Urban underground track structure vibration rapid prediction method based on energy method

Abstract

The invention discloses an urban underground track structure vibration rapid prediction method based on an energy method. The method comprises the following steps: constructing vibration parameters of a track structure; calculating strain energy and kinetic energy of the track structure; elastic potential energy among all components of the track structure is calculated based on the virtual spring model, and an internal connection energy functional of the track structure is formed; according to the vibration parameters, external force is applied to the steel rail position, work of the external force is obtained, and a total energy functional is formed by combining strain energy, kinetic energy and an energy functional of the track structure; performing functional variation on the total energy to obtain a motion characteristic equation of the track structure, inputting a vibration frequency range required to be predicted to obtain a generalized amplitude, and calculating vibration response of each component based on the generalized amplitude; based on a framework of an energy method, a virtual spring model is utilized to simulate a direct constraint condition of a track structure, a displacement shape function and a boundary condition are decoupled, a displacement field of the track structure can be simply and quickly represented, and analysis and prediction precision is comprehensively improved.

Description

Technical Field

[0001] The invention relates to the technical field of transportation engineering, and in particular to a method for quickly predicting vibration of an urban underground rail structure based on an energy method. Background Art

[0002] Subways can effectively ease urban traffic congestion and facilitate people's travel, but the environmental vibration problem caused by them is becoming increasingly prominent. The main methods for predicting subway environmental vibration include field measurement method, semi-analytical method, numerical method, hybrid method, etc.

[0003] Field measurements can obtain vibration data of the subway system in actual operation, avoiding assumptions and deviations in theoretical analysis or simulation, and accurately reflecting the actual situation. In addition, it can provide accurate vibration data, which is convenient for quantitative analysis, and then further vibration control and optimization design. However, field measurements require special equipment, technicians and time, and may involve interference or suspension of subway operations, thereby increasing measurement costs and time. In addition, subway vibration is affected by many factors, and a single field measurement may only reflect the vibration situation at a specific time and conditions, and cannot fully understand all possible changes and performance under different circumstances.

[0004] Numerical methods, such as finite element method, have great advantages in calculating complex periodic structures and are a good analytical method, but the calculation accuracy depends on the reasonableness of mesh division. This means that even a small change to the model must be re-meshed, which seriously affects the calculation efficiency during geometric parameter analysis and optimization.

[0005] The semi-analytical method can reduce the amount of calculation by simplifying the analytical steps, thereby improving the solution efficiency. The energy method is a typical energy method that has the characteristic of transforming the functional boundary value problem into an extreme value problem, which is conducive to the solution of the coupling problem. This also has a great advantage in the calculation of complex track structures. However, there are few energy methods that use energy methods to predict the vibration of urban underground track structures. Summary of the invention

[0006] In view of the deficiencies of the prior art, the present invention provides a method for quickly predicting the vibration of an urban underground rail structure based on an energy method, which aims to solve the problems mentioned in the background technology.

[0007] To achieve the above object, the present invention provides the following technical solution: a method for rapid prediction of urban underground rail structure vibration based on energy method, comprising the following steps: Step S1: constructing vibration parameters of the track structure, including displacement and rotation angle of the rail and the invert, displacement and rotation angle of the track plate, and axial displacement, circumferential displacement, and normal displacement of the tunnel; Step S2: obtaining the strain energy and kinetic energy of the rails, fasteners, track plates, inverts and tunnels according to the material parameters of the track structure and the energy method calculation principle; Step S3: The connection between the components of the track structure is simulated by a virtual spring model, and according to the displacement difference of each component, the connection between the components of the track structure is converted into the elastic potential energy of the virtual spring to form the internal connection energy functional of the track structure; Step S4: According to the vibration parameters of the track structure constructed in step S1, an excitation force is applied at the rail position, and the external force work of the excitation force is obtained, and 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 are combined to form a total energy functional; Step S5: The motion characteristic equation of the track structure is obtained by varying the total energy functional, and the vibration frequency range to be predicted is input to obtain the generalized amplitude of each part of the track, and the vibration response of each component is calculated based on the generalized amplitude.

