# Research on the Coupled Bending-Torsional Flutter Mechanism for Ideal Plate
## 1 Introduction  
Large-span suspension bridges typically exhibit low structural damping and stiffness, making them highly sensitive to both static and dynamic wind loads. Large-span bridge structures can face static wind instability when subjected to static wind loads, which may compromise their load-bearing capacity. Additionally, under dynamic loads, bridges with blunt bodies as the main beam section are susceptible to vortex-induced vibrations, flutter, galloping, and buffeting. These phenomena can adversely affect the structure's fatigue life and overall safety. In 1940, the Tacoma Narrows Suspension Bridge collapsed as a result of wind-induced vibration, thereby initiating a period of research into the wind resistance of modern long-span bridges and leading to the establishment of the discipline of bridge wind engineering. In recent years, with the in-depth development of the research on the flutter performance of bridges, scholars in the field of bridge wind resistance have carried out extensive research on the flutter performance of super-large-span suspension bridges [1], flutter critical wind speed prediction [2, 3], box girder aerodynamic shape optimization [4], post-flutter characteristics [5, 6, 7], nonlinear flutter [8], and bridge flutterperformance in complex wind environments [9; 10]. Compared with the previous research results on flutter, the accuracy and method of flutter critical wind speed identification, flutter morphology analysis, and actual complex wind environments are considered in more detail and comprehensively. However, in order to analyze and solve the flutter problem of long-span bridges in essence, it is necessary to carry out a mechanism analysis of flutter. Since the inception of bridge wind engineering, two dominant theories have emerged to explain the vibration mechanism of large-span suspension bridges: the \"stiffness-driven\" and the \"torsional negative damping drive\" theories. [PERSON] proposed the \"stiffness-driven\" mechanism theory for flutter in streamlined and non-streamlined box girder sections, as well as the \"torsional negative damping-driven\" mechanism theory. The \"stiffness-driven\" mechanism theory suggests that the streamlined box beam section may experience classical aeroelastic flutter similar to that of an aircraft wing. In other words, the stiffness effects caused by high-speed airflow will alter the vertical bending and torsional vibration frequencies of the structure, ultimately coupling them into a unified flutter frequency, thereby driving vibration divergence. The theory of \"negative damping driven by torsional flutter\" suggests that non-streamlined cross-sections are more prone to experiencing torsional flutter. In other words, flowing air creates a negative damping effect on the cross-section's torsional motion. When the critical wind speed for flutter is reached, the air's negative damping overcomes the structure's inherent positive damping, leading to vibration divergence. In the \"torsional negative damping drive\" mechanism, the decoupling of bending-torsion-coupled vibration modes and the source of torsional mode aerodynamic negative damping become the primary issues to be addressed in this theory. For this purpose, various solution methods and theories have been proposed, such as the \"step-by-step solution\" method, the \"incentive feedback\" mechanism theory, and the \"coupled flutter closure solution\" theory. [PERSON] [11; 12; 13; 14; 15; 16; 17], a Japanese scholar, employs a \"step-by-step\" approach to investigate the mechanism of vibration-induced instability. His research specifically centers on modal decoupling in relation to torsional vibrations. [PERSON] [12] conducted wind tunnel experiments to investigate flutter. He subjected both rectangular and H-shaped cross-sections to forced twisting and measured their surface-unsteady pressures. Using a step-by-step approach, he identified the torsional modal aerodynamic damping. His research revealed that the mechanism behind torsional flutter is fundamentally similar to coupled flutter.