Flutter Control of Active Aerodynamic Flaps Mounted on Streamlined Bridge Deck Fairing Edges: An Experimental Study

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Introduction
As a divergent vibration phenomenon that could cause the catastrophic collapse of bridges, futter should be carefully examined and improved to meet the wind environment requirements at the site of interest [1].Te aerodynamic force of a bridge deck immersed in the airfow that drives the futter vibration of the deck is dominated by the fow separation and reattachment around the deck surface [2].Customarily, the optimization of the fow pattern around the deck using passive aerodynamic countermeasures, such as tailoring the confguration [3] or installing a stabilizer [4], is believed to be the most convenient and cost-efective approach to improving the futter performance of bridges.However, the futter capacity using passive aerodynamic countermeasures cannot satisfy the design requirements with the continuous increase in the span length and fexibility of bridges [5][6][7][8].Passive aerodynamic countermeasures are only efective at a specifc wind attack angle and wind speed range, which are challenging for the survival of long-span bridges in extreme wind environments, such as strong typhoons featuring high wind speeds, large angles of attack, and high turbulence intensities [9][10][11].Most passive aerodynamic countermeasures are bridge deck or vibration phenomena-dependent, indicating that an aerodynamically helpful countermeasure for a certain bridge deck or wind-induced vibration phenomena could be futile for other decks or vibrations, and even reduce the futter performance [12].Moreover, the potential efects of global warming on increasing the frequency and intensity of tropical cyclones require a sufciently high critical futter wind speed for longspan bridges to be constructed in coastal regions [13,14].
To enhance the futter stability of long-span bridges and endure the challenges of a complicated and volatile wind environment, an active aerodynamic control algorithm is proposed by installing actively controlled faps around the surface of the deck.Tese faps can adjust the real-time state of motion according to the vibration of the bridge to improve futter stability by optimizing the fow pattern around the bridge deck.Teoretically, the introduction of the active aerodynamic control strategy allows the long-span bridge to withstand threats from complicated wind environments as well as various wind-induced vibrations and adapt diferent geometries of the deck.Ostenfeld and Larsen [15] frst proposed two types of active control faps, whose schematic diagrams are shown in Figures 1(a) and 1(b).Active control faps were combined (Figure 1(a)) or separated (Figure 1(b)) with the bridge deck using a mechanical transmission system inside the box girder.Although the efects of applying the novel strategy in their study have not been fully verifed, it provides a blueprint for subsequent research on the active aerodynamic control of bridges.Since then, the application of active faps to bridges has been thoroughly investigated.
Arco et al. [20][21][22][23] studied the efect of wing plates as aerodynamic accessories on the wind stability of box girder suspension bridges.And it is confrmed that the use of active aerodynamic accessories can be a very efective tool for solving the wind stability problem of very large-span bridges in the future.
Kobayashi and Nagaoka [24], who designed an alternative model with actively controlled faps, performed a wind tunnel test using an artifcially designated gain coeffcient, in which the critical futter wind speed of the section doubled as much as the original one.Fujino and Wilde [16,25] proposed an active control surface at the bottom of a bridge section model, as shown in Figure 1(c).A pair of control surfaces, installed on both sides of the bottom of the deck and connected to the internal pendulum through a pull wire, would drive the windward and leeward faps to move during the relative motion between the pendulum and main girder and suppress deck futter.Omenzetter et al. [17,26,27] proposed an active control model, as shown in Figure 1(d).Numerical simulations were performed to study the infuence of control surfaces with diferent widths on futter control, and the results indicated that the critical futter wind speed of the deck could increase by 202% when the width of the control surface reached 3 m.
Teoretical studies based on diferent active control models have been conducted by some scholars.Huynh et al. [28] assumed that the fow pattern of the deck would not interfere with the faps, and vice versa, because of the long distance between them.Ten, the futter theory of deck-fap systems was developed by superimposing their aerodynamic forces.Tis theory was improved by Nissen et al. [29], who considered the mutual interference between the deck and faps.Phan and Kobayshi [30] studied the theoretical model of a pair of passive control fns and established a self-excited force model of the main girder and faps, whose efectiveness was verifed numerically.Bera et al. [31][32][33] studied constant-gain feedback control and variable-gain feedback control, assuming that the deck and faps did not interfere with each other.Te results demonstrated that variable gain feedback control was more efective.Guo et al. [18] studied the active control, focusing mainly on the infuence of the frequency and phase of the fap vibration on the futter of the deck using the model shown in Figure 1(e) and the openloop control method.In accordance with previous research, Li et al. [34][35][36] established the framework of the closed-loop feedback control theory for the active control system and proved the validity of the theory numerically.In addition, the unavailability of the open-loop control and the necessity of closed-loop control through the section model wind tunnel test were confrmed.In their test, a better control result was achieved when the vibration phase of the windward fap was 90 °behind the deck and the leeward fap was 90 °ahead of the deck, and the critical futter wind speed was simultaneously increased by 33%.Zhan and Liao [19] studied the futter control efect of the active fap model using computational fuid dynamics theory considering the structure displayed in Figure 1(f ).Te results also demonstrated that the futter control efect was better when the movement phase of the windward fap was behind the movement of the deck, and the movement phase of the leeward fap was ahead of the movement of the deck.
Te unsteady aerodynamic analysis theory of the combined fap originated from the aviation feld, and the wing futter is controlled by the movement of the fap [37].Based on this theory, Kwon et al. [38][39][40][41] introduced control theory and perfected the framework of bridge futter control based on combined faps through theoretical derivation and numerical simulation.However, these studies are based on the theory of plate with control surfaces and cannot be directly applied to bridge sections with control surfaces.
Many scholars have made a great contribution to investigating the characteristics of active control, constructing relevant theoretical frameworks, realizing control processes in wind tunnels, and improving control algorithms.Te research history of the active control of bridge futter is summarized in Table 1.Numerical simulation methods are used in most current active control studies, and there are limited bridge futter control cases through wind tunnel tests.Restricted by technical conditions, the present parameters are mostly used to control movable faps, and realtime feedback control experimental research is still scarce.Te efectiveness of extant active control measures still requires confrmation by more experimental studies.Moreover, many previous studies used separate faps, assuming that the movement of the faps would not interfere with the fow feld of the deck, which simplifed the analysis of the aerodynamic theory.
Following the pioneering studies, this study equipped the deck with a pair of movable faps near the edge of the streamlined box girder wind fairing and constructed a closed-loop active control system to perform experiments on 2 Structural Control and Health Monitoring the futter performance of the deck-fap system with the realtime feedback control method.It should be noted that the control parameters may change for diferent types of bridge sections.Te purpose of this study is to preliminarily verify the futter control method and control efect of the deck-fap system through the test method and provide a basis for future control theory.