[0008] Furthermore, according to the calculation principle of the energy method, the displacement and rotation angle of the rail and the invert are expressed by combining the shape function and the time-dependent generalized amplitude: ; In the formula, It represents the combination of rail and invert. , Indicates rails, Indicates an inverted arch; express The number of shape functions, i.e., the truncation coefficient; express The shape function vector of ; express The generalized magnitude vector of the displacement; express The generalized magnitude vector of the rotation angle; express The transpose of express The transpose of express The displacement at any position of ; express of The coordinates of the direction; express The corner at any position of ; express The number of the directional shape function; express The displacement unknown coefficients; express The corner unknown coefficients; Indicates time; express No. A shape function.

[0009] Furthermore, according to the calculation principle of the energy method, the track plate only considers the out-of-plane vibration, and the displacement of the track plate and the rotation angle in the x and y directions are expressed by combining the shape function and the time-related generalized amplitude: ; In the formula, Indicates the vertical bending of the track plate at any position. Direction angle and Combination of direction angles; , Indicates the vertical bending of the track plate at any position. Indicates track plate Direction angle, Indicates track plate Direction angle; and Respectively represent track slabs Direction and The number of directional shape functions; and Respectively represent track slabs Direction and The coordinates of the direction; express The shape function vector of ; and Respectively of Direction and Direction type function vector; represents the Klaunec product; Represents the vertical bending at any position of the track slab; Indicates the position of the track plate Angle of direction; Indicates the position of the track plate Angle of direction; Indicates the number of the shape function in the y direction; The vertical bending of the track slab A generalized amplitude; Indicates track plate Direction A generalized amplitude; Indicates track plate Direction A generalized amplitude; The vertical bending of the track slab shape function; Indicates track plate Direction shape function; Indicates track plate Direction shape function; express The transpose of express The transpose of express The transpose of The generalized magnitude vector representing the vertical bending of the track slab; Indicates track plate The generalized magnitude vector of the direction angle; Indicates track plate The generalized magnitude vector of the direction angle; The shape function vector representing the vertical bending of the track slab; Indicates track plate The shape function vector of the direction angle; Indicates track plate The shape function vector for the direction rotation.

[0010] Furthermore, according to the calculation principle of the energy method, the axial displacement of any point on the tunnel is Circumferential displacement and normal displacement It is expressed by combining shape functions and time-dependent generalized amplitudes: ; In the formula, and Represent the number of shape functions in the axial and circumferential directions respectively; , and They represent the generalized amplitudes in the axial, circumferential and normal directions at any point in the tunnel respectively; , Respectively represent the tunnel Directional and polar angle coordinates; Indicates the axial direction of the tunnel shape function; Indicates the tunnel circumference shape function; The tunnel normal shape function; express The transpose of express The transpose of express The transpose of , and represent the generalized magnitude vectors in the axial, circumferential and normal directions of the tunnel respectively; represents the shape function vector of the tunnel axis; Represents the shape function vector of the tunnel circumference; The shape function vector representing the tunnel normal; It represents the combination of axial, circumferential and normal directions of the tunnel; , Indicates the axial direction of the tunnel; Indicates the tunnel circumference; represents the normal direction of the tunnel; Indicates along the tunnel Shape functions for directional motion; Indicates along the tunnel Shape functions for directional motion.

[0011] Furthermore, the strain energy and kinetic energy of the rail and invert are expressed as: ; ; In the formula, and express strain energy and kinetic energy; and express The stiffness matrix and mass matrix of ; , , , , and They are The elastic modulus, section moment of inertia, shear coefficient, shear modulus, cross-sectional area, and mass density of , The generalized magnitude vector representing the displacement of the rail; The generalized magnitude vector representing the rotation angle of the rail; The generalized magnitude vector representing the displacement of the invert; The generalized magnitude vector representing the rotation angle of the invert; Indicates the symbol of the differential element; express The conjugate transpose of ; express Derivative with respect to time; express Derivative with respect to time; express Derivative with respect to time; express Derivative with respect to time; is the length of the track structure.

[0012] Furthermore, considering the strain energy and kinetic energy of the track plate under bending vibration, it can be expressed as: ; ; In the formula, and represent the strain energy and kinetic energy of the track plate respectively; It represents the bending stiffness of the track slab; Indicates the width of the track plate; , , , , and They represent the shear coefficient, shear modulus, cross-sectional area, mass density, Poisson's ratio and thickness of the track slab respectively; and denote the stiffness matrix and mass matrix of the track slab respectively; represents the symbol of partial derivative; express Derivative with respect to time; express Derivative with respect to time; express The derivative with respect to time.

