Train-track coupling vibration analysis method and device, computer equipment, storage medium and computer program product

Abstract

The invention relates to a train-rail coupling vibration analysis method and device, computer equipment, a storage medium and a computer program product. The method comprises the following steps: dividing a plurality of track units, constructing a motion differential equation of a steel rail in the track units in a frequency domain, and solving to obtain a relation between longitudinal displacement and frequency on nodes at two ends of the steel rail and a relation between stress and frequency on the nodes at the two ends of the steel rail; constructing a steel rail dynamic stiffness matrix for the steel rails in the rail units, and constructing a rail dynamic stiffness matrix for the rail units according to the steel rail dynamic stiffness matrix; assembling to obtain an integral dynamic stiffness matrix of the train-rail system, and constructing a transfer matrix of a rail; and determining a track irregularity excitation power spectrum, and solving to obtain a dynamic response index of the train-track system on each degree of freedom under the track irregularity excitation power spectrum according to the transfer matrix and the overall dynamic stiffness matrix. By adopting the method, the calculation efficiency of the dynamic response index can be improved.

Description

Technical Field

[0001] The present application relates to the field of simulation analysis, and in particular to a train-track coupling vibration analysis method, apparatus, computer equipment, storage medium, and computer program product. Background Art

[0002] Heavy-haul railways play a vital role in my country's freight system. Due to their long train formations, heavy-haul trains pose a significant computational challenge and time-consuming challenge when performing train-track coupled dynamics simulation analysis. Therefore, optimizing existing modeling and calculation methods is crucial to significantly improve computational efficiency while ensuring accuracy.

[0003] Traditional heavy-haul train-track coupled vibration analysis primarily uses time-domain analysis, with solutions typically consisting of intra-time-step iteration and full-process iteration. These methods require multiple iterations to ensure model convergence, resulting in low computational efficiency. Summary of the Invention

[0004] Based on this, it is necessary to provide a train-track coupling vibration analysis method, device, computer equipment, storage medium and computer program product to address the above technical problems.

[0005] In a first aspect, the present application provides a train-track coupled vibration analysis method. The method comprises:

[0006] The track in the train-track system is divided into multiple track units. Based on the theoretical differential motion equation of the beam structure, the differential motion equation of the rail in the track unit is constructed in the frequency domain. The differential motion equation is solved to obtain the relationship between the longitudinal displacement and frequency at the nodes at both ends of the rail, as well as the relationship between the force and frequency at the nodes at both ends of the rail.

[0007] Based on the relationship between the longitudinal displacement and the frequency, and the relationship between the force and the frequency, a rail dynamic stiffness matrix for the rails in the track unit is constructed, and a track dynamic stiffness matrix for the track unit is constructed based on the rail dynamic stiffness matrix;

[0008] Assembling the track dynamic stiffness matrix of each track unit and the train dynamic stiffness matrix of the train in the train-track system to obtain an overall dynamic stiffness matrix of the train-track system, and constructing a transfer matrix of the track based on the track dynamic stiffness matrix of each track unit;

[0009] The track irregularity excitation power spectrum is determined, and based on the transfer matrix and the overall dynamic stiffness matrix, the dynamic response index of the train-track system in each degree of freedom under the track irregularity excitation power spectrum is solved.

[0010] In one embodiment, the method further includes:

[0011] constructing a mass matrix, a damping matrix, and a static stiffness matrix for the train based on the mass blocks included in the train and the degrees of freedom of the mass blocks;

[0012] The train dynamic stiffness matrix is ​​constructed based on the mass matrix, the damping matrix and the static stiffness matrix.

[0013] In one embodiment, constructing a track dynamic stiffness matrix for the track unit according to the rail dynamic stiffness matrix includes:

[0014] For any degree of freedom of the rail, determining the force condition of the rail in the track unit at the degree of freedom in the frequency domain;

[0015] For any degree of freedom of any mass block included in the track unit, determining a force condition of the mass block on the degree of freedom in the frequency domain;

[0016] The force conditions of each degree of freedom of the rail are integrated with the rail dynamic stiffness matrix, and the integrated rail dynamic stiffness matrix and the force conditions of each degree of freedom of each mass block in the track unit are integrated to obtain the track dynamic stiffness matrix of the track unit.

[0017] In one embodiment, constructing the transfer matrix of the track based on the track dynamic stiffness matrix of each track unit includes:

[0018] Constructing the node dynamic stiffness matrices corresponding to the nodes at both ends of the track unit respectively;

[0019] Based on the node dynamic stiffness matrix and the track dynamic stiffness matrix, constructing the relationship between the displacements of the nodes at both ends of the track unit when the wave propagates in the track, and constructing the relationship between the forces on the nodes at both ends of the track unit;

[0020] The transfer matrix of the track is constructed based on the relationship between the displacements of the nodes at both ends of the track unit and the relationship between the forces at the nodes at both ends of the track unit.

[0021] In one embodiment, the solving of the dynamic response index of the train-track system in each degree of freedom under the track irregularity excitation power spectrum based on the transfer matrix and the overall dynamic stiffness matrix includes:

[0022] determining at least one sampling frequency corresponding to the track irregularity excitation power spectrum;

[0023] For any of the sampling frequencies, a track irregularity excitation signal corresponding to each wheel of the train at the sampling frequency is constructed, and based on the track irregularity excitation signal, the transfer matrix and the overall dynamic stiffness matrix, the dynamic response index of the train-track system in each degree of freedom at the sampling frequency is solved.

[0024] In one embodiment, the solving of the dynamic response index of the train-track system in each degree of freedom at the sampling frequency based on the track irregularity excitation signal, the transfer matrix, and the overall dynamic stiffness matrix includes:

[0025] Determining, based on the track irregularity excitation signal and the transfer matrix, displacements caused by the track irregularity excitation signal at nodes on both sides of each track unit;

[0026] determining a dynamic load on the train-track system based on the displacement;

[0027] Based on the dynamic load and the overall dynamic stiffness matrix, a displacement matrix of the train-track system is obtained, and a dynamic response index of the train-track system in each degree of freedom at the sampling frequency is extracted from the displacement matrix.

