A system dynamic analysis method suitable for mountainous railway continuous rigid frame bridge

Abstract

The application discloses a system dynamic analysis method suitable for a mountainous railway continuous rigid frame bridge, relates to the field of bridge system dynamic analysis, and comprises the following steps: S1, dividing a high-pier bridge into a plurality of sub-beam units along the longitudinal direction, establishing corresponding dynamic control equations, and constructing a forced motion equation of the high-pier bridge based on all the dynamic control equations; S2, respectively constructing a train subsystem motion equation, a steel rail subsystem motion equation and a track slab subsystem motion equation; S3, calculating the interaction forces of various contact interfaces through equivalent stiffness and damping parameters; S4, obtaining the dynamic responses of each subsystem through a numerical integration method; and S5, evaluating the vibration response of the high-pier bridge-track system and the safety of the vehicle by using the dynamic responses of each subsystem. The application can improve the modeling and solving efficiency while ensuring the accuracy, completely describes the three-way coupling dynamic action of the vehicle-track-bridge system, and provides a basis for evaluating the smoothness and safety of train operation.

Description

Technical Field

[0001] This application relates to the field of bridge system dynamic analysis, and in particular to a system dynamic analysis method applicable to continuous rigid frame railway bridges in mountainous areas. Background Technology

[0002] As high-speed railways extend into challenging mountainous areas, the application of high-pier, long-span, continuous rigid frame bridges is increasing. These bridge structures exhibit high flexibility, significant cross-sectional changes, and complex boundary conditions. When trains pass, they easily generate complex vehicle-track-bridge coupled dynamic interactions, directly affecting bridge vibration, train running stability, and operational safety. Existing vehicle-track-bridge dynamic analyses largely rely on finite element modeling, which suffers from cumbersome modeling, numerous degrees of freedom, and low computational efficiency when dealing with continuous rigid frame high-pier bridges, making it difficult to meet the requirements for parametric and multi-condition analysis.

[0003] Meanwhile, traditional models typically focus on lateral and vertical vibrations, while neglecting the longitudinal wheel-rail interaction and its transmission between the rail, track slab, and bridge. The interaction between the track slab and its interface with the bridge is also often simplified, making it difficult to accurately reflect in-plane and out-of-plane coupled vibrations, shear deformation, discrete fastener arrangement, and local stress differences. Summary of the Invention

[0004] In view of this, this application provides a system dynamic analysis method applicable to continuous rigid frame bridges in mountainous railways, which improves modeling and solution efficiency while ensuring accuracy, fully describes the three-dimensional coupled dynamic action of the vehicle-track-bridge system, and provides a basis for evaluating the smoothness and safety of train operation.

[0005] A system dynamic analysis method applicable to continuous rigid frame railway bridges in mountainous areas includes: S1. Divide the high-pier bridge into multiple sub-beam units along the longitudinal direction and establish the corresponding dynamic control equations. Based on all the dynamic control equations, construct the forced motion equations of the high-pier bridge. S2. Based on multi-rigid-body dynamics, spatial beam theory, and Mindlin thick plate theory, the motion equations of the train subsystem, rail subsystem, and track slab subsystem are constructed respectively. S3. Calculate the interaction forces at each contact interface using equivalent stiffness and damping parameters; The contact interfaces include: the contact interface between the high pier bridge and the track slab, the contact interface between the track slab and the rail, and the contact interface between the rail and the wheel. S4. Calculate the loads on the high-pier bridge, track slab, rail and train based on all interaction forces, and obtain the dynamic response of each subsystem through numerical integration method; S5. Evaluate the vibration response of the high-pier bridge-track system using the dynamic response of each subsystem.

[0006] In one possible technical solution, the forced motion equations for constructing a high-pier bridge in S1 include: S101. Divide the high-pier bridge into multiple sub-beam units along the longitudinal direction. The main beam is divided according to the span direction and the pier is divided according to the height direction. Each sub-beam unit is equivalent to a beam unit with a uniform cross section. S102. For each sub-beam element, establish dynamic control equations based on beam vibration theory, including: longitudinal vibration equation, transverse bending vibration equation, vertical bending vibration equation and torsional vibration equation. S103. Expand the displacement field of the sub-beam element and discretize the continuous control equations into finite-dimensional algebraic equations. S104. Solve the finite-dimensional algebraic equations using the Galerkin method, and construct the mass matrix, stiffness matrix, and damping matrix by combining the boundary conditions and the continuity conditions between sub-elements. Assemble the mass matrix, stiffness matrix, and damping matrix to form the overall dynamic matrix of the bridge.

[0007] S105. Based on the weighted orthogonality principle of modes and considering the damping ratio, the forced motion equations of high-pier bridges are obtained.

[0008] In one possible technical solution, the steps described in S2 for constructing the motion equations of the train subsystem, the rail subsystem, and the track slab subsystem based on multi-rigid-body dynamics, spatial beam theory, and Mindlin thick plate theory respectively include: S201. Discretize the train into a multi-rigid-body system consisting of the car body, bogies and wheelsets, and establish its dynamic equations, which include the elastic and damping effects of the primary suspension and the secondary suspension. S202. Establish the dynamic equations for the longitudinal, transverse, vertical, and torsional vibrations of the rail, respectively. S203. Based on Mindlin's thick plate theory, establish the dynamic equations of the track plate, including in-plane motion equations and out-of-plane motion equations.

[0009] In one possible technical solution, constructing the motion equations of the train subsystem includes: A single train consists of multiple carriages, and each carriage is further divided into multiple subsystems, including: a body subsystem, two frame subsystems, and four wheelset subsystems. The equation of motion for a single carriage is as follows: ; In the formula, M v The mass matrix of the vehicle subsystem; X v This represents the displacement vector of the vehicle subsystem; F fv =[F fc ,F ft1 ,Fft2 ,F fw1 ,F fw2 ,F fw3 ,F fw4 ] T F represents the vector sum of the six internal and external forces acting on the rigid sub-components of the vehicle body, frame, and wheelsets; mv =[F mc ,F mt1 ,F mt2 ,F mw1 ,F mw2 ,F mw3 ,F mw4 ] T These represent the sum of the gravity of the rigid sub-components of the vehicle body, frame, and wheelsets, and the inertial deflection force vectors caused by navigating curves; The motion equations of the train subsystem are obtained by combining the motion equations of all the carriages. The dynamic equations include the elastic and damping effects of the primary and secondary suspension systems.

