Fast prediction method for vibration of urban underground track structure based on energy method

Abstract

The present invention discloses a rapid prediction method for the vibration of urban underground track structures based on the energy method, comprising the following steps: constructing the vibration parameters of the track structure; calculating the strain energy and kinetic energy of the track structure; calculating the elastic potential energy between the components of the track structure based on the virtual spring model to form the internal connection energy functional of the track structure; applying an external force at the position of the rail according to the vibration parameters and obtaining the work done by the external force, and combining the strain energy, kinetic energy and energy functional of the track structure to form the total energy functional; variationalizing the total energy functional to obtain the motion characteristic equation of the track structure, inputting the required vibration frequency range to be predicted to obtain the generalized amplitude, and calculating the vibration response of each component based on the generalized amplitude; based on the framework of the energy method, the present invention uses the virtual spring model to simulate the direct constraint conditions of the track structure, decouples the displacement shape function from the boundary conditions, can simply and quickly represent the displacement field of the track structure, and comprehensively improves the accuracy of analysis and prediction.

Description

Technical Field

[0001] The present invention relates to the technical field of transportation engineering, and specifically to a rapid prediction method for urban underground track structure vibration based on the energy method. Background Technique

[0002] The subway can effectively relieve urban traffic congestion and facilitate people's travel, but the resulting environmental vibration problems are becoming increasingly prominent. The main methods for predicting subway environmental vibration include the field measurement method, semi-analytical method, numerical method, hybrid method, etc.

[0003] Field measurement can obtain the vibration data of the subway system during actual operation, avoiding the assumptions and deviations in theoretical analysis or simulation, and can accurately reflect the actual situation. Moreover, it can provide accurate vibration data for quantitative analysis, and then for further vibration control and optimization design. However, field measurement requires special equipment, technical personnel and time, and may involve interference or suspension of subway operation, thus increasing the measurement cost and time. In addition, subway vibration is affected by various factors, and a single field measurement may only reflect the vibration situation at a certain specific time and condition, and cannot comprehensively understand all possible changes and performances under different conditions.

[0004] For the numerical method, taking the finite element method as an example, the finite element method has great advantages in calculating complex periodic structures and is a useful analysis method, but the calculation accuracy depends on the reasonable degree of mesh division. This means that even a small change to the model requires re-meshing, seriously affecting the calculation efficiency during geometric parameter analysis and optimization.

[0005] The semi-analytical method can reduce the amount of calculation through simplified analytical steps, thereby improving the solution efficiency. Using the energy method is a typical one with the characteristic of transforming the functional boundary value problem into an extreme value problem, which is beneficial to the solution of coupling problems. This also has great advantages in the calculation of complex track structures. However, at present, there are few studies on using the energy method to predict the vibration of urban underground track structures. Summary of the Invention

[0006] Aiming at the deficiencies of the prior art, the present invention provides a rapid prediction method for urban underground track structure vibration based on the energy method, aiming to solve the problems mentioned in the background technique.

[0007] To achieve the above object, the present invention provides the following technical solution: A rapid prediction method for urban underground track structure vibration based on the energy method, including the following steps:

[0008] Step S1: Construct the vibration parameters of the track structure, including the displacements and rotations of the rail and invert, the displacements and rotations of the track slab, and the axial displacement, circumferential displacement and normal displacement of the tunnel;

[0009] Step S2: Obtain the strain energy and kinetic energy of the rail, fasteners, track slab, inverted arch, and tunnel according to the material parameters of the track structure and the calculation principle of the energy method;

[0010] Step S3: Simulate the connections between the components of the track structure using a virtual spring model, and convert the connections between the components of the track structure into the elastic potential energy of the virtual spring based on the displacement differences between the components, forming the internal connection energy functional of the track structure;

[0011] Step S4: Apply an excitation force at the rail position according to the vibration parameters of the track structure constructed in Step S1, and obtain the external work done by the excitation force. Combine the strain energy and kinetic energy of the track structure obtained in Step S2 and the internal connection energy functional of the track structure obtained in Step S3 to form the total energy functional;

[0012] Step S5: Variate the total energy functional to obtain the motion characteristic equation of the track structure, input the required predicted vibration frequency range, obtain the generalized amplitudes of each part of the track, and calculate the vibration responses of each component based on the generalized amplitudes.

[0013] Furthermore, according to the calculation principle of the energy method, represent the displacements and rotations of the rail and the inverted arch by combining the shape functions and the generalized amplitudes related to time:

[0014] ;

[0015] In the formula, represents the combination of the rail and the inverted arch, , represents the rail, represents the inverted arch; represents the number of shape functions of i.e., the truncation coefficient; represents the shape function vector of represents the generalized amplitude vector of the displacement of represents the generalized amplitude vector of the rotation angle of represents the transpose of represents the transpose of represents the displacement at any position of represents the coordinate in the direction of represents the rotation angle at any position of represents the number of the shape function in the represents the th unknown coefficient of displacement; denote the th unknown coefficient of rotation angle; denote time; denote the th shape function.

