frame was supported by the vertical spring
k_{h}
which allowed only vertical motion of the frame including the model. The model was elastically supported by the pitching springs
k_{α}
from the frame allowing only a pitching oscillation about the hinge H
_{α}
. The sway motion and rotation of the frame was inhibited by the connecting rods AC and BD. Thus, the model has two degrees of freedom for oscillation, heaving and pitching.
The flaps were installed at both sides of the model. The leadingedge flap had a driving arm FH
_{γ}
to its end of the rotation axis. During a heaving motion of the modelmounted system, this driving arm did not move. The vertical tie bar EF connected a given point of the driving arm and the frame. The pitching motion of the flap was given by the pitching motion of the bridge deck through the driving arm. The trailingedge flap also had the same mechanism. Thus, the control manner described in the above section (
Figure 2
) was realized in this model.
The magnitudes of the flaps depend on the location of the tie bars
EF
and
JK
as follows:
If
l
_{1}
=
l
_{2}
and
G
=(
cb
+
l
_{1}
)/
l
_{1}
, the controlling angles are
β
=
Gα
,
γ
=
Gα
.
 3.2 Windinduced responses
The static aerodynamic force coefficient by the flap was measured in a smooth wind when the angle of attack regarding the bridge deck was 0 degrees. The slopes of the coefficients (derivatives) of the lift and moment among these angles were estimated and compared with the theoretical values on the flat plate. The results are shown in
Table 2
. A very small lift slope was obtained from the trailing edge flap. It is assumed to be caused by the following reasons. The trailing flap was in a region of the separated flow different from the theoretical flow. In addition, the small gap between the deck and the flap also reduced the aerodynamic force by the flap resulting in the differences between the theoretical and measured values as provided in
Table 2
. The damping ratios and the frequencies of the model with the controlling flaps were measured in still air. The results are listed in
Table 3
. These values changed when factor G changed.
The control effect of the flap control with several values of
G
was investigated in a turbulent flow. The results are shown in
Figure 7
. In this figure, the numerical results are also included. The numerical simulation was made with parameters listed in
Table 3
figure,
ω_{α}
is the natural frequency of
ω_{α}
pitching motion in the case of without control (see
Table 1
);
b
is the halfwidth of the bridge deck. In the experiment case, the flutter motion of the model without control took place at
U
/(
ω_{α}b
)=3.69. This speed could be improved to
U_{cr}
/(
ω_{α}b
)=6.46 by the flap control with
G
= 5.0. The flap motion by
G
= 1.4 was considered to give a reversed
Damping and frequency in smooth flow test
Damping and frequency in smooth flow test
Reduced wind speed at G on flutter control
Time trace, without control G = 0 and with control G = 1.4 at U/(ω_{α}b)=2.77
effect for the aerodynamic response. The experimental result by
G
= 1.4 demonstrated that flutter appeared at a slightly lower speed compared with the case without control. The trends for the numerical and simulation results are similar.
An example for the time history of the response in regard to the model with and without flap control in a turbulent flow is shown in
Figures 8

10
. The time experiment responses
Time trace, without control G = 0 and with control G = 1.4 at U/(ω_{α}b)=3.38
Time trace, without control G = 0 and with control G = 5 at U/(ω_{α}b) = 3.38
of heaving and pitching for the reverse effect are shown in
Figure 8
. For the case without control,
G
= 0; and for the case with control
G
= 1.4. The reduced wind speed was
U_{cr}
/
(ω_{α}b
) = 2.77. The amplitudes increased after controlling. The reversed effect happened.
ω_{α}
was the value at
G
= 1.4. At the flutter speed for the case of
G
= 1.4, as shown in
Figure 9
, the responses increased exponentially. If the value of
G
= 5,nse amplitudes decreased after controlling as shown in
Figure 10
. The pitching motion completely reduced, but the heaving motion was not affected. If the model was controlled, the flaps were driven with the amplification factor of
G
= 5.0, the pitching response was suppressed, however almost no effect on the heaving response was observed.
4. The threedimensional modebymode analysis
 4.1 The aeroelastic full model of suspension bridge
To apply the flap control to a full span bridge, the following problems were to be investigated. First, the relation between the install flap length and the critical wind speed was examined. Second, the influence of the vibration mode number on the control result was checked. Finally, the buffeting response of the suspension bridge was also tested.
In this numerical study, a simple span suspension bridge with span length
l
= 3000 m was employed, which is shown in
Figure 11
. Details of the bridge model and the flaps for
The suspension bridge model
Structural properties of the full bridge model and the flaps
Structural properties of the full bridge model and the flaps
this simulation are listed in
Table 4
. The damping ratios of bending and torsion modes were assumed to be 0.01 for the modebymode study. This assumption was made for all modes.
The vibration mode shapes and the natural frequencies of bending and torsion motion were obtained by using the finite element method (AbdelGhaffar, 1976). The results are shown in
Table 5
for the first twenty modes. The lateral motion was neglected in this study.
 4.2 Simulation of turbulent wind field
The simulation of the vertical turbulent wind field acting on the bridge was carried out. The spectra representation method was used (Paola 1998, Deodatis 1996, Cao
et al
. 2000, Ding
et al
. 2006) with the VonKarman’s spectrum formula. The Davenport’s coherence function (Davenport 1968) between point
j
and
m
was obtained by the following formula:
Sample of vertical turbulent wind at U = 10 m/s
In which
ω
is the frequency;
λ
is the dimensionless decay factor; Δ
_{jm}
=Δ
j

