Flutter Derivative Research Articles

Flutter control optimization for a 5000 m suspension bridge with active aerodynamic flaps: a CFD-enabled strategy

The theoretical model of self-excited aerodynamic force with four active flaps attached to the deck is proposed in this study for a 5000 m bridge with a broad slotted section. The feedback control method is introduced with details, and the phase lag and gain coefficient of the flaps according to the vertical and torsional motions of the deck are used as key control parameters. And the assumption of linear superposition of the aerodynamic forces of the deck and flaps is verified by computational fluid dynamics (CFD) simulations. On this basis, considering the angle of attack (AoA) effect, the flutter derivative of the deck-flap system is identified utilizing forced vibration technology in different states, and the flutter analysis is carried out by the two-dimensional two-degree-of-freedom method. The results indicate that the feedback control based on the vertical and torsional motions can effectively increase the critical wind speed of flutter of the system, and the effective control range of each flap is revealed. Besides, it is found that there is almost no stable phase lag which can make the critical state of the system be improved at all AoAs, the AoA has a great influence on the control parameters. Finally, the control effects using the double flaps are compared, and using the flaps on the outside of the deck gives a better flutter improvement than that using the flaps on the inside of the deck. Flutter performance of deck-flaps system can be improved even more by using more flaps. When the four flaps are activated, the flutter performance of the system can be improved by 45.13% under the unity gain coefficient.

Nonlinear flutter of a narrow truss girder

The nonlinear flutter of a narrow railway truss girder are studied through free vibration and forced vibration wind tunnel tests. Four functions are used to fit the structural amplitude-varying damping ratio under no wind condition, and the fourth-order polynomial has higher goodness-of-fit. The narrow truss girder shows nonlinear flutter of a velocity-restricted type without amplitude hysteresis. At large angles of attack (AoA), the vertical participation and the AoA have little effect on the onset wind speed and amplitude. While at medium AoAs, even a low vertical participation level will increase the onset wind speed and decrease the amplitude. At small AoAs, there is no vibration observed. The flutter derivative A2* is positive only in the limited reduced wind speed and amplitude range. The nonlinearity of self-excited lifting moment is higher only in the limited reduced wind speed and amplitude range, while that of self-excited lift is weaker and becomes stronger with the increasing amplitude and reduced wind speed. The negative aerodynamic damping is mainly provided by the flutter derivative A2* at small or large AoAs. Generally, the nonlinear flutter response is mainly controlled by the first-order component of the self-excited force, and the influence of high-order components can be ignored.

Analysis of Span-Directional Coherence Function and Buffeting Response of a Long-Span Natural Gas Pipeline Suspension Bridge under a Turbulent Wind Field

A long-span natural gas pipeline suspension bridge is prone to buffeting under the action of a turbulent wind field. In order to accurately calculate the buffeting response of the structure under a turbulent wind field, the 1 : 15 segment model wind tunnel test is used to obtain the aerodynamic coefficient and flutter derivative of the bridge deck structure. According to the test results, the buffeting force coherence functions under five different span-directional spacing are fitted. The results show that the buffeting force coherence function corresponding to different wind attack angles has a peak at the corresponding wing grid vibration frequency in the low-frequency region; when the spacing increases to r = 0.51 m or above, the amplitude of coherence function decreases significantly; for the spacing of r = 0.17 m , the buffeting force coherence functions in different directions are obviously different but the corresponding coherence functions of resistance, lift, and torque show a similar curve trend between different wind attack angles. Based on the Scanlan buffeting force correction model, the buffeting response under the reference wind speed of 30.1 m/s is analyzed in the frequency domain and compared with the wind tunnel test results of the whole bridge. The results show that the buffeting response calculated in this paper is in good agreement with the wind tunnel test results of the whole bridge and the buffeting response law is consistent. The maximum value of vertical buffeting response is located near the 1/4 span, and the maximum values of lateral and torsional response are located in the middle of the span. The lateral buffeting displacement response is significantly greater than the vertical buffeting displacement response. Under different wind attack angles, the vertical, lateral, and torsional buffeting displacement responses of the bridge deck structure increase nonlinearly with the increase of wind speed.