[0010] In one possible technical solution, the dynamic equations for the longitudinal, transverse, vertical, and torsional vibrations of the rail established in step S202 include: Based on the spatial beam theory, the longitudinal, transverse, vertical and torsional forces of the rail are analyzed, and the motion equations and mode shapes in the Cartesian coordinate system are established. ; ; in, and These represent the longitudinal acceleration and displacement of the rail in the modal coordinate system, respectively. It is the elastic modulus; For rail density; This is the modal truncation number; This refers to the length of the rail. Total number of fasteners; The longitudinal resistance of the i-th fastener; This refers to the longitudinal displacement of the rail at the fastener location; For fastener mileage; This represents the total number of wheels; For the j-th wheel-rail interaction lateral component; This represents the longitudinal displacement of the rail at the wheel position. For wheel mileage; and These represent the torsional acceleration and displacement of the rail in the modal coordinate system, respectively. For the torsional stiffness of the rail; The polar moment of inertia of the rail; For rail density; This represents the longitudinal torsional moment of the i-th fastener; This refers to the torsional displacement at the location of the rail fastener. The j-th wheel-rail interaction torque about the X-axis; and These represent the lateral acceleration and displacement of the rail in the modal coordinate system, respectively. Let Z be the moment of inertia of the rail about the Z-axis; The j-th wheel-rail interaction transverse component; This refers to the lateral displacement at the location of the rail fastener; The lateral resistance of the i-th fastener; and These represent the vertical acceleration and displacement of the rail in the modal coordinate system, respectively. Let Y be the moment of inertia of the rail about the Y-axis; This refers to the vertical displacement at the location of the rail fastener; The vertical resistance of the i-th fastener; This refers to the longitudinal displacement of the rail; This is the longitudinal vibration mode of the rail; This represents the longitudinal modal cutoff number of the rail; This represents the cutoff number for the torsional mode of the rail. This represents the vertical modal cutoff number of the rail; Transverse modal cutoff number of rail; This is the torsional vibration mode of the rail. This represents the vertical displacement of the rail. This refers to the lateral displacement of the rail. This refers to the torsional displacement of the rail. This represents the vertical displacement of the rail. This refers to the lateral displacement of the rail.

[0011] In one possible technical solution, the in-plane motion equations in the track slab dynamic equations established based on Mindlin thick plate theory in S203 include: ; in, The in-plane vibration acceleration of the track plate in modal coordinates; This refers to the in-surface vibration damping coefficient of the track slab. The frequency of vibration within the track slab surface; The vibration velocity within the track slab surface in modal coordinates; This represents the in-plane vibration displacement of the track slab in modal coordinates. Let Y be the modal order in the corresponding direction of Y; Let Y be the modal order in the corresponding direction of Y; The total number of fasteners under a single rail on a track slab; and The longitudinal force of the fasteners at the left and right rails; and The lateral force of the fasteners at the left and right rails; This refers to the longitudinal vibration mode within the track slab surface. This refers to the lateral vibration mode within the track slab surface. This refers to the longitudinal reaction force of the support layer under the track slab. This refers to the lateral reaction force of the support layer under the track slab. The out-of-plane motion equations include: ; in, The out-of-plane vibration acceleration of the track plate in modal coordinates; The frequency of the track slab's out-of-plane vibration; The out-of-plane vibration velocity of the track slab in modal coordinates; This represents the out-of-plane vibration displacement of the track slab in modal coordinates. This refers to the vertical vibration mode of the track slab. and The vertical force of the fasteners at the left and right rails; This represents the supporting force along the z-axis under the plate.

[0012] In one possible technical solution, step S3 involves calculating the interaction forces at each contact interface using equivalent stiffness and damping parameters, including: S301. Determine the spatial position of each wheelset in the bridge coordinate system based on the train speed and time step, calculate the dynamic response at the corresponding wheel and the rail node, determine the wheel-rail contact relationship based on the dynamic response at the corresponding wheel and the rail node, and further calculate the interaction force at the rail-wheel contact interface. S302. Obtain the dynamic response of the track slab-rail, calculate the relative displacement and relative velocity between each interface node, and calculate the interaction force of each track slab-rail contact interface through the spring-damping element. S303. In the modeling of the interface between the track slab and the bridge, a discrete node array is used to replace the traditional continuous surface spring model. Spring-damping elements are independently established for each node to calculate the interaction force at the contact interface between the high-pier bridge and the track slab.

[0013] In one possible technical solution, the discrete node array is a multi-row node array arranged horizontally.

[0014] In one possible technical solution, the interaction force at the rail-wheel contact interface includes the rail-wheel normal force and the creep force. The normal force is based on the contact compression and is calculated using Hertz normal contact force. The wheel-rail creep force is calculated based on Kallker's nonlinear creep theory, and the longitudinal relative velocity between the wheel and the rail at the actual contact point is improved in the calculation of the wheel-rail longitudinal creep force.

[0015] This application provides a system dynamic analysis method applicable to continuous rigid frame railway bridges in mountainous areas, which has the following technical advantages compared with the prior art: By employing an innovative element lumped method, the dynamic equations of high-pier bridges were analyzed. This method can solve the problem of low computational efficiency of commonly used finite element programs, significantly reduce the computational degrees of freedom and computational scale while ensuring computational accuracy. It also improves the adaptability of bridge structures with variable cross sections, high piers, and complex boundary conditions.

[0016] By establishing the motion equations of the train, rail, and track slab subsystems based on multi-rigid-body dynamics, spatial beam theory, and Mindlin thick plate theory, and considering the in-plane and out-of-plane coupled vibrations of the track slab, the accuracy of the dynamic response calculation of the track system is improved. At the same time, the longitudinal dynamic load of the train can be transferred to the high-pier bridge system. This overcomes the problem of traditional track models neglecting shear deformation and high-frequency response, making the model more suitable for complex dynamic excitation conditions.

[0017] By acquiring the dynamic response of the interface nodes of high-pier bridge-track slab, track slab-rail, and rail-wheelset, and calculating the interface interaction forces based on equivalent stiffness and damping parameters, the interface forces can be calculated quickly, thereby improving the overall calculation efficiency. Attached Figure Description

[0018] To more clearly illustrate the technical solutions in the embodiments of this application or the prior art, the drawings used in the embodiments will be briefly introduced below. Obviously, the drawings described below are only some embodiments of this application. For those skilled in the art, other drawings can be obtained based on these drawings without creative effort.

[0019] Figure 1 This is a schematic diagram of the unit lumped method provided in the embodiments of this application; Figure 2 This is a force diagram of the vehicle model provided in the embodiments of this application; Figure 3 A comparison chart of the first 100 natural frequencies and errors of the bridge provided in the embodiment of this application and the ANSYS finite element model; Figure 4 This is a comparison diagram of the representative order bridge vibration modes provided in the unit lumped method and the ANSYS finite element model in the embodiments of this application; Figure 5 Here are comparison diagrams of the dynamic response of bridge structures provided in the embodiments of this application: Figure 5 a represents the top of pier number 5; Figure 5 b is the measuring point on the main beam; Figure 6This is a time history diagram of longitudinal wheel-rail force at a representative wheel-rail contact point provided in the embodiments of this application; Figure 7 This is a flowchart of a system dynamic analysis method for continuous rigid frame railway bridges in mountainous areas, provided in an embodiment of this application. Detailed Implementation

[0020] The technical solutions of the embodiments of this application will be clearly and completely described below with reference to the accompanying drawings. Obviously, the described embodiments are only some embodiments of this application, and not all embodiments. Based on the embodiments of this application, all other embodiments obtained by those skilled in the art without creative effort are within the scope of protection of this application.

