A wind-vehicle-bridge coupling vibration analysis method for long-span arch bridges under non-stationary fluctuating wind field

Abstract

The application relates to a wind-vehicle-bridge coupling vibration analysis method of a large-span arch bridge under non-stationary fluctuating wind field, which comprises the following steps: analyzing the spatial characteristics of a mountain valley wind field, simulating the multi-point non-stationary fluctuating wind field of the arch bridge by using a coherence function; quantifying the arch rib-arch upper stand column-main beam multi-component aerodynamic interference effect by CFD numerical simulation, obtaining the main beam and train aerodynamic force coefficients, and calculating the non-stationary wind load of the vehicle-bridge considering the interference; based on the ANSYS and SIMPACK joint simulation, a vehicle-bridge rigid-flexible coupling model is constructed, the above wind load and track irregularity excitation are applied, and a wind-vehicle-bridge coupling model is established; a plurality of wind speed and vehicle speed working conditions are simulated, train safety indexes and bridge dynamic responses are extracted, and a wind speed-vehicle speed control boundary for train safe operation is demarcated; the analysis method can truly reflect the non-stationary wind characteristics of a mountain valley, fully consider the aerodynamic interference of the multi-component of the arch bridge, and realize fine modeling of the vehicle-bridge rigid-flexible coupling.

Description

Technical Field

[0001] This invention belongs to the field of dynamic response analysis of bridge structures, and relates to a method for wind-vehicle-bridge coupled vibration analysis under non-stationary pulsating wind fields of long-span arch bridges. Background Technology

[0002] Railway transportation, with its advantages of large capacity, high efficiency, and low energy consumption, has become a core link supporting the coordinated development of regional economies. As my country's high-speed railway network extends into mountainous areas, long-span arch bridges, due to their excellent terrain adaptability and load-bearing capacity, have become the mainstream bridge type for mountain railways. However, the dramatic undulations of mountain valleys result in wind fields at bridge sites exhibiting strong non-stationarity, spatial heterogeneity, and non-Gaosterianism, fundamentally different from the uniform and stable wind fields in plains areas. When trains run on bridge sections, they simultaneously bear crosswind loads and bridge-coupled vibrations, significantly increasing the risk of derailment and overturning. Therefore, conducting refined wind-train-bridge coupled vibration analysis is of great engineering significance for ensuring the safe operation of mountain railways.

[0003] Currently, some progress has been made in the analysis methods for wind-vehicle-bridge coupled vibration, but the following technical shortcomings still urgently need to be addressed:

[0004] (1) The wind load values ​​ignore the aerodynamic interference between key components of the arch bridge, resulting in insufficient input accuracy of the coupled model. Existing wind-vehicle-bridge coupling studies mostly use the aerodynamic coefficients of the isolated main beam model to calculate the wind load of the bridge. This method is still applicable to lightweight and flexible structures such as suspension bridges and cable-stayed bridges, but there is a complex spatial combination of arch ribs, arch columns and main beams in large-span upper-bearing arch bridges in mountainous areas. The traditional method completely ignores the synergistic aerodynamic interference effect of the arch ribs and arch columns at different positions on the main beam, resulting in the distortion of the aerodynamic coefficient values ​​of the main beam. Correlation analysis shows that after considering the aerodynamic interference of multiple components such as arch ribs, arch columns and main beams, the drag coefficient, lift coefficient and moment coefficient of the main beam may change significantly compared with the isolated main beam section; if the aerodynamic coefficients of the isolated main beam are still used directly to calculate the wind load of the bridge, it may cause the wind load input to deviate from the actual stress state, which is difficult to meet the requirements of refined wind-vehicle-bridge coupled vibration analysis.

[0005] (2) Wind field simulations often use steady, pulsating winds, which do not match the actual non-steady wind environment in mountainous areas. In some existing wind-vehicle-arch bridge coupling analyses, the traditional steady, pulsating winds are still used to simulate the wind field at the bridge site, ignoring the non-steady characteristics of the wind field in mountainous canyons, such as time-varying average wind speeds and strong gusts, which are caused by topographic constraints. This assumption of stability leads to deviations between the simulated wind speed time history and the actual bridge wind environment, resulting in errors in the calculation of the vehicle-bridge coupled dynamic response and failing to truly reflect the extreme vibration response of vehicles and bridges under strong non-steady winds.

[0006] (3) There is a lack of refined modeling methods for vehicle-bridge rigid-flexible coupling in mountainous arch bridges. Most existing wind-vehicle-bridge coupling studies focus on long-span suspension bridges and cable-stayed bridges, while coupled vibration analysis of long-span arch bridges in mountainous areas is relatively scarce. At the same time, traditional modeling methods often simplify the bridge as a pure rigid body or use simple beam elements, failing to fully consider the fine coupling between the flexible characteristics of long-span arch bridges and the multibody dynamics of trains, and rarely integrate non-stationary wind loads and track irregularities, resulting in low model fidelity.

[0007] (4) The wind speed-vehicle speed control boundary for safe train operation is not clearly defined. Due to insufficient accuracy of wind load, distortion of wind field simulation, and simplification of coupling models, existing analysis methods are unable to accurately quantify the coupling amplification effect of non-stationary wind speed and train speed on the system's dynamic response. In particular, the lack of revelation on the nonlinear distortion law of core sensitive indicators such as wheel load reduction rate in strong wind ranges makes it impossible to scientifically define the critical wind speed threshold for ensuring safe train operation at different vehicle speeds, which brings difficulties to the operation control and early warning of mountain railways in strong wind environments.

[0008] In summary, existing technologies lack a vibration analysis method that can accurately reflect the characteristics of non-stationary winds in mountainous canyons, fully consider the aerodynamic interference of multiple components of arch bridges, and achieve precise modeling of the rigid-flexible coupling between the vehicle and the bridge. This leads to inaccurate safety assessments of trains crossing bridges and ambiguous speed limit boundaries. Therefore, it is urgent to establish a wind-vehicle-bridge coupled vibration analysis method under non-stationary pulsating wind fields to overcome the aforementioned technical challenges. Summary of the Invention

[0009] In view of this, in order to solve the problems that existing technologies generally have in the analysis of wind-vehicle-bridge coupled vibration, such as ignoring the non-stationary characteristics of mountain valleys in wind field simulation, ignoring the aerodynamic interference of multiple components of the arch bridge in wind load values, and lacking fine-grained rigid-flexible modeling of the vehicle-bridge coupling model, which leads to unclear definition of the wind speed-vehicle speed control boundary for safe train operation, this invention provides a wind-vehicle-bridge coupled vibration analysis method under non-stationary pulsating wind field of a long-span arch bridge.

