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

Authors: Zilong Wang; Lin Zhao; Hanlin Chen; Genshen Fang; Ke Li; Yaojun Ge
Journal: Structural Control and Health Monitoring
 (2023)

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## Abstract

Traditional aerodynamic measures with fixed geometric shapes make it difficult to relieve the wind-induced vibration issues under the continuous expansion of bridge spans. Active aerodynamic control measures would be an alternative way to improve the wind vibration performance of long-span bridges. In this study, a bridge flutter control method based on active flaps was proposed, and the main girder-active flap-suspended aeroelastic model was designed to study the control effect. A pair of active flaps was installed on both sides of the box girder near the lateral fairings, and the movement signal of the main girder was detected using internal sensors. The two flaps move relative to the deck according to the preset equations of motion. This improved the flutter stability of the system with high effectiveness compared to traditional passive aerodynamic measures. By adjusting the gain coefficient in the motion function of the flaps and the phase difference between the flaps and the deck, the changing characteristics of the critical flutter wind speed were revealed with a deeper understanding of the relationship between the phase differences and gain coefficient of the active flaps and main girder. Studies have shown that the flutter performance of the deck-flap system is significantly affected by the phase difference and gain coefficient on both sides of the aerodynamic flap. The control law of the windward flap is relatively stable, but the control effect of the leeward flap is greatly affected by the movement of the windward flap.


