A method and system for calculating coupled dynamics of high-speed train vehicles and tracks

Abstract

This invention relates to the field of rail transit technology, and more particularly to a method and system for calculating the coupled dynamics of high-speed rail vehicles and tracks. The constitutive equations of the fractional-order Zener model are transformed into a CA mortar layer model, and its numerator operator is expressed as a discrete series to obtain the target CA mortar layer model. The ground reaction force of the CA mortar layer acting on the track slab is then obtained. Based on the ground reaction force of the CA mortar layer acting on the track slab, the vibration differential equations of the track slab are constructed. The vibration differential equations of the track slab are transformed into a set of ordinary differential equations in modal coordinates containing fractional-order terms using the Ritz method. Taking a half-car model as the vehicle subsystem, the vibration differential equations of the rail in modal coordinates and the set of ordinary differential equations in modal coordinates containing fractional-order terms of the track slab are integrated into the track subsystem to construct the coupled dynamics model of the high-speed rail vehicle and track. This invention effectively improves the accuracy of vehicle-track coupled dynamics calculations.

Description

Technical Field

[0001] This invention relates to the field of rail transit technology, and in particular to a method and system for calculating the coupled dynamics of high-speed train vehicles and tracks. Background Technology

[0002] The high-speed rail vehicle-track coupled dynamics model is a core technical tool for evaluating the safety, stability, and service performance of high-speed rail operations and track structures. Based on the coupling mechanism between the two core modules of the vehicle and track, the high-speed rail vehicle-track coupled dynamics model achieves accurate simulation of the dynamic response during high-speed rail operation, providing crucial computational support for engineering practice and theoretical research such as ballastless track structure design, vehicle operating parameter optimization, and fault mechanism analysis.

[0003] The existing high-speed rail vehicle-track coupled dynamics model is a classic coupled calculation model widely used in the dynamic analysis of high-speed rail ballastless tracks. Its core architecture includes two major subsystems: the vehicle and the track. The vehicle sub-model adopts a 10-DOF model, covering the vertical and pitch motions of the car body, bogies, and wheelsets. The suspension system is simplified as a spring-damped parallel structure. In the track sub-model, the rail is regarded as a continuous Euler-Bernoulli beam, the track slab is modeled as a finite-length free beam, and the mechanical behavior of the CA (cement emulsion asphalt) mortar layer under the rail is described by the traditional Kelvin viscoelastic model. The dynamic coupling between the vehicle and the track is based on the wheel-rail nonlinear Hertzian theory. The system dynamic equations are solved by the direct integration method, which can output key response indicators such as rail displacement, car body acceleration, and wheel-rail force, meeting the basic requirements of conventional dynamic analysis.

[0004] Although existing high-speed rail vehicle-track coupled dynamics models have a mature application foundation, their constitutive characterization accuracy for the CA mortar layer under the track is significantly insufficient, directly restricting the overall model's computational accuracy and adaptability to operating conditions. The Kelvin viscoelastic model used in existing models can only describe the simple linear viscoelastic behavior of materials, failing to accurately reflect the true mechanical properties of CA mortar in actual high-speed rail operation scenarios, especially its temperature-dependent viscoelastic characteristics over a wide temperature range. High-speed rail lines operate across regions and experience significant seasonal temperature variations. CA mortar undergoes significant mechanical state changes with temperature fluctuations, exhibiting rigid, solid-like mechanical properties at low temperatures and flexible, fluid-like rheological properties at high temperatures. The Kelvin model completely fails to characterize this crucial temperature response law, resulting in large deviations between the model's dynamic response calculations under complex temperature conditions and actual operating conditions. This makes it difficult to accurately reflect the dynamic behavior of ballastless track structures in real operating environments and limits the model's application expansion in complex scenarios such as extreme temperatures and long-term service. Summary of the Invention

[0005] Therefore, the technical problem to be solved by the present invention is to overcome the shortcomings of the existing Kelvin viscoelastic model, which can only describe the simple linear viscoelastic behavior of materials and cannot fit the real mechanical properties of CA mortar in the actual operation scenario of high-speed rail, resulting in poor calculation accuracy of the high-speed rail vehicle-track coupled dynamics model.

[0006] To address the aforementioned technical problems, this invention provides a method for calculating the coupled dynamics of high-speed train vehicles and tracks, comprising: The constitutive equations of the fractional Zener model are transformed into the CA mortar layer model; The molecular operator in the CA mortar layer model is expressed as a discrete series form to obtain the target CA mortar layer model. Based on the target CA mortar layer model, the ground reaction force of the CA mortar layer acting on the track slab is obtained; Based on the ground reaction force of the CA mortar layer acting on the track slab, the bending stiffness, vertical displacement, mass of the track slab, the length of a single track slab, and the vertical force transmitted from the rail to the track slab through the fasteners, the vibration differential equation of the track slab is constructed. The vibration differential equation of the track slab is transformed into a system of ordinary differential equations in the modal coordinates of the track slab containing fractional terms by performing modal coordinate transformation using the Ritz method. Using a half-car model as the vehicle subsystem, the vibration differential equations of the rail in modal coordinates and the ordinary differential equations of the track slab in modal coordinates containing fractional terms are integrated into the track subsystem. Based on the wheel-rail nonlinear Hertzian contact theory, a high-speed rail vehicle-track coupled dynamic model is constructed, and dynamic numerical calculations are performed on this high-speed rail vehicle-track coupled dynamic model.

[0007] Preferably, the method for converting the constitutive equation of the fractional Zener model into a CA mortar layer model includes: The stress in the constitutive equation of the fractional Zener model is replaced by the ground reaction force of the CA mortar layer acting on the track slab, and the strain in the constitutive equation of the fractional Zener model is replaced by the vertical displacement of the track slab, thus obtaining the CA mortar layer model.

