# Flutter Mechanism of a Thin Plate Considering Attack Angles
## 3 Flutter Calculation Results for the Thin Plate  
Given the size of the forced vibration device used to test flutter derivatives and the size of the wind tunnel, the length of the flat plate model used in this experiment is 1.1 m. Considering the requirement for blockage ratio after model vibration in the experiment and the size of the forced vibration device, the width of the model is 0.4 m, with a thickness of 0.01 m and a width-to-thickness ratio of 40. The model satisfies the thin plate dimensions in an engineering sense. Additionally, to approximate ideal plate characteristics and reduce flow separation at the plate's leading edge, the model's front and rear edges are rounded with a fillet radius of 10 mm. The cross-section of the model is shown in Figure 3.  
Three different sets of dynamic parameters are used for calculations to evaluate the flutter performance of the thin plate at various angles of attack. Additionally, free-vibration wind tunnel tests are conducted using the same dynamic parameters to obtain flutter test results for comparison with the calculated results, thus validating the accuracy of the calculations. The calculation parameters are shown in Table 1, where \(\alpha_{u}\) represents the initial wind angle of attack, and \(\omega_{a0}/\omega_{h0}\) is the torsional frequency ratio. Figure 3: Section of thin plate (unit: mm). Figure 2: Flow diagram of torsional branch flutter analysis. It should be noted that, in recent years, observations of wind characteristics in the bridge location area have shown that under the influence of natural winds in mountainous areas, the wind attack angle is often greater than 3\({}^{\circ}\), while under the influence of typhoons, the average wind attack angle can reach 7\({}^{\circ}\). In addition, at negative angles of attack, the lift coefficient of the bridge is generally negative, and the aerodynamic force acting on the bridge is downward, which makes the bridge more stable. Generally speaking, the flutter performance at negative angles of attack is significantly better than that at positive angles of attack. Therefore, this article studies the flutter performance of thin flat plates under unfavorable wind attack angles, namely positive attack angles. Flutter derivatives are crucial parameters that characterize the flutter characteristics of bridge cross-sections and are indispensable in flutter calculations. The flutter derivatives of a thin plate are identified at various angles of attack using a forced vibration test device, with the test results shown in Figures 4 and 5. The results indicate that, at a 0\({}^{\circ}\) angle of attack, several critical flutter derivatives--\(A_{1}^{*}\), \(A_{2}^{*}\), \(A_{3}^{*}\), and \(H_{3}^{*}\)--are consistent with the theoretical solutions for an ideal plate. Other non-critical derivatives, except \(H_{1}^{*}\), exhibit certain discrepancies from the theoretical values, with absolute values exceeding those of the ideal plate. Flutter derivatives vary with changes in the wind angle of attack. Notably, \(H_{2}^{*}\), \(H_{4}^{*}\), \(A_{2}^{*}\), and \(A_{4}^{*}\) show significant changes, especially at a 7\({}^{\circ}\) angle of attack, where the curves for \(A_{2}^{*}\) and \(A_{4}^{*}\) exhibit a reversal in trend with the reduced wind speed. Variations in flutter derivatives can significantly impact the flutter characteristics of the cross-section, including flutter wind speed, flutter modes (amplitude ratio, phase difference, and critical wind speed), and driving mechanisms. The comparison between the flutter calculation results and the wind tunnel test results is shown in Table 2. The results demonstrate that the calculated flutter wind speed and frequency match the experimental results, with a maximum calculation error of only 5%. This significant accuracy validates the flutter derivatives as fundamental parameters for revealing the flutter mechanisms of a thin plate at different angles of attack.