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## Body

## 1. Introduction

As a divergent vibration phenomenon that could cause the catastrophic collapse of bridges, flutter should be carefully examined and improved to meet the wind environment requirements at the site of interest [1]. The aerodynamic force of a bridge deck immersed in the airflow that drives the flutter vibration of the deck is dominated by the flow separation and reattachment around the deck surface [2]. Customarily, the optimization of the flow pattern around the deck using passive aerodynamic countermeasures, such as tailoring the configuration [3] or installing a stabilizer [4], is believed to be the most convenient and cost-effective approach to improving the flutter performance of bridges. However, the flutter capacity using passive aerodynamic countermeasures cannot satisfy the design requirements with the continuous increase in the span length and flexibility of bridges [5–8]. Passive aerodynamic countermeasures are only effective at a specific 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 [9–11]. 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 flutter performance [12]. Moreover, the potential effects of global warming on increasing the frequency and intensity of tropical cyclones require a sufficiently high critical flutter wind speed for long-span bridges to be constructed in coastal regions [13, 14].To enhance the flutter 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 flaps around the surface of the deck. These flaps can adjust the real-time state of motion according to the vibration of the bridge to improve flutter stability by optimizing the flow pattern around the bridge deck. Theoretically, 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 different geometries of the deck. Ostenfeld and Larsen [15] first proposed two types of active control flaps, whose schematic diagrams are shown in Figures 1(a) and 1(b). Active control flaps 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 effects of applying the novel strategy in their study have not been fully verified, it provides a blueprint for subsequent research on the active aerodynamic control of bridges. Since then, the application of active flaps to bridges has been thoroughly investigated.Figure 1
Development of active aerodynamic control measures. (a) Combined active flap model [15]. (b) Separate active flap model [15]. (c) Separate active flap model [16]. (d) Combined active flap model [17]. (e) Separate active flap model [18]. (f) Combined active flap model [19].
(a)(b)(c)(d)(e)(f)Arco et al. [20–23] studied the effect of wing plates as aerodynamic accessories on the wind stability of box girder suspension bridges. And it is confirmed that the use of active aerodynamic accessories can be a very effective 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 flaps, performed a wind tunnel test using an artificially designated gain coefficient, in which the critical flutter 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 flaps to move during the relative motion between the pendulum and main girder and suppress deck flutter. Omenzetter et al. [17, 26, 27] proposed an active control model, as shown in Figure 1(d). Numerical simulations were performed to study the influence of control surfaces with different widths on flutter control, and the results indicated that the critical flutter wind speed of the deck could increase by 202% when the width of the control surface reached 3 m.Theoretical studies based on different active control models have been conducted by some scholars. Huynh et al. [28] assumed that the flow pattern of the deck would not interfere with the flaps, and vice versa, because of the long distance between them. Then, the flutter theory of deck-flap systems was developed by superimposing their aerodynamic forces. This theory was improved by Nissen et al. [29], who considered the mutual interference between the deck and flaps. Phan and Kobayshi [30] studied the theoretical model of a pair of passive control fins and established a self-excited force model of the main girder and flaps, whose effectiveness was verified numerically. Bera et al. [31–33] studied constant-gain feedback control and variable-gain feedback control, assuming that the deck and flaps did not interfere with each other. The results demonstrated that variable gain feedback control was more effective. Guo et al. [18] studied the active control, focusing mainly on the influence of the frequency and phase of the flap vibration on the flutter of the deck using the model shown in Figure 1(e) and the open-loop control method. In accordance with previous research, Li et al. [34–36] 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 confirmed. In their test, a better control result was achieved when the vibration phase of the windward flap was 90° behind the deck and the leeward flap was 90° ahead of the deck, and the critical flutter wind speed was simultaneously increased by 33%. Zhan and Liao [19] studied the flutter control effect of the active flap model using computational fluid dynamics theory considering the structure displayed in Figure 1(f). The results also demonstrated that the flutter control effect was better when the movement phase of the windward flap was behind the movement of the deck, and the movement phase of the leeward flap was ahead of the movement of the deck.The unsteady aerodynamic analysis theory of the combined flap originated from the aviation field, and the wing flutter is controlled by the movement of the flap [37]. Based on this theory, Kwon et al. [38–41] introduced control theory and perfected the framework of bridge flutter control based on combined flaps 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. The research history of the active control of bridge flutter is summarized in Table1. Numerical simulation methods are used in most current active control studies, and there are limited bridge flutter control cases through wind tunnel tests. Restricted by technical conditions, the present parameters are mostly used to control movable flaps, and real-time feedback control experimental research is still scarce. The effectiveness of extant active control measures still requires confirmation by more experimental studies. Moreover, many previous studies used separate flaps, assuming that the movement of the flaps would not interfere with the flow field of the deck, which simplified the analysis of the aerodynamic theory.Table 1
Research progress on the active flutter control of bridges.
AuthorTimeInterference effect of flaps on the flow fieldInstallation positionResearch methodImproved ratio of the flutter wind speedControl strategyGain coefficientOstenfeld1992Ignored/consideredWind fairing/below the deckTheoretical analysis———Kabayashi1992IgnoredAbove the deckWind tunnel tests100%No feedback4Fujino1999IgnoredBelow the deckWind tunnel tests96%No feedback1.5Omenzetter2000ConsideredWind fairingNumerical analysis202%No feedback4Huynh2001IgnoredBelow the deckNumerical analysis50%No feedback2Liu2002IgnoredBelow the deckNumerical analysis50%No feedback1Nissen2004IgnoredBelow the deckTheoretical analysis———Guo2013IgnoredAbove the deckTests/theoretical analysis80%No feedback1Li2017IgnoredAbove the deckTests/theoretical/numerical analysis33%Real-time feedback2Zhan2018ConsideredWind fairingNumerical analysis60%No feedback18Liu2018ConsideredWind fairingWind tunnel tests70%No feedback1Chen2020ConsideredWind fairingWind tunnel tests25%Real-time feedback4Following the pioneering studies, this study equipped the deck with a pair of movable flaps near the edge of the streamlined box girder wind fairing and constructed a closed-loop active control system to perform experiments on the flutter performance of the deck-flap system with the real-time feedback control method. It should be noted that the control parameters may change for different types of bridge sections. The purpose of this study is to preliminarily verify the flutter control method and control effect of the deck-flap system through the test method and provide a basis for future control theory.