[0008] Preferably, the CA mortar layer model is as follows: , in, CA mortar layer applied to the track slab location ,time The foundation reaction force, , , , For elastic modulus, The viscosity coefficient is... Let be the order of the fractional derivative. Indicates time, Indicates the position of the track slab. This indicates finding the partial derivative. For the track slab in position ,time The vertical displacement.

[0009] Preferably, the target CA mortar layer model is as follows: , in, CA mortar layer applied to track slab ,time The foundation reaction force, , , , , , , For elastic modulus, The viscosity coefficient is... Let be the order of the fractional derivative. Indicates time, Indicates the time step. Indicates the position of the track slab. This indicates finding the partial derivative. For the track slab in position ,time The vertical displacement, This is the Grunwald coefficient. , For gamma function, For historical integration steps, For historical step indexing, For the track slab in position ,time The vertical displacement, CA mortar layer applied to track slab ,time The ground reaction force.

[0010] Preferably, the vibration differential equation of the track slab is: , in, Let be the bending stiffness of the track slab. For the track slab at time The Modal coordinates, For the track slab in position The first First-order modal displacement, Indicates the position of the track slab. The length of a single track slab, express Position The fourth derivative, For the quality of the track slab, for The second derivative, , , , , , , For elastic modulus, The viscosity coefficient is... Let be the order of the fractional derivative. For historical integration steps, For historical step indexing, Indicates time, Indicates the time step. Indicates the position of the track slab. CA mortar layer applied to track slab ,time The foundation reaction force, Here, u represents the number of fastener support points on a single track slab, and u is the index of the fastener support point on a single track slab. For the force at the u-th fastener support point on a single track plate, For the track slab in position The first First-order modal displacement, Let u be the position of the u-th fastener support point on a single track plate. This is the Grunwald coefficient. , For gamma function, For the track slab at time The Modal coordinates.

[0011] Preferably, the system of ordinary differential equations for the modal coordinates of the track slab containing fractional terms is as follows: , in, For the quality of the track slab, For the track slab at time The Modal coordinates, for The second derivative, Let be the bending stiffness of the track slab. The length of a single track slab, , , , , , , For elastic modulus, The viscosity coefficient is... Let be the order of the fractional derivative. For historical integration steps, For historical step indexing, Indicates time, Indicates the time step. For free beam coefficients, This is the Grunwald coefficient. , For gamma function, CA mortar layer applied to track slab ,time The foundation reaction force, Indicates the position of the track slab. Here, u represents the number of fastener support points on a single track slab, and u is the index of the fastener support point on a single track slab. For the force at the u-th fastener support point on a single track plate, For the track slab in position The first First-order modal displacement, Let be the position of the u-th fastener support point on a single track plate.

[0012] Preferably, the process of obtaining the vibration differential equation of the rail in modal coordinates includes: The vibration differential equation of the rail is transformed into the vibration differential equation of the rail in modal coordinates using the Ritz method.

[0013] Preferably, the vibration differential equation of the rail in modal coordinates is: , in, For the rails at all times The Modal coordinates, Indicates the modal order of the orbit. for The second derivative, where E is the elastic modulus of the track slab. Let be the mass per unit length of the rail, and l be the length of the rail. Let N be the moment of inertia of the rail section about the Y-axis, and let N represent the total number of fastener supports within the length l. Let u be the reaction force at the u-th fastener support point on the rail. For the track in position The first First-order modal displacement, Let U be the coordinates of the u-th fastener support point on the rail. For the track in position The first First-order modal displacement, Let r be the position of the r-th wheelset on the track. For the r-th wheel pair at time... The force, Indicates the time.

[0014] Preferably, the method for performing dynamic numerical calculations on the high-speed rail vehicle-track coupled dynamic model includes: A fast explicit integration method is used to perform numerical dynamic calculations on the coupled dynamic model of high-speed train vehicle-track.

[0015] This invention also provides a high-speed rail vehicle-track coupled dynamics calculation system, comprising: The model building module is used to transform the constitutive equations of the fractional Zener model into the CA mortar layer model; The discrete module is used to represent the molecular operators in the CA mortar layer model as discrete series form to obtain the target CA mortar layer model. The foundation reaction force acquisition module is used to acquire the foundation reaction force of the CA mortar layer acting on the track slab based on the target CA mortar layer model. The vibration differential equation construction module is used to construct the vibration differential equation of the track slab based on the foundation reaction force of the CA mortar layer acting on the track slab, the bending stiffness of the track slab, the vertical displacement, the mass, the length of a single track slab, and the vertical force transmitted from the rail to the track slab through the fasteners. The transformation module is used to perform modal coordinate transformation on the vibration differential equation of the track slab using the Ritz method, transforming it into a system of ordinary differential equations in the modal coordinates of the track slab containing fractional terms. The computation module is used to integrate the vibration differential equations of the rail in modal coordinates and the set of ordinary differential equations of the track slab in modal coordinates containing fractional terms into the track subsystem, taking the half-car model as the vehicle subsystem; based on the wheel-rail nonlinear Hertzian contact theory, a high-speed rail vehicle-track coupled dynamic model is constructed, and dynamic numerical calculations are performed on the high-speed rail vehicle-track coupled dynamic model.

