## 5 Conclusions  
This paper considers the effects of nonlinear damping and structural stiffness on the limit cycle response of a long-span suspension bridge's flutter systems under supercritical  
Figure 19: The Hopf bifurcation results of the calculation examples in Table 2, the relationship between the amplitude of the LCO and the wind speed, including stable and unstable LCO.  
Figure 18: The Hopf bifurcation results of the calculation examples in Table 2, the relationship between the amplitude of the LCO and the wind speed.  
conditions, focusing on analyzing the influence of strong nonlinear damping and cubic stiffness on limit cycle oscillation at the flutter critical state. The nonlinear flutter self-excited force model proposed by [PERSON] is considered, and the flutter derivatives obtained from wind tunnel tests are fitted using the least squares method to establish the nonlinear flutter self-excited force equation.  
The flutter equation is transformed into state-space equations, and the flutter critical state is solved by analyzing the changes in matrix eigenvalues and obtaining the critical wind speed value. By using the Jacobian matrix of the state-space equations, the changes in matrix eigenvalues at the critical point are analyzed, and the supercritical Hopf bifurcation of the flutter system is demonstrated through vibration responses. Furthermore, the results of the four cases in Table 2 show that the flutter critical values for the systems are the same, indicating that supercritical Hopf bifurcation occurs at the same critical value. Finally, through a comparative analysis of the four case studies, the relationship between the limit cycle amplitude and wind speed under supercritical conditions is analyzed, demonstrating the effective flutter suppression and energy dissipation effects of cubic damping and stiffness.  
The above studies indicate that in terms of engineering practicality, the flutter occurring in long-span suspension bridges is predictable and regular, displaying LCOs with amplitudes increasing as wind speed rises. To mitigate the detrimental effects of flutter on the bridge, increasing vertical bending and torsional damping of the bridge is an effective approach. By employing tuned mass dampers to apply strong nonlinear damping in both vertical bending and torsion motion dimensions, the amplitude of flutter oscillations in the bridge can be significantly reduced, thereby minimizing further damage caused by the displacement and deformation of the bridge. With the advancement of deep learning and machine learning, our research will leverage artificial neural network algorithms to conduct investigations, facilitating the prediction and evaluation of the vibrational response of bridges under aerodynamic forces.  
Conceptualization, J.L. and F.W.; methodology, J.L. and F.W.; software, J.L.; validation, J.L. and Y.Y.; writing--original draft preparation, J.L.; writing--review and editing, J.L., F.W. and Y.Y.; visualization, J.L.; supervision, F.W.; project administration, F.W.; funding acquisition, F.W. All authors have read and agreed to the published version of the manuscript.  
This research was supported by the Key Laboratory of Disaster Forecast and Control in Engineering (Jinan University), MOE of China.  
Not applicable.  
The data presented in this study are available on request from the corresponding author.  
The authors declare no conflict of interest.