Comparisons of bridges flutter derivatives and generalized ones

The causes of the nonlinearity of self-excited aerodynamic force of bridge are interpreted from such two aspects as amplitude and wind velocity. The concept of “generalized flutter derivative” is proposed, and its physical meaning is illustrated. The graphs of the generalized flutter derivatives of plate and Sutong Bridge section model are plotted. The characteristics of all generalized flutter derivatives are compared and analyzed, and their superiorities are verified. The results indicate that the physical meaning of generalized flutter derivatives are more explicit compared to the traditional ones. It is more convenient to understand the nonlinearity properties of self-excited aerodynamic force of bridge according to the generalized flutter derivatives graphs with the wind velocity as the horizontal coordinate. __________ Translated from Journal of Wuhan University of Technology (Transportation Science and Engineering), 2008, 32(4): 607–610 [译自: 武汉理工大学学报(交通科学与工程版)]     

Multimode flutter of long-span cable-stayed bridge based on 18 experimental aeroelastic derivatives

This paper presents flutter analysis of a super-long span (central span 1020 m) cable-stayed bridge based on experimentally extracted 18 flutter derivatives. In the analysis, the aeroelastic forces to be used in the equation of motion utilize rational function approximation of flutter derivatives. The equation of motion is finally recast into aeroelastically modified modal state space form, which yields an unsymmetric eigenvalue problem. The complex eigenvalues thus obtained are used to identify the critical flutter condition. The numerical analysis is performed on a three-dimensional (3D) finite element model of the bridge. The results show that critical flutter velocity based on theoretical flutter derivatives is exceptionally higher than that based on experimental flutter derivatives.     

Identification of aerodynamic parameters for eccentric bridge section model

The equations of motion for bridge deck section model elastically suspended in wind tunnel are formulated about mass center of the system using the Lagrangian approach, accommodating both the elasticity and damping eccentricities in the formulation. The Subsection Extended-Order Iterative Least Square (SEO-ILS) algorithm is developed in the state space for direct identification of system matrices from free vibration data of section model obtained from wind tunnel testing. The flutter derivatives can be extracted straightforwardly from the difference in the system matrices identified at zero wind velocity and at a specific wind velocity, respectively. By making use of complex modal decomposition technique, a procedure is employed to correct the system matrix at zero wind velocity considering both eccentricities. The proposed method is applied to identify the flutter derivatives of a thin plate section model and the section model of a suspension bridge. The results show a favorable agreement between the flutter derivatives of a thin plate obtained with the proposed method and those derived from the analytic formulae. The identified direct flutter derivatives of the suspension bridge section model also are in good agreements with those obtained using Scanlan's method. It is shown that the use of the corrected system matrix at zero wind velocity leads to better accuracy in identifying the flutter derivatives especially at high reduced wind velocity than using the original system matrix, and the eccentricity is found to have more influence on the cross flutter derivatives than on the direct flutter derivatives.     

Comparisons of bridges flutter derivatives and generalized ones

The causes of the nonlinearity of self-excited aerodynamic force of bridge are interpreted from such two aspects as amplitude and wind velocity. The concept of “generalized flutter derivative” is proposed, and its physical meaning is illustrated. The graphs of the generalized flutter derivatives of plate and Sutong Bridge section model are plotted. The characteristics of all generalized flutter derivatives are compared and analyzed, and their superiorities are verified. The results indicate that the physical meaning of generalized flutter derivatives are more explicit compared to the traditional ones. It is more convenient to understand the nonlinearity properties of self-excited aerodynamic force of bridge according to the generalized flutter derivatives graphs with the wind velocity as the horizontal coordinate.     

Flutter mechanism for rectangular prisms in smooth and turbulent flow

The aim of the present work is to clarify the flutter mechanism for suspended long span bridges via a parametric analysis on flutter instability for a set of given deck profiles. Several wind tunnel tests in the DIC-CRIACIV boundary layer wind tunnel (BLWT) have been carried out on spring suspended section models such as rectangular cylinders of different slenderness ratios B/D=5 and 12.5, where B is the longitudinal length of the prism and D is the height of the prism. The main experimental parameters needed for examining whether a given bridge profile is flutter-prone below a certain mean wind velocity are the flutter derivatives, so a system identification procedure (combined system identification method, CSIM) has been developed to extract simultaneously all flutter derivatives from two degrees of freedom (2DoF) section model test results (coupled vertical-torsional free vibration tests). The parametric analysis includes the investigation on (1) the effects of model dynamic properties on BLWT test results, (2) the consequence of turbulence on bridge stability, (3) the possible definition of an aerodynamic stability performance index (β) for rectangular cylinders for designing purposes.     

