Dynamic Dimensionality Reduction Method for Rail-Sleeper / Track Slab-Bridge System Model

Abstract

The present invention discloses a dynamic dimensionality reduction method for a rail-sleeper / track slab-bridge system model, comprising: S1, splitting the rail-sleeper / track slab-bridge system into three substructures: a rail structure, a sleeper structure / track slab structure, and a bridge structure, and constructing finite element theoretical models of the three substructures respectively; S2, performing dimensionality reduction on the bridge structure finite element theoretical model to obtain a finite element theoretical model of the bridge structure after dimensionality reduction; S3, assembling and simplifying the rail structure finite element theoretical model, the sleeper structure / track slab structure finite element theoretical model, and the finite element theoretical model of the bridge structure after dimensionality reduction to obtain a dynamic dimensionality reduction model of the rail-sleeper / track slab-bridge system; S4, calculating the actual response of the rail-sleeper / track slab-bridge system. This method is suitable for dynamic dimensionality reduction of a rail-sleeper / track slab-bridge three-layer structural system model, significantly improving dimensionality reduction effect and computational efficiency while ensuring accuracy.

Description

Technical Field

[0001] The present invention relates to the technical field of railway engineering structure dynamic modeling, and in particular to a dynamic dimension reduction method for a rail-sleeper / track slab-bridge system model. Background Art

[0002] Railway bridge-track systems consist of bridge and track structures, with the track structure further divided into rails, connectors, sleepers / track slabs, and roadbed layers. The most commonly used methods for dynamic modeling of existing railway bridge-track systems include the finite element method and theoretical methods. The finite element method is widely used due to its versatility. There are also some relatively mature commercial finite element software packages available on the market that can be used to model bridge-track systems with a very high degree of sophistication. However, as the scale and structural complexity of bridge-track systems increase, the efficiency of calculations using the finite element method or software modeling methods decreases significantly, and the performance requirements for computers become more stringent. Therefore, experts and scholars from various countries are considering applying some model dynamic dimensionality reduction methods, such as modal superposition and dynamic substructuring, to the dynamic dimensionality reduction modeling of bridge-track systems. The advantage of this modeling method is that it can characterize the real physical characteristics of the system through low-order modes, thereby greatly reducing the model dimension. In particular, the efficiency of parameter analysis of this type of model is significantly better than that of the finite element model. However, this method requires that each substructure must be linear, and the high-frequency vibration components of the substructure have little impact on the overall structure. Therefore, generally only the bridge structure is reduced in dimension, and the track structure is modeled in its entirety.

[0003] Typically, if only the vertical dynamic response of the bridge and track structure is of interest, and if good contact between the track and bridge is assumed, the entire track structure can be considered a linear structure. In this case, the free interface modal synthesis method can be used to establish a dynamic dimension reduction model of the bridge-track system. However, when analyzing the longitudinal dynamic response of the bridge-track system, or considering the dynamic response of a ballastless track with the track slab debonded, the track structure cannot be considered as a linear entity. It must be split into a three-layer structure system of rails, sleepers, and track slabs, based on the nonlinear connection locations. The free interface modal synthesis method can then be used to reduce the dimension of the dynamic model.

[0004] Although the free interface modal synthesis method can theoretically be used to reduce the dynamic dimensionality of multi-substructure connected system models, existing research involving the free interface modal synthesis method's theoretical derivation and application scenarios are generally demonstrated using cases involving two connected substructures. Few studies have derived theoretical formulas suitable for connecting three or more substructures. As a result, its application typically requires only the entire system to be decomposed into two substructures, and nonlinear elastic connections cannot exist within the two substructures. This is disadvantageous for three-layer structural systems such as rails, sleepers, track slabs, and bridges, because the rails are connected to the sleepers / track slabs via fasteners, and the sleepers / track slabs are connected to the bridge via the roadbed layer. Nonlinear elastic connection conditions may exist between these connecting layers, such as the longitudinal resistance of the fasteners and the vertical compression but not tension characteristics of the isolation layer. These are often simulated using nonlinear spring elements, and the three-layer structural system cannot be further simplified to a two-layer structure.

[0005] Therefore, it is urgent to develop a dynamic dimension reduction method suitable for the three-layer structural system model of rail-sleeper / track slab-bridge. Summary of the Invention

[0006] In response to the shortcomings of the existing technology, the present invention provides a dynamic dimension reduction method for the rail-sleeper / track slab-bridge system model, which can accurately and efficiently calculate and analyze the dynamic response of the three-layer structure of rails, sleepers / track slabs and bridges.