Active Control System
Te concept of the gain coefcient was introduced to monitor the displacement and angular velocity of the deck.After phase translation and amplitude amplifcation, the movement mode of the deck was applied to the faps.Te deck is the feedback subject because it is controlled by the movement of the faps, and its feedback is shown as vibration attenuation owing to suppression or vibration divergence due to promotion.Ten, the motion of the deck is observed by adjusting the corresponding fap motion parameters until suitable parameters for the vibration control of the deck are obtained.Te aforementioned process enables the formation of a complete closed-loop control loop.Te motion of the deck in the critical state can be regarded as a single-frequency sinusoidal motion whose torsional motion is given by where A α , ω α , and φ α are the vibration amplitude of the deck, circular frequency of the deck motion, and the initial phase of the deck motion, respectively.Te torsional velocity and torsional acceleration of the deck motion are represented by the following equations, respectively: Te phase diference of faps and the gain coefcient from the amplitude of the main girder to the amplitude of the faps were selected as the control parameters to derive the fap motion function, as shown in equations ( 4)- (6).Te parameters φ l α and φ t α are the phase diference between the windward-leeward faps and torsional motion of the deck, respectively.Te gain coefcient G is the magnifcation of the vibration amplitude of the faps relative to that of the deck, refecting the degree of interference.Te torsional motion function of the windward fap is as follows:  Expanding and substituting equation (2) with equation (1), we get the following equation: Similarly, the motion function form of the leeward fap can be obtained as follows: During the test, the motions of the deck were measured by sensors, and the movement of the fap was determined using equations ( 5) and (6).A schematic diagram of the phase diference and gain coefcient during the movement of the main girder and faps is shown in Figure 2. Te control signal of the fap is input to generate the corresponding movement by internal monitors, and the evaluation function is set to determine the divergence and suppression of the deck, refecting the movement of the deck at the next moment.Te feedback control loop of the deck-faps system is shown in Figure 3.