## 2. Active Control System

The concept of the gain coefficient was introduced to monitor the displacement and angular velocity of the deck. After phase translation and amplitude amplification, the movement mode of the deck was applied to the flaps. The deck is the feedback subject because it is controlled by the movement of the flaps, and its feedback is shown as vibration attenuation owing to suppression or vibration divergence due to promotion. Then, the motion of the deck is observed by adjusting the corresponding flap motion parameters until suitable parameters for the vibration control of the deck are obtained. The aforementioned process enables the formation of a complete closed-loop control loop.The motion of the deck in the critical state can be regarded as a single-frequency sinusoidal motion whose torsional motion is given by(1)αt=Aα⋅sinωαt+φα,where Aα,ωα, and φα are the vibration amplitude of the deck, circular frequency of the deck motion, and the initial phase of the deck motion, respectively. The torsional velocity and torsional acceleration of the deck motion are represented by the following equations, respectively:(2)α˙t=ωαAα⋅cosωαt+φα,(3)α¨t=−ωα2Aα⋅sinωαt+φα.The phase difference of flaps and the gain coefficient from the amplitude of the main girder to the amplitude of the flaps were selected as the control parameters to derive the flap motion function, as shown in equations (4)–(6). The parameters φαl and φαt are the phase difference between the windward-leeward flaps and torsional motion of the deck, respectively. The gain coefficient G is the magnification of the vibration amplitude of the flaps relative to that of the deck, reflecting the degree of interference. The torsional motion function of the windward flap is as follows:(4)βlt=GAαsinωαt+φα+φαl.Expanding and substituting equation (2) with equation (1), we get the following equation:(5)βlt=Gαtcosφαl+Gωαα˙tsinφαl.Similarly, the motion function form of the leeward flap can be obtained as follows:(6)βtt=Gαtcosφαt+Gωαα˙tsinφαt.During the test, the motions of the deck were measured by sensors, and the movement of the flap was determined using equations (5) and (6). A schematic diagram of the phase difference and gain coefficient during the movement of the main girder and flaps is shown in Figure 2. The control signal of the flap 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, reflecting the movement of the deck at the next moment. The feedback control loop of the deck-flaps system is shown in Figure 3.Figure 2
Schematic of the phase difference and gain coefficient.Figure 3
Feedback control of the deck-flaps system.


## 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 flaps, was developed before the experiment. The movement of the deck and flaps was collected by the monitoring system in real time. The flap was connected to the connecting rod through a hinge. The flap, 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. The rotation of the two flaps was independently controlled by two sets of steering gears to ensure the independent movement of each flap.The dimensions of the model were 500 mm × 392 mm × 35 mm, in which the width of the deck, the width of the flap, and the thickness of the flap were 322 mm, 35 mm, and 4 mm, respectively. The other details are shown in Figure4. The mass of the model and the mass moment of inertia are 2.44 kg and 0.045 kg·m2, respectively, while the vertical and torsional frequencies are 0.98 Hz and 1.77 Hz, respectively. The vertical and torsional damping ratios are 2.1% and 0.75%, respectively.Figure 4
Geometric parameters of the segment model (unit: mm).Figure5 shows the internal structure of the model. The 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. The maximum wind speed could reach 18 m/s, and the turbulence was less than or equal to 1% in the uniform flow field. Figure 6 shows the layout of the model in the wind tunnel.Figure 5
Structure of the model and its internal frame work.Figure 6
Deck-flaps system installation in the TJ-5 wind tunnel. (a) Front view. (b) Top view.
(a)(b)The constant-gain output method was adopted in the test to explore the control law of the active flaps. First, the critical flutter wind speed of the deck was obtained with the flaps in the fixed state, and then the entire control experiment was performed under the critical wind speed of the deck.Different output motions of the flaps affected the vibration of the deck differently. The motion mode of the flaps, which can improve the flutter performance of the deck, can be determined by changing the parameters of their feedback motion. The rotation amplitude amplification (gain coefficient) of the flaps and the combination of the phase difference of the flaps were selected as testing parameters during the wind tunnel experiments. The control variable method was adopted to study the influence of the vibration of the deck under the critical wind speed by varying each parameter.The experimental control analysis was conducted under uniform flow 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. The model diverged near the reduced wind speed of 11.8 m/s.