[0021] To make the above-mentioned objectives, features and advantages of this application more apparent and understandable, the application will be further described in detail below with reference to the accompanying drawings and specific embodiments.

[0022] like Figure 7 As shown, a system dynamic analysis method applicable to continuous rigid frame railway bridges in mountainous areas includes: S1. Divide the high-pier bridge into multiple sub-beam units along the longitudinal direction and establish the corresponding dynamic control equations. Based on all the dynamic control equations, construct the forced motion equations of the high-pier bridge. S2. Based on multi-rigid-body dynamics, spatial beam theory, and Mindlin thick plate theory, the motion equations of the train subsystem, rail subsystem, and track slab subsystem are constructed respectively. S3. Calculate the interaction forces at each contact interface using equivalent stiffness and damping parameters; The contact interfaces include: the contact interface between the high pier bridge and the track slab, the contact interface between the track slab and the rail, and the contact interface between the rail and the wheel. S4. Calculate the loads on the high-pier bridge, track slab, rail and train based on all interaction forces, and obtain the dynamic response of each subsystem through numerical integration method; S5. Evaluate the vibration response of the high-pier bridge-track system using the dynamic response of each subsystem.

[0023] One possible design is that the forced motion equations for the high-pier bridge in S1 include: S101. Divide the high-pier bridge into multiple sub-beam units along the longitudinal direction. The main beam is divided according to the span direction and the pier is divided according to the height direction. Each sub-beam unit is equivalent to a beam unit with a uniform cross section. Specifically, the high-pier continuous rigid frame bridge is divided into main beams and piers. Taking the main beam as an example (the motion equations of the piers have a similar form, so they will not be elaborated here), it is further divided into finite independent sub-beams with equal cross-sections. These sub-beams are interconnected using continuous internal force conditions. Figure 1 As shown.

[0024] S102. For each sub-beam element, establish dynamic control equations based on beam vibration theory, including: longitudinal vibration equation, transverse bending vibration equation, vertical bending vibration equation and torsional vibration equation. Specifically, each independent sub-beam is considered as an Euler-Bernoulli beam, and the model's degrees of freedom are { u b ( x , t ), v b ( x , t ), w b ( x , t )} represents longitudinal, lateral, and vertical displacements, and its angular displacement can be expressed as { θ bx ( x , t ), θ by ( x , t ), θ bz ( x , t The relation is )} θ by ( x , t )= v b,x ( x , t ), θ bz ( x , t )= w b,x ( x , t ). No. J The governing equations of the segment beam are: ; In the formula, E bJ and G bJ They are the first J Segmental elastic modulus and shear modulus; ρ bJ andA bJ These are its density and cross-sectional area, respectively. I ybJ and I zbJ Indicates the beam section with respect to Y shaft and Z The inertia of the axis; I tbJ and I 0bJ They are the first J Torsional inertia and polar moment of inertia of the beam section; f xJ , f yJ , f zJ , m xJ} indicates along X , Y and Z The shaft acts on the first J External forces on the segmental beam, and surrounding X The torque of the shaft.

[0025] S103. Expand the displacement field of the sub-beam element and discretize the continuous control equations into finite-dimensional algebraic equations. Specifically, the displacement field of the sub-beam element is expanded to derive the relationship between internal forces and displacements, in order to subsequently derive the boundary conditions and internal force continuity conditions: ; In the formula, { T Jx , M Jx} are respectively the first J Axial force and torque of the segmental beam; Q Jy , M Jy} are respectively Y Shear force and bending moment of the shaft; Q Jz , M Jz} are respectively Z Shear force and bending moment of the shaft; , , , These are the vertical bending, transverse bending, shear, and torsional stiffness of the J-th beam, respectively.

[0026] S104. Solve the finite-dimensional algebraic equations using the Galerkin method, and construct the mass matrix, stiffness matrix, and damping matrix by combining the boundary conditions and the continuity conditions between sub-elements. Assemble the mass matrix, stiffness matrix, and damping matrix to form the overall dynamic matrix of the bridge.

[0027] In this application, dimensionless coordinates are introduced and the beam structure is assumed to undergo simple harmonic motion. After substituting the governing equations into the corresponding form, the mode shape function is expanded using Chebyshev polynomials and discretized based on the Galerkin method. Furthermore, additional second- or fourth-order modes are extended based on the determined number of truncated modes, as follows: ; In the formula, M Jn ，N Jn ，R Jn ，S Jn It can be an arbitrary constant that can be determined by the equation of motion. The beam section K also has a similar expression; you only need to replace J with K. Represents Chebyshev polynomials; This indicates the longitudinal displacement of beam segment J; Indicates the Jth segment of the beam Lateral displacement; This represents the vertical displacement of beam segment J; N represents the torsional displacement of beam segment J; x , N y , N z , N t These represent the modal cutoff numbers for longitudinal, lateral, vertical, and torsional motions, respectively.

[0028] In this embodiment, the control equations are normalized by introducing dimensionless coordinates. This not only eliminates the influence of the difference in dimensions of different physical quantities on the stability of numerical calculations, but also allows the structural geometric parameters and material parameters to be expressed on a unified scale, thereby improving the universality of the equations.

[0029] Using Chebyshev polynomials as basis functions for the mode shape function expansion can improve the accuracy of displacement field approximation, especially in the boundary region; reduce the number of expansion terms required, thereby reducing the computational scale; and improve the model's adaptability to complex structures (such as high piers, variable cross-section beams, and non-classical boundary constraints).

[0030] By introducing the Chebyshev weighted orthogonality rule, the matrix form of the motion equations for the entire beam segment can be obtained: ; ; In the formula, { , , , } are the longitudinal, vertical, transverse, and torsional stiffness matrices of the entire beam, respectively; , , , } are the longitudinal, vertical, transverse, and torsional extended stiffness matrices of the entire beam, respectively; , , , } are the longitudinal, vertical, transverse, and torsional mass matrices of the entire beam, respectively; , , , } are the longitudinal, vertical, transverse, and torsional extended stiffness matrices of the entire beam, respectively; It is a user-defined operator; Modal cutoff number; Dimensionless coordinates; Weight; Find the β derivative of the Chebyshev polynomial; Find the γ derivative of the Chebyshev polynomial.

[0031] By assembling the lumped matrix of key elements using the boundary conditions of the entire beam and the continuity conditions of each sub-beam segment, taking the longitudinal motion equation as an example, all other directions have a similar format: ; ; ; ; ; In the formula, the lumped matrix of key units V x and W x The dimensions are 2 N × N x N and 2 N ×2 N M and M* are the mass matrices of the longitudinal vibration of the entire beam.