Deck-Flaps Model and Wind Tunnel Tests
An active control model system, including a section model and software system that controlled the movement of the faps, was developed before the experiment.Te movement of the deck and faps was collected by the monitoring system in real time.Te fap was connected to the connecting rod through a hinge.Te fap, which was driven by the steering gear in the box girder, could freely rotate around the axis formed by the hinges on the two rods.Te rotation of the two faps was independently controlled by two sets of steering gears to ensure the independent movement of each fap.
Te dimensions of the model were 500 mm × 392 mm × 35 mm, in which the width of the deck, the width of the fap, and the thickness of the fap were 322 mm, 35 mm, and 4 mm, respectively.Te other details are shown in Figure 4. Te mass of the model and the mass moment of inertia are 2.44 kg and 0.045 kg•m 2 , respectively, while the vertical and torsional frequencies are 0.98 Hz and 1.77 Hz, respectively.Te vertical and torsional damping ratios are 2.1% and 0.75%, respectively.
Figure 5 shows the internal structure of the model.Te test was conducted in a TJ-5 wind tunnel as shown in Figure 6, whose dimensions were 1.5 m × 1.8 m × 10 m at the working position.Te maximum wind speed could reach 18 m/s, and the turbulence was less than or equal to 1% in the uniform fow feld.Figure 6 shows the layout of the model in the wind tunnel.
Te constant-gain output method was adopted in the test to explore the control law of the active faps.First, the critical futter wind speed of the deck was obtained with the faps in the fxed state, and then the entire control experiment was performed under the critical wind speed of the deck.
Diferent output motions of the faps afected the vibration of the deck diferently.Te motion mode of the faps, which can improve the futter performance of the deck, can be determined by changing the parameters of their feedback motion.Te rotation amplitude amplifcation (gain coefcient) of the faps and the combination of the phase difference of the faps were selected as testing parameters during the wind tunnel experiments.Te control variable method was adopted to study the infuence of the vibration of the deck under the critical wind speed by varying each parameter.
Te experimental control analysis was conducted under uniform fow with a wind attack angle of 0 °.When the wind speed increased, the displacement response of the deck was recorded by a laser displacement sensor.Te model diverged near the reduced wind speed of 11.8 m/s.

