A method for fast solution of transient dynamics of a bogie with cracked load-bearing structure

Abstract

The application discloses a kind of quick solution method for the transient dynamics of bogie containing crack bearing structure, belongs to bogie structure dynamic analysis technical field, it includes: obtaining the three-dimensional geometric model of bearing structure, crack geometric dimension and the position coordinate size of crack;Divide bearing structure finite element model, crack surface is carried out grid division processing;On the surface of bearing structure and crack surface, select the main degree of freedom node of finite element model;Substructure analysis is carried out to bearing structure finite element model;Obtain the mass matrix, damping matrix and stiffness matrix of bearing structure finite element model after reducing degree of freedom;Crack surface node is simulated by using one-way nonlinear stiffness unit and is constrained;Processing constraint boundary condition;Processing excitation load;Solve the second-order differential equation set of dynamics, judge whether crack surface occurs contact or not.The application can consider the opening and closing effect of crack, obtain the transient dynamics response of bearing structure containing crack under external load, and give consideration to solving precision and efficiency.

Description

Technical Field

[0001] This invention relates to a method for solving the transient dynamics of a load-bearing structure, and in particular to a method for rapidly solving the transient dynamics of a bogie with cracks, belonging to the field of bogie structural dynamic analysis technology. Background Technology

[0002] The bogie connects the car body and wheelsets, serving as a support and guide; it is a key load-bearing structure of rail vehicles. With service life, cracks can develop within the structure, reducing its load-bearing capacity and significantly impacting train safety. Crack damage introduces contact nonlinearity, drastically reducing model solution efficiency. To obtain the dynamic response of a load-bearing structure with damage under transient loads, a fast solution method or system for the transient dynamics of cracked structures needs to be developed, combining accuracy and efficiency. Currently, methods for analyzing the dynamic response of cracked load-bearing structures mainly fall into three categories: analytical methods, semi-analytical methods, and numerical methods. Analytical methods are typically suitable for simplified geometric structures and ideal material models under small deformation conditions, such as the Halpin-Tsai model, the Mori-Tanaka model, and incremental models. These methods can introduce microscopic damage mechanisms during modeling. However, analytical methods are limited by problem size and boundary conditions, resulting in a narrow range of applicability. Semi-analytical methods have certain advantages in handling nonlinear response problems. These methods characterize the micro-to-macro mechanical behavior transformation process of materials at a local scale through explicit relationships. Typical methods include the unit cell method and transformation field analysis. Compared with analytical methods, semi-analytical methods can, to some extent, account for the nonlinear deformation and complex damage evolution processes of materials. Numerical methods, on the other hand, connect the microstructure and macroscopic mechanical behavior of materials by constructing homogenized models, and are applicable to arbitrary geometries and nonlinear material properties. Common numerical methods include the unit cell model, the representative volume element method, and the fast Fourier transform method. These methods have strong expressive power in capturing the influence of microstructure evolution on overall mechanical behavior.

[0003] However, the above methods are mostly used for small-sized samples. For actual engineering problems of large-scale structures, the calculation is extremely time-consuming, which limits the application of dynamic analysis of crack-bearing structures.

[0004] Crack damage is one of the most significant failure modes in bogie structures, and its application in practical engineering remains challenging. For example, in the service of high-speed trains, effectively assessing the impact of crack damage on the overall vehicle system's dynamic safety performance is a difficult problem. Although direct damage modeling methods offer high accuracy, they are computationally extremely time-consuming, making them unsuitable for practical engineering applications within large-scale vehicle-track coupled dynamics models.

[0005] Relevant patent document CN111209620A discloses a method for predicting the residual load-bearing capacity and crack propagation path of a cracked structure based on LSTM-cGAN. In the training phase, the strength of a structure with cracks of varying degrees and its crack propagation path under loading conditions are first obtained through finite element analysis or field measurements. Based on a conditional producer-adversarial network (CAN) model and long short-term memory (LSTM) method, four deep neural networks are simultaneously trained: a generator network (G), a decision network (D), an LSTM network for processing time series data, and a convolutional neural network (CNN) for determining the strength of the cracked structure. After training, the crack propagation history measured in the field is input into the generator network G and the LSTM network to obtain the corresponding predictions of the structural strength and crack propagation path. CN120633208A discloses a method for solving the dynamic model of gear pitting based on microscopic surface morphology, including establishing a tooth surface morphology and pitting defect model; sampling the selected gear tooth surface morphology and collecting nonlinear excitation parameters inside the gear transmission system; establishing a gear tooth surface morphology mathematical model based on fractal dimension and fractal roughness; establishing a mixed elastohydrodynamic lubrication coupling analysis mathematical model of pitting gears with rough surfaces based on the relationship between the radius of curvature, dynamic load, and oil film entrainment speed with the meshing position under different surface roughness; and solving the nonlinear dynamic mathematical model of gears with rough surfaces.