Comparative and sensitivity study of flutter derivatives of selected bridge deck sections,Part 2: Implications on the aerodynamic stability of long-span bridges

This study is the continuation of a comprehensive investigation on section-model aeroelastic coefficients for bridge decks (flutter derivatives) extracted from wind tunnel section-model tests. The original motivation emerged from the United States—Japan Benchmark Study on Bridge Flutter Derivatives, which promoted a series of systematic comparisons of experimental data extracted by two laboratories (Iowa State University, USA and Public Works Research Institute, Japan) as well as previous results available in the literature. Comparisons, which included both streamlined and bluff deck girder models, were summarized in a companion paper [Sarkar P, Caracoglia L, Haan FL, Sato H, Murakoshi J. Comparative and sensitivity study of flutter derivatives of selected bridge deck sections. Part 1: Analysis of inter-laboratory experimental data. Eng Struct 2009;31(1):159–69]. Differences in the flutter derivatives were mainly attributed to: distinct experimental methods in the wind tunnel (free or forced vibration methods), intrinsic variability between different laboratory environments and effects of amplitude dependency in the tests (for bluff sections).In this paper, a sensitivity study was performed to examine the implications of the perceived dissimilarities among flutter-derivative data sets discussed in [Sarkar P, Caracoglia L, Haan FL, Sato H, Murakoshi J. Comparative and sensitivity study of flutter derivatives of selected bridge deck sections. Part 1: Analysis of inter-laboratory experimental data. Eng Struct 2009;31(1):159–69], on the aeroelastic instability of long-span bridges.Numerical analyses were conducted to evaluate flutter instability boundaries of a set of long-span bridge configurations. Both single-mode and coupled-mode instability were considered, depending on the cross section type and characteristics. It is concluded that uncertainty in flutter derivatives occurring as a result of extraction method or intrinsic variability between different laboratories from negligibly small values to as much as fifty percent, as observed in [Sarkar P, Caracoglia L, Haan FL, Sato H, Murakoshi J. Comparative and sensitivity study of flutter derivatives of selected bridge deck sections. Part 1: Analysis of inter-laboratory experimental data. Eng Struct 2009;31(1):159–69], do not affect the variability in the predicted critical velocity in a proportional way. However, differences in the resulting critical velocities have been observed and estimated from as small as five percent to more than thirty percent, heavily depending on the type of bridge, the simulated conditions and type of instability, either dominated by a single mode or influenced by modal coupling.     

风雨联合作用下大跨桥梁颤振稳定性试验研究

针对风雨联合作用下的大跨桥梁颤振稳定性,以一开槽双箱梁桥梁为研究对象,通过在大气边界层风洞中搭建的风雨联合作用试验系统,完成基于自由振动法的节段模型颤振试验。通过分析不同雨强下该桥梁主梁的颤振导数以及颤振临界风速,进而获取降雨对大跨桥梁颤振稳定性的影响规律。试验结果显示:颤振导数随雨强变化而变化,其中体现扭转气动阻尼特性的颤振导数变化较为显著,随雨强增大,颤振临界风速先增大后减小。试验结果表明:降雨对大跨桥梁的颤振导数以及颤振临界风速均有一定影响。     

根据随机交通状况修正横截面尺寸的大跨桥梁的风致性能

现有的交通状况对细长大跨桥梁(SLB)面板主要影响有两类:1)桥梁横截面尺寸发生改变,这可能会反过来改变颤振导数及作用在桥梁上的风致气动弹性荷载;2)作用于桥梁上的附加动力荷载,包括来自于车辆的动力相互作用。与外部动力荷载——车辆相比,通过改变桥梁横截面尺寸来研究其影响是很少见的。在桥面板上分布车辆模型,在风洞实验室模拟随机交通流对按比例制作的桥梁截面模型进行试验。在风洞试验中通过改变桥梁的横截面尺寸来获得不同的颤振导数,目前的研究是从数值上评估其对大跨桥梁的风致性能,如气动弹性性能、风致响应和潜在疲劳累积性能的影响。     