Flutter Reliability Analysis of Xiangshan Harbor Highway Cable-Stayed Bridges in Service

With the development of bridge structures towards being light weight and having a large span , the overall flexibility, and, hence, wind sensitivity, of the bridge increases. Flutter is one of the pivotal factors considered in the design and operation stage for long-span cable-stayed bridges due to its devastating impact, often intrigued by relatively low instability caused by wind speed. This paper presents a reliability theory-based numerical analysis on bridge flutter stability and its influence law of key parameters using a real bridge, the Xiangshan Harbor highway cable-stayed bridge in China. The analysis starts with creating a full scale of finite element model for the bridge in service to calculate the flutter derivative and time-dominated combining rational function in order to obtain the critical-flutter wind speed, and then the aerodynamic self-excited forces on the bridge and flutter time-history response are calculated to identify the flutter critical wind speed. Further, the influence of key parameters for flutter reliability, including the stiffness of the main girder, wire breaking rate, damping ratio and cable breakage location are analyzed comprehensively to achieve the change law of critical flutter wind speed with these parameters. Considering the uncertainty of the actual parameters, these parameters are taken as random variables, and the reliability index and failure probability of bridge flutter are calculated according to their probability distribution and the Latin hypercube sampling method. On this basis, a few suggestions are put forward for flutter risk-control during the service of this cable-stayed bridge, which can further enhance the design theory for long-span flexible bridges.

Computational Fluid Dynamics Based Kriging Prediction on Flutter Derivatives of Flat Steel Box Girders

An investigation on the flutter derivative prediction of flat steel box girders is carried out based on CFD simulations. Firstly, by taking the flat steel girder section of Qingshan Yangtze River Bridge as the basic section and considering its width and height as the design variables of cross-section shape, the design domain of cross-section shape is defined by controlling the possible variation range of cross-section design variables. A small number of cross-sections are selected for the calculation of aerodynamic forces by CFD simulations. Secondly, according to the aerodynamic lift and moment time-histories of these steel box girders, of which the flutter derivatives are identified by the least square method. Next, these selected cross-section shape design parameters are used as the inputs, and the flutter derivatives obtained from CFD simulations are used as the outputs to train Kriging models. To improve the prediction accuracy of Kriging models, a modified method of model training is presented. Finally, the flutter derivatives of other cross-sections in the design domain are predicted by using the trained Kriging models, and the predicted flutter derivatives are verified by CFD simulations. It is feasible to directly predict the flutter derivatives of steel box girders by Kriging models.

Bayesian spectral density approach for identification of bridge section’s flutter derivatives operated in turbulent flow

This study presents a Bayesian spectral density approach for identification of flutter derivatives of bridge sections utilizing buffeting displacement responses, where the wind tunnel test is conducted in turbulent flow. Different from traditional time-domain approaches (e.g., least square method and stochastic subspace identification), the newly-proposed approach is operated in frequency domain. Based on the affine invariant ensemble sampler algorithm, Markov chain Monte-Carlo sampling is employed to accomplish the Bayesian inference. The probability density function of flutter derivatives is modeled based on complex Wishart distribution, where probability serves as the measure. By the Bayesian spectral density approach, the most probable values and corresponding posterior distributions (determined by conditions of measured signals) of each flutter derivative can be obtained at the same time. Firstly, numerical simulations are conducted and the identified results are accurate. Secondly, thin plate model, flutter derivatives of which have theoretical solutions, is chosen to be tested in turbulent flow for the sake of verification. The identified results of thin plate model are consistent with the theoretical solutions. Thirdly, the center-slotted girder model, which is widely-utilized long-span bridge sections in current engineering practice, is employed to investigate the applicability of the proposed approach on a general bridge section. For the center-slotted girder model, the flutter derivatives are also extracted by least square method in uniform flow to cross validate the newly-proposed approach. The identified results by two different approaches are compatible.