## 4. Phase Difference of the Flap Motion and Flutter Performance

During the control process, the gain coefficientG = 2 in the motion function of the two flaps remained unchanged. Thus, the influence of the change in the phase difference on flutter control can be highlighted. The phase differences φαl and φαt of the windward and leeward flaps were changed from 0° to 360° at intervals of 45°. The rotation direction of the flaps was defined as positive in accordance with the torsion direction of the deck and negative in contrast. The flutter control of the deck under 64 types of combined movement of flaps was studied. The test conditions are listed in Table 2. The improved ratio of flutter performance is defined as follows:(7)η=Uij−UcrUcr,where Ucr is the flutter critical wind speed of system without the flaps control; and Uij is the flutter critical wind speed of system when the phase difference of the windward and leeward side flaps is equal to i and j, respectively.Table 2
Phase difference and gain coefficient for flutter active tests.
Phase difference of the windward flapPhase difference of the leeward flapGain coefficientφαl=0°−360°φαt=0°−360°G=2Δφαl=45°Δφαt=45°The experimental results demonstrate the significant effect of the change in the movement phase difference of the flaps on the flutter performance of the deck. The displacement variation of the deck before and after the control was compared using a set of control parameters with improvement and deterioration effects. Among all the combined cases,φαl = 270° and φαt = 45° improved the flutter performance, whereas the opposite case is φαl = 90° and φαt = 180°. The corresponding relationship between the displacement and wind speed of the system under the above two working conditions is shown in Figure 7.Figure 7
Flutter control effect under different flap phase differences.Under the combined working conditions of different phase differences, the relationship between the critical flutter wind speed and the phase differences of the two flaps is shown in Figure8(a). When φαl≥180°, the critical flutter wind speed of the system increases by up to 17.3%. In contrast, when φαl<180°, the flutter performance of the deck does not improve significantly by the combined movement of the two flaps, and the maximum reduction in the critical flutter wind speed is −14.8%.Figure 8
Effect of the flap phase difference combination on the flutter performance when the gain coefficientG = 2.
(a)(b)When the gain coefficientG was 2, the optimal combination of the phase difference of the flaps was φαl=270° and φαt=90°, while the worst combination was φαl = 90° and φαt = 45°, which can increase the critical flutter wind speed of the system by 17.3% and reduce the critical flutter wind speed of the girder by −14.8%, respectively.A counter map of the influence of the phase difference on the flutter wind speed was drawn to illustrate the improved ratio of critical flutter wind speed caused by the windward and leeward flaps, as shown in Figure8(b). The color intensities in the figure represent the degree of improvement or decline in the flutter performance. The map is divided into four regions according to the degree to which the phase difference combination improves the flutter performance of the system. These are the “strong improvement area,” “minor improvement area,” “minor decline area,” and “strong decline area” of flutter performance, which are represented by I, II, III, and IV, respectively.In area I,180°<φαl<360°, 0°<φαt<180°, the critical flutter wind speed of the system increased significantly, indicating that the effective control process requires the deck torsional motion to be in the same direction as that of the leeward flap and directed opposite to that of the windward flap. The greatest improvement in the critical flutter wind speed of the system occurred when the phase difference of the windward flap was maintained between 225° and 270°, while that of the leeward flap was maintained between 45° and 90°.In area II, where180°<φαl<360° and 180°<φαt<360°, the critical flutter wind speed of the system increased slightly, indicating that the combination of the windward and leeward flaps continued to benefit the flutter stability of the system. However, when the flutter stability of the system is 180°<φαt<360°, this combination can neither cooperate with the windward flaps to strongly interfere with the flow field around the deck to improve the flutter performance of the system nor dramatically reduce the flow field structure around the system.In area III,0°<φαl<180°, 180°<φαt<360°, the critical flutter wind speed of the system is slightly reduced. The areas III and II differed in the different values of the windward flap, indicating that the phase difference of the windward flap determined the direction of the improvement or deterioration of the flutter performance of the system. Both the windward and leeward flaps can work on the deck, but since the windward and leeward flaps are located upstream and downstream of the deck, respectively, the former directly impacts the flow field around the entire girder, while the latter only functions in the region near the wake of the girder. In addition, the leeward flap has a smaller influence.In area IV,0°<φαl<180°, 0°<φαt<180°, the critical flutter wind speed of the system is significantly reduced. From area III to area IV, only the phase difference of the leeward flap changes, indicating that when 0°<φαl<180°, the flutter performance of the system deteriorates regardless of the value of φαt.The movement of the flaps on both sides corresponding to the phase difference combination in area III and area IV deteriorated the flow field 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 flutter wind speed of the system. In particular, when the motion of the deck was in the same direction as that of the windward flap with a phase difference between 45° and 90° and in the opposite direction to that of the leeward flap with a phase difference in the same range, the critical flutter wind speed of the system decreased drastically.The above analysis demonstrates that the windward flap phase differenceφαl determines whether the flutter performance of the system improves or declines; the leeward flap phase difference φαt determines the extent of improvement or decline.To study the influence of the two flaps on the flutter performance separately, we fixed the phase difference between the torsional motion of the deck and one of the flaps and investigated the influence of the movement of the other, and vice versa. The different curves representing this variation are shown in Figure9.Figure 9