[0016] Furthermore, according to the calculation principle of the energy method, only the out-of-plane vibration of the track slab is considered, and the displacement and the rotation angles in the x and y directions of the track slab are expressed by combining the shape function and the time-related generalized amplitude:

[0017] ;

[0018] In the formula, denote the combination of the vertical bending, the rotation angle in the direction and the rotation angle in the direction at any position of the track slab; denote the vertical bending at any position of the track slab, denote the track slab direction rotation angle, denote the track slab direction rotation angle; and respectively denote the number of shape functions in the direction and the direction of the track slab; and respectively denote the coordinates in the direction and the direction of the track slab; denote shape function vector; and respectively denote the direction and the direction shape function vectors of denote the Kronecker product; denote the vertical bending at any position of the track slab; denote the rotation angle in the direction at any position of the track slab; denote the rotation angle in the direction at any position of the track slab; denote the number of the shape function in the y direction; denote the th generalized amplitude of the vertical bending of the track slab; denote the track slab direction A generalized amplitude; Indicates the track slab In the direction of the A generalized amplitude; The representing the vertical bending of the track slab Shape function; Indicates the track slab In the direction of the Shape function; Indicates the track slab In the direction of the Shape function; Indicates Transpose of; Indicates Transpose of; Indicates Transpose of; Indicates the generalized amplitude vector of the vertical bending of the track slab; Indicates the track slab Generalized amplitude vector of the angular rotation in the direction of; Indicates the track slab Generalized amplitude vector of the angular rotation in the direction of; Indicates the shape function vector of the vertical bending of the track slab; Indicates the track slab Shape function vector of the angular rotation in the direction of; Indicates the track slab Shape function vector of the angular rotation in the direction of.

[0019] Furthermore, according to the calculation principle of the energy method, the axial displacement 、circumferential displacement and normal displacement at any point on the tunnel are expressed by combining the shape function and the time-dependent generalized amplitude:

[0020] ;

[0021] In the formula, and respectively represent the number of shape functions in the axial and circumferential directions; 、 and respectively represent the generalized amplitudes in the axial, circumferential, and normal directions at any point on the tunnel; 、 respectively represent the direction coordinate and polar angle coordinate of the tunnel; Represents the th shape function in the axial direction of the tunnel; Represents the th shape function in the circumferential direction of the tunnel; Represents the th shape function in the normal direction of the tunnel Shape function; Denote The transpose of; Denote The transpose of; Denote The transpose of; , And Respectively denote the generalized amplitude vectors in the axial, circumferential and normal directions of the tunnel; Denote the shape function vector in the axial direction of the tunnel; Denote the shape function vector in the circumferential direction of the tunnel; Denote the shape function vector in the normal direction of the tunnel; Denote the combination of the axial, circumferential and normal directions of the tunnel; , Denote the axial direction of the tunnel; Denote the circumferential direction of the tunnel; Denote the normal direction of the tunnel; Denote along the tunnel Direction of motion of the shape function; Denote along the tunnel Direction of motion of the shape function.

[0022] Furthermore, the strain energy and kinetic energy of the rail and invert are expressed as:

[0023] ;

[0024] ;

[0025] In the formula, And Denote The strain energy and kinetic energy of; And Denote The stiffness matrix and mass matrix of; , , , , And Are respectively The elastic modulus, moment of inertia of the cross-section, shear coefficient, shear modulus, cross-sectional area and mass density of; The column vector , Denote the generalized amplitude vector of the displacement of the rail; Denote the generalized amplitude vector of the rotation angle of the rail; Denote the generalized amplitude vector of the displacement of the invert; Denote the generalized amplitude vector of the rotation angle of the invert; Denote the differential element symbol; Denote Conjugate transpose; Denote Derivative with respect to time; Denote Derivative with respect to time; Denote Derivative with respect to time; Denote Derivative with respect to time; is the length of the track structure.

[0026] Furthermore, considering the strain energy and kinetic energy under the bending vibration of the track slab, which are expressed as:

[0027] ;

[0028] ;

[0029] In the formula, and respectively denote the strain energy and kinetic energy of the track slab; denotes the bending stiffness of the track slab; denotes the width of the track slab; , , , , and respectively denote the shear coefficient, shear modulus, cross-sectional area, mass density, Poisson's ratio and thickness of the track slab; and respectively denote the stiffness matrix and mass matrix of the track slab; denotes the partial derivative symbol; Denote Derivative with respect to time; Denote Derivative with respect to time; Denote Derivative with respect to time.

[0030] Furthermore, considering the tunnel as a Kirchhoff-Love shell, the strain energy and kinetic energy of the tunnel vibration are expressed as:

[0031] ;

[0032] ;

[0033] In the formula, and respectively denote the strain energy and kinetic energy of the tunnel; , and respectively denote the radius, density and thickness of the tunnel; denotes the transpose of, is the tunnel strain vector; is the Hooke's operator of the tunnel; denotes the derivative of the normal displacement of the tunnel with respect to time; denotes the derivative of the axial displacement of the tunnel with respect to time; denotes the derivative of the circumferential displacement of the tunnel with respect to time; and denote the stiffness matrix and mass matrix of the tunnel respectively.