Aeroelastic stability of a twin-box deck: Comparison of different procedures to assess the effect of geometric details

Focus of this paper is the comparison of three different approaches to assess the aerodynamic stability of twin-box decks: experimental free-motion tests, numerical simulations using quasi-steady theory, and numerical simulations using flutter derivatives. To this end, these approaches are used to compare three deck configurations: a nominal twin-box deck configuration, a variation that includes a different walkway porosity, and a variation that consists in a different shape of the twin-box internal corner. Even though these configurations are geometrically very similar, their aeroelastic behaviour is quite different in terms of critical flutter speed, eigenvalue/eigenvector trend and static deformation as a function of wind speed.It is shown that free-motion tests provide the reference aeroelastic solution for each configuration, but do not provide any insight on the reason of the differences; simulations based on aerodynamic flutter derivative coefficients provide accurate results and insights, but only if unsteady coefficients are measured over all the mean angles of attack of interest; quasi-steady simulations, based on static coefficients slopes, can define quite precisely the aeroelastic behaviour in the nearby of flutter, provided that torsional aeroelastic damping is measured experimentally. Each method has therefore advantages and disadvantages that are highlighted and discussed throughout the paper.

Uncertainty analysis of bridge flutter considering dependence and sampling error in flutter derivative measurements

Analyzing the uncertainty of flutter propagated from different sources especially flutter derivatives is important for the risk assessment of long-span bridges. The uncertainty of flutter derivatives is often characterized based on the experimental data. Dependence and sampling error would therefore exist which may influence the onset prediction of bridges. This paper devotes to developing a framework towards the uncertainty analysis of bridge flutter considering the dependence and sampling error in flutter derivative measurements. In this framework, the dependence among different flutter derivatives is identified with Spearman’s rho instead of conventional Pearson’s product-moment correlation coefficient. A modified procedure based on bootstrap is proposed to characterize the sampling error by representing distribution parameters of flutter derivatives as interval variables. Regarding the probability-interval hybrid uncertainty, a double-layer analysis is employed. In the inner layer, probability of onset velocity is estimated using the point estimate-based probability analysis. In the outer one, the distribution envelope is obtained by the genetic algorithm-assisted interval analysis. Application of the framework to reliability assessment of bridge against flutter is also presented. In the numerical study, the influence of dependence and sampling error is verified in detail. By this framework, the distribution envelope of onset velocity or the interval of reliability index against flutter are expected to contain the true estimates. Consequently, the potential risk with respect to bridge flutter can be assessed in a conservative way.

Limitations of the flutter derivative model

Flutter derivatives identified from transient (free-decay) vibrations might not be suitable for the analysis of bridge flutter. The application of transient flutter derivatives in the flutter analysis relies on two assumptions: (1) transient flutter derivatives and steady-state flutter derivatives are equivalent and (2) aeroelastic effects are superposable. Both assumptions are challenged in this article. It is shown through transient vibration tests that (1) the aeroelastic-coupling between heaving and rotational motions may switch from one pattern to another as the wind speed varies and (2) some of the transient flutter derivatives may be time-varying. The former implies that the predicted flutter type based on transient flutter derivatives may not be unconditionally consistent with the experimentally observed flutter type; the latter implies the transient flutter derivatives may be physically different from the steady-state flutter derivatives. These two issues undermine the basic assumptions of the flutter analysis of bridges. A possible corollary to this study is that if free vibration is used to predict bridge flutter, we should resort to the steady-state (flutter state) vibration instead of the transient vibration of the sectional model. A revision to the aeroelastic force model is proposed to facilitate the discussion.