Phase Difference of the Flap Motion and Flutter Performance
During the control process, the gain coefcient G � 2 in the motion function of the two faps remained unchanged.Tus, the infuence of the change in the phase diference on futter control can be highlighted.Te phase diferences φ l α and φ t α of the windward and leeward faps were changed from 0 °to 360 °at intervals of 45 °.Te rotation direction of the faps was defned as positive in accordance with the torsion direction of the deck and negative in contrast.Te futter control of the deck under 64 types of combined movement of faps was studied.Te test conditions are listed in Table 2. Te improved ratio of futter performance is defned as follows: where U cr is the futter critical wind speed of system without the faps control; and U ij is the futter critical wind speed of system when the phase diference of the windward and leeward side faps is equal to i and j, respectively.Te experimental results demonstrate the signifcant efect of the change in the movement phase diference of the faps on the futter performance of the deck.Te displacement variation of the deck before and after the control was compared using a set of control parameters with improvement and deterioration efects.Among all the combined cases, φ l α � 270 °and φ t α � 45 °improved the futter performance, whereas the opposite case is φ l α � 90 °and φ t α � 180 °.Te corresponding relationship between the displacement and wind speed of the system under the above two working conditions is shown in Figure 7.
Under the combined working conditions of diferent phase diferences, the relationship between the critical futter wind speed and the phase diferences of the two faps is shown in Figure 8(a).When φ l α ≥ 180 °, the critical futter wind speed of the system increases by up to 17.3%.In contrast, when φ l α < 180 °, the futter performance of the deck does not improve signifcantly by the combined movement of the two faps, and the maximum reduction in the critical futter wind speed is −14.8%.
When the gain coefcient G was 2, the optimal combination of the phase diference of the faps was φ l α � 270 °4 Structural Control and Health Monitoring and φ t α � 90 °, while the worst combination was φ l α � 90 °and φ t α � 45 °, which can increase the critical futter wind speed of the system by 17.3% and reduce the critical futter wind speed of the girder by −14.8%, respectively.
A counter map of the infuence of the phase diference on the futter wind speed was drawn to illustrate the improved ratio of critical futter wind speed caused by the windward and leeward faps, as shown in Figure 8(b).Te color intensities in the fgure represent the degree of improvement or decline in the futter performance.Te map is divided into four regions according to the degree to which the phase diference combination improves the futter performance of the system.Tese are the "strong improvement area," "minor improvement area," "minor decline area," and "strong decline area" of futter performance, which are represented by I, II, III, and IV, respectively.
In area I, 180 °< φ l α < 360 °, 0 °< φ t α < 180 °, the critical futter wind speed of the system increased signifcantly, indicating that the efective control process requires the deck torsional motion to be in the same direction as that of the leeward fap and directed opposite to that of the windward fap.Te greatest improvement in the critical futter wind speed of the system occurred when the phase diference of the windward fap was maintained between 225 °and 270 °, while that of the leeward fap was maintained between 45 °and 90 °.
In area II, where 180 °< φ l α < 360 °and 180 °< φ t α < 360 °, the critical futter wind speed of the system increased slightly, indicating that the combination of the windward and leeward faps continued to beneft the futter stability of the system.However, when the futter stability of the system is 180 °< φ t α < 360 °, this combination can neither cooperate