[0028] In a second aspect, the present application further provides a train-track coupling vibration analysis device. The device comprises:

[0029] A first construction module is configured to divide the track in the train-track system into a plurality of track units, construct a motion differential equation for the rails in the track units in the frequency domain based on a theoretical motion differential equation for a beam structure, and solve the motion differential equation to obtain a relationship between the longitudinal displacement and frequency at the nodes at both ends of the rail, as well as a relationship between the force and frequency at the nodes at both ends of the rail;

[0030] a second construction module, configured to construct a rail dynamic stiffness matrix for the rails in the track unit based on the relationship between the longitudinal displacement and the frequency, and the relationship between the force and the frequency, and to construct a track dynamic stiffness matrix for the track unit based on the rail dynamic stiffness matrix;

[0031] a third construction module, configured to assemble the track dynamic stiffness matrix of each track unit and the train dynamic stiffness matrix of the train in the train-track system to obtain an overall dynamic stiffness matrix of the train-track system, and construct a transfer matrix of the track based on the track dynamic stiffness matrix of each track unit;

[0032] A solution module is used to determine the track irregularity excitation power spectrum, and to solve the dynamic response index of the train-track system in each degree of freedom under the track irregularity excitation power spectrum based on the transfer matrix and the overall dynamic stiffness matrix.

[0033] In one embodiment, the apparatus further comprises:

[0034] a fourth construction module, configured to construct a mass matrix, a damping matrix, and a static stiffness matrix for the train based on the mass blocks included in the train and the degrees of freedom of the mass blocks;

[0035] A fifth construction module is used to construct the train dynamic stiffness matrix based on the mass matrix, the damping matrix and the static stiffness matrix.

[0036] In one embodiment, the second building block is further configured to:

[0037] For any degree of freedom of the rail, determining the force condition of the rail in the track unit at the degree of freedom in the frequency domain;

[0038] For any degree of freedom of any mass block included in the track unit, determining a force condition of the mass block on the degree of freedom in the frequency domain;

[0039] The force conditions of each degree of freedom of the rail are integrated with the rail dynamic stiffness matrix, and the integrated rail dynamic stiffness matrix and the force conditions of each degree of freedom of each mass block in the track unit are integrated to obtain the track dynamic stiffness matrix of the track unit.

[0040] In one embodiment, the third building block is further configured to:

[0041] Constructing the node dynamic stiffness matrices corresponding to the nodes at both ends of the track unit respectively;

[0042] Based on the node dynamic stiffness matrix and the track dynamic stiffness matrix, constructing the relationship between the displacements of the nodes at both ends of the track unit when the wave propagates in the track, and constructing the relationship between the forces on the nodes at both ends of the track unit;

[0043] The transfer matrix of the track is constructed based on the relationship between the displacements of the nodes at both ends of the track unit and the relationship between the forces at the nodes at both ends of the track unit.

[0044] In one embodiment, the solution module is further configured to:

[0045] determining at least one sampling frequency corresponding to the track irregularity excitation power spectrum;

[0046] For any of the sampling frequencies, a track irregularity excitation signal corresponding to each wheel of the train at the sampling frequency is constructed, and based on the track irregularity excitation signal, the transfer matrix and the overall dynamic stiffness matrix, the dynamic response index of the train-track system in each degree of freedom at the sampling frequency is solved.

[0047] In one embodiment, the solution module is further configured to:

[0048] Determining, based on the track irregularity excitation signal and the transfer matrix, displacements caused by the track irregularity excitation signal at nodes on both sides of each track unit;

[0049] determining a dynamic load on the train-track system based on the displacement;

[0050] Based on the dynamic load and the overall dynamic stiffness matrix, a displacement matrix of the train-track system is obtained, and a dynamic response index of the train-track system in each degree of freedom at the sampling frequency is extracted from the displacement matrix.

[0051] In a third aspect, the present application further provides a computer device, wherein the computer device comprises a memory and a processor, wherein the memory stores a computer program, and the processor implements any of the above methods when executing the computer program.

[0052] In a fourth aspect, the present application further provides a computer-readable storage medium having a computer program stored thereon, wherein the computer program implements any of the above methods when executed by a processor.

[0053] In a fifth aspect, the present application further provides a computer program product, comprising a computer program, which implements any of the above methods when executed by a processor.

[0054] The above-mentioned train-track coupled vibration analysis method, device, computer equipment, storage medium and computer program product solve the theoretical motion differential equation of the beam structure in the frequency domain, construct the dynamic stiffness matrix of the rail in the form of an analytical solution, and synthesize the dynamic stiffness matrix of the rail and the track unit to obtain the track dynamic stiffness matrix in the form of an analytical solution; on this basis, the known train dynamic stiffness matrix and the dynamic stiffness matrices of each track are assembled to obtain the overall dynamic stiffness matrix in the form of an analytical solution, and the transfer matrix in the form of an analytical solution is constructed based on the track dynamic stiffness matrix. Finally, the track irregularity excitation power spectrum is used to calculate the dynamic response index of each degree of freedom of the train-track system under the track irregularity excitation power spectrum under the constraints of the transfer matrix and the overall dynamic stiffness matrix. The embodiment of the present application can calculate the analytical solution of the dynamic response index without the need for iterative calculation, which can improve the calculation efficiency of the dynamic response index. BRIEF DESCRIPTION OF THE DRAWINGS

[0055] Figure 1 1 is a flow chart of a train-track coupled vibration analysis method according to an embodiment;

[0056] Figure 2 is a schematic diagram of a track model in one embodiment;

[0057] Figure 3 is a schematic diagram of a train model in one embodiment;

[0058] Figure 4 is a structural block diagram of a train-track coupled vibration analysis device in one embodiment;

[0059] Figure 5 FIG. 1 is a diagram showing the internal structure of a computer device in one embodiment. DETAILED DESCRIPTION

[0060] In order to make the purpose, technical solutions and advantages of this application more clear, the following further describes this application in detail with reference to the accompanying drawings and embodiments. It should be understood that the specific embodiments described herein are only used to explain this application and are not intended to limit this application.

