Simulation Method for Wheel-Rail Interaction of Curved Tracks Based on Coordinated Modeling of Dual Coordinate Systems

Abstract

The present invention discloses a method for simulating wheel-rail interaction of a curved track based on collaborative modeling of dual coordinate systems, including: establishing a train model according to physical parameters of the train; generating a track model according to control points of the curved track and physical parameters of the track; using a mobile coordinate system and a fixed coordinate system to establish kinematic equations of the train model and the track model, wherein the mobile coordinate system is used to describe vehicle motion, and the fixed coordinate system is used to describe track motion, and the kinematic equations under the dual coordinate systems are established through the kinematic mapping relationship between the mobile coordinate system and the fixed coordinate system; extracting the contour lines of the wheel and the track, and establishing a contact coordinate system under the mobile coordinate system based on the geometric contact of the contour lines, and solving the wheel-rail contact force in the kinematic equation and the train response under the curved track under the contact coordinate system; drawing the obtained train response and wheel-rail contact force into curves, and analyzing the calculation results. The present invention can realize efficient simulation of complex working conditions such as curved tracks.

Description

Technical Field

[0001] The present invention relates to the technical field of dynamic analysis, and in particular to a simulation method for wheel-rail interaction of a curved track based on co-modeling of dual coordinate systems. Background Technique

[0002] The dynamic analysis of wheel-rail interaction is a complex problem in the engineering field, and its strong non-linearity and high computational cost characteristics have long restricted the simulation accuracy and efficiency. The existing research methods are mainly divided into two categories: the substructure method and the integration method. The substructure method regards the vehicle and the track as independent subsystems and realizes coupling through external forces. Although solution technologies such as explicit integration and predictor-corrector algorithms have been developed, it is necessary to frequently handle the deformation coordination and force balance between subsystems, and it relies on multi-platform data exchange, resulting in a cumbersome calculation process; the integration method adopts a unified modeling framework, treats the wheel-rail contact force as an internal force, and realizes the coupling of geometric relations and mechanical responses through special contact units. Although the substructure coordination problem is avoided, the existing models are mostly limited to the uniform linear motion condition, and it is difficult to accurately simulate non-linear behaviors such as multi-point contact and wheel flange collision in complex scenarios such as curved tracks. Both methods have significant limitations in terms of adaptability to non-uniform motion. Traditional models often need to preset ideal track geometry, resulting in insufficient engineering practicability. In recent years, although research has tried to combine multi-body dynamics and finite element methods to construct a three-dimensional contact model and introduce Hertz contact theory and Polach creep formula to improve the calculation accuracy, it still faces challenges such as insufficient universality of kinematic description and low computational efficiency in complex working conditions.

[0003] The invention patent with the patent announcement number CN 117078812 B discloses a three-dimensional animation simulation method, storage medium and device for a rail transit train. The method includes: obtaining a vector map of the rail transit line according to the point-line-plane data of the rail transit line, and measuring the geographical locations of each subway station and the positions between stations; transmitting the vector map of the rail transit line into the Map class of the map engine for WebGIS-end developers, and given the initial coordinates and the map container, constructing a digital twin map of the rail transit, and constructing a running scenario of the rail transit train; performing a simulation linear animation of the rail transit train in the running scenario of the rail transit train according to the real-time speed of the rail transit train running on the track, the geographical locations of each subway station and the positions between stations; initializing the physical world in the running scenario of the rail transit train; performing simulation calculations on the initialized world and synchronously updating them into the simulation linear animation of the rail transit train.

[0004] The invention patent with the patent publication number CN 118192301 B discloses a simulation method for wheel-rail anti-skid and anti-spin adhesion control considering a drive system, which includes the following steps: First, establish a longitudinal dynamics model of a heavy-haul train considering the locomotive drive system through a first simulation platform; Second, establish a wheel-rail anti-skid and anti-spin adhesion control model through a second simulation platform; Then, conduct a co-simulation of the longitudinal dynamics model of the heavy-haul train in the first simulation platform and the wheel-rail anti-skid and anti-spin adhesion control model in the second simulation platform, perform data exchange at set time nodes, and complete the adhesion control process during the data exchange. The present invention can accurately consider the influence of the drive system on wheel-rail adhesion control and complete the adhesion control process through real-time co-simulation of dynamics and adhesion control.

[0005] Traditional wheel-rail interaction simulation methods need to frequently handle the deformation coordination and force balance between moving components (vehicles) and fixed components (tracks / bridges) in the dynamic analysis of the train-track system, resulting in a cumbersome calculation process and easy introduction of errors. At the same time, relying on multi-platform collaboration (such as commercial software SIMPACK, ABAQUS, etc.) and data exchange requirements increases the implementation cost and flexibility limitations. Therefore, there is an urgent need to develop a simulation method for wheel-rail interaction on curved tracks based on co-modeling in dual coordinate systems. Summary of the Invention

[0006] In view of this, the purpose of the present invention is to propose a simulation method for wheel-rail interaction on curved tracks based on co-modeling in dual coordinate systems. Describe the vehicle motion through a moving coordinate system and represent the track constraints through a fixed coordinate system. Separate the dynamic responses of moving components and fixed components through the dual coordinate systems, support dynamic contact modeling of curved tracks, construct a new type of wheel-rail contact unit to uniformly handle contact geometry and mechanical responses, and can be applicable to both uniform linear motion conditions and curved track motion conditions. And based on an open-source finite element platform (such as OpenSees), a full-process modeling is realized, reducing the dependence on commercial software, improving the portability and cost-effectiveness of the method, and achieving efficient simulation of complex conditions such as curved tracks.