[0007] To this end, the present invention adopts the following technical solutions:

[0008] A dynamic dimension reduction method for a rail-sleeper / track slab-bridge system model comprises the following steps:

[0009] S1, split the rail-sleeper / track slab-bridge system into three substructures: rail structure, sleeper structure / track slab structure, and bridge structure; obtain the mass matrix, damping matrix, and stiffness matrix of each substructure respectively and construct the finite element theoretical model of the rail structure, the finite element theoretical model of the sleeper structure / track slab structure, and the finite element theoretical model of the bridge structure;

[0010] S2, the acceleration of the bridge structure ,speed and displacement respectively expressed in generalized coordinates, substituting the generalized coordinates into the bridge structure finite element theoretical model for dimension reduction to obtain a finite element theoretical model of the bridge structure after dimension reduction;

[0011] S3, deriving a conversion relationship between the actual response of each substructure and the response under the generalized coordinates after dimensionality reduction; then assembling the finite element theoretical model of the rail structure, the finite element theoretical model of the sleeper structure / track slab structure, and the finite element theoretical model of the bridge structure after dimensionality reduction and simplifying them using the conversion relationship to obtain a dynamic dimensionality reduction model of the rail-sleeper / track slab-bridge system;

[0012] S4, solving the response of the dynamic dimension reduction model of the rail-sleeper / track slab-bridge system in generalized coordinates by a numerical method, and then calculating the actual response of the rail-sleeper / track slab-bridge system by the coordinate transformation relationship.

[0013] In the above step S1:

[0014] The physical parameters of the rail structure, sleeper structure / track slab structure, and bridge structure are input into the finite element analysis software to calculate the mass matrix, damping matrix, and stiffness matrix of each substructure.

[0015] The finite element theoretical model of the rail structure is as follows:

[0016] (1)

[0017] Where, 、 and are the mass matrix, damping matrix and stiffness matrix of the rail structure respectively, 、 and are the acceleration, velocity and displacement of the rail structure, is the internal force vector acting on the connection node of the rail structure, is the transpose of the identity matrix of the position of the degree of freedom of the connection node in the rail structure. The unit length of the rail structure is consistent with the spacing between the fasteners so that is a non-singular square matrix, is the external excitation vector of the rail structure;

[0018] The finite element theoretical model of the sleeper structure / track plate structure is as follows:

[0019] (2)

[0020] Where, 、 and are the mass matrix, damping matrix and stiffness matrix of the sleeper structure / track plate structure respectively, 、 and are the acceleration, velocity and displacement of the sleeper structure / track plate structure respectively, is the internal force vector acting on the connection node of the sleeper structure / track slab structure, is the transpose of the identity matrix of the position of the degrees of freedom of the connection nodes in the sleeper structure / track slab structure, is the external excitation vector of the sleeper structure / track slab structure;

[0021] The finite element theoretical model of the bridge structure is as follows:

[0022] (3)

[0023] Where, 、 and are the mass matrix, damping matrix and stiffness matrix of the bridge structure respectively, 、 and are the acceleration, velocity and displacement of the bridge structure, is the internal force vector acting on the connection node of the bridge structure, is the transpose of the identity matrix of the position of the degree of freedom of the connection node in the bridge structure, is the external excitation vector of the bridge structure.

[0024] In the above method, S2 includes the following steps:

[0025] S2-1, using the free interface modal synthesis method to convert the acceleration of the bridge structure ,speed and displacement Expanded in generalized coordinates, they are expressed as:

[0026] (4)

[0027] (5)

[0028] (6)

[0029] Where, is the main mode of the bridge structure, the main mode include order natural frequency, Is the main mode The corresponding generalized coordinates are, and They are Second and first derivatives with respect to time; is the inertial attached mode of the bridge structure. Since the bridge structure does not contain rigid body modes, the inertial attached mode The calculation formula is: ; It is the habitual mode The corresponding generalized coordinates are, and They are Second and first derivatives with respect to time;

[0030] S2-2, replace formula (4) Substituting formula (6) into formula (3), we can obtain the finite element theoretical model of the bridge structure after dimension reduction:

[0031] (7)

[0032] in:

[0033] ;

[0034] ;

[0035] ;

[0036] Where, 、 and are the mass matrix, damping matrix and stiffness matrix of the bridge structure in generalized coordinates respectively.

[0037] In the above method, S3 includes the following steps:

[0038] S3-1, based on the interlayer elastic connection relationship between two adjacent substructures in the rail-sleeper / track slab-bridge system, the coordination relationship between the connection node displacements and the relationship between the connection interface forces of the two adjacent substructures are obtained:

[0039] (8)

[0040] (9)

[0041] Where, is the displacement of the connection node of the sleeper structure / track slab structure; is the displacement of the node on the sleeper structure / track slab structure connected to the rail structure; is the displacement of the nodes on the sleeper structure / track slab structure connected to the bridge structure; is the displacement of the connection node of the rail structure; is the displacement of the connection node of the bridge structure; is the relative displacement difference vector of the connection node between the sleeper structure / track plate structure and the rail structure; is the relative displacement difference vector of the connection nodes between the sleeper structure / track plate structure and the bridge structure; is the node force at the connection interface of the sleeper structure / track plate structure; The node force connecting the sleeper structure / track plate structure with the rail structure; The node force connecting the sleeper structure / track slab structure with the bridge structure; is the nodal force at the connection interface of the rail structure; is the nodal force at the connection interface of the bridge structure;

[0042] The following transformation relationship is obtained based on the displacement of the connection node of each substructure and the displacement of each substructure:

[0043] (10)

[0044] (11)

[0045] (12)