Effect of the phase difference between the two flaps on flutter performance of the system. (a) Phase difference control effect of the leading flap (trailing flap fixed). (b) Phase difference control effect of the trailing flap (leading flap fixed).
(a)(b)Each curve has a noticeable roughly horizontal “S” shape in the first chart, which illustrates that the flutter performance of the system changes in a similar manner under the effect of the phase difference of the windward flap when the phase difference of the leeward flap is fixed. Hence, when 90°<φαl<180°, the critical flutter 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 φαl grows. When 180°<φαl<270°, the critical flutter 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 φαl grows.In contrast to the windward flap, the influence law of the leeward flap on the flutter performance of the deck exhibits no consistent trend: whenφαl = 0°,45°,90°,315°,360°, and 0°<φαt<90°, the critical flutter wind speed of the system decreased gradually, reaching the minimum value at 90°. When 90°<φαt<270°, the critical flutter wind speed of the system exceeds that of the uncontrolled level. When 135°<φαt<270°, the critical flutter wind speed of the system first decreases, then increases, and finally decreases as the phase difference φαt between the leeward flap and the deck increases. At φαt = 90°, the critical flutter wind speed of the system is at its minimum. The critical flutter wind speed of the system was maximum when φαt = 270°.The contributions of the windward and leeward flaps to the flutter performance of the system were different under the simultaneous action of the two flaps. The influence coefficient of the phase difference was defined to determine how the flaps exert a greater influence on the flutter performance of the deck. The corresponding expressions are as follows:(8)FL=RL¯RL¯+RT¯×100%,(9)FT=RT¯RL¯+RT¯×100%,where FL and FT are the influence coefficients of the phase difference between the windward and leeward flaps, respectively, which satisfy FL+FT=1; RL¯ is the average correlation coefficient between the influence curve of the windward flap phase difference and the system flutter performance under different leeward flap phase differences. RT¯ is similar to RL¯, and they are calculated using the following expressions:(10)RL¯=∑i=0,j=i+1i=360,j=360RLij360,(11)RT¯=∑i=0,j=i+1i=360,j=360RTij360,where RLij is the correlation coefficient between the phase difference of the windward flap and the flutter performance of the system when the leeward flap phase differences are i and j, respectively. Similarly, RTij is the correlation coefficient between the leeward flap phase difference and the flutter performance of the system when the windward flap phase differences are i and j. i and j were evaluated at 45° intervals from 0° to 360°. RLij and RTij were calculated using the expressions:(12)RLij=EVLi−VLi¯VLj−VLj¯σVLiσVLj,(13)RTij=EVTi−VTi¯VTj−VTj¯σVTiσVTj,where VLi and VLj represent the critical flutter wind speed when the leeward flap phase differences are i and j, respectively, with the windward flap phase difference varying from 0° to 360° at intervals of 45°. VTi¯ and VTj¯ are the mean values of VLi and VLj, respectively. The critical flutter wind speed of the system corresponding to the leeward flap phase difference from 0° to 360° at intervals of 45° were determined when the windward flap phase differences were i and j. VTi¯ and VTj¯ represent the mean values of VLi and VLj, respectively, while E and σ are the expectation operator and standard deviation operator, respectively.The correlation coefficientsRLij and RTij of the response between the windward and leeward flaps are shown in Figure 10. The upper left area of the figure is the influence area of the phase difference of the windward flap, which represents the correlation coefficient of the phase difference of the leeward flap to the system critical flutter wind speed curve when the corresponding horizontal and vertical coordinates of the phase difference of the windward flap are considered. The lower right area is the leeward flap phase difference influence area, which represents the correlation coefficient of the leeward flap phase difference and the system critical flutter wind speed curve when the corresponding horizontal and vertical coordinates of the leeward flap phase difference are considered. In the influence region of the phase difference of the windward flap, the values of RTij are relatively low. Most of them exhibit a negative or weak correlation, which indicates that the influence of the phase difference of the leeward flap on the critical flutter wind speed can be significantly affected by the phase difference of the windward flap.Figure 10
Correlation between the flap phase difference and critical flutter wind speed.However, the values ofRTij are relatively high in the leeward flap phase influence area, most of which are close to the strong correlation of 1, which illustrates that the phase difference of the leeward flap contributes little to the critical flutter wind speed of the system affected by the phase difference of the windward flap.In the leeward flap phase influence area, the high values ofRTij with most of them approximately 1, indicate that the effect of the phase difference φαl of the windward flap on the critical flutter wind speed of the system is less affected by the phase difference φαt of the leeward flap.According to equations (8) and (9), the influence coefficients FL and FT of the two flaps were 76.7% and 23.3%, respectively. It can be deduced that the windward flap plays a dominant role, whereas the leeward flap is the secondary factor when both flaps are controlled simultaneously.