[0007] In order to achieve the above technical objectives, the technical solutions adopted by the present invention are as follows:

[0008] A simulation method for wheel-rail interaction on curved tracks based on co-modeling in dual coordinate systems, including the following steps:

[0009] Step 1: Establish a train model according to the physical parameters of the train; Generate curve track control points through a curve track control point generation program, and then generate a track model according to the curve track control points and track physical parameters;

[0010] Step 2: Use a dual coordinate system based on a mobile coordinate system and a fixed coordinate system to establish the kinematic equations of the train model and the track model, where the mobile coordinate system is used to describe the vehicle motion, and the fixed coordinate system is used to describe the track motion, and the kinematic equations of the train model and the track model in the dual coordinate system are jointly established through the kinematic mapping relationship between the mobile coordinate system and the fixed coordinate system;

[0011] Step 3: According to the model information corresponding to the train model and the track model, the contour lines of the wheels and the tracks are extracted, and a contact coordinate system is established in the moving coordinate system based on the geometric contact of the contour lines. The wheel-rail contact force in the kinematic equation and the train response on the curved track are solved in the contact coordinate system.

[0012] Step 4: Plot the obtained train response and wheel-rail contact force into a curve, and analyze the calculation results.

[0013] Furthermore, the step 1 specifically includes:

[0014] Step 11: determine the train physical parameters and track physical parameters corresponding to the train and track to be calculated, wherein the train physical parameters include the train model, and the track physical parameters include the track model, curve track style, and curve function type;

[0015] Step 12: Establish a train model based on actual calculation examples and train physical parameters;

[0016] Step 13, write a curve track control point generation program in Matlab, and make a user input interface and an image generation interface;

[0017] Step 14: Input the geometric parameters corresponding to the track on the user input interface, generate a program drawing and generate the curve track control points according to the curve track control point generation program;

[0018] Step 15: Establish a track model based on the actual calculation example, the curved track control points and the track physical parameters, and display it on the image generation interface;

[0019] Step 16: Save the model information corresponding to the train model and track model into the target folder.

[0020] Furthermore, the curve track control point generation program includes the following modules:

[0021] Parametric input module: supports users to input track straight segment length, transition curve length, circular curve length, circular curve radius, track gauge and node spacing;

[0022] Control point generation module: Based on the transition curve and arc integration algorithm, the coordinates of the track centerline control points are automatically calculated, and the coordinates of the inner and outer track control points are generated through the normal offset algorithm;

[0023] Dynamic visualization module: Real-time rendering of the three-dimensional track model, supporting synchronous display and local magnification of the center line, inner track, and outer track.

[0024] Furthermore, the mathematical expressions of the transition curve and the circular curve are as follows:

[0025] The transition curve is a curve with a curvature that varies linearly with the arc length and is used to connect the straight line segment and the circular curve segment. Its curvature is defined as:

[0026] (1)

[0027] where denotes the curvature at the arc length , R is the target circle radius, L s is the total length of the transition curve;

[0028] The expression of the transition curve in the form of Fresnel integral is expressed as a parametric equation:

[0029] (2)

[0030] where and respectively represent the and x coordinates of the control points of the transition curve at the arc length y , is the tangent angle, is the integration independent variable; the tangent angle at the arc length satisfies:

[0031] (3)

[0032] where A is the transition curve parameter, representing the curvature change rate;

[0033] In actual programming, Taylor expansion or numerical integration is used to approximately simulate the transition curve:

[0034] (4)

[0035] The parametric equation of the circular curve is expressed as:

[0036] (5)

[0037] where and respectively represent the and x coordinates of the control points of the circular curve at the arc lengthand y coordinate;

[0038] According to track gauge and curvature direction, the inner rail control point coordinates and the outer rail control point coordinates are the normal offsets of the track centerline control point coordinates, specifically expressed as:

[0039] (6)

[0040] in, Indicates the coordinates of the inner track control points, represents the coordinates of the outer track control points, represents the coordinates of the track centerline control point, Represents the centerline unit normal vector and points to the center side of the circular curve.

[0041] Furthermore, the step 2 specifically includes:

[0042] Step 21, the vehicle model is regarded as a moving part in a moving coordinate system, and the track model is regarded as a stationary part in a fixed coordinate system;

[0043] Step 22: Construct a dual coordinate system collaborative system based on the mobile coordinate system and the fixed coordinate system, establish the kinematic equation in the fixed coordinate system, discretize the kinematic equation in the fixed coordinate system and combine it with the displacement, velocity, and acceleration mapping relationship in the mobile coordinate system to form a unified correction force term, and establish the kinematic equations of each part of the train model and the track model in the mobile coordinate system and the fixed coordinate system.

[0044] Furthermore, the step 22 specifically includes:

[0045] Step 221: According to the D'Alembert principle, the basic formula of the kinematic equations of the train model and the track model in a fixed coordinate system is:

[0046] (7)

[0047] Among them, M represents the mass matrix of the train model and the track model, C represents the damping matrix, represents the internal force matrix of the train model and the track model, represents the relative displacement of the contact point between the train wheel and the track, F represents the external load matrix, and They represent the second and first derivatives of the displacement u in the fixed coordinate system, i.e., acceleration and velocity;

[0048] Step 222: Use the Newmark-beta method to discretize equation (7):

[0049] (8)

[0050] Among them, n and n +1 respectively represent the current time step and the next time step in the discretized solution process, represents the discretized time step size, and respectively represent the parameters of the Newmark-beta method, with values , ;

[0051] Rewrite formula (7) as:

[0052] (9)

[0053] (10)

[0054] Among them, is the unbalanced force of this kinematic equation, and the solution obtained by setting it to zero is the displacement response of the system at the current moment; represents the external load term;

[0055] Step 223, there are the following mapping relationships between the displacement, velocity, and acceleration in the moving coordinate system and the displacement, velocity, and acceleration in the fixed coordinate system:

[0056] (11)

[0057] (12)

[0058] (13)

[0059] Among them, u, and respectively represent the displacement, velocity, and acceleration in the fixed coordinate system; , and respectively represent the displacement, velocity, and acceleration of the object in the moving coordinate system; , and respectively represent the displacement, velocity, and acceleration of the moving coordinate system itself, as reference parameters;

[0060] Step 224, Discretize formulas (11)-(13) and substitute them into (8) to obtain:

[0061] (14)

[0062] Then according to

[0063] (15)

[0064] in, i and i +1 indicates the current iteration step and the next iteration step in the solution process;

[0065] as well as

[0066] (16)

[0067] get:

[0068] (17)

[0069] Substituting formula (17) into formulas (9)-(10), we obtain:

[0070] (18)

[0071] in:

[0072] (19)

[0073] because This term is only related to external damping and has nothing to do with internal damping. This term is related to the external load term Forming a correction force term under a curved track , realizing the unification of the form of the kinematic equations in the mobile coordinate system and the fixed coordinate system, thereby achieving the simultaneous kinematic equations.

