Numerical Simulation and Aerodynamic Energy Analysis of Limit Cycle Flutter of a Bridge Deck

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1 Proceedings of te 018 World Transport Convention Beijing, Cina, June 18-1, 018 Numerical Simulation and Aerodynamic Energy Analysis of Limit Cycle Flutter of a Bridge Deck Xuyong Ying State Key Laboratory of Safety and Healt for In-Service Long Span Bridges, Jiangsu Transportation Institute Co. Ltd. No. 00 Cengxin Ave., Nanjing, P.R. Cina yingxuyong@otmail.com Zen Sun State Key Laboratory of Safety and Healt for In-Service Long Span Bridges, Jiangsu Transportation Institute Co. Ltd. No. 00 Cengxin Ave., Nanjing, P.R. Cina sz1@jsti.com ABSTRACT Fuyou Xu Scool of Civil Engineering, Dalian University of Tecnology No. Linggong Road, Dalian, P.R. Cina fuyouxu@otmail.com Based on ANSYS FLUENT, a fluid-structure interaction (FSI) model as been developed to investigate te limit cycle flutter (LCF) caracteristics of a bridge deck. Its accuracy is verified by simulating te critical flutter state of one tin plate section wit teoretical solutions. Te unsteady flows around an elastically supported deck at different wind speeds and structural damping ratios are simulated. Te LCF penomenon of te bridge deck is well captured by te present numerical model. Te simulated results indicate tat te LCF amplitude increases gradually wit te wind speed until te divergent flutter occurs. Te structural damping as remarkable influence on te critical wind speed and te LCF amplitude. Te caracteristics of aerodynamic energy input and dissipation during LCF are detailedly investigated. Te total mecanical energy varies bounded during LCF, and it remains uncanged in a vibration period. For te LCF occurrence, te work done by aerodynamic forces and te dissipated energy by structural damping sould cancel out. Te developed numerical simulation approac can elp to serve as a building block for developing an overall analysis framework for investigating te LCF caracteristics of long-span bridges. KEYWORDS: Limit cycle flutter; CFD; Bridge deck; Aerodynamic energy; Unsteady flow. 1 INTRODUCTION Flutter analysis is classically based on Scanlan's linear self-excited forces model (Scanlan and Tomko, 1971), in wic te bridge deck undergoes armonic vibration wit net zero damping at critical flutter state. According to te Scanlan's model, te self-excited forces are te linear functions of deck motions and flutter derivatives. Actually, te self-excited forces inevitably contain nonlinear components due to te bluff configurations and large amplitudes, and terefore, nonlinear aeroelastic penomenon may be observed for some cases of bridge flutter. For numerous wind tunnel tests on bluff and/or streamlined deck sections wit large angle of attack, wen te wind speed is sufficiently ig, te decks experience limit cycle oscillation (LCO), rater tan te divergent flutter (Xu and Cen, 008; Long, 010; Zu and Gao, 015). Tis kind of LCO can be classified as te limit cycle flutter (LCF), in wic te deck reaces steady-state vibration for a specific wind speed. Zang 1

2 (007) presented a single-degree-of-freedom (SDOF) nonlinear aerodynamic model, by wic te penomenon of LCF was well explained. Zu and Gao (015) carried out a series of wind tunnel tests on several typical bridge deck section models to investigate teir caracteristics of LCF. Te results sowed tat te LCF is coupled vertical and torsional vibration wit a single frequency. Te usefulness of energy concept in investigating te flutter penomenon is well known. In recent years many researces on explaining te flutter mecanism ave been conducted from te energy standpoint. In order to gain insigt into te understanding of multimode flutter and its generation mecanism, te aerodynamic coupling among modes and its damping or exciting roles are investigated from te viewpoint of system energy by Cen et al. (000). Tey concluded tat te dominant modes for most coupled flutter are te fundamental symmetric vertical and torsional modes. Te coupled self-excited forces acting on te central span are te main source of negative damping tat leads to flutter. Bendiksen (004) presented an analysis of te LCO of aircraft wings in order to investigate te mecanisms of