[0032] The matrix form of the above whole beam segment motion equations is transformed from a non-square matrix to a square matrix form, enabling this method to be used to calculate the mode shapes and frequencies of free vibration of beam structures. ; In the formula, matrix K x and M x The dimension is N x N × N x N ;K* x and M* x The dimension is N x N ×2 N ; Matrix K y and M y The dimension is N y N × N y N The dimensions of K*y and M*y are N y N ×4 N ; Matrix K z and M z The dimension is N z N × N z N The dimensions of K*z and M*z are: N z N ×4 N ; Matrix K t and M t The dimension is N t N × N t N ; The dimensions of K*t and M*t are N t N ×2 N； The longitudinal vibration frequency of the entire beam; The transverse vibration frequency of the entire beam; The vertical vibration frequency of the entire beam; The torsional vibration frequency of the entire beam; Longitudinal modal coefficients; Transverse modal coefficients; Vertical modal coefficients; Torsional modal coefficient; Longitudinal extension modal coefficients; Lateral extension modal coefficients; Vertical extension modal coefficients; Torsional extended modal coefficients; S105. Based on the weighted orthogonality principle of modes and considering the damping ratio, the forced motion equations of high-pier bridges are obtained.

[0033] ; In the formula, k 1, k 2, k 3, k Take 1 from each of the 4 options. N x ,1~ N y ,1~ N z ,1~ N t ; , , , These represent the longitudinal, lateral, vertical, and torsional accelerations of a high-pier bridge in a canonical coordinate system, respectively. , , , These represent the longitudinal, lateral, vertical, and torsional velocities of a high-pier bridge in a canonical coordinate system, respectively. , , , These represent the longitudinal, lateral, vertical, and torsional displacements of a high-pier bridge in a canonical coordinate system, respectively. , , , These are the longitudinal, transverse, vertical, and torsional damping coefficients of high-pier bridges; , , , These are the longitudinal, transverse, vertical, and torsional vibration modes of high-pier bridges; , , , These are the longitudinal force, lateral force, vertical force, and torque experienced by a high-pier bridge. , , , These are the longitudinal, lateral, vertical, and torsional modal masses of the high-pier bridge.

[0034] This embodiment innovatively presents the forced vibration equation for a variable cross-section beam, which can effectively provide a general expression for the main beam and piers of a continuous rigid frame high-pier bridge with different variable cross-sections. It aims to parametrically model any pier height, bridge span, and cross-sectional shape, breaking through the limitation of the traditional finite element method of "one bridge, one construction".

[0035] The forced motion equations of the main beam of the high-pier continuous rigid frame bridge have been obtained above. By repeating the above steps, the forced vibration equations of each pier can be obtained. Then, by adding three-dimensional high-stiffness springs to the bottom and top of the beam, they can be coupled into the motion equations of the high-pier continuous rigid frame bridge system. ; In the formula, {M g C g , K g} and {M p C p , K p} represents the mass, damping, and stiffness matrix of the beam and pier, X b and X p F represents the displacement response of the main girder and each pier; b and F p This represents the external forces borne by the main beam and each pier.

[0036] In this embodiment, the main beam and each pier are strongly coupled using high-stiffness three-dimensional springs, thereby realizing the parameterized expression of the forced vibration equation of the continuous rigid frame high-pier bridge and breaking through the limitations of traditional finite element calculation of the efficiency of the continuous rigid frame bridge.

[0037] In step S1, the high-pier bridge is divided into multiple uniform cross-section sub-beam elements, and the dynamic control equations of each sub-element are established based on beam vibration theory. Simultaneously, the displacement field is expanded using Chebyshev polynomials or other orthogonal functions, and discretized using the Galerkin method to construct the overall bridge dynamic matrix. This approach solves the problems of large computational scale, complex modeling, and low computational efficiency in existing technologies that rely on the finite element method for modeling high-pier bridges, resulting in the following technical effects: (1) By combining sub-beam element division with function expansion, the continuous variable cross-section structure is transformed into a finite-dimensional algebraic system, which significantly reduces the system's degrees of freedom while ensuring computational accuracy. (2) It avoids the complex mesh division and repetitive modeling process in the traditional finite element method, thereby improving modeling efficiency and computational stability. (3) It can be applied to bridge structures with variable cross-sections, high piers, and complex boundary conditions, thereby increasing the applicability of the method. (4) It achieves efficient construction of bridge dynamic equations through a unified matrix assembly form, which facilitates coupling analysis with the track and train subsystems.

[0038] As one possible design, the steps described in S2 for constructing the motion equations of the train subsystem, rail subsystem, and track slab subsystem based on multi-rigid-body dynamics, spatial beam theory, and Mindlin thick plate theory respectively include: S201. Discretize the train into a multi-rigid-body system consisting of the car body, bogies and wheelsets, and establish its dynamic equations, which include the elastic and damping effects of the primary suspension and the secondary suspension. S202. Establish the dynamic equations for the longitudinal, transverse, vertical, and torsional vibrations of the rail, respectively. S203. Based on Mindlin's thick plate theory, establish the dynamic equations of the track plate, including in-plane motion equations, out-of-plane motion equations, and shear deformation effects.

[0039] One possible design is that the equations of motion for constructing the train subsystem include: High-speed trains typically consist of multiple motor cars and trailer cars arranged in formation. A coupler buffer system, simulated by nonlinear spring-damping elements, dissipates vibration energy between adjacent cars. A single car is usually considered a multi-rigid-body system, comprising one car body subsystem, two frame subsystems, and four wheelset subsystems. Figure 2 As shown. The wheelset-frame and body-frame subsystems are connected by primary and secondary suspension systems, respectively, to buffer vibrations between sub-components. Based on d'Alembert's principle, the vehicle's equation of motion can typically be characterized as a second-order ordinary differential equation consisting of inertial terms, suspension system impedance terms, and wheel-rail interaction terms: The equation of motion for a single carriage is as follows: ; In the formula, M v The mass matrix of the vehicle subsystem; X v This represents the displacement vector of the vehicle subsystem; F fv =[F fc ,F ft1 ,F ft2 ,F fw1 ,F fw2 ,F fw3 ,F fw4 ] T F represents the vector sum of the six internal and external forces acting on the rigid sub-components of the vehicle body, frame, and wheelsets; mv =[F mc ,F mt1 ,F mt2 ,F mw1 ,F mw2 ,F mw3 ,F mw4 ] TThese represent the sum of the gravity of the rigid sub-components of the vehicle body, frame, and wheelsets, and the inertial deflection force vectors caused by navigating curves; The motion equations of the train subsystem are obtained by combining the motion equations of all the carriages. The dynamic equations include the elastic and damping effects of the primary and secondary suspension systems.