[0074] Furthermore, the step 3 specifically includes:

[0075] Step 31: according to the model information corresponding to the train model and the track model, use dense point linear fitting to fit the contours of the wheel and the track;

[0076] Step 32: Establish a contact coordinate system in the moving coordinate system based on the geometric contact of the contour line, wherein the contact coordinate system x Axis direction and moving coordinate system x The contact coordinate system is defined according to the embedding point of the wheel and the track. y Axis direction; through the contact coordinate system x Axis and y The cross product of the axes gives the contact coordinate system z Axis direction;

[0077] Step 33: Use the Newton-Raphson implicit iteration method to calculate the n The first step under the time i The displacement of the iteration step;

[0078] Step 34: Determine whether convergence is achieved. If not, proceed to Step 35; if so, continue to check whether all time steps are completed. If yes, record the response at each time step; if not, after moving to the next time step, return to Step 33.

[0079] Step 35: Calculate the velocity at the n time step and the i iteration step using the Newmark-beta method.

[0080] Step 36: Calculate the wheel-rail normal contact force using the non-linear Hertz formula and calculate the wheel-rail tangential contact force using the Polach formula.

[0081] Step 37: Convert the wheel-rail normal contact force and the wheel-rail tangential contact force to obtain the internal force of the wheel-rail coupling element.

[0082] Step 38: Update the kinematic equation according to the internal force of the wheel-rail coupling element.

[0083] Step 39: Move to the next iteration step and return to Step 33.

[0084] Furthermore, in Step 36, calculating the wheel-rail normal contact force using the non-linear Hertz formula specifically includes:

[0085] Based on the maximum embedding depth of the profiles of the wheel and the track , and calculate the wheel-rail normal contact force using the non-linear Hertz formula.

[0086] The wheel-rail normal contact force is obtained by the following formula:

[0087] (20)

[0088] where represents the wheel-rail normal contact force, K represents the wheel-rail normal contact force coefficient, which is related to the curvature and elastic modulus of the wheel and rail, and 1.5 represents the power value.

[0089] Furthermore, in Step 36, calculating the wheel-rail tangential contact force using the Polach formula specifically includes:

[0090] Define the wheel-rail creep ratio as a physical quantity related to the relative velocity between the wheel and the track. The wheel-rail creep ratio is calculated as follows:

[0091] (21)

[0092] where represents x the translational creep ratio in the direction,y The translational creepage rate in the direction, z represents the rotational creepage rate of rotation about the axis, and x respectively represent the relative velocity difference between the wheel contact point and the rail contact point in the y direction, represents the relative velocity difference between the wheel contact point and the rail contact point in the rotational direction about the z axis, represents the traveling speed;

[0093] The Polach formula describes the relationship between the wheel-rail tangential contact force and the creepage rate, and the Polach formula is expressed as:

[0094] (22)

[0095] Wherein, represents the corrected translational creepage rate related to and ; F tr and F ys respectively represent the corrected tangential force and the tangential force generated by pure spin, and respectively represent x the wheel-rail tangential contact force in the y direction and the wheel-rail tangential contact force in the

[0096] Furthermore, in step 3, solving for the wheel-rail contact force and the train response under a curved track in the contact coordinate system specifically includes:

[0097] Writing a program for calculating the wheel-rail contact force in the open-source finite element platform OpenSees, including the non-linear Hertz formula for calculating the wheel-rail normal contact force and the Polach formula for calculating the wheel-rail tangential contact force. The open-source finite element platform OpenSees also comes with the Newton-Raphson implicit iteration method and the Newmark-beta method;

[0098] Calculating and solving for the wheel-rail contact force and the train response under a curved track in the kinematic equation in the open-source finite element platform OpenSees through the Newton-Raphson implicit iteration method, the Newmark-beta method, the non-linear Hertz formula, and the Polach formula.

[0099] Adopting the above technical solution, compared with the prior art, the beneficial effects of the present invention are:

[0100] 1. Dual - coordinate system decoupling modeling: Using a moving - fixed dual - coordinate system collaborative architecture to separate vehicle dynamic responses from track constraints, avoiding the complexity of frequent coordination between moving and fixed components in traditional methods, improving computational efficiency, and can be extended to dynamic contact modeling of curved tracks.

[0101] 2. Full - process open - source integration: Based on open - source platforms such as OpenSees, unified solution of wheel - rail contact elements and mechanical responses is achieved, eliminating the dependence on commercial software such as SIMPACK and ABAQUS, avoiding cross - platform interactions, enhancing the portability of the method, and improving computational efficiency.

[0102] 3. Universal simulation of complex working conditions: Construct a new type of wheel - rail contact element (contact coordinate system) that is compatible with scenarios such as uniform straight - line, curved tracks, and wheel - flange collisions, accurately depicting the non - linear behavior of multi - point contact, and having strong engineering applicability. Brief Description of the Drawings

[0103] In order to more clearly illustrate the technical solutions in the embodiments of the present invention or the prior art, the following will briefly introduce the drawings required for use in the description of the embodiments or the prior art. Obviously, the following - described drawings are only some embodiments of the present invention. For those of ordinary skill in the art, without creative efforts, other drawings can be obtained based on these drawings.

