Bridge section large-amplitude nonlinear self-excitation modeling method based on flow field memory effect

Abstract

The invention relates to a bridge section large-amplitude nonlinear self-excitation force modeling method based on a flow field memory effect, and belongs to the technical field of bridge engineering. Comprising the following steps that S1, a self-excitation aerodynamic force model of a bridge section is established, and the self-excitation aerodynamic force model is formed by superposition of a linear aerodynamic force component and a nonlinear aerodynamic force component; the linear aerodynamic component is calculated by adopting a linear model based on a flutter derivative; the nonlinear aerodynamic component is described through a first-order kinetic equation to represent a memory effect of a flow field on a bridge section movement process; s2, acquiring aerodynamic force time history data of the bridge section under the target amplitude; and S3, based on the aerodynamic force time history data, identifying parameters in the self-excitation aerodynamic force model and coefficients in the first-order kinetic equation corresponding to the nonlinear aerodynamic force component. By means of the method, the problem that in the prior art, a bridge section large-amplitude nonlinear aerodynamic model is indefinite in physical significance is solved.

Description

Technical Field

[0001] This invention relates to a method for modeling large-amplitude nonlinear self-excited forces in bridge cross sections based on flow field memory effect, belonging to the field of bridge engineering technology. Background Technology

[0002] As a key component of modern transportation infrastructure, the wind resistance stability, especially flutter stability, of bridges directly affects the safety and durability of the structure. Flutter is a divergent self-excited vibration caused by the interaction between airflow and structural vibration, and its occurrence often leads to catastrophic consequences. Therefore, establishing an accurate and reliable flutter self-excited force model is the theoretical foundation for bridge flutter analysis and wind-resistant design.

[0003] Currently, the linear self-excited force model based on flutter derivatives, introduced by Professor Scanlan, is widely used in the engineering field. This model linearly describes the aerodynamic lift and lift moment acting on the bridge cross-section through a set of flutter derivatives related to the reduction frequency. This model is based on two fundamental assumptions: small-amplitude vibration and constant incoming flow angle of attack. Under small-amplitude vibration conditions, this model has high accuracy, and due to its simple form and ease of solving in conjunction with the structural motion equations, it has become a classic method in bridge wind-resistant design.

[0004] However, with the continuous increase in bridge span, the structural flexibility increases significantly, and the vibration amplitude under strong winds no longer satisfies the assumption of being "small." Extensive wind tunnel tests and field observations show that under large amplitude conditions, the aerodynamic forces acting on the bridge cross-section exhibit significant nonlinear characteristics. These include the presence of higher harmonic components in the aerodynamic time history, variations in aerodynamic damping with amplitude, and the potential for "soft flutter" (limit cycle oscillation) with stable amplitude. At this point, the Scanlan model, based on linear assumptions, can no longer accurately describe the true behavior of aerodynamic forces, and its predictions may contain significant errors, failing to meet the requirements of refined wind-resistant design for long-span bridges.

[0005] To overcome the limitations of linear models, scholars both domestically and internationally have proposed various nonlinear aerodynamic models. For example, some researchers have constructed nonlinear expressions based on transient angles of attack and their rates of change; others have used the van der Bohr equation to describe soft flutter phenomena; still others have used Volterra functional series or Taylor series expansions of the equilibrium positions and superposition of different harmonic components to fit nonlinear aerodynamic forces. These models have, to some extent, improved the ability to describe aerodynamic responses under large amplitudes.

[0006] However, a deeper analysis of these existing nonlinear models reveals a common problem: they are mostly based on mathematical fitting of experimental data or borrowing from known nonlinear equations, with their modeling approach focusing on finding mathematical functions that match the data curves. These models typically lack characteristic variables that clearly reflect the physical mechanisms of the flow field, and the physical relationship between model parameters and flow structures (such as the generation, evolution, and shedding of vortices) is ambiguous. In fact, the nonlinear nature of aerodynamics stems from the strong fluid-structure interaction between the structure and its surrounding flow field under large-amplitude vibrations, particularly the continuous influence of the large-scale "moving vortex system" excited by structural motion on the pressure distribution on the cross-sectional surface—the fluid's "memory effect." Existing models fail to effectively embed this key physical mechanism into their framework, resulting in unclear physical meaning and limited extrapolation and mechanistic explanation capabilities under different flow conditions.

[0007] Therefore, there is an urgent need in this field for a new nonlinear aerodynamic modeling method that can both inherit the reliability of the classical linear model under small amplitudes and clearly characterize the "memory effect" of the flow field dominated by the moving vortex system under large amplitudes and the resulting nonlinear aerodynamic characteristics, thereby providing a more accurate and clearer theoretical tool for the wind resistance stability analysis of long-span bridges. Summary of the Invention

[0008] The technical problem to be solved by this invention is to provide a modeling method for large amplitude nonlinear self-excited forces of bridge cross sections based on flow field memory effect, which solves the problem of unclear physical meaning of large amplitude nonlinear aerodynamic models of bridge cross sections in the prior art.

[0009] The technical problem to be solved by this invention is achieved by the following technical solution:

[0010] A method for modeling large-amplitude nonlinear self-excited forces in bridge cross-sections based on flow field memory effect includes the following steps:

[0011] S1. Establish a self-excited aerodynamic model of the bridge section, wherein the self-excited aerodynamic model is composed of the superposition of linear aerodynamic components and nonlinear aerodynamic components;

[0012] The linear aerodynamic components are calculated using a linear model based on flutter derivatives.

[0013] The nonlinear aerodynamic component is described by a first-order dynamic equation. The structure of this equation makes the current value of the nonlinear aerodynamic component depend on both the current motion state of the bridge section and the historical evolution of the nonlinear aerodynamic component, thereby characterizing the memory effect of the flow field on the motion history of the bridge section.

[0014] S2. Obtain the aerodynamic time history data of the bridge cross section under the target amplitude;

[0015] S3. Based on the aerodynamic time history data, identify the parameters in the self-excited aerodynamic model, including the flutter derivative in the linear aerodynamic component and the coefficients in the first-order dynamic equation corresponding to the nonlinear aerodynamic component.

[0016] The present invention is further configured such that the step of determining the nonlinear aerodynamic component includes:

[0017] The nonlinear aerodynamic component is regarded as the output of a dynamic system, and the state update equation of the dynamic system is a first-order differential equation or its first-order discrete-time difference equation.

[0018] The structure of the state update equation is as follows: the current value of the nonlinear aerodynamic component is equal to the product of a coefficient representing memory decay and its previous value, plus a nonlinear function value with the motion state variable of the bridge section at the current moment as input.

[0019] The present invention is further configured such that the nonlinear function is a polynomial function relating to the displacement and velocity of the bridge cross-section.