Torsional vibration signal:
α, α, α, ... Structural Control and Health Monitoring with the windward faps to strongly interfere with the fow feld around the deck to improve the futter performance of the system nor dramatically reduce the fow feld structure around the system.In area III, 0 °< φ l α < 180 °, 180 °< φ t α < 360 °, the critical futter wind speed of the system is slightly reduced.Te areas III and II difered in the diferent values of the windward fap, indicating that the phase diference of the windward fap determined the direction of the improvement or deterioration of the futter performance of the system.Both the windward and leeward faps can work on the deck, but since the windward and leeward faps are located upstream and downstream of the deck, respectively, the former directly impacts the fow feld around the entire girder, while the latter only functions in the region near the wake of the girder.In addition, the leeward fap has a smaller infuence.
In area IV, 0 °< φ l α < 180 °, 0 °< φ t α < 180 °, the critical futter wind speed of the system is signifcantly reduced.From area III to area IV, only the phase diference of the leeward fap changes, indicating that when 0 °< φ l α < 180 °, the futter performance of the system deteriorates regardless of the value of φ t α .Te movement of the faps on both sides corresponding to the phase diference combination in area III and area IV deteriorated the fow feld structure around the deck, possibly because the movement rhythm contributed to the formation of adverse periodic aerodynamic forces around the deck at critical wind speed, which reduced the critical futter wind speed of the system.In particular, when the motion of the deck was in the same direction as that of the windward fap with a phase diference between 45 °and 90 °and in the opposite direction to that of the leeward fap with 6 Structural Control and Health Monitoring a phase diference in the same range, the critical futter wind speed of the system decreased drastically.
Te above analysis demonstrates that the windward fap phase diference φ l α determines whether the futter performance of the system improves or declines; the leeward fap phase diference φ t α determines the extent of improvement or decline.
To study the infuence of the two faps on the futter performance separately, we fxed the phase diference between the torsional motion of the deck and one of the faps and investigated the infuence of the movement of the other, and vice versa.Te diferent curves representing this variation are shown in Figure 9.
Each curve has a noticeable roughly horizontal "S" shape in the frst chart, which illustrates that the futter performance of the system changes in a similar manner under the efect of the phase diference of the windward fap when the phase diference of the leeward fap is fxed.Hence, when 90 °< φ l α < 180 °, the critical futter wind speed of the system is lower than that of the system in the absence of control.However, it decreases before 90 °and increases after 90 °as the value of φ l α grows.When 180 °< φ l α < 270 °, the critical futter wind speed of the system is higher than that of the system in the absence of control; however, 270 °appears to be the turning point, increasing before it and decreasing after it as the value of φ l α grows.
In contrast to the windward fap, the infuence law of the leeward fap on the futter performance of the deck exhibits no consistent trend: when φ l α � 0 °,45 °,90 °,315 °,360 °, and 0 °< φ t α < 90 °, the critical futter wind speed of the system decreased gradually, reaching the minimum value at 90 °.When 90 °< φ t α < 270 °, the critical futter wind speed of the system exceeds that of the uncontrolled level.When 135 °< φ t α < 270 °, the critical futter wind speed of the system frst decreases, then increases, and fnally decreases as the phase diference φ t α between the leeward fap and the deck increases.At φ t α � 90 °, the critical futter wind speed of the system is at its minimum.Te critical futter wind speed of the system was maximum when φ t α � 270 °.
Te contributions of the windward and leeward faps to the futter performance of the system were diferent under the simultaneous action of the two faps.Te infuence coefcient of the phase diference was defned to determine how the faps exert a greater infuence on the futter performance of the deck.Te corresponding expressions are as follows: where F L and F T are the infuence coefcients of the phase diference between the windward and leeward faps, respectively, which satisfy F L + F T � 1; R L is the average correlation coefcient between the infuence curve of the windward fap phase diference and the system futter performance under diferent leeward fap phase diferences.R T is similar to R L , and they are calculated using the following expressions: where R Lij is the correlation coefcient between the phase diference of the windward fap and the futter performance of the system when the leeward fap phase diferences are i and j, respectively.Similarly, R Tij is the correlation coefcient between the leeward fap phase diference and the futter performance of the system when the windward fap phase diferences are i and j. i and j were evaluated at 45 °intervals from 0 °to 360 °.R Lij and R Tij were calculated using the expressions:   Structural Control and Health Monitoring where V Li and V Lj represent the critical futter wind speed when the leeward fap phase diferences are i and j, respectively, with the windward fap phase diference varying from 0 °to 360 °at intervals of 45 °.V Ti and V Tj are the mean values of V Li and V Lj , respectively.Te critical futter wind speed of the system corresponding to the leeward fap phase diference from 0 °to 360 °at intervals of 45 °were determined when the windward fap phase diferences were i and j.V Ti and V Tj represent the mean values of V Li and V Lj , respectively, while E and σ are the expectation operator and standard deviation operator, respectively.Te correlation coefcients R Lij and R Tij of the response between the windward and leeward faps are shown in Figure 10.Te upper left area of the fgure is the infuence area of the phase diference of the windward fap, which represents the correlation coefcient of the phase diference of the leeward fap to the system critical futter wind speed curve when the corresponding horizontal and vertical coordinates of the phase diference of the windward fap are considered.Te lower right area is the leeward fap phase diference infuence area, which represents the correlation coefcient of the leeward fap phase diference and the system critical futter wind speed curve when the corresponding horizontal and vertical coordinates of the leeward fap phase diference are considered.In the infuence region of the phase diference of the windward fap, the values of R Tij are relatively low.Most of them exhibit a negative or weak correlation, which indicates that the infuence of the phase diference of the leeward fap on the critical futter wind speed can be signifcantly afected by the phase diference of the windward fap.
However, the values of R Tij are relatively high in the leeward fap phase infuence area, most of which are close to the strong correlation of 1, which illustrates that the phase diference of the leeward fap contributes little to the critical futter wind speed of the system afected by the phase difference of the windward fap.
In the leeward fap phase infuence area, the high values of R Tij with most of them approximately 1, indicate that the efect of the phase diference φ l α of the windward fap on the critical futter wind speed of the system is less afected by the phase diference φ t α of the leeward fap.According to equations ( 8) and ( 9), the infuence coefcients F L and F T of the two faps were 76.7% and 23.3%, respectively.It can be deduced that the windward fap plays a dominant role, whereas the leeward fap is the secondary factor when both faps are controlled simultaneously.