[0006] While traditional finite element methods offer high accuracy for detailed modeling of cracks and defects, they are extremely time-consuming and can only handle small-sized samples, failing to address practical engineering problems involving large-scale structures. Therefore, the aforementioned techniques do not resolve the inherent trade-off between accuracy and efficiency in existing technologies. Summary of the Invention

[0007] The purpose of this invention is to provide a rapid solution method for the transient dynamics of a bogie with cracked load-bearing structure. This method can consider the opening and closing effects of cracks, obtain the transient dynamic response of the bogie with cracked load-bearing structure under external loads, evaluate the impact of the bogie with cracked load-bearing structure on the dynamic performance of the system, and transform the original structural dynamics problem with contact nonlinearity into an equivalent multibody dynamics problem with nonlinear stiffness. This method balances solution accuracy and efficiency, thus solving the problem in existing technologies where solution accuracy and efficiency cannot be simultaneously achieved.

[0008] To solve the above-mentioned technical problems, the technical solution adopted by the present invention is as follows:

[0009] A rapid solution method for transient dynamics of a bogie bearing structure with cracks (or a rapid solution method for transient dynamics of a bogie bearing structure with cracks in a rail vehicle) is provided, and its technical solution includes the following steps:

[0010] S1: Obtain the three-dimensional geometric model, material elastic model, material Poisson's ratio, material density, crack geometry, and the location coordinates of the crack in the load-bearing structure of the bogie;

[0011] S2: Divide the finite element model of the load-bearing structure and perform mesh generation on the crack surface to establish the crack surface mesh;

[0012] S3: Select the master degree of freedom nodes of the finite element model on the surface of the load-bearing structure and the crack surface;

[0013] S4: Perform substructure analysis on the finite element model of the load-bearing structure and compare it with the modal results of the original model. If the relative error of the natural frequency of the substructure analysis model exceeds the set threshold, the master degree of freedom nodes need to be readjusted until the error meets the accuracy requirements.

[0014] S5: Obtain the mass matrix, damping matrix, and stiffness matrix of the finite element model of the load-bearing structure after reducing the degrees of freedom;

[0015] S6: Constraint simulation of crack surface nodes is performed using unidirectional nonlinear stiffness elements;

[0016] S7: Handling constraint boundary conditions. If the degree of freedom of the bearing structure in a certain direction is constrained, the corresponding rows and columns are removed from the mass matrix, damping matrix, stiffness matrix and load vector to obtain a system of differential equations. The size of the system of differential equations is then compressed to further improve the model calculation efficiency.

[0017] S8: Handles excitation loads and provides boundary conditions for handling excitation loads;

[0018] S9: Solve the compressed differential equations from step S7, determine whether the crack surface is in contact, adjust the contact stiffness, and iteratively solve the dynamic equation response for each time step.

[0019] In the above technical solutions, the preferred technical solution is that, in step S2, the method of establishing the crack surface mesh is that the crack surface is composed of two layers of mesh with completely overlapping nodes, and no node fusion is performed between the nodes. This simulates the crack opening and closing effect of the unidirectional nonlinear stiffness element, which can both provide crack surface support force and prevent crack surfaces from penetrating each other.

[0020] In the above technical solution, a preferred technical solution may also include step S5 comprising:

[0021] S5.1: Using the substructure analysis method, the entire load-bearing structure is condensed into a super-element to obtain the mass matrix, damping matrix, and stiffness matrix of the main degree-of-freedom nodes; the structural dynamic equations are:

[0022] (1),

[0023] (2),

[0024] (3),

[0025] (4),

[0026] (5),

[0027] (6),

[0028] (7),

[0029] in, M , C and K These are the mass matrix, damping matrix, and stiffness matrix of the fully free-degree-of-freedom model, respectively. , and These are the acceleration matrix, velocity matrix, and displacement matrix of the fully free model, respectively. F The load matrix of the fully free model. and These are the main degree-of-freedom displacement matrices and the secondary degree-of-freedom displacement matrices of the reduced degree-of-freedom model, respectively. 、 、 、 T and are submatrices of the stiffness matrix of the fully free model. I These are the matrix transpose operator and the identity matrix, respectively. T 1 The transformation matrix between the main degree of freedom displacement and the secondary degree of freedom displacement. This is the transformation matrix between the structural full degrees of freedom and the master degrees of freedom. M m , C m and K m These are the mass matrix, damping matrix, and stiffness matrix of the reduced-degree-of-freedom model, respectively. and These are the acceleration and velocity matrices of the reduced-degree-of-freedom model, respectively. Fm To reduce the load matrix of the degree-of-freedom model.