[0061] In one embodiment, Figure 1 As shown, a train-track coupled vibration analysis method is provided. This embodiment uses the method applied to a server as an example. It is understood that the method can also be applied to a terminal, or to a system including a terminal and a server, and implemented through the interaction between the terminal and the server. In this embodiment, the method includes the following steps:

[0062] Step 102: Divide the track in the train-track system into multiple track units. Based on the theoretical differential motion equation of the beam structure, construct the differential motion equation of the rail in the track unit in the frequency domain. Then solve the differential motion equation to obtain the relationship between the longitudinal displacement and frequency at the nodes at both ends of the rail, and the relationship between the force and frequency at the nodes at both ends of the rail.

[0063] In the embodiment of the present application, the track can be regarded as an infinitely circulating periodic structure. Figure 2 As shown, this is an example of a track unit. Figure 2 The middle track unit consists of two adjacent sleepers, the rails between the sleepers and the ballast below the sleepers. The rails, sleepers and ballast are connected by fasteners. It should be noted that those skilled in the art can also adopt different methods according to the different track structures and modeling accuracy requirements. Figure 2 There are different ways to construct track units, for example, for ballastless track, you can Figure 2 The ballast is removed from the track unit, and the ballast can be further layered modeled for scenes with higher modeling accuracy requirements. The embodiment of the present application does not specifically limit the construction method of the track unit.

[0064] When constructing the dynamic stiffness matrix of the track (hereinafter referred to as the track dynamic stiffness matrix), the dynamic stiffness matrix of the rail (hereinafter referred to as the rail dynamic stiffness matrix) can be first obtained in the form of an analytical solution for the rail, and then other structures in the track unit are simplified into mass blocks and integrated into the rail dynamic stiffness matrix to obtain the track dynamic stiffness matrix.

[0065] When modeling the rail, the rail can be considered as a beam structure that satisfies the Euler-Bernoulli beam element assumption. The theoretical motion differential equation of the beam structure under this assumption can then be used to describe the motion of the rail. This allows the relationship between the displacement and force of the rail to be obtained, and the rail dynamic stiffness matrix can then be obtained based on the definition of the dynamic stiffness matrix. The theoretical motion differential equation of the beam structure that satisfies the Euler-Bernoulli beam element assumption is shown in Formula (1):

[0066] Formula (1)

[0067] Where E is the elastic modulus of the beam; I is the bending moment of inertia; m is the mass per unit length of the beam; y is the vertical deflection; and c is the damping coefficient of the structure.

[0068] Based on this formula, the differential equation of motion of the rail in the time domain can be constructed, and then the differential equation of motion in the time domain can be converted to the frequency domain to obtain the analytical solution of the formula. The physical parameters of the rail may include the elastic modulus, bending moment of inertia, mass per unit length, and damping coefficient involved in formula (1). The above parameters can be obtained through experiments, etc. After constructing the differential equation of motion of the rail based on the physical parameters of the rail and formula (1), the differential equation of motion is converted to the frequency domain to obtain formula (2):

[0069] Formula (2)

[0070] Where s is the Laplace operator; Y is the longitudinal displacement of the point x on the rail in the frequency domain; and λ is a parameter related to the rail frequency.

[0071] The analytical solution of formula (2) is formula (3)

[0072] Formula (3)

[0073] Among them, A1-A4 are unknown coefficients, and l is the total length of the rail in the rail unit.

[0074] Since the forces and displacements of the rail at the two end nodes will be derived in the subsequent calculation process, the rail dynamic stiffness matrix can be expressed here using the forces and displacements of the rail at the nodes. Based on formula (3), the displacements of the rail at the two end nodes (for x=0 and x=1, respectively) can be written as formula (4):

[0075] Formula (4)

[0076] Among them, u e is the displacement matrix, Q is a 4×4 matrix, A is a 1×4 matrix containing A1-A4, Y(0) and Y(L) are Figure 2 The longitudinal displacements of the left and right nodes of the rail unit, Y'(0) and Y'(L) are the first-order derivatives of Y(0) and Y(L), that is, the displacement velocities of the left and right nodes of the rail unit.

[0077] Since the shear force and bending moment in the beam are the third-order derivative and second-order derivative of the displacement, respectively, the force on the rail at the two end nodes can be written as formula (5):

[0078] Formula (5)

[0079] Among them, p e is the force matrix, R is a 4×4 matrix, A is a 1×4 matrix containing A1-A4, Q(0) and Q(L) are Figure 2The shear forces of the left and right nodes of the middle rail unit are M(0) and M(L) respectively Figure 2 The bending moments of the left and right nodes of the middle rail unit.

[0080] In step 104, a rail dynamic stiffness matrix for the rail in the rail unit is constructed based on the relationship between the longitudinal displacement and the frequency and the relationship between the force and the frequency, and a track dynamic stiffness matrix for the rail unit is constructed according to the rail dynamic stiffness matrix.

[0081] In the embodiment of the present application, as shown in formula (six), since the product of the rail dynamic stiffness matrix and the displacement matrix is the force matrix, the rail dynamic stiffness matrix can be expressed as the product of the matrix R in formula (five) and the inverse matrix of the matrix Q in formula (four):

[0082] Formula (six)

[0083] wherein K(ω) is the rail stiffness dynamic matrix, and the elements thereof can be written in the form of formula (seven):

[0084] Formula (seven)

[0085] wherein A = e -λl , B = sin λl, C = cos λl, and D = e -λ(x-l) .

[0086] After obtaining the analytical form expression of the rail dynamic stiffness matrix, other mass blocks obtained by modeling the rail unit are integrated into the rail dynamic stiffness matrix to obtain the track dynamic stiffness matrix. The other mass blocks specifically include which and the degrees of freedom of each mass block can be set by a person skilled in the art according to actual needs. For example, when the rail unit is modeled in the manner as shown in formula (one), the other mass blocks can include the sleepers and ballast respectively located below the left and right nodes of the rail. The integration method can adopt any existing dynamic stiffness matrix integration method, and the embodiment of the present application does not make specific limitation thereto. Figure 2

[0087] In one embodiment, the track dynamic stiffness matrix for the rail unit is constructed according to the rail dynamic stiffness matrix, comprising:

[0088] For any degree of freedom of the rail, the force on the degree of freedom of the rail in the frequency domain is determined;

[0089] For any degree of freedom of any mass block included in the rail unit, the force on the degree of freedom of the mass block in the frequency domain is determined;

[0090] ​The force conditions of each degree of freedom of the rail are integrated with the rail dynamic stiffness matrix, and the integrated rail dynamic stiffness matrix is ​​integrated with the force conditions of each degree of freedom of each mass block in the track unit to obtain the track dynamic stiffness matrix of the track unit.