[0016] Compared with the prior art, the above-described technical solution of the present invention has the following advantages:

[0017] The present invention discloses a method and system for calculating the coupled dynamics of high-speed rail vehicles and tracks. The present invention takes into account the advantage of the fractional Zener model in accurately characterizing the temperature-dependent viscoelastic characteristics of CA mortar from solid-like to fluid-like over a wide temperature range. However, the fractional derivatives need to be discretized using the Grünwald-Letnikov scheme, which results in problems such as large amounts of historical data storage and recursive calculations, and high computational costs. Furthermore, the vehicle-track coupled dynamics system itself is a complex multibody system, and directly introducing it would significantly increase the numerical solution time, making it difficult to adapt to the real-time simulation and analysis requirements of actual engineering. To this end, this invention first uses the Lenwald-Laitnikov scheme to discretize the fractional derivative, transforming the complex fractional derivative into a weighted summation of historical data. Then, through algebraic operations, an explicit expression for the reaction force of the CA mortar layer foundation at the current moment is derived. This explicit expression is substituted into the vibration differential equation of the track slab, replacing the original simple force terms. The Ritz method is then applied to perform modal coordinate transformation on the reconstructed differential equation, transforming the partial differential equation containing spatial variables and fractional historical terms into a system of modal coordinate ordinary differential equations containing only time variables, thus eliminating the computational overhead of coupling between the spatial domain and the fractional derivative. This transformation unifies the 10-DOF equations of the vehicle, the track slab equations using the fractional Zener model, the rail equations, and the mathematical form of the wheel-rail nonlinear Hertzian contact force into a set of ordinary differential equations. Ultimately, this allows the entire complex coupled system to be solved using numerical integration methods. This not only overcomes the technical difficulties of low computational efficiency and incompatibility with the coupled system after the introduction of the fractional Zener model, but also organically combines the refined mechanical modeling of the CA mortar layer with the vehicle-track coupled dynamics system. This allows the accurate representation of the true mechanical properties of the CA mortar by the fractional Zener model to be integrated into the dynamic response calculation of the coupled system, thus improving the computational accuracy of the high-speed rail vehicle-track coupled dynamics model from the perspective of material constitutive modeling. Attached Figure Description

[0018] To make the content of this invention easier to understand, the invention will be further described in detail below with reference to specific embodiments and accompanying drawings, wherein:

[0019] Figure 1 It is a traditional high-speed rail vehicle-track coupled dynamics calculation model.

[0020] Figure 2 This is a flowchart illustrating a high-speed rail vehicle-track coupling dynamics calculation method according to the present invention.

[0021] Figure 3 It is a CA mortar layer model that adopts a fractional Zener model.

[0022] Figure 4 This is a comparison diagram of track slab displacements for different high-speed rail vehicle-track coupled dynamic models.

[0023] Figure 5 This is a comparison chart of rail acceleration and displacement for different high-speed rail vehicle-track coupled dynamic models.

[0024] Figure 6 This is a comparison chart of rail displacements for different high-speed rail vehicle-track coupled dynamic models.

[0025] Figure 7 This is a comparison chart of rail acceleration and displacement for different high-speed rail vehicle-track coupled dynamic models.

[0026] Figure 8 This is a comparison diagram of the CA mortar layer forces under different high-speed rail vehicle-track coupled dynamic models.

[0027] Figure 9 The creep compliance curves of the high-speed rail vehicle-track coupled dynamic model of this invention, corresponding to CA mortar, were obtained in a uniaxial compression experiment under different temperature conditions.

[0028] Figure 10 These are comparison diagrams of track slab displacement in the high-speed rail vehicle-track coupled dynamics model of this invention under different temperature conditions.

[0029] Figure 11 This is a comparison diagram of track slab acceleration in the high-speed rail vehicle-track coupled dynamics model of this invention under different temperature conditions.

[0030] Figure 12 These are comparison diagrams of rail displacement in the high-speed rail vehicle-track coupled dynamics model of this invention under different temperature conditions.

[0031] Figure 13 This is a comparison diagram of rail acceleration in the high-speed rail vehicle-track coupled dynamics model of this invention under different temperature conditions. Detailed Implementation

[0032] The present invention will be further described below with reference to the accompanying drawings and specific embodiments, so that those skilled in the art can better understand and implement the present invention. However, the embodiments described are not intended to limit the present invention.

[0033] like Figure 1 As shown, Figure 1 This is a traditional high-speed rail vehicle-track coupled dynamics calculation model, which describes the mechanical behavior of the CA (cement emulsion asphalt) mortar layer under the track using the traditional Kelvin viscoelastic model. However, the Kelvin model used can only describe simple viscoelastic behavior and cannot accurately reflect the temperature-dependent viscoelastic characteristics of this material in the wide temperature range of actual high-speed rail operation, such as the transformation of mechanical properties from solid-like at low temperatures to fluid-like at high temperatures.

[0034] Fractional-order Zener (FDZ) models have certain advantages in describing such complex viscoelastic behavior. By leveraging the temperature correlation of fractional-order FDZ models, they can accurately characterize the wide-temperature-range mechanical properties of CA mortar. However, this model has not yet been truly embedded into vehicle-track coupled dynamics systems. In actual calculations, the CA mortar layer of the track sub-model still uses traditional integer-order models to solve for the dynamic response. This is because fractional-order derivative models, when handling non-integer-order differential operations (e.g., based on the Riemann-Liouville definition), typically require discretization using the Grunwald-Letnikov scheme. This process involves storing large amounts of historical data and recursive calculations, significantly increasing computational costs. Vehicle-track coupled dynamics itself is a complex multibody system; introducing fractional-order models would further increase the numerical solution time, hindering real-time simulation and analysis in engineering practice.

[0035] Furthermore, the development of vehicle-track coupled dynamics research has evolved from independent modeling of vehicles and tracks to holistic coupled analysis. For a long time, research in this field has focused primarily on the damage behavior of fastener systems or CA mortar, while relatively little attention has been paid to the refined mechanical modeling of components such as CA mortar. It is precisely due to the dual constraints of computational efficiency and previous research orientation that fractional derivative methods have not been directly and widely applied in vehicle-track coupled dynamics systems, leading to their neglect.