[0046] Where, is the identification matrix of the positions of the degrees of freedom of the connection nodes in the rail structure; is the identification matrix of the position of the degrees of freedom of the connection nodes in the bridge structure; is the identification matrix of the positions of the degrees of freedom of the connection nodes in the sleeper structure; for The identity matrix of the positions of the degrees of freedom associated with the rail structure in , for The identity matrix of the positions of the degrees of freedom associated with the bridge structure;

[0047] Substituting formula (6) and formula (10) to (12) into formula (8), we can obtain the explicit expression of the coordination relationship between the displacements of the connection nodes of two adjacent substructures:

[0048] (13)

[0049] Formula (13) can be further simplified by and express ,pass 、 and express , and obtain the simplified explicit expression:

[0050] (14)

[0051] (15)

[0052] S3-2, according to formula (14), (15) and formula (6), the displacements of the three substructures are obtained and generalized coordinates and generalized coordinates The coordinate transformation relationship between them is:

[0053] (16)

[0054] Where, and are the displacements of the three substructures and the transformation matrices between the two generalized coordinates, respectively:

[0055] , (17)

[0056] Where, is the identity matrix;

[0057] S3-3, assemble the finite element theoretical model of the rail structure, the finite element theoretical model of the sleeper structure / track plate structure, and the finite element theoretical model of the bridge structure after dimension reduction to obtain:

[0058] (18)

[0059] Substitute formula (16) into formula (18) and multiply both sides of the equation by Combined with formula (9), the dynamic dimension reduction model of the rail-sleeper / track plate-bridge system is obtained:

[0060] (19)

[0061] in,

[0062] , ,

[0063] Where, is the linear spring stiffness between the rail structure and the sleeper structure / track plate structure, is the linear spring stiffness between the sleeper structure / track slab structure and the bridge structure; for The first derivative with respect to time, The general form of nonlinear internal forces between rail structure and sleeper structure / track slab structure; for The first derivative with respect to time, A general form for expressing the nonlinear internal forces between the sleeper structure / track slab structure and the bridge structure.

[0064] In the above step S4, the calculation formula for the actual response of the dynamic dimension reduction model of the rail-sleeper / track slab-bridge system is:

[0065] (20)

[0066] (twenty one)

[0067] (twenty two).

[0068] Compared with the prior art, the present invention has the following beneficial effects:

[0069] 1. Based on the free interface modal method, the present invention derives the theoretical formula for system dynamic dimensionality reduction when three substructures, namely the rail structure, the sleeper structure / track plate structure, and the bridge structure, are connected. The formula is successfully applied to the dynamic dimensionality reduction of the refined model of the rail-sleeper / track plate-bridge system, expanding the application scenarios of the free interface modal method. At the same time, it also provides a new approach for the refined modeling and dynamic dimensionality reduction calculation and analysis of the bridge-track system.

[0070] 2. Compared with the existing finite element model, the dynamic dimension reduction method of the rail-sleeper / track plate-bridge system of the present invention significantly improves the dimension reduction effect and calculation efficiency while ensuring accuracy.

[0071] 3. The dynamic dimension reduction model of the rail-sleeper / track slab-bridge system established by the present invention can better analyze the dynamic characteristics of the system when a single-frequency harmonic external excitation is applied, and the actual response calculated has a high accuracy. This method is also applicable to analyzing the situation when other types of external excitations are applied to the rail-sleeper / track slab-bridge system.

[0072] 4. The dynamic dimensionality reduction method of the present invention is applicable to various railway bridges with ballastless and ballasted tracks. By using this dynamic dimensionality reduction method, a spatially refined model of the rail-sleeper / track slab-bridge system is established. This method not only takes into account the linear or nonlinear connection characteristics between track layers, but also analyzes the dynamic response of each degree of freedom in the three-dimensional space of the rail-sleeper / track slab-bridge system, which is conducive to its application and promotion in actual engineering. BRIEF DESCRIPTION OF THE DRAWINGS

[0073] Figure 1 It is a structural diagram of the rail-sleeper-bridge system;

[0074] Figure 2 The displacement response curves of the finite element model established by commercial software and the dynamic dimension reduction model established by the method of the present invention at the same node in the embodiment of the present invention are as follows;

[0075] Figure 3 The acceleration response curve at the same node of the finite element model and the refined dimensionality reduction model in the embodiment of the present invention;

[0076] Figure 4 for Figure 2 and Figure 3Fourier transform curves of displacement response and acceleration response of the refined dimensionality reduction model.

[0077] In the picture: 1. Rails; 2. Sleepers; 3. Bridge. DETAILED DESCRIPTION

[0078] The technical solution of the present invention is described in detail below with reference to the accompanying drawings and embodiments.

[0079] In this invention, the rail-sleeper / track slab-bridge system is divided into the rail structure, the sleeper / track slab structure, and the bridge structure. In ballasted track, the rails and sleepers are connected by fasteners, and the sleepers and bridge are connected by ballast. In ballastless track, the rails and track slab are connected by fasteners, and the track slab and bridge are connected by CA mortar or self-compacting concrete infill. In the rail-sleeper / track slab-bridge system, the interaction between any two connected structures is elastic.