energy transfer from te fluid to te structure. He pointed out tat energy is extracted from te airstream at a sufficient rate to produce rapidly increasing amplitudes in te subsonic flutter. Te LCO was te rule rater tan te divergent flutter in te transonic region, because of te strong aerodynamic nonlinearities induced by moving socks on te wing surface. Furtermore, an aeroelastic mode energy diagram for bending-torsion flutter was presented for explaining te occurrence of LCO in te transonic region. Liu et al. (015) numerically simulated te flutter of a box girder by a fluid-structure calculation strategy and dynamic mes tecnique. Results indicated tat te windward nozzle of te box girder was te main pneumatic energy absorption area. Te flutter stability criterion was also provided from te viewpoint of energy balance. Based on energy analyses, a large amount of researces ave been performed to investigate te flutter mecanisms and control of bridge decks, e.g. Körlin and Starossek (004), Hua and Cen (008), Zang et al. (011), and Li et al. (015). In recent years, te performances of te divergent flutter for bridge decks ave been well studied. However, te LCF penomenon as received less attention. Te intensive investigations on te LCF are of great significance to construct more accurate bridge flutter teory and judging criterion for flutter instability. In present paper, numerical simulation and aerodynamic energy analysis of LCF of a bridge deck are conducted using CFD tecniques. Section presents te fluid-structure interaction (FSI) model and solving strategy. Section 3 describes te structural parameter and geometry of te deck section used in present study. Section 4 focuses on te verification of te present numerical model by simulating te critical flutter state of a tin plate section. Section 5 analyzes te LCF response of te bridge deck for various wind speeds. Te caracteristics of aerodynamic energy input and dissipation during LCF are investigated in Section 6. NUMERICAL MODEL DESCRIPTIO.1 Governing Equations for Fluid Te incompressible, unsteady, -D air flow wit moving boundaries can be modeled by means of te Reynolds Averaged Navier-Stokes (RANS) equations. For te numerical aeroelastic simulation tat contains dynamic meses, an important requirement is te accurate simulation for te interactions between air flow and moving deck section. In te present study, te governing equations are given in Arbitrary Lagrange-Euler (ALE) formulation, wic accommodates te moving boundaries and any subsequent deformation of te underlying discrete mes. By introducing te grid velocity u mj of te moving mes, te ALE formulation for mass and momentum of incompressible fluid may be expressed as ρ ρ( uj umj ) + = 0 (1a) t x were u i or u j, ρ, ρu ρ( uj umj ) u i i p ui + = + ( µ eff ) + Si t x x x x j j i j j (1b) x and p are te fluid velocity components, te fluid density, te Cartesian j spatial coordinates and te fluid pressure, respectively; S i denotes te additional momentum source

3 contributions, if any; µ eff is te effective viscosity wic includes laminar and turbulent contributions (Hassan et al., 010). In te RANS approac, te turbulence viscosity is modeled by te SST k ω model (Menter, 1994). Ying et al. (01) conducted a compreensive simulation of unsteady flow around rectangular cylinders by using different RANS approaces, and te SST k ω model is found to be te best coice among various RANS models and as accuracy enoug to be suitable for practical problems. Furter details on te SST turbulence model implementation may be referred to te reference by Menter (1994). Te ANSYS FLUENT adopted in tis study uses te finite-volume metod (FVM) to solve te fluid governing equations. Te ALE formulation of governing equation enables te conservative fluid calculations wit mes adaptation in time. Te discretization metod of governing RANS equations remains uncanged from te general application of FVM (Scneider and Raw, 1987). Te second order implicit sceme and upwind sceme are used for te time and spatial discretization, respectively. SIMPLE (semi-implicit pressure linked equations) algoritm is used for solving te discretized governing equations. Te time-step is set as Δt=0.001s.. Computational Domain and Mes