[0091] In the embodiment of the present application, the force conditions of each degree of freedom of the rail in the rail unit can be determined by considering the influence of the inertial force and damping force provided by the longitudinal structure of the rail unit on each degree of freedom of the rail, and the force conditions of each other mass block in the rail unit in each degree of freedom can be determined by also considering the inertial force and damping force between each mass block in the longitudinal structure of the rail unit and considering the force balance inside the rail unit. Then, based on the force conditions of each degree of freedom of the rail in the rail unit, each element in the rail dynamic stiffness matrix is ​​updated, and the updated rail dynamic stiffness matrix is ​​embedded in the track dynamic stiffness matrix. Then, according to the force conditions of each other mass block in each degree of freedom, other elements in the track dynamic stiffness matrix are constructed to obtain the track dynamic stiffness matrix.

[0092] For example, with respect to the degree of freedom corresponding to the longitudinal displacement of the left node in the rail, the rail is also affected by the damping force of the fastener between the rail and the sleeper, as well as the inertia force of the sleeper. Therefore, the force applied to the rail in this degree of freedom is superimposed on the elements in the rail dynamic stiffness matrix corresponding to the longitudinal force exerted by the rail itself on the left node of the rail when the left node of the rail is displaced in the longitudinal direction, to obtain an updated element. Similar methods can be used to update the elements in the rail dynamic stiffness matrix for other degrees of freedom of the rail, and this will not be described in detail in the embodiments of the present application.

[0093] For example, taking the rail dynamic matrix in formula (VII) as an example, after integrating the force conditions of the rail in the track unit, the force conditions of other mass blocks in the track unit and the rail dynamic stiffness matrix, the track dynamic stiffness matrix in formula (VIII) can be obtained:

[0094] Formula (8)

[0095] Among them, the elements without the t subscript are the original elements in the rail dynamic stiffness matrix, and the elements with the t subscript are the elements obtained by updating the elements in the rail dynamic stiffness matrix, or the elements constructed based on other mass blocks in the track unit. sym indicates that the matrix is ​​symmetric. The specific elements are shown in formula (9):

[0096] Formula (9)

[0097] Among them, reference Figure 2 As shown, c r1 is the damping coefficient of the fastener between rail and sleeper, k r1is the stiffness of the fastener between the rail and the sleeper, m1 is the mass of the mass block representing the sleeper, c r2 is the damping coefficient of the fastener between the sleeper and the ballast, k r2 is the stiffness of the fasteners between the sleeper and the ballast, and m2 is the mass of the mass block representing the ballast.

[0098] Step 106: Assemble the track dynamic stiffness matrix of each track unit and the train dynamic stiffness matrix of the train in the train-track system to obtain the overall dynamic stiffness matrix of the train-track system, and construct the track transfer matrix based on the track dynamic stiffness matrix of each track unit;

[0099] In the embodiment of the present application, a model can be built for the train in the train-track system to be analyzed, and the train can be represented as a plurality of masses with mutual forces, and the directions in which each mass can move (i.e., the degrees of freedom of the mass) can be determined. Figure 3 As shown, this is an example of train modeling. Figure 3 A train can be modeled as a structure consisting of three mass blocks: the carbody, the bogie, and the wheels, connected by fasteners. The subscript c represents carbody parameters, the subscript t represents bogie parameters (t1 and t2 are used to distinguish between different bogies), and the subscript w represents wheel parameters (w1, w2, w3, and w4 are used to distinguish between different wheels). θ represents the steering angle, z represents the vertical displacement, M represents the mass, J represents the moment of inertia, lc represents the distance from the wheel to the carbody's center of mass, lt represents the distance from the wheel to the bogie's center of mass, and k represents the stiffness.

[0100] After modeling the train, the train's dynamic stiffness matrix (hereinafter referred to as the train dynamic stiffness matrix) can be constructed based on the degrees of freedom of each mass. The train dynamic stiffness matrix characterizes the train's stiffness characteristics under dynamic loads. The jth element in the i-th row of the matrix represents the dynamic force required to be applied to the i-th degree of freedom when the j-th degree of freedom of the train undergoes unit displacement.

[0101] After constructing the train's mass matrix, damping matrix, and static stiffness matrix, the train's dynamic stiffness matrix can be constructed based on the train's motion equations, including:

[0102] Based on the mass blocks and degrees of freedom of each mass block included in the train, a mass matrix, a damping matrix and a static stiffness matrix for the train are constructed;

[0103] The train dynamic stiffness matrix is ​​constructed based on the mass matrix, damping matrix and static stiffness matrix.

[0104] In the embodiment of the present application, the train dynamic stiffness matrix can be constructed based on the train's equation of motion. The train's equation of motion can be a general equation of motion in structural dynamics, as shown in formula (10), which is the frequency domain form of the equation of motion:

[0105] Formula (10)

[0106] Among them, K vd is the train dynamic stiffness matrix, X v is the displacement matrix of the train, M v is the mass matrix of the train, where the physical meaning of the jth element in the i-th row is the inertial force at the i-th degree of freedom when the j-th degree of freedom produces unit acceleration; C v is the damping matrix of the train, where the physical meaning of the jth element in the i-th row is the damping force at the i-th degree of freedom when the j-th degree of freedom produces unit acceleration; K v is the static stiffness matrix of the train, F v is the dynamic load of the train. ω is the frequency of the dynamic load.

[0107] The mass matrix, damping matrix and static stiffness matrix can all be derived by performing dynamic theoretical analysis on the mass block. The embodiments of this application do not limit how to construct these three matrices, and any existing method is applicable to the embodiments of this application.