[0080] Example 1

[0081] This embodiment takes ballasted track as an example to illustrate the dynamic dimension reduction method of its rail-sleeper-bridge system model. The structure of the rail-sleeper-bridge system is as follows: Figure 1 As shown, it includes rails 1, sleepers 2 and bridge 3. In this embodiment, the two rails 1 are both 30 m long seamless rails, the sleepers 2 include 51 2.5 m long concrete sleepers, and the bridge 3 is a 30 m span concrete single-hole simply supported box girder with a fastener spacing of 0.6 m. The rails and sleepers are connected by 51 pairs of fasteners, and the sleepers and bridge are connected by ballast. The effects of the fasteners and ballast are simulated by spring damping units. The rails, sleepers and bridge are all simulated by Timoshenko beams. A single-frequency harmonic external excitation is applied vertically (in the Z-axis direction) at the end of a seamless rail. , The unit of is Newton (N), t is time, and the unit of t is second (s).

[0082] The dynamic dimension reduction method of the rail-sleeper-bridge system model of this embodiment includes the following steps:

[0083] S1, split the rail-sleeper-bridge system into three substructures, namely: rail structure, sleeper structure and bridge structure. First, obtain the mass matrix, damping matrix and stiffness matrix of each substructure, and then construct the finite element theoretical model of each substructure separately, namely: rail structure finite element theoretical model, sleeper structure finite element theoretical model and bridge structure finite element theoretical model.

[0084] Specifically, the physical parameters of the above three substructures are input into the finite element analysis software, and the mass matrix, damping matrix and stiffness matrix corresponding to the three substructures are calculated; where:

[0085] The physical parameters of the rail structure include: rail elastic modulus, rail density, rail Poisson's ratio and rail cross-sectional area;

[0086] The physical parameters of the sleeper structure include: sleeper elastic modulus, sleeper density, sleeper Poisson's ratio and sleeper cross-sectional area;

[0087] The physical parameters of bridge structures include: bridge elastic modulus, bridge density, bridge Poisson's ratio and bridge cross-sectional area.

[0088] S1-2, based on the mass matrix, damping matrix and stiffness matrix of the three substructures of rail structure, sleeper structure and bridge structure, establish the finite element theoretical model of rail structure, the finite element theoretical model of sleeper structure and the finite element theoretical model of bridge structure respectively. Among them:

[0089] The finite element theoretical model of rail structure is as follows:

[0090] (1)

[0091] Where, 、 and are the mass matrix, damping matrix and stiffness matrix of the rail structure respectively; 、 and are the acceleration, velocity and displacement of the rail structure respectively; is the internal force vector acting on the connection node of the rail structure; is the identification matrix of the position of the degree of freedom of the connection node in the rail structure; the unit length of the rail structure is consistent with the fastener spacing, so that is a non-singular square matrix, is the external excitation vector of the rail structure.

[0092] The finite element theoretical model of the sleeper structure is as follows:

[0093] (2)

[0094] Where, 、 and are the mass matrix, damping matrix and stiffness matrix of the sleeper structure respectively; 、 and are the acceleration, velocity and displacement of the sleeper structure respectively; is the internal force vector acting on the connection node of the sleeper structure; is the identification matrix of the position of the degrees of freedom of the connection nodes in the sleeper structure; is the external excitation vector of the sleeper structure.

[0095] The finite element theoretical model of the bridge structure is as follows:

[0096] (3)

[0097] Where, 、 and are the mass matrix, damping matrix and stiffness matrix of the bridge structure respectively; 、 and are the acceleration, velocity and displacement of the bridge structure respectively; is the internal force vector acting on the connection node of the bridge structure; is the identification matrix of the position of the degrees of freedom of the connection nodes in the bridge structure; is the external excitation vector of the bridge structure.

[0098] In this embodiment, the damping matrix of the rail structure , the damping matrix of the sleeper structure and the damping matrix of the bridge structure Rayleigh damping is used; external excitation vector , external excitation vector is 0 vector, external excitation vector When establishing the finite element theoretical model of the rail structure, the finite element theoretical model of the sleeper structure, and the finite element theoretical model of the bridge structure, the connection nodes in each substructure are numbered regularly to facilitate the generation of the identification matrix.

[0099] S2, the acceleration of the bridge structure ,speed and displacement The generalized coordinates are respectively expressed, and the generalized coordinates are substituted into the bridge structure finite element theoretical model for dimensionality reduction to obtain the bridge structure finite element theoretical model after dimensionality reduction. The specific steps are as follows:

[0100] S2-1, using the free interface modal synthesis method to convert the acceleration of the bridge structure ,speed and displacement Expand them respectively in generalized coordinates to obtain their expansions:

[0101] (4)

[0102] (5)

[0103] (6)

[0104] Where, is the main mode of the bridge structure, include order natural frequency; Is the main mode The corresponding generalized coordinates; and They are Second and first derivatives with respect to time; is the inertial attached mode of the bridge structure. Since the bridge structure does not contain rigid body modes, the inertial attached mode The calculation formula is: ; It is the habitual mode The corresponding generalized coordinates are, and They are Second and first derivatives with respect to time.