Experimental and numerical studies on flutter stability of a closed box girder accounting for ground effects

The aerodynamic loads that the bridge deck experiences as a result of interactions with the ground are not well studied when bridges are constructed at low heights from the ground. Flutter stability is the primary goal in the modern bridge wind-resistant design. Therefore, it is necessary to investigate the ground effects on the flutter stability and its influence mechanism. The present study investigated the ground effects on the flutter stability of a closed box girder based on wind tunnel tests and numerical simulations method. First, we presented a vibration measurement system for wind-tunnel test to simulate static ground effects on the bridge girder. Second, we investigated the system torsional damping ratios, the flutter derivatives, and the flutter critical wind speeds and their variation laws correlated with the heights from the ground in two flow fields (two different ground conditions) through a series of wind tunnel tests. The outcomes showed the following: that there is an approximate linear relationship between the height from the ground and the flutter critical wind speed in the uniform flow field, but this does not appear in the turbulent flow field; that the flutter critical wind speed dramatically decreases when the height from the ground is reduced in the uniform flow field; that the ground effects increase the absolute values of H*3 and A*1, thus resulting in faster growth in negative aerodynamic damping induced by the coupling effects of the torsional and heaving motions; and that the flutter derivative A*2 shows a tendency from negative to positive and ground effects make this tendency occur in advance at +3°wind attack angle. Finally, we applied the numerical simulations method to study the ground effect mechanism on the flutter stability. The calculated flutter critical wind speeds are in agreement with test results. The numerical results showed that existence of ground boundary layer generally reduces the flow speed in the region between the deck and the ground and increases both the positive pressure on the stagnation point of the lower surface and the local negative pressure on the upper surface. As a result, the positive and negative pressure increase as the height from the ground decreases, leading to large aerodynamic forces on the section, the flutter critical wind speed consequently decreases.

Sensitivity Analysis of Geometrical Parameters on the Aerodynamic Performance of Closed-Box Girder Bridges.

In this study, the influence of two critical geometrical parameters (i.e., angles of wind fairing, α; and lower inclined web, β) in the aerodynamic performance of closed-box girder bridges was systematically investigated through conducting a theoretical analysis and wind tunnel testing using laser displacement sensors. The results show that, for a particular inclined web angle β, a closed-box girder with a sharper wind fairing angle of α = 50° has better flutter and vortex-induced vibration (VIV) performance than that with α = 60°, while an inclined web angle of β = 14° produces the best VIV performance. In addition, the results from particle image velocimetry (PIV) tests indicate that a wind fairing angle of α = 50° produces a better flutter performance by inducing a single vortex structure and a balanced distribution of the strength of vorticity in both upper and lower parts of the wake region. Furthermore, two-dimensional three-degrees-of-freedom (2D-3DOF) analysis results demonstrate that the absolute values of Part A (with a reference of flutter derivative A2*) and Part D (with a reference of A1*H3*) generally decrease with the increase of β, while the change of the participation level of heaving degrees of freedom (DOF) in torsion-dominated coupled flutter initially increases, reaches its peak, and then decreases with the increase of β.

Effect of stabilizer on flutter stability of truss girder suspension bridges

An aerodynamic optimization measure of the flutter stability of long-span suspension bridges with truss girder is presented in this paper. At first, the improvement of several kinds of central stabilizers and horizontal stabilizers on flutter stability is examined through series of section model and full aeroelastic model wind tunnel tests. Subsequently, the flutter derivatives of the truss girder with and without stabilizer are identified based on two degrees of freedom coupling free vibration method. Furthermore, based on the identified flutter derivatives, the critical flutter velocities of the truss girder section with and without stabilizer are analyzed through two dimensional flutter analysis method and the critical flutter velocities of the full bridge with and without stabilizer are analyzed through three dimensional method. Afterwards, the influence of each flutter derivative on the flutter stability of the truss girder is investigated. The results indicate that central upper stabilizer can effectively increase the critical flutter velocity of the truss girder. In contrast, the central lower stabilizer and horizontal stabilizer have less influence. Setting up central upper stabilizer leads to an obvious decrease in the value of the flutter derivatives A2* and H2*, while the flutter derivatives H1*, H4*, A1* and A3* are little influenced. The two dimensional and three dimensional flutter analysis results agree well with the sectional model and full model wind tunnel test results respectively.