[0013] Furthermore, considering the tunnel as a Kirchhoff-Love shell, the strain energy and kinetic energy of 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; express The transpose of is the tunnel strain vector; is the Hookean operator of the tunnel; represents the time derivative of the tunnel normal displacement; represents the time derivative of the tunnel axial displacement; represents the time derivative of the tunnel circumferential displacement; and represent the stiffness matrix and mass matrix of the tunnel respectively.

[0014] Furthermore, the specific process of step S3 is as follows: The virtual spring is used to simulate the space between the fastener and the track plate. The elastic potential energy of the fastener is obtained through the displacement difference between the fastener and the track plate, which is expressed as: ; In the formula, Indicates that the qth fastener is on the rail and the track plate The coordinates of the direction, Indicates the number of fasteners; Indicates the fastener on the track plate coordinate; Indicates the stiffness of the fastener; represents the stiffness matrix of the fastener; represents the elastic potential energy of the fastener; Indicates Rail displacement at each fastener; Indicates Track plate displacement at each fastener; The support layer between the track plate and the invert is simulated by virtual springs. The elastic potential energy of the support layer between the track plate and the invert is obtained through the displacement difference between the track plate and the invert, which is expressed as: ; In the formula, Indicates that virtual springs are used to simulate the surface stiffness between support layers; represents the stiffness matrix between the track slab and the invert; represents the elastic potential energy of the supporting layer between the track slab and the invert; Indicates the vertical displacement of any position of the track plate; It represents the vertical displacement of any position of the invert; The connection between the tunnel and the invert is simulated by virtual springs. Multiple groups of virtual line springs are used. Each group of virtual line springs is connected along the tangential and normal directions of the tunnel respectively. The elastic potential energy between the tunnel and the invert is obtained by the displacement difference between the tunnel and the invert, which is expressed as: ; In the formula, and represent the virtual spring stiffness in the tangential and normal directions between the tunnel and the invert, respectively; Indicates The angle of the tunnel where the virtual line spring of the group is connected; The stiffness matrix representing the connection between the tunnel and the invert; represents the elastic potential energy of the connection between the tunnel and the invert; Indicates the total number of virtual line springs; represents the circumferential displacement of the tunnel at the virtual line spring connection between the p-th group of tunnels and the invert; represents the tunnel normal displacement at the virtual line spring connection between the p-th group of tunnels and the invert; The soil is considered as a virtual spring outside the tunnel and simulated. The elastic potential energy of the soil is expressed as: ; In the formula, , , They represent the stiffness of the axial, circumferential and normal virtual springs used to simulate the soil outside the tunnel; represents the stiffness matrix of the soil; represents the elastic potential energy of the soil.

[0015] Furthermore, the specific process of step S4 is as follows: On the rails An excitation force is applied at the position, and the work done by the external force of the excitation force is expressed as: ; In the formula, Indicates the magnitude of the excitation force applied; ; Indicates the incentive point displacement vector; Indicates the incentive point The shape function vector of the rail at ; It represents the work done by the exciting force applied on the rail; represents the time variable; Indicates the incentive point Displacement of the rail at Considering the work done by external force as pseudo potential energy, the total energy functional under excitation is obtained and expressed in 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.

[0016] Furthermore, the specific process of step S5 is as follows: according to the Lagrange equation , solve the total energy functional and obtain the characteristic equation of motion of the orbital structure: ; In the formula, represents the circular frequency; By scanning the circular frequency in the characteristic equation of motion of the orbital structure , and we get the column vector ,Will The displacement and rotation angle of the rail and the invert, the displacement and rotation angle of the track plate and the axial displacement, circumferential displacement and normal displacement equations of the tunnel in step S1 are introduced to obtain the vibration displacement of any position of the track structure.

[0017] Compared with the existing technology, the present invention has the following beneficial effects:

[0018] (1) Based on the framework of the energy method, the present invention uses a virtual spring model to simulate the connection conditions between the various components of the track structure, decouples the displacement shape function, and can simply and quickly represent the displacement field of the track structure. The connection conditions are converted into the elastic potential energy of the virtual spring and coupled to the total energy functional, which comprehensively improves the accuracy of analysis and prediction and simplifies the calculation process.