[0020] The present invention is further configured such that: in step S3, identifying the coefficients in the first-order dynamic equation corresponding to the nonlinear aerodynamic component specifically includes:

[0021] S31. Identify the flutter derivative based on the dynamic response time history data under small amplitude;

[0022] S32. Subtract the linear aerodynamic components calculated using the flutter derivative from the aerodynamic time history data under the target amplitude to separate the nonlinear aerodynamic time history data;

[0023] S33. Based on the nonlinear aerodynamic time history data and the corresponding bridge cross-section motion time history data, the coefficients in the first-order dynamic equation are identified by a system identification method.

[0024] The present invention is further configured such that: in step S33, before using the system identification method, the first-order dynamic equation is first converted into a discrete-time difference equation.

[0025] The present invention is further configured such that, in step S33, the system identification method is:

[0026] Stepwise regression analysis was used, with the nonlinear aerodynamic time history data as the dependent variable and the product terms of different orders of the bridge cross-section motion state variables as candidate independent variables, to screen and determine the polynomial expression form and order of the nonlinear function.

[0027] Based on the determined polynomial expression, the least squares method is used to make a preliminary estimate of all unknown parameters in the first-order dynamic equation;

[0028] Based on the initial estimates obtained, the parameters are optimized and identified using the recursive maximum likelihood method, and white noise sequence is introduced during the identification process to correct the model error.

[0029] The present invention is further configured such that: in step S2, the aerodynamic time history data is obtained through computational fluid dynamics numerical simulation; the numerical simulation includes:

[0030] Dynamic mesh technology is used to simulate the forced vibration of a bridge cross section at a predetermined amplitude and frequency;

[0031] The unsteady flow field was solved using the SST turbulence model;

[0032] The scale of the near-wall mesh satisfies To ensure the accuracy of boundary layer resolution;

[0033] The time step is set so that each vibration cycle contains at least 100 calculation steps.

[0034] The present invention is further configured such that the target amplitude satisfies one of the following conditions:

[0035] The vertical vibration amplitude of the bridge cross section shall not be less than 0.5% of its cross-sectional width;

[0036] The amplitude of torsional vibration of the bridge cross section shall not be less than 3 degrees.

[0037] At this amplitude, through numerical simulation or experimental observation, a moving vortex system can be identified, which is directly caused by structural vibration, has a scale that is significantly correlated with the amplitude, and dominates the flow field structure near the cross section.

[0038] The present invention is further configured such that: the self-excited aerodynamic model is used to calculate the aerodynamic lift and / or aerodynamic lift moment acting on the bridge cross section.

[0039] The beneficial effects of this invention are:

[0040] 1. This invention, for the first time, introduces the "memory effect" of flow field on the motion history of bridge cross-sections into the modeling of nonlinear self-excited forces. It describes the nonlinear aerodynamic components through first-order dynamic equations, ensuring that their current values ​​depend not only on the current motion state of the cross-section but also on its historical evolution. This modeling method directly corresponds to the continuous evolution and influence of large-scale "moving vortices" excited by structural vibrations under large amplitudes in the flow field. It explains the root cause of nonlinear aerodynamic forces from a physical mechanism perspective, overcoming the problems of existing nonlinear models that rely heavily on mathematical fitting and lack clear flow physics meaning. This enhances the model's mechanistic explanatory power and extrapolation applicability.

[0041] 2. This invention employs a superposition model structure of "linear components + nonlinear components." The linear component follows the classic flutter derivative model, ensuring consistency with existing theories under small amplitude conditions and maintaining the model's engineering inheritance and reliability. The nonlinear component is independently described by first-order dynamic equations, enabling flexible capture of nonlinear characteristics such as high-frequency harmonics and amplitude-dependent damping under large-amplitude vibrations. This structure retains the simplicity and practicality of the linear model while significantly improving the modeling accuracy and adaptability under large amplitude conditions by introducing physically meaningful nonlinear dynamic processes.

[0042] 3. Verification through CFD numerical simulations and wind tunnel tests shows that the model established in this invention significantly outperforms traditional linear models in predicting aerodynamic time histories, spectral characteristics, and hysteresis curves under large amplitude conditions. Especially within the large amplitude range where the vertical amplitude is ≥0.5% of the cross-sectional width or the torsional amplitude is ≥3°, this model accurately captures the higher harmonic components and nonlinear amplitude-frequency characteristics of aerodynamic forces, providing a more reliable theoretical tool for nonlinear flutter response analysis, limit cycle oscillation prediction, and wind resistance safety assessment of bridge structures under strong winds. Attached Figure Description

[0043] Figure 1 This is a flowchart illustrating the overall steps of the present invention.

[0044] Figure 2 This is a structural diagram of the bridge segment model in this invention.

[0045] Figure 3 This is a structural diagram of the box girder model in this invention.

[0046] Figure 4 This is the area for numerical calculation of the box girder cross-section in this invention.

[0047] Figure 5 The y-axis of the near-wall mesh of the box girder cross-section in this invention. + value.

[0048] Figure 6 This refers to the flow characteristics around the box girder cross-section in a static state in this invention.

[0049] Figure 7 This is the time history of the aerodynamic force under the conditions of 0.05m amplitude and 9.5Hz frequency of the present invention.

[0050] Figure 8 This is the time history of the aerodynamic force under the conditions of 0.15m amplitude and 3.5Hz frequency of the present invention.

[0051] Figure 9 This invention describes the frequency-amplitude characteristics of aerodynamic forces under the conditions of 0.15m amplitude and 3.5Hz frequency.

[0052] Figure 10 This invention relates to the time history and frequency amplitude characteristics of aerodynamic forces under the conditions of 0.15m amplitude and 9.5Hz frequency.

[0053] Figure 11 This invention relates to the time history and frequency characteristics of aerodynamic forces under the conditions of amplitude 0.25m and frequency 3.5Hz.

[0054] Figure 12 This invention relates to the time history and frequency characteristics of aerodynamic forces under the conditions of amplitude 0.25m and frequency 9.5Hz.

[0055] Figure 13 This invention relates to the relationship between aerodynamic force and displacement under 3.5Hz and 9.5Hz conditions, respectively.

[0056] Figure 14 This is a schematic diagram of the four phase points in this invention.

[0057] Figure 15 This invention describes the flow field and pressure field around a box girder under the conditions of an amplitude of 0.05m and a frequency of 9.5Hz.

[0058] Figure 16 This invention describes the flow field and pressure field around a box girder under the conditions of an amplitude of 0.25m and a frequency of 9.5Hz.

[0059] Figure 17 This invention is based on an amplitude of 5 0 The time history of aerodynamic forces at a frequency of 3.5 Hz.

[0060] Figure 18 This invention is based on an amplitude of 5 0 The time history and frequency characteristics of aerodynamic forces at a frequency of 3.5 Hz.

[0061] Figure 19 This invention is based on an amplitude of 20. 0 The time history and frequency characteristics of aerodynamic forces at a frequency of 3.5 Hz.