[0010] To achieve the above objectives, the present invention provides the following technical solution:

[0011] A method for analyzing the coupled vibration of a long-span arch bridge under non-stationary pulsating wind fields involving wind, vehicles, and the bridge includes the following steps:

[0012] S1. Analysis of wind load on a long-span arch bridge in mountainous areas under non-stationary pulsating wind; specifically including: S11. Analyzing the spatial characteristics of wind fields in mountainous canyons, using the harmonic synthesis method and introducing a non-stationary modulation function, and using a coherence function to simulate the non-stationary pulsating wind field at different spatial locations of the arch bridge, generating non-stationary pulsating wind speed time histories at multiple points along the windward and vertical directions of the entire bridge; S12. Based on CFD numerical simulation, calculating the influence of mutual aerodynamic interference between the main beam, arch rib, and arch column on the aerodynamic characteristics of the main beam, obtaining the static three-component force coefficients of the main beam considering aerodynamic interference under different wind attack angles, and obtaining the train aerodynamic coefficients through CFD numerical simulation of the train model; S13. Combining the non-stationary pulsating wind field simulated in step S11 and the aerodynamic coefficients of the bridge obtained in step S12, calculating the non-stationary wind loads on the bridge at different locations and wind speeds under aerodynamic interference, as well as the non-stationary wind loads on the train;

[0013] S2. Wind-vehicle-bridge coupled vibration analysis under non-stationary pulsating wind fields; specifically including: S21. Constructing a refined finite element model of the bridge using ANSYS APDL software, and establishing a train dynamics model in SIMPACK software. Then, integrating the finite element model of the bridge with the train dynamics model in SIMPACK, and constructing a vehicle-bridge rigid-flexible coupling model by establishing wheel-rail contact relationship and inputting track irregularities; S22. Applying the non-stationary wind load obtained in step S13 as an external excitation to the vehicle-bridge rigid-flexible coupling model constructed in step S21, and establishing a non-stationary wind-vehicle-bridge coupled model based on joint modeling and simulation of SIMPACK and ANSYS; S23. Obtaining the dynamic response of the vehicle and bridge under different wind speeds and vehicle speeds based on the non-stationary wind-vehicle-bridge coupled model constructed in step S22, and conducting traffic safety analysis.

[0014] The beneficial effects of this invention are as follows:

[0015] 1. The wind-vehicle-bridge coupled vibration analysis method for long-span arch bridges under non-stationary pulsating wind fields disclosed in this invention analyzes the spatial characteristics of wind fields in mountain canyons, introduces non-stationary modulation functions and combines them with coherent functions to simulate the multi-point non-stationary pulsating wind fields of arch bridges, overcomes the shortcomings of traditional stationary wind simulation that does not match the actual wind environment in mountainous areas, and can truly reflect the time-varying characteristics and spatial heterogeneity of wind fields in the bridge site area, providing high-precision wind environment input for wind-vehicle-bridge coupled vibration analysis.

[0016] 2. The wind-vehicle-bridge coupled vibration analysis method for long-span arch bridges under non-stationary pulsating wind fields disclosed in this invention quantifies the multi-component collaborative aerodynamic interference effect between the arch rib, the arch column and the main beam through CFD numerical simulation, obtains the aerodynamic coefficient of the main beam considering aerodynamic interference, and calculates the non-stationary wind load at different locations of the bridge accordingly. This solves the problem of wind load value distortion caused by ignoring the aerodynamic interference of key components of the arch bridge in existing methods, and significantly improves the precision of wind load calculation.

[0017] 3. The wind-vehicle-bridge coupled vibration analysis method for long-span arch bridges under non-stationary pulsating wind fields disclosed in this invention is based on joint simulation of ANSYS and SIMPACK. It adopts the Guyan reduction method for bridge modal reduction, uses the FlexModal module to realize the rigid-flexible coupling of vehicle and bridge through virtual rigid bodies, and simultaneously integrates non-stationary wind load and track irregularity excitation to construct a high-fidelity non-stationary wind-vehicle-bridge coupled dynamic model, which breaks through the limitations of insufficient consideration of bridge flexibility characteristics and external excitation in traditional modeling.

[0018] 4. The wind-vehicle-bridge coupled vibration analysis method for long-span arch bridges under non-stationary pulsating wind fields disclosed in this invention can accurately capture the nonlinear distortion law of train safety indicators in strong wind sections by systematically carrying out multiple sets of wind speed and vehicle speed combinations for working condition simulation. It can scientifically delineate the wind speed limits that allow passage under different train speeds, clarify the wind speed-vehicle speed control boundary for safe train operation, and provide a reliable theoretical basis for operation management and early warning in mountainous railways under strong wind conditions.

[0019] 5. The wind-vehicle-bridge coupled vibration analysis method for long-span arch bridges under non-stationary pulsating wind fields disclosed in this invention organically combines aerodynamic interference effects, non-stationary wind field characteristics, and vehicle-bridge rigid-flexible coupling mechanism to form a complete wind-vehicle-bridge coupled vibration analysis method under non-stationary pulsating wind fields. It is not only applicable to long-span arch bridges in mountainous areas, but can also provide a general technical framework for the assessment of railway bridge traffic safety under similar complex wind environments.

[0020] Other advantages, objectives, and features of the invention will be set forth in part in the description which follows, and in part will be apparent to those skilled in the art from the following examination, or may be learned from practice of the invention. The objectives and other advantages of the invention can be realized and obtained through the following description. Attached Figure Description

[0021] To make the objectives, technical solutions, and advantages of the present invention clearer, the preferred embodiments of the present invention will be described in detail below with reference to the accompanying drawings, wherein:

[0022] Figure 1This is a flowchart of the wind-vehicle-bridge coupled vibration analysis method for long-span arch bridges under non-stationary pulsating wind fields according to the present invention.

[0023] Figure 2 The following is a graph showing the calculation results of the vehicle's operational safety indicators under various working conditions in the example embodiment. Figure 2 (a) is a comparison chart of the calculated derailment coefficients. Figure 2 (b) is a comparison chart of the calculation results of wheel load reduction rate. Figure 2 (c) is a comparison chart of the calculation results of the wheel-rail lateral force;

[0024] Figure 3 The time-varying average wind speed curve is shown in the example when the mean time-varying average wind speed of the non-stationary wind is 10 m / s.

[0025] Figure 4 The following are the calculation results of the vehicle running stability index under various operating conditions in the example embodiment, where... Figure 4 (a) is a comparison chart of the calculated results of the vertical acceleration of the vehicle body. Figure 4 (b) is a comparison chart of the calculated results of the vehicle body's lateral acceleration. Figure 4 (c) is a comparison chart of the calculation results for the vertical stability of the vehicle body. Figure 4 (d) is a comparison chart of the calculation results for the lateral stability of the vehicle body;

[0026] Figure 5 This is a schematic diagram of the maximum vertical displacement of the bridge under certain working conditions in the example embodiment;

[0027] Figure 6 The diagram shows the maximum response of the bridge under various working conditions in the example. Detailed Implementation

[0028] The following specific examples illustrate the implementation of the present invention. Those skilled in the art can easily understand other advantages and effects of the present invention from the content disclosed in this specification. The present invention can also be implemented or applied through other different specific embodiments, and various details in this specification can also be modified or changed based on different viewpoints and applications without departing from the spirit of the present invention.