[0019] (2) The present invention utilizes the decoupling processing of the displacement shape function to simplify the track structure analysis, and can quickly calculate the displacement field of the track structure, thereby improving the convenience of prediction. The virtual spring model is used to simulate the connection between the components, so that the complex connection conditions can be converted into an easy-to-handle elastic potential energy form, thereby reducing the impact of complex boundary conditions on the calculation model.

[0020] (3) Compared with the prior art, the present invention significantly simplifies the connection condition processing of the track structure by introducing a virtual spring model, solving the problem that it is difficult to effectively represent and calculate complex connection conditions in traditional semi-analytical methods. Existing semi-analytical technologies often have problems such as poor adaptability and high computational complexity when processing different tunnel structures. The method provided by the present invention can adapt to a variety of tunnel structures and has strong versatility and flexibility, which enables the present invention to be widely used in different engineering projects, greatly improving the computational efficiency and reducing the difficulty of engineering implementation. BRIEF DESCRIPTION OF THE DRAWINGS

[0021] Figure 1 The figure is a flow chart of the method of the present invention.

[0022] Figure 2 This is a schematic diagram of the front view of the track structure.

[0023] Figure 3 A side view of the track structure.

[0024] Figure 4 This is a schematic diagram of the connections between the various components of the track structure.

[0025] Figure 5It is a comparison diagram of rail vibration displacement between the method of the present invention and the finite element method.

[0026] Figure 6 It is a comparison diagram of track plate vibration displacement using the method of the present invention and the finite element method.

[0027] Figure 7 It is a comparison diagram of the invert vibration displacement using the method of the present invention and the finite element method.

[0028] Figure 8 This is a comparison diagram of tunnel vibration displacement between the method of the present invention and the finite element method. DETAILED DESCRIPTION

[0029] like Figure 1 As shown, the present invention provides a technical solution: a rapid prediction method for urban underground rail structure vibration based on energy method:

[0030] Step S1: constructing vibration parameters of the track structure, including displacement and rotation angle of the rail and the invert, displacement and rotation angle of the track plate, and axial displacement, circumferential displacement, and normal displacement of the tunnel.

[0031] like Figure 2-Figure 4 As shown in the figure, the track structure includes rails, fasteners, track slabs, inverts and tunnels; the rails are described by Timoshenko beams, the fasteners are described by springs, the track slabs below the fasteners are described by Minderlin slabs, the inverts below the track slabs are described by Timoshenko beams, the tunnels are described by cylindrical shells, and the surrounding soil is described as being in elastic contact with the tunnel; the vertical displacement and rotation angle of the rails are expressed as ; The vertical displacement and rotation angle of the invert are expressed as ; Vertical displacement of track slab, Direction angle and The direction angles are ; The axial displacement, circumferential displacement and normal displacement of a point on the tunnel wall are expressed as ; The length of the track structure is , the fastener spacing is ; The width and thickness of the track plate are The radius and thickness of the tunnel are .

[0032] According to the calculation principle of the energy method, the displacement and rotation angle of the rail and the invert are expressed by combining the shape function and the generalized amplitude related to time: (1); In the formula, It represents the combination of rail and invert. , Indicates rails, Indicates an inverted arch; express The number of shape functions, i.e., the truncation coefficient; express The shape function vector of ; express The generalized magnitude vector of the displacement; express The generalized magnitude vector of the rotation angle; express The transpose of express The transpose of express The displacement at any position of ; express of Coordinates of the direction; express The corner at any position of ; express The number of the directional shape function; express The displacement unknown coefficients; express The corner unknown coefficients; Indicates time; express No. A shape function.

[0033] According to the calculation principle of the energy method, the track plate only considers the out-of-plane vibration, and the displacement of the track plate and the rotation angle in the x and y directions are expressed by combining the shape function and the generalized amplitude related to time: (2); In the formula, Indicates the vertical bending of the track plate at any position. Direction angle and Combination of direction angles; , Indicates the vertical bending of the track plate at any position. Indicates track plate Direction angle, Indicates track plate Direction angle; and Respectively represent track slabs Direction and The number of directional shape functions; and Respectively represent track slabs Direction and Coordinates of the direction; express The shape function vector of ; and Respectively of Direction and Direction type function vector; represents the Klaunec product; Represents the vertical bending at any position of the track slab; Indicates the position of the track plate Angle of direction; Indicates the position of the track plate Angle of direction; Indicates the number of the shape function in the y direction; The vertical bending of the track slab A generalized amplitude; Indicates track plate Direction A generalized amplitude; Indicates track plate Direction A generalized amplitude; The vertical bending of the track slab shape function; Indicates track plate Direction shape function; Indicates track plate Direction shape function; express The transpose of express The transpose of express The transpose of The generalized magnitude vector representing the vertical bending of the track slab; Indicates track plate The generalized magnitude vector of the direction angle; Indicates track plate The generalized magnitude vector of the direction angle; The shape function vector representing the vertical bending of the track slab; Indicates track plate The shape function vector of the direction angle; Indicates track plate The shape function vector for the direction rotation.