[0062] Figure 20 This invention is based on an amplitude of 30. 0 The time history and frequency characteristics of aerodynamic forces at a frequency of 3.5 Hz.

[0063] Figure 21 This invention relates to the relationship between aerodynamic torque and displacement under 3.5Hz and 6.5Hz conditions, respectively.

[0064] Figure 22 This invention is based on an amplitude of 15. 0 The flow field and pressure field around the box girder at a frequency of 3.5Hz.

[0065] Figure 23 This invention is based on an amplitude of 30. 0 The flow field and pressure field around the box girder at a frequency of 3.5Hz. Detailed Implementation

[0066] To facilitate a clear understanding of the technical means, creative features, objectives, and effects of this invention, the following description, in conjunction with specific illustrations, further elucidates the invention. This embodiment uses a typical box girder cross-section as an example to illustrate the complete implementation process of the bridge cross-section large-amplitude nonlinear self-excited force modeling method based on flow field memory effect. Those skilled in the art will understand that the method and model are equally applicable to bridge structures with other cross-sectional forms.

[0067] In bridge wind-resistant design, the classic Scanlan linear self-excited force model is based on the assumption of small amplitude. However, as the bridge span increases, the vibration amplitude of the structure under strong winds increases significantly, and the aerodynamic forces exhibit strong nonlinear characteristics, such as higher harmonic components and limit cycle oscillations. In this case, the linear model has a large prediction error. Existing nonlinear models are mostly based on mathematical fitting and lack clear flow physics mechanisms, making it difficult to reveal the essential connection between nonlinear aerodynamic forces and the evolution of the flow field structure.

[0068] To address the aforementioned issues, this embodiment provides a modeling method for large-amplitude nonlinear self-excited forces in bridge cross-sections that is based on the flow field memory effect and has clear physical meaning.

[0069] like Figure 1 As shown, a method for modeling large-amplitude nonlinear self-excited forces in bridge cross-sections based on flow field memory effect includes the following steps:

[0070] S1. Establish a self-excited aerodynamic model of the bridge cross section.

[0071] Based on previous research, this invention proposes that two types of vortex systems exist around the vibrating main beam cross-section: free vortex shedding and motion-induced vortices. Free vortex shedding is formed due to the viscosity of the fluid causing the airflow to separate and reattach after passing through the blunt body cross-section; this type of vortex system is mainly determined by the aerodynamic shape of the cross-section. Motion-induced vortex systems are generated due to the instability of the fluid shear layer on the model surface caused by the model's motion; therefore, they are closely related to the model's motion state and are also a major factor affecting aerodynamic characteristics.

[0072] When the model is under small amplitude and small angle of attack, the scale of the moving vortex system caused by the model vibration is small, the free vortex shedding in the flow field plays a dominant role, the fluid-structure interaction between the moving vortex system and the vibrating structure is weak, and the nonlinear characteristics of the aerodynamic force are not significant. At this time, the aerodynamic force acting on the vibration section can be approximated as a linear simplification, and it is considered that the aerodynamic force is only related to the aerodynamic shape of the model.

[0073] However, when the model vibrates significantly, the motion results in large-scale moving vortices, leading to a strong fluid-structure interaction between the moving vortices and the vibrating structure, and causing the aerodynamic forces to exhibit significant nonlinear characteristics. As the moving vortices move from the front to the rear of the model, they continuously interact strongly with the vibrating structure. This means that the aerodynamic forces acting on the model surface are not only related to the model's current motion state but also closely related to its motion history—a phenomenon known as the fluid's "memory effect."

[0074] Based on the above understanding of the aerodynamic force formation mechanism, this embodiment, after drawing on the aerodynamic force modeling method in the aviation field, decomposes the self-excited aerodynamic force acting on the vibration cross-section into a superposition of linear aerodynamic force components and nonlinear aerodynamic force components.

[0075] Among them, such as Figure 2 As shown, the linear aerodynamic components are calculated using the Scanlan linear self-excited force model based on flutter derivatives. Its form is consistent with classical theory, ensuring the model's correctness under small amplitudes. Its lift... and lift torque The expression is:

[0076]

[0077]

[0078] In the formula: It is air density; It is the average wind speed of the incoming flow; It is the width of the bridge deck; , It is a dimensionless flutter derivative; , It refers to the displacement of vertical and torsional vibrations; It is a dimensionless reduced frequency.

[0079] pneumatic lift For example, its expression is:

[0080]

[0081] In the formula: , The linear aerodynamic forces acting on the model surface during the vibration of the main beam are represented here by the flutter derivative. , , It can be described in a form consistent with the classic Scanlan linear self-excited force model.

[0082] In this embodiment, the nonlinear aerodynamic components are... Considered as the output of a dynamic system, the state update equation of this dynamic system can be expressed as a first-order differential equation. This equation describes the influence of the memory effect of the moving vortex system on the motion history of the main beam in the flow field on the self-excited aerodynamic forces. Where: Represents dimensionless time. , Functions related to the model's motion.

[0083] To make the nonlinear function , To specify this and fit complex nonlinear relationships, this embodiment defines it as a polynomial function of the displacement and velocity of the bridge cross-section. This polynomial function is flexible in form and capable of approximating various nonlinear relationships. Specifically, for nonlinear functions related to vertical motion... Adopting the approach regarding and Expand the Taylor series:

[0084]

[0085] Similarly, for nonlinear functions related to torsional motion Adopting the approach regarding and Taylor series expansion:

[0086]

[0087] In the above formula: , Represents the coefficients in the Taylor series expansion; , The displacement and velocity of the main beam's vertical movement; , The displacement and velocity of the main beam during torsional motion. Let be the total order of the polynomial. The velocity term uses... and The form is to ensure that the function value is zero when the velocity is zero, which is consistent with physical intuition.

[0088] therefore It can be written as the following expression:

[0089]

[0090] Using polynomial functions to describe the mapping relationship between motion states and nonlinear aerodynamic forces gives the model a strong nonlinear fitting capability. This is achieved by adjusting the order. This allows for flexible control of model complexity, balancing fitting accuracy with the risk of overfitting. Furthermore, the polynomial form facilitates mathematical processing and parameter identification.

[0091] Similarly, the aerodynamic lift moment acting on the cross-section of the vibrating main beam under large amplitude conditions can be written out. Calculation formula:

[0092]

[0093] In the formula: , This represents the linear component of the aerodynamic torque and can be represented by the flutter derivative. , , , This approximates the description, and its form is consistent with the classic Scanlan linear self-excited force model.

[0094] In this embodiment, the nonlinear aerodynamic torque component is... Considered as the output of a dynamic system, the state update equation of this dynamic system can be expressed as a first-order differential equation. This reflects the memory effect of the flow field on the motion history of the main beam cross-section. Represents dimensionless time. , This represents a function related to the motion of the main beam; its specific expression can be found in the aerodynamic lift function. China regarding , It uses the Taylor series expansion.