[0105] Main modes of bridge structures The order of The value of is generally much smaller than the degree of freedom of the finite element theoretical model of the bridge structure, thereby achieving dimensionality reduction of the bridge structure. =5.

[0106] S2-2, replace formula (4) Substituting formula (6) into formula (3), we can obtain the finite element theoretical model of the bridge structure after dimension reduction:

[0107] (7)

[0108] in:

[0109] ;

[0110] ;

[0111] ;

[0112] Where, 、 and are the mass matrix, damping matrix and stiffness matrix of the bridge structure in generalized coordinates respectively;

[0113] S3: Derive the conversion relationship between the actual response of each substructure (displacement, velocity, and acceleration in physical coordinates) and the response in generalized coordinates after dimensionality reduction. Then, assemble the finite element theoretical model of the rail structure, the finite element theoretical model of the sleeper structure, and the finite element theoretical model of the bridge structure after dimensionality reduction and simplify them using the conversion relationship to obtain a dynamic dimensionality reduction model of the rail-sleeper-bridge system. The specific steps are as follows:

[0114] S3-1, based on the interlayer elastic connection relationship between two adjacent substructures in the rail-sleeper-bridge system, the coordination relationship between the displacements of the connection nodes of the two adjacent substructures and the relationship between the connection interface forces are obtained; based on the displacement of the connection nodes of each substructure and the displacement of each substructure, a conversion relationship between the two displacements is obtained; this conversion relationship and the expansion of the displacement of the bridge structure obtained in S2-1 in generalized coordinates are substituted into the coordination relationship and the relationship between the connection interface forces to obtain an explicit expression for the coordination relationship between the displacements of the connection nodes of the two adjacent substructures, and then a simplified explicit expression is obtained. Specifically as follows:

[0115] The coordination relationship between the displacements of the connection nodes of two adjacent substructures and the relationship between the connection interface forces is:

[0116] (8)

[0117] (9)

[0118] Where, is the displacement of the connection node of the sleeper structure; is the displacement of the node on the sleeper structure connected to the rail structure; is the displacement of the node on the sleeper structure connected to the bridge structure; is the displacement of the connection node of the rail structure; is the displacement of the connection node of the bridge structure; is the relative displacement difference vector of the connection node between the sleeper structure and the rail structure; is the relative displacement difference vector of the connection node between the sleeper structure and the bridge structure; is the nodal force at the connection interface of the sleeper structure; is the node force connecting the sleeper structure and the rail structure; is the node force connecting the sleeper structure and the bridge structure; is the nodal force at the connection interface of the rail structure; is the nodal force at the connection interface of the bridge structure;

[0119] According to the displacement of the connection node of each substructure and the displacement of each substructure, the conversion relationship between the two displacements is obtained:

[0120] (10)

[0121] (11)

[0122] (12)

[0123] Where: is the identification matrix of the positions of the degrees of freedom of the connection nodes in the rail structure; is the identification matrix of the position of the degrees of freedom of the connection nodes in the bridge structure; is the identification matrix of the positions of the degrees of freedom of the connection nodes in the sleeper structure; for The identity matrix of the positions of the degrees of freedom associated with the rail structure in , for The identity matrix of the positions of the degrees of freedom associated with the bridge structure;

[0124] Substituting formula (6) and formula (10) to (12) into formula (8), we can obtain the explicit expression of the coordination relationship between the displacements of the connection nodes of two adjacent substructures:

[0125] (13)

[0126] Formula (13) can be further simplified by and express ,pass 、 and express , and obtain the simplified explicit expression:

[0127] (14)

[0128] (15)

[0129] Where, ;

[0130] S3-2, according to formula (14), (15) and formula (6), the displacements of the three substructures are obtained and generalized coordinates and generalized coordinates The coordinate transformation relationship between:

[0131] (16)

[0132] Where, and are the displacements of the three substructures and the transformation matrices between the two generalized coordinates, respectively:

[0133] ,

[0134] (17)

[0135] Where, is the identity matrix;

[0136] S3-3, assemble the rail structure finite element model, the sleeper structure finite element model and the finite element theoretical model of the bridge structure after dimensionality reduction, that is, assemble formula (1), formula (2) and formula (7) and merge them into a matrix form, which is:

[0137] (18)

[0138] Substitute formula (16) into formula (18) and multiply both sides of the equation by Combined with formula (9), the dynamic dimension reduction model of the rail-sleeper-bridge system is obtained:

[0139] (19)

[0140] in,

[0141] , ,

[0142] Where, is the linear spring stiffness between the rail structure and the sleeper structure, is the linear spring stiffness between the sleeper structure and the bridge structure; for The first derivative with respect to time, The general form of nonlinear internal forces between rail structure and sleeper structure; for The first derivative with respect to time, The general form of nonlinear internal forces between the sleeper structure and the bridge structure.

[0143] S4. Calculate the actual response of the rail-sleeper-bridge system: The response of the dynamic dimension reduction model of the rail-sleeper-bridge system in generalized coordinates is obtained by numerically solving formula (19). Then, the actual response of the dynamic dimension reduction model of the rail-sleeper-bridge system is calculated by combining the coordinate transformation relationship of formula (16), that is, the response in the physical coordinate system, as shown in formulas (19), (20) and (21):

[0144] (20)

[0145] (twenty one)

[0146] (twenty two).