[0030] In the above technical solution, a preferred technical solution may also be that, in step S6, the method of using unidirectional nonlinear stiffness elements to constrain the crack surface nodes is as follows:

[0031] In terms of crack contact treatment, unidirectional nonlinear stiffness elements are used to simulate the main degree-of-freedom nodes of the crack surface. When the crack contact surface closes, the stiffness elements are activated to provide compressive support, thereby preventing geometric penetration of the structure.

[0032] When the crack contact surface separates, the stiffness element fails and no longer provides the reaction force of the supporting force, simulating the unconstrained motion during the crack opening process.

[0033] In the above technical solution, a preferred technical solution may also be that, in step S7, the method for handling the constraint boundary conditions is as follows:

[0034] In terms of handling constraint boundary conditions, if the degree of freedom of the bearing structure in a certain direction is constrained, then the displacement, velocity, acceleration and external load in that direction are all zero.

[0035] By removing the corresponding rows and columns from the mass matrix, damping matrix, stiffness matrix, and load vector, the size of the differential equation system is reduced.

[0036] In the above technical solution, a preferred technical solution may also be that, in step S8, processing the excitation load and providing the boundary conditions for processing the excitation load includes:

[0037] In terms of external load handling, the excitation effect of external forces on structural nodes along specific directions is considered. The excitation load is applied to the corresponding components of the load vector in the form of time history. Dynamic excitation under any complex boundary conditions is flexibly introduced. Dynamic excitation includes impact load, harmonic excitation, and may also include measured load history.

[0038] In the above technical solution, a preferred technical solution may also be that, in step S9, solving the compressed differential equation system in step S7 is done using the Newmark-β method to solve the second-order dynamic differential equation system, with given δ and β adjustment parameters, and the introduction of virtual damping of the system to improve the stability of the solution; in step S9, the method for determining whether the crack surface has made contact, adjusting the contact stiffness, and iteratively solving the dynamic equation response for each time step is to determine whether the crack contact surface has become embedded in each step of the integral iteration, and if it has, update the load vector and derive the transient dynamic response of the displacement, velocity, and acceleration of the main nodes of the finite element model of the structure at each time step; step S9 further includes:

[0039] S9.1: Assume that during the time interval [ t , t The internal acceleration [+Δt] changes linearly, based on the fundamental assumption that...

[0040] (8),

[0041] (9),

[0042] in, , and for t Displacement, velocity, and acceleration at time t. , and They are respectively t +Δ t Displacement, velocity, and acceleration at time Δ t For time step, δ and β To adjust the parameters;

[0043] S9.2: Each step of the integration process should satisfy the following: t +Δ t The dynamic equations at time points are used to determine whether crack embedding occurs at the crack contact surface.

[0044] (10)

[0045] (11),

[0046] in, For the external load matrix, To account for the load matrix at crack surface contact, and These represent the displacements between the nodes on the crack surface. k For connection stiffness;

[0047] S9.3: Will and use , and Perform representation

[0048] (12)

[0049] (13)

[0050] S9.4: In The equivalent stiffness and load balance equations at any time are as follows:

[0051] (14)

[0052] (15)

[0053] (16)

[0054] in, and These are the equivalent stiffness matrix and the equivalent load matrix, respectively.

[0055] In the above technical solution, a preferred technical solution may also include step S9 further comprising:

[0056] S9.5: A rapid solution method for transient dynamics of a bogie with cracked load-bearing structure, implemented in a computer program format. The calculation method is as follows:

[0057] A. Initial Calculation

[0058] ①Give the mass matrix M Damping matrix C and stiffness matrix K ,

[0059] ② Provide the initial values ​​of displacement, velocity, and acceleration. , and ,

[0060] ③ Select time step ,parameter δ and β ,make

[0061] (17)

[0062] ④ Calculate the equivalent stiffness matrix (18)

[0063] B. Calculation of each time increment

[0064] ① Determine whether the crack surfaces are in contact and calculate the nodal loads.

[0065] (11),

[0066] ②Calculation The equivalent load matrix at time t.

[0067] (19)

[0068] ③ Solve Displacement at any moment

[0069] (14)

[0070] ④ Solve acceleration and velocity at any moment

[0071] (20),

[0072] (twenty one).

[0073] This invention provides a rapid solution method for the transient dynamics of a bogie bearing structure with cracks. The method includes: establishing a finite element model of the structure containing the crack, where the crack contact nonlinearity does not need to be considered; selecting principal degree-of-freedom nodes based on the modal characteristics of the bearing structure, and approximating the original model with a small number of representative nodes to reduce the degrees of freedom; performing substructure modal analysis on the reduced model and comparing the modal results with those of the original model; if the relative error of its natural frequencies exceeds a set threshold, the principal degree-of-freedom nodes need to be readjusted until the error meets the accuracy requirements. Substructure analysis was performed to derive the mass matrix, damping matrix, and stiffness matrix for subsequent dynamic solutions. A crack contact model was established, with crack surfaces connected by unidirectional stiffness elements to allow crack opening while applying a penetration penalty term to prevent crack embedding. A system of second-order differential equations with the mass matrix, damping matrix, and stiffness matrix as coefficients was constructed. Structural constraints, external loads, and contact boundary conditions were handled. The compressed differential equations were numerically solved using the Newmark-β time-domain integration method to obtain the transient dynamic responses of the structure under external excitation, including displacement, velocity, and acceleration.