Arrangement Te computational domain and boundary conditions are scematically sown in Figure 1 for a two dimensional x-y slice. Due to large vibration amplitudes may be concerned, te wide rectangular computational domain is set as m. Te boundaries are sufficiently far away from te sections so as to eliminate te flow obstacle effect on te inflow and outflow boundary conditions. At te inflow boundary, te flow wit a low turbulence intensity of 0.5% is used. A non-slip condition is used for te section surfaces and te top and bottom surface of te domain. It is assumed tat te flow is fully developed at te outlet boundary. Figure 1: Computational domain and boundary conditions (unit: m). Considering te mes number sould be as low as possible to obtain ig efficiency in computation, te ybrid grids are used for te wole computational domain. Te structured quadrangular grid is generated for te zone in te vicinity of deck section, wile te unstructured triangular grid is adopted for te outside zone. Te eigt of m is selected for te first layer close to deck surface and te total number of mes cells is 86136, wic is adequate to resolve te velocity and viscous sublayer. A view of meses near te surface of te deck section is depicted in Figure. Figure : Mes near te surface te deck section. 3

4 For fluid-structure interaction (FSI) problems, te section model is subjected to eaving and/or torsional vibration, and te dynamic mes tecnique is employed to fulfill tis target. Te present wole computational domain is decomposed into two sub-domains tose similar to te early work by Fransos and Bruno (006), as sown in Figure. Te inner oval-saped region is rigidly connected wit te sectional model and move syncronously, wile te mes in outer region deforms at eac iteration time-step. Te spring-based smooting tecnique is adopted to accommodate te deformation of dynamic mes. Not any deformation occurs for te rigid mes region and teir quality remains uncanged trougout te wole calculation process, wic is a benefit especially for solving te viscous sub-layer tat close to te section surface. Te more detailed descriptions for dynamic mes tecnique may be found in (Xu et al., 014)..3 Governing Equations for Deck Motion For a -D rigid section, its planar vibrations can be described in terms of tree displacement components, i.e., p, and α, defined at te sear center. p and are te translational motion displacement in te x and y directions as sown in Figure 3, respectively, and α is te torsional displacement about te sear center. Te aeroelastic drag force, lift force, and twist moment induced by lateral motion ave been generally regarded as insignificant to bridge flutter. Tus, a typical -D section model wit -DOF, i.e., and α, is used in te formulation of present aeroelastic simulation. Te torsional displacement α is taken positive clockwise. Figure 3: Structural reference system. Te deck model is assigned mass (m, I), stiffness (k, k α) and viscous damping coefficients (c, c α), and te motion equations for te section model can be expressed as mt & () + ct & () + kt () = FL() t (a) Iα& () t + c & αα() t + kαα() t = MT () t (b) were t & (), t & (), t () are te vertical acceleration, velocity, and displacement, respectively; α& () t, α& () t, α () t are te torsional acceleration, velocity, and displacement, respectively; FL () t and MT () t are te aerodynamic lift and twist moment acting on te section model, respectively. Te fourt-order Runge-Kutta metod is used to discretely solve te structural motion Eq. (). For te vertical motion, te vertical motion Eq. (a) can be expressed as a first order differential equation: f( &, ) = FL()/ t m ξω t &() ωt () (3) were k ω = = π f (4) m c ξ = (5) km were ω, f, and ξ denote te natural circular frequency, te natural frequency, and te damping ratio of vertical bending, respectively. Te velocity t &() and displacement t () can be obtained by discretizing Eq. (a): t t & ( + t) = t &() + ( K1 + K + K3 + K4) (6) 6 4