[0104] Figure 1 It is the execution flowchart of a wheel - rail interaction simulation method for curved tracks based on dual - coordinate system collaborative modeling provided by an embodiment of the present invention.

[0105] Figure 2 It is the flowchart of the pre - processing link provided by an embodiment of the present invention.

[0106] Figure 3 It is the flowchart of the finite - element analysis link provided by an embodiment of the present invention.

[0107] Figure 4 It is the schematic diagram of a wheel set and a curved track provided by an embodiment of the present invention.

[0108] Figure 5 It is the GUI interface diagram of the program for generating control points of a curved track provided by an embodiment of the present invention.

[0109] Figure 6 It is the wheel - rail cross - section and wheel - rail contour line diagram provided by an embodiment of the present invention.

[0110] Figure 7 It is the diagram of the lateral force and lateral displacement of a wheel set at a speed of 10 m / s provided by an embodiment of the present invention.

[0111] Figure 8It is the graph of the wheel set lateral force and lateral displacement at a speed of 15 m / s provided by the embodiment of the present invention. Detailed implementation manners

[0112] The present invention will be further described in detail below in conjunction with the accompanying drawings and embodiments. It should be specifically noted that the following embodiments are only used to illustrate the present invention, but do not limit the scope of the present invention. Similarly, the following embodiments are only partial embodiments of the present invention rather than all embodiments. All other embodiments obtained by those of ordinary skill in the art without creative efforts belong to the scope of protection of the present invention.

[0113] Please refer to Figures 1-8 , a method for simulating wheel-rail interaction of a curved track based on co-modeling of double coordinate systems according to the present invention, includes the following steps:

[0114] Step 1: Establish a train model according to the physical parameters of the train; generate curve track control points according to the curve track control point generation program, and then generate an orbit model according to the curve track control points and the orbit physical parameters;

[0115] In this embodiment, the specific steps of Step 1 include:

[0116] Step 11: Determine the train physical parameters and orbit physical parameters corresponding to the train and orbit to be calculated. The train physical parameters include the train type, and the orbit physical parameters include the orbit type, curve track style, and curve function type;

[0117] Step 12: Establish a train model according to the actual example and the train physical parameters;

[0118] Step 13: Write a curve track control point generation program in Matlab (Matrix Laboratory, a high-performance numerical calculation and visualization software widely used in engineering, science, mathematics, and finance, etc.), and make a user input interface and an image generation interface;

[0119] Step 14: Input the geometric parameters corresponding to the orbit on the user input interface, draw a graph according to the curve track control point generation program, and generate curve track control points;

[0120] Step 15: Establish an orbit model according to the actual example, curve track control points, and orbit physical parameters, and display it on the image generation interface;

[0121] Step 16: Store the model information corresponding to the train model and the orbit model in the target folder.

[0122] Preferably, the curve track control point generation program includes the following modules:

[0123] Parametric Input Module: Supports users to input the lengths of straight track segments, transition curve lengths, circular curve lengths, circular curve radii, gauge, and node spacing;

[0124] Control Point Generation Module: Based on the integral algorithm of transition curve and circular arc, automatically calculates the coordinates of the control points on the track centerline, and generates the coordinates of the inner rail control points and the outer rail control points of the track through the normal offset algorithm;

[0125] Dynamic Visualization Module: Real-time renders the 3D model of the track, and supports the synchronous display and local magnification of the centerline, inner rail, and outer rail.

[0126] In the preprocessing stage, a program for generating control points of the curved track is developed based on Matlab to achieve accurate parametric modeling of the transition curve (clothoid) in the curved track, ensure the continuous smoothness of the curvature gradient between the straight segment, transition curve segment, and circular curve segment, eliminate geometric mutations in traditional manual modeling, and provide high-fidelity track geometry input for wheel-rail dynamic contact simulation.

[0127] By integrating the functions of track geometric parameter input, control point algorithm generation, and 3D visualization, the automatic output of the coordinates of the high-precision track centerline, inner rail, and outer rail is realized. This part can be divided into three modules: parametric input module, control point generation module, and dynamic visualization module. Among them, the parametric input module: supports users to input the core parameters such as the lengths of straight track segments (L1, L2), transition curve lengths (Ls1, Ls2), circular curve length (Lc), circular curve radius (R), gauge (D), and node spacing (ds), and has a built-in data verification and exception handling mechanism to check the input parameter range to ensure the accuracy of the input parameters. The control point generation module: based on the integral algorithm of the transition curve (clothoid) and circular arc, automatically calculates the coordinates of the track centerline, and generates the inner and outer rail control points through the normal offset algorithm to ensure the smooth transition between the transition curve and the circular curve. The dynamic visualization module: uses the Matlab graphics engine to real-time render the 3D model of the track, supports the synchronous display and local magnification of the centerline, inner rail, and outer rail, and provides an interface for exporting node coordinates.

[0128] Preferably, the mathematical expressions of the transition curve and the circular curve (the core algorithms for control point generation) are as follows:

[0129] The transition curve is a curve whose curvature changes linearly with the arc length, and is used to connect the straight segment and the circular curve segment to ensure smooth curvature transition. Its curvature is defined as:

[0130] (1)

[0131] Where refers to the arc length at which the curvature is located, R is the target circle radius,L s is the total length of the transition curve;

[0132] The expression of the transition curve is expressed in the form of Fresnel integral as a parametric equation:

[0133] (2)

[0134] Where, and respectively represent the and x coordinates of the control points of the transition curve at the arc length y , is the tangent angle, is the integral independent variable; the tangent angle at the arc length satisfies:

[0135] (3)

[0136] Where, A is the transition curve parameter, characterizing the curvature change rate;

[0137] In actual programming, Taylor expansion or numerical integration (such as Simpson's method) is used to approximately simulate the transition curve:

[0138] (4)

[0139] The parametric equation of the circular curve is expressed as:

[0140] (5)

[0141] Where, and respectively represent the and x coordinates of the control points of the circular curve at the arc length y ;

[0142] According to the gauge and the curvature direction, the coordinates of the inner rail control point and the outer rail control point are the normal offsets of the coordinates of the track center line control point, and are specifically expressed as:

[0143] (6)

[0144] Where, represents the coordinates of the inner rail control point, represents the coordinates of the outer rail control point, represents the coordinates of the track center line control point, represents the unit normal vector of the center line and points to the center side of the circular curve.