Gain Coefficient and Flutter Performance
According to the test results, the windward fap played a leading role when the two side faps were controlled synchronously.To understand the mechanism by which the gain coefcient G afected the futter control efect, the phase diference between the torsional motion of the deck and the leeward fap was maintained at 0 °.In the case of the windward fap, the phase diference was still evaluated from 0 °to 360 °at intervals of 45 °.Te gain coefcient G was calculated from 1 to 9 at an interval of 1, which was maintained between the windward and leeward faps throughout the entire process.Te test results are presented in Figure 11.
Te experimental results demonstrate that the phase diference either improves or deteriorates the futter performance of the system, and its extent is determined by the gain coefcient G.In addition, the futter performance of the system improved by 25.6% at the best and declined by −26.6% at its worst.To refect the magnitude of the gain coefcient in detail, two groups of typical phase-diference combination values under three diferent futter performances were selected for the test.Tese values are listed in Table 3 and the corresponding results are presented in Figure 12.
Te general laws of the infuence of the gain coefcient G on the futter performance of the system can be stated as follows: when the gain coefcient G is very small, the movement of the two faps has little efect on the futter performance of the system; as the gain coefcient G increases, both the improvement range under favorable conditions and the deterioration range under unfavorable conditions of the futter performance increase.However, this characteristic is not valid when the phase diference combination of the two faps has little efect on the futter control of the system.When G is too high, the futter performance of the system tends to worsen, and there is an optimal value of G. Te reason for these laws may be that when G is small, the faps possess a small range of motion and hence, cause little infuence on the fow feld around the deck; when G is too high, the motion amplitude of the faps is too high, which increases the overall wind resistance of the system, destroys the streamlined confguration, and deteriorates the futter performance.
In the test, the optimal value of G was 4, which is consistent with the optimal gain coefcient of 4-8 in the study by Omenzetter et al. [17].It was also consistent with those given by Phan [42], where the gain coefcients of the windward and leeward faps were 4 and 3, respectively.However, it was slightly higher than that of Li et al. [34], whose gain coefcient was approximately 2. Tis is because the design gain coefcient in Li's investigation was small, so the test model chose a larger gain coefcient range to avoid this shortcoming.
Structural Control and Health Monitoring