[0034] Furthermore, the specific process of step S3 is as follows:

[0035] The fastener and the track slab are simulated by a virtual spring. Through the displacement difference between the fastener and the track slab, the elastic potential energy of the fastener is obtained, which is expressed as:

[0036] ;

[0037] In the formula, denotes the coordinate of the q-th fastener on the rail and the track slab in the direction, denotes the number of fasteners; denotes the coordinate of the fastener on the track slab denotes the stiffness of the fastener; denotes the stiffness matrix of the fastener; denotes the elastic potential energy of the fastener; denotes the rail displacement at the -th fastener; denotes the track slab displacement at the -th fastener;

[0038] The supporting layer between the track slab and the inverted arch is simulated by a virtual spring. Through the displacement difference between the track slab and the inverted arch, the elastic potential energy of the supporting layer between the track slab and the inverted arch is obtained, which is expressed as:

[0039] ;

[0040] In the formula, denotes the surface stiffness of the supporting layer simulated by the virtual spring; denotes the stiffness matrix between the track slab and the inverted arch; denotes the elastic potential energy of the supporting layer between the track slab and the inverted arch; denotes the vertical displacement at any position of the track slab; denotes the vertical displacement at any position of the inverted arch;

[0041] The connection between the tunnel and the inverted arch is simulated by virtual springs. Multiple groups of virtual wire springs are used, and each group of virtual wire springs is connected along the tangential and normal directions of the tunnel respectively. The elastic potential energy between the tunnel and the inverted arch is obtained through the displacement difference between them, which is expressed as:

[0042] ;

[0043] In the formula, and respectively represent the tangential and normal virtual spring stiffnesses between the tunnel and the inverted arch; represents the angle of the tunnel at the connection of the -th group of virtual wire springs; represents the stiffness matrix of the connection between the tunnel and the inverted arch; represents the elastic potential energy of the connection between the tunnel and the inverted arch; represents the total number of groups of virtual wire springs; represents the circumferential displacement of the tunnel at the connection of the p-th group of virtual wire springs between the tunnel and the inverted arch; represents the normal displacement of the tunnel at the connection of the p-th group of virtual wire springs between the tunnel and the inverted arch;

[0044] The soil body is considered as a virtual spring outside the tunnel and simulated. The elastic potential energy of the soil body is expressed as:

[0045] ;

[0046] In the formula, , , respectively represent the stiffnesses of the virtual springs used to simulate the axial, circumferential, and normal directions of the soil body outside the tunnel; represents the stiffness matrix of the soil body; represents the elastic potential energy of the soil body.

[0047] Furthermore, the specific process of step S4 is as follows:

[0048] Apply an excitation force at the rail . The external work done by the excitation force is expressed as:

[0049] ;

[0050] In the formula, represents the magnitude of the applied excitation force; ; represents the displacement vector of the excitation point ; represents the shape function vector of the rail at the excitation point ; represents the work done by the excitation force applied to the rail; represents the time variable; Denote the excitation point the displacement of the rail at the position;

[0051] Consider the work done by the external force as a pseudo-potential energy, obtain the total energy functional under the excitation and express it with the Lagrangian:

[0052] ;

[0053] In the formula, denotes the Lagrangian; denotes the kinetic energy of the rail; denotes the kinetic energy of the invert; denotes the elastic potential energy of the rail; denotes the elastic potential energy of the invert; denotes the mass matrix of the rail; denotes the mass matrix of the invert; denotes the stiffness matrix of the rail; denotes the stiffness matrix of the invert; denotes the stiffness matrix of the tunnel.

[0054] Furthermore, the specific process of step S5 is as follows: According to the Lagrange equation , solve the total energy functional to obtain the motion characteristic equation of the track structure:

[0055] ;

[0056] In the formula, denotes the circular frequency;

[0057] By scanning the circular frequency in the motion characteristic equation of the track structure, obtain the column vector , and substitute into the displacement and rotation angle of the rail and invert, the displacement and rotation angle of the track slab, and the axial displacement, circumferential displacement, and normal displacement equations of the tunnel in step S1 to obtain the vibration displacement at any position of the track structure.

[0058] Compared with the existing technologies, the present invention has the following beneficial effects:

[0059] (1) Based on the framework of the energy method, the present invention uses a virtual spring model to simulate the connection conditions between the components of the track structure, decouples the displacement shape function, can simply and quickly represent the displacement field of the track structure, converts the connection conditions into the elastic potential energy of the virtual spring, and couples it into the total energy functional, comprehensively improving the accuracy of analysis and prediction and simplifying the calculation process.

