A Study on Post-Flutter Characteristics of a Large-Span Double-Deck Steel Truss Main Girder Suspension Bridge

Abstract

To investigate the nonlinear flutter characteristics of long-span suspension bridges under different deck ancillary structures and configurations, including those with and without a central wind-permeable zone, as well as to analyze the hysteresis phenomenon of a subcritical flutter and elucidate the mechanisms leading to the occurrence of nonlinear flutter, this paper studies first the post-flutter characteristics of the torsion single-degree-of-freedom (SDOF) test systems and vertical bending–torsion two-degree-of-freedom (2DOF) test systems under different aerodynamic shape conditions are further analyzed, and the role of the vertical vibration in coupled nonlinear flutter is discussed. The results indicate that better flutter performance is achieved in the absence of bridge deck auxiliary structures with a central wind-permeable zone. The participation of vertical vibrations in the post-flutter vibration increases with the increase in wind speed, reducing the flutter performance of the main girder. Furthermore, the hysteresis phenomenon in the subcritical flutter state is observed in the wind tunnel experiment, and its evolution law and mechanism are discussed from the perspective of amplitude-dependent damping. Finally, the vibration-generating mechanism of the limit oscillation ring is elaborated in terms of the evolution law of the post-flutter vibration damping.

1. Introduction

With economic and social development and increasing transportation demands, bridge structures are evolving towards ultra-long spans. Ultra-long suspension bridge structures are flexible and highly sensitive to wind-induced vibrations, especially in strong and extreme wind environments, where bridges may experience strong vibrations that can cause damage. For example, under strong turbulent wind speeds, bridges may experience flutter [1]. Typhoons with non-Gaussian and non-stationary characteristics may cause wind-induced vibration problems such as bridge flutter and vibration [2]. Bridge flutter is an aerodynamic instability phenomenon due to the interaction between wind and bridge structures. It is the most hazardous form of vibration and must be strictly prevented in the design of long-span bridges.

In 1940, the old Tacoma Narrows Bridge in the United States experienced prolonged significant vibrations and eventually collapsed due to strong winds of 17 m/s, which was significantly lower than the design wind speed [3]. Initially, researchers attributed its collapse to vortex-induced vibrations. However, further investigation revealed that the vibrations were due to aeroelastic flutter, a self-excited vibration induced by the wind. In recognition of the harm of flutter, flutter prevention has received increasing attention. To date, bridge design criteria for wind resistance have been based on the linear flutter theory established by Scanlan [4,5]. This theory suggests that bridge flutter is a divergent vibration occurring as incoming wind speed exceeds the critical flutter wind speed. According to this theory, the goal in the design stage is to prevent the occurrence of bridge flutter. However, in order to meet the requirements of linear flutter prevention standards, the design and construction costs of large-span bridges will increase substantially [6]. For example, the Akashi-Kaikyo Bridge in Japan has implemented a 14 m high central stabilizing girder to achieve flutter prevention standards.

Due to structural and aerodynamic nonlinearities, many bridge sections exhibit limit cycle oscillation (LCO) behavior rather than the divergent “hard flutter” as suggested by the linear theory, and this behavior is also known as “soft flutter” [7,8]. Engineering practice and relevant studies have shown the inconsistency between the linear flutter theory and the actual bridge flutter vibration. The damage to the old Tacoma Narrows Bridge lasted for nearly 70 min and exhibited large-amplitude anti-symmetric torsional oscillations, with no increase in torsional amplitude after reaching 19° [9]. In addition, torsional vibrations with a maximum amplitude of 6° occurred in 1951 on the Golden Gate Bridge in the United States, which were also considered an example of the soft flutter phenomenon. As early as 1963, Selberg pointed out that a well-constructed large-span bridge can sustain vibrations for several hours in a torsional amplitude of 6°, which is due to the influence of factors such as nonlinear aerodynamic forces. The characteristic of bridge fluttering oscillation is LCO [10]. Therefore, if the flutter protection can be based on post-flutter theory, it is expected to significantly reduce the wind resistance design requirements of bridges, lower construction costs, and provide a broad development space for large-span bridges. In order to achieve these expectations, a deeper understanding of the post-flutter characteristics of typical main-beam sections for large-span bridges is needed.