[0036] When the traditional Kelvin viscoelastic model describes the mechanical behavior of the CA (cement emulsion asphalt) mortar layer under the track, the track slab is regarded as a finite-length free beam, and its corresponding vibration differential equation is as follows: , in, Let be the bending stiffness of the track slab. For the track slab in position ,time The vertical displacement, This indicates finding the partial derivative. Indicates time, Indicates the position of the track slab. For the quality of the track slab, The length of a single track slab, , These represent the distributed damping and distributed stiffness per unit length of the lower CA mortar layer (traditional Kelvin model), respectively. This refers to the number of fastener support points on a single track slab. This is the Dirac function.

[0037] The above vibration differential equations are transformed into a system of ordinary differential equations in modal coordinates using the Ritz method, as follows: , in, For the quality of the track slab, For the track slab at time The Modal coordinates, for The second derivative, The length of a single track slab, , These represent the distributed damping and distributed stiffness per unit length of the lower CA mortar layer (traditional Kelvin model), respectively. Let be the bending stiffness of the track slab. This is the free beam coefficient (constant). For the track slab in position The first First-order modal displacement, Let u be the position of the u-th fastener support point on a single track plate. .

[0038] To address the shortcomings of traditional integer-order viscoelastic models in fitting accuracy over a wide temperature range and the difficulty in quantifying the temperature-dependent transformation of CA mortar from a solid-like to a fluid-like state, a fractional-order viscoelastic model is introduced to more accurately characterize the complex mechanical behavior of materials in actual temperature fields, significantly improving the consistency between the constitutive model and the actual response.

[0039] Reference Figure 2 As shown, this embodiment provides a method for calculating the coupled dynamics of high-speed rail vehicles and tracks, aiming to more accurately reflect the temperature-dependent viscoelastic properties of the CA mortar layer, and on this basis improve the accuracy of dynamic response analysis of the vehicle-track system, including: like Figure 3 As shown, Figure 3 This is a CA mortar layer model using a fractional Zener model.

[0040] Step S1: Transform the constitutive equation of the fractional Zener model into the CA mortar layer model; In this embodiment, specifically, the method for transforming the constitutive equation of the fractional Zener model into the CA mortar layer model includes: The stress in the constitutive equation of the fractional Zener model is replaced by the ground reaction force of the CA mortar layer acting on the track slab, and the strain in the constitutive equation of the fractional Zener model is replaced by the vertical displacement of the track slab, thus obtaining the CA mortar layer model.

[0041] The constitutive equation of the fractional Zener model is: , in, For stress, In response, Let be the order of the fractional derivative. The sign for differentiation.

[0042] The CA mortar layer model is as follows: , in, CA mortar layer applied to the track slab location ,time The foundation reaction force, , , , For elastic modulus, The viscosity coefficient is... Let be the order of the fractional derivative. Indicates time, Indicates the position of the track slab. Denotes the fractional derivative. For the track slab in position ,time The vertical displacement.

[0043] Step S2: Express the molecular operators in the CA mortar layer model as discrete series form to obtain the target CA mortar layer model; Based on the definition of fractional derivatives, this invention uses the Grünwald-Letnikov scheme to represent the molecular operator in the CA mortar layer model as a discrete series: , , The target CA mortar layer model is as follows: , in, CA mortar layer applied to track slab ,time The foundation reaction force, , , , , , , For elastic modulus, The viscosity coefficient is... Let be the order of the fractional derivative. Indicates time, Indicates the time step. Indicates the position of the track slab. Represents fractional partial derivatives. For the track slab in position ,time The vertical displacement, This is the Grunwald coefficient. , For gamma function, For historical integration steps, For historical step indexing, For the track slab in position ,time The vertical displacement, CA mortar layer applied to track slab ,time The ground reaction force.

[0044] Step S3: Based on the target CA mortar layer model, obtain the ground reaction force of the CA mortar layer acting on the track slab; Step S4: Based on the ground reaction force of the CA mortar layer acting on the track slab, the bending stiffness, vertical displacement, mass, length of a single track slab, and the vertical force transmitted from the rail to the track slab through the fasteners, construct the vibration differential equation of the track slab. The vibration differential equation of the track slab is: , Applying the CA mortar layer to the foundation reaction force of the track slab Substitute the expression into the equation and multiply both sides. Then, the entire length of the track slab was inspected. Integral, utilizing modal orthogonality and the properties of the Dirac function: , , in, for The derivative of For containing The function.

[0045] get: , in, Let be the bending stiffness of the track slab. For the track slab at time The Modal coordinates, For the track slab in position The first First-order modal displacement, Indicates the position of the track slab. The length of a single track slab, express Position The fourth derivative, For the quality of the track slab, for The second derivative, , , , , , , For elastic modulus, The viscosity coefficient is... Let be the order of the fractional derivative. For historical integration steps, For historical step indexing, Indicates time, Indicates the time step. Indicates the position of the track slab. CA mortar layer applied to track slab ,time The foundation reaction force, Here, u represents the number of fastener support points on a single track slab, and u is the index of the fastener support point on a single track slab. For the force at the u-th fastener support point on a single track plate, For the track slab in position The first First-order modal displacement, Let u be the position of the u-th fastener support point on a single track plate. This is the Grunwald coefficient. , For gamma function, For the track slab at time The Modal coordinates.

[0046] Step S5: Perform modal coordinate transformation on the vibration differential equation of the track slab using the Ritz method, transforming it into a system of ordinary differential equations in the modal coordinates of the track slab containing fractional terms; Using the Ritz method: , , The system of ordinary differential equations for the modal coordinates of the track slab containing fractional terms is as follows: , in, For the quality of the track slab, For the track slab at time The Modal coordinates, for The second derivative, Let be the bending stiffness of the track slab. The length of a single track slab, , , , , , , For elastic modulus, The viscosity coefficient is... Let be the order of the fractional derivative. For historical integration steps, For historical step indexing, Indicates time, Indicates the time step. For free beam coefficients, This is the Grunwald coefficient. , For gamma function, CA mortar layer applied to track slab ,time The foundation reaction force, Indicates the position of the track slab. Here, u represents the number of fastener support points on a single track slab, and u is the index of the fastener support point on a single track slab. For the force at the u-th fastener support point on a single track plate, For the track slab in position The first First-order modal displacement, Let be the position of the u-th fastener support point on a single track plate.