m
 is the horizontal distance from point j to
m
. △ is the distance interval between successive points. These parameters are listed in
Table 6
.
Figure 12
presents the first 1000 sec samples of the vertical turbulent wind field at point
P
1 (left support),
P
2, P50 (midpoint),
P
51,
P
100 and
P
101 (right support). The mean wind speed in this case was 10 m/s.
 4.3 Numerical analysis
The bridge deck motions were assumed by the combination of one bending mode and one torsion mode as follows:
where
ϕ_{b}
(
x
),
ϕ_{t}
(
x
) are the mode shapes of bending and torsion, respectively;
g_{b}
(
t
),
g_{t}
(
t
) are the generalized coordinates of bending and torsion, respectively;
x
is the spanwise coordinate. In this study, the first three modes of
The first twenty natural frequencies of the bridge
The first twenty natural frequencies of the bridge
Characteristic values of the wind field simulation
Characteristic values of the wind field simulation
bending and torsion were examined. The lateral mode was not considered. The response calculation was conducted based on the modal analysis with the second order equations of motion composed of one of the bending modes and one of the torsion modes.
Referring to Eqs. (1), the equations of motion in modal analysis become
where
I
_{0b}
and
I
_{0t}
are the bending and torsion generalized inertias, respectively
The buffeting forces and the selfexcited forces are the sum of the forces on the bridge deck and the flaps.
U
(
x
,
t
) is assumed to be constant with time and space. The vertical turbulent wind
w
(
x
,
t
) has a spanwise correlation. The flap with length △
l_{F}
is installed at
x_{j}
,
j
= 1, 2...
N
, where
N
is the divided number of the span.
By replacing
α
in Eqs. (5) by Eqs. (9), with
X
=
G
and
Y
=
G
, the flap angles at position
x
and at time
t
are driven by the following equations:
In Eqs. (12), the coefficients
G
= 5 is used for the control case. The three methods of the flap arrangement as shown in
Figure 13
are employed here. In method 1, the flap is installed partially at the center of the span. The installed portion corresponds to the hoop of the first natural vibration mode shape. The locations of the flap of method 2 and 3 correspond to the hoops of the natural vibration modes for the second and the third, respectively.
Bridge deck motions are assumed by the combination of one bending mode and one torsion mode, bending and torsion of bridge deck are defined by the Eqs. (9) and (10). If the flap angles are derived by Eqs. (12), the equations of motion for the full span of the bridge are:
where
Flap arrangement.
Critical speeds – without flap
Critical speeds – without flap
where
ψ_{F}
is a forcing function taking the value of 1 where flap is installed and is 0 otherwise.
 4.4 Numerical results
At first, the critical speed of the bridge deck without controlling was checked. The result is shown in
Table 7
. In which, modes(
ij
) means the combination of the i
^{th}
bending mode and the
j^{th}
torsion mode. The term
ω
_{α1}
is the torsion natural frequency of the first mode. For the mode combinations (11), (22) and (33), the critical phenomenon was flutter. In addition, the divergence motion took place in the mode combinations (12), (21), (23) and (32). In the design of a bridge, only the lowest critical speed, in this case the mode combinations (11), is important. However, in this study, all the mode combinations shown in
Table 7
were investigated.
The critical speeds were checked for the arrangement methods with a variation of the flap length ratio, and modes(
ij
). The flap length ratio
FLR
is defined by the ratio of the flap length on one side and the span length. The results are depicted in
Figure 14
. In case of modes(11) without control, the flutter speed was 1.26. After controlling, the flaps installed for about 50% of the span length could increase the flutter speeds of the modes(11) up to 2.28, approximately 1.8 times compared to without control. From the results of this analysis and the plots in
Figure 14
, it was determined that the flaps could improve the critical speed of the bridge. They controlled the flutter speed well.
The arrangement method 1 could increase the flutter speeds of the modes(22) up to 4.19, approximately 2.34 times compared to without control. It was also effective for divergence. Modes(12), (21), (23), (32) were the divergence cases. Modes(12) and (32) were of the same effect, because of the same pitching mode 2. In the real case, the arrangement method 2 is more realistic since the length of the hung cable was long enough to produce the relation motion between the flap and bridge, as discussed above. This method also increased the flutter speed and the divergence speed. The same effect could be seen in the arrangement method 3.
5. Conclusion and Discussion
A mechanically turning flap to control the aerodynamic stability was studied with the quasisteady force approach. We have demonstrated that the quasisteady force is easy to be applied in the time domain. In the twodimensional study, the experimental force coefficients and vertical wind gust were used to both simulate and find the best control. The amplification factor
G
was investigated in the viewpoint of the large flutter speed and the small buffeting response. The following remarks were addressed:

 The quasisteady force was reasonably accepted to
Flutter speed improvement by FLR with different modes

investigate the efficiency of the amplification factorG. The control manner for the leadingedge flap and the trainingedge flap such asγ=Gαandβ=Gαcan be easily set up in a real case. The positive values ofGcan be applied to suppress the buffeting response and improve the flutter speed or to improve the aerodynamic stability of the bridge deck. The negative values ofGhad the reverse effect. The numerical results using the experimental frequencies and damping ratios were compared to the experiment cases to check the efficiency of the flap control and to validate the quasisteady force method.

 In a turbulent flow, control byG= 5.0 also improved the flutter speed approximately 2 times and suppressed the divergence phenomenon up to its wind speed. Buffeting in the pitching motion was effectively suppressed but the heaving motion was not.
The quasisteady force method was also applied for full span bridge to check the efficiency of the flap control. Effects of the flap location, the flap length, and the mode shapes on the control were investigated. The modebymode technique of coupled motions was considered. The flutter speed and the buffeting response for with and without the control were computed. The parameters used in the numerical analysis were the same with those in the twodimensional cases. The following results were found by the modebymode technique:

 The flaps installed for about 50% of the span length could increase the flutter speed of the modes(11) up to approximately 1.8 times.

 For divergence, the passive flap control was proven to be effective.
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