## 8 Discussions  
The main purpose and contribution of this study was to propose three improvement measures based on the traditional linear [PERSON]'s flutter model and the least square principle, thus providing a new way of thinking for flutter derivative identification. Firstly, a new method of determining weighting factors in the least square objective function is proposed. Secondly, the original ABC algorithm is improved and applied to flutter derivative identification. Thirdly, the uncertainty of flutter derivatives is quantified by the bootstrap method. Since the model in this paper is based on traditional assumptions and needs further improvement in some aspects, the relevant problems are listed below for further discussion and improvement in future studies. 1. The first issue is the aeroelastic coupling effect of vertical and torsional modes at non-zero wind speed. For a 2-DOF sectional model system, there are always two natural modes (vertical and torsional) at any wind speed, and their frequencies and mode shapes vary with wind speed. Thus, each of the vertical and torsional motion in the free-decay vibration tests at non-zero wind speed contains components of both frequencies, and therefore, there are two sets of eight frequency-dependent flutter derivatives (i.e., 16 unknown parameters) to be identified. However, these two sets of eight flutter derivatives cannot be uniquely identified if only the free vibration response of a 2-DOF bridge sectional model at one wind speed is given. An approximation is always made in most traditional identification methods as well as  
Figure 14: Convergence lines of weighting factors for experimental data. Figure 13: Histogram comparison of flutter derivatives identified from 100 bootstrap samples and 10 original data sets (\(U\) = 8.6 m/s). (**a**) 10 original data sets; (**b**) 100 bootstrap samples. our method (Equations (5) and (6)) that the flutter derivatives related to the vertical motion are dominated by the first frequency component and contrarily those related to the torsional motion are dominated by the second frequency component. This approximation may lead to error or scattering of identified flutter derivatives. This implied approximation in the identification of flutter derivatives has been deeply discussed by [PERSON] and [PERSON] [36]. Their parametric study shows that excellent agreement was obtained despite the approximation in the self-excited forces being made, which supports the efficacy of this approximation. 2. The second issue is the nonlinear vibration characteristics of the bridge deck sectional model. The identification of flutter derivatives involves a step of extraction of the stiffness and damping coefficients in still air, which include both mechanical and aerodynamic components. In most traditional identification methods as well as our method, the mechanical components in still air are conventionally assumed constants, while the aerodynamic components in still air are neglected. These assumptions will inevitably lead to some error of the identified flutter derivatives. Actually, the mechanical damping ratios and natural frequencies of the spring-suspended sectional model system vary to some extent with the change of the oscillating amplitude [37], and the aerodynamic damping ratio and frequency in still air vary with the vibration amplitude as well [38; 39]. It was found that the aerodynamic components were more sensitive to the vibration amplitude than the mechanical components. 3. Treatment of the data at high wind speed especially near flutter onset is a challenging and difficult issue in flutter derivative identification. In the higher wind speed cases, especially in the cases of wind speeds close to the critical point of flutter, the second frequency component is absolutely dominant in both the vertical and torsional responses, and the first frequency component almost vanishes.