[0047] Step S6: Using the half-car model as the vehicle subsystem, integrate the vibration differential equations of the rail in modal coordinates and the ordinary differential equations of the track slab in modal coordinates containing fractional terms into the track subsystem; based on the wheel-rail nonlinear Hertzian contact theory, construct a high-speed rail vehicle-track coupled dynamic model, and perform dynamic numerical calculations on the high-speed rail vehicle-track coupled dynamic model.

[0048] In this embodiment, the high-speed rail vehicle-track coupled dynamics model specifically includes the following modules: the vehicle module adopts a 10-DOF half-vehicle model; the track module includes rails, track slabs, CA mortar layer, and fastener system; wherein, the viscoelastic behavior of the CA mortar layer is described by a fractional-order Zener model (FDZ model); the wheel-rail contact module is based on nonlinear Hertzian contact theory; and the excitation module considers track irregularity excitation. For numerical solution, a fast explicit integration method is used for calculation.

[0049] This invention uses a half-vehicle model for dynamic analysis. This model considers the following degrees of freedom: the vehicle's buoyancy and pitching motion (…). , ), the heave and pitching motion of the front and rear bogies ( , ; , ), and the vertical displacement of the four wheelsets ( , =1-4). Therefore, the system has a total of 10 degrees of freedom, and its equations of motion are expressed in matrix form as follows:

[0050] ,

[0051] in, , , These are the vehicle's mass matrix, damping matrix, and stiffness matrix, respectively. , , These are the generalized acceleration, velocity, and displacement vectors, respectively. The wheel-rail force vector. This refers to the vertical displacement of the vehicle body. This refers to the nose-dive displacement of the vehicle body. This represents the vertical heave displacement of the front bogie. This refers to the nose-dive angular displacement of the front bogie. This refers to the vertical heave displacement of the rear bogie. This refers to the nose-dive angular displacement of the rear bogie. For the first Vertical displacement of each wheelset This is the index for the wheel pair.

[0052] In this embodiment, specifically, the process of obtaining the vibration differential equation of the rail in modal coordinates includes:

[0053] The rail is simplified using an Euler-Bernoulli beam model, and its corresponding vibration differential equation is as follows: , Where E is the elastic modulus of the track slab. Let be the moment of inertia of the rail section about the Y-axis. For the position of the rail ,time The vertical displacement, This indicates finding the partial derivative. This refers to the mass per unit length of the rail. Let u be the reaction force at the u-th fastener support point on the rail. Let N represent the force acting on the r-th wheelset, l be the rail length, and N represent the total number of fastener support points within the length l. , For the first The coordinates of each fastener support point (support point) For the spacing between fasteners, This is the Dirac function.

[0054] The vibration differential equation of the rail is transformed into the vibration differential equation of the rail in modal coordinates using the Ritz method.

[0055] The differential equation of the rail vibration in modal coordinates is: , in, For the rails at all times The Modal coordinates, Indicates the modal order of the orbit. for The second derivative, where E is the elastic modulus of the track slab. Let be the mass per unit length of the rail, and l be the length of the rail. Let N be the moment of inertia of the rail section about the Y-axis, and let N represent the total number of fastener supports within the length l. Let u be the reaction force at the u-th fastener support point on the rail. For the track in position The first First-order modal displacement, Let U be the coordinates of the u-th fastener support point on the rail. For the track in position The first First-order modal displacement, Let r be the position of the r-th wheelset on the track. For the r-th wheel pair at time... The force, Indicates the time.

[0056] Vertical wheel-rail contact force Determined using nonlinear Hertzian contact theory: , in, The track contact constant, This refers to the elastic compression between the wheel and the rail.

[0057] , in, For the first Each round in Vertical displacement at time t, Let r be the position of the r-th wheelset on the track. In order to be in The displacement of the rail below the r-th wheelset at time r. express The vertical track is constantly uneven.

[0058] This invention employs a fast explicit integration method (Zhai method) to perform time-domain integration on the coupled dynamics model of high-speed rail vehicle-track. The recursive formula is as follows: , in, For vertical displacement, subscript These represent the current time step. and the previous time step , Indicates to Differentiate, Indicates to Differentiate, , These are independent parameters that control the characteristics of the integration method.

[0059] This invention addresses the problem that traditional integer-order viscoelastic models lack fitting accuracy over a wide temperature range and struggle to quantify the temperature-dependent transformation of CA mortar from a solid-like to a fluid-like state. It introduces a fractional-order viscoelastic model to more accurately characterize the complex mechanical behavior of materials in actual temperature fields, significantly improving the consistency between the constitutive model and the actual response.

[0060] Breaking through the limitations of existing fractional-order models which are only used for isolated material parameter analysis or fasteners, this paper embeds high-precision fractional-order viscoelastic constitutive relations into the CA mortar layer of the track sub-model of the vehicle-track coupled dynamics system, realizing the mechanical correlation between materials and the system, and enabling accurate material-level characterization to directly serve reliable prediction of system dynamic response.

[0061] Through modeling and analysis, the differences in the impact of material property changes on various components in the system are clarified, and the different mechanisms of direct and indirect effects are distinguished, thereby providing more targeted theoretical references and optimization ideas for overall structural design, material selection, and long-term maintenance management.

[0062] In this embodiment, specifically, the method for performing dynamic numerical calculations on the high-speed rail vehicle-track coupled dynamic model includes: A fast explicit integration method is used to perform numerical dynamic calculations on the coupled dynamic model of high-speed train vehicle-track.