## 5. Gain Coefficient and Flutter Performance

According to the test results, the windward flap played a leading role when the two side flaps were controlled synchronously. To understand the mechanism by which the gain coefficientG affected the flutter control effect, the phase difference between the torsional motion of the deck and the leeward flap was maintained at 0°. In the case of the windward flap, the phase difference was still evaluated from 0° to 360° at intervals of 45°. The gain coefficient G was calculated from 1 to 9 at an interval of 1, which was maintained between the windward and leeward flaps throughout the entire process. The test results are presented in Figure 11.Figure 11
Effect of the windward flap phase difference and gain coefficientG on the critical flutter wind speed.The experimental results demonstrate that the phase difference either improves or deteriorates the flutter performance of the system, and its extent is determined by the gain coefficientG. In addition, the flutter performance of the system improved by 25.6% at the best and declined by −26.6% at its worst. To reflect the magnitude of the gain coefficient in detail, two groups of typical phase-difference combination values under three different flutter performances were selected for the test. These values are listed in Table 3 and the corresponding results are presented in Figure 12.Table 3
Three types of typical phase difference combinations.
Phase difference combined working conditionsPhase differenceGain coefficientGCase A: Improve systemφαl=225°, φαt=0°, φαl=270°, φαt=0°G=1−9,ΔG=1Case B: No effectφαl=180°, φαt=0°, φαl=135°, φαt=45°Case C: Worsen systemφαl=45°, φαt=45°, φαl=90°, φαt=45°Figure 12
Gain coefficient and flutter performance.The general laws of the influence of the gain coefficientG on the flutter performance of the system can be stated as follows: when the gain coefficient G is very small, the movement of the two flaps has little effect on the flutter performance of the system; as the gain coefficient G increases, both the improvement range under favorable conditions and the deterioration range under unfavorable conditions of the flutter performance increase. However, this characteristic is not valid when the phase difference combination of the two flaps has little effect on the flutter control of the system. When G is too high, the flutter performance of the system tends to worsen, and there is an optimal value of G. The reason for these laws may be that when G is small, the flaps possess a small range of motion and hence, cause little influence on the flow field around the deck; when G is too high, the motion amplitude of the flaps is too high, which increases the overall wind resistance of the system, destroys the streamlined configuration, and deteriorates the flutter performance.In the test, the optimal value ofG was 4, which is consistent with the optimal gain coefficient of 4–8 in the study by Omenzetter et al. [17]. It was also consistent with those given by Phan [42], where the gain coefficients of the windward and leeward flaps were 4 and 3, respectively. However, it was slightly higher than that of Li et al. [34], whose gain coefficient was approximately 2. This is because the design gain coefficient in Li’s investigation was small, so the test model chose a larger gain coefficient range to avoid this shortcoming.