5 were ( t) ( ) t ( + t) = t () + t t &() + K1 + K + K (7) 3 6 FL () t K1 = ξω t () ωt () m (8a) F () L () K ξω t K1 ω t () = + + t () m (8b) F ( ) L () t t t t K3 ξω t () K ω = t () t () K m 4 (8c) ( t) FL () t t K4 = ξω t () K3 ω t () t t () K & m & (8d) Te discretized Eq. (6)~(7) can be solved wit te initial conditions of t= &( 0) = 0 and t= ( 0) = 0. Te torsional motion Eq. (b) for te section model can also be solved by te same metod, wic is omitted erein for brevity..4 Te Fluid-structure Coupling Strategy In te FSI analysis for aeroelastic problems, it is significant to satisfy te geometrical compatibility and te equilibrium conditions on te interface between te fluid and te structure (Zang and Hisada, 004). For tis purpose, te strong coupling metod or te weak coupling metod as often been used to solve te FSI system. For te strong coupling metod, sometimes referred to as directly coupling or simultaneous solution by some researcers, te variables of te coupled system is solved and corrected simultaneously. However, te use of strong coupling metod is limited due to te less flexibility between te fluid and structure solver in te time integration. It also needs to modify te existing fluid dynamic solver and structure dynamic solver. Instead of directly solving te coupling system of Eq. (1) and (), a weak coupling metod is used to solve tis fluid-structure coupling problem, by wic te fluid and structural governing equations are sequentially solved and te interface conditions are satisfied in an iterative manner. Terefore, te existing solvers for fluid and structure can be easily adopted, and te flexibility between te fluid and structure solver is maintained. Furtermore, te weak coupling metod requires less storage and computational time compared wit te strong coupling metod. Te flow cart of te computational process is sown in Figure 4. Initially, te fully developed flow around te stationary deck section can be obtained. Meanwile, te current flow condition provides te basis for te subsequent fluid-structure coupling simulation. At tat moment, te corresponding lift force and torsional moment may be calculated by integrating te pressures and sear forces along te section surfaces. Ten te calculated aerodynamic forces are employed as an input into te structural motion solver to predict te displacement and velocity of te deck section. Tis velocity is ten used to determine te new position at te next time-step. Te ANSYS FLUENT re-meses te computational domain in response to new geometry configuration. A series of aeroelastic problems can be efficiently simulated by tis fluid-structure coupling strategy. 5

6 Figure 4: Flow cart of te computational process. 3 DESCRIPTION OF DECK SECTION AND SIMULATION CASES Two representative sections are included in tis study, and teir scaled sections are depicted in Figure 5. Te first section is a tin flat plate, wose teoretical value of critical flutter speed is available, by wic te validity and accuracy of te present numerical model can be verified. Te second section originated from a streamlined cross-section of a steel box girder of te Taouayu selfancored suspension bridge (main span lengt: 406 m) in Cina. Te aeroelastic tests for tis deck section were conducted using te spring-supported section model in te wind tunnel laboratory by Long (010). Table 1 sows te parameters of te section model in te wind tunnel tests. As sown in Long (010), te LCF penomenon was observed at te +5 initial attack angle for te section model witout andrails. At +5 initial attack angle wit different wind speed U 0, te wind-induced response of te deck model can be calculated by te numerical simulations, were te deck model is allowed to vibrate under te specified structural parameters as given in Table 1. Te torsional and vertical responses (include displacement and velocity of te motion) are monitored to evaluate te vibration performances of te deck section. 6