[0147] In this embodiment, the physical parameters of the rail-sleeper-bridge system are shown in Table 1. Modal analysis is performed on the dynamic dimension reduction model of the rail-sleeper-bridge system to obtain the natural frequencies of each order.

[0148] Table 1

[0149] Physical parameters Numerical Rail elastic modulus <![CDATA[2.06×10 11 Well]]> Rail density <![CDATA[7850 kg / m 3 ]]> Poisson's ratio of rail 0.3 Rail cross-sectional area <![CDATA[0.0077 m 2 ]]> Sleeper elastic modulus <![CDATA[3.45×10 10 Well]]> Sleeper density <![CDATA[2600 kg / m 3 ]]> Poisson's ratio of sleeper 0.2 Sleeper cross-sectional area <![CDATA[0.042 m 2 ]]> Bridge elastic modulus <![CDATA[3.45×10 10 Well]]> Bridge density <![CDATA[2420 kg / m 3 ]]> Poisson's ratio of bridge 0.2 Bridge cross-sectional area <![CDATA[8.98 m 2 ]]> Fastener X-direction stiffness <![CDATA[4.7×10 6 N / m]]> Fastener Y-direction stiffness <![CDATA[5×10 7 N / m]]> Fastener Z-direction stiffness <![CDATA[3.5×10 7 N / m]]> Fastener RX direction stiffness <h2 style=";text-align:left;direction:ltr"><![CDATA[5×10 <h2 style=";text-align:left;direction:ltr"> 10 <h2 style=";text-align:left;direction:ltr"> Nm / rad]]><h2 style=";text-align:left;direction:ltr"> Fastener stiffness in RY direction <h2 style=";text-align:left;direction:ltr"><![CDATA[5×10 <h2 style=";text-align:left;direction:ltr"> 10 <h2 style=";text-align:left;direction:ltr"> Nm / rad]]><h2 style=";text-align:left;direction:ltr"> Fastener RZ direction stiffness <h2 style=";text-align:left;direction:ltr"><![CDATA[5×10 <h2 style=";text-align:left;direction:ltr"> 10 <h2 style=";text-align:left;direction:ltr"> Nm / rad]]><h2 style=";text-align:left;direction:ltr"> Ballast X-direction stiffness <![CDATA[1.4×10 8 N / m]]> Ballast Y-direction stiffness <![CDATA[1.4×10 8 N / m]]> Ballast Z-direction stiffness <![CDATA[1.5×10 9 N / m]]> Ballast RX direction stiffness <h2 style=";text-align:left;direction:ltr"><![CDATA[2.6×10 <h2 style=";text-align:left;direction:ltr"> 12 <h2 style=";text-align:left;direction:ltr"> Nm / rad]]><h2 style=";text-align:left;direction:ltr"> Ballast stiffness in RY direction <h2 style=";text-align:left;direction:ltr"><![CDATA[2.6×10 <h2 style=";text-align:left;direction:ltr"> 12 <h2 style=";text-align:left;direction:ltr"> Nm / rad<h2 style=";text-align:left;direction:ltr"><!-- 11 --> ]]><h2 style=";text-align:left;direction:ltr"> Ballast RZ direction stiffness <h2 style=";text-align:left;direction:ltr"><![CDATA[2.6×10 <h2 style=";text-align:left;direction:ltr"> 12 <h2 style=";text-align:left;direction:ltr"> Nm / rad]]><h2 style=";text-align:left;direction:ltr">

[0150] The natural frequencies and relative errors of the dynamic dimension reduction model of the rail-sleeper-bridge system in this embodiment and the finite element model of the rail-sleeper-bridge system established using existing commercial software at different orders are shown in Table 2:

[0151] Table 2

[0152] Natural frequency Finite element model, Hz Dynamic dimensionality reduction model, Hz Relative error, % Stage 1 6.64 6.64 0 Stage 2 18.18 18.19 0.06 Tier 3 21.71 21.71 0 Tier 4 26.32 26.31 0.04 Tier 5 27.82 27.82 0 Tier 6 32.12 32.12 0 Tier 7 32.68 32.72 0.12 Tier 8 32.81 32.81 0 Tier 9 36.35 36.35 0 Level 10 36.92 38.22 3.52

[0153] As can be seen from Table 2, under the premise of retaining only the first five natural frequencies of the bridge structure, the first ten natural frequencies of the dynamic dimensionality reduction model of the rail-sleeper-bridge system of this embodiment are in good agreement with the results of the finite element model established by the existing software, proving the accuracy of the dimensionality reduction method of the present invention.

[0154] In this embodiment, the displacement in the actual response of the dynamic dimension reduction model of the rail-sleeper-bridge system is compared with the calculation results of the finite element model established by existing commercial software. Figure 2 As shown in the figure, the comparison between the acceleration and the finite element model calculation results is as follows: Figure 3 As shown. Figure 2 and Figure 3 It can be seen that when k=5 in this embodiment, the results of the dimensionality reduction model and the finite element model are basically consistent, indicating that the accuracy requirement is met when k is 5. When using the present invention in other application scenarios, the value of k can also be determined by comparing with the results of the finite element model.