[0029] Example

[0030] The method for analyzing the coupled vibration of a long-span arch bridge under non-stationary pulsating wind fields involving wind, vehicles, and the bridge includes the following steps:

[0031] S1. Wind load analysis of a long-span arch bridge in a mountainous area under non-stationary pulsating wind conditions; specifically including:

[0032] S11. Analyze the spatial characteristics of wind field in mountain canyons, adopt the harmonic synthesis method and introduce non-stationary modulation function, and combine Davenport coherence function to calculate the non-stationary pulsating wind field at different spatial locations of the simulated arch bridge, and generate non-stationary pulsating wind speed time histories at multiple points of the entire bridge in the downwind and vertical directions.

[0033] The non-stationary fluctuating wind speed is composed of the superposition of the time-varying average wind speed and the non-stationary fluctuating wind speed; the non-stationary modulation function adopts a Gaussian form, and its time history distribution characteristics show a single-peak shape with high in the middle and low at both ends. The value reaches the peak value of 1 at the midpoint of the time history and drops to the minimum value of 0.1 at the beginning and end times; the spatial correlation analysis adopts the Davenport coherence function and ignores the correlation between the three-dimensional wind speeds in different directions, simplifying the three-dimensional wind field into three independent one-dimensional random processes.

[0034] Wind speed, as a three-dimensional spatial vector, has x-components along the bridge, y-components across the bridge, and z-components vertical, with significant spatial correlations among these components. When using MATLAB to simulate fluctuating wind speeds at specific points on a spatial structure, it is necessary to consider both the positional correlation between that point and its surroundings, as well as the spatial correlations of each wind speed component. However, the coupled analysis of these dual correlations significantly increases the complexity and computational cost of the simulation. Given that current theories of wind speed spatial correlation simulation cannot fully guarantee the authenticity and reliability of the results, a simplification strategy is adopted. Based on wind field characteristics and engineering application requirements, fluctuating winds in atmospheric turbulence can be considered as a four-dimensional stochastic wind speed field that varies with time t and spatial location (x, y, z). Based on the spatial continuity of the wind field, the discretization concept of the finite element method can be used to discretize the continuous wind field within the bridge area. Considering the weak correlation of three-dimensional wind speeds in nature, this correlation is usually ignored in engineering practice, further supporting the rationality of simplifying the three-dimensional wind field into three independent one-dimensional stochastic processes. Subsequent analysis only needs to focus on the correlation characteristics of a single direction, effectively reducing the difficulty of three-dimensional wind field simulation.

[0035] A typical three-dimensional wind field in space can be represented by the following formula:

[0036]

[0037] In the formula: The crosswind speed is the wind speed across the bridge. It is a transverse bridge; Vertical; These represent the fluctuating wind speeds in the vertical, downwind, and crosswind directions, respectively. For time. The spatial correlation of pulsating winds is related to spatial spacing; here, the Davenport coherence function is used.

[0038] Harmonic synthesis (WAWS) is currently the mainstream method for simulating fluctuating wind, and it can generally be divided into stationary fluctuating wind simulation and non-stationary fluctuating wind simulation. The expression for stationary fluctuating wind is generally as follows:

[0039]

[0040] In the formula: To stabilize wind speed, The average wind speed, This refers to the pulsating wind speed.

[0041] When simulating steady fluctuating wind, it is generally necessary to first determine its two-sided power spectral density matrix. Its description is as follows:

[0042]

[0043] In the formula: It is the cross-spectral density function; This is the Davenport coherence function model, whose values ​​depend on the spatial location between simulation points. The two-sided power spectral density matrix... By performing Cholesky matrix decomposition, we can obtain:

[0044]

[0045] The description is as follows: Let it be its conjugate transpose.

[0046]

[0047] The elements in the equation satisfy the following relationship:

[0048]

[0049]

[0050] Among the non-diagonal elements Represented in polar coordinates:

[0051]

[0052] In the formula:

[0053] Then through the Decomposition yields:

[0054]

[0055] In the formula: For dual-index frequency; For the cutoff frequency; yes The phase angle on.

[0056]

[0057] However, the meteorological conditions of long-span arch bridges in mountainous areas are generally complex, and the wind environment is more unstable compared to plains. Therefore, it is difficult to simulate the wind environment of long-span arch bridges in mountainous areas using steady-state pulsating wind; non-steady-state pulsating wind is needed to simulate its non-steady characteristics. The expression for non-steady-state pulsating wind is generally as follows:

[0058]

[0059] In the formula: For non-steady wind speeds, The time-varying average wind speed, This refers to non-stationary pulsating wind speeds.

[0060] The simulation of non-stationary fluctuating wind first requires converting the power spectrum of stationary fluctuating wind into that of non-stationary fluctuating wind using a modulation function. The evolving power spectral density of non-stationary fluctuating wind is as follows:

[0061]

[0062] in: It is a non-stationary power spectral function; It is a deterministic non-uniform time-frequency modulation function; The autospectral density function of a zero-mean stationary random process;

[0063] In real-world environments, non-stationary fluctuating wind speeds at different points in space exhibit spatial correlation. Generally, the closer two points are, the stronger the correlation. The spatial correlation function... The expression is:

[0064]

[0065] In the formula: The attenuation factor transforms the above equation into an expression for the angular frequency:

[0066]

[0067] For wind speed simulations at different altitudes, the vertical distribution of wind speed within the atmospheric boundary layer follows an exponential law, and the average wind speed at different altitudes can be calculated using the following formula.

[0068]

[0069] In the formula: It is the surface roughness coefficient; The reference wind speed is at a height of 10m. This represents the actual height of the observation point above the ground.

[0070] The arrangement of wind field simulation points on the bridge includes: simulation points are set at the four division points of the main beam segment on the arch, simulation points are set at the four division points of the continuous beam segment, simulation points are set at preset intervals along the bridge direction of the arch rib segment, and simulation points are set at the four, three, or two division points of the arch column according to its different heights. A total of no less than 100 wind speed simulation points are set up on the entire bridge to discretize the wind field of the entire bridge.

[0071] S12. Based on the Reynolds Time-averaged Simulation (RANS) method combined with the k-ω SST two-equation turbulence model, the CFD numerical simulation was used to calculate the influence of the mutual aerodynamic interference between the main beam, arch rib, and arch column on the aerodynamic characteristics of the main beam. The static three-component force coefficients of the main beam considering aerodynamic interference under different wind attack angles were obtained, and the aerodynamic coefficients of the train were calculated by CFD numerical simulation of the train model.