[0108] Taking into account the internal force balance of the train-track system, the displacement of the train wheels is the same as the displacement of the track, and the displacement and force of two adjacent track units at the shared node are also the same, the track dynamic stiffness matrix and the train dynamic stiffness matrix of each track unit can be assembled to obtain the overall dynamic stiffness matrix by combining the aforementioned formula (10) and the formula (3) expressing the longitudinal displacement at any position on the rail. This embodiment of the application does not elaborate on how to assemble the overall dynamic stiffness matrix. Any method for assembling the dynamic stiffness matrices of different parts of a physical system is applicable to the embodiment of the application.

[0109] Furthermore, Formula (3) can be modified based on the aforementioned Formula (4) so ​​that the overall dynamic stiffness matrix does not include the unknown coefficients A1-A4. Since the matrix A in Formula (4) is the product of the displacement matrix and the inverse matrix of the matrix Q, Formula (3) can be transformed into Formula (11):

[0110] Formula (11)

[0111] Where N is a 4×1 matrix The product of the inverse matrix of the matrix Q is called the interpolation function vector. The expression of each element in the interpolation function vector can be found in formula (12):

[0112] Formula (12)

[0113] The track transfer matrix is ​​used to characterize the wave propagation characteristics on the track. It can be obtained by constructing the relationship between the displacement and force of the nodes on the left and right sides of the track. The steps include:

[0114] Construct the nodal dynamic stiffness matrices corresponding to the nodes at both ends of the track unit respectively;

[0115] Based on the node dynamic stiffness matrix and the track dynamic stiffness matrix, the relationship between the displacement of the nodes at both ends of the track unit when the wave propagates in the track is constructed, and the relationship between the forces on the nodes at both ends of the track unit is constructed;

[0116] The transfer matrix of the track is constructed based on the relationship between the displacements of the nodes at both ends of the track unit and the relationship between the forces at the nodes at both ends of the track unit.

[0117] In the embodiment of the present application, the dynamic equation describing the relationship between the displacement and force of the nodes on the left and right sides of the track can be shown in Formula (13):

[0118] Formula (13)

[0119] Among them, K LL and K RR Represents the dynamic stiffness matrix of the left and right nodes of the track element, respectively. Figure 2 As shown, the nodes of the track unit are Figure 2 The system includes rail nodes, sleepers and ballast below the rail nodes, and intermediate fasteners between rails, sleepers and ballast. The dynamic stiffness matrix of the track unit node can be constructed by analyzing the various mass blocks involved in the track unit node. K LR and K RL Both represent the track dynamic stiffness matrix. L and u R are the displacement matrices of the left and right nodes of the track element, p L and p R are the force matrices of the left and right nodes of the track element respectively.

[0120] Therefore, the transfer matrix can be expressed as formula (14):

[0121] Formula (14)

[0122] Where T(ω) is the transfer matrix. When solving the dynamic response later, the transfer matrix can be used to constrain the relationship between the displacement and force at the left and right nodes of the track element.

[0123] In step 108, the track irregularity excitation power spectrum is determined, and the dynamic response indicators of the train-track system at each degree of freedom under the track irregularity excitation power spectrum are solved according to the transfer matrix and the overall dynamic stiffness matrix.

[0124] In the embodiments of the present application, the virtual excitation method is used to simulate the track irregularity, and the power spectrum is used as the input in the virtual excitation method. The track irregularity excitation power spectrum can be selected according to actual requirements. A plurality of frequencies can be selected as sampling frequencies from the frequency range covered by the track irregularity excitation power spectrum for analysis. Specifically,

[0125] The at least one sampling frequency corresponding to the track irregularity excitation power spectrum is determined.

[0126] For any sampling frequency, the track irregularity excitation signal corresponding to each wheel of the train at the sampling frequency is constructed, and the dynamic response indicators of the train-track system at each degree of freedom under the sampling frequency are solved based on the track irregularity excitation signal, the transfer matrix and the overall dynamic stiffness matrix.

[0127] In the embodiments of the present application, the sampling frequencies can be selected according to actual requirements, for example, a plurality of frequencies corresponding to high energy density in the power spectrum can be selected as the sampling frequencies, and the embodiments of the present application do not limit how to select the sampling frequencies.

[0128] For each sampling frequency, the sampling frequency can be substituted into the foregoing formulas to obtain the overall dynamic stiffness matrix and the transfer matrix of the train-track system under the sampling frequency. Then the dynamic load of the train-track system can be analyzed. In the embodiments of the present application, the dynamic load of the train-track system based on the track irregularity is mainly analyzed. The track irregularity excitation signal received by each wheel of the train at the sampling frequency can be constructed based on the sampling frequency to determine the dynamic load. Specifically,

[0129] Based on the track irregularity excitation signal and the transfer matrix, the displacement caused by the track irregularity excitation signal at the nodes on both sides of each track unit is determined.

[0130] Based on the displacement, the dynamic load of the train-track system is determined.

[0131] Based on the dynamic load and the overall dynamic stiffness matrix, the displacement matrix of the train-track system is solved, and the dynamic response indicators of the train-track system at each degree of freedom under the sampling frequency are extracted from the displacement matrix.

[0132] In the embodiments of the present application, the train model in the formula (XIII) is taken as an example, and the track irregularity excitation signal of each wheel can be seen from the formula (XV): Figure 1

[0133] ​Formula (15)

[0134] Where, t1=0, t2=2l t / v,t3=2l c / v,t4=2(l t +l c ) / v, v is the train speed, l t and l c The meaning is as shown in the above embodiment. Figure 1 Description, t1 to t4 are used to represent the time points when the four wheels of the train pass through the track unevenness. η (ω) = S η (Ω) / v, Ω is the spatial frequency of the virtual excitation, S η (ω) is the power spectrum of track irregularity excitation.

[0135] Assuming that the train and the track satisfy linear Hertz contact, the contact force between each wheel and the track can be shown in formula (16):

[0136] Formula (16)

[0137] Among them, f i is the contact force of the i-th wheel (the embodiment of the present application is based on Figure 1 The train model in is analyzed, so i can be 1, 2, 3, 4, representing Figure 1 ), k represents the wheel-rail contact stiffness of linear Hertz contact, y i represents the displacement of the rail at the wheel-rail contact point, Indicates the track irregularity excitation signal, y wi Represents the displacement of the wheel at the wheel-rail contact point.