In recent years, scholars have repeatedly observed typical post-flutter limit cycle oscillation phenomena through section model free vibration in wind tunnel tests. Main-beam sections with pronounced bluff body characteristics typically exhibited “soft fluttering” phenomena, i.e., single-degree-of-freedom (SDOF) torsional vibrations or weak bending–torsional coupled vibrations. There was no obvious critical wind speed point where the vibration amplitude sharply increased. Gao et al. [11] observed significant large-amplitude LCOs in the section model during wind tunnel tests on a double-sided ribbed main-beam section. By determining the aerodynamic parameters, it was concluded that the mechanism of LCOs was due to the negative aerodynamic damping provided by the linear term as an energy source driving the increase in vibration amplitude. The streamlined main-beam sections also exhibited significant bluff body effects at high angles of attack. Zhang et al. [12] studied the flutter characteristics of streamlined main beams at high angles of attack through wind tunnel tests. It was concluded that the steady-state oscillations correspond to a state where the energy absorbed by the linear damping and the energy dissipated by the nonlinear damping remain in equilibrium during oscillations. Li et al. [13] investigated the entire nonlinear flutter process of growth stages, LCOs, and decay stages for different aerodynamic configurations of the leading edge by means of section model experiments. In the presence of significant disturbances, LCOs were observed at wind speeds below the flutter critical wind speed, i.e., hysteresis response of nonlinear flutter [14,15,16,17,18]. In an experimental study of flutter, a thin rectangular cross-section exhibits similar flutter characteristics. Researchers observed hysteresis phenomena in the limit cycle flutter amplitudes of rectangular sections [19,20] and streamlined box beam sections [21,22]. The hysteresis phenomenon usually occurs in the critical state between the static and fluttering behavior of a nonlinear system and is represented by the Hopf bifurcation [23]. Zhang et al. [24] conducted wind tunnel tests with section models and full-bridge aeroelastic models to investigate the post-flutter characteristics of Π-shaped beams. The test results showed that the evolution time of the limit cycle oscillation amplitude decreased with increasing wind speed. Some scholars have also investigated the post-flutter characteristics of large-span bridges through full-bridge aeroelastic model testing [25,26,27,28]. With the continuous development of computer technology, computational fluid dynamics (CFD) has also been applied to study LCOs in post-flutter [29,30].

The linear flutter theory can only solve the flutter critical problem of aerodynamic self-excited systems [31,32]. In order to accurately predict the post-flutter response of large-span bridges, many scholars have proposed nonlinear self-excited force models for typical bridge sections [13,33,34,35,36]. The current research on the nonlinear flutter characteristics of large-span bridges and the development of nonlinear self-excited force models have greatly enhanced the understanding of the sophisticated post-flutter behavior and the formation mechanisms of bridges. However, existing studies on nonlinear post-flutter of large-span bridges primarily focus on typical steel box beam sections, while relatively few studies have been conducted on steel truss main girder sections. Truss main girders are widely used in large-span bridges owing to their strong crossing capacity, excellent stability, and high durability. Wu et al. [23] comprehensively investigated the intrinsic time-varying nonlinear properties and the actual energy feedback mechanism of the nonlinear flutter of trusses in LCOs, as well as the subcritical Hopf bifurcations, through wind tunnel tests on a section model. Double-deck truss girders offer more strength and stability than single-deck ones, making them ideal for long-span structures. They are commonly used for their ability to handle high traffic on two levels, especially in combination highway–railroad bridges and busy transportation hubs. Sun et al. [37] obtained effective aerodynamic measures to suppress the soft flutter of double-deck trusses through a series of wind tunnel tests on the section model. They also experimentally verified that increasing the structural damping ratio can effectively suppress single-degree-of-freedom torsional soft flutter. However, the nonlinear post-flutter characteristics of large-span double-deck steel truss main girders were rarely studied, making it necessary to conduct comprehensive explorations and analyses.

The subject of this study is a super-long-span double-deck truss suspension bridge. Based on section model wind tunnel tests, the post-flutter responses of the main girder under the 2DOF test system with vertical and torsional coupling are investigated. The influence of bridge deck accessories (e.g., railings and maintenance tracks and the central wind-permeable zone in the lower deck) on the nonlinear flutter characteristics of the double-deck truss bridge is also studied. Furthermore, by examining the differences and variations in the flutter response of the double-deck truss bridge under different aerodynamic shapes, the mechanisms behind the limit cycle flutter of the steel truss main girder section are revealed. The remaining sections of this paper are organized as follows: Section 1 introduces the design of the wind tunnel experiments, and Section 2 analyzes the nonlinear flutter characteristics of the double-deck steel truss girder under different aerodynamic shapes. The participation of vertical vibrations in flutter is discussed, and the “subcritical hysteresis phenomenon” is further analyzed to elucidate the evolution patterns of frequency and damping in the wind speed and amplitude space during post-flutter. The conclusions are described in Section 3.