[0095] The structure of the equation makes the current value of the nonlinear aerodynamic component depend on both the current motion state of the bridge cross section and the historical evolution of the nonlinear aerodynamic component.

[0096] This dependence is intended to characterize the memory effect of the flow field on the motion history of the bridge cross section. Specifically, it reflects the continuous influence of the large-scale "motion vortex system" excited by the large amplitude vibration of the main beam on the surface pressure of the cross section during its evolution.

[0097] By decomposing aerodynamic forces into explicit linear and nonlinear components, and introducing first-order dynamic equations with explicit physical meaning (memory effect) to describe the nonlinear components, the established model can not only accurately fit nonlinear aerodynamic data under large amplitude, but also explain the root cause of nonlinearity from the perspective of fluid-structure interaction mechanism, thereby enhancing the physical transparency and extrapolation reliability of the model.

[0098] The self-excited aerodynamic model established in this embodiment is specifically used to calculate the aerodynamic lift and / or aerodynamic lift moment acting on the bridge cross section.

[0099] For aerodynamic lift Its complete model consists of a linear lift term. With nonlinear lift term The sum of the aerodynamic lift moments. Its complete model consists of linear torque terms. With nonlinear torque term The sum of the two. They have similar model structures but different parameters and need to be identified separately.

[0100] This model can be directly used to couple structural dynamic equations to perform time-domain analysis of nonlinear flutter of bridge sections under large amplitude conditions, predict their vibration response, and determine stability (such as whether limit cycles occur), thus providing a more accurate theoretical tool for the wind-resistant design and safety assessment of bridges.

[0101] Moreover, the aerodynamic model established in this embodiment is well compatible with the existing flutter self-excited force model. When the main beam section is under small amplitude, the nonlinear terms in the established self-excited force model have a small impact, and the aerodynamic model will automatically degenerate into a linear flutter self-excited force model.

[0102] S2. Obtain the aerodynamic time history data of the bridge section under the target large amplitude.

[0103] Obtained through wind tunnel forced vibration tests or computational fluid dynamics numerical simulations.

[0104] S3. Based on the aerodynamic time history data, identify the parameters in the self-excited aerodynamic model.

[0105] As can be seen from the aforementioned formula, the parameters to be identified in the established aerodynamic model include flutter derivative. , , , ; , , , Dimensionless time , as well as In the function , In the function .

[0106] In summary, the parameters that need to be identified include two parts: the flutter derivative in the linear aerodynamic components. , , , ; , , , .

[0107] The coefficients in the first-order dynamic equations corresponding to the nonlinear aerodynamic components, such as dimensionless time... , as well as In the function , In the function

[0108] In practical numerical calculations or parameter identification, to facilitate the use of discrete-time series data, it is necessary to first discretize the continuous first-order dynamic equations into discrete-time difference equations using methods such as the backward Euler method. Therefore, the form of the first-order discrete-time difference equation is:

[0109]

[0110] In the above formula, make the following parameter substitution, let:

[0111]

[0112]

[0113]

[0114] The above formula then becomes as follows:

[0115]

[0116] This equation shows that the current value of the nonlinear aerodynamic component This is equal to a coefficient that characterizes memory decay. Compared to its previous value The product, plus a motion state variable of the bridge cross section at the current moment. , , , The input is the value of the nonlinear function.

[0117] In the transformed difference equation, all variables are discrete-time series data, which can be directly obtained from the sampling results of experiments or CFD simulations. This allows standard discrete-time system identification algorithms (such as least squares method and maximum likelihood method) to be applied directly, greatly facilitating the automated and programmed identification of parameters.

[0118] In this embodiment, identifying the coefficients in the first-order dynamic equation corresponding to the nonlinear aerodynamic components specifically includes the following steps:

[0119] S31. Identify flutter derivatives based on dynamic response time history data under small amplitude.

[0120] Short-term excitation was applied to a segmental model of elastic suspension in a wind tunnel to obtain the dynamic response time history of the main beam section at different equivalent wind velocities under small amplitude conditions. System identification methods, such as the least squares method, were then used to identify the flutter derivative. , , , .

[0121] CFD numerical simulation methods can also be used to calculate and identify the aerodynamic forces of small-amplitude forced vibrations.

[0122] To obtain high-precision aerodynamic time history data at the target amplitude, this embodiment employs computational fluid dynamics numerical simulation. The specific numerical simulation process includes: using dynamic mesh technology to simulate the forced vibration of the bridge cross-section at a predetermined amplitude and frequency, i.e., controlling the motion of the bridge cross-section boundaries in the computational domain, such as vertical vibration. .

[0123] The shear stress transport (SST) turbulence model is used to solve the unsteady flow field. This model can simulate separated flows well and is suitable for flow around bluff bodies such as bridge cross-sections. The computational domain needs to be set large enough to reduce the influence of boundaries. The inlet is set as a velocity inlet boundary, the outlet as a pressure outlet boundary, and the cross-sectional wall is set as a no-slip boundary condition.

[0124] To ensure analytical accuracy of near-wall flow, the thickness of the first mesh layer must be strictly controlled during mesh generation, ensuring that the scale of the near-wall mesh meets the dimensionless wall distance requirement. This ensures that the mesh is located within the viscous sublayer, thereby accurately capturing the flow details within the boundary layer.

[0125] The time step size needs to meet the requirements of computational stability and time resolution, and should generally contain at least 100 computational steps within each vibration cycle. Where T is the oscillation period. This ensures both the stability of unsteady calculations and a sufficiently detailed description of the aerodynamic forces changing over time. During the calculation, the pressure-velocity coupling uses the SIMPLE algorithm, and the discretization scheme is a second-order upwind scheme, until the iterative residuals converge to a set threshold, such as... .

[0126] The CFD numerical simulation method can efficiently acquire aerodynamic data under various large amplitude and multi-frequency combinations under controllable and repeatable conditions, and can simultaneously obtain detailed flow field information, which greatly facilitates model verification and mechanism analysis.

[0127] S32. Obtain aerodynamic time history data of vertical forced vibration of the main beam section segment model under large amplitude conditions through wind tunnel forced vibration tests on the main beam section segment model or by using CFD dynamic mesh numerical calculation methods. From the aerodynamic time history data under the target large amplitude In the middle, subtract the linear aerodynamic components calculated using the flutter derivative identified in the previous step. This allows for the separation of purely nonlinear aerodynamic time history data. .

[0128] Right now: .

[0129] S33. Based on the separated nonlinear aerodynamic time history data and the bridge cross-section motion time history data measured simultaneously. , , , The coefficients in the first-order dynamic equation (discrete difference equation form) are identified using a system identification method. , , .