Direct identification of flutter derivatives and aerodynamic admittances of bridge decks

Flutter derivatives and aerodynamic admittances provide basis of predicting the critical wind speed in flutter and buffeting analysis of long-span cable-supported bridges. In this paper, one popular stochastic system identification technique, covariance-driven stochastic subspace identification (SSI in short), is first presented for estimation of the flutter derivatives and aerodynamic admittances of bridge decks from their random responses in turbulent flow. Numerical simulations of an ideal thin plate are adopted to extract these aerodynamic parameters to evaluate the applicability of the present method. Then wind tunnel tests of a streamlined thin plate model and a Π type blunt bridge section model were conducted in turbulent flow and the flutter derivatives and aerodynamic admittances are determined by the SSI technique. The identified aerodynamic parameters are compared with the theoretical ones and the results indicate the applicability of the current method.     

Estimation of torsional-flutter probability in flexible bridges considering randomness in flutter derivatives

Torsional-flutter instability is an aeroelastic phenomenon of interest to the bridge engineer, corresponding to a torsionally unstable vibration regime of the deck driven by wind excitation and appearing beyond a certain critical wind velocity. In this study a method for the derivation of the flutter probability for long-span bridges with bluff decks is proposed.In the first part of this study the deterministic problem is addressed. In contrast with the classical solution method in the frequency domain based on a numerical procedure for assessing the critical wind velocity, a single-mode “closed-form” algorithm for the derivation of the critical velocity was investigated. A polynomial representation of the aeroelastic-loading coefficients (flutter derivatives), necessary for torsional-flutter analysis, was utilized.In the second part an algorithm for estimating the torsional-flutter probability was developed, considering randomness in bridge properties, and flutter derivatives in particular due to their preeminent role in torsional-flutter velocity estimation.Experimental errors in the extraction of flutter derivatives from wind tunnel tests were analyzed. The “closed-form” algorithm, developed in the first part, allowed for a direct numerical solution of the flutter probability in a simple way.The torsional-flutter probability for three simulated bridge models with rectangular closed-box and truss-type girder deck was numerically determined. A set of experimental data, available from the literature, was employed. The simulations enabled the validation of the proposed algorithm.     

桥梁颤振的随机有限元分析

在众多桥梁颤振分析的实验加理论方法中，ＰＫ－Ｆ法是一种通用性很强的等效颤振分析方法。本文以１８个气动导数的桥梁颤振分析ＰＫ－Ｆ法为基础，将随机有限元法应用于随机桥梁颤振分析中，着重研究自激气动力或气动导数的随机影响，建立了桥梁颤振分析的随机有限元法──随机有限元颤振分析方法，用该方法对江阴长江大桥的随机颤振进行了分析。     

Identification of flutter derivatives of bridge decks with free vibration technique

The flutter derivatives of bridge decks can be determined in a unique manner on condition that the complex modal parameters of the system at one reduced frequency are obtained. Based on the idea, a new method of identifying the flutter derivatives of bridge decks is proposed and it can overcome some shortcoming of the existing method and extend the applicability of the free vibration technique at high wind velocity. The identified results have agreements with the target ones of an ideal thin-plate section and those of a thin-plate section measured by the forced vibration technique. The proposed method is reliable and effective to extract the flutter derivatives from coupled free vibration.     

Parametric study on flutter derivatives of bridge decks

The method for identification of flutter derivatives of bridge decks developed by the authors is first briefly described in this paper. To investigate the effects of dynamic parameters of a bridge deck model on the flutter derivatives a test of the sectional Jiangyin Bridge deck's models with different dynamic parameters was carried out in a boundary layer wind tunnel using the present identification method. In both smooth and simulated turbulent flow conditions, a plate model and the sectional deck model of the Jiangyin Bridge were tested to further survey the effects of turbulence on flutter derivatives. The identified results tend to indicate that the effects of parameters of the model and turbulence on the flutter derivatives are negligible.     