7 Figure 5: Sectional model of tin plate and Taouayu Bridge deck (unit: mm). Table 1: Parameters of te section model. Parameters Symbols Tin Plate Bridge Deck Widt of te section model B (m) Heigt of te section model H (m) Mass of te section model m (kg/m) Mass moment of te section model I (kg.m /m) Natural frequency of vertical motion f (Hz) Natural frequency of torsional motion f α (Hz) Damping ratio of vertical motion ξ Damping ratio of torsional motion ξ α VERIFICATION OF THE PRESENT FSI MODEL Figure 6 sows te calculated time istory of dynamic response for te tin plate section. Te dynamic response of te tin plate almost decay to zero after initial perturbation at U 0 = 19.5 m/s, wile te divergent vibration occurs at U 0 = m/s. It indicates tat te critical flutter speed of te tin plate is between 19.5~19.30 m/s according to te simulated results. Te teoretical analysis results from complex model metod and te results from present simulations are compared in Table. As can be seen, te numerically simulated critical flutter speed and frequency are in good agreement wit te teoretical solutions, wic verifies te accuracy of te present numerical model Dynamic response Dynamic response Torsinal displacement α Torsinal displacement α Vertical displacement /B Vertical displacement /B Time (s) Time (s) (a) U 0 = 19.5 m/s, f=.145 Hz (b) U 0 = 19.3 m/s, f=.143 Hz Figure 6: Time istory of dynamic response, tin plate section

8 Table : Critical flutter wind speed and frequency of te tin plate. Calculation metod Critical flutter wind speed (m/s) Flutter frequency (Hz) Complex model metod Numerical simulation 19.5~ ~ ANALYSIS OF WIND-INDUCED RESPONSE For te case of initial attack angle α 0=+5, te wind-induced response of deck section under different wind speeds can be calculated by conducting a series of numerical simulation. Te torsional and vertical responses (include displacement and velocity of te motion) are monitored to evaluate te wind-induced vibration caracteristics of te bridge deck. Te time istory of torsional and vertical displacements of te deck section for different wind speeds can be seen in Figure 7. (1) At U 0=9 m/s, te motion type is attenuated vibration wit te dynamic response converging to equilibrium position after initial perturbation. Te vertical motion ave large aerodynamic damping wen te wind speed approaces te critical flutter wind speed, results in te rapid attenuation of vertical displacement. () Wen te wind speed reaces U 0=1 m/s, te torsional and vertical displacements increase gradually wit flow time in te beginning, and ten te motion canges from divergent vibration to steady-state vibration. Te maximum of te amplitude keep bounded bot for torsional and vertical motion, wic indicates tat te LCF occurs at U 0=1 m/s. Te aerodynamic energy balance during LCF will be detailed discussed in te following section. (3) For te wind speed U 0=13.5 m/s, te wind-induced displacements rapidly increase wit flow time, wic sows te divergent flutter occurs. It also indicates tat te deck motion continuously extracts energy from te flow field for tis case. Deck displacements α /B W M t (s) (a) U 0 = 9 m/s Deck displacements α /B W M -0.6 Mecanical energy (J) Deck displacements α /B W M t (s) (b) U 0 = 1 m/s t (s) (c) U 0 = 13.5 m/s Figure 7: Time istory of dynamic response and mecanical energy, ξ 0= Te structural damping is one key factor in determining te critical flutter speed of bridge decks (Cen and Kareem, 004). Wit different structural damping ratios, te torsional and vertical displacements at different wind speeds are numerically calculated, and te corresponding steady Mecanical energy (J) Mecanical energy (J) 8

9 amplitudes are sown in Figure 8. Te critical flutter wind speed increases wit te structural damping ratio, wic indicates tat te structural damping is very efficient in controlling te LCF of bridge decks. At te same wind speed, te steady amplitude can be reduced by increasing te structural damping ratio. For different structural damping ratios, te increasing rates of torsional amplitude wit wind speed remain essentially te same. However, for te vertical amplitude, te increasing rate is lower for te case of lower structural damping ratio. Te numerically simulated critical wind speed of LCF is fairly close to tat of te experimental result, bot are around 11 m/s. However, at te same wind speed, te simulated torsional amplitudes are relatively larger tan tat of te experimental results. Tese discrepancies may be ascribed to te following causes: (a) a quasi 3D model was used in wind tunnel tests, and a D model was used in te numerical simulations. Te spanwise correlation effects exist in te experiments, wile it s not considered in te numerical simulations; (b) te structural nonlinearities in stiffness and damping are unavoidable for te experimental model, wic