Flutter characteristics of twin-box girder bridges with vertical central stabilizers

It has been well known that Vertical Central Stabilizers (VCS) have the potential of improving flutter performance of long-span bridges. However, the fundamental flutter mechanisms of VCS are still not fully understood so far. In this study, a series of wind-tunnel tests involving the combination of six representative heights and four types of VCS were conducted to fundamentally investigate the influence of VCS on flutter performance of twin-box girders with various Slot Width Ratios (SWRs). Experimental results show that the flutter instability of 20% SWR is significantly sensitive to the height change of VCS, whereas the VCS have little effect on the flutter performance for 80% and 100% SWR. In addition, the results from Two-Dimensional Three Degree of freedom (2D-3DOF) flutter analysis demonstrates that aerodynamic damping Part A with reference of flutter derivative A2∗ makes the greatest contribution to the flutter instability for a 0.8h/H VCS, while the role of Part D with reference of A1∗H3∗ becomes critical for a short VCS (i.e. the ratio of h/H is less than 0.2). Besides, the results of Computational Fluid Dynamics simulation indicate that the geometry of VCS could potentially influence the transforming vortices’ structures and pressure distribution under the central slotting. Finally, the modified Selberg formula presented in this study has the capability of predicting the critical flutter speeds of twin-box girders with various SWRs and VCS.

Insight into coupled forced vibration method to identify bridge flutter derivatives

The flutter derivatives of bridge decks can be efficiently identified using the experimentally and/or numerically coupled forced vibration method. This paper addresses the issue of inherent requirement for adopting different frequencies of three modes in this method. The aerostatic force components and the inertia of force and moment are mathematically proved to exert no influence on identification results if the signal length (t) is integer (n=1,2,3...) times of the least common multiple (T) of three modal periods. It is one important contribution to flutter derivatives identification theory and engineering practice in this study. Therefore, it is unnecessary to worry about the determination accuracy of aerostatic force and inertia of force and moment. The influences of signal length, amplitude, and frequency ratio on flutter derivative are thoroughly investigated using a bridge example. If the signal length t is too short, the extraction results may be completely wrong, and particular attention should be paid to this issue. The signal length t=nT (<TEX>$n{\geq}5$</TEX>) is strongly recommended for improving parameter identification accuracy. The proposed viewpoints and conclusions are of great significance for better understanding the essences of flutter derivative identification through coupled forced vibration method.

Effects of Exponentially Modified Sinusoidal Oscillation and Amplitude on Bridge Deck Flutter Derivatives

With the use of a computational fluid dynamics (CFD) numerical simulation, in this study an unusual exponentially modified sinusoidal (EMS) oscillation was forcibly imposed on deck section to simulate amplitude-varying self-excited forces. This approach facilitated the investigation of the influence of EMS transient oscillation and amplitude on self-excited forces and flutter derivatives. Study results show that for an ideal thin plate, the sensitivities of H2*, H4*, A2*, and A4* to oscillation decaying/diverging ratio (ξ) were higher than those of H1*, H3*, A1*, and A3*. With different amplitudes, ξ, and reduced wind velocities, the self-excited forces of one flat plate and two bluff bridge decks were numerically calculated and the flutter derivatives were extracted and compared with the theoretical and experimental results. If |ξ| < 0.1, its influence was almost negligible, by which the traditional sinusoidal oscillation technique was validated. The heaving and torsional amplitudes had different influences on each flutter derivative. The influences were also related to deck cross sections and reduced wind velocities. The proposed numerical simulation of EMS oscillation enables a better understanding of the influence of oscillation amplitude and ξ on bridge deck flutter derivatives.

Higher-Order Self-Excited Drag Forces on Bridge Decks

AbstractNonlinear features concerning self-excited drag forces induced by the vertical and torsional motions for typical deck sections are investigated comprehensively in this study. The self-excited drag forces are calculated using computational fluid dynamics (CFD)-based numerical simulations. In a conventional analysis framework, the self-excited drag force is modeled as a linear function of the structural motions. However, the simulation results from CFD indicate that in many instances, the second-order (nonlinear) component of the self-excited drag force is more significant than the first-order (linear) component. To enhance the modeling fidelity of a conventional aeroelastic analysis framework on the basis of the semiempirical flutter derivative concept, a nonlinear mathematical model for characterizing both the first- and second-order components is developed to better quantify the self-excited forces and more accurately extract the flutter derivatives. Its efficacy and superiority compared with the...