## 6. Flutter Performance under the Independent Control of Two Flaps

The experiment was conducted under the premise of only one side flap control to study the change in the flutter performance of the two flaps. The gain coefficientG and the phase difference between the torsional movements of the flap and deck φα continued to control the movement of the single flap. The system’s flutter wind speed was tested, and the results are summarized by the curved surface, as shown in Figure 13.Figure 13
Effect of the phase difference and gain coefficient of the flaps on the flutter performance of the system.The combined value of the phase difference and gain coefficient that produces the greatest change ratio on the flutter performance during the test is shown in Table4. The fluctuation of the critical flutter wind surface in the figure indicates that the flutter performance of the system depends on the changes in the two parameters.Table 4
Typical combination values for the single flap control effect.
FlapsPhase difference (°)Gain coefficientImproved ratio (%)Windward flap270412.82459−27.2Leeward flap4558.209−31.6The influence of the windward and leeward flaps on the flutter performance of the system under independent control was partitioned as shown in Figure14, and the partition parameters are shown in Table 5. The phase difference of the flutter performance of the system under the independent control of the windward flap was between 180° and 360°, which is consistent with the results obtained under the simultaneous control of the two flaps. This proves that the phase difference of the windward flap φαl affects the flutter performance of the system significantly. The phase difference range of the flutter performance improvement is between 0° and 180° under the independent control of the leeward flap, which is different from that under the simultaneous control of two flaps. When the system was controlled only by the leeward flap, the flow field of the system was only affected by the leeward flap. Therefore, the flutter performance of the system differed significantly from that of the two simultaneously controlled flaps.Figure 14
Influence areas of the phase difference and gain coefficient of the flaps on flutter performance.Table 5
Parameter values of the influence area on the flutter performance.
Influence areaGain coefficientGPhase differenceφαS11 ≤G ≤ 7180° ≤φαl ≤ 360°S27 <G ≤ 90° <φαl ≤ 180°S37 <G ≤ 90° <φαl ≤ 180°S41 ≤G ≤ 7180° ≤φαl ≤ 360°T11 ≤G ≤ 6.50° ≤φαt ≤ 180°T26.5 <G ≤ 9180° <φαt ≤ 360°T36.5 <G ≤ 9180° <φαt ≤ 360°T41 ≤G ≤ 6.50° ≤φαt ≤ 180°When the gain coefficientG was between 4 and 5, the flutter performance could be improved by the movement of both the windward and leeward flaps. When the gain coefficient was too high, the flutter performance of the system declined, which is consistent with the results of the simultaneous control of both the flaps.The gain coefficientG and phase difference φα were fixed to investigate the effect of another parameter on the flutter performance of the system. The effect of the phase difference φα on the flutter performance of the system with different values of the gain coefficient G is shown in Figure 15. When the gain coefficient is 1, the change in the phase difference φα can be neglected, possibly because when the gain coefficient is small, the motion amplitude of the flap is small, and the flow field interference has little effect on the deck.Figure 15
Influence effect of the phase difference and gain coefficients.When the gain coefficients were 3, 5, 7, and 9, the critical flutter wind speed of the system presented an “S” curve as the phase difference of the windward flap increased, namely, the same trend as the simultaneous control of the two flaps. The critical flutter wind speed of the system presented a reverse “S” curve as the phase difference of the leeward flap increased. When 45°≤φαt≤135°, the critical flutter wind speed of the system fluctuated slightly. The phase difference did not influence the critical flutter wind speed of the system significantly.Figure16 shows the influence curve of the gain coefficient G corresponding to different values of the phase difference φα on the critical flutter wind speed of the system. The optimal gain coefficient is 3–5, which is the same as that obtained when the two flaps are controlled simultaneously.Figure 16
Influence effect about gain coefficient and different phase differences.Both the gain coefficientG and phase difference φα significantly affect the flutter performance of the system. To quantitatively describe the leading coefficients of G and φα in the process of simultaneous change, according to equations (8)–(13), the influence coefficients of G and φαl are 48.1% and 51.9%, respectively. The influence coefficients of G and φαt are 48.5% and 51.5%, respectively. The results demonstrate that the phase difference parameter φα and gain coefficient G have the same effect on the flutter performance of the deck.