were verified in some studies (Zu and Gao, 015; Lee and LeBlanc, 1986); owever, tese nonlinearities can be perfectly excluded for te numerical model; (c) te model geometry and te wall conditions inevitably ave sligtly differences between experimental tests and numerical simulations. α 0 (deg.) Num., ξ 0 =0 Num.,ξ 0 =0.005 Num.,ξ 0 =0.01 Num.,ξ 0 =0.0 Exp., ξ (m) ξ 0 =0 ξ 0 =0.005 ξ 0 =0.01 ξ 0 = U 0 (m/s) U 0 (m/s) Figure 8: Effect of structural damping on steady-state amplitude: (a) torsional; (a) vertical 6 AERODYNAMIC ENERGY ANALYSIS Let W denote te work done by aerodynamic forces over a specified time interval [ t1, t ], A t W = W + W = M () t α() t F () t () t & + & dt (9) A α t T L 1 were W α, W are te work done by aerodynamic moment and lift, respectively. Te structural damping always does negative work during LCF. Te energy dissipated by structural damping can be expressed as t WD = c () t c () t dt t αα + & & (10) 1 During vibration, te sum of te kinetic and potential energy of te deck section is called its total mecanical energy, and it can be expressed as were E M, K { [ α ] [ ] [ ] } & α α E () t = E () t + E () t = 0.5 I &() t + m t () + k () t + k t () (11) E, M K P E P are te total mecanical energy, kinetic and potential energy of te deck section, respectively. Figure 7 sows te time istory of mecanical energy at different wind speed. At U 0 = 9m/s, wit te attenuation of deck motion, te initial mecanical energy is dissipated by te air flow effectively. For te case of U 0 = 13m/s, te mecanical energy rapidly increase wit flow time, and result in te occurrence of divergent flutter. Wen te LCF occurs at U 0 = 1m/s, te mecanical energy of te deck section increases initially over a finite time interval, and ten it tends to be steady. Te total mecanical energy of te deck section can also be written in Hamiltonian form as s [ (, &, )] (1) E = qp Lq q t M i i i i i= 1 9

10 were Lq ( i, q& i, t) = EK EP is te Lagrangian and p i = L q i is te generalized momentum corresponding to te generalized aerodynamic coordinate q i ; s is te degree of freedom. Since te unsteady aerodynamic forces are non-conservative, te generalized forces acting on te deck section include aerodynamic forces and damping forces, wic can be expressed as d L L Qi ( = ) (13) dt q & q Taking E M wit respect to time, s s s de M d L L L d L L d = ( q& i ) q& i q& i = q& i ( ) = qq & i i = ( WA + WD) (14) dt i= 1 dt q& i qi q& i i= 1 dt q& i qi i= 1 dt Te above equation simply expresses te energy conservation law for te fluid-structure system: EM E0 = WA + WD (15) were E 0 is te initial mecanical energy of te deck section at zero reference point, for example, at t 1=0. Wen integrating over one period of vibration, te energy conservation law becomes EM = WA + WD (16) In te above equation, EM, WA and WD denote mecanical energy increment, aerodynamic work and damping dissipation energy over LCF period. According to Eq. (16), te work done by te aerodynamic forces are partially dissipated by te structural damping, and te remaining work goes into increasing te total mecanical energy of te deck section. Figure 9 sows te time istory of EM, WA and WD at different wind speeds. Let us firstly focus on te divergent flutter case as sown in Figure 9b. W A remains positive value, wic indicates te fluid-structure system always absorbs energy from te air flow during vibration. Te value of WA and W D increase over time. However, te work done by te aerodynamic forces is muc larger tan te damping dissipation energy, i.e., WA + WD> 0, wic lead to occurrence of divergent vibration and eventual failure. For te case of LCF, WD increases over time till te vibration reaces steady state. WA also remains positive value during vibration. In te time interval [, 10], te work done by te aerodynamic forces is larger tan te damping dissipation energy; so te deck section experiences divergent vibration during tis process. Te value of WA first increases ten decreases over time, and it reaces a maximum at t 6s. Altoug te motion amplitude continues to increase in te time interval [6, 10], te aerodynamic work WA decreases over time. At t 10s, te total energy absorbed from te air flow are effectively dissipated by te structural damping, i.e., EM = WA + WD= 0, wic indicates te LCF occurs. Tus, te LCF only occurs if te total mecanical energy of te deck section remains uncanged in a vibration period. From te energy standpoint, we conclude tat WA + WD= 0 Limit Cycle Flutter (17) is a necessary condition for LCF E M W A W D E M W A W D i i Energy (J) Energy (J) t (s) (a) U 0 = 1 m/s Figure 9: Time istory of M t (s) (b) U 0 = 13.5 m/s W, W, ξ 0= E, A D 10

