## 1 Introduction  
With the continuous progress of engineering research and technology, bridge spans are increasing and have broken through 2000 m class, which makes bridges more sensitive to wind and may cause more severe vibration of the main beam [1]. Flutter belongs to large amplitude self-excited vibration, which can cause catastrophic damage. The self-excited vibration is related to the self-excited aerodynamic forces acting on the main beam, and the self-excited aerodynamic forces are related to the structure shape [2, 3] and its motion form [4, 5, 6]. Therefore, the aerodynamic system of the bridge deck with an apparent blunt body shape may have significant nonlinear characteristics under large amplitude motion. At this time, the nonlinear effect of self-excited aerodynamic forces cannot be ignored. The linear self-excited forces model proposed by [PERSON] [7] is utilized as the basis of the current flutter calculation theory, which is based on the assumption of minor disturbance and small amplitude. In fact, due to the influence of aerodynamic forces nonlinearity and structural nonlinearity, the post-flutter behavior of bridge deck may not be divergent vibration, but enter a limit cycle oscillations (LCO) state of stable amplitude. This nonlinear flutter phenomenon, also known as soft flutter, has been found by many scholars in tests [8, 9, 10, 11, 12, 13, 14, 15]. [PERSON] et al. [3] observed the obvious LCO when studying the aerodynamic characteristics of two-edge girders, and considered that the geometric shapes sensitively affect the flutter stability. When studying the aerodynamic response of a rigid flat and rectangular steel plate of chord length and thickness ratio 1:23.3, LCO under coupled-mode flutter was discovered in the wind tunnel by [PERSON] et al. [12], which appeared below the critical flutter velocity. The flutter performance test of streamlined box deck section under a large amplitude was carried out by [PERSON] et al. [16], and the opposite law was found in the variation in heaving and pitching amplitude ratio of coupled LCO and divergent vibration with amplitude. With the development of computer technology and CFD, numerical simulation, as a cost-effective method, is widely used in the study of nonlinear aerodynamic forces [17; 18; 19; 20]. [PERSON] et al. [2] investigated the amplitude dependence of self-excited aerodynamic forces in small amplitude motion state, and the results showed that the flutter derivative was more sensitive to the motion amplitude in the sharp-edge configuration. [PERSON] et al. [21] also performed amplitude dependence studies through two-dimensional numerical simulation. They identified that the pitching amplitude of 10\({}^{\circ}\) can be used as the critical value to distinguish the flow pattern of the flow field around the section. [PERSON] et al. [22] investigated the characteristics of nonlinear aerodynamic forces varying with amplitude, frequency and the angle of attack under single degree of freedom motion. They found that the effect of amplitude on the nonlinear components of pitching motion is generally higher than that of heaving motion. The energy budget analysis of the post-flutter LCO was carried out by [PERSON] et al. [23], using the CFD numerical simulation method. The results show that the higher-order components of the self-excited forces contribute insignificantly in the energy input for the post-flutter LCO of the streamlined box deck. Most of the above analyses of nonlinear self-excited aerodynamic forces involve two issues. Firstly, the current research on nonlinear self-excited aerodynamic forces mainly focuses on flat plate or streamlined box bridge decks, but their nonlinear characteristics are not significant compared with the severely blunted bridge decks. Secondly, the nonlinear characteristics of self-excited aerodynamic forces in heaving motion are weak, and the heaving amplitude range selected in relevant studies is not large enough.