[0130] This parameter identification method decouples the identification of linear and nonlinear parameters, reducing the overall identification difficulty. First, a mature method is used to determine the linear component. Then, a refined model is performed on the separated nonlinear data, ensuring that the accuracy of the linear component is not interfered with by nonlinearity, while also making the identification target of the nonlinear parameters clearer.

[0131] To ensure the accuracy and reliability of parameter identification, this embodiment employs a combined system identification method. First, stepwise regression analysis is used with nonlinear aerodynamic time history data. The dependent variable is the motion state variable of the bridge cross-section. , , , Different orders of product terms were used as candidate independent variables, and significance tests were performed.

[0132] By gradually introducing or eliminating variables, the nonlinear function is screened and finally determined. and The polynomial expression and its optimal order This avoids underfitting or overfitting that might result from manually pre-setting the order.

[0133] Then, based on the determined polynomial expression, the least squares method is used to evaluate all unknown parameters in the discrete difference equation. , , A preliminary estimate is made. The least squares method is computationally efficient and can provide a good initial value for subsequent optimization.

[0134] Finally, using the preliminary least squares estimates obtained in the previous step as the initial values ​​for iteration, the recursive maximum likelihood method is employed to optimize and identify the parameters. The recursive maximum likelihood method can handle more general noise assumptions and typically yields estimates with better statistical properties than the least squares method. During the identification process, introducing a white noise sequence to correct model errors can further improve the accuracy and robustness of parameter estimation.

[0135] To clearly define the applicable scope of "large amplitude", this embodiment provides quantitative conditions for the target amplitude.

[0136] When the vertical vibration amplitude of the bridge section is not less than 0.5% of its section width B;

[0137] Alternatively, if the torsional vibration amplitude of the bridge cross section is not less than 3 degrees, it is considered to meet the large amplitude condition.

[0138] At this amplitude threshold, observations through the aforementioned CFD numerical simulations or wind tunnel tests reveal that "moving vortices," directly induced by structural vibrations and whose scale is significantly correlated with amplitude, begin to form and gradually dominate the flow field structure near the cross-section. For example, when the vertical amplitude increases from 0.05m (≈0.25%B, assuming B=20m) to 0.25m (≈1.25%B), large-scale vortices (moving vortices) generated by the model's motion become very prominent in the flow field, rather than simply being "free vortex shedding" determined by the inherent aerodynamic shape of the cross-section. This strong coupling between the moving vortex system and structural vibration is the physical root cause of the significant nonlinear characteristics of the aerodynamic forces.

[0139] By clearly defining the amplitude threshold, a clear and operable criterion is provided for the application scenarios of this method. Engineers can determine whether and when to apply this nonlinear model based on the expected vibration level of the target bridge, thus enhancing the engineering practicality of the method.

[0140] like Figure 3 As shown, this embodiment takes the box girder cross section as an example and calculates the aerodynamic forces acting on the surface of the main beam when the box girder cross section undergoes vertical forced vibration and torsional forced vibration under different amplitudes and different vibration frequencies.

[0141] Establishing a CFD computational model: The RANS method provided by the commercial software Fluent was used in the numerical simulation. The computational domain is set as follows for the two-equation model: Figure 4 As shown.

[0142] The minimum grid size near the wall was 0.0002 m during computation. The computational domain used a block-structured grid with 145,000 grid cells. The computational parameters were set as follows: momentum, turbulent kinetic energy, and energy dissipation were discretized using a second-order upwind scheme; the pressure-velocity coupling used the SIMPLE algorithm; a split solver was used; and a second-order implicit computational mode was selected. Boundary conditions were set as follows: velocity inlet with an initial inlet wind speed of 10 m / s and turbulence intensity of 0.5%; pressure outlet; symmetrical boundary conditions were set at the upper and lower ends of the computational domain; and no-slip wall conditions were used on the surface. The computational time step was set to... The residual calculated in each iteration is less than The calculation results are considered convergent. Based on the calculated Reynolds number, the mesh is adjusted so that the first mesh layer near the wall is located within the viscous sublayer. Along the width direction of the model like Figure 5 As shown.

[0143] like Figure 6 As shown, this embodiment first calculates the flow field characteristics of the box girder section when it is stationary with an initial angle of attack of 0 degrees. The Q criterion is used to identify the vortex characteristics in the flow field.

[0144] like Figure 6 As shown in (a), vortices V1 and V3 exist near the windward side nozzle, while a more pronounced vortex V4 exists at the tail nozzle. Overall, the box girder cross-section has a good streamline shape, and the free vortices formed after the airflow passes through the box girder cross-section are all relatively small in scale. Figure 6 As shown in (b), the pressure distribution on the model surface is relatively uniform, except that the pressure is greater near the wind-facing nozzle on the windward side.

[0145] Verify the effectiveness of the aerodynamic model:

[0146] To verify the effectiveness of the proposed aerodynamic model, this embodiment employs CFD numerical simulation combined with dynamic mesh technology to calculate the flow characteristics and aerodynamic properties of the box girder cross-section under different amplitudes and frequencies. The vibration equation of the box girder is:

[0147]

[0148] In the formula: For amplitude, The frequency is the vibration frequency.

[0149] This embodiment calculates 12 working conditions for the box girder with vertical amplitudes of 0.05m, 0.15m, and 0.25m, and vibration frequencies of 3.5Hz, 6.5Hz, 9.5Hz, and 12.5Hz.

[0150] like Figure 7The figure shows the aerodynamic time history acting on the box girder when the amplitude is 0.05 m and the vibration frequency is 9.5 Hz. It can be seen from the figure that under this condition, the Scanlan linear self-excited force model can fit the CFD calculation results very well. This indicates that under small amplitude conditions, the moving vortex system caused by the model vibration is weak, the fluid-structure interaction between the model and the moving vortex system is weak, and the nonlinear characteristics of the aerodynamic forces are not prominent.

[0151] like Figure 8 and Figure 9 As shown, the time histories of aerodynamic lift and lift moment on the model surface are obtained when the amplitude is 0.15 m and the vibration frequency is 3.5 Hz. When the vertical amplitude of the box girder reaches 0.15 m, the aerodynamic forces acting on the model surface exhibit strong nonlinear characteristics.

[0152] from Figure 8 As can be seen, due to the nonlinear characteristics of aerodynamic forces, there is a significant difference between the fitted values ​​of the Scanlan self-excited force model and the CFD calculation results. However, the nonlinear self-excited force model established in this embodiment matches the CFD calculation results quite well, which demonstrates the effectiveness of the model in this embodiment.

[0153] Under this operating condition, the aerodynamic forces exhibit significant nonlinear characteristics. Both the aerodynamic lift and lift torque contain high-order harmonic components, which are integer multiples of the fundamental frequency, with the second and third harmonic components being dominant. Furthermore, the frequency-amplitude characteristics of the aerodynamic forces calculated in this embodiment match well with those calculated by CFD; however, the Scanlan linear self-excited force model can only represent the fundamental frequency characteristics of the vibration.