[0074] The beneficial effects of adopting the technical solution of the present invention are as follows:

[0075] 1. This invention establishes a rapid solution method for transient dynamics of a bogie bearing a crack based on a coupled method of "substructure analysis + multibody dynamics solution". It replaces all nodes of the structural finite element model with a small number of master degree-of-freedom nodes and uses unidirectional nonlinear stiffness elements to simulate crack contact behavior, which can take into account the crack opening and closing effects. This technical solution transforms the original structural dynamics problem with contact nonlinearity into an equivalent multibody dynamics problem with nonlinear stiffness, which takes into account both solution accuracy and efficiency and has good embeddability and engineering applicability.

[0076] 2. This invention significantly shortens the transient dynamic solution time for crack-bearing structures. It can integrate multibody dynamics models to obtain the transient dynamic response of vehicles, engineering equipment, civil engineering structures, etc., under crack damage, and conduct risk and reliability assessments. It is applicable to the reliability assessment of crack damage in full-size engineering structures such as vehicles, engineering equipment, and civil engineering structures. Through dynamic response solutions, it can achieve comprehensive adaptation to service conditions and extreme conditions, improving the operational reliability and performance stability of engineering structures under harsh service conditions.

[0077] 3. Furthermore, the solution method of this invention changes the traditional approach to solving transient dynamics of cracked structures, providing a new technical path for risk assessment of crack damage in complex systems. This method can not only be directly applied to structural reliability assessment, but also provide theoretical and methodological support for online monitoring of structural health under service conditions for crack damage in engineering structures.

[0078] In summary, the solution method of this invention considers the opening and closing effects of cracks, obtains the transient dynamic response of the bogie with cracks under external loads, evaluates the impact of the bogie with cracks on the system's dynamic performance, and transforms the original structural dynamics problem with contact nonlinearity into an equivalent multibody dynamics problem with nonlinear stiffness. It balances solution accuracy and efficiency, solving the problem in existing technologies where solution accuracy and efficiency cannot be simultaneously achieved. Attached Figure Description

[0079] Figure 1 This is a flowchart (block diagram) of the rapid solution method (algorithm) for transient dynamics of the bogie bearing structure with cracks in this invention.

[0080] Figure 2 This is a flowchart illustrating the rapid solution method for transient dynamics of a bogie with cracked load-bearing structure in this invention.

[0081] Figure 3 This is a schematic diagram of the structure of a high-speed train bogie.

[0082] Figure 4 This is a finite element model of the side beam of a high-speed train bogie.

[0083] Figure 5 This is a time history diagram of the excitation load.

[0084] Figure 6 This is a diagram of the free modal test of the side beam of the bogie.

[0085] Figure 7 This is a comparison diagram of the finite element method and the solution method of this invention in terms of the displacement (vertical displacement) of the side beam nodes of the bogie.

[0086] Figure 8 This is a comparison diagram of the finite element method and the solution method of this invention in terms of the side beam node velocity (vertical velocity) of the bogie.

[0087] Figure 9 This is a comparison diagram of the finite element method and the solution method of this invention in terms of the acceleration (vertical acceleration) of the side beam nodes of the bogie.

[0088] Figure 10 This is a comparison diagram of the power spectrum of the side beam nodes (vertical displacement power spectrum) of the bogie using the finite element method and the solution method of this invention.

[0089] Figure 11 This is a comparison diagram of the finite element method and the solution method of this invention in terms of the velocity power spectrum (vertical velocity power spectrum) of the side beam nodes of the bogie.

[0090] Figure 12 This is a comparison diagram of the acceleration power spectrum (vertical acceleration power spectrum) of the side beam nodes of the bogie between the finite element method and the solution method of this invention.

[0091] Figure 13 This is a comparison of the acceleration (vertical acceleration) response of the side beam of the bogie with and without delaminated crack damage (no damage).

[0092] Figure 14 This is a comparison of the acceleration power spectrum (vertical acceleration power spectrum) of the side beam of the bogie with and without delaminated crack damage (no damage). Detailed Implementation

[0093] To make the objectives, technical solutions, and advantages of this invention clearer, the technical solutions of this invention will be clearly and completely described below in conjunction with embodiments. Obviously, the described embodiments are only some, not all, of the embodiments of this invention. Based on these embodiments, all other embodiments obtained by those skilled in the art without creative effort are within the scope of this invention.