[0040] One possible design is that the dynamic equations for the longitudinal, transverse, vertical, and torsional vibrations of the rail, respectively, established in S202, include: Based on the spatial beam theory, the longitudinal, transverse, vertical and torsional forces of the rail are analyzed, and the motion equations and mode shapes in the Cartesian coordinate system are established. ; ; in, and These represent the longitudinal acceleration and displacement of the rail in the modal coordinate system, respectively. It is the elastic modulus; For rail density; This is the modal truncation number; This refers to the length of the rail. Total number of fasteners; The longitudinal resistance of the i-th fastener; This refers to the longitudinal displacement of the rail at the fastener location; For fastener mileage; This represents the total number of wheels; For the j-th wheel-rail interaction lateral component; This represents the longitudinal displacement of the rail at the wheel position. For wheel mileage; and These represent the torsional acceleration and displacement of the rail in the modal coordinate system, respectively. For the torsional stiffness of the rail; The polar moment of inertia of the rail; For rail density; This represents the longitudinal torsional moment of the i-th fastener; This refers to the torsional displacement at the location of the rail fastener. The j-th wheel-rail interaction torque about the X-axis; and These represent the lateral acceleration and displacement of the rail in the modal coordinate system, respectively. Let Z be the moment of inertia of the rail about the Z-axis; The j-th wheel-rail interaction transverse component; This refers to the lateral displacement at the location of the rail fastener; The lateral resistance of the i-th fastener; and These represent the vertical acceleration and displacement of the rail in the modal coordinate system, respectively. Let Y be the moment of inertia of the rail about the Y-axis; This refers to the vertical displacement at the location of the rail fastener; The vertical resistance of the i-th fastener; This refers to the longitudinal displacement of the rail; This is the longitudinal vibration mode of the rail; This represents the longitudinal modal cutoff number of the rail; This represents the cutoff number for the torsional mode of the rail. This represents the vertical modal cutoff number of the rail; Transverse modal cutoff number of rail; This is the torsional vibration mode of the rail. This represents the vertical displacement of the rail. This refers to the lateral displacement of the rail. This refers to the torsional displacement of the rail. This represents the vertical displacement of the rail. This refers to the lateral displacement of the rail. The innovative introduction of the longitudinal vibration expression of the beam enables the transmission of the longitudinal wheel-rail force of the train to the track and the continuous rigid frame high pier bridge, breaking through the limitation of traditional vehicle-rail-bridge systems that do not consider longitudinal vibration.

[0041] One possible design is that, in S203, the in-plane motion equations of the track slab dynamics equations based on Mindlin thick plate theory include: ; in, The in-plane vibration acceleration of the track plate in modal coordinates; This refers to the in-surface vibration damping coefficient of the track slab. The frequency of vibration within the track slab surface; The vibration velocity within the track slab surface in modal coordinates; This represents the in-plane vibration displacement of the track slab in modal coordinates. Let Y be the modal order in the corresponding direction of Y; Let Y be the modal order in the corresponding direction of Y; The total number of fasteners under a single rail on a track slab; and The longitudinal force of the fasteners at the left and right rails; and The lateral force of the fasteners at the left and right rails; This refers to the longitudinal vibration mode within the track slab surface. This refers to the lateral vibration mode within the track slab surface. This refers to the longitudinal reaction force of the support layer under the track slab. This refers to the lateral reaction force of the support layer under the track slab. The out-of-plane motion equations include: ; in, The out-of-plane vibration acceleration of the track plate in modal coordinates; The frequency of the track slab's out-of-plane vibration; The out-of-plane vibration velocity of the track slab in modal coordinates; This represents the out-of-plane vibration displacement of the track slab in modal coordinates. This refers to the vertical vibration mode of the track slab. and The vertical force of the fasteners at the left and right rails; This represents the supporting force along the z-axis under the plate.

[0042] In this embodiment, the Mindlin medium-thick plate theory is introduced to transfer the triaxial vibration energy of the rail to the track slab and the continuous rigid frame high pier bridge. This breaks through the limitations of the traditional track slab being considered as a rigid body in the lateral direction and not considering longitudinal vibration, thus realizing the refined modeling of the track slab.

[0043] One possible design is that, in S3, the interaction forces at each contact interface are calculated using equivalent stiffness and damping parameters, including: S301. Determine the spatial position of each wheelset in the bridge coordinate system based on the train speed and time step, calculate the dynamic response at the corresponding wheel and the rail node, determine the wheel-rail contact relationship based on the dynamic response at the corresponding wheel and the rail node, and further calculate the interaction force at the rail-wheel contact interface. The wheel-rail spatial motion relationship at each moment is simulated using the trace method to obtain the real-time coordinates of the wheel and rail tread in the Cartesian coordinate system.

[0044] ; ; In the formula, δ w It is the wheel contact angle; d w It is the abscissa of the wheel tread in the wheelset coordinate system; R w It is the nominal rolling circle radius; when it is the left wheel, the formula " "and" Take the upper symbol; if it is the right side, take the lower symbol. Wheelset side roll; The wheel shakes its head; Wheelset lateral movement; l x , l y ,l z It is the direction cosine; S302. Obtain the dynamic response of the track slab-rail, calculate the relative displacement and relative velocity between each interface node, and calculate the interaction force of each track slab-rail contact interface through the spring-damping element. Specifically, the dynamic response parameters at the interface nodes between the track slab and the rail are obtained, including information such as displacement and velocity. The relative displacement and relative velocity between the two are calculated based on the corresponding nodes. On this basis, the connection relationship between the track slab and the rail is equivalent to a spring-damping unit. The interface interaction force is calculated through the relative displacement and relative velocity. The interaction force is determined by the stiffness term and the damping term, thereby establishing the dynamic transmission relationship between the track slab and the rail.

[0045] S303. In the modeling of the interface between the track slab and the bridge, a discrete node array is used to replace the traditional continuous surface spring model. Spring-damping elements are independently established for each node to calculate the interaction force at the contact interface between the high-pier bridge and the track slab.

[0046] In the modeling of the track slab and bridge interface, a discrete node array is used to represent the interface, replacing the traditional continuous surface spring model. Specifically, multiple rows of nodes are arranged laterally, preferably five rows, corresponding to two rows of nodes at the fastener locations, two rows of nodes at the two edges of the track slab, and one row of nodes in the middle of the track slab. Independent spring-damping elements are established at each node to achieve discretized calculation of the interface interaction forces. This interaction can be expressed by the relative displacement of the force transmission nodes on the lower surface of the track slab and the bridge deck. ; ; ; ; Track slab height, Height from the center of the bridge to the bridge deck; Bridge displacement; Vertical displacement of the track slab; Vertical displacement of the bridge; Lateral displacement of the bridge; Longitudinal modes of bridge; Vertical velocity of track slab bridge vertical velocity bridge longitudinal displacement Lateral displacement of bridge Bridge torsional displacement bridge lateral speed Bridge torsional speed; Vertical velocity of the track slab; Longitudinal velocity of the bridge; Lateral speed of the track slab; In-situ vibration mode number of the track slab Out-of-plane vibration mode number of track slab In-deck vibration mode number of the bridge; Out-of-plane vibration mode number of bridge; F sxij , F syij , F szij This represents the interaction force between the track slab and the continuous rigid frame high pier bridge. k sa , k sh , k sv They represent XYZ Stiffness of the underside support of the directional track slab c sa , c sh , c sv It is the corresponding damping coefficient; e bj This represents the lateral distance between the force transmission point on the upper surface of the bridge deck and the center of the bridge's torsion. N cy The number of columns representing the force transmission points of the bridge deck along the transverse direction is 5 in this application; u s , v s , w s Represents the neutral plane of a rectangular plate along X , Y , Z Translational motion of the axis; θ sx , θ sy They represent circumference respectively. X and Y Angular displacement in direction.