[0063] This invention describes the mechanical behavior of CA mortar layers by introducing a fractional derivative constitutive model and improves the coupled vehicle-track dynamics model, resulting in the following effects: 1. Significantly improves the prediction accuracy of the dynamic response of track structures, especially for the CA mortar layer and the track slab in direct contact. The FDZ model replaces the traditional Kelvin model to describe the viscoelasticity of the CA mortar. The resulting technical benefits: The FDZ model, with its fractional-order differential operator, can more accurately characterize the temperature sensitivity of the CA mortar. This makes the model of this invention closer to the actual mechanical response of the material when calculating track slab displacement, acceleration, and internal forces in the CA mortar layer, avoiding prediction bias caused by oversimplification in traditional models.

[0064] 2. This invention can more realistically reveal the internal stress state of the CA mortar layer under train loads, thereby more effectively assessing its damage and fatigue risks. Through a more precise mechanical description, the model of this invention reveals the greater internal forces borne by the CA mortar layer in actual operation, providing a more reliable and conservative early warning basis for judging the risk of damage, cracking, or failure under long-term cyclic loading.

[0065] 3. The influence of temperature on dynamic response was studied. The FDZ model was used to describe the change in CA mortar properties with temperature. The key parameters were all temperature-dependent and substituted into the vehicle-track coupled dynamic equations for solution. Technical advantages: Traditional models cannot accurately characterize temperature dependence, leading to deviations between the analysis results and actual conditions. This invention can more accurately simulate the dynamic response of the track system under different ambient temperatures, from low to high temperatures.

[0066] To verify the effectiveness of the high-speed train vehicle-track coupling dynamics calculation method of the present invention, the present invention uses the following system parameters: Table 1 shows the basic parameters of the high-speed train, and Table 2 shows the basic parameters of the track.

[0067] Table 1

[0068] Table 2

[0069] Based on the parameters in Table 1, the matrix motion equations of the 10-DOF half-car model are established. The rail adopts the Euler beam model, and the Ritz method is applied. The first NM order modes are taken to construct a set of ordinary differential equations for the rail slab modal coordinates containing fractional terms.

[0070] The track slab adopts a finite-length free beam model. Different constitutive models of the CA mortar layer use different equations, namely the ordinary differential equations in modal coordinates corresponding to the Kelvin viscoelastic model and the ordinary differential equations in modal coordinates of the track slab containing fractional terms in this invention.

[0071] Using nonlinear Hertzian contact theory, The constant is taken as the wear-type wheel, and the US Level 6 track spectrum is used to generate time-domain random vertical irregularity samples.

[0072] Two independent vehicle-track coupled models were established under identical vehicle parameters (Table 1), track parameters (Table 2), and operating conditions (speed 200 km / h, same track irregularity samples). A fast explicit integration method was used to integrate the coupled system in the time domain, calculating and outputting key dynamic response indicators, including: time histories of track slab mid-span displacement and acceleration, time histories of rail displacement and acceleration, and time histories of internal forces in the CA mortar layer.

[0073] like Figure 4 , Figure 5 As shown, Figure 4 This is a comparison diagram of track slab displacements for different high-speed rail vehicle-track coupled dynamic models. Figure 5 This is a comparison chart of rail acceleration and displacement for different high-speed rail vehicle-track coupled dynamic models.

[0074] Under the same excitation, the FDZ model predicted the largest peak displacement of the track slab, while the Kelvin model predicted a relatively smaller peak displacement. The track slab acceleration and displacement trends were consistent. This indicates that traditional models may significantly underestimate the vibration intensity.

[0075] like Figure 6 , Figure 7 As shown, Figure 6 This is a comparison chart of rail displacements for different high-speed rail vehicle-track coupled dynamic models. Figure 7 This is a comparison chart of rail acceleration and displacement for different high-speed rail vehicle-track coupled dynamic models.

[0076] Similar to the track slab, the FDZ model predicts a relatively large peak rail displacement, but the range of rail displacement variation predicted by both models is smaller compared to the track slab displacement. The rail acceleration curves are quite similar for both models, with the peak acceleration of the FDZ model only slightly higher than that of the Kelvin model.

[0077] like Figure 8 As shown, Figure 8 This is a comparison diagram of the CA mortar layer forces under different high-speed rail vehicle-track coupled dynamic models.

[0078] The FDZ model predicts a larger force in the CA mortar layer than the Kelvin model. Traditional integer-order models, due to constitutive defects, will seriously underestimate the actual stress state of the layer, which may lead to potential safety risks.

[0079] This embodiment also sets up different temperature conditions to verify the method of the present invention: Set a set of ambient temperature conditions: -20℃, 0℃, 20℃.

[0080] For the above working conditions, this paper uses the FDZ model to fit the creep compliance curves of CA mortar obtained in uniaxial compression tests at different temperatures. The fitted curves are shown below. Figure 9 As shown, Figure 9 The creep compliance curves of the high-speed rail vehicle-track coupled dynamic model of this invention, corresponding to CA mortar, were obtained in a uniaxial compression experiment under different temperature conditions.

[0081] according to Figure 9 The fitting results shown are used to extract parameter values ​​corresponding to different temperatures. Subsequently, these parameters are input into the model, as shown in Table 3, which contains the fitting parameters of the FDZ model.

[0082] Table 3

[0083] Under identical operating conditions, the vehicle-track coupled dynamics model described in this invention was run for the three different temperature conditions described above, and the results of the comparative analysis of the three conditions are as follows: like Figure 10 , Figure 11 As shown, Figure 10 These are comparison diagrams of track slab displacement in the high-speed rail vehicle-track coupled dynamics model of this invention under different temperature conditions. Figure 11 This is a comparison diagram of track slab acceleration in the high-speed rail vehicle-track coupled dynamics model of this invention under different temperature conditions.

[0084] As the ambient temperature rises, the displacement time history curve of the track slab shifts upwards overall, and its peak displacement increases significantly. Consistent with the displacement trend, the acceleration response of the track slab also increases with increasing temperature.

