## 1 Introduction  
In recent years, long-span suspension bridges have gradually developed towards more than 2000 m, and the structure of the suspension bridge has become more slender and flexible [1]. As a typical aerodynamic instability phenomenon, the flutter of a long-span suspension bridge has been paid more attention and has even been a key factor in the design of a \"super\" long-span suspension bridge. The initial research on bridge flutter originated after the shocking collapse of the Tacoma Narrows Bridge in 1940. As a result of this tragedy, wind load effects emerged as an important potential topic for researchers to investigate. As a significant breakthrough, the theory of bridge flutter derivatives was proposed by [PERSON] in 1971 to describe the relationship between the unsteady aerodynamic self-excited forces and the motion states when the flutter occurred [2]. The self-excited forces were regarded as a linear function of the motion states in the theory of flutter derivatives. The flutter derivatives were corresponding linear coefficients, which needed to be identified by experiments or numerical simulations [3, 4, 5, 6]. The linearized self-excited force model was further improved to be available for higher degrees of freedom, and lots of identification methods were developed to obtain the flutter derivatives, which have been successfully applied to evaluate the flutter critical wind speed [7, 8]. To suppress the flutter instability and improve the flutter critical wind speed, a series of passive and active aerodynamic countermeasures, including central slot [9, 10], leading edge [11], central vertical stabilizer [12, 13] and controllable winglets [14, 15, 16] et al., were employed to make some fixed or adjustable modifications on the aerodynamic configuration of the bridge deck. Over the past few decades, nonlinear flutter behavior in the postflutter limit cycle oscillations (LCOs) has gradually drawn the researchers' attention [17, 18, 19, 20, 21, 22, 23, 24, 25]. Different from the classical flutter, the vibration did not tend to diverge but maintain a LCO with a stable amplitude when the inflow velocity exceeded a specific critical value under a big angle of attack or for a bluff aerodynamic configuration [26, 27, 28]. The typical theory of bridge flutter derivatives is not suitable to describe the nonlinear flutter phenomenon. In fact, [PERSON]'s self-excited force model is a linearized simplification of a dynamic system at the critical flutter state with a small amplitude. Nevertheless, the aerodynamic nonlinear effect of self-excited force under large vibration amplitude cannot be ignored. [PERSON] et al. [29] and [PERSON] et al. [30] pointed out that there existed nonlinear components dependent on vibration amplitude in the self-excited force during the flutter process. [PERSON] et al. [17] made a detailed nonlinear analysis of postcritical states of prismatic bodies. A combination of Rayleigh or Van Por and Duffing types of differential equations were introduced to describe the nonlinear postflutter LCO. [PERSON] et al. [19] investigated the nonlinear flutter instability of a two-degree-of-freedom flat plate with wind tunnel tests. A coupled-mode LCO occurred when the wind speed exceeded the critical velocity. Interestingly, a nonlinear hysteresis phenomenon was observed, i.e., a stable LCO was maintained even though the wind speed was from a higher value to a lower value than the critical velocity. A torsional postcritical LCO with a slight heave\(-\)torsion coupling effect was observed for an open twin-side-girder bridge deck in spring-suspended sectional model tests [28]. Based on the measured self-excited forces, a nonlinear polynomial flutter model was proposed and the third-order angular velocity term was regarded as a stabilizing factor to limit the divergent vibration and make the dynamic system come into an LCO. [PERSON] et al.