11 7 CONCLUSIONS Based on ANSYS FLUENT, a fluid-structure interaction (FSI) model as been developed to investigate te LCF caracteristics of a bridge deck. Te numerically simulated critical flutter speed and frequency of a tin plate section exibit good agreement wit teoretical solutions, by wic te applicability and accuracy of te numerical approac is validated. Te LCF responses of te bridge deck were also successfully simulated. Te LCF amplitude gradually increases wit wind speed, and eventually te divergent flutter occurs wen te wind speed reaces a specific value. Te critical wind speed of LCF can be significantly increased by increasing te structural damping. However, te structural damping as negligible effect on te frequency of LCF. Te total mecanical energy varies bounded during LCF, and it remains uncanged in a vibration period. Wen te wind speed is sufficient ig, te work done by te aerodynamic forces is larger tan te damping dissipation energy, and te deck section experiences divergent vibration; LCF occurs wen te aerodynamic work done by air flow on te deck section is equal to te dissipated energy by structure damping. REFERENCES Bendiksen, O.O., Transonic Limit Cycle Flutter/LCO. Proc. Of 45t AIAA/ASME/ASCE/AHS/ASC Structures, Structural Dynamics & Materials Conference, California, 004. Cen, X., Matsumoto, M., Kareem, A., Aerodynamic Coupling Effects on Flutter and Buffeting of Bridge. Journal of Engineering Mecanics, 16(1): 17-6, 000. Cen, X., and Kareem, A., Efficacy of tuned mass dampers for bridge flutter control. Journal of Engineering Mecanics, 130(1): , 004. Fransos, D., and Bruno, L., Determination of te aeroelastic transfer functions for streamlined bodies by means of a Navier Stokes solver. Matematical and Computer Modelling, 43(5), , 006. Hua, X.G., Cen, Z.Q., Full-order and multimode flutter analysis using ANSYS. Finite Elements in Analysis & Design, 44(9): , 008. Körlin, R., Starossek, U., Active mass dampers for flutter control of bridges. Proc. of International Conference on Flow-induced Vibrations, Paris, 004. Lee, B.H.K., LeBlanc, P., Flutter analysis of a two-dimensional airfoil wit cubic nonli-near restoring force. Aeronautical Note NAE-AN-36, NRC No. 6438, National Researc Council Canada, Li, K., Ge, Y.J., Guo, Z.W., Zao, L., Teoretical framework of feedback aerodynamic control of flutter oscillation for long-span suspension bridges by te twin-winglet system. Journal of Wind Engineering & Industrial Aerodynamics, 145: , 015. Liu Z.J., Yang, T.X., Ge, Y.J., Zang, W., Numerical simulation of box-girders flutter and pneumatic energy analysis. Engineering Mecanics, 3(9): 58-67, 015. (in Cinese) Long, F., Wind-resistant performance analysis and experimental investigation on long-span selfancored suspension bridge. Dalian University of Tecnology, Cina, 010. (in Cinese) Menter, F. R., Two-equation eddy-viscosity turbulence models for engineering applications. AIAA Journal, 3(8): , Xu, F.Y., Cen, A.R., 3-D flutter analysis of Sutong bridge. Engineering Mecanics, 5(8): , 008. (in Cinese). Scanlan, R. H., Tomko, J., Air foil and bridge deck flutter derivatives. Journal of Engineering Mecanics, 97(6), , Scneider, G., and Raw, M., Control volume finite-element metod for eat transfer and fluid flow using colocated variables 1. computational procedure. Numerical Heat Transfer, Part A Applications, 11(4), , Xu, F. Y., Ying, X. Y., Zang, Z., Tree-degree-of-freedom coupled numerical tecnique for extracting 18 aerodynamic derivatives of Bridge Decks. Journal of Structural Engineering, 140(11), 014. Ying, X.Y., Xu, F.Y., Zang, Z., Numerical simulation and visualization of flow around rectangular bluff bodies." Proc. of BBAA 7 Int. Colloquium on Bluff Bodies Aerodynamics and Applications, Sangai, Cina,

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