[0145] According to the above formula, a curve track control point growth program is written in Matlab, and the following is generated: Figure 5 The GUI interface (Graphical User Interface) shown in the figure is divided into left and right columns. The left side is the parameter input area and the right side is the track visualization area, which supports real-time interactive operations. The left panel contains the track geometry parameter input box, the generate button and the parameter description label. The right panel dynamically displays the track centerline, the curves of the inner and outer rails, and supports zooming, panning and partial zooming.

[0146] Step 2: Use a dual coordinate system based on a mobile coordinate system and a fixed coordinate system to establish the kinematic equations of the train model and the track model, where the mobile coordinate system is used to describe the vehicle motion, and the fixed coordinate system is used to describe the track motion, and the kinematic equations of the train model and the track model in the dual coordinate system are jointly established through the kinematic mapping relationship between the mobile coordinate system and the fixed coordinate system;

[0147] During the modeling stage, a collaborative modeling system of mobile coordinate system and fixed coordinate system is constructed. The mobile coordinate system describes the vehicle motion, and the fixed coordinate system represents the track geometric constraints. The kinematic mapping relationship is used to realize the simultaneous solution of the vehicle-track subsystem motion equations in the two coordinate systems, avoiding deformation coordination and multi-platform data exchange in the substructure method.

[0148] In this embodiment, step 2 specifically includes:

[0149] Step 21, the vehicle model is regarded as a moving part in a moving coordinate system, and the track model is regarded as a stationary part in a fixed coordinate system;

[0150] Step 22: Based on the mobile coordinate system and the fixed coordinate system, a dual coordinate system coordination system is constructed, and the kinematic equations in the fixed coordinate system are established. The kinematic equations in the fixed coordinate system are discretized and combined with the displacement, velocity, and acceleration mapping relationship in the mobile coordinate system to form a unified correction force term, and the kinematic equations of the train model and the track model in the mobile coordinate system and the fixed coordinate system are established. By establishing the kinematic mapping relationship between the mobile coordinate system and the fixed coordinate system, the kinematic equations are derived and combined, and the applicability of this method is extended from a simplified uniform linear motion scenario to a curved track simulation.

[0151] Preferably, the step 22 specifically includes:

[0152] Step 221: According to the D'Alembert principle, the basic formula of the kinematic equations of the train model and the track model in a fixed coordinate system is:

[0153] (7)

[0154] Among them, \(M\) represents the mass matrix of the train model and the track model, \(C\) represents the damping matrix, represents the internal force matrix of the train model and the track model, represents the relative displacement of the contact point between the train wheel and the track, \(F\) represents the external load matrix, and respectively represent the second and first derivatives of the displacement \(u\) in the fixed coordinate system, that is, acceleration and velocity;

[0155] Step 222: Since the wheel-rail interaction involves strong nonlinearity, the Newmark-beta method (the Newmark-β method is a numerical integration method for solving structural dynamics equations) is usually used to discretize equation (7):

[0156] (8)

[0157] Among them, n and n \(+1\) respectively represent the current time step and the next time step in the discrete solution process, represents the discrete time step, and respectively represent the parameters of the Newmark-beta method, with values , ;

[0158] Rewrite formula (7) as:

[0159] (9)

[0160] (10)

[0161] Among them, is the unbalanced force of this kinematic equation, and the solution obtained by setting it to zero is the displacement response of the system at the current moment; represents the external load term; has no specific meaning and is used for substitution in formula (9) to prevent formula (10) from being too long.

[0162] Step 223: The displacements, velocities, and accelerations in the moving coordinate system have the following mapping relationships with the displacements, velocities, and accelerations in the fixed coordinate system:

[0163] (11)

[0164] (12)

[0165] (13)

[0166] Among them, \(u\), and respectively represent the displacement, velocity, and acceleration in the fixed coordinate system; 、 and respectively represent the displacement, velocity, and acceleration of the object in the moving coordinate system; 、 and respectively represent the displacement, velocity, and acceleration of the moving coordinate system itself, serving as reference parameters;

[0167] Step 224: Discretize equations (11)-(13) and substitute them into (8) to obtain:

[0168] (14)

[0169] Then, according to

[0170] (15)

[0171] wherein, i and i +1 represent the current iteration step and the next iteration step in the solution process;

[0172] and

[0173] (16)

[0174] to obtain:

[0175] (17)

[0176] Substitute equation (17) into equations (9)-(10) to obtain:

[0177] (18)

[0178] wherein:

[0179] (19)

[0180] Since this term is only related to the external damping and has nothing to do with the internal damping, while this term forms a correction force term under the curve track with the external load term to achieve the unification of the kinematic equation forms in the moving coordinate system and the fixed coordinate system, so as to simultaneously solve the kinematic equations to obtain the system response, realizing the unified modeling of the train and the track and avoiding the repeated iteration problem in the substructure method.

[0181] ​Step 3: According to the model information corresponding to the train model and the track model, extract the contour lines of the wheel and the track, and establish a contact coordinate system in the moving coordinate system based on the geometric contact situation of the contour lines. Solve the wheel-rail contact force and the train response under the curved track in the contact coordinate system for the kinematic equation;

[0182] In the solution stage, a three-dimensional wheel-rail contact element under the finite element framework is designed, that is, a contact coordinate system is established; and the Newton-Raphson implicit iteration method (Newton-Raphson method), the Newmark-beta method (the Newmark-β method is a numerical integration method for solving structural dynamics equations), the non-linear Hertz formula and the Polach (a theoretical model for calculating the longitudinal creep force of wheel-rail, widely used in the dynamic analysis of railway vehicles) formula are integrated in the open-source finite element platform OpenSees (Open System for Earthquake Engineering Simulation, an open-source software framework for earthquake engineering simulation, mainly used for the non-linear analysis of structures and geotechnical engineering). The contact coordinate system is directly embedded into the solution process of the dual coordinate system to support the simulation of complex working conditions such as multi-point contact and wheel flange collision under the curved track.