[0085] like Figure 12 , Figure 13 As shown, Figure 12 These are comparison diagrams of rail displacement in the high-speed rail vehicle-track coupled dynamics model of this invention under different temperature conditions. Figure 13 This is a comparison diagram of rail acceleration in the high-speed rail vehicle-track coupled dynamics model of this invention under different temperature conditions.

[0086] As temperature rises, the displacement time history curve and peak value of the rail show a slight increasing trend, but the change is much smaller than that of the track slab. The acceleration time history curve of the rail shows a slight increase in peak value with increasing temperature, but the change in peak value is extremely insignificant.

[0087] This second embodiment provides a high-speed rail vehicle-track coupled dynamics calculation system, including: The model building module is used to transform the constitutive equations of the fractional Zener model into the CA mortar layer model; The discrete module is used to represent the molecular operators in the CA mortar layer model as discrete series form to obtain the target CA mortar layer model. The foundation reaction force acquisition module is used to acquire the foundation reaction force of the CA mortar layer acting on the track slab based on the target CA mortar layer model. The vibration differential equation construction module is used to construct the vibration differential equation of the track slab based on the foundation reaction force of the CA mortar layer acting on the track slab, the bending stiffness of the track slab, the vertical displacement, the mass, the length of a single track slab, and the vertical force transmitted from the rail to the track slab through the fasteners. The transformation module is used to perform modal coordinate transformation on the vibration differential equation of the track slab using the Ritz method, transforming it into a system of ordinary differential equations in the modal coordinates of the track slab containing fractional terms. The computation module is used to integrate the vibration differential equations of the rail in modal coordinates and the set of ordinary differential equations of the track slab in modal coordinates containing fractional terms into the track subsystem, taking the half-car model as the vehicle subsystem; based on the wheel-rail nonlinear Hertzian contact theory, a high-speed rail vehicle-track coupled dynamic model is constructed, and dynamic numerical calculations are performed on the high-speed rail vehicle-track coupled dynamic model.

[0088] Those skilled in the art will understand that embodiments of this application can be provided as methods, systems, or computer program products. Therefore, this application can take the form of a completely hardware embodiment, a completely software embodiment, or an embodiment combining software and hardware aspects. Furthermore, this application can take the form of a computer program product embodied on one or more computer-usable storage media (including but not limited to disk storage, CD-ROM, optical storage, etc.) containing computer-usable program code.

[0089] This application is described with reference to flowchart illustrations and / or block diagrams of methods, apparatus (systems), and computer program products according to embodiments of this application. It will be understood that each block of the flowchart illustrations and / or block diagrams, and combinations of blocks in the flowchart illustrations and / or block diagrams, can be implemented by computer program instructions. These computer program instructions can be provided to a processor of a general-purpose computer, special-purpose computer, embedded processor, or other programmable data processing apparatus to produce a machine, such that the instructions, which execute via the processor of the computer or other programmable data processing apparatus, generate instructions for implementing the flowchart... Figure 1 One or more processes and / or boxes Figure 1 A device that provides the functions specified in one or more boxes.

[0090] These computer program instructions may also be stored in a computer-readable storage medium that can direct a computer or other programmable data processing device to function in a particular manner, such that the instructions stored in the computer-readable storage medium produce an article of manufacture including instruction means, which are implemented in a process Figure 1 One or more processes and / or boxes Figure 1 The function specified in one or more boxes.

[0091] These computer program instructions may also be loaded onto a computer or other programmable data processing equipment to cause a series of operational steps to be performed on the computer or other programmable equipment to produce a computer-implemented process, thereby providing instructions that execute on the computer or other programmable equipment for implementing the process. Figure 1 One or more processes and / or boxes Figure 1 The steps of the function specified in one or more boxes.

[0092] Obviously, the above embodiments are merely illustrative examples for clear explanation and are not intended to limit the implementation. Those skilled in the art will recognize that other variations or modifications can be made based on the above description. It is neither necessary nor possible to exhaustively list all possible implementations here. However, obvious variations or modifications derived therefrom are still within the scope of protection of this invention.

Claims

1. A method for calculating the coupled dynamics of high-speed train vehicles and tracks, characterized in that, include: The constitutive equations of the fractional Zener model are transformed into the CA mortar layer model; The molecular operator in the CA mortar layer model is expressed as a discrete series form to obtain the target CA mortar layer model. Based on the target CA mortar layer model, the ground reaction force of the CA mortar layer acting on the track slab is obtained; Based on the ground reaction force of the CA mortar layer acting on the track slab, the bending stiffness, vertical displacement, mass of the track slab, the length of a single track slab, and the vertical force transmitted from the rail to the track slab through the fasteners, the vibration differential equation of the track slab is constructed. The vibration differential equation of the track slab is transformed into a system of ordinary differential equations in the modal coordinates of the track slab containing fractional terms by performing modal coordinate transformation using the Ritz method. Using a half-car model as the vehicle subsystem, the vibration differential equations of the rail in modal coordinates and the ordinary differential equations of the track slab in modal coordinates containing fractional terms are integrated into the track subsystem. Based on the wheel-rail nonlinear Hertzian contact theory, a high-speed rail vehicle-track coupled dynamic model is constructed, and dynamic numerical calculations are performed on this high-speed rail vehicle-track coupled dynamic model.

2. The method for calculating the coupled dynamics of high-speed train vehicles and tracks according to claim 1, characterized in that, The method for transforming the constitutive equation of the fractional Zener model into the CA mortar layer model includes: The stress in the constitutive equation of the fractional Zener model is replaced by the ground reaction force of the CA mortar layer acting on the track slab, and the strain in the constitutive equation of the fractional Zener model is replaced by the vertical displacement of the track slab, thus obtaining the CA mortar layer model.