[0072] The CFD numerical simulation employs the SIMPLE algorithm and QUICK spatial discretization scheme. The computational domain inlet uses a velocity inlet boundary, the outlet uses a pressure outlet boundary, the upper and lower boundaries use periodic boundaries, and the main beam model uses a non-slip wall boundary. By mixing structured and unstructured meshes, an unstructured mesh refinement region is created around the bridge structure, while a structured mesh is used in the far field region to balance computational accuracy and efficiency.

[0073] The analysis of the aerodynamic interference between the main beam, arch rib, and arch column includes: selecting the main beam cross-section under the joint interference of the arch column and arch rib at different heights, and the main beam cross-section under different vertical distances between the arch rib and the main beam; calculating the static three-component force coefficients of the uninterrupted isolated main beam cross-section and each interfering cross-section under different wind attack angles.

[0074] The aerodynamic coefficients of a train include at least the drag coefficient, lift coefficient, and moment coefficient.

[0075] S13. Combining the non-stationary pulsating wind field simulated in step S11 and the aerodynamic coefficients of the vehicle and bridge obtained in step S12, calculate the non-stationary wind loads on the bridge at different locations and wind speeds under aerodynamic disturbances, as well as the non-stationary wind loads on the train.

[0076] The calculation of non-stationary wind loads on bridges only considers the time-varying average wind load and the buffeting force caused by non-stationary pulsating wind. The time-varying average wind load is calculated based on the time-varying average wind speed and the static three-component force coefficients considering aerodynamic disturbances, while the buffeting force is calculated based on the non-stationary pulsating wind speed and the aerodynamic admittance function.

[0077] The time-varying average wind load is expressed by the following formula:

[0078]

[0079] In the formula: , , These represent the lift, drag, and torque generated by wind loads, respectively. , , These are the static force coefficients of the structure (lift coefficient, drag coefficient, and moment coefficient), and the angle of attack is... The function; The time-varying average wind speed; The width of the structural cross-section; Segment length; For air density, it is generally taken as .

[0080] The buffeting force caused by non-stationary pulsating wind is calculated using the following formula:

[0081]

[0082] In the formula: , , These are the non-stationary buffeting lift, drag, and torque, respectively. , , These are the derivatives of the corresponding static lift coefficient, drag coefficient, and moment coefficient with respect to the effective angle of attack, respectively. , , , , , It is the aerodynamic admittance function; , This is to simulate non-stationary fluctuating wind speeds in the downwind and vertical directions.

[0083] The main girder wind load segments are divided into three categories: main girder segments subjected to aerodynamic interference from both the arch rib and the arch column, main girder segments subjected to aerodynamic interference from the arch rib alone, and main girder segments without interference. Corresponding static three-component force coefficients are assigned to each type of segment to achieve differentiated and accurate calculation of wind loads at different locations of the entire bridge.

[0084] The calculation of non-steady wind loads on the train adopts quasi-steady-state theory, neglecting higher-order fluctuation terms of downwind and upwind pulsations. The combined wind speed and wind deflection angle are obtained by vector synthesis of natural wind and train speed. Based on this combined wind speed and the train's aerodynamic coefficient, the drag, lift, and moment acting on the train's center of gravity are calculated. The combined wind speed acting on the train... The calculation formula is as follows:

[0085]

[0086] In the formula: The train speed; The angle of natural wind direction; To synthesize the wind deflection angle, where The expression is as follows:

[0087]

[0088] The calculation method for non-steady wind loads on trains is similar to that for bridges, but when transforming the load into wind speed, the direction of travel is considered. and vertical wind direction The pulsating wind speeds are all much smaller than the composite wind speeds. Referring to the quasi-steady-state theory proposed by Professor Li Yongle, and for ease of calculation, neglecting its higher-order pulsation terms, the calculation formula for the non-stationary wind load acting on the train is as follows:

[0089]

[0090] In the formula: , , These are the resistance, lift, and torque acting at the train's center of gravity, respectively. , , These represent the train's drag coefficient, lift coefficient, and moment coefficient, respectively, and the combined wind deflection angle. The function; The area of ​​the train's windward surface; This refers to the height of the train's center of gravity above the ground.

[0091] S2. Wind-vehicle-bridge coupled vibration analysis under non-stationary pulsating wind fields: A refined finite element model of the bridge is constructed using ANSYS APDL software, and a train dynamics model is established in SIMPACK software. The bridge finite element model and the train dynamics model are then integrated in SIMPACK to construct a vehicle-bridge rigid-flexible coupled model. The vehicle-bridge wind load obtained in step S1 is applied as an external excitation to the vehicle-bridge coupled model to construct the wind-vehicle-bridge coupled model. Specifically:

[0092] S21. Use ANSYS APDL software to construct a refined finite element model of the bridge, and establish a train dynamics model in SIMPACK software. Then integrate the finite element model of the bridge with the train dynamics model in SIMPACK. By establishing the wheel-rail contact relationship and inputting track irregularities, construct a vehicle-bridge rigid-flexible coupling model.

[0093] S211. Establish a refined finite element model of the bridge in ANSYS APDL, use the Guyan reduction method for modal reduction, and generate super element files containing mass and stiffness matrices, including .sub and .cdb format files; the Guyan reduction method divides the bridge's degrees of freedom into principal degrees of freedom and dependent degrees of freedom, and eliminates dependent degrees of freedom based on static transformation to obtain a reduced equilibrium equation containing only principal degrees of freedom.

[0094] The equation of motion for the train is described by the following formula:

[0095]

[0096] In the formula: , , These are the mass, damping, and stiffness matrices of the train, respectively. , , These are vector matrices representing the train's acceleration, velocity, and displacement, respectively. The motivation received by the train.

[0097] Since SIMPACK cannot directly recognize the model, the bridge model created in ANSYS APDL needs to undergo substructure analysis to simplify it into a superelement containing only mass, stiffness, and modal matrices. The equations of motion for the bridge based on the finite element model can be expressed as follows:

[0098]

[0099] In the formula: , , These are the mass, damping, and stiffness matrices of the bridge, respectively. Let be the displacement vector matrix of the nodes; The external excitation experienced by the bridge.

[0100] The basic calculation principle of Guyan's reduction method is as follows:

[0101] The basic static equation of motion of the structure can be expressed as:

[0102]

[0103] In the formula: The overall stiffness matrix of the structure. Let the nodal displacement column vector be... This represents the column vector of external forces at the nodes. When using a static transformation method to reduce the degrees of freedom, the model's degrees of freedom are divided into principal degrees of freedom. Degrees of freedom and subordinate Two types, based on the master-slave relationship of degrees of freedom, can be expanded into the following basic equation:

[0104]

[0105] Decomposing the above equation according to principal and subordinate degrees of freedom, we obtain two independent equilibrium equations:

[0106]

[0107]

[0108] For subordinate degrees of freedom By solving the equilibrium equations, the mapping relationship between dependent degrees of freedom and master degrees of freedom can be derived:

[0109]

[0110] This gives us the conversion relationship between principal and dependent degrees of freedom. Substituting this equation into the equilibrium equation for the principal degree of freedom, we can eliminate the dependent degrees of freedom and obtain the equilibrium equation containing only the principal degree of freedom:

[0111]

[0112] This equation can be further simplified to a reduced equilibrium equation form:

[0113]

[0114] In the formula: This is the reduced structural stiffness matrix. The reduced equivalent nodal load column vector is used to derive the reduced equations of motion, which contain only the set of master degrees of freedom. The mass matrix and damping matrix of the structure are both reduced using the same coordinate transformation method.