[0183] The present invention uses the Newton-Raphson implicit iteration method to synchronously iterate the combined motion equations of the moving / fixed system response discretized by the newmark-beta method, and combines the smooth track data generated by the preprocessing to significantly improve the convergence speed of the non-linear contact problem.

[0184] In this embodiment, the specific steps of step 3 include:

[0185] Step 31: The open-source finite element platform OpenSees retrieves the model information from the target folder. According to the model information corresponding to the train model and the track model, use the dense point linear fitting to fit the contour lines of the wheel and the track, and obtain the Figure 6 schematic diagram as shown;

[0186] Step 32: Establish a contact coordinate system in the moving coordinate system based on the geometric contact situation of the contour lines. The contact coordinate system is used to calculate the wheel-rail contact force; wherein, the x axis direction of the contact coordinate system is consistent with the x axis direction of the moving coordinate system; according to the embedding points of the wheel and the track ( P i , P i+1 , assuming no deformation), define the y axis direction of the contact coordinate system; through the x axis and yThe cross product of the axes (the cross product is a concept in vector algebra that describes a vector perpendicular to the plane containing two vectors in three-dimensional space) is used to obtain the z axis direction of the contact coordinate system;

[0187] Step 33: Use the Newton-Raphson implicit iteration method to calculate the displacement at the n time step for the i iteration step; n and i are positive integers, and n and i The maximum values of are given by the user. When the time step n reaches the maximum value, the program calculation is completed; when the iteration step i is less than the given maximum value and the program converges, it enters the next iteration step. If the iteration step i reaches the given maximum value and has not converged, the program convergence fails, the calculation ends, and the user needs to perform a convergence check and modify the relevant parameters.

[0188] Step 34: Determine whether it has converged. If not, enter Step 35; if so, continue to check whether all time steps are completed. If so, record the response at each time step; if not, after entering the next time step, return to Step 33;

[0189] Step 35: Use the Newmark-beta method to calculate the velocity at the n time step for the i iteration step;

[0190] Step 36: Use the non-linear Hertz formula to calculate the wheel-rail normal contact force, and use the Polach formula to calculate the wheel-rail tangential contact force;

[0191] Step 37: Convert the wheel-rail normal contact force and the wheel-rail tangential contact force to obtain the internal force of the wheel-rail coupling element;

[0192] Step 38: Update the kinematic equation according to the internal force of the wheel-rail coupling element;

[0193] Step 39: Enter the next iteration step and return to Step 33.

[0194] Preferably, in Step 36, using the non-linear Hertz formula to calculate the wheel-rail normal contact force specifically includes:

[0195] The wheel-rail normal contact force is related to the maximum embedding depth of the contour lines of the wheel and the track. The calculation of the wheel-rail contact force will be briefly introduced below. The present invention uses the non-linear Hertz method to calculate the wheel-rail normal contact force. This method assumes that the wheel and the track are rigid bodies that do not deform but have virtual mutual penetration. The contour lines of the wheel and the track are linearly fitted with dense points to find the maximum embedding depth between the two. Based on the maximum embedding depth of the contour lines of the wheel and the track , and the non-linear Hertz formula is used to calculate the wheel-rail normal contact force;

[0196] The wheel-rail normal contact force is obtained by the following formula:

[0197] (20)

[0198] where, represents the wheel-rail normal contact force, K represents the wheel-rail normal contact force coefficient, which is related to the curvature and elastic modulus of the wheel and the rail, and 1.5 represents the power value.

[0199] Furthermore, in step 36, the Polach formula is used to calculate the wheel-rail tangential contact force, specifically including:

[0200] The Polach formula can more accurately describe the relationship between the wheel-rail tangential contact force (longitudinal and lateral) and the wheel-rail creepage rate by introducing the friction saturation effect and a simplified assumption of the contact patch stress distribution. This method believes that the wheel-rail tangential contact force is closely related to the wheel-rail creepage rate, and the wheel-rail creepage rate is defined as a physical quantity related to the relative speed between the wheel and the track. The wheel-rail creepage rate is calculated as follows:

[0201] (21)

[0202] where, represents x the translational creepage rate in the direction, y represents the translational creepage rate in the z direction, and respectively represent the relative velocity differences at the wheel contact point and the track contact point in the x and y directions, represents the relative velocity difference at the wheel contact point and the track contact point in the rotational direction around the z axis, represents the traveling speed;

[0203] The Polach formula describes the relationship between the wheel-rail tangential contact force and the creepage rate, and the Polach formula is expressed as:

[0204] (22)

[0205] Among them, represents the corrected translational creepage rate related to and The relevant corrected translational creepage rate, F tr and F ys represent the corrected tangential force and the tangential force generated by pure spin respectively, and represent respectively x The wheel-rail tangential contact force in the direction of y The wheel-rail tangential contact force in the direction of

[0206] Preferably, in step 3, solving the wheel-rail contact force and the train response in the kinematic equation in the contact coordinate system specifically includes:

[0207] Writing a program for calculating the wheel-rail contact force in the open-source finite element platform OpenSees, including the non-linear Hertz formula for calculating the wheel-rail normal contact force and the Polach formula for calculating the wheel-rail tangential contact force. The open-source finite element platform OpenSees also comes with the Newton-Raphson implicit iteration method and the Newmark-beta method;

[0208] Calculating and solving the wheel-rail contact force and the train response in the kinematic equation in the open-source finite element platform OpenSees through the Newton-Raphson implicit iteration method, the Newmark-beta method, the non-linear Hertz formula and the Polach formula.