Flutter Performance under the Independent Control of Two Flaps
Te experiment was conducted under the premise of only one side fap control to study the change in the futter performance of the two faps.Te gain coefcient G and the phase diference between the torsional movements of the fap and deck φ α continued to control the movement of the single fap.Te system's futter wind speed was tested, and the results are summarized by the curved surface, as shown in Figure 13.Te combined value of the phase diference and gain coefcient that produces the greatest change ratio on the futter performance during the test is shown in Table 4. Te fuctuation of the critical futter wind surface in the fgure indicates that the futter performance of the system depends on the changes in the two parameters.
Te infuence of the windward and leeward faps on the futter performance of the system under independent control was partitioned as shown in Figure 14, and the partition parameters are shown in Table 5. Te phase difference of the futter performance of the system under the independent control of the windward fap was between 180 °and 360 °, which is consistent with the results obtained under the simultaneous control of the two faps.Tis proves that the phase diference of the windward fap φ l α afects the futter performance of the system signifcantly.Te phase diference range of the futter performance improvement is between 0 °and 180 °under the independent control of the leeward fap, which is diferent from that under the simultaneous control of two faps.When the system was controlled only by the leeward fap, the fow feld of the system was only afected by the leeward fap.Terefore, the futter performance of the system difered signifcantly from that of the two simultaneously controlled faps.
When the gain coefcient G was between 4 and 5, the futter performance could be improved by the movement of  both the windward and leeward faps.When the gain coefcient was too high, the futter performance of the system declined, which is consistent with the results of the simultaneous control of both the faps.Te gain coefcient G and phase diference φ α were fxed to investigate the efect of another parameter on the futter performance of the system.Te efect of the phase diference φ α on the futter performance of the system with diferent values of the gain coefcient G is shown in Figure 15.When the gain coefcient is 1, the change in the phase diference φ α can be neglected, possibly because when the gain coefcient is small, the motion amplitude of the fap is small, and the fow feld interference has little efect on the deck.
When the gain coefcients were 3, 5, 7, and 9, the critical futter wind speed of the system presented an "S" curve as the phase diference of the windward fap increased, namely, the   Structural Control and Health Monitoring same trend as the simultaneous control of the two faps.Te critical futter wind speed of the system presented a reverse "S" curve as the phase diference of the leeward fap increased.When 45 °≤ φ t α ≤ 135 °, the critical futter wind speed of the system fuctuated slightly.Te phase diference did not infuence the critical futter wind speed of the system signifcantly.
Figure 16 shows the infuence curve of the gain coefcient G corresponding to diferent values of the phase difference φ α on the critical futter wind speed of the system.Te optimal gain coefcient is 3-5, which is the same as that obtained when the two faps are controlled simultaneously.
Both the gain coefcient G and phase diference φ α signifcantly afect the futter performance of the system.To quantitatively describe the leading coefcients of G and φ α in the process of simultaneous change, according to equations ( 8)-( 13), the infuence coefcients of G and φ l α are 48.1% and 51.9%, respectively.Te infuence coefcients of G and φ t α are 48.5% and 51.5%, respectively.Te results demonstrate that the phase diference parameter φ α and gain coefcient G have the same efect on the futter performance of the deck.

Numerical Simulation
Te method is used to simulate the parameter combination φ l α � 270 °, φ t α � 90 °to further verify the control efect of the active wing on the system futter and analyze the reasons.
Te scale of the bridge model during the numerical simulation was the same as in the wind tunnel test; the incoming wind fow angle of attack was 0 °.Te natural frequency and damping ratio were also the same as those in the wind tunnel test.Te computation domain size was set as [−7B, 10B] × [−5B, 5B] during simulation (B is the width of the fap-deck model), and y + < 1 in near-wall regions which was according to the requirement of the numerical calculation.Figure 17 shows the grid division of the computational domain and some key boundary conditions.Te freestream turbulence intensity is set at 0.5%, and the turbulent viscosity ratio is set at 5, a typical value for the low turbulence wind tunnels.50 layers of structural grids are generated on the wall of the deck and faps.And a certain distance is reserved between the faps and the main girder to ensure the deformation of the grid during the movement of the faps.Te simulation frst makes the system futter under the futter critical wind speed, and the fap motion is turned on when the torsional displacement exceeds a certain level.Te torsional vibration time history of the system during the simulation is shown in Figure 18.