2. Wind Tunnel Test of the Section Model

2.1. Background

The Yanji Yangtze River Bridge is located at the junction of Huanggang City and Ezhou City in Hubei Province, China. The bridge arrangement is shown in Figure 1A with three spans (550 + 1860 + 450 m). The main span (1860 m) is a four-cable, double-deck steel truss girder suspension structure. The main cables are designed with four different cable inclinations. The span of the inner main cable is (550 + 1860 + 450) m, with a vertical sag ratio of f/L = 1/13.058 and a transverse spacing of 35 m; the span of the outer main cable is (510 + 1860 + 410) m, with a vertical sag ratio of f/L = 1/12.130 and a transverse spacing of 41 m. The center distance between the two main cables on one side is 3 m, and the hanger spacing is 9.0 m. The main truss of the steel girder is a Warren truss structure, and the cross-section of the main girder is shown in Figure 1B,C. The truss height is 9.5 m, with a standard section length of 9 m and a center distance of 35 m between the two main truss girders. The upper and lower chord members and web members all have a box-shaped cross-section. The bridge deck is a double-layer structure with six lanes in each direction on the upper deck and four lanes in each direction on the lower deck. Orthotropic steel bridge decks are adopted for both the upper and lower deck systems.

2.2. Wind Tunnel Test Setup

The wind tunnel test of the section model with the double-deck truss main girder was conducted in the high-speed test section of the boundary layer wind tunnel laboratory at Changsha University of Science and Technology. The test section measures 21.0 m (length) × 4.0 m (width) × 3.0 m (height), with a continuously adjustable test wind speed of up to 45 m/s. The turbulence intensity of the uniform flow field is less than 0.5%, meeting test requirements. A scaled section model with a scale ratio of 1:60 was designed and fabricated. The model has a length (L) of 1.58 m, a width (B) of 0.6 m, and a height (H) of 0.158 m. The arrangement of the section model for the wind tunnel test is shown in Figure 2A. To ensure sufficient rigidity of the rigid section model, its skeleton was constructed using a stainless-steel frame. In addition, high-quality PVC panels were used to simulate the outer shape of the main girder for geometric similarity. End plates were installed at both ends of the model to ensure the two-dimensional uniformity of the flow field. The auxiliary components (e.g., maintenance tracks, railings, and water channels) were made of ABS panels, and the shape and air permeability of the railings and guardrails were simulated. During the experiment, four laser displacement sensors were used to measure the vibration displacement response of the section model. To ensure the accuracy of the wind tunnel test results, the equipment used in the experiment was specialized, high-precision instrumentation. The Australian TFI Cobra system was set up at the bridge model’s front for accurate wind speed and direction readings, with 0.3 m/s and 1-degree precision and 16-bit sampling. The displacement sensor was placed at the four corners of the cross-sectional model with a sampling frequency of 500 Hz and was matched with a professional level data acquisition system using the DH5922N dynamic testing system, which is manufactured by Jiangsu Donghua Testing Technology Co., Ltd. and originating in Taizhou, China.

To investigate the bending–torsional coupling characteristics of nonlinear flutter and the influence of the vertical degrees of freedom, an SDOF torsional test system allowing twisting and a 2DOF system allowing bending–vertical and torsional vibrations were designed, as shown in Figure 2B. This apparatus consists of the section model, springs, and restraining wires. The section model (labeled as 4) was suspended in the wind tunnel laboratory by springs (labeled as 7), rigid end bars (labeled as 3), and end axes (labeled as 6). Four horizontal restraining wires (labeled as 1) and four vertical restraining wires (labeled as 2) were used to restrict the lateral and vertical movements of the model. The end plate (labeled as 5) reduces airflow disturbance and ensures the uniformity of the flow field. Horizontally tensioned restraining wires restrict the lateral displacement of the section model, thus avoiding excessive tilting of the springs caused by large lateral displacements at high wind speeds. Similarly, vertically tensioned restraining wires limit the vertical motion of the model. When both restraining wires (labeled as 1 and 2) are applied, the model only performs SDOF torsional motion, representing the SDOF torsional test system. After removing the vertical restraining wires (labeled as 2), the section model converts into a 2DOF bending–torsion coupled system. By comparing and analyzing the post-flutter responses of the two systems, the role of the vertical degrees of freedom in the nonlinear flutter of the double-deck truss section can be further revealed.