[0094] Example 1: As Figure 1 , Figure 2 , Figure 3 , Figure 4 , Figure 5 , Figure 6 , Figure 7 , Figure 8 , Figure 9 , Figure 10 , Figure 11 , Figure 12 , Figure 13 , Figure 14 As shown, the rapid solution method for transient dynamics of a bogie with cracked load-bearing structure according to the present invention includes the following steps:

[0095] S1: Obtain the three-dimensional geometric model of the bogie's load-bearing structure, the material elastic model, the material Poisson's ratio, the material density, the crack geometry, and the location coordinates of the crack in the load-bearing structure.

[0096] S2: Divide the finite element model of the load-bearing structure and perform mesh generation on the crack surface to establish the crack surface mesh.

[0097] In step S2, the method for establishing the crack surface mesh is that the crack surface is composed of two layers of meshes with completely overlapping nodes. No node fusion is performed between the nodes. This simulates the crack opening and closing effects of unidirectional nonlinear stiffness elements, which can both provide support force for the crack surface and prevent the crack surfaces from penetrating each other.

[0098] S3: Select the master degree of freedom nodes of the finite element model on the surface of the load-bearing structure and the crack surface.

[0099] The original finite element model contains 9216 nodes. In this invention, 314 main nodes are selected on the surface and layers of the carbon fiber beam to reduce the structural degrees of freedom, and the model is compressed by 29.35 times.

[0100] S4: Perform substructure analysis on the finite element model of the load-bearing structure and compare it with the modal results of the original model. If the relative error of the natural frequency of the substructure analysis model exceeds the set threshold, the master degree of freedom nodes need to be readjusted until the error meets the accuracy requirements.

[0101] S5: Obtain the mass matrix, damping matrix, and stiffness matrix of the finite element model of the load-bearing structure after reducing the degrees of freedom. Step S5 further includes:

[0102] S5.1: Using the substructure analysis method, the entire load-bearing structure is condensed into a super-element to obtain the mass matrix, damping matrix, and stiffness matrix of the main degree-of-freedom nodes; the structural dynamic equations are:

[0103] (1),

[0104] (2),

[0105] (3),

[0106] (4),

[0107] (5),

[0108] (6),

[0109] (7),

[0110] in, M , C and K These are the mass matrix, damping matrix, and stiffness matrix of the fully free-degree-of-freedom model, respectively. , and These are the acceleration matrix, velocity matrix, and displacement matrix of the fully free model, respectively.F The load matrix of the fully free model. and These are the main degree-of-freedom displacement matrices and the secondary degree-of-freedom displacement matrices of the reduced degree-of-freedom model, respectively. 、 、 、 T and are submatrices of the stiffness matrix of the fully free model. I These are the matrix transpose operator and the identity matrix, respectively. T 1 The transformation matrix between the main degree of freedom displacement and the secondary degree of freedom displacement. This is the transformation matrix between the structural full degrees of freedom and the master degrees of freedom. M m , C m , K m These are the mass matrix, damping matrix, and stiffness matrix of the reduced-degree-of-freedom model, respectively. and These are the acceleration and velocity matrices of the reduced-degree-of-freedom model, respectively. Fm To reduce the load matrix of the degree-of-freedom model.

[0111] S6: Constraint simulation of crack surface nodes is performed using unidirectional nonlinear stiffness elements. The method for constraining the crack surface nodes using unidirectional nonlinear stiffness elements in step S6 is as follows:

[0112] In terms of crack contact treatment, unidirectional nonlinear stiffness elements are used to simulate the main degree-of-freedom nodes of the crack surface. When the crack contact surface closes, the stiffness elements are activated to provide compressive support, thereby preventing geometric penetration of the structure.

[0113] When the crack contact surface separates, the stiffness element fails and no longer provides the reaction force of the supporting force, simulating the unconstrained motion during the crack opening process.

[0114] S7: Handling Constraint Boundary Conditions. If the degree of freedom of the load-bearing structure in a certain direction is constrained, the corresponding rows and columns are removed from the mass matrix, damping matrix, stiffness matrix, and load vector to obtain a system of differential equations. The size of this system of differential equations is then compressed to further improve the model's computational efficiency. The method for handling constraint boundary conditions in step S7 is as follows:

[0115] In terms of handling constraint boundary conditions, if the degree of freedom of the bearing structure in a certain direction is constrained, then the displacement, velocity, acceleration and external load in that direction are all zero.

[0116] By removing the corresponding rows and columns from the mass matrix, damping matrix, stiffness matrix, and load vector, the size of the differential equation system is reduced.