## 7. Numerical Simulation

The CFD method is used to simulate the parameter combinationφαl = 270°, φαt = 90° to further verify the control effect of the active wing on the system flutter and analyze the reasons.The scale of the bridge model during the numerical simulation was the same as in the wind tunnel test; the incoming wind flow angle of attack was 0°. The natural frequency and damping ratio were also the same as those in the wind tunnel test. The computation domain size was set as [−7B, 10B] × [−5B, 5B] during simulation (B is the width of the flap-deck model), andy+ < 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. The 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 flaps. And a certain distance is reserved between the flaps and the main girder to ensure the deformation of the grid during the movement of the flaps. The simulation first makes the system flutter under the flutter critical wind speed, and the flap motion is turned on when the torsional displacement exceeds a certain level. The torsional vibration time history of the system during the simulation is shown in Figure 18.Figure 17
Computational domain meshing.Figure 18
Time history of torsional displacement of the system under flutter critical wind speed.The vibration starts to decay when the torsional displacement exceeds 8° and the control is turned on, which indicates that the combination of the phase difference parameters of the flaps obtained by the experiment is valid.The pressure distribution of the flow field is extracted from four moments in a vibration period before and after the opening control, as shown in Figure19.Figure 19
Pressure distribution of the flow field (unit: pascal).The four moments before the start of flutter control are a, b, c, and d, and the four moments after turning on the flap control are A, B, C, and D. The results show that there are obvious pressure changes on the upper and lower surfaces of the windward flap, while the leeward flap is located in the wake area of the trailing edge of the box girder, where the flow velocity is low and the force generated by the flap is much smaller than that of the windward flap. This explains why the leeward flap are not so important for the flutter 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 flaps respectively form pressure zones in opposite directions due to the existence of the phase difference and generate control moments opposite to the direction of torsional velocity, which are very helpful for flutter control.The combined active aerodynamic flaps will not only directly generate aerodynamic force on the system but also affect the flow pattern of the system itself, which should be the reason for the greater control potential compared with the separate active aerodynamic flaps. More flow fields analysis will be discussed further in follow-up research.


## 8. Conclusions

In this study, the phase difference and gain coefficient of the flaps on the flutter performance of the system were explored by introducing real-time feedback for a streamlined box girder with a pair of flaps on either side of the deck. The flutter control method of the active flaps improved the flutter performance of the deck by adjusting the movement of the flap in real time according to the motion state of the deck. The phase difference combination of the two flaps and the gain coefficient affected the flutter performance of the system significantly. The conclusions are summarized as follows:(1)
In the combined control of two flaps, the windward flap played a dominant role. Its phase difference determined whether the flutter performance of the system improved or worsened, while the leeward flap played a subordinate role and determined the extent of this control effect;(2)
As the gain coefficientG increased, the critical flutter wind speed of the system first increased and then decreased. The optimal value of G lay between 3 and 5, and the flutter performance of the system declined when its value exceeded 7;(3)
When the phase differences between the windward and leeward flaps and the deck were 270° and 90°, respectively, the flutter performance of the system was optimized, and the critical flutter wind speed increased by 25.6%;(4)
The flutter performance of the system improved when the phase difference of the windward flapφαl was between 180° and 360°. Its influence curve on the critical flutter wind speed of the system was “S”-shaped. The optimal value of φαl 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 affect the flutter control law of active flaps. And when the phase difference parameter between the flaps and the deck is not selected properly, the motion of the aerodynamic flaps will worsen the flutter performance of the system. Therefore, some mechanical measures should be supplemented in the actual case to ensure the reliability of the movement of the flaps according to the determined phase difference. Active aerodynamic flap control based on real-time feedback can be used as the final guarantee for bridge safety. More potential risks will be analyzed in future work.



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*Source: 9970603-2023-02-16.xml*