Different wind tunnel test conditions are shown in

Table 1. Parameters for the section model wind tunnel test.

Test Conditions	$\mathit{m}$ (kg/m)	$\mathit{I}$ (kg·m2/m)	${\mathit{f}}_{\mathit{h}}$ (Hz)	${\mathit{f}}_{\mathit{\alpha }}$ (Hz)	${\mathbf{\xi }}_{\mathit{h}}$(%)	${\mathbf{\xi }}_{\mathit{\alpha }}$ (%)	Torsion–Bending Ratio	Degrees of Freedom	Bridge Deck Accessory Facilities	Central Wind-Permeable Zone
A1	27.37	10.947	1.322	2.541	0.223–0.546	0.168–0.460	1.922	2DOF	No	Yes
A2	27.37	10.777	/	2.561	/	0.194–0.687	/	SDOF	No	Yes
B1	19.444	14.119	1.186	2.238	0.234–0.625	0.156~0.804	1.887	2DOF	Yes	Yes
B2	19.444	14.012	/	2.246	/	0.159~0.497	/	SDOF	Yes	Yes
C1	30.72	13.467	1.285	2.291	0.211–0.510	0.165–0.513	1.783	2DOF	No	No
C2	30.72	13.020	/	2.330	/	0.161–0.500	/	2DOF	No	No

, with conditions A1, B1, and C1 for the 2DOF system and conditions A2, B2, and C2 for the SDOF system. Conditions B1 and B2 include additional bridge deck auxiliary structures (such as railings and guardrails) to study the influence of bridge deck accessories. Conditions C1 and C2 involve the aerodynamic shape change of the bridge section by closing the central wind-permeable zone to examine the effect of aerodynamic shape. In order to ensure relatively stable damping of the model under sustained large-amplitude oscillations, the method of attaching rubber band damping to the spring proposed by Li [13] was adopted. In Table 1, $m$ and $I$ represent the unit length mass and mass moment of inertia of the section model, respectively; $f_h$ and $f_{\alpha }$ are the vertical and torsional natural frequencies obtained from free decay vibration responses under the windless conditions, respectively; and ${\mathrm{\xi }}_h$ and ${\mathrm{\xi }}_{\alpha }$ represent the vertical and torsional damping ratios, respectively. Since the damping ratio has amplitude-dependent properties, a range of damping ratio values for a certain range of amplitudes is also provided in the table. Finally, the inflow condition is a uniform flow, and the initial wind angle of attack for all test conditions is 0°.

3. Post-Flutter Response Characteristics

3.1. LCO Characteristics

In the wind tunnel free vibration tests, this paper investigates the flutter responses of the double-deck truss section model under different aerodynamic shapes, damping ratios, flexural–torsional ratios, and degrees of freedom. Under various operating conditions, the section model exhibits soft flutter phenomena, with a distinct critical point known as the critical flutter wind speed (Ucr). It is one of the most important criteria for the wind-resistant design of bridges. Regarding the definition of this critical point, the highway and bridge wind-resistance design standard (JTG/T3360-01-2018) [38] clearly defines it as follows: when the amplitude of torsional vibration exceeds 0.5°, it is considered that flutter has occurred.

In order to obtain nonlinear aeroelastic responses with large vibration amplitudes over a certain wind speed range, the section model was subjected to external vertical bending excitation, torsional excitation, or coupled excitation of large vibration amplitude at different test speeds. The original displacement signals obtained in the test were tested using laser displacement sensors arranged on both sides of the segment model, as shown in Figure 2A. During test condition A1, both the vertical displacement and torsional displacement of the model gradually decayed to a very small random vibration state when the incoming wind speed was lower than the critical wind speed (Ucr), as shown in Figure 3. Figure 4 further illustrates the spectral analysis results corresponding to the displacement responses in Figure 3. It can be observed that the predominant frequencies of the vertical and torsional displacements are inconsistent, where the predominant frequency of the torsional displacement is 2.55 Hz, and that of the vertical displacement is 1.33 Hz. Significant changes in frequency and damping can be observed when compared with the windless case. Similar to the windless case, the frequency and damping of the section model still exhibited amplitude-dependent characteristics under low wind speeds, with decreasing frequency and increasing damping with the increase in the vibration amplitude, as shown in Figure 5.