[0034] According to the calculation principle of the energy method, the axial displacement of any point on the tunnel is Circumferential displacement and normal displacement It is expressed by combining shape functions and time-dependent generalized amplitudes: (3); In the formula, and Represent the number of shape functions in the axial and circumferential directions respectively; , and They represent the generalized amplitudes in the axial, circumferential and normal directions at any point in the tunnel respectively; , Respectively represent the tunnel Directional and polar angle coordinates; Indicates the axial direction of the tunnel shape function; Indicates the tunnel circumference shape function; The tunnel normal shape function; express The transpose of express The transpose of express The transpose of , and represent the generalized magnitude vectors in the axial, circumferential and normal directions of the tunnel respectively; represents the shape function vector of the tunnel axis; Represents the shape function vector of the tunnel circumference; The shape function vector representing the tunnel normal; It represents the combination of axial, circumferential and normal directions of the tunnel; , Indicates the axial direction of the tunnel; Indicates the tunnel circumference; represents the normal direction of the tunnel; Indicates along the tunnel Shape functions for directional motion; Indicates along the tunnel Shape functions for directional motion.

[0035] Step S2: The strain energy and kinetic energy of the rails, fasteners, track plates, inverts and tunnels are obtained based on the material parameters of the track structure and the energy method calculation principle.

[0036] The strain energy and kinetic energy of the rail and invert are expressed as: (4); (5); In the formula, and express strain energy and kinetic energy; and express The stiffness matrix and mass matrix of ; , , , , and They are The elastic modulus, section moment of inertia, shear coefficient, shear modulus, cross-sectional area, and mass density of , The generalized magnitude vector representing the displacement of the rail; The generalized magnitude vector representing the rotation angle of the rail; The generalized magnitude vector representing the displacement of the invert; The generalized magnitude vector representing the rotation angle of the invert; Indicates the symbol of the differential element; express The conjugate transpose of ; express Derivative with respect to time; express Derivative with respect to time; express Derivative with respect to time; express The derivative with respect to time.

[0037] Considering the strain energy and kinetic energy of the track plate under bending vibration, it can be expressed as: (6); (7); In the formula, and represent the strain energy and kinetic energy of the track plate respectively; It represents the bending stiffness of the track slab; Indicates the width of the track plate; , , , , and They represent the shear coefficient, shear modulus, cross-sectional area, mass density, Poisson's ratio and thickness of the track slab respectively; and denote the stiffness matrix and mass matrix of the track slab respectively; represents the symbol of partial derivative; express Derivative with respect to time; express Derivative with respect to time; express The derivative with respect to time.

[0038] Considering the tunnel as a Kirchhoff-Love shell, the strain energy and kinetic energy of tunnel vibration are expressed as: (8); (9); 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; express The transpose of is the tunnel strain vector; is the Hookean operator of the tunnel; represents the time derivative of the tunnel normal displacement; represents the time derivative of the tunnel axial displacement; represents the time derivative of the tunnel circumferential displacement; and represent the stiffness matrix and mass matrix of the tunnel respectively.

[0039] Step S3: The connection between the components of the track structure is simulated by a virtual spring model, and according to the displacement difference of each component, the connection between the components of the track structure is converted into the elastic potential energy of the virtual spring to form the internal connection energy functional of the track structure.

[0040] The virtual spring is used to simulate the connection between the fastener and the track plate. The elastic potential energy of the fastener can be obtained through the displacement difference between the fastener and the track plate, which is expressed as: (10); In the formula, Indicates that the qth fastener is on the rail and the track plate The coordinates of the direction, Indicates the number of fasteners, there are 9 fasteners in a cell; Indicates the fastener on the track plate coordinate; Indicates the stiffness of the fastener; represents the stiffness matrix of the fastener; represents the elastic potential energy of the fastener; Indicates Rail displacement at each fastener; Indicates Track plate displacement at each fastener.