[0155] The Fourier transform curve of the actual response calculated by the dynamic dimension reduction model of the rail-sleeper-bridge system in this embodiment is as follows: Figure 4 As shown in the figure, the peak values ​​of the Fourier spectra of displacement and acceleration responses are both 12 Hz, which is consistent with the frequency of the single-frequency harmonic external excitation applied to the rail-sleeper-bridge system. This proves that the model can accurately reflect the frequency domain characteristics of the system response under single-frequency harmonic external excitation.

[0156] In this embodiment, the order of dimensionality reduction is =5, meaning that after dimensionality reduction, the finite element theoretical model of the bridge structure only retains the first five natural frequencies of the bridge structure. At this point, the total bridge model dimension is 1827, and the connection node degrees of freedom are 1530, effectively reducing the bridge model dimension by 1827-1530-5 = 292. The solution time using the dynamic dimensionality reduction method of the present invention is 965 seconds, while the solution time for the finite element model established using existing commercial software is 1032 seconds, a total reduction of 67 seconds. The results show that the dimensionality reduction effect and computational efficiency of the present invention are improved compared to existing methods, but not significantly. This is because the concrete single-span simply supported box girder structure selected in this embodiment is relatively simple and is modeled using beam elements. When modeling the same bridge structure using shell elements or solid elements, even if the element geometry remains the same, the total bridge model dimension will increase significantly (by several dozen times or even more). However, the total number of connection nodes does not change due to the choice of element type, meaning the connection node degrees of freedom do not increase (and may even be reduced by half for solid elements). Furthermore, the order k is generally small, which ensures that the dimensionality of the reduced model does not increase excessively. In this case, the effect and computational efficiency of dimensionality reduction using the method of the present invention will be significantly improved.

[0157] In addition, for bridges with longer spatial scales or more complex structural types, the effect of dimensionality reduction and improvement of computational efficiency using the method of the present invention will be more significant.

[0158] Example 2

[0159] This embodiment takes ballastless track as an example to illustrate the dynamic dimension reduction method of its rail-track plate-bridge system model.

[0160] A dynamic dimension reduction method for a rail-track plate-bridge system model is basically the same as that in Example 1, except that:

[0161] Replacement of sleepers suitable for ballasted tracks in the rail-sleeper-bridge system with track slabs suitable for ballastless tracks;

[0162] The physical parameters of the sleeper structure in S1 are replaced by the physical parameters of the track plate structure, which include: track plate elastic modulus, track plate density, track plate Poisson's ratio and track plate cross-sectional area;

[0163] The linear spring stiffness between the rail structure and the sleeper structure in S3-3 is Replace the linear spring stiffness between the rail structure and the track plate structure with the linear spring stiffness between the sleeper structure and the bridge structure. Replaced with a linear spring stiffness between the track slab structure and the bridge structure.

Claims

1. A dynamic dimension reduction method for a rail-sleeper / track slab-bridge system model, characterized in that: The following steps are involved: S1, split the rail-sleeper / track slab-bridge system into three substructures: rail structure, sleeper structure / track slab structure, and bridge structure; obtain the mass matrix, damping matrix, and stiffness matrix of each substructure respectively and construct the finite element theoretical model of the rail structure, the finite element theoretical model of the sleeper structure / track slab structure, and the finite element theoretical model of the bridge structure; S2, the acceleration of the bridge structure ,speed and displacement respectively expressed in generalized coordinates, substituting the generalized coordinates into the bridge structure finite element theoretical model for dimension reduction to obtain a finite element theoretical model of the bridge structure after dimension reduction; S3, derive the conversion relationship between the actual response of each substructure and the response under the generalized coordinates after dimensionality reduction; Then, the rail structure finite element theoretical model, the sleeper structure / track slab structure finite element theoretical model and the finite element theoretical model of the bridge structure after dimensionality reduction are assembled and simplified using the conversion relationship to obtain a dynamic dimensionality reduction model of the rail-sleeper / track slab-bridge system; S4, solving the response of the dynamic dimension reduction model of the rail-sleeper / track slab-bridge system in generalized coordinates by a numerical method, and then calculating the actual response of the rail-sleeper / track slab-bridge system by the coordinate transformation relationship.