[0117] S8: Process the excitation load and provide the boundary conditions for processing the excitation load. Step S8, processing the excitation load and providing the boundary conditions for processing the excitation load, includes:

[0118] In terms of external load handling, the excitation effect of external forces on structural nodes along specific directions is considered. The excitation load is applied to the corresponding components of the load vector in the form of time history. Dynamic excitation under any complex boundary conditions is flexibly introduced. Dynamic excitation includes impact load, harmonic excitation, and may also include measured load history.

[0119] S9: Solve the compressed differential equations from step S7, determine whether contact has occurred at the crack surface, adjust the contact stiffness, and iteratively solve the dynamic equation response for each time step. In step S9, solving the compressed differential equations from step S7 uses the Newmark-β method to solve the second-order dynamic differential equations. Given δ and β adjustment parameters, virtual damping of the system is introduced to improve the stability of the solution. The method for determining whether contact has occurred at the crack surface, adjusting the contact stiffness, and iteratively solving the dynamic equation response for each time step in step S9 involves determining whether the crack contact surface has become embedded at each step of the integral iteration. If embedding occurs, the load vector is updated, and the transient dynamic response of the displacement, velocity, and acceleration of the principal nodes of the finite element model of the structure at each time step is derived. Step S9 further includes:

[0120] S9.1: Assume that during the time interval [ t , t The internal acceleration [+Δt] changes linearly, based on the fundamental assumption that...

[0121] (8),

[0122] (9),

[0123] in, , and for t Displacement, velocity, and acceleration at time t. , and for t +Δ t Displacement, velocity, and acceleration at time Δ t For time step, δ and β To adjust the parameters, the solution time step is set to 2 × 10. - 5 s, δ =0.5, β =1.2.

[0124] S9.2: Each step of the integration process should satisfy the following: t +Δ t The dynamic equations at time points are used to determine whether crack embedding occurs at the crack contact surface.

[0125] (10)

[0126] (11),

[0127] in, For the external load matrix, To account for the load matrix at crack surface contact, and These represent the displacements between the nodes on the crack surface. k For connection stiffness;

[0128] S9.3: Will and use , and To represent,

[0129] (12)

[0130] (13)

[0131] S9.4: In The equivalent stiffness and load balance equations at any time are as follows:

[0132] (14)

[0133] (15)

[0134] (16)

[0135] in, and These are the equivalent stiffness matrix and the equivalent load matrix, respectively;

[0136] S9.5: The calculation method for the rapid solution of transient dynamics of a bogie with cracked load-bearing structure in a computer program implementation format is as follows:

[0137] A. Initial Calculation

[0138] ①Give the mass matrix M Damping matrix C and stiffness matrixK ,

[0139] ② Provide the initial values ​​of displacement, velocity, and acceleration. , and ,

[0140] ③ Select time step ,parameter δ and β ,make

[0141] (17)

[0142] ④ Calculate the equivalent stiffness matrix (18)

[0143] B. Calculation of each time increment

[0144] ① Determine whether the crack surfaces are in contact and calculate the nodal loads.

[0145] (11),

[0146] ②Calculation The equivalent load matrix at time t.

[0147] (19)

[0148] ③ Solve Displacement at any moment

[0149] (14)

[0150] ④ Solve acceleration and velocity at any moment

[0151] (20)

[0152] (twenty one).

[0153] A comparison is now made between the traditional finite element method and the proposed solution method:

[0154] The solution method proposed in this invention was used to model and perform dynamic simulation analysis on the side beam of a bogie with crack delamination damage, such as... Figure 3 As shown, Figure 3 This is a schematic diagram of the structure of a high-speed train bogie. Figure 3 In the attached diagram, reference numeral 1 indicates the side beam of the bogie, which is a carbon fiber side beam. The support seats at both ends of the model are subject to fully constrained boundary conditions, such as... Figure 4 As shown, Figure 4Figure 2 in the diagram represents the load F, where F = 10000[sin(10×2π× t )+ sin(20×2π× t ) + sin(30×2π× t [ ]; Figure 3 indicates full constraint; Figure 4 indicates layered structure; Figure 5 indicates master node; Figure 6 indicates response node. A dynamic load containing multi-frequency components is applied to the middle of the side beam, such as Figure 5 As shown, a through-crack is introduced at the spring support location of the side beam to simulate typical excitations experienced by the vehicle during operation, characterizing the impact of crack delamination damage on structural integrity. Simultaneously, the dynamic response of the mid-node of the side beam is extracted to analyze the influence of damage on the structural response characteristics.

[0155] First, the model was compressed in terms of degrees of freedom. The original finite element model contained 9216 nodes. 314 main nodes were selected on the surface of the carbon fiber side beam and the sub-layers to reduce the structural degrees of freedom. The model was compressed by 29.35 times.