[0115] S212. A multibody dynamics model of the train is established in SIMPACK. Using SIMPACK's FlexModal module, the reduced bridge flexible body model is coupled with the train multibody model through virtual rigid bodies and kinematic constraints. The wheel-rail interaction is defined by combining wheel-rail contact geometry, Hertz elastic contact theory, and Kalker's simplified creep theory. Track irregularity excitations generated based on the US track spectrum (levels 5-6, suitable for 200 km / h lines) are imported to construct the vehicle-bridge rigid-flexible coupling model and its boundary conditions. The virtual rigid body and flexible body are connected by hinges, and kinematic constraints are defined between the movement markers on the virtual rigid body and flexible body to realize the transfer of train loads and bridge deformation. The wheel-rail contact geometry is characterized using equivalent taper combined with discrete profile curves.

[0116] Hertz's elastic contact theory assumes that the contact surfaces are smooth and frictionless, that only normal forces exist at the contact interface, that the magnitude of the contact force is directly related to the geometry of the contact area, and that the contact behavior occurs only within an elliptical region in a plane, exhibiting significant stress concentration characteristics. Its core calculation formula is:

[0117]

[0118] In the formula: The contact force is the normal force between the wheel and the rail. The contact constant between the wheel and rail; This refers to the relative compression of the wheel-rail system due to its elasticity.

[0119] Kalker's simplified creep theory, along with its corresponding numerical calculation method (FASTSIM), is used to calculate the wheel-rail tangential contact force. This method is compatible with the solution framework of the three-dimensional rolling Hertz contact theory and is well-suited for extreme conditions with large creep and large spin, combining high computational efficiency with the solution accuracy required for engineering applications. The core formula for calculating creep force is as follows:

[0120]

[0121]

[0122]

[0123] In the formula: It represents half of the minor and major semi-axis of the contact patch ellipse. For the shear modulus of rails and wheels, Vertically, It is horizontal. is the spin rate.

[0124] Track irregularities can be viewed as a stationary, ergodic stochastic process, and samples can be generated using the power spectral density function. Track irregularities are typically categorized into horizontal, vertical, directional, and gauge irregularities. Typical track spectra include those of the United States, Germany, and China's mainline railways. The US track spectrum (including elevation, directional, horizontal, and gauge irregularities) is suitable for lines with speeds of 160 km / h or higher. This study uses this track spectrum to simulate track irregularities. The relevant formulas for the numerical simulation are as follows:

[0125] The track is uneven.

[0126]

[0127] The track direction is not smooth:

[0128]

[0129] Uneven track level and gauge:

[0130]

[0131] in Orbital power spectral density; For safety factors, a value between 0.25 and 1 is used, with 0.25 generally chosen for safety. The spatial spectrum of irregular orbits; , For the cutoff frequency; , is the roughness constant.

[0132] S22. Apply the non-stationary wind load obtained in step S13 as an external excitation to the vehicle-bridge rigid-flexible coupling model constructed in step S21 to establish a non-stationary wind-vehicle-bridge coupling model based on SIMPACK and ANSYS joint modeling and simulation.

[0133] Specifically, this includes: treating train wind load as a concentrated force applied to the center of mass of each train car, defining time history data through the input function of SIMPACK, and implementing loading using force elements; treating bridge wind load as a concentrated force applied to the corresponding mass points of the bridge finite element model, and achieving efficient application by batch modifying background data through self-written code; non-stationary wind load is the vehicle-bridge wind load calculated considering the aerodynamic interference effect of key components of the arch bridge, including time-varying average wind load and buffeting force caused by non-stationary pulsating wind.

[0134] The formula for wind-vehicle-bridge coupled vibration is shown below:

[0135]

[0136] In the formula: These are the mass matrix, damping matrix, and stiffness matrix, respectively, with subscripts... They represent trains and bridges, respectively. External force, subscript These represent the bridge's self-weight, the wheel-rail contact force acting on the bridge, the static wind force, the buffeting force, and the self-excited force, respectively, with subscripts... These represent the train's own weight, the wheel-rail contact force acting on the train, and the static wind force, respectively.

[0137] S23. Based on the non-stationary wind-vehicle-bridge coupled model constructed in step S22, obtain the dynamic response of the vehicle and bridge under different wind speeds and vehicle speeds, and conduct traffic safety analysis.

[0138] Based on the established wind-vehicle-bridge coupling model, to study the dynamic response of the vehicle-bridge system under non-stationary wind conditions, a multi-condition wind-vehicle-bridge coupling model with different wind speeds and vehicle speeds was established. The specific conditions were set as follows: train speeds of 80 km / h, 100 km / h, 120 km / h, 140 km / h, 160 km / h, 180 km / h, and 200 km / h; and the mean time-varying wind speeds of 0 m / s, 10 m / s, 15 m / s, 20 m / s, 25 m / s, and 30 m / s, respectively. Since the train crossing time is relatively short, the entire wind speed range was divided into segments of 30 seconds. The dynamic response of the vehicle-bridge system was solved for each time segment. The system response was most unfavorable when the time-varying wind reached its peak. Therefore, only the period with the largest non-stationary time-varying wind was selected for wind-vehicle-bridge coupling analysis to obtain the dynamic response of the vehicle-bridge under each condition.

[0139] I. Analysis of the following vehicle operation safety indicators under non-stationary pulsating wind loads considering aerodynamic disturbances

[0140] Safety control of train operation under non-stationary wind conditions is one of the core bases for the operation and management of mountain railway lines, and also a key focus of the dynamics research on wind-train-bridge coupled systems. This embodiment focuses on exploring the influence mechanism of two key parameters, non-stationary wind speed and train speed, on various safety indicators, quantifying the train safety reserve level under different operating conditions, and clarifying the wind speed control threshold for safe train operation in high wind conditions. The calculation results of the train safety indicators under various operating conditions are as follows: Figure 2 As shown, where Figure 2 (a) is a comparison chart of the calculated derailment coefficients. Figure 2 (b) is a comparison chart of the calculation results of wheel load reduction rate. Figure 2 (c) is a comparison chart of the calculation results of the wheel-rail lateral force.