According to the results of the segment model test, it can be concluded that when the incoming wind speed exceeds the critical wind speed (Ucr), both the torsional and vertical displacements of the section model gradually increase from small vibrations, as shown in Figure 6. The increasing rate of amplitude is initially fast and then decreases until reaching a stable amplitude. Then, the amplitude stops increasing and remains at a constant level, indicating the onset of LCO, and the amplitude at this stage is referred to as the steady-state amplitude. Figure 7 further provides spectral analysis results corresponding to the displacement responses in Figure 6. It can be observed that the predominant frequencies of the vertical and torsional displacements are consistent at 2.43 Hz, slightly lower than the torsional structural frequency of the section model under windless conditions (2.54 Hz). This result indicates that typical torsional mode limit cycle vibrations occur in the section with significant vertical displacement. In addition, both the vertical and torsional displacement responses contain high harmonics (e.g., the second, third, and fourth harmonics), which are caused by aerodynamic nonlinearity. However, because of their very small amplitudes, the system response can still be approximated as a simple harmonic oscillation containing only the fundamental frequency component.

It can be observed from Figure 4A and Figure 5A that the frequency of post-flutter vibrations in bridge structures is marginally lower than that of the torsional free vibration frequency in a windless state. The occurrence of this phenomenon can be attributed to the fact that post-flutter vibrations in bridge structures are not confined to simple torsional motion. Instead, they manifest as a sophisticated form of bending–torsional coupling vibrations.

3.2. The Influence of Aerodynamic Shape on Flutter Behavior

3.2.1. The Influence of Bridge Deck Auxiliary Structures on Flutter Performance

The influence of bridge deck auxiliary structures (e.g., railings, guardrails, and maintenance lanes) on the flutter performance of double-deck truss bridge sections can be preliminarily analyzed based on different flutter responses between the completed bridge and the bridge under construction. The section model test results, as shown in Figure 8, indicate that a bridge deck with bridge deck auxiliary structures (test condition B1) has a lower critical flutter wind speed and a greater likelihood of encountering flutter than that without bridge deck auxiliary structures (test condition A1). Under the same wind speed, the steady-state amplitude of the bridge is higher at the completion stage (with auxiliary structures) than at the construction phase (without auxiliary structures). The auxiliary facilities increase the turbulence level of the flow field and allow for adequate separation of the flow field. Consequently, the nonlinear characteristics of self-excitation forces are enhanced. As each deck in the double-deck truss is equipped with bridge deck auxiliary structures, it is significantly influenced by the aerodynamic shape, leading to a more significant increase in self-excitation forces.

3.2.2. The Influence of the Central Wind-Permeable Zone on Flutter Performance

Wind tunnel tests of the section model with and without wind-permeable zones were conducted for SDOF and 2DOF systems. A schematic diagram of the deck segment with a central wind-permeable zone is shown in Figure 9. The variations in the torsional and vertical bending steady-state amplitudes with wind speed are depicted in Figure 10, respectively. With an increasing incoming wind speed, significant LCO occurs under all four conditions, and the steady-state amplitudes in both vertical and torsional directions increase significantly. The critical flutter wind speed is higher for the system with a central wind-permeable zone than that without a central wind-permeable zone. When the wind speed is greater than the critical flutter wind speed (Ucr), the 2DOF system without a central wind-permeable zone has larger steady-state amplitudes for both the torsional and vertical bending motions than that with a central wind-permeable zone at the same wind speed. Therefore, for the double-deck truss bridge section without bridge deck auxiliary structures, a central wind-permeable zone can effectively improve the flutter performance of the bridge.

As shown in Figure 11, the steady-state amplitudes of torsional motion between the SDOF and 2DOF systems with and without a central wind-permeable zone are compared. It can be observed that the critical flutter wind speed of the SDOF system is higher than that of the 2DOF system. When the wind speed is greater than the critical flutter wind speed (Ucr), the steady-state amplitude of the SDOF system is smaller than that of the 2DOF system. Moreover, the steady-state amplitude of both systems increases with increasing wind speed. Further analysis reveals that the increasing trends of the torsional steady-state amplitudes are similar for systems with and without a central wind-permeable zone. The difference in torsional steady-state amplitudes between the two conditions remains within a relatively stable range. In terms of the vertical bending steady-state amplitude, its increasing rate is higher for the case with a central wind-permeable zone than without a central wind-permeable zone. The difference in vertical bending steady-state amplitudes between the two conditions increases with increasing wind speed. Based on the above analysis, vertical vibration plays an important role in flutter, and the degree of vertical vibration involved in flutter varies with wind speed.