[0138] The contact force of each wheel is transmitted to each track unit on the track. Based on the aforementioned transfer matrix, the displacement of the left and right nodes of each track unit under the contact force of each wheel can be calculated. Then, the displacement of the wheel at the wheel-rail contact point can be further calculated by interpolating the function vector and the displacement of the left and right nodes of the track unit where the wheel is located, as shown in Formula (17):

[0139] Formula (17)

[0140] Wherein, N is the interpolation function vector in the above embodiment, u 0,i Refers to the displacement of the left node of the track unit where wheel i is located, u l,i It refers to the displacement of the right node of the track unit where wheel i is located.

[0141] Combined with the overall dynamic stiffness matrix obtained from the above integration, the frequency domain motion equation of the train-track system can be written as formula (18):

[0142] Formula (18)

[0143] Among them, K g (ω) is the overall dynamic stiffness matrix, X g (ω) is the displacement matrix of the train-track system, F g (ω) is the dynamic load of the train-track system, considering the internal force balance of the train-track system and the dynamic load imposed on the train-track system by track irregularities, F g (ω) is formula (19):

[0144] Formula (19)

[0145] Where k refers to the wheel-rail contact stiffness of the wheel-rail contact, and N is the interpolation function vector in the above formulas (11) and (12).

[0146] Furthermore, the steps of calculating displacement based on the transfer matrix can be simplified based on the symplectic method. Without considering external forces, when the simple harmonic wave passes through the track element, the forces and displacements on the left and right sides of the track element satisfy formula (XX):

[0147] Formula (20)

[0148] Here, λ is usually a complex number and is called the wave propagation constant.

[0149] The propagation problem of simple harmonic waves in orbit can be expressed as an eigenvalue problem, that is, formula (21):

[0150] Formula (21)

[0151] It can be shown that the transfer matrix satisfies the symplectic orthogonality relation, so if λ is an eigenvalue of T(ω), then 1 / λ is also an eigenvalue of T(ω). The transfer matrix is ​​a 2n×2n matrix (n is the degree of freedom of the orbital unit) with a total of 2n eigenvalues. Therefore, the 2n eigenvalues ​​of T(ω) can be divided into two groups according to formula (XXII):

[0152] Formula (22)

[0153] The eigenvector can be written as formula (XXIII):

[0154] Formula (23)

[0155] According to the eigenvector, the displacement relationship of the nodes on the left and right sides of the track unit can be written as formula (XXIV):

[0156] Formula (24)

[0157] Where A, B, C, and D are all n×n square matrices. m and n are vectors to be solved, which can be determined by the continuity condition of displacement and the balance condition of force. k represents the interface position of wave propagation, k≥1 means the wave propagates forward, and k≤-1 means the wave propagates backward. u L,ij is the displacement caused by the track irregularity excitation signal of the i-th wheel at the left node of the j-th track unit, u R,ij is the displacement caused by the track irregularity excitation signal of the i-th wheel at the right node of the j-th track unit.

[0158] The train-track coupling vibration analysis method provided in the embodiment of the present application solves the theoretical motion differential equation of the beam structure in the frequency domain, constructs the dynamic stiffness matrix of the rail in the form of an analytical solution, and synthesizes the dynamic stiffness matrix of the rail and the track unit to obtain the track dynamic stiffness matrix in the form of an analytical solution; on this basis, the known train dynamic stiffness matrix and the dynamic stiffness matrices of each track are assembled to obtain the overall dynamic stiffness matrix in the form of an analytical solution, and the transfer matrix in the form of an analytical solution is constructed based on the track dynamic stiffness matrix. Finally, the track irregularity excitation power spectrum is used to calculate the dynamic response index of each degree of freedom of the train-track system under the track irregularity excitation power spectrum under the constraints of the transfer matrix and the overall dynamic stiffness matrix. The embodiment of the present application can calculate the analytical solution of the dynamic response index without the need for iterative calculation, which can improve the calculation efficiency of the dynamic response index.

[0159] It should be understood that, although the various steps in the flowcharts involved in the various embodiments described above are displayed in sequence according to the instructions of the arrows, these steps are not necessarily executed in sequence in the order indicated by the arrows. Unless otherwise specified herein, there is no strict order restriction on the execution of these steps, and these steps can be executed in other orders. Moreover, at least a portion of the steps in the flowcharts involved in the various embodiments described above can include multiple steps or multiple stages, and these steps or stages are not necessarily executed and completed at the same time, but can be executed at different times, and the execution order of these steps or stages is not necessarily to be carried out in sequence, but can be executed in turn or alternately with other steps or at least a portion of steps or stages in other steps.

[0160] Based on the same inventive concept, embodiments of the present application also provide a train-track coupling vibration analysis device for implementing the aforementioned train-track coupling vibration analysis method. The solution provided by this device is similar to the solution described in the aforementioned method. Therefore, the specific limitations of one or more of the following embodiments of the train-track coupling vibration analysis device can be found in the aforementioned limitations of the train-track coupling vibration analysis method and will not be further elaborated here.

[0161] In one embodiment, Figure 4 As shown, a train-track coupled vibration analysis device 400 is provided, comprising: a first building module 402, a second building module 404, a third building module 406, and a solution module 408, wherein:

[0162] A first construction module 402 is configured to divide the track in the train-track system into a plurality of track units, construct a motion differential equation for the rails in the track units in the frequency domain based on a theoretical motion differential equation for a beam structure, and solve the motion differential equation to obtain a relationship between the longitudinal displacement and frequency at the nodes at both ends of the rail, as well as a relationship between the force and frequency at the nodes at both ends of the rail;

[0163] A second construction module 404 is configured to construct a rail dynamic stiffness matrix for the rails in the track unit based on the relationship between the longitudinal displacement and the frequency, and the relationship between the force and the frequency, and to construct a track dynamic stiffness matrix for the track unit based on the rail dynamic stiffness matrix;

[0164] A third construction module 406 is configured to assemble the track dynamic stiffness matrix of each track unit and the train dynamic stiffness matrix of the train in the train-track system to obtain an overall dynamic stiffness matrix of the train-track system, and construct a transfer matrix of the track based on the track dynamic stiffness matrix of each track unit;

[0165] The solution module 408 is used to determine the track irregularity excitation power spectrum, and solve the dynamic response index of the train-track system in each degree of freedom under the track irregularity excitation power spectrum based on the transfer matrix and the overall dynamic stiffness matrix.