[0060] (2) The present invention simplifies the analysis of the track structure by decoupling the displacement shape functions, enabling rapid calculation of the displacement field of the track structure and enhancing the convenience of prediction. By simulating the connections between components using the virtual spring model, complex connection conditions can be transformed into an elastic potential energy form that is easy to handle, reducing the impact of complex boundary conditions on the calculation model.

[0061] (3) Compared with the prior art, the present invention significantly simplifies the handling of connection conditions of the track structure by introducing the virtual spring model, solving the problem that complex connection conditions are difficult to effectively represent and calculate in traditional semi-analytical methods. Existing semi-analytical techniques often have problems such as poor adaptability and high computational complexity when dealing with different tunnel structure forms. In contrast, the method provided by the present invention can adapt to various tunnel structure forms, has strong versatility and flexibility, enabling the present invention to be widely applied in different engineering projects, greatly improving the calculation efficiency and reducing the difficulty of project implementation. BRIEF DESCRIPTION OF THE DRAWINGS

[0062] Figure 1 It is a flowchart of the method of the present invention.

[0063] Figure 2 It is a front view schematic diagram of the track structure.

[0064] Figure 3 It is a side view schematic diagram of the track structure.

[0065] Figure 4 It is a schematic diagram of the connections between components of the track structure.

[0066] Figure 5 It is a comparison diagram of the rail vibration displacement between the method of the present invention and the finite element method.

[0067] Figure 6 It is a comparison diagram of the track slab vibration displacement between the method of the present invention and the finite element method.

[0068] Figure 7 It is a comparison diagram of the invert vibration displacement between the method of the present invention and the finite element method.

[0069] Figure 8 It is a comparison diagram of the tunnel vibration displacement between the method of the present invention and the finite element method. DETAILED DESCRIPTION OF THE INVENTION

[0070] As Figure 1 shown, the present invention provides a technical solution: a rapid prediction method for the vibration of urban underground track structures based on the energy method:

[0071] Step S1: Construct the vibration parameters of the track structure, including the displacements and rotations of the rail and the invert, the displacements and rotations of the track slab, and the axial displacement, circumferential displacement, and normal displacement of the tunnel.

[0072] As Figures 2 - 4 shown, the track structure includes rails, fasteners, track slabs, inverted arches and tunnels; the rails are described by Timoshenko beams, the fasteners are described by springs, the track slab under the fasteners is considered as a Mindlin plate for description, the inverted arch under the track slab is considered as a Timoshenko beam, the tunnel is considered as a cylindrical shell, and the surrounding soil is described as elastically contacting the tunnel; the vertical displacement and rotation angle of the rails are respectively expressed as ; the vertical displacement and rotation angle of the inverted arch are respectively expressed as ; the vertical displacement of the track slab, the rotation angle in the direction and the rotation angle in the direction are respectively ; the axial displacement, circumferential displacement and normal displacement of a point on the tunnel wall are respectively expressed as ; the length of the track structure is ; the fastener spacing is ; the width and thickness of the track slab are respectively .

[0073] According to the calculation principle of the energy method, the displacements and rotation angles of the rails and inverted arches are expressed by combining the shape functions and the generalized amplitudes related to time:

[0074] (1);

[0075] In the formula, represents the combination of the rails and the inverted arch, , represents the rails, represents the inverted arch; represents the number of shape functions, that is, the truncation coefficient; represents the shape function vector of ; represents the generalized amplitude vector of the displacement of ; represents the generalized amplitude vector of the rotation angle of ; represents the transpose of ; represents the transpose of ; represents the displacement at any position of ; represents the coordinate in the direction of ; represents the rotation angle at any position of ; represents the th unknown coefficient of displacement; denote the th unknown coefficient of rotation angle; denote time; denote the th shape function.

[0076] According to the calculation principle of the energy method, only the out-of-plane vibration of the track slab is considered, and the displacement of the track slab and the rotation angles in the x and y directions are expressed by combining the shape function and the generalized amplitude related to time:

[0077] (2);

[0078] In the formula, denote the combination of the vertical bending, direction rotation angle and direction rotation angle at any position of the track slab; , denote the vertical bending at any position of the track slab, denote the track slab direction rotation angle, denote the track slab direction rotation angle; and respectively denote the number of shape functions in the direction and direction of the track slab; and respectively denote the coordinates in the direction and direction of the track slab; denote the shape function vector of and respectively denote of direction and direction shape function vectors of denote the Kronecker product; denote the vertical bending at any position of the track slab; denote the direction rotation angle at any position of the track slab; denote the direction rotation angle at any position of the track slab; denote the number of the shape function in the y direction; denote the th generalized amplitude of the vertical bending of the track slab; denote the track slab the a generalized amplitude; representing the track slab in the direction of the a generalized amplitude; the shape function representing the vertical bending of the track slab; representing the track slab in the direction of the shape function; representing the track slab in the direction of the shape function; representing the transpose of; representing the transpose of; representing the transpose of; the generalized amplitude vector representing the vertical bending of the track slab; representing the track slab in the direction of the the generalized amplitude vector of the angular rotation; representing the track slab in the direction of the the generalized amplitude vector of the angular rotation; the shape function vector representing the angular rotation of the track slab; representing the track slab in the direction of the