[0141] The results show that the changes in the three train operation safety indicators are mainly affected by average wind speed and train speed. The overall trend is that the higher the wind speed and the faster the train speed, the higher the indicator values ​​and the worse the train operation safety. Among these, non-steady wind plays a dominant role; at the same train speed, all three indicators increase significantly with increasing wind speed. This is because the lateral aerodynamic force exerted by crosswinds on the car body directly changes the wheel-rail stress state, increasing the lateral force on the wheelset and causing an asymmetrical distribution of the vertical load on the wheelset. The higher the wind speed, the more significant this aerodynamic effect, and the higher the risk of derailment. Train speed plays a more significant role in amplifying the crosswind effect: at the same wind speed, increased train speed enhances the car body vibration response under track irregularities and also changes the speed of the incoming airflow relative to the train, thus strengthening the impact of aerodynamic loads. Combined with the crosswind effect, this further increases the risk of derailment and the wheel load reduction rate.

[0142] When the average wind speed during non-steady winds is between 0 and 20 m / s, the three indicators show relatively stable changes with vehicle speed, exhibiting a monotonically increasing trend. The dynamic response of the wind-vehicle-bridge system demonstrates a clear linear characteristic without significant abnormal fluctuations. Within this range, all indicators meet the specified limits: the maximum derailment coefficient is 0.37206, far below the limit of 0.8; the maximum wheel load reduction rate is 0.89409, close to but not exceeding the limit of 0.9; and the maximum wheel-rail lateral force is 17.2617 kN, less than half of the 41 kN limit, indicating that the overall performance is within a safe and controllable range.

[0143] When the average time-varying wind speed of non-steady winds reaches 25 m / s or higher, the monotonic variation of the three indicators with vehicle speed exhibits varying degrees of fluctuation and distortion, and the nonlinear characteristics of the system's dynamic response are significantly enhanced. The derailment coefficient and wheel-rail lateral force no longer maintain a stable increasing trend in the low and medium speed wind ranges, but instead show a fluctuating state without a clear linear pattern. For example, at a wind speed of 25 m / s, the derailment coefficient reaches its maximum value under all operating conditions at a vehicle speed of 140 km / h, while the wheel-rail lateral force reaches its peak at 100 km / h and does not continue to increase with increasing vehicle speed; when the average time-varying wind speed of non-steady winds is 30 m / s, the fluctuations of these two indicators are even more pronounced, and there is no clear monotonic relationship between the values ​​at different vehicle speeds, with a significant weakening of regularity.

[0144] Wheel load reduction rate is the most sensitive indicator to the coupling effect of strong winds and high train speeds, and it is also the first of the three control indicators to exceed the safety limit. With a time-varying average wind speed of 25 m / s in non-steady winds, when the train speed reaches 140 km / h or higher, the wheel load reduction rate exceeds the standard limit of 0.9. A wheel load reduction rate of 1 indicates that one wheel has derailed. However, it can also be observed that when the train derails, the derailment coefficient still does not exceed the limit. This is because when the SIMPACK software calculates the derailment coefficient, if the vertical force between the wheel and rail is 0, the derailment coefficient will also be automatically set to 0. When the time-varying average wind speed in non-steady winds is 30 m / s, the wheel load reduction rate corresponding to train speeds of 120 km / h and above has exceeded the limit. When the train speed reaches 180-200 km / h, and the time-varying average wind speed in non-steady winds is greater than 10 m / s, the wheel load reduction rate exceeds the limit across the entire wind speed range, and the train no longer meets the basic conditions for safe operation. The disordered and excessive phenomena of the above indicators in the strong wind range indicate that when the average time-varying wind speed of non-steady wind reaches 25 m / s or above, the strong pulsating characteristics of non-steady wind, the aerodynamic interference between bridges and trains, and the mutual influence of wind-induced vibration of bridges and vehicle vibration are fully stimulated. The superposition and cancellation relationship between multiple excitations becomes highly complex, and the dynamic response of the system is no longer linearly dominated by wind speed and vehicle speed. This leads to a significant distortion in the change pattern of safety indicators and a sharp increase in the risk of train operation.

[0145] The time-varying average wind speed curve when the mean wind speed of non-stationary wind is 10 m / s is as follows: Figure 3 As shown in the figure, this illustrates an envelope / pulse-type wind condition where the time-varying average wind speed changes over time. It shows that the wind speed first increases and then decreases within 0–1000 s, reaching a peak around 500 s. Taking the most unfavorable 30 s period near the peak applied to the vehicle-axle system, the short-term average wind speed during this period is approximately 15 m / s. Combined with… Figure 2 The results from multiple wind speed conditions indicate that when the wind speed exceeds or reaches approximately 15 m / s, some safety indicators approach or exceed the limits.

[0146] II. Analysis of Train Operation Stability Indicators under Non-stationary Pulsating Wind Loads Considering Aerodynamic Disturbances

[0147] Under non-steady wind loads, considering the aerodynamic interference between the arch rib, the arch column, and the main beam, and taking into account the dynamic response of the trailer, the calculation results of the trailer safety indicators for each working condition are as follows: Figure 4 As shown, where Figure 4 (a) is a comparison chart of the calculated results of the vertical acceleration of the vehicle body. Figure 4 (b) is a comparison chart of the calculated results of the vehicle body's lateral acceleration. Figure 4 (c) is a comparison chart of the calculation results for the vertical stability of the vehicle body. Figure 4 (d) is a comparison chart of the calculation results of the vehicle body's lateral stability.

[0148] As shown in the figure, the four stability indicators generally exhibit a characteristic that the higher the average wind speed of the non-steady wind and the faster the train speed, the larger the indicator value and the worse the train's stability. The lateral and vertical vibration acceleration of the car body generally show a macro trend that the higher the average crosswind speed and the faster the train speed, the larger the indicator value. In the low wind speed range, both acceleration indicators show a stable monotonically increasing pattern with increasing train speed. When the average time-varying wind speed of the non-steady wind reaches 20 m / s or above, the monotonically increasing pattern with train speed disappears, exhibiting non-monotonic fluctuation characteristics. In this calculation across the entire operating condition range, the maximum values ​​of the lateral and vertical vibration accelerations of the car body are 1.06 m / s² and 1.44 m / s², respectively, both less than the 2.5 m / s² limit specified in GB / T 5599-2019 "Specifications for Evaluation and Testing of Dynamic Performance of Locomotives and Rolling Stock," with no instances of exceeding the limits.

[0149] The lateral and vertical stability indices of the vehicle body also generally show a macro trend of increasing with wind speed and vehicle speed. Among them, the maximum value of the lateral stability of the vehicle body is 2.76 across all operating conditions, which is always less than the specification limit of 3.0, and all operating conditions meet the specification requirements. When the average wind speed of the time-varying wind is 0 m / s and 10 m / s in non-steady wind conditions, the lateral stability of the vehicle body shows a stable monotonically increasing law with the increase of train speed, with significant linear change characteristics. When the average wind speed of the time-varying wind is 15 m / s in non-steady wind conditions, the increasing law is partially disrupted, and it shows a small non-monotonic fluctuation with the increase of vehicle speed, but it still maintains an overall increasing trend. When the average wind speed of the time-varying wind reaches 20 m / s and above, the monotonically increasing law with the change of vehicle speed completely disappears, showing significant non-monotonic fluctuation characteristics. At a wind speed of 30 m / s, it reaches a peak at a vehicle speed of 160 km / h, and then continues to decline with the increase of vehicle speed. The maximum lateral stability of the vehicle body under all working conditions is 2.76, which is less than the specification limit of 3.0, and there are no cases of exceeding the limit.