To further investigate the role of vertical vibration in flutter, with reference to the idea of constructing quiver pattern vectors [39], further vertical involvement maps were drawn to analyze the vertical vibration involvement of conditions C1 and A1, as shown in Figure 12. The vertical participation coefficient R was defined as the ratio of the root-mean-square of the vertical bending amplitude to that of the torsional amplitude when the main beam experiences soft flutter at the same wind speed. A larger value of R indicates a higher degree of vertical vibration participation. The data collected under high-amplitude torsional excitation were chosen to analyze the vertical participation of the 2DOF system with and without a central wind-permeable zone. The experimental results show that when the incoming wind speed is greater than the flutter critical wind speed (Ucr), the vertical vibration participation coefficient, R, increases as the incoming wind speed increases. Overall, the vertical vibration after flutter is generally higher in the absence of a central wind-permeable zone.

3.3. Post-Flutter “Hysteresis Phenomenon of a Subcritical Flutter”

As shown in Figure 13A,B, a phenomenon known as the “subcritical flutter hysteresis phenomenon” appears in condition A2 during the experimental process. Specifically, the steady-state amplitudes varied with different initial excitations at the same incoming wind speed, or “soft flutter” occurred under large initial excitations, while random and irregular vibrations were observed without initial excitations. This phenomenon is referred to as “subcritical flutter hysteresis”. It is worth noting that all instances of “subcritical flutter hysteresis” occurred in the absence of bridge deck auxiliary structures. Based on this observation, it can be concluded that bridge deck auxiliary systems are an important factor influencing the hysteresis phenomenon. Figure 13C,D show the phase diagrams of the model under the central wind-permeable zone conditions for wind speeds of 8.91 m/s and 9.23 m/s. When a large initial excitation is applied, the amplitude of the model gradually decreases to the outermost red circle. In contrast, the amplitude of the model gradually increases to the inner red circle in the absence of an initial excitation. The difference in amplitude between the inner and outer red circles represents the amplitude discrepancy. As the incoming wind speed increases, the steady-state amplitude of the outermost red circle and the unstable steady-state amplitude of the inner dashed red circle gradually increase, with a gradual overlap between them. This phenomenon represents a sophisticated nonlinear flutter bifurcation, often referred to as the multi-stable amplitude phenomenon. Within a specific amplitude range where the structure exhibits zero total damping, the amplitude of the limit cycle oscillation is contingent upon the initial excitation conditions. Consequently, distinct initial excitations can lead to varied amplitudes in the limit cycle oscillation.

The vibration processes at the two wind speeds were identified, and the overall system damping ratios were obtained, as shown in Figure 13E,F. When the section model starts vibrating without initial excitation, both models gradually transition to an LCO state with smaller steady-state amplitudes in region A (“unstable steady-state amplitude stage”). However, the LCO state is unstable at this point. When a large amplitude initial excitation is applied to the model, the trend of the damage ratio change is Show in stage C, which centers stage LC (“stable steady state amplitude stage”), and the amplitude gradu-ally decays until reaching the stable LCO state. As the incoming wind speed increases, stage B gradually weakens until the model exhibits typical post-flutter phenomena. Due to the narrow torsional amplitude range in stage B, no additional unstable steady-state amplitudes were observed.