[0166] In one embodiment, the apparatus further comprises:

[0167] a fourth construction module, configured to construct a mass matrix, a damping matrix, and a static stiffness matrix for the train based on the mass blocks included in the train and the degrees of freedom of the mass blocks;

[0168] The fifth construction module is used to construct the train dynamic stiffness matrix based on the mass matrix, the damping matrix and the static stiffness matrix.

[0169] In one embodiment, the second building module 404 is further configured to:

[0170] For any degree of freedom of the rail, determining the force condition of the rail in the track unit at the degree of freedom in the frequency domain;

[0171] For any degree of freedom of any mass block included in the track unit, determining a force condition of the mass block on the degree of freedom in the frequency domain;

[0172] The force conditions of each degree of freedom of the rail are integrated with the rail dynamic stiffness matrix, and the integrated rail dynamic stiffness matrix and the force conditions of each degree of freedom of each mass block in the track unit are integrated to obtain the track dynamic stiffness matrix of the track unit.

[0173] In one embodiment, the third building block 406 is further configured to:

[0174] Constructing the node dynamic stiffness matrices corresponding to the nodes at both ends of the track unit respectively;

[0175] Based on the node dynamic stiffness matrix and the track dynamic stiffness matrix, constructing the relationship between the displacements of the nodes at both ends of the track unit when the wave propagates in the track, and constructing the relationship between the forces on the nodes at both ends of the track unit;

[0176] The transfer matrix of the track is constructed based on the relationship between the displacements of the nodes at both ends of the track unit and the relationship between the forces at the nodes at both ends of the track unit.

[0177] In one embodiment, the solution module 408 is further configured to:

[0178] determining at least one sampling frequency corresponding to the track irregularity excitation power spectrum;

[0179] For any of the sampling frequencies, a track irregularity excitation signal corresponding to each wheel of the train at the sampling frequency is constructed, and based on the track irregularity excitation signal, the transfer matrix and the overall dynamic stiffness matrix, the dynamic response index of the train-track system in each degree of freedom at the sampling frequency is solved.

[0180] In one embodiment, the solution module 408 is further configured to:

[0181] Determining, based on the track irregularity excitation signal and the transfer matrix, displacements caused by the track irregularity excitation signal at nodes on both sides of each track unit;

[0182] determining a dynamic load of the train-track system based on the displacement;

[0183] Based on the dynamic load and the overall dynamic stiffness matrix, a displacement matrix of the train-track system is obtained, and a dynamic response index of the train-track system in each degree of freedom at the sampling frequency is extracted from the displacement matrix.

[0184] Each module in the above-mentioned apparatus may be implemented in whole or in part by software, hardware, or a combination thereof. Each module may be embedded in or independent of a processor in a computer device in the form of hardware, or may be stored in a memory in the computer device in the form of software, so that the processor can call and execute the operations corresponding to each module.

[0185] In one embodiment, a computer device is provided. The computer device may be a server, and its internal structure diagram may be as follows: Figure 5 As shown. The computer device includes a processor, memory, and a network interface connected via a system bus. The processor of the computer device is used to provide computing and control capabilities. The memory of the computer device includes a non-volatile storage medium and an internal memory. The non-volatile storage medium stores an operating system and a computer program. The internal memory provides an environment for the operation of the operating system and computer program in the non-volatile storage medium. The network interface of the computer device is used to communicate with an external terminal via a network connection. When executed by the processor, the computer program implements a train-track coupled vibration analysis method.

[0186] Those skilled in the art will understand that Figure 5 The structure shown in the figure is only a block diagram of a part of the structure related to the solution of the present application, and does not constitute a limitation on the computer device to which the solution of the present application is applied. The specific computer device may include more or fewer components than shown in the figure, or combine certain components, or have a different component arrangement.

[0187] In one embodiment, a computer device is provided, including a memory and a processor. The memory stores a computer program, and the processor implements the steps in the above method embodiments when executing the computer program.

[0188] In one embodiment, a computer-readable storage medium is provided, on which a computer program is stored. When the computer program is executed by a processor, the steps in the above-mentioned method embodiments are implemented.

[0189] In one embodiment, a computer program product is provided, including a computer program, which implements the steps in the above method embodiments when executed by a processor.

[0190] It should be noted that the user information (including but not limited to user device information, user personal information, etc.) and data (including but not limited to data used for analysis, stored data, displayed data, etc.) involved in this application are all information and data authorized by the user or fully authorized by all parties.

[0191] Those skilled in the art will appreciate that all or part of the processes in the above-mentioned embodiments can be implemented by instructing the relevant hardware through a computer program. The computer program can be stored in a non-volatile computer-readable storage medium. When the computer program is executed, it can include the processes of the above-mentioned embodiments. In particular, any reference to memory, database, or other media used in the embodiments provided in this application can include at least one of non-volatile and volatile memory. Non-volatile memory can include read-only memory (ROM), magnetic tape, floppy disk, flash memory, optical memory, high-density embedded non-volatile memory, resistive random access memory (ReRAM), magnetic random access memory (MRAM), ferroelectric random access memory (FRAM), phase change memory (PCM), graphene memory, etc. Volatile memory can include random access memory (RAM) or external cache memory, etc. By way of illustration and not limitation, RAM can take various forms, such as static random access memory (SRAM) or dynamic random access memory (DRAM). The databases involved in the various embodiments provided herein may include at least one of a relational database and a non-relational database. Non-relational databases may include, but are not limited to, distributed databases based on blockchains. The processors involved in the various embodiments provided herein may be, but are not limited to, general-purpose processors, central processing units (CPUs), graphics processing units (GPUs), digital signal processors (DSPs), programmable logic devices (PLDs), data processing logic devices based on quantum computing, and the like.

[0192] The technical features of the above embodiments can be combined arbitrarily. In order to make the description concise, not all possible combinations of the technical features in the above embodiments are described. However, as long as there is no contradiction in the combination of these technical features, they should be considered to be within the scope of this specification.