Effect of relative amplitude on bridge deck flutter

Self-excited wind forces on a bridge deck can be non-linear even when the vibration amplitude of the body is small. This phenomenon is evaluated in this paper. Experiments detecting the nonlinearity are performed first, with the concept of “relative amplitude”, i.e. the amplitude of the externally triggered free vibration relative to the envelope of the ambient response of an elastically supported rigid sectional model. Two types of sectional model, a twin-deck bluff model (model A) and a partially streamlined box girder model (model B) are tested with two extreme cases of relative amplitude. Based on the flutter derivatives of model B, a flutter boundary prediction is subsequently carried out on a cable-supported bridge to manifest the changes of critical flutter wind velocity due to different relative amplitudes. The effect of relative amplitude on flutter derivatives and on the flutter boundary reveals, from the structural point of view, a complex relationship between the self-excited forces and the “structural vibration noise” due to turbulence that is inherent in the interaction of the ambient wind with the structure. Although the aeroelastic forces are linear when the body motion due to an external trigger is not affected significantly by this turbulence, they are postulated to be nonlinear when this “vibration noise” cannot be neglected.     

风驱雨作用下桥梁主梁颤振节段模型试验研究

针对风驱雨作用下桥梁主梁的颤振问题,依据风驱雨作用和主梁振动特点,给出了分别考虑雨滴冲击和表面积水后的降雨相似关系,并探讨了其选取原则。选取大跨度桥梁较常采用的典型断面,通过节段模型试验模拟了风驱雨对主梁断面的颤振导数和颤振发生过程的影响。试验结果表明:主梁断面的颤振气动导数随雨强的变化无明显规律,各导数的变化量值相当,随风速增加,降雨引起的导数变化有所加大,但基本没有改变其随风速变化的整体趋势,试验雨强120mm/h时,模型颤振临界风速会有20%～30%左右的提高,但考虑雨强相似比后可以认为降雨对桥梁主梁的风致颤振失稳特征的影响基本可以忽略。     

桥梁气动导纳识别的阶跃函数拟合法

基于桥梁主梁断面气动导数、阶跃函数与气动导纳之间的关系,提出一种获取气动导纳的阶跃函数拟合法。首先根据紊流风场中的抖振响应,识别桥梁结构气动导数和等效气动导纳,然后由气动导数可拟合得到阶跃函数,并根据阶跃函数系数计算得到竖向脉动风对应的气动导纳,最后结合等效气动导纳计算水平向脉动风对应的气动导纳。阶跃函数拟合法直接根据抖振响应完成了桥梁断面完整气动导纳的识别,实例研究表明,该方法对于桥梁断面气动导纳识别而言是可行的。     

Dynamic response of a long span suspension bridge and running safety of a train under wind action

A dynamic analysis model of a wind-train-bridge system is established. The wind excitations of the system are the buffeting and self-excited forces simulated in time domain using measured aerodynamic coefficients and flutter derivatives. The proposed formulations are then applied to a long rail-cum-road suspension bridge. The dynamic responses of the bridge and the train under wind action are analyzed. The results show that the lateral and rotational displacements of the bridge are dominated by wind, while the vertical by the gravity loading of the moving train. The running safeties of the train vehicles are much affected by wind. Under wind conditions of 30–40 m/s, the offload factors, derail factors and overturn factors of the train vehicles exceed the safety allowances, to which great attention should be paid. Translated from Engineering Mechanics, 2006, 23(2): 103–110 [译自: 工程力学]     

Relationships among aerodynamic admittance functions, flutter derivatives and static coefficients for long-span bridges

Wind actions on long-span bridges are commonly considered as the superimposition of buffeting forces and self-excited forces, depending on the aerodynamic admittance functions and on the flutter derivatives, respectively. Since bridge deck sections are bluff bodies, the aerodynamic admittance functions and the flutter derivatives have to be determined experimentally by wind tunnel tests. This paper introduces a generalized quasi-static theory, defining new relationships among the flutter derivatives and the aerodynamic admittance functions. All the relationships are theoretically verified for the zero circular frequency; based upon experimental results, the validation of the relationships among the flutter derivatives is also provided for non-zero values of the frequency.