3.4. Evolution Law of Post-Flutter Frequency and Damping

Figure 14 shows the evolution of the negative structural damping ratio $-{\xi }_{\alpha }(A_{\alpha })$ and the aerodynamic damping ratio ${\xi }_{wind}(A_{\alpha })$ with the torsional amplitude and wind speed under condition C1. The aerodynamic damping first increases and then decreases with increasing wind speed, achieving a transition from positive damping to negative damping. When the wind speed is below the flutter critical wind speed, the aerodynamic damping remains greater than $-{\xi }_{\alpha }(A_{\alpha })$ within the vibration amplitude range. This result indicates that a stable state is achieved with positive total damping of the system. After applying a certain initial excitation to the system, the vibration eventually decays to zero. However, when the incoming wind speed exceeds the critical flutter wind speed, the curves of the aerodynamic damping ratio and the negative structural damping ratio intersect at different wind speeds, at which the total damping of the system is zero. The intersection points represent the steady-state amplitude of the flutter at various wind speeds after the flutter occurs (indicated by the red intersection line in the figure, denoted as LC). When the amplitude of the section model is smaller than LC (${\xi }_{wind}(A_{\alpha })$ < $-{\xi }_{\alpha }(A_{\alpha })$), the system is in an unstable state with a total damping below zero. The amplitude gradually increases until it reaches LC. When the amplitude is larger than LC (${\xi }_{wind}(A_{\alpha })$ > $-{\xi }_{\alpha }(A_{\alpha })$), the vibration system exhibits periodic oscillations with the total damping above zero, and the amplitude gradually decreases until reaching LC.

As the wind speed increases, the steady-state amplitude shows an increasing trend, which is due to the decrease in the overall aerodynamic damping ratio. As a result, the intersection of ${\xi }_{wind}(A_{\alpha })$ with $-{\xi }_{\alpha }(A_{\alpha })$ needs to be achieved at larger amplitudes, thus increasing the steady-state amplitude of the flutter with increasing wind speed. Based on the overall wind speed and amplitude space in the figure, the surface of the aerodynamic damping ratio intersects with the surface of the negative structural damping ratio, generating an intersection line (indicated by the red line in Figure 14). This line represents the stable boundary of the system in the wind speed and amplitude space, demonstrating that the amplitude of flutter increases with increasing wind speed, which is the characteristic of “soft flutter”. The side with total damping greater than zero represents the stable region of the system, while the side with total damping less than zero represents the unstable region. As can be seen from Figure 14B, the system frequency decreases with increasing wind speed. At the same wind speed, the frequency slightly decreases with increasing torsional amplitude, indicating a minor variation in amplitude characteristics.

4. Conclusions

In this study, the soft flutter characteristics of a large-span double-layer truss bridge under different aerodynamic conditions were studied by conducting large-amplitude tests on a rigid section model. In addition, a detailed analysis was conducted to investigate the impact of various factors on the soft flutter characteristics. The mechanism of the “subcritical flutter hysteresis phenomenon,” the frequency of LCOs, and the evolution of damping during post-flutter were also discussed. The main conclusions are as follows:

• The double-deck steel truss section model exhibits typical LCOs for all test conditions. Under the 2DOF system, the vibration is dominated by the torsional mode with bending–torsional coupling vibration. The vertical bending mode branch and torsional mode branch are separate, and their dominant frequencies are consistent. Influenced by factors such as the aerodynamic shape of the bridge, the participation degree of the vertical vibration near the critical flutter wind speed is uncertain. However, the participation degree increases with increasing incoming wind speed in the post-flutter phase. These research results make up for the lack of research on the nonlinear flutter characteristics of double-deck trusses with four main cables and provide reference for the study of flutter performance of large-span bluff body sections such as π-shaped main beams and steel box beams.
• The bridge deck auxiliary structures have a significant effect on the critical flutter wind speed of the double-deck steel truss section model. Compared to the system without bridge deck auxiliary structures, the system with bridge deck auxiliary structures has a lower critical flutter wind speed and is more prone to soft flutter. After flutter occurs, the steady-state amplitude of the bridge is also influenced by the bridge deck auxiliary structures. Under the same wind speed, the steady-state amplitude is higher in the presence of bridge deck auxiliary structures. The nonlinear limit cycle flutter of the double-deck truss rigid section model in the wind tunnel test is due to the nonlinear dependence of the structural damping and aerodynamic damping on the vibration amplitude. The generation of steady-state amplitudes of LCOs at different wind speeds and the determination of specific amplitudes depend on the competition between structural damping and aerodynamic damping. They vary with amplitude until reaching a balance where they offset each other.
• The “subcritical flutter hysteresis” was observed during the post-flutter stage. The presence of bridge deck auxiliary structures significantly contributes to the occurrence of this hysteresis phenomenon. The different steady-state amplitudes caused by different initial excitations are attributed to the characteristics of the modal damping amplitude variation. This phenomenon can be attributed to the zeros of the total system damping at different amplitudes, where only one of the zeros corresponds to an amplitude that stabilizes the LCO. This research result provides a reference mechanism explanation for the nonlinear flutter of various types of bridges.