[0150] The vertical stability of the train body is more sensitive to changes in train speed. Within the time-varying average wind speed range of 0 m / s and 10 m / s, the monotonically increasing trend with increasing speed is more pronounced, with only a slight decrease at 200 km / h, and the overall upward trend remains unchanged. When the time-varying average wind speed is 15 m / s, the fluctuation range of the indicator with increasing speed increases, but the overall upward trend is still maintained. When the time-varying average wind speed reaches 20 m / s... At speeds of s and above, the monotonically increasing trend with vehicle speed completely disappears, and the trend becomes significantly erratic. When the average time-varying wind speed of the non-steady wind is 25 m / s, the speed drops significantly at 200 km / h, decreasing by 17.1% compared to the 180 km / h condition. When the average time-varying wind speed of the non-steady wind is 30 m / s, the trend is most erratic, with the index reaching a maximum of 3.02 at 160 km / h, exceeding the standard compliance limit of 3.0. All other conditions meet the standard requirements.

[0151] III. Dynamic Response Analysis of Arch Bridge Structure under Non-stationary Pulsating Wind with Consideration of Aerodynamic Disturbances

[0152] To accurately extract the most unfavorable response of the main girder under various working conditions, we first identified the points where the main girder displacement changes the most under each working condition, and then extracted their vertical and lateral displacements and accelerations for bridge response analysis. During this process, we found that the points where the main girder displacement changes the most differed under each working condition. Taking wind speeds of 80 km / h, 100 km / h, and 120 km / h as examples, the displacement multiplied by 1000 when the entire bridge reaches its most unfavorable state is as follows: Figure 5 As shown.

[0153] Maximum bridge response under various working conditions, such as Figure 6As shown in the figure, the main factor affecting the dynamic response of the bridge is the average wind speed of the non-stationary wind. At the same wind speed, excluding a few special cases, the lateral and vertical displacements and lateral and vertical accelerations of the main girder only show a slight increasing trend with the increase of train speed. This pattern indicates that for long-span arch bridges, the proportion of the moving live load of the train relative to the self-weight of the bridge structure is extremely low. The fluctuation of the moving load effect caused by changes in train speed is far smaller than the static gust displacement effect and pulsating wind-induced vibration effect caused by non-stationary crosswinds. Therefore, the contribution of train speed to the overall dynamic response of the main girder is limited, and the average wind speed of the non-stationary wind is the core parameter that determines the response level of the main girder. The maximum vertical displacement of the main beam under all operating conditions was 57.5627 mm, which is only 29.2% of the limit of 197 mm in the "Railway Bridge Inspection Specification" (Railway Transport

[2004] No. 120); the maximum lateral displacement of the main beam under all operating conditions was 46.7288 mm, which is only 53.3% of the limit of 87.6 mm in the "Railway Bridge Inspection Specification". Both displacement indicators did not reach the limit of the specification, which fully demonstrates that the bridge structure has sufficient vertical and lateral stiffness under the target operating conditions and the wind-induced deformation is completely within the controllable range. Regarding acceleration indicators, the maximum vertical acceleration of the main beam under all working conditions is 2.94697 m / s², which does not reach the limit of 3.5 m / s² in the "Railway Bridge Inspection Specification". The lateral acceleration of the main beam reaches the maximum value of 1.369 m / s² under the most unfavorable combination of train speed of 200 km / h and time-varying average wind speed of 30 m / s under non-steady wind conditions. Although it is close to the limit of 1.4 m / s² in the "Railway Bridge Inspection Specification", it still does not exceed the qualified range specified in the specification and fully meets the vibration comfort and dynamic safety requirements of the bridge structure during the operation stage.

[0154] Finally, it should be noted that the above embodiments are only used to illustrate the technical solutions of the present invention and are not intended to limit it. Although the present invention has been described in detail with reference to preferred embodiments, those skilled in the art should understand that modifications or equivalent substitutions can be made to the technical solutions of the present invention without departing from the spirit and scope of the present invention, and all such modifications or substitutions should be covered within the scope of the claims of the present invention.

Claims

1. A method for analyzing the coupled vibration of a long-span arch bridge under non-stationary pulsating wind fields involving wind, a vehicle, and a bridge, characterized in that... Includes the following steps: S1. Analysis of wind load on a long-span arch bridge in mountainous areas under non-stationary pulsating wind; specifically including: S11. Analyzing the spatial characteristics of wind fields in mountainous canyons, using the harmonic synthesis method and introducing a non-stationary modulation function, and using a coherence function to simulate the non-stationary pulsating wind field at different spatial locations of the arch bridge, generating non-stationary pulsating wind speed time histories at multiple points along the windward and vertical directions of the entire bridge; S12. Based on CFD numerical simulation, calculating the influence of mutual aerodynamic interference between the main beam, arch rib, and arch column on the aerodynamic characteristics of the main beam, obtaining the static three-component force coefficients of the main beam considering aerodynamic interference under different wind attack angles, and obtaining the train aerodynamic coefficients through CFD numerical simulation of the train model; S13. Combining the non-stationary pulsating wind field simulated in step S11 and the aerodynamic coefficients of the bridge obtained in step S12, calculating the non-stationary wind loads on the bridge at different locations and wind speeds under aerodynamic interference, as well as the non-stationary wind loads on the train; S2. Wind-vehicle-bridge coupled vibration analysis under non-stationary pulsating wind fields; specifically including: S21. Constructing a refined finite element model of the bridge using ANSYS APDL software, and establishing a train dynamics model in SIMPACK software. Then, integrating the finite element model of the bridge with the train dynamics model in SIMPACK, and constructing a vehicle-bridge rigid-flexible coupling model by establishing wheel-rail contact relationship and inputting track irregularities; S22. Applying the non-stationary wind load obtained in step S13 as an external excitation to the vehicle-bridge rigid-flexible coupling model constructed in step S21, and establishing a non-stationary wind-vehicle-bridge coupled model based on joint modeling and simulation of SIMPACK and ANSYS; S23. Obtaining the dynamic response of the vehicle and bridge under different wind speeds and vehicle speeds based on the non-stationary wind-vehicle-bridge coupled model constructed in step S22, and conducting traffic safety analysis.