[0193] The above-described embodiments merely represent several implementation methods of the present application. While the descriptions are relatively specific and detailed, they should not be construed as limiting the scope of the present application. It should be noted that a person of ordinary skill in the art may make various modifications and improvements without departing from the spirit of the present application, and these modifications and improvements fall within the scope of protection of the present application. Therefore, the scope of protection of the present application shall be determined by the appended claims.

Claims

1. A train-track coupled vibration analysis method, characterized in that: The method comprises: The track in the train-track system is divided into multiple track units. Based on the theoretical differential motion equation of the beam structure, the differential motion equation of the rail in the track unit is constructed in the frequency domain. The differential motion equation is solved to obtain the relationship between the longitudinal displacement and frequency at the nodes at both ends of the rail, as well as the relationship between the force and frequency at the nodes at both ends of the rail. Based on the relationship between the longitudinal displacement and the frequency, and the relationship between the force and the frequency, a rail dynamic stiffness matrix for the rails in the track unit is constructed, and a track dynamic stiffness matrix for the track unit is constructed based on the rail dynamic stiffness matrix; Assembling the track dynamic stiffness matrix of each track unit and the train dynamic stiffness matrix of the train in the train-track system to obtain an overall dynamic stiffness matrix of the train-track system, and constructing a transfer matrix of the track based on the track dynamic stiffness matrix of each track unit; The track irregularity excitation power spectrum is determined, and based on the transfer matrix and the overall dynamic stiffness matrix, the dynamic response index of the train-track system in each degree of freedom under the track irregularity excitation power spectrum is solved.

2. The method according to claim 1, characterized in that The method further comprises: constructing a mass matrix, a damping matrix, and a static stiffness matrix for the train based on the mass blocks included in the train and the degrees of freedom of the mass blocks; The train dynamic stiffness matrix is ​​constructed based on the mass matrix, the damping matrix and the static stiffness matrix.

3. The method according to claim 1, characterized in that The constructing of a track dynamic stiffness matrix for the track unit according to the rail dynamic stiffness matrix includes: For any degree of freedom of the rail, determining the force condition of the rail in the track unit at the degree of freedom in the frequency domain; For any degree of freedom of any mass block included in the track unit, determining a force condition of the mass block on the degree of freedom in the frequency domain; The force conditions of each degree of freedom of the rail are integrated with the rail dynamic stiffness matrix, and the integrated rail dynamic stiffness matrix and the force conditions of each degree of freedom of each mass block in the track unit are integrated to obtain the track dynamic stiffness matrix of the track unit.

4. The method according to claim 1, wherein The constructing of the track transfer matrix based on the track dynamic stiffness matrix of each track unit includes: Constructing the node dynamic stiffness matrices corresponding to the nodes at both ends of the track unit respectively; Based on the node dynamic stiffness matrix and the track dynamic stiffness matrix, constructing the relationship between the displacements of the nodes at both ends of the track unit when the wave propagates in the track, and constructing the relationship between the forces on the nodes at both ends of the track unit; The transfer matrix of the track is constructed based on the relationship between the displacements of the nodes at both ends of the track unit and the relationship between the forces at the nodes at both ends of the track unit.

5. The method according to claim 1, wherein The step of solving the dynamic response index of the train-track system in each degree of freedom under the track irregularity excitation power spectrum based on the transfer matrix and the overall dynamic stiffness matrix includes: determining at least one sampling frequency corresponding to the track irregularity excitation power spectrum; For any of the sampling frequencies, a track irregularity excitation signal corresponding to each wheel of the train at the sampling frequency is constructed, and based on the track irregularity excitation signal, the transfer matrix and the overall dynamic stiffness matrix, the dynamic response index of the train-track system in each degree of freedom at the sampling frequency is solved.

6. The method according to claim 5, characterized in that The step of solving the dynamic response index of the train-track system in each degree of freedom at the sampling frequency based on the track irregularity excitation signal, the transfer matrix, and the overall dynamic stiffness matrix includes: Determining, based on the track irregularity excitation signal and the transfer matrix, displacements caused by the track irregularity excitation signal at nodes on both sides of each track unit; determining a dynamic load of the train-track system based on the displacement; Based on the dynamic load and the overall dynamic stiffness matrix, a displacement matrix of the train-track system is obtained, and a dynamic response index of the train-track system in each degree of freedom at the sampling frequency is extracted from the displacement matrix.

7. A train-track coupled vibration analysis device, characterized in that: The device comprises: A first construction module is configured to divide the track in the train-track system into a plurality of track units, construct a motion differential equation for the rails in the track units in the frequency domain based on a theoretical motion differential equation for a beam structure, and solve the motion differential equation to obtain a relationship between the longitudinal displacement and frequency at the nodes at both ends of the rail, as well as a relationship between the force and frequency at the nodes at both ends of the rail; a second construction module, configured to construct a rail dynamic stiffness matrix for the rails in the track unit based on the relationship between the longitudinal displacement and the frequency, and the relationship between the force and the frequency, and to construct a track dynamic stiffness matrix for the track unit based on the rail dynamic stiffness matrix; a third construction module, configured to assemble the track dynamic stiffness matrix of each track unit and the train dynamic stiffness matrix of the train in the train-track system to obtain an overall dynamic stiffness matrix of the train-track system, and construct a transfer matrix of the track based on the track dynamic stiffness matrix of each track unit; A solution module is used to determine the track irregularity excitation power spectrum, and to solve the dynamic response index of the train-track system in each degree of freedom under the track irregularity excitation power spectrum based on the transfer matrix and the overall dynamic stiffness matrix.

8. A computer device comprising a memory and a processor, wherein the memory stores a computer program, wherein: When the processor executes the computer program, the steps of the method according to any one of claims 1 to 6 are implemented.

9. A computer-readable storage medium having a computer program stored thereon, characterized in that: When the computer program is executed by a processor, the steps of the method according to any one of claims 1 to 6 are implemented.

10. A computer program product comprising a computer program, characterized in that When the computer program is executed by a processor, the steps of the method according to any one of claims 1 to 6 are implemented.

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