[0079] According to the calculation principle of the energy method, the axial displacement circumferential displacement and the normal displacement of any point on the tunnel are expressed by combining the shape function and the time - related generalized amplitude:

[0080] (3);

[0081] wherein, and respectively represent the number of shape functions in the axial and circumferential directions; , and respectively represent the generalized amplitudes in the axial, circumferential and normal directions of any point on the tunnel; , respectively represent the direction coordinate and the polar angle coordinate of the tunnel; represents the shape function of the axial direction of the tunnel; shape function of the circumferential direction of the tunnel; Shape function; Denote The transpose of; Denote The transpose of; Denote The transpose of; , And Respectively represent the generalized amplitude vectors in the axial, circumferential and normal directions of the tunnel; Denote the shape function vector in the axial direction of the tunnel; Denote the shape function vector in the circumferential direction of the tunnel; Denote the shape function vector in the normal direction of the tunnel; Denote the combination of the axial, circumferential and normal directions of the tunnel; , Denote the axial direction of the tunnel; Denote the circumferential direction of the tunnel; Denote the normal direction of the tunnel; Denote along the tunnel The shape function of the moving direction; Denote along the tunnel The shape function of the moving direction.

[0082] Step S2: Obtain the strain energy and kinetic energy of the rail, fastener, track slab, invert and tunnel according to the material parameters of the track structure and the calculation principle of the energy method.

[0083] The strain energy and kinetic energy of the rail and invert are expressed as:

[0084] (4);

[0085] (5);

[0086] In the formula, And Denote The strain energy and kinetic energy of; And Denote The stiffness matrix and mass matrix of; , , , , And Are respectively The elastic modulus, moment of inertia of the cross section, shear coefficient, shear modulus, cross-sectional area and mass density of; The column vector , Denote the generalized amplitude vector of the displacement of the rail; Denote the generalized amplitude vector of the rotation angle of the rail; Generalized amplitude vector representing the displacement of the invert Generalized amplitude vector representing the rotation angle of the invert Symbol representing an infinitesimal element Represents Conjugate transpose of Represents Derivative with respect to time Represents Derivative with respect to time Represents Derivative with respect to time Represents Derivative with respect to time.

[0087] Considering the strain energy and kinetic energy under the bending vibration of the track slab, expressed as:

[0088] (6);

[0089] (7);

[0090] In the formula, and Respectively represent the strain energy and kinetic energy of the track slab; Represents the bending stiffness of the track slab; Represents the width of the track slab; , , , , and Respectively represent the shear coefficient, shear modulus, cross-sectional area, mass density, Poisson's ratio and thickness of the track slab; and Respectively represent the stiffness matrix and mass matrix of the track slab; Symbol representing partial derivative; Represents Derivative with respect to time; Represents Derivative with respect to time; Represents Derivative with respect to time.

[0091] Considering the tunnel as a Kirchhoff-Love shell, the strain energy and kinetic energy of the tunnel vibration, expressed as:

[0092] (8);

[0093] (9);

[0094] In the formula, and Respectively represent the strain energy and kinetic energy of the tunnel; , and respectively represent the radius, density, and thickness of the tunnel; represents the transpose of is the tunnel strain vector; is the Hooke's operator of the tunnel; represents the derivative of the normal displacement of the tunnel with respect to time; represents the derivative of the axial displacement of the tunnel with respect to time; represents the derivative of the circumferential displacement of the tunnel with respect to time; and respectively represent the stiffness matrix and mass matrix of the tunnel.

[0095] Step S3: Simulate the connections between the components of the track structure using a virtual spring model, and convert the connections between the components of the track structure into the elastic potential energy of the virtual springs according to the displacement differences of the components, forming the internal connection energy functional of the track structure.

[0096] Simulate the connection between the fastener and the track slab using a virtual spring. Through the displacement difference between the fastener and the track slab, the elastic potential energy of the fastener can be obtained, expressed as:

[0097] (10);

[0098] In the formula, represents the coordinate of the q-th fastener on the rail and the track slab in the direction, represents the number of fasteners. There are 9 fasteners in one cell; represents the coordinate of the fastener on the track slab; represents the stiffness of the fastener; represents the stiffness matrix of the fastener; represents the elastic potential energy of the fastener; represents the displacement of the rail at the -th fastener; represents the displacement of the track slab at the

[0099] Simulate the support layer between the track slab and the inverted arch using a virtual spring. Through the displacement difference between the track slab and the inverted arch, the elastic potential energy of the support layer between the track slab and the inverted arch can be obtained, expressed as:

[0100] (11);

[0101] In the formula, represents the surface stiffness of the support layer simulated by the virtual spring; represents the stiffness matrix between the track slab and the inverted arch; Represents the elastic potential energy of the supporting layer between the track slab and the inverted arch; Represents the vertical displacement at any position of the track slab; Represents the vertical displacement at any position of the inverted arch.