[0156] Then, a comparative analysis of simulated and experimental modal characteristics was conducted. The free modes of the bogie's side beams were tested using the impact test method, such as... Figure 6 As shown, Figure 6 In the attached figures, 7 represents a fixed point, 8 represents a flexible rope, and 9 represents a hammer. Hammer 9 was used to strike the object, and a modal comparison was made with the reduced-degree-of-freedom model. See Table 1 for the comparison results between the experimental and simulated modes.

[0157] Table 1

[0158]

[0159] Table 1 shows that the constructed degree-of-freedom reduction model is consistent with the experimental data in terms of modal frequencies, with a maximum relative error of 6.59%. This result verifies that the degree-of-freedom compression strategy based on substructure modeling has good accuracy and provides a foundation for subsequent large-scale coupled dynamics simulations.

[0160] The dynamic response of the mid-node of the carbon fiber side beam under dynamic load was compared and analyzed using both the finite element method and the proposed degree-of-freedom reduction method. The calculation results include the time histories of displacement, velocity, and acceleration of the node, as well as its power spectral density function, such as... Figure 7 , Figure 8 , Figure 9 , Figure 10 , Figure 11 , Figure 12 As shown in the figure. The results show that the time-domain response waveforms of nodal displacement, velocity, and acceleration are highly consistent, and the frequency position and energy distribution of the power spectrum are consistent with the finite element calculation results.

[0161] In terms of computational efficiency, the finite element method takes 165,200 seconds, while the reduced-degree-of-freedom model constructed in this paper only takes 1,353 seconds, improving computational efficiency by 122 times.

[0162] A dynamic analysis was performed on the bogie side beams to determine whether they contained crack delamination damage, such as... Figure 13 , Figure 14 As shown, for a side beam without delamination damage (no damage), due to its constant overall stiffness, the system exhibits linear characteristics and produces a stable harmonic response under harmonic excitation load. However, when delamination damage exists in the structure, the crack opening and closing process causes periodic changes in the system stiffness, leading to a nonlinear evolution of the dynamic response. When the crack opens, the structural stiffness decreases, the system is in a softened state, and the response amplitude increases significantly; while when the crack closes, the system returns to a rigid state, and the response is suppressed. This stiffness modulation effect induces a nonlinear modulation of the original harmonic excitation load, resulting in additional harmonic components in the frequency spectrum. This provides theoretical and methodological support for online monitoring of structural health under service conditions for crack damage in engineering structures.

[0163] In summary, the solution method of this invention considers the opening and closing effects of cracks, obtains the transient dynamic response of the bogie with cracks under external loads, evaluates the impact of the bogie with cracks on the system's dynamic performance, and transforms the original structural dynamics problem with contact nonlinearity into an equivalent multibody dynamics problem with nonlinear stiffness. It balances solution accuracy and efficiency, solving the problem in existing technologies where solution accuracy and efficiency cannot be simultaneously achieved.

Claims

1. A method for rapid solution of transient dynamics of a bogie bearing structure with cracks, characterized in that... It includes the following steps: S1: Obtain the three-dimensional geometric model, material elastic model, material Poisson's ratio, material density, crack geometry, and the location coordinates of the crack in the load-bearing structure of the bogie; S2: Divide the finite element model of the load-bearing structure and perform mesh generation on the crack surface to establish the crack surface mesh; The method for establishing the crack surface mesh is that the crack surface is composed of two layers of meshes with completely overlapping nodes, without node fusion, to simulate the crack opening and closing effects of unidirectional nonlinear stiffness elements. S3: Select the master degree of freedom nodes of the finite element model on the surface of the load-bearing structure and the crack surface; S4: Perform substructure analysis on the finite element model of the load-bearing structure and compare it with the modal results of the original model. If the relative error of the natural frequency of the substructure analysis model exceeds the set threshold, the master degree of freedom nodes need to be readjusted until the error meets the accuracy requirements. S5: Obtain the mass matrix, damping matrix, and stiffness matrix of the finite element model of the load-bearing structure after reducing the degrees of freedom; S6: Constraint simulation of crack surface nodes is performed using unidirectional nonlinear stiffness elements; The method for constraining and simulating crack surface nodes using uniaxial nonlinear stiffness elements is as follows: In terms of crack contact treatment, unidirectional nonlinear stiffness elements are used to simulate the main degree-of-freedom nodes of the crack surface. When the crack contact surface closes, the stiffness elements are activated to provide compressive support. When the crack contact surface separates, the stiffness element fails and no longer provides the reaction force of the supporting force, simulating the unconstrained motion during the crack opening process; S7: Handling constraint boundary conditions. If the degree of freedom of the bearing structure in a certain direction is constrained, the corresponding rows and columns are removed from the mass matrix, damping matrix, stiffness matrix and load vector to obtain a system of differential equations, and the size of the system of differential equations is reduced. S8: Handles excitation loads and provides boundary conditions for handling excitation loads; S9: Solve the compressed differential equations from step S7, determine whether the crack surface is in contact, adjust the contact stiffness, and iteratively solve the dynamic equation response for each time step.