2. The wind-vehicle-bridge coupled vibration analysis method as described in claim 1, characterized in that, Step S11 specifically involves simplifying the three-dimensional wind field into three independent one-dimensional stochastic processes. Typically, the spatial three-dimensional wind field is represented by the following formula: In the formula: The crosswind speed is the wind speed across the bridge. It is a transverse bridge; Vertical; These represent the fluctuating wind speeds in the vertical, downwind, and crosswind directions, respectively. For time; The expression for non-stationary pulsating wind is: In the formula: For non-steady wind speeds, The time-varying average wind speed, This refers to non-stationary pulsating wind speeds; The simulation of non-stationary fluctuating wind first requires converting the power spectrum of stationary fluctuating wind into that of non-stationary fluctuating wind using a modulation function. The evolving power spectral density of non-stationary fluctuating wind is as follows: in: It is a non-stationary power spectral function; It is a deterministic non-uniform time-frequency modulation function; The autospectral density function of a zero-mean stationary random process; In real-world environments, non-stationary fluctuating wind speeds at different points in space exhibit spatial correlation. This correlation is stronger the closer the two points are to each other, and the spatial correlation function... The expression is: In the formula: The attenuation factor transforms the above equation into an expression for the angular frequency: For wind speed simulations at different altitudes, the vertical distribution of wind speed within the atmospheric boundary layer follows an exponential law, and the average wind speed at different altitudes can be calculated using the following formula: In the formula: It is the surface roughness coefficient; The reference wind speed is at a height of 10m. This represents the actual height of the observation point above the ground.

3. The wind-vehicle-bridge coupled vibration analysis method as described in claim 1, characterized in that, Step S12 specifically involves: the computational domain of the CFD numerical simulation adopts a velocity inlet boundary, a pressure outlet boundary, and periodic boundaries for the upper and lower boundaries; the main beam model adopts a non-slip wall boundary; by mixing structured and unstructured meshes, an unstructured mesh refinement region is created around the bridge structure, and a structured mesh is used in the far field region to balance computational accuracy and efficiency. The analysis of the aerodynamic interference between the main beam, arch rib, and arch column includes: selecting the main beam cross-section under the joint interference of the arch column and arch rib at different heights, and the main beam cross-section under different vertical distances between the arch rib and the main beam; calculating the static three-component force coefficients of the uninterrupted isolated main beam cross-section and each interfering cross-section under different wind attack angles. The aerodynamic coefficients of a train include at least the drag coefficient, lift coefficient, and moment coefficient.

4. The wind-vehicle-bridge coupled vibration analysis method as described in claim 1, characterized in that, In step S13, the calculation of the non-stationary wind load on the bridge only considers the time-varying average wind load and the buffeting force caused by the non-stationary pulsating wind; wherein, the time-varying average wind load is calculated based on the time-varying average wind speed and the static three-component force coefficient considering aerodynamic disturbances, and the buffeting force is calculated based on the non-stationary pulsating wind speed and the aerodynamic admittance function. The time-varying average wind load is expressed by the following formula: In the formula: , , These represent the lift, drag, and torque generated by wind loads, respectively. , , These are the lift coefficient, drag coefficient, and moment coefficient, respectively, and represent the angle of attack. The function; The time-varying average wind speed; The width of the structural cross-section; Segment length; air density; The buffeting force caused by non-stationary pulsating wind is calculated using the following formula: In the formula: , , These are the non-stationary buffeting lift, drag, and torque, respectively. , , These are the derivatives of the corresponding lift coefficient, drag coefficient, and moment coefficient with respect to the effective angle of attack, respectively. , , , , , It is the aerodynamic admittance function; , This is to simulate non-stationary fluctuating wind speeds in the downwind and vertical directions.

5. The wind-vehicle-bridge coupled vibration analysis method as described in claim 4, characterized in that, In step S13, the calculation of the non-steady wind load on the train includes the resultant wind speed acting on the train. The calculation formula is as follows: In the formula: The train speed; The angle of natural wind direction; To synthesize the wind deflection angle, where The expression is as follows: The calculation of non-stationary wind load on trains neglects higher-order pulsation terms. The formula for calculating non-stationary wind load acting on trains is as follows: In the formula: , , These are the resistance, lift, and torque acting at the train's center of gravity, respectively. , , These represent the train's drag coefficient, lift coefficient, and moment coefficient, respectively, and the combined wind deflection angle. The function; The area of ​​the train's windward surface; This refers to the height of the train's center of gravity above the ground.

6. The wind-vehicle-bridge coupled vibration analysis method as described in claim 1, characterized in that, Step S21 specifically includes: S211. Establish a refined finite element model of the bridge in ANSYS APDL, use the Guyan reduction method for modal reduction, and generate super element files containing mass and stiffness matrices, including .sub and .cdb format files; the Guyan reduction method divides the bridge's degrees of freedom into principal degrees of freedom and dependent degrees of freedom, and eliminates dependent degrees of freedom based on static transformation to obtain a reduced equilibrium equation containing only principal degrees of freedom. S212. Establish a train multibody dynamics model in SIMPACK. Using SIMPACK's FlexModal module, couple the reduced bridge flexible body model with the train multibody model through virtual rigid bodies and kinematic constraints. Combine the wheel-rail contact geometry, Hertz elastic contact theory, and Kalker's simplified creep theory to define wheel-rail interaction and import track irregularity excitation to construct a car-bridge rigid-flexible coupling model and its boundary conditions. The virtual rigid body and flexible body are connected by hinges, and kinematic constraints are defined between the movement markers on the virtual rigid body and the flexible body to realize the transfer of train load and bridge deformation.

7. The wind-vehicle-bridge coupled vibration analysis method as described in claim 6, characterized in that, The reduced equilibrium equation in step S211 is: In the formula: This is the reduced structural stiffness matrix. The reduced equivalent nodal load column vectors are obtained by using the same coordinate transformation method to reduce the mass matrix and damping matrix of the structure.

8. The wind-vehicle-bridge coupled vibration analysis method as described in claim 1, characterized in that, The formula for the wind-vehicle-bridge coupled vibration in step S22 is as follows: In the formula: These are the mass matrix, damping matrix, and stiffness matrix, respectively, with subscripts... They represent trains and bridges, respectively. External force, subscript These represent the bridge's self-weight, the wheel-rail contact force acting on the bridge, the static wind force, the buffeting force, and the self-excited force, respectively, with subscripts... These represent the train's own weight, the wheel-rail contact force acting on the train, and the static wind force, respectively.

9. A computer-readable storage medium having a computer program stored thereon, characterized in that, When the program is executed by the processor, it implements the wind-vehicle-bridge coupled vibration analysis method for long-span arch bridges under non-stationary pulsating wind fields as described in any one of claims 1 to 8.

10. An electronic device, characterized in that, The method includes a memory, a processor, and a computer program stored in the memory and executable on the processor. When the processor executes the program, it implements the wind-vehicle-bridge coupled vibration analysis method under non-stationary pulsating wind fields of long-span arch bridges as described in any one of claims 1 to 8.

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