[0047] One possible design is that the discrete node array consists of multiple rows of nodes arranged horizontally.

[0048] In this embodiment, a discrete node array is used instead of the traditional continuous surface spring model. Multiple rows of nodes (preferably five rows, including two rows of nodes at the fastener positions, two rows of nodes at the edge of the track slab, and one row of nodes in the middle of the track slab) are arranged laterally at the interface between the track slab and the bridge. This effectively solves the problem that the continuous surface spring model cannot reflect local force differences and edge effects, resulting in the following technical effects: it can more finely depict the force distribution in the edge and middle areas of the track slab, improving the accuracy of interface force calculation; it can reflect the influence of the discrete arrangement of fasteners on the force transmission path, enhancing the physical consistency of the model; at the same time, it transforms the interface force from a continuous distribution form to a discrete node form, avoiding complex integral calculations and achieving a balance between calculation accuracy and efficiency.

[0049] One possible design is that the interaction forces at the rail-wheel contact interface include the rail-wheel normal force and the creep force; The normal force is based on the contact compression and is calculated using Hertz normal contact force. The normal force is solved using Hertz's nonlinear elastic contact theory, while the creep force is solved using Kalker's rolling contact theory. Nonlinear corrections are made to make it applicable to arbitrary creep rates and small spin conditions.

[0050] Because the longitudinal degrees of freedom of the vehicle and rail are increased, corresponding degrees of freedom need to be added in the solution of wheel-rail creep rate: ; ; ; In the formula, , , These represent longitudinal, transverse, and spin creep rates, respectively. The symbol “(L,R)” takes the top side when it is L and the bottom side when it is R; , , The longitudinal and lateral relative velocities of the wheel and rail, and the relative angular velocity around the X-axis, are given in the contact patch coordinate system. This is a coordinate system transformation matrix, used to transform from the absolute coordinate system to the contact patch coordinate system; , , These represent the longitudinal, lateral, and vertical relative velocities of the left and right wheels and rails in the absolute coordinate system, respectively. , , These represent the relative angular velocities of the left and right wheels and rails around the X, Y, and Z axes in the absolute coordinate system, respectively. , , These represent the relative angular velocities of the wheel around the X, Y, and Z axes in the absolute coordinate system, respectively. , , , respectively, represent the radius vector from the wheelset's center of mass to the wheel-rail contact point in the absolute coordinate system; hr represents the vertical distance from the contact point to the rail's center of torsion; er represents the abscissa of the contact point in the rail coordinate system; The radius of the curve under the current wheel.

[0051] In this embodiment, by correcting the longitudinal relative velocity at the wheel-rail contact point, the creep rate calculation can reflect the actual contact state, thereby improving the calculation accuracy of the longitudinal creep force of the wheel and rail, enhancing the reliability of the dynamic response analysis under conditions such as train acceleration and braking, and ensuring the reasonable transmission of longitudinal forces in the vehicle-rail-bridge system, thus improving the accuracy and applicability of the overall coupled dynamic analysis.

[0052] The wheel-rail creep force is calculated based on Kallker's nonlinear creep theory, and the longitudinal relative velocity between the wheel and the rail at the actual contact point is improved in the calculation of the wheel-rail longitudinal creep force.

[0053] creep coefficient f ji The creep force is determined by the rolling circle radius, the rail curvature radius at the contact point, and the normal force obtained from Hertzian normal contact theory. f ji The relationship between them is as follows: ; In the formula, F cx , F cy These are the X and Y components of the creep force, respectively; M cz This represents the wheel-rail spin creep torque about the Z-axis in the contact patch coordinate system; Longitudinal creep coefficient; Lateral creep coefficient; Vertical creep coefficient; Nonlinear correction is employed to make it widely applicable to cases with arbitrary creep rates and small spins, defined as: ; In the formula, This represents the creep force and torque in the contact patch coordinate system after nonlinear correction; Correction factor; coefficient of friction; Horizontal creep force; Corrected horizontal creep force; The transpose matrix is ​​transformed to the absolute coordinate system to obtain the final forces and moments used for structural vibration.

[0054] One possible design is that in step S4: (1) The interaction forces of the wheel-rail interface, the rail-track slab interface and the track slab-continuous rigid frame high pier bridge interface are respectively converted into the equivalent external loads of the train, rail, track slab and bridge subsystem. (2) Based on the mass matrix, stiffness matrix and damping matrix of each subsystem, and in combination with the equivalent external load, establish the dynamic equations of the train, rail, track slab and bridge. (3) Discretize the dynamic response analysis process into multiple time steps, and solve the dynamic equations of each subsystem using the integral method: ; In the formula, n Representing the integral step sequence, X, V, and A represent the displacement, velocity, and acceleration matrices of the vehicle-track-bridge system, respectively, and Δ t Represents the integration step size. ψ and φ It is the integral coefficient, and when the initial step is 0, it has the "self-starting" property of integration.

[0055] (4) Update the displacement, velocity and acceleration response of each subsystem in each time step, and feed the update results back to the interface interaction calculation to realize the iterative solution of the coupled dynamic response of the vehicle-track-bridge system.

[0056] One possible design is that in step S5: (1) Based on the dynamic response results, extract the vibration response of the high-pier bridge and track system, including bridge displacement, acceleration and track structure vibration response; (2) Calculate the train operation safety and comfort indicators based on the train dynamic response, including derailment coefficient, wheel load reduction rate and car body acceleration, etc. (3) Compare and analyze the vibration response index and train operation status index with the preset limit or standard, as shown in Table 1; Table 1

[0057] (4) Based on the comparison results, determine the vibration state of the high-pier bridge-track system and the safety and comfort of train operation, and output the evaluation results.

[0058] To verify the accuracy of the element lumped method described in this application for solving the modal characteristics of high-pier continuous rigid frame bridges, this embodiment takes a 120 m high-pier continuous rigid frame bridge as the object, establishes a bridge dynamic model using the element lumped method, and uses the ANSYS BEAM188 / Timoshenko finite element model as the reference model. Both models use the same bridge geometry, cross-section, materials, boundary conditions, and mass parameters, and respectively solve for the first 100 natural frequencies and representative mode shapes of the bridge.