[0154] like Figure 10 As shown, to analyze the influence of vibration frequency on aerodynamic properties, the aerodynamic time history acting on the cross-section is given when the amplitude is 0.15 m and the vibration frequency is 9.5 Hz. From Figure 10 As can be seen in (b), compared to Figure 9 (a) With the increase of vibration frequency, the proportion of the third harmonic component in the aerodynamic lift becomes more prominent, indicating that the nonlinear characteristics of the aerodynamic force are enhanced. Therefore, the aerodynamic lift fitting value of the Scanlan self-excited force model under this condition differs more significantly from the CFD calculation result. In this embodiment, the calculation result of the aerodynamic force model agrees well with the CFD calculation value. Figure 10 (a).

[0155] like Figure 11 As shown in (a), the aerodynamic time history is given when the vertical amplitude is 0.25 m and the vibration frequency is 3.5 Hz. When the vertical amplitude increases to 0.25 m, the aerodynamic characteristics acting on the vibration cross-section change significantly. From Figure 11As shown in (b), under this operating condition, the third harmonic component of the aerodynamic lift has become the dominant component. When the amplitude is 0.15 m and the vibration frequency is 3.5 Hz, the second harmonic component accounts for a larger proportion of the aerodynamic force. Therefore, with the increase of amplitude, the nonlinear characteristics of the aerodynamic force become more pronounced. The calculation results of the linear self-excited force model differ significantly from the CFD calculation results, especially in the aerodynamic lift torque. However, the calculation results of the aerodynamic force model in this embodiment still agree well with the CFD calculation values.

[0156] like Figure 12 As shown in (a), the aerodynamic time history is displayed when the vertical amplitude is maintained at 0.25 m and the vibration frequency is adjusted to 9.5 Hz. From Figure 12 As can be seen in (b), as the vibration frequency increases, the proportion of the third harmonic component in the aerodynamic lift increases rapidly, indicating that the nonlinearity of the aerodynamic lift is further aggravated. Therefore, the difference between the linear self-excited force model and the CFD calculation results is already very large, while the aerodynamic model in this embodiment can fit the CFD calculation results well.

[0157] In summary, the nonlinear characteristics of aerodynamic forces become more pronounced as the vertical amplitude and vibration frequency increase.

[0158] like Figure 13 As shown, the curves depicting the relationship between aerodynamic lift and vertical displacement at different amplitudes with vibration frequencies of 3.5 Hz and 9.5 Hz are presented. It can be seen from the figure that for vertical vibration, the direction of rotation of the curve remains constant within different periods (counterclockwise in the figure), indicating that the way the aerodynamic force generated by vertical vibration consumes the system's energy does not change within one period, which is significantly different from torsional vibration. Furthermore, comparing the curve characteristics at the two frequencies reveals that the local curvature of the curve increases with increasing vibration frequency, indicating an enhancement of the nonlinear characteristics of the aerodynamic force.

[0159] The above comparative analysis examines the influence of the amplitude and frequency of vertical vibration on the nonlinear characteristics of aerodynamic forces, such as... Figure 14 As shown, the following section explores the relationship between the nonlinearity of aerodynamic forces and the evolution of vortices in the flow field by examining the spatiotemporal distribution characteristics of the flow and pressure fields. This embodiment employs a phase averaging method to process the flow field at the same spatial location; the four phase points are as follows: Figure 14 As shown in t1, t2, t3, and t4, 20 flow fields were selected at each phase for averaging.

[0160] like Figure 15As shown, the flow and pressure field distributions at four phases are presented when the amplitude is 0.05 m and the vibration frequency is 9.5 Hz. When the box girder is in equilibrium (at t1), separate vortices V1 and V3 form near the windward side of the model, while a larger vortex V5 forms at the leeward side windward side, although its intensity is relatively low. Therefore, the influence of vortices V1 and V3 on the pressure distribution characteristics of the model surface is greater than that of vortex V5.

[0161] When the box girder moves from the equilibrium position t1 to the maximum vertical displacement t2, the vortices on the upper surface of the model disappear, and only smaller vortices V3 and V4 are generated on its lower surface. At this time, the pressure distribution characteristics on the model surface are similar to those of the box girder when it is at rest. Figure 6 (b)).

[0162] When the model returns to the equilibrium position t3 from the positive maximum displacement t2, the vortex sizes on the model surface are relatively small, and the pressure distribution on the model surface is relatively uniform, with a larger pressure amplitude at the windward side nozzle. However, when the model moves from the equilibrium position t3 to the negative maximum displacement t4, a larger and stronger vortex V6 forms near the windward side nozzle on the upper surface of the model. Within the influence range of this vortex, the local pressure distribution on the model surface undergoes a large curvilinear change.

[0163] Overall, when the vertical amplitude is 0.05m, the moving vortices (vortex V5 and vortex V6) caused by the model motion are small in scale, and the vortex characteristics in the flow field are similar to those of the box girder in the static state. The interaction between the moving vortex and the vibrating section is weak, so the nonlinear characteristics of the aerodynamic force are not obvious.

[0164] like Figure 16 As shown, the distribution characteristics of the flow field and pressure field around the box girder at the four phase points mentioned above are given when the vertical amplitude is 0.25m and the vibration frequency is 9.5Hz.

[0165] When the model is at equilibrium position t1, a large-scale vortex V1 with strong vorticity is formed near the windward side of the model, which is similar to the flow field characteristics at position t1 with an amplitude of 0.05m. Figure 15 (a) By comparison, it can be seen that the vortex is generated by the large-amplitude vibration of the model. The vortex V1 causes a large increase in the amplitude of the pressure distribution near the windward side nozzle, which changes the uniformity of the pressure distribution on the model surface. This reflects the strong fluid-structure interaction between the moving vortex and the vibrating structure, resulting in strong nonlinear characteristics of the aerodynamic forces.

[0166] When the model moves from the equilibrium position t1 to the positive maximum displacement t2, a large-scale vortex V5′ is generated on the lower surface of the model near the tail, which is similar to the flow field characteristics at position t2 with an amplitude of 0.05m. Figure 15(b) In comparison, it can be seen that the vortex is a large-scale moving vortex system formed by the large amplitude of the model. The vortex V5′ generates a strong suction effect near the tail of the lower surface of the model, which causes the amplitude of the distributed pressure in this area to increase sharply and the local nonlinearity of the aerodynamic force to be enhanced.

[0167] When the model moves from its positive maximum position t2 back to its equilibrium position t3, a large-scale vortex V1′ forms near the windward side of the model. This vortex is a motion vortex system caused by the large-amplitude vibration of the model. V1′ causes the pressure distribution on the upper surface near the windward side to exhibit a curvilinear change, which enhances the nonlinearity of the local pressure distribution in this region. When the model moves from its equilibrium position t3 to its negative maximum displacement t4, a large-scale motion vortex V6′ forms on the upper surface of the model near the tail. This causes the pressure distribution on the model surface near this vortex to exhibit a curvilinear distribution with a large amplitude.