[0102] The connection between the tunnel and the inverted arch is simulated by virtual springs. Multiple groups of virtual wire springs are used, and each group of virtual wire springs is connected along the tangential and normal directions of the tunnel respectively; through the displacement difference between the tunnel and the inverted arch, the elastic potential energy between the tunnel and the inverted arch is obtained, which is expressed as:

[0103] (12);

[0104] In the formula, and respectively represent the tangential and normal virtual spring stiffnesses between the tunnel and the inverted arch; Represents the angle of the tunnel at the connection of the th group of virtual wire springs; Represents the stiffness matrix of the connection between the tunnel and the inverted arch; Represents the elastic potential energy of the connection between the tunnel and the inverted arch; Represents the total number of groups of virtual wire springs; Represents the circumferential displacement of the tunnel at the connection of the virtual wire spring between the th group of the tunnel and the inverted arch;

[0105] The soil body is considered as a virtual spring outside the tunnel and simulated. The elastic potential energy of the soil body can be expressed as:

[0106] (13);

[0107] In the formula, , , respectively represent the stiffnesses of the virtual springs for simulating the axial, circumferential and normal directions of the soil body outside the tunnel; Represents the stiffness matrix of the soil body; Represents the elastic potential energy of the soil body.

[0108] Step S4: According to the vibration parameters of the track structure constructed in Step S1, an excitation force (wheel-rail interaction force) is applied at the position of the rail, and the external work done by the excitation force is obtained. Combining the strain energy and kinetic energy of the track structure obtained in Step S2 and the internal connection energy functional of the track structure obtained in Step S3, a total energy functional is formed.

[0109] Assume that an excitation force is applied at the rail position. Then the external work done by this excitation force can be expressed as:

[0110] (14);

[0111] In the formula, represents the magnitude of the applied excitation force; ; represents the excitation point displacement vector; represents the excitation point shape function vector of the rail at the position; represents the work done by the excitation force applied to the rail; represents the time variable; represents the excitation point displacement of the rail at the position.

[0112] Considering the work done by the external force as the pseudo-potential energy, the total energy functional under the excitation is obtained and expressed by the Lagrangian:

[0113] (15);

[0114] In the formula, represents the Lagrangian; represents the kinetic energy of the rail; represents the kinetic energy of the inverted arch; represents the elastic potential energy of the rail; represents the elastic potential energy of the inverted arch; represents the mass matrix of the rail; represents the mass matrix of the inverted arch; represents the stiffness matrix of the rail; represents the stiffness matrix of the inverted arch; represents the stiffness matrix of the tunnel.

[0115] Step S5: Variate the total energy functional to obtain the motion characteristic equation of the track structure, input the required predicted vibration frequency range, obtain the generalized amplitudes of each part of the track, and calculate the vibration responses of each component based on the generalized amplitudes.

[0116] According to the Lagrangian equation , solving the total energy functional, the motion characteristic equation of the track structure can be obtained:

[0117] (16);

[0118] In the formula, represents the circular frequency.

[0119] By scanning the circular frequency in the motion characteristic equation of the track structure, the column vector can be obtained. Substituting Substituting the displacement and rotation angle of the rail and invert, the displacement and rotation angle of the track slab, and the axial displacement, circumferential displacement, and normal displacement equations of the tunnel into the equations, the vibration displacement at any position of the track structure can be obtained.

[0120] To verify the accuracy of this embodiment, the same material parameters are taken, and the method of the present invention and the finite element method are used for calculation to obtain the track vibration results as Figures 5 - 8 shown, and the results of the two are in good agreement.

[0121] Although the embodiments of the present invention have been shown and described, it will be understood by those of ordinary skill in the art that various changes, modifications, substitutions, and variations can be made to these embodiments without departing from the principles and spirit of the present invention, and the scope of the present invention is defined by the appended claims and their equivalents.