[0059] Depend on Figure 3 It can be seen that the first 100 natural frequencies calculated by the method of this application are in high agreement with the ANSYS finite element results. The maximum relative error between the two is 2.403e-05%, the average relative error is 4.652e-06%, and the maximum absolute difference is 3.339e-06 Hz, indicating that the method of this application can accurately obtain the natural frequencies of high-pier continuous rigid frame bridges.

[0060] Depend on Figure 4 It can be seen that when the 1st, 20th, and 50th modes are selected as representative modes for mode shape comparison, the deformation patterns of the main beam and piers obtained by the element lumped method and the ANSYS finite element model are basically consistent. The above results show that the method of this application can effectively solve the modes of high-pier continuous rigid frame bridges under parametric modeling conditions, and can serve as the basis for the bridge subsystem of subsequent vehicle-track-bridge coupled dynamic response analysis.

[0061] To further verify the rationality of this method in solving the dynamic response of bridge structures, the results of the element lumped method were compared with the reference results of ANSYS (BEAM188) co-simulation by MATLAB-ANSYS. The lateral displacement of the top of pier No. 5 and the lateral bending displacement of the measuring point of the main beam were selected as the structural side response indicators.

[0062] Depend on Figure 5 It can be seen that the lateral displacement at the top of pier No. 5 calculated by the element lumped method is consistent with the reference results of ANSYS (BEAM188) in terms of time history variation trend and response magnitude, with a peak difference of approximately 1.61%. The two results for the lateral bending displacement of the main beam measuring points also basically overlap, with a peak difference of approximately 0.75%. The above results show that this method can not only complete the modal solution of the bridge, but also reasonably reflect the dynamic response of the bridge structure under the action of train passage.

[0063] To verify the computational capability of this method for longitudinal power transmission paths, a train was selected to pass through a 120-m high continuous rigid frame bridge at 300 km / h. The longitudinal wheel-rail force time history at the wheel-rail contact point was extracted and compared with the control results that do not consider the longitudinal interaction between wheel and rail.

[0064] Depend on Figure 6It can be seen that when the train passes the bridge at 300 km / h, the longitudinal force calculated by this method is no longer zero, but fluctuates significantly with the train's movement, representing a maximum absolute value of approximately 5.42 kN at the wheel-rail contact point. Without considering wheel-rail longitudinal interaction, this longitudinal response degenerates to zero, making it difficult to reflect the longitudinal dynamic transmission between the train, track, and bridge. These results indicate that this model can describe the wheel-rail longitudinal interaction and its transmission to the track slab and bridge system, demonstrating better applicability in longitudinal dynamic response analysis.

[0065] After obtaining the dynamic responses of the vehicle, wheel-rail, and bridge, the Sperling stability index was further calculated based on the lateral and vertical accelerations of the vehicle body. The driving safety index was then calculated based on the lateral and vertical forces of the wheel and rail, wheel lift, relative lateral displacement of the wheel and rail, vehicle body acceleration, and bridge vertical acceleration. For ease of comparison, each evaluation index was compared with its corresponding limit or reference value. The summary results of the driving stability and safety evaluation indices are shown in Table 2.

[0066] Table 2

[0067] Table 2 shows that the lateral, vertical, and combined Sperling indices of the car body are 1.42, 1.69, and 1.72, respectively, all below 2.5, indicating that the evaluation results are all qualified. Regarding safety, the derailment coefficient is 0.239, the wheel load reduction rate is approximately 0.700, the maximum axle lateral force is 18.34 kN, the peak lateral and vertical accelerations of the car body are 0.159 m / s² and 0.148 m / s², respectively, and the peak vertical acceleration of the bridge is 0.414 m / s², all meeting the corresponding limit requirements. The wheel-rail lateral relative displacement range is -10.49 to 11.33 mm, within the limit range of -57 to 38.875 mm, and the maximum wheel lift is 0.089 mm, below the 28 mm limit, indicating that the evaluation results are all qualified. These results demonstrate that this method can output the vehicle, wheel-rail, and bridge responses required for stability and safety evaluation, and form the basis for operational evaluation when trains pass over high-pier bridges.

[0068] The technical features of the above embodiments can be combined in any way. For the sake of brevity, not all possible combinations of the technical features in the above embodiments are described. However, as long as there is no contradiction in the combination of these technical features, they should be considered to be within the scope of this specification.

[0069] This document uses specific examples to illustrate the principles and implementation methods of this application. The descriptions of the above embodiments are only for the purpose of helping to understand the methods and core ideas of this application. Furthermore, those skilled in the art will recognize that, based on the ideas of this application, there will be changes in the specific implementation methods and application scope. Therefore, the content of this specification should not be construed as a limitation of this application.

Claims

1. A system dynamic analysis method applicable to continuous rigid frame railway bridges in mountainous areas, characterized in that, include: S1. Divide the high-pier bridge into multiple sub-beam units along the longitudinal direction and establish the corresponding dynamic control equations. Based on all the dynamic control equations, construct the forced motion equations of the high-pier bridge. S2. Based on multi-rigid-body dynamics, spatial beam theory, and Mindlin thick plate theory, the motion equations of the train subsystem, rail subsystem, and track slab subsystem are constructed respectively. S3. Calculate the interaction forces at each contact interface using equivalent stiffness and damping parameters; The contact interfaces include: the contact interface between the high pier bridge and the track slab, the contact interface between the track slab and the rail, and the contact interface between the rail and the wheel. S4. Calculate the loads on the high-pier bridge, track slab, rail and train based on all interaction forces, and obtain the dynamic response of each subsystem through numerical integration method; S5. The vibration response of the high-pier bridge-track system and the vehicle safety are evaluated by utilizing the dynamic response of each subsystem.

2. The system dynamic analysis method for continuous rigid frame railway bridges in mountainous areas according to claim 1, characterized in that, The forced motion equations for the high-pier bridge constructed in S1 include: S101. Divide the high-pier bridge into multiple sub-beam units along the longitudinal direction. The main beam is divided according to the span direction and the pier is divided according to the height direction. Each sub-beam unit is equivalent to a beam unit with a uniform cross section. S102. For each sub-beam element, establish dynamic control equations based on beam vibration theory, including: longitudinal vibration equation, transverse bending vibration equation, vertical bending vibration equation and torsional vibration equation. S103. Expand the displacement field of the sub-beam element and discretize the continuous control equations into finite-dimensional algebraic equations. S104. Solve the finite-dimensional algebraic equations using the Galerkin method, and construct the mass matrix, stiffness matrix, and damping matrix by combining the boundary conditions and the continuity conditions between sub-elements. Assemble the mass matrix, stiffness matrix, and damping matrix to form the overall dynamic matrix of the bridge. S105. Based on the weighted orthogonality principle of modes and considering the damping ratio, the forced motion equations of high-pier bridges are obtained.