[0168] In summary, by comparing the spatial distribution characteristics of the flow field and pressure field around the box girder when the amplitude is 0.05m and 0.25m, it can be seen that when the amplitude is small, the scale of the moving vortex system caused by the model motion is small, and the flow field is mainly controlled by free vortex shedding. Therefore, the fluid-structure interaction between the vortex and the vibrating structure is weak, and the nonlinear characteristics of the aerodynamic force are not significant.

[0169] When the model vibrates significantly, the model's motion will form a large-scale moving vortex system in the flow field. The flow field is mainly controlled by the moving vortex system. The strong fluid-structure interaction between the moving vortex system and the structural vibration leads to the aerodynamic forces within the influence range of the moving vortex system having obvious nonlinear characteristics. The nonlinear aerodynamic forces in the local area of ​​the model surface ultimately make the overall aerodynamic forces acting on the model surface exhibit nonlinear characteristics.

[0170] like Figure 17 As shown, the aerodynamic time history of the model surface under torsional amplitude of 50° and vibration frequency of 3.5Hz is presented. It can be seen from the figure that, due to the small amplitude, the nonlinear characteristics of the aerodynamic forces are not obvious, and the Scanlan linear self-excited force model can well fit the CFD calculation results.

[0171] like Figure 18 As shown, the time histories of aerodynamic forces acting on the model surface are presented when the amplitude is 150° and the vibration frequency is 3.5Hz. When the torsional amplitude of the box girder reaches 150°, the aerodynamic forces acting on the model surface exhibit certain nonlinear characteristics. From... Figure 18 As can be seen in (a), the aerodynamic lift and lift torque fitted by the Scanlan self-excited force model are different from the CFD calculation results. However, the nonlinear self-excited force model established in this embodiment matches the CFD calculation results quite well, which shows the effectiveness of the model in this embodiment.

[0172] From the frequency response diagram of aerodynamic lift moment ( Figure 18 (b) It can be seen that higher harmonic components appear in the frequency components of the aerodynamic torque, mainly the second and third harmonic components, which are also the main components of the aerodynamic nonlinear model.

[0173] like Figure 19 As shown, the aerodynamic time history acting on the model surface is presented when the torsional amplitude is 200° and the vibration frequency is 3.5 Hz. Under this condition, the nonlinear characteristics of the aerodynamic forces are more pronounced, and the frequency components of the aerodynamic lift torque contain second and third higher-order harmonic components. Figure 19 (b)), from Figure 19 As can be seen in (a), the calculation results of the Scanlan linear self-excited force model deviate significantly from the CFD calculation values, while the model in this embodiment matches the CFD calculation results very well.

[0174] like Figure 20 As shown, the aerodynamic time history acting on the model surface is presented when the torsional amplitude is 300° and the vibration frequency is 3.5Hz. When the torsional amplitude continues to increase to 300°, the nonlinear characteristics of the aerodynamic force become very significant. Under this condition, the Scanlan self-excited force calculation results deviate greatly from the CFD calculation values, and the linear self-excited force model is no longer applicable. However, the nonlinear self-excited force model established in this embodiment agrees quite well with the CFD calculation results. Figure 20 (a)). As the amplitude increases, the aerodynamic force exhibits strong nonlinearity, and the proportion of the third harmonic component in the spectrum of the aerodynamic torque increases significantly. Figure 20 (b)).

[0175] In summary, as the torsional amplitude increases, the nonlinear characteristics of aerodynamic forces become more pronounced, rendering the linear self-excited force model inapplicable. In contrast, the nonlinear aerodynamic force model established in this embodiment effectively reflects the nonlinear characteristics of aerodynamic forces.

[0176] like Figure 21As shown in the figure, the relationship between aerodynamic torque and torsional displacement under different torsional amplitudes at vibration frequencies of 3.5Hz and 6.5Hz is presented, also known as the hysteresis curve. It can be seen from the figure that for torsional vibration, when the amplitude is small (less than 15 degrees in this example), the rotation direction of the hysteresis curve remains unchanged in different periods; in this example, it is counterclockwise, indicating that the aerodynamic force always does negative work on the system. However, when the amplitude is large (30 degrees in this example), the aerodynamic force may do negative work (consuming energy) or positive work (inputting energy into the system) within one period. Therefore, under large torsional amplitudes, the vibration instability of the system manifests as limit cycle oscillations, reflecting the nonlinear characteristics of aerodynamic forces. Unlike linear flutter, where the system amplitude increases infinitely after reaching a certain wind speed, causing a sudden loss of system stability, this does not exhibit these characteristics.

[0177] To illustrate the influence of torsional amplitude on aerodynamic nonlinearity from the flow field perspective, the distribution characteristics of the flow field and pressure field around the model are given below when the torsional amplitude is 150° and 300° and the vibration frequency is 3.5 Hz. Similar to the flow field analysis during vertical vibration, only the flow field and pressure field at four phase points are given here.

[0178] like Figure 22 As shown, the flow and pressure fields around the box girder are presented when the torsional amplitude is 15 degrees and the vibration frequency is 3.5 Hz. When the model is at the equilibrium position t1, small-scale vortices V1 and V3 are generated near the windward side of the model, while a larger-scale vortex V2′ is formed on the upper surface near the tail windward side. These vortices cause the pressure near the tail windward side to be non-uniformly distributed. When the model moves from the equilibrium position t1 to the positive maximum displacement t2 (here, counterclockwise rotation is taken as positive), the large-scale vortices on the model surface disappear, and only small-scale vortices V1 and V3 are generated near the windward side windward side. When the model returns to the equilibrium position t3 from the positive maximum displacement t2, there are no large-scale vortices around the model, and the pressure distribution characteristics on the model surface are close to those when the model is stationary. When the model moves to the negative maximum displacement t4, a larger-scale vortex V1 is generated near the windward side windward side on the upper surface of the model, resulting in a large pressure peak and non-uniform distribution in this region. Overall, when the torsional amplitude of the model is 150, the scale of the moving vortex system generated by the model vibration is small, the fluid-structure interaction between the moving vortex system and the structural vibration is weak, and the aerodynamic force has certain nonlinear characteristics, but they are not very significant.

[0179] like Figure 23 As shown, the flow field and pressure field around the box girder are given when the torsional amplitude is 300 and the vibration frequency is 3.5 Hz. When the torsional amplitude of the model increases to 300, the nonlinearity of the aerodynamic force becomes very significant.