[0209] Step 4: Plot the obtained train response and wheel-rail contact force into curves and analyze the calculation results.

[0210] After calculation, the present invention obtains the response of a single wheel set passing through a curve track example. The curve track is composed of a straight section, a 50-meter-long transition curve section and a 50-meter-long circular curve section respectively, and the curve radius is 200 meters. The corresponding curve track control points are generated by the Matlab curve track control point generation program, and the corresponding tcl file is generated and placed in the corresponding folder for OpenSees to call. The wheels adopt the ML95 model, and the tracks adopt the UIC50 model.

[0211] Such as Figure 7The figure shows the lateral force and lateral displacement diagram obtained when the wheel set advances at a speed of 10 m / s. It can be seen that under the action of the centrifugal force, the wheel set has a slight deflection, but generally still moves in a snake-like motion along the center line.

[0212] As Figure 8 The figure shows the lateral force and lateral displacement diagram obtained when the wheel set advances at a speed of 15 m / s. It can be seen that under the action of the centrifugal force, the wheel set has a serious deflection, which leads to continuous contact between the wheel flange and the track. At the same time, the continuous contact between the wheel flange and the track also causes a sudden change in the lateral force.

[0213] The above are only some embodiments of the present invention, and thus do not limit the protection scope of the present invention. Any equivalent device or equivalent process transformation made by using the content of the specification and drawings of the present invention, or directly or indirectly applied in other related technical fields, shall be equally included in the patent protection scope of the present invention.

Claims

1. A simulation method for wheel-rail interaction of a curved track based on co-modeling of dual coordinate systems, characterized in that The steps include: Step 1: Establish a train model according to the physical parameters of the train; generate curved track control points according to the curved track control point generation program, and then generate a track model according to the curved track control points and the track physical parameters; the curved track control point generation program includes the following modules: Parametric input module: supports users to input track straight segment length, transition curve length, circular curve length, circular curve radius, track gauge and node spacing; Control point generation module: Based on the transition curve and arc integration algorithm, the coordinates of the track centerline control points are automatically calculated, and the coordinates of the inner and outer track control points are generated through the normal offset algorithm; Dynamic visualization module: real-time rendering of the track 3D model, supporting simultaneous display and partial zoom of the center line, inner rail and outer rail; The mathematical expressions of the transition curve and the circular curve are as follows: A transition curve is a curve whose curvature changes linearly with the arc length. It is used to connect straight line segments and circular curve segments. Its curvature is defined as: (1) Among them, refers to the curvature at the arc length , R is the target circle radius, L s is the total length of the transition curve; The expression of the transition curve is expressed in the form of a parametric equation using the Fresnel integral form: (2) Among them, and respectively represent the at the arc length of the easement curve control point x and y coordinates, is the tangent angle, is the integral independent variable; the tangent angle at the arc length satisfies: (3) Among them, A is the easement curve parameter, representing the curvature change rate; In actual programming, Taylor expansion or numerical integration is used to approximate the relaxation curve: (4) The parametric equation of the circular curve is expressed as: (5) Among them, and respectively represent the and x coordinates of the control points of the circular curve at the arc length y ; According to the gauge and the curvature direction, the inner rail control point coordinates and the outer rail control point coordinates are the normal offsets of the track centerline control point coordinates, and are specifically expressed as: (6) Among them, represents the coordinates of the inner rail control points, represents the coordinates of the outer rail control points, represents the coordinates of the control points on the track center line, represents the unit normal vector of the center line and points to the center side of the circular curve; Step 2: Use a dual coordinate system based on a mobile coordinate system and a fixed coordinate system to establish the kinematic equations of the train model and the track model, where the mobile coordinate system is used to describe the vehicle motion, and the fixed coordinate system is used to describe the track motion, and the kinematic equations of the train model and the track model in the dual coordinate system are jointly established through the kinematic mapping relationship between the mobile coordinate system and the fixed coordinate system; Step 3: According to the model information corresponding to the train model and the track model, the contour lines of the wheels and the tracks are extracted, and a contact coordinate system is established in the moving coordinate system based on the geometric contact of the contour lines. The wheel-rail contact force in the kinematic equation and the train response on the curved track are solved in the contact coordinate system. Step 4: Plot the obtained train response and wheel-rail contact force into a curve, and analyze the calculation results.

2. The simulation method for wheel-rail interaction of a curved track based on dual coordinate system collaborative modeling according to claim 1, characterized in that The step 1 specifically includes: Step 11: determine the train physical parameters and track physical parameters corresponding to the train and track to be calculated, wherein the train physical parameters include the train model, and the track physical parameters include the track model, curve track style, and curve function type; Step 12: Establish a train model based on actual calculation examples and train physical parameters; Step 13, write a curve track control point generation program in Matlab, and make a user input interface and an image generation interface; Step 14: Input the geometric parameters corresponding to the track on the user input interface, generate a program drawing and generate the curve track control points according to the curve track control point generation program; Step 15: Establish a track model based on the actual calculation example, the curved track control points and the track physical parameters, and display it on the image generation interface; Step 16: Save the model information corresponding to the train model and track model into the target folder.

3. The simulation method for wheel-rail interaction of a curved track based on dual coordinate system collaborative modeling according to claim 1, characterized in that, The step 2 specifically includes: Step 21, the vehicle model is regarded as a moving part in a moving coordinate system, and the track model is regarded as a stationary part in a fixed coordinate system; Step 22: Construct a dual coordinate system collaborative system based on the mobile coordinate system and the fixed coordinate system, establish the kinematic equation in the fixed coordinate system, discretize the kinematic equation in the fixed coordinate system and combine it with the displacement, velocity, and acceleration mapping relationship in the mobile coordinate system to form a unified correction force term, and establish the kinematic equations of each part of the train model and the track model in the mobile coordinate system and the fixed coordinate system.