Some new insights into the identification of bridge deck flutter derivatives

In the identification of flutter derivatives using coupled free vibration technique, the influences of implied approximations are assessed in this study. For a thin flat plate section and three representative deck sections of long-span bridges, KH1∗, KH2∗, KA1∗, KA2∗, K2H3∗, K2H4∗, K2A3∗, and K2A4∗ are demonstrated to be dependent on reduced wind velocity, especially at lower reduced wind velocities. This is contrary to the commonly used assumption of these parameters being constants. Kh is conventionally used to quantify H1∗, H4∗, A1∗, A4∗, and Kα for H2∗, A2∗, H3∗, A3∗. This means the direct components of aeroelastic motions (h1,α2) are comparatively predominant than the incentive cross components (h2,α1). However, this approximation is only valid for weak modal coupling cases at lower wind velocities. Novel perspectives are provided regarding the subtle underlying complexities in extracting flutter derivatives using coupled free vibration technique. The accuracy in identifying flutter derivatives can be effectively improved by setting close frequencies of coupled modes. The mechanical damping ratio, mass, and mass moment of inertia can be artificially increased to postpone the occurrence of flutter and extend the reduced wind velocity region to acquire sufficient flutter derivatives for flutter analysis. This study may serve as a building block for improving the understanding of flutter derivative identification from coupled free vibration responses, and practical strategies are proposed to improve the identification accuracy.

A modification to the flutter derivative model

There is an ambiguity in the original equations for the identification of flutter derivatives. The analysis in this paper shows that, for the two degrees of freedom (2DOF) model, these derivatives in these equations are, in fact, three-dimensional functions instead of one dimensional functions of reduced frequency. Alternative governing equations are proposed in this paper to solve this problem. These new equations “decouple” the aeroelastic coupling effect by approximated modeling strategy. The number of independent aeroelastic parameters is reduced from eight to five. Nonlinear nature of the aeroelastic coupling effect is unveiled. The nonlinear identification method is presented to retrieve the new set of aeroelastic parameters. Wind tunnel experiments were conducted to show the effectiveness of the newly proposed equations.

THE EFFECT OF THE PRESSURE DISTRIBUTION CHARACTERISTICS OF PLATA SURFACE ON THE FLUTTER DERIVATIVES AND VIBRATION FORM

The aerodynamic derivatives were expressed by the fluctuation pressure on the plate surface through rational simplification. The influence of fluctuation pressure distribution characteristics on the aerodynamic derivatives was studied by using of numerical simulation method based on the fluid-solid loosely coupled calculation strategy and existing fluid software user defined function (UDF). The influence of aerodynamic derivatives on the model vibration and aerodynamic coupling phenomenon were explained at micro level. The study shows that aerodynamic coupling flutter derivative which produced different action on the flutter depended on the model surface fluctuating pressure distribution. So the main factor of model flutter was determined by the fluctuating pressure distribution. The main component of fluctuation pressure was drifted to the model windward side in the flutter critical state, which caused the center of torsional vibration move forward and not coincide with the geometric torsion level.

A reliability study for the Messina Bridge with respect to flutter phenomena considering uncertainties in experimental and numerical data

The First-Order Reliability Method is applied to bridge flutter instability considering uncertainties in structural damping, flutter derivatives and extreme wind velocity. The baseline is to set each flutter derivative value as a random variable for computing the failure probability by flutter. The flutter speed is obtained by solving the dynamic equilibrium equation. Hence performance function cannot be expressed explicitly. Reliability sampling methods are not considered in this paper because it would require a large computational time. A modification of Hasofer-Lind–Rackwitz-Fiessler algorithm is proposed for avoiding numerical oscillations. Finally, the methodology proposed is applied to the Messina bridge project.