2. The dynamic dimension reduction method according to claim 1, characterized in that: In S1: The physical parameters of the rail structure, sleeper structure / track slab structure, and bridge structure are input into the finite element analysis software to calculate the mass matrix, damping matrix, and stiffness matrix of each substructure. The finite element theoretical model of the rail structure is as follows: (1) Where, 、 and are the mass matrix, damping matrix and stiffness matrix of the rail structure respectively, 、 and are the acceleration, velocity and displacement of the rail structure, is the internal force vector acting on the connection node of the rail structure, is the transpose of the identity matrix of the position of the degree of freedom of the connection node in the rail structure. The unit length of the rail structure is consistent with the spacing between the fasteners so that is a non-singular square matrix, is the external excitation vector of the rail structure; The finite element theoretical model of the sleeper structure / track plate structure is as follows: (2) Where, 、 and are the mass matrix, damping matrix and stiffness matrix of the sleeper structure / track plate structure respectively, 、 and are the acceleration, velocity and displacement of the sleeper structure / track plate structure respectively, is the internal force vector acting on the connection node of the sleeper structure / track slab structure, is the transpose of the identity matrix of the position of the degrees of freedom of the connection nodes in the sleeper structure / track slab structure, is the external excitation vector of the sleeper structure / track slab structure; The finite element theoretical model of the bridge structure is as follows: (3) Where, 、 and are the mass matrix, damping matrix and stiffness matrix of the bridge structure respectively, 、 and are the acceleration, velocity and displacement of the bridge structure, is the internal force vector acting on the connection node of the bridge structure, is the transpose of the identity matrix of the position of the degree of freedom of the connection node in the bridge structure, is the external excitation vector of the bridge structure.

3. The dynamic dimension reduction method according to claim 2, characterized in that: S2 includes the following steps: S2-1, using the free interface modal synthesis method to convert the acceleration of the bridge structure ,speed and displacement Expanded in generalized coordinates, they are expressed as: (4) (5) (6) Where, is the main mode of the bridge structure, the main mode include order natural frequency, Is the main mode The corresponding generalized coordinates are, and They are Second and first derivatives with respect to time; is the inertial attached mode of the bridge structure. Since the bridge structure does not contain rigid body modes, the inertial attached mode The calculation formula is: ; It is the habitual mode The corresponding generalized coordinates are, and They are Second and first derivatives with respect to time; S2-2, replace formula (4) Substituting formula (6) into formula (3), we can obtain the finite element theoretical model of the bridge structure after dimension reduction: (7) in: ； ； ； Where, 、 and are the mass matrix, damping matrix and stiffness matrix of the bridge structure in generalized coordinates respectively.

4. The dynamic dimension reduction method according to claim 3, characterized in that: S3 includes the following steps: S3-1, based on the interlayer elastic connection relationship between two adjacent substructures in the rail-sleeper / track slab-bridge system, the coordination relationship between the connection node displacements and the relationship between the connection interface forces of the two adjacent substructures are obtained: （8） （9） Where, is the displacement of the connection node of the sleeper structure / track slab structure; is the displacement of the node on the sleeper structure / track slab structure connected to the rail structure; is the displacement of the nodes on the sleeper structure / track slab structure connected to the bridge structure; is the displacement of the connection node of the rail structure; is the displacement of the connection node of the bridge structure; is the relative displacement difference vector of the connection node between the sleeper structure / track plate structure and the rail structure; is the relative displacement difference vector of the connection nodes between the sleeper structure / track plate structure and the bridge structure; is the node force at the connection interface of the sleeper structure / track plate structure; The node force connecting the sleeper structure / track plate structure with the rail structure; The node force connecting the sleeper structure / track slab structure with the bridge structure; is the nodal force at the connection interface of the rail structure; is the nodal force at the connection interface of the bridge structure; The following transformation relationship is obtained based on the displacement of the connection node of each substructure and the displacement of each substructure: （10） （11） （12） Where, is the identification matrix of the positions of the degrees of freedom of the connection nodes in the rail structure; is the identification matrix of the position of the degrees of freedom of the connection nodes in the bridge structure; is the identification matrix of the positions of the degrees of freedom of the connection nodes in the sleeper structure; for The identity matrix of the positions of the degrees of freedom associated with the rail structure in , for The identity matrix of the positions of the degrees of freedom associated with the bridge structure; Substituting formula (6) and formula (10) to (12) into formula (8), we can obtain the explicit expression of the coordination relationship between the displacements of the connection nodes of two adjacent substructures: （13） Formula (13) can be further simplified by and express ,pass 、 and express , and obtain the simplified explicit expression: （14） （15） S3-2, according to formula (14), (15) and formula (6), the displacements of the three substructures are obtained and generalized coordinates and generalized coordinates The coordinate transformation relationship between them is: (16) Where, and are the displacements of the three substructures and the transformation matrices between the two generalized coordinates, respectively: , （17） Where, is the identity matrix; S3-3, assemble the finite element theoretical model of the rail structure, the finite element theoretical model of the sleeper structure / track plate structure, and the finite element theoretical model of the bridge structure after dimension reduction to obtain: （18） Substitute formula (16) into formula (18) and multiply both sides of the equation by Combined with formula (9), the dynamic dimension reduction model of the rail-sleeper / track plate-bridge system is obtained: （19） in, ， ， Where, is the linear spring stiffness between the rail structure and the sleeper structure / track plate structure, is the linear spring stiffness between the sleeper structure / track slab structure and the bridge structure; for The first derivative with respect to time, The general form of nonlinear internal forces between rail structure and sleeper structure / track slab structure; for The first derivative with respect to time, A general form for expressing the nonlinear internal forces between the sleeper structure / track slab structure and the bridge structure.

5. The dynamic dimension reduction method according to claim 1, characterized in that: The calculation formula for the actual response of the dynamic dimension reduction model of the rail-sleeper / track slab-bridge system in S4 is: （20） （21） （22）。

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