Infuence area
Te vibration starts to decay when the torsional displacement exceeds 8 °and the control is turned on, which indicates that the combination of the phase diference parameters of the faps obtained by the experiment is valid.
Te pressure distribution of the fow feld is extracted from four moments in a vibration period before and after the opening control, as shown in Figure 19.Structural Control and Health Monitoring much smaller than that of the windward fap.Tis explains why the leeward fap are not so important for the futter control of the box girder.
It is worth noting that when the system moves to the equilibrium position (b and B), the upper and lower surfaces of the windward and leeward faps respectively form pressure zones in opposite directions due to the existence of the phase diference and generate control moments opposite to the direction of torsional velocity, which are very helpful for futter control.
Te combined active aerodynamic faps will not only directly generate aerodynamic force on the system but also afect the fow pattern of the system itself, which should be the reason for the greater control potential compared with the separate active aerodynamic faps.More fow felds analysis will be discussed further in follow-up research.

Conclusions
In this study, the phase and gain coefcient of the faps on the futter performance of the system were explored by introducing real-time feedback for a streamlined box girder with a pair of faps on either side of the deck.Te futter control method of the active faps improved the futter performance of the deck by adjusting the movement of the fap in real time according to the motion state of the deck.Te phase diference combination of the two faps and the gain coefcient afected the futter performance of the system signifcantly.Te conclusions are summarized as follows: (1) In the combined control of two faps, the windward fap played a dominant role.Its phase diference determined whether the futter performance of the 14 Structural Control and Health Monitoring system improved or worsened, while the leeward fap played a subordinate role and determined the extent of this control efect; (2) As the gain coefcient G increased, the critical futter wind speed of the system frst increased and then decreased.Te optimal value of G lay between 3 and 5, and the futter performance of the declined when its value exceeded 7; (3) When the phase diferences between the windward and leeward faps and the deck were 270 °and 90 °, respectively, the futter performance of the system was optimized, and the critical futter wind speed increased by 25.6%; (4) Te futter performance of the system improved when the phase diference of the windward fap φ l α was between 180 °and 360 °.Its infuence curve on the critical futter wind speed of the system was "S"shaped.Te optimal value of φ l α was obtained between 270 °and 315 °, while the suboptimal value was between 45 °and 90 °.
Apart from this, the wind angle of attack, turbulence intensity, and other factors will afect the futter control law of active faps.And when the phase diference parameter between the faps and the deck is not selected properly, the motion of the aerodynamic faps will worsen the futter performance of the system.Terefore, some mechanical measures should be supplemented in the actual case to ensure the reliability of the movement of the faps according to the determined phase diference.Active aerodynamic fap control based on real-time feedback can be used as the fnal guarantee for bridge safety.More potential risks will be analyzed in future work.

Figure 2 :
Figure 2: Schematic of the phase diference and gain coefcient.

Figure 3 :
Figure 3: Feedback control of the deck-faps system.

Figure 10 :
Figure 10: Correlation between the fap phase diference and critical futter wind speed.

Figure 11 :
Figure 11: Efect of the windward fap phase diference and gain coefcient G on the critical futter wind speed.

Figure 13 :
Figure 13: Efect of the phase diference and gain coefcient of the faps on the futter performance of the system.

Figure 14 :
Figure 14: Infuence areas of the phase diference and gain coefcient of the faps on futter performance.

Table 1 :
Research progress on the active futter control of bridges.

Table 2 :
Phase diference and gain coefcient for futter active tests.

Table 3 :
Tree types of typical phase diference combinations.

Table 4 :
Typical combination values for the single fap control efect.

Table 5 :
Parameter values of the infuence area on the futter performance.