2. The rapid solution method for transient dynamics of a bogie with cracked load-bearing structure according to claim 1, characterized in that... Step S5 includes: S5.1: Using the substructure analysis method, the entire load-bearing structure is condensed into a super-element to obtain the mass matrix, damping matrix, and stiffness matrix of the main degree-of-freedom nodes; the structural dynamic equation is: （1）， （2）， （3）， （4）， （5）， （6）， （7）， in, M, C and K These are the mass matrix, damping matrix, and stiffness matrix of the fully free-degree-of-freedom model, respectively. , and These are the acceleration matrix, velocity matrix, and displacement matrix of the fully free model, respectively. F The load matrix of the fully free model. and These are the main degree-of-freedom displacement matrices and the secondary degree-of-freedom displacement matrices of the reduced degree-of-freedom model, respectively. , , , T and are submatrices of the stiffness matrix of the fully free model. I These are the matrix transpose operator and the identity matrix, respectively. T 1 The transformation matrix between the main degree of freedom displacement and the secondary degree of freedom displacement. This is the transformation matrix between the structural full degrees of freedom and the master degrees of freedom. M m , C m and K m These are the mass matrix, damping matrix, and stiffness matrix of the reduced-degree-of-freedom model, respectively. and These are the acceleration and velocity matrices of the reduced-degree-of-freedom model, respectively. F m To reduce the load matrix of the degree-of-freedom model.

3. The rapid solution method for transient dynamics of a bogie with cracked load-bearing structure according to claim 1, characterized in that... In step S7, the method for handling the constraint boundary conditions is as follows: In terms of handling constraint boundary conditions, if the degree of freedom of the bearing structure in a certain direction is constrained, then the displacement, velocity, acceleration and external load in that direction are all zero. By removing the corresponding rows and columns from the mass matrix, damping matrix, stiffness matrix, and load vector, the size of the differential equation system is reduced.

4. The rapid solution method for transient dynamics of a bogie with cracked load-bearing structure according to claim 1, characterized in that... In step S8, processing the excitation load and providing the boundary conditions for processing the excitation load include: In terms of external load handling, the excitation effect of external forces on structural nodes along specific directions is considered. The excitation load is applied to the corresponding components of the load vector in the form of a time history, introducing dynamic excitation under any complex boundary conditions. Dynamic excitation includes impact load and harmonic excitation.

5. The rapid solution method for transient dynamics of a bogie with cracked load-bearing structure according to claim 2, characterized in that... In step S9, the compressed differential equations from step S7 are solved using the Newmark-β method to solve the second-order differential equations of dynamics. and Adjusting parameters and introducing virtual damping of the system; In step S9, the method of determining whether the crack surface has contact, adjusting the contact stiffness, and iteratively solving the dynamic equation response for each time step is to determine whether the crack contact surface has embedded in each step of the integral iteration. If it has embedded in, the load vector is updated, and the displacement, velocity, and acceleration transient dynamic response of the main node of the finite element model of the structure at each time step is derived. Step S9 includes: S9.1: Assume that during the time interval [ t , t +Δ t The internal acceleration changes linearly, based on the fundamental assumption that... （8）， （9）， in, , and for t Displacement, velocity, and acceleration at time t. , , for Displacement, velocity, and acceleration at time t. For time step, and To adjust the parameters; S9.2: Each step of the integration process should satisfy the following: The dynamic equations at time points are used to determine whether crack embedding occurs at the crack contact surface. （10）， （11）， in, For the external load matrix, To account for the load matrix at crack surface contact, and These represent the displacements between the nodes on the crack surface. k For connection stiffness; S9.3: Will and use , and To express, （12）， （13）， S9.4: In The equivalent stiffness and load balance equations at any time are as follows: （14）, (15) (16)， in, and These are the equivalent stiffness matrix and the equivalent load matrix, respectively.

6. The rapid solution method for transient dynamics of a bogie with cracked load-bearing structure according to claim 5, characterized in that... Step S9 also includes: S9.5: A rapid solution method for transient dynamics of a bogie with cracked load-bearing structure, implemented in a computer program format. The calculation method is as follows: A. Initial Calculation ①Give the mass matrix M Damping matrix C and stiffness matrix K , ② Provide the initial values ​​of displacement, velocity, and acceleration. , and , ③ Select time step ,parameter and ,make （17）， ④ Calculate the equivalent stiffness matrix (18); B. Calculation of each time increment ① Determine whether the crack surfaces are in contact and calculate the nodal loads. （11）， ②Calculation The equivalent load matrix at time t. （19）， ③ Solve Displacement at any moment （14）， ④ Solve acceleration and velocity at any moment （20）， （21）。

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