# Effects of the Configuration of Trailing Edge on the Flutter of an Elongated Bluff Body
## 1 Introduction  
Flutter is a phenomenon of aerodynamic instability caused by the interaction of fluid and the vibration of structure, which was first found on thin wings. The research of wind-induced vibration of bridge decks began in 1940. Tacoma Narrows Bridge of Washington State, USA, which had just been completed and opened to traffic for 4 months, experienced severe torsional vibration under a low-speed wind of 19 m/s. After more than an hour of vibration, the bridge collapsed. Since then, the wind-induced vibration of the bridge decks has been highly reported, and a large number of research results have been obtained after 80 years of continuous in-depth research. [PERSON] in 1948 [1] regarded the deck of the truss stiffening beam, commonly used in suspension bridges as a flat plate, and ignored the air force on the hollow truss. It was proposed that the [PERSON]'s unsteady aerodynamic theory [2] can be used to predict the vibration of bridge. However, the critical flutter wind speed, calculated based on [PERSON]'s unsteady aerodynamic theory, is much greater than the wind speed when the Tacoma Narrows Bridge collapsed. [PERSON] et al. [3] also used [PERSON]'s function to solve the critical flutter wind speed of the bridge, but for a specific actual bridge section. A reduction factor obtained by wind tunnel tests was introduced to modify the critical flutter wind speed. [PERSON] et al. [4] believed that it is difficult to obtain an aerodynamic expression similar to [PERSON]'s function for bridges with bluff-body section. Therefore, it is recommended to use aerodynamic derivatives to establish a small-amplitude, unsteady aerodynamic model of bridges in which the self-excited aerodynamic force of the main girder section of bridges is approximated by a linear function of the two-dimensional section's motion state vector composed of flutter derivatives, and the determination of the aerodynamic derivatives requires a special wind tunnel test on segment model. The vertical and torsional self-excited aerodynamic force are obtained:  
\[L=\frac{1}{2}\rho U^{2}B\left[KH_{1}^{*}\frac{\dot{h}}{U}+KH_{2}^{*}\frac{B \dot{\alpha}}{U}+K^{2}H_{3}^{*}\alpha+K^{2}H_{4}^{*}\frac{h}{B}\right] \tag{1}\]  
\[M=\frac{1}{2}\rho U^{2}B\left[KA_{1}^{*}\frac{\dot{h}}{U}+KA_{2}^{*}\frac{B \dot{\alpha}}{U}+K^{2}A_{3}^{*}\alpha+K^{2}A_{4}^{*}\frac{h}{B}\right] \tag{2}\]  
where \(L\) and \(M\) are the self-excited aerodynamic forces in the vertical and torsional directions of the main girder of the bridge, respectively; \(H_{i}^{*}\), \(A_{i}^{*}\), and \(i=1,2,3,4\), are the flutter derivative of the main girder; \(\rho\) is the density of air; \(B\) is the width of the bridge; \(U\) is the average wind speed; \(K\) is the converted frequency, \(K=\omega B/U\), where \(\omega\) is the circular frequency; \(h\) and \(\alpha\) are the displacements in the vertical and torsional directions respectively; and \(\dot{h}\) and \(\dot{\alpha}\) are the speeds in the vertical and torsional directions respectively. Since all the motion parameters that have an influence on the self-excited aerodynamic force are considered, this method has strong universality. Therefore, [PERSON]'s flutter theory has been widely used in the design of the flutter stability of bridge structures and has become a classic theory of bridge flutter. In the research of the mechanism of aerodynamic coupling of flutter, [PERSON] et al. [5; 6; 7] proposed the use of step-by-step analysis to analyze bridge flutter. This method divides the analysis of flutter into torsional branch and vertical branch. In each branch, the coupling effects of torsion and vertical motion are considered separately. Not only can the change in system's damping and frequency with wind speed be obtained, but it can also reflect the role of an aerodynamic derivative in the change of vibration characteristics and the flutter derivatives' impact on a system's damping. Based on [PERSON]'s linear flutter theory, [PERSON] et al.