4. The simulation method for wheel-rail interaction of a curved track based on dual coordinate system collaborative modeling according to claim 3, characterized in that, The step 22 specifically includes: Step 221: According to the D'Alembert principle, the basic formula of the kinematic equations of the train model and the track model in a fixed coordinate system is: (7) Among them, \(M\) represents the mass matrix of the train model and the track model, \(C\) represents the damping matrix, represents the internal force matrix of the train model and the track model, represents the relative displacement of the contact point between the train wheel and the track, \(F\) represents the external load matrix, and respectively represent the second and first derivatives of the displacement \(u\) in the fixed coordinate system, that is, acceleration and velocity; Step 222: Use the Newmark-beta method to discretize equation (7): (8) Among them, n and n +1 represent the current time step and the next time step in the discrete solution process respectively, represents the discrete time step size, and represent the parameters of the Newmark-beta method respectively, with values of , ; Rewrite formula (7) as: (9) (10) wherein, is the unbalanced force of the kinematic equation, and the solution obtained by setting it to zero is the displacement response of the system at the current moment; represents the external load term; Step 223, the displacement, velocity and acceleration in the mobile coordinate system have the following mapping relationships with the displacement, velocity and acceleration in the fixed coordinate system: (11) (12) (13) Among them, u, and represent displacement, velocity, and acceleration in the fixed coordinate system respectively; , and represent displacement, velocity, and acceleration of the object in the moving coordinate system respectively; , and represent displacement, velocity, and acceleration of the moving coordinate system itself respectively, as reference parameters; Step 224: Discretize formulas (11)-(13) and substitute them into (8) to obtain: (14) According to (15) Among them, i and i +1 represents the current iteration step and the next iteration step in the solution process; as well as (16) get: (17) Substituting formula (17) into formulas (9)-(10), we obtain: (18) in: (19) because This term is only related to external damping and has nothing to do with internal damping. This term is related to the external load term Forming a correction force term under a curved track , realizing the unification of the form of the kinematic equations in the mobile coordinate system and the fixed coordinate system, thereby achieving the simultaneous kinematic equations.

5. The simulation method for wheel-rail interaction of a curved track based on dual coordinate system collaborative modeling according to claim 1, characterized in that, The step 3 specifically includes: Step 31: according to the model information corresponding to the train model and the track model, use dense point linear fitting to fit the contours of the wheel and the track; Step 32: Establish a contact coordinate system in the moving coordinate system based on the geometric contact situation of the contour line. Among them, the x axis direction of the contact coordinate system is consistent with the x axis direction of the moving coordinate system; define the y axis direction of the contact coordinate system according to the embedding point of the wheel and the track; obtain the x axis and y axis cross product of the contact coordinate system to obtain the z axis direction of the contact coordinate system; Step 33. Use the Newton-Raphson implicit iteration method to calculate the displacement at the n time step for the i iteration step. Step 34, determine whether it has converged, if not, proceed to step 35; if yes, continue to check whether all time steps are completed, if yes, record the response of each time step; if not, proceed to the next time step and return to step 33; Step 35. Calculate the velocity at the n time step for the i iteration step using the Newmark-beta method; Step 36: Use the nonlinear Hertz formula to calculate the wheel-rail normal contact force, and use the Polach formula to calculate the wheel-rail tangential contact force; Step 37, converting the wheel-rail normal contact force and the wheel-rail tangential contact force to obtain the wheel-rail coupling unit internal force; Step 38: Update the kinematic equation according to the internal force of the wheel-rail coupling unit; Step 39, enter the next iteration step and return to step 33.

6. The simulation method for wheel-rail interaction of a curved track based on dual coordinate system collaborative modeling according to claim 5, wherein In step 36, the nonlinear Hertz formula is used to calculate the wheel-rail normal contact force, which specifically includes: Based on the maximum embedding depth of the wheel and rail contour lines , and the non-linear Hertz formula is used to calculate the wheel-rail normal contact force; The wheel-rail normal contact force is obtained by the following formula: (20) Among them, represents the wheel-rail normal contact force, K represents the wheel-rail normal contact force coefficient, which is related to the curvature and elastic modulus of the wheel-rail, and 1.5 represents the power value.

7. The simulation method for the wheel-rail interaction of a curved track based on dual coordinate system collaborative modeling according to claim 5, characterized in that In step 36, the wheel-rail tangential contact force is calculated using the Polach formula, which specifically includes: The wheel-rail creep rate is defined as a physical quantity related to the relative speed between the wheel and the rail. The wheel-rail creep rate is calculated as follows: （21） Among them, represents x the translational creepage rate in the direction, y represents the translational creepage rate in the z direction, and respectively represent the relative velocity differences at the wheel contact point and the track contact point in the x and y directions, represents the relative velocity difference at the wheel contact point and the track contact point in the rotational direction around the z axis, represents the traveling speed; The Polach formula describes the relationship between the wheel-rail tangential contact force and the creep rate. The Polach formula is expressed as: （22） Among them, denote the corrected translational creepage rates related to and respectively, F tr and F ys denote the corrected tangential force and the tangential force generated by pure spin respectively, and denote respectively x the wheel-rail tangential contact force in the y direction and the wheel-rail tangential contact force in the 8. The simulation method for the wheel-rail interaction of a curved track based on dual coordinate system collaborative modeling according to claim 1, wherein The step 3 of solving the wheel-rail contact force and the train response on the curved track in the kinematic equation in the contact coordinate system specifically includes: Write a program to calculate the wheel-rail contact force in the open source finite element platform OpenSees, including the nonlinear Hertz formula for calculating the wheel-rail normal contact force and the Polach formula for calculating the wheel-rail tangential contact force. The open source finite element platform OpenSees also comes with the Newton-Raphson implicit iteration method and the Newmark-beta method. In the open-source finite element platform OpenSees, the wheel-rail contact force and the train response on the curved track in the kinematic equation are calculated and solved by the Newton-Raphson implicit iteration method, the Newmark-beta method, the non-linear Hertz formula, and the Polach formula.

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