Claims

1. A rapid prediction method for the vibration of urban underground track structures based on the energy method, characterized in that, It includes the following steps: Step S1: Construct the vibration parameters of the track structure, including the displacements and rotations of the rail and invert, the displacements and rotations of the track slab, and the axial displacement, circumferential displacement, and normal displacement of the tunnel; Step S2: Obtain the strain energy and kinetic energy of the rail, fastener, track slab, invert, and tunnel according to the material parameters of the track structure and the calculation principle of the energy method; Step S3: Simulate the connections between the components of the track structure with a virtual spring model, and convert the connections between the components of the track structure into the elastic potential energy of the virtual spring according to the displacement differences between the components, forming the internal connection energy functional of the track structure; Step S4: Apply an excitation force at the rail position according to the vibration parameters of the track structure constructed in Step S1, and obtain the external work done by the excitation force. Combine the strain energy and kinetic energy of the track structure obtained in Step S2 and the internal connection energy functional of the track structure obtained in Step S3 to form the total energy functional; Step S5: Variate the total energy functional to obtain the motion characteristic equation of the track structure, input the required predicted vibration frequency range, obtain the generalized amplitudes of each part of the track, and calculate the vibration responses of each component based on the generalized amplitudes; The specific process of Step S3 is as follows: Simulate the fastener and the track slab with a virtual spring, and obtain the elastic potential energy of the fastener through the displacement difference between the fastener and the track slab, which is expressed as: ； In the formula, represents the elastic potential energy of the fastener; represents the stiffness of the fastener; represents the rail displacement at the th fastener; represents the slab displacement at the th fastener; represents the coordinates of the qth fastener in the direction on the rail and slab, represents the number of fasteners; represents the conjugate transpose of represents the stiffness matrix of the fastener; represents a column vector, represents the generalized amplitude vector of the rail displacement, represents the generalized amplitude vector of the rail rotation angle, represents the generalized amplitude vector of the vertical bending of the slab, represents the direction rotation angle of the slab; represents the direction rotation angle of the slab; represents the generalized amplitude vector of the invert displacement, represents the generalized amplitude vector of the invert rotation angle, , and respectively represent the generalized amplitude vectors in the axial, circumferential and normal directions of the tunnel; Simulate the support layer between the track slab and the invert with a virtual spring, and obtain the elastic potential energy of the support layer between the track slab and the invert through the displacement difference between the track slab and the invert, which is expressed as: ； In the formula, represents the elastic potential energy of the supporting layer between the track slab and the inverted arch; represents the surface stiffness between the supporting layers simulated by a virtual spring; is the length of the track structure; represents the vertical displacement at any position of the track slab; represents the vertical displacement at any position of the inverted arch; represents the differential symbol; represents the stiffness matrix between the track slab and the inverted arch; Simulate the connection between the tunnel and the invert with a virtual spring, and use multiple groups of virtual wire springs. Each group of virtual wire springs is connected along the tangential and normal directions of the tunnel respectively; obtain the elastic potential energy between the tunnel and the invert through the displacement difference between the tunnel and the invert, which is expressed as: ； In the formula, represents the elastic potential energy of the connection between the tunnel and the inverted arch; represents the total number of groups of virtual line springs; and respectively represent the tangential and normal virtual spring stiffnesses between the tunnel and the inverted arch; represents the circumferential displacement of the tunnel at the connection of the p-th group of virtual line springs between the tunnel and the inverted arch; represents the angle of the tunnel at the connection of the group of virtual line springs; represents the normal displacement of the tunnel at the connection of the p-th group of virtual line springs between the tunnel and the inverted arch; represents the stiffness matrix of the connection between the tunnel and the inverted arch; Consider the soil mass as a virtual spring outside the tunnel and simulate it. The elastic potential energy of the soil mass is expressed as: ； In the formula, represents the elastic potential energy of the soil mass; , and respectively represent the axial displacement, circumferential displacement and normal displacement of any point on the tunnel; , , respectively represent the stiffnesses of the virtual springs for simulating the axial, circumferential and normal directions of the soil mass outside the tunnel; , respectively represent the direction coordinate and polar angle coordinate of the tunnel; represents the stiffness matrix of the soil mass.

2. The rapid prediction method for vibration of urban underground track structure based on energy method according to claim 1, wherein: According to the calculation principle of the energy method, express the displacements and rotations of the rail and the invert by combining the shape function and the time-related generalized amplitude; ； In the formula, represents the combination of the rail and the inverted arch, , represents the rail, represents the inverted arch; represents the number of shape functions of , that is, the truncation coefficient; represents the shape function vector of ; represents the generalized amplitude vector of the displacement of ; represents the generalized amplitude vector of the rotation angle of ; represents the transpose of ; represents the transpose of ; represents the displacement at any position of ; represents the coordinate in the direction of ; represents the rotation angle at any position of ; represents the number of the shape function in the direction; represents the th unknown coefficient of the displacement; represents the th unknown coefficient of the rotation angle; represents time; represents the th shape function of .