[0180] When the model is at equilibrium position t1, large-scale moving vortices V2′ and V5 form in the flow field near the model's tail. These vortices cause a sharp increase in pressure amplitude in the tail region, disrupting the uniformity of pressure distribution on the model surface. This is similar to the flow field at position t1 when the amplitude is 150. Figure 22 (a)) There are significant differences.

[0181] When the model moves from the equilibrium position t1 to the positive maximum displacement t2, a large-scale vortex V1 is formed on the windward side, which is different from the flow field at position t2 when the amplitude is 150. Figure 22 (b) The vortex is larger in scale, causing significant changes in local pressure near the windward side.

[0182] When the model returns to the equilibrium position t3 from the maximum positive displacement t2, large-scale moving vortices V4′ and V6 are generated in the flow field on the lower side of the model tail. This is similar to the flow field at position t3 when the amplitude is 150. Figure 22 (c) There is a very large difference, and the vortex significantly changes the pressure distribution characteristics at the tail nozzle of the model, making the aerodynamic forces in the local area of ​​the model more nonlinear.

[0183] When the model moves from the equilibrium position t3 to the negative maximum displacement t4, a large-scale vortex V2′ is generated on the upper surface of the windward side of the model, which is similar to the flow field at position t4 with an amplitude of 150. Figure 22 Compared to (d), the vortex has a wider range of influence and more significantly alters the uniformity of aerodynamic distribution.

[0184] The above comparative analysis shows that, for torsional vibration, when the amplitude is small, the vortex system generated by the model motion is small in scale and has a small nonlinear effect on aerodynamic forces. When the large amplitude motion can excite a large-scale moving vortex system, the moving vortex system and the structural vibration have a strong coupling effect, which makes the pressure distribution in the vortex-affected area have strong nonlinear characteristics. The local nonlinearity of the model surface area ultimately leads to the overall aerodynamic force acting on the model surface having significant nonlinear characteristics.

[0185] The implementation principle of this invention is as follows:

[0186] This invention physically characterizes the memory effect brought about by the evolution of a "moving vortex system" under large amplitude by decomposing the self-excited force into linear components and nonlinear components described by first-order dynamic equations. It provides a parameter identification process combining stepwise regression, least squares, and recursive maximum likelihood methods.

[0187] The excellent performance of the model under large amplitude was verified through systematic CFD numerical simulation, and the physical root cause of nonlinear aerodynamic forces was revealed at the flow field level. This method has both clear physical significance and good engineering applicability, providing an effective tool for refined nonlinear flutter analysis of long-span bridges.

[0188] The foregoing has shown and described the basic principles, main features, and advantages of the present invention. Those skilled in the art should understand that the present invention is not limited to the above embodiments, and various changes and modifications can be made without departing from the spirit and scope of the invention, all of which fall within the scope of protection claimed by the present invention. The scope of protection of the present invention is defined by the appended claims and their equivalents.

Claims

1. A method for modeling large-amplitude nonlinear self-excited forces in bridge cross-sections based on flow field memory effect, characterized in that, Includes the following steps: S1. Establish a self-excited aerodynamic model of the bridge section, wherein the self-excited aerodynamic model is composed of the superposition of linear aerodynamic components and nonlinear aerodynamic components; The linear aerodynamic components are calculated using a linear model based on flutter derivatives. The nonlinear aerodynamic component is described by a first-order dynamic equation. The structure of this equation makes the current value of the nonlinear aerodynamic component depend on both the current motion state of the bridge section and the historical evolution of the nonlinear aerodynamic component, thereby characterizing the memory effect of the flow field on the motion history of the bridge section. S2. Obtain the aerodynamic time history data of the bridge cross section under the target amplitude; S3. Based on the aerodynamic time history data, identify the parameters in the self-excited aerodynamic model, including the flutter derivative in the linear aerodynamic component and the coefficients in the first-order dynamic equation corresponding to the nonlinear aerodynamic component.

2. The method as described in claim 1, characterized in that, The steps for determining the nonlinear aerodynamic components include: The nonlinear aerodynamic component is regarded as the output of a dynamic system, and the state update equation of the dynamic system is a first-order differential equation or its first-order discrete-time difference equation. The structure of the state update equation is as follows: the current value of the nonlinear aerodynamic component is equal to the product of a coefficient representing memory decay and its previous value, plus a nonlinear function value with the motion state variable of the bridge section at the current moment as input.

3. The method as described in claim 2, characterized in that, The nonlinear function is a polynomial function relating to the displacement and velocity of the bridge cross-section.

4. The method as described in claim 1, characterized in that, In step S3, identifying the coefficients in the first-order dynamic equations corresponding to the nonlinear aerodynamic components specifically includes: S31. Identify the flutter derivative based on the dynamic response time history data under small amplitude; S32. Subtract the linear aerodynamic components calculated using the flutter derivative from the aerodynamic time history data under the target amplitude to separate the nonlinear aerodynamic time history data; S33. Based on the nonlinear aerodynamic time history data and the corresponding bridge cross-section motion time history data, the coefficients in the first-order dynamic equation are identified by a system identification method.

5. The method as described in claim 4, characterized in that, In step S33, before using the system identification method, the first-order dynamic equation is first converted into a discrete-time difference equation.

6. The method as described in claim 4, characterized in that, In step S33, the system identification method is as follows: Stepwise regression analysis was used, with the nonlinear aerodynamic time history data as the dependent variable and the product terms of different orders of the bridge cross-section motion state variables as candidate independent variables, to screen and determine the polynomial expression form and order of the nonlinear function. Based on the determined polynomial expression, the least squares method is used to make a preliminary estimate of all unknown parameters in the first-order dynamic equation; Based on the initial estimates obtained, the parameters are optimized and identified using the recursive maximum likelihood method, and white noise sequence is introduced during the identification process to correct the model error.

7. The method as described in claim 1, characterized in that, In step S2, the aerodynamic time history data is obtained through computational fluid dynamics numerical simulation; the numerical simulation includes: Dynamic mesh technology is used to simulate the forced vibration of a bridge cross section at a predetermined amplitude and frequency; The unsteady flow field was solved using the SST turbulence model; The scale of the near-wall mesh satisfies To ensure the accuracy of boundary layer resolution; The time step is set so that each vibration cycle contains at least 100 calculation steps.

8. The method according to claim 1, characterized in that, The target amplitude satisfies one of the following conditions: The vertical vibration amplitude of the bridge cross section shall not be less than 0.5% of its cross-sectional width; The amplitude of torsional vibration of the bridge cross section shall not be less than 3 degrees; At this amplitude, through numerical simulation or experimental observation, a moving vortex system can be identified, which is directly caused by structural vibration, has a scale that is significantly correlated with the amplitude, and dominates the flow field structure near the cross section.

9. The method according to any one of claims 1-8, characterized in that, The self-excited aerodynamic model is used to calculate the aerodynamic lift and / or aerodynamic lift moment acting on the bridge cross section.

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