[0041] The support layer between the track plate and the invert is simulated by virtual springs. The elastic potential energy of the support layer between the track plate and the invert is obtained through the displacement difference between the track plate and the invert, which is expressed as: (11); In the formula, Indicates that virtual springs are used to simulate the surface stiffness between support layers; represents the stiffness matrix between the track slab and the invert; represents the elastic potential energy of the supporting layer between the track slab and the invert; Indicates the vertical displacement of any position of the track plate; Represents the vertical displacement at any position of the invert.

[0042] The connection between the tunnel and the invert is simulated by virtual springs. Multiple groups of virtual line springs are used, where each group of virtual line springs is connected along the tangential and normal directions of the tunnel respectively. The elastic potential energy between the tunnel and the invert is obtained by the displacement difference between the tunnel and the invert, which is expressed as: (12); In the formula, and represent the virtual spring stiffness in the tangential and normal directions between the tunnel and the invert, respectively; Indicates The angle of the tunnel where the virtual line spring of the group is connected; The stiffness matrix representing the connection between the tunnel and the invert; represents the elastic potential energy of the connection between the tunnel and the invert; Indicates the total number of virtual line springs; represents the circumferential displacement of the tunnel at the virtual line spring connection between the p-th group of tunnels and the invert; represents the tunnel normal displacement at the virtual line spring connection between the p-th group of tunnels and the invert.

[0043] Considering the soil as a virtual spring outside the tunnel and simulating it, the elastic potential energy of the soil can be expressed as: (13); In the formula, , , They represent the stiffness of the axial, circumferential and normal virtual springs used to simulate the soil outside the tunnel; represents the stiffness matrix of the soil; represents the elastic potential energy of the soil.

[0044] Step S4: According to the vibration parameters of the track structure constructed in step S1, an excitation force (wheel-rail force) is applied at the rail position, and the external force work of the excitation force is calculated. 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 are combined to form a total energy functional.

[0045] Assume that on the rail If an excitation force is applied at , the work done by the external force of the excitation force can be expressed as: (14); In the formula, Indicates the magnitude of the excitation force applied; ; Indicates the incentive point displacement vector; Indicates the incentive point The shape function vector of the rail at ; It represents the work done by the exciting force applied on the rail; represents the time variable; Indicates the incentive point Displacement of the rail.

[0046] Considering the work done by external force as pseudo potential energy, the total energy functional under excitation is obtained and expressed in Lagrangian: (15); 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.

[0047] Step S5: The motion characteristic equation of the track structure is obtained by varying the total energy functional, and the vibration frequency range to be predicted is input to obtain the generalized amplitude of each part of the track, and the vibration response of each component is calculated based on the generalized amplitude.

[0048] According to Lagrange equation , solving the total energy functional, we can obtain the characteristic equation of motion of the orbital structure: (16); In the formula, Represents circular frequency.

[0049] By scanning the circular frequency in the characteristic equation of motion of the orbital structure , we can get the column vector ,Will By substituting the displacement and rotation angle of the rail and the invert, the displacement and rotation angle of the track plate and the axial displacement, circumferential displacement and normal displacement equations of the tunnel in step S1, the vibration displacement of any position of the track structure can be obtained.

[0050] 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 to calculate and obtain the track vibration results as follows: Figure 5-Figure 8 As shown, the two results are in good agreement.

[0051] Although embodiments of the present invention have been shown and described, it will be appreciated by those skilled in the art that various changes, modifications, substitutions and variations may be made to the embodiments without departing from the principles and spirit of the present invention, and that the scope of the present invention is defined by the appended claims and their equivalents.