3. The system dynamic analysis method for continuous rigid frame railway bridges in mountainous areas according to claim 1, characterized in that, The steps described in S2 for constructing the motion equations of the train subsystem, rail subsystem, and track slab system based on multi-rigid-body dynamics, spatial beam theory, and Mindlin thick plate theory respectively include: S201. Discretize the train into a multi-rigid-body system consisting of the car body, bogies and wheelsets, and establish its dynamic equations, which include the elastic and damping effects of the primary suspension and the secondary suspension. S202. Establish the dynamic equations for the longitudinal, transverse, vertical, and torsional vibrations of the rail, respectively. S203. Based on Mindlin's thick plate theory, establish the dynamic equations of the track plate, including in-plane motion equations and out-of-plane motion equations.

4. The system dynamic analysis method for continuous rigid frame railway bridges in mountainous areas according to claim 3, characterized in that, The construction of the train subsystem motion equations includes: A single train consists of multiple carriages, and each carriage is further divided into multiple subsystems, including: a body subsystem, two frame subsystems, and four wheelset subsystems. The equation of motion for a single carriage is as follows: ； In the formula, M v The mass matrix of the vehicle subsystem; X v This represents the displacement vector of the vehicle subsystem; F fv =[F fc , F ft1 , F ft2 , F fw1 , F fw2 , F fw3 , F fw4 ] T F represents the vector sum of the six internal and external forces acting on the rigid sub-components of the vehicle body, frame, and wheelsets; mv =[F mc , F mt1 , F mt2 , F mw1 , F mw2 , F mw3 , F mw4 ] T These represent the sum of the gravity of the rigid sub-components of the vehicle body, frame, and wheelsets, and the inertial deflection force vectors caused by navigating curves; The motion equations of the train subsystem are obtained by combining the motion equations of all the carriages. The dynamic equations include the elastic and damping effects of the primary and secondary suspension systems.

5. The system dynamic analysis method for continuous rigid frame railway bridges in mountainous areas according to claim 3, characterized in that, The dynamic equations for the longitudinal, transverse, vertical, and torsional vibrations of the rail established in S202 include: Based on the spatial beam theory, the longitudinal, transverse, vertical and torsional forces of the rail are analyzed, and the motion equations and mode shapes in the Cartesian coordinate system are established. ； ； in, and These represent the longitudinal acceleration and displacement of the rail in the modal coordinate system, respectively. It is the elastic modulus; For rail density; This is the modal truncation number; This refers to the length of the rail. Total number of fasteners; The longitudinal resistance of the i-th fastener; This refers to the longitudinal displacement of the rail at the fastener location; For fastener mileage; This represents the total number of wheels; For the j-th wheel-rail interaction lateral component; This represents the longitudinal displacement of the rail at the wheel position. For wheel mileage; and These represent the torsional acceleration and displacement of the rail in the modal coordinate system, respectively. For the torsional stiffness of the rail; The polar moment of inertia of the rail; For rail density; This represents the longitudinal torsional moment of the i-th fastener; This refers to the torsional displacement at the location of the rail fastener. The j-th wheel-rail interaction torque about the X-axis; and These represent the lateral acceleration and displacement of the rail in the modal coordinate system, respectively. Let Z be the moment of inertia of the rail about the Z-axis; The j-th wheel-rail interaction transverse component; This refers to the lateral displacement at the location of the rail fastener; The lateral resistance of the i-th fastener; and These represent the vertical acceleration and displacement of the rail in the modal coordinate system, respectively. Let Y be the moment of inertia of the rail about the Y-axis; This refers to the vertical displacement at the location of the rail fastener; The vertical resistance of the i-th fastener; This refers to the longitudinal displacement of the rail; This is the longitudinal vibration mode of the rail; This represents the longitudinal modal cutoff number of the rail; This represents the cutoff number for the torsional mode of the rail. This represents the vertical modal cutoff number of the rail; This represents the number of transverse modal cutoffs for the rail. This is the torsional vibration mode of the rail. This represents the vertical displacement of the rail. This refers to the lateral displacement of the rail. This refers to the torsional displacement of the rail. This represents the vertical displacement of the rail. This refers to the lateral displacement of the rail.

6. The system dynamic analysis method for continuous rigid frame railway bridges in mountainous areas according to claim 3, characterized in that, In the S203, the in-plane motion equations of the track slab dynamics equations established based on Mindlin thick plate theory include: ； in, The in-plane vibration acceleration of the track plate in modal coordinates; This refers to the in-surface vibration damping coefficient of the track slab. The frequency of vibration within the track slab surface; The vibration velocity within the track slab surface in modal coordinates; This represents the in-plane vibration displacement of the track slab in modal coordinates. Let Y be the modal order in the corresponding direction of Y; Let Y be the modal order in the corresponding direction of Y; The total number of fasteners under a single rail on a track slab; and The longitudinal force of the fasteners at the left and right rails; and The lateral force of the fasteners at the left and right rails; This refers to the longitudinal vibration mode within the track slab surface. This refers to the lateral vibration mode within the track slab surface. This refers to the longitudinal reaction force of the support layer under the track slab. This refers to the lateral reaction force of the support layer under the track slab. The out-of-plane motion equations include: ； in, The out-of-plane vibration acceleration of the track plate in modal coordinates; The frequency of the track slab's out-of-plane vibration; The out-of-plane vibration velocity of the track slab in modal coordinates; This represents the out-of-plane vibration displacement of the track slab in modal coordinates. This refers to the vertical vibration mode of the track slab. and The vertical force of the fasteners at the left and right rails; This represents the supporting force along the z-axis under the plate.

7. The system dynamic analysis method for continuous rigid frame railway bridges in mountainous areas according to claim 1, characterized in that, In step S3, the interaction forces at each contact interface are calculated using equivalent stiffness and damping parameters, including: S301. Determine the spatial position of each wheelset in the bridge coordinate system based on the train speed and time step, calculate the dynamic response at the corresponding wheel and the rail node, determine the wheel-rail contact relationship based on the dynamic response at the corresponding wheel and the rail node, and further calculate the interaction force at the rail-wheel contact interface. S302. Obtain the dynamic response of the track slab-rail, calculate the relative displacement and relative velocity between each interface node, and calculate the interaction force of each track slab-rail contact interface through the spring-damping element. S303. In the modeling of the interface between the track slab and the bridge, a discrete node array is used to replace the traditional continuous surface spring model. Spring-damping elements are independently established for each node to calculate the interaction force at the contact interface between the high-pier bridge and the track slab.

8. The system dynamic analysis method for continuous rigid frame railway bridges in mountainous areas according to claim 7, characterized in that, The discrete node array is composed of multiple rows of nodes arranged horizontally.

9. A system dynamic analysis method for continuous rigid frame railway bridges in mountainous areas according to claim 7, characterized in that, The interaction forces at the rail-wheel contact interface include the rail-wheel normal force and the creep force; The normal force is based on the contact compression and is calculated using Hertz normal contact force. The wheel-rail creep force is calculated based on Kallker's nonlinear creep theory, and the longitudinal relative velocity between the wheel and the rail at the actual contact point is improved in the calculation of the wheel-rail longitudinal creep force.

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