3. A method for calculating the coupled dynamics of a high-speed train vehicle and track according to claim 1 or 2, characterized in that, The CA mortar layer model is as follows: ， in, CA mortar layer applied to the track slab location ,time The foundation reaction force, , , , For elastic modulus, The viscosity coefficient is... Let be the order of the fractional derivative. Indicates time, Indicates the position of the track slab. This indicates finding the partial derivative. For the track slab in position ,time The vertical displacement.

4. The method for calculating the coupled dynamics of high-speed train vehicles and tracks according to claim 1, characterized in that, The target CA mortar layer model is as follows: ， in, CA mortar layer applied to track slab ,time The foundation reaction force, , , , , , , For elastic modulus, The viscosity coefficient is... Let be the order of the fractional derivative. Indicates time, Indicates the time step. Indicates the position of the track slab. This indicates finding the partial derivative. For the track slab in position ,time The vertical displacement, This is the Grunwald coefficient. , For gamma function, For historical integration steps, For historical step indexing, For the track slab in position ,time The vertical displacement, CA mortar layer applied to track slab ,time The ground reaction force.

5. The method for calculating the coupled dynamics of high-speed train vehicles and tracks according to claim 1, characterized in that, The vibration differential equation of the track slab is: ， in, Let be the bending stiffness of the track slab. For the track slab at time The First modal coordinates, For the track slab in position The first First-order modal displacement, Indicates the position of the track slab. The length of a single track slab, express Position The fourth derivative, For the quality of the track slab, for The second derivative, , , , , , , For elastic modulus, The viscosity coefficient is... Let be the order of the fractional derivative. For historical integration steps, For historical step indexing, Indicates time, Indicates the time step. Indicates the position of the track slab. CA mortar layer applied to track slab ,time The foundation reaction force, Here, u represents the number of fastener support points on a single track slab, and u is the index of the fastener support point on a single track slab. For the force at the u-th fastener support point on a single track plate, For the track slab in position The first First-order modal displacement, Let u be the position of the u-th fastener support point on a single track plate. This is the Grunwald coefficient. , For gamma function, For the track slab at time The Modal coordinates.

6. The method for calculating the coupled dynamics of high-speed train vehicles and tracks according to claim 1, characterized in that, The system of ordinary differential equations for the modal coordinates of the track slab containing fractional terms is as follows: ， in, For the quality of the track slab, For the track slab at time The First modal coordinates, for The second derivative, Let be the bending stiffness of the track slab. The length of a single track slab, , , , , , , For elastic modulus, The viscosity coefficient is... Let be the order of the fractional derivative. For historical integration steps, For historical step indexing, Indicates time, Indicates the time step. For free beam coefficients, This is the Grunwald coefficient. , For gamma function, CA mortar layer applied to track slab ,time The foundation reaction force, Indicates the position of the track slab. Here, u represents the number of fastener support points on a single track slab, and u is the index of the fastener support point on a single track slab. For the force at the u-th fastener support point on a single track plate, For the track slab in position The first First-order modal displacement, Let be the position of the u-th fastener support point on a single track plate.

7. The method for calculating the coupled dynamics of high-speed train vehicles and tracks according to claim 1, characterized in that, The process of obtaining the differential equation of rail vibration in modal coordinates includes: The vibration differential equation of the rail is transformed into the vibration differential equation of the rail in modal coordinates using the Ritz method.

8. A method for calculating the coupled dynamics of a high-speed train vehicle and track according to claim 1 or 7, characterized in that, The differential equation of the rail vibration in modal coordinates is: ， in, For the rails at all times The First modal coordinates, Indicates the modal order of the orbit. for The second derivative, where E is the elastic modulus of the track slab. Let be the mass per unit length of the rail, and l be the length of the rail. Let N be the moment of inertia of the rail section about the Y-axis, and let N represent the total number of fastener supports within the length l. Let u be the reaction force at the u-th fastener support point on the rail. For the track in position The first First-order modal displacement, Let U be the coordinates of the u-th fastener support point on the rail. For the track in position The first First-order modal displacement, Let r be the position of the r-th wheelset on the track. For the r-th wheel pair at time... The force, Indicates the time.

9. The method for calculating the coupled dynamics of high-speed train vehicles and tracks according to claim 1, characterized in that, Methods for performing numerical calculations of the dynamics of high-speed rail vehicle-track coupled dynamics models include: A fast explicit integration method is used to perform numerical dynamic calculations on the coupled dynamic model of high-speed train vehicle-track.

10. A high-speed rail vehicle-track coupled dynamics calculation system, characterized in that, include: The model building module is used to transform the constitutive equations of the fractional Zener model into the CA mortar layer model; The discrete module is used to represent the molecular operators in the CA mortar layer model as discrete series form to obtain the target CA mortar layer model. The foundation reaction force acquisition module is used to acquire the foundation reaction force of the CA mortar layer acting on the track slab based on the target CA mortar layer model. The vibration differential equation construction module is used to construct the vibration differential equation of the track slab based on the foundation reaction force of the CA mortar layer acting on the track slab, the bending stiffness of the track slab, the vertical displacement, the mass, the length of a single track slab, and the vertical force transmitted from the rail to the track slab through the fasteners. The transformation module is used to perform modal coordinate transformation on the vibration differential equation of the track slab using the Ritz method, transforming it into a system of ordinary differential equations in the modal coordinates of the track slab containing fractional terms. The computation module is used to integrate the vibration differential equations of the rail in modal coordinates and the set of ordinary differential equations of the track slab in modal coordinates containing fractional terms into the track subsystem, taking the half-car model as the vehicle subsystem; based on the wheel-rail nonlinear Hertzian contact theory, a high-speed rail vehicle-track coupled dynamic model is constructed, and dynamic numerical calculations are performed on the high-speed rail vehicle-track coupled dynamic model.

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