Claims

1. A rapid prediction method for urban underground rail structure vibration based on energy method, characterized in that: The steps include: Step S1: constructing vibration parameters of the track structure, including displacement and rotation angle of the rail and the invert, displacement and rotation angle of the track plate, and axial displacement, circumferential displacement, and normal displacement of the tunnel; Step S2: obtaining the strain energy and kinetic energy of the rails, fasteners, track plates, inverts and tunnels according to the material parameters of the track structure and the energy method calculation principle; Step S3: The connection between the components of the track structure is simulated by a virtual spring model, and according to the displacement difference of each component, the connection between the components of the track structure is converted into the elastic potential energy of the virtual spring to form the internal connection energy functional of the track structure; Step S4: According to the vibration parameters of the track structure constructed in step S1, an excitation force is applied at the rail position, and the external force work of the excitation force is obtained, and 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 are combined to form a total energy functional; Step S5: The motion characteristic equation of the track structure is obtained by varying the total energy functional, and the vibration frequency range to be predicted is input to obtain the generalized amplitude of each part of the track, and the vibration response of each component is calculated based on the generalized amplitude.

2. The method for rapid prediction of urban underground rail structure vibration based on energy method according to claim 1 is characterized by: According to the calculation principle of the energy method, the displacement and rotation angle of the rail and the invert are expressed by combining the shape function and the generalized amplitude related to time: ； In the formula, It represents the combination of rail and invert. , Indicates rails, Indicates an inverted arch; express The number of shape functions, i.e., the truncation coefficient; express The shape function vector of ; express The generalized magnitude vector of the displacement; express The generalized magnitude vector of the rotation angle; express The transpose of express The transpose of express The displacement at any position of ; express of The coordinates of the direction; express The corner at any position of ; express The number of the directional shape function; express The displacement unknown coefficients; express The corner unknown coefficients; Indicates time; express No. A shape function.

3. The method for rapid prediction of urban underground rail structure vibration based on energy method according to claim 1 is characterized by: According to the calculation principle of the energy method, the track plate only considers the out-of-plane vibration, and the displacement of the track plate and the rotation angle in the x and y directions are expressed by combining the shape function and the generalized amplitude related to time: ； In the formula, Indicates the vertical bending of the track plate at any position. Direction angle and Combination of direction angles; , Indicates the vertical bending of the track plate at any position. Indicates track plate Direction angle, Indicates track plate Direction angle; and Respectively represent track slabs Direction and The number of directional shape functions; and Respectively represent track slab Direction and The coordinates of the direction; express The shape function vector of ; and Respectively of Direction and Direction type function vector; represents the Klaunec product; It represents the vertical bending at any position of the track slab; Indicates the position of the track plate Angle of direction; Indicates the position of the track plate Angle of direction; Indicates the number of the shape function in the y direction; The vertical bending of the track slab A generalized amplitude; Indicates track plate Direction A generalized amplitude; Indicates track plate Direction A generalized amplitude; The vertical bending of the track slab shape function; Indicates track plate Direction shape function; Indicates track plate Direction shape function; express The transpose of express The transpose of express The transpose of The generalized magnitude vector representing the vertical bending of the track slab; Indicates track plate The generalized magnitude vector of the direction angle; Indicates track plate The generalized magnitude vector of the direction angle; The shape function vector representing the vertical bending of the track slab; Indicates track plate The shape function vector of the direction angle; Indicates track plate The shape function vector for the direction rotation.

4. The method for rapid prediction of urban underground rail structure vibration based on energy method according to claim 3 is characterized by: According to the calculation principle of the energy method, the axial displacement of any point on the tunnel is Circumferential displacement and normal displacement It is expressed by combining shape functions and time-dependent generalized amplitudes: ； In the formula, and Represent the number of shape functions in the axial and circumferential directions respectively; , and They represent the generalized amplitudes in the axial, circumferential and normal directions at any point in the tunnel respectively; , Respectively represent the tunnel Directional and polar angle coordinates; Indicates the axial direction of the tunnel shape function; Indicates the tunnel circumference shape function; The tunnel normal shape function; express The transpose of express The transpose of express The transpose of , and represent the generalized magnitude vectors in the axial, circumferential and normal directions of the tunnel respectively; represents the shape function vector of the tunnel axis; Represents the shape function vector of the tunnel circumference; The shape function vector representing the tunnel normal; It represents the combination of axial, circumferential and normal directions of the tunnel; , Indicates the axial direction of the tunnel; Indicates the tunnel circumference; represents the normal direction of the tunnel; Indicates along the tunnel Shape functions for directional motion; Indicates along the tunnel Shape functions for directional motion.

5. The method for rapid prediction of urban underground rail structure vibration based on energy method according to claim 4 is characterized in that: The strain energy and