3. The rapid prediction method for the vibration of urban underground track structures based on the energy method according to claim 2, characterized in that: According to the calculation principle of the energy method, only consider the out-of-plane vibration of the track slab, and express the displacement and the rotations in the x and y directions of the track slab by combining the shape function and the time-related generalized amplitude; ； In the formula, represents the combination of the vertical bending, the direction angle of rotation, and the direction angle of rotation at any position of the track slab; , represents the vertical bending at any position of the track slab, represents the direction angle of rotation of the track slab, represents the direction angle of rotation of the track slab; and respectively represent the number of shape functions in the direction and the direction of the track slab; and respectively represent the coordinates in the direction and the direction of the track slab; represents the shape function vector of; and respectively represent the of direction and the direction shape function vectors; represents the Kronecker product; represents the vertical bending at any position of the track slab; represents the direction angle of rotation at any position of the track slab; represents the direction angle of rotation at any position of the track slab; represents the number of the shape function in the y direction; represents the th generalized amplitude of the vertical bending of the track slab; represents the th th generalized amplitude in the direction of the track slab; represents the th th generalized amplitude in the direction of the track slab; represents the th th shape function in the direction of the track slab; represents the th represents the transpose of; represents the transpose of; represents the transpose of; The shape function vector representing the vertical bending of the track slab; Represents the track slab The shape function vector of the angular rotation in the direction; The shape function vector of the angular rotation in the 4. The rapid vibration prediction method for urban underground track structure based on the energy method according to claim 3, characterized in that: According to the calculation principle of the energy method, the axial displacement of any point on the tunnel , circumferential displacement and normal displacement are expressed by combining the shape function and the time-related generalized amplitude: ； In the formula, and represent the number of shape functions in the axial and circumferential directions respectively; , and represent the generalized amplitudes in the axial, circumferential, and normal directions of any point on the tunnel respectively; represents the th shape function in the axial direction of the tunnel; represents the th shape function in the circumferential direction of the tunnel; represents the th shape function in the normal direction of the tunnel; represents transpose; represents transpose; represents transpose; represents the shape function vector in the axial direction of the tunnel; represents the shape function vector in the circumferential direction of the tunnel; represents the shape function vector in the normal direction of the tunnel; represents the combination of the axial, circumferential, and normal directions of the tunnel; , represents the axial direction of the tunnel; represents the circumferential direction of the tunnel; represents the normal direction of the tunnel; represents the shape function moving along the tunnel direction; represents the shape function moving along the tunnel direction.

5. The rapid prediction method for the vibration of urban underground track structure based on the energy method according to claim 4, characterized in that: The strain energy and kinetic energy of the rail and the invert are expressed as: ； ； Wherein, and represent the strain energy and kinetic energy of; and represent the stiffness matrix and mass matrix of; , , , , and are respectively the elastic modulus, moment of inertia of cross-section, shear coefficient, shear modulus, cross-sectional area and mass density of; represents the derivative with respect to time; represents the derivative with respect to time; represents the derivative with respect to time; represents the derivative with respect to time.

6. The rapid vibration prediction method for urban underground track structures based on the energy method according to claim 5, characterized in that: Consider the strain energy and kinetic energy under the bending vibration of the track slab, which is expressed as: ； ； Wherein, and respectively represent the strain energy and kinetic energy of the track slab; represents the flexural stiffness of the track slab; represents the width of the track slab; 、 、 、 、 and respectively represent the shear coefficient, shear modulus, cross-sectional area, mass density, Poisson's ratio and thickness of the track slab; and respectively represent the stiffness matrix and mass matrix of the track slab; represents the partial derivative symbol; represents the derivative with respect to time; represents the derivative with respect to time; represents the derivative with respect to time.

7. The rapid vibration prediction method for urban underground track structures based on the energy method according to claim 6, characterized in that: Consider the tunnel as a Kirchhoff-Love shell, and the strain energy and kinetic energy of the tunnel vibration are expressed as: ； ； In the formula, and represent the strain energy and kinetic energy of the tunnel respectively; , and represent the radius, density and thickness of the tunnel respectively; represents the transpose of, is the tunnel strain vector; is the Hooke operator of the tunnel; represents the derivative of the normal displacement of the tunnel with respect to time; represents the derivative of the axial displacement of the tunnel with respect to time; represents the derivative of the circumferential displacement of the tunnel with respect to time; and represent the stiffness matrix and mass matrix of the tunnel respectively.

8. The rapid prediction method for the vibration of urban underground track structure based on the energy method according to claim 7, characterized in that: The specific process of Step S4 is as follows: On the rail An excitation force is applied, and the external work done by the excitation force is expressed as: ； In the formula, represents the magnitude of the applied excitation force; ; represents the excitation point displacement vector; represents the excitation point shape function vector of the rail at the point; represents the work done by the excitation force applied to the rail; represents the time variable; represents the displacement of the rail at the excitation point ; Consider the external work done as a pseudo-potential energy, obtain the total energy functional under the excitation and express it with the Lagrangian; ； In the formula, represents the Lagrangian; represents the kinetic energy of the rail; represents the kinetic energy of the invert; represents the elastic potential energy of the rail; represents the elastic potential energy of the invert; represents the mass matrix of the rail; represents the mass matrix of the invert; represents the stiffness matrix of the rail; represents the stiffness matrix of the invert; represents the stiffness matrix of the tunnel.

9. The rapid prediction method for the vibration of urban underground track structure based on the energy method according to claim 8, characterized in that: The specific process of step S5 is as follows: According to the Lagrange equation , solve the total energy functional to obtain the motion characteristic equation of the orbital structure: ； In the formula, represents the angular frequency; By scanning the circular frequency in the motion characteristic equation of the track structure , a column vector is obtained. Substitute into the displacement and rotation angle equations of the rail and invert arch, the displacement and rotation angle equations of the track slab, and the axial displacement, circumferential displacement, and normal displacement equations of the tunnel in step S1 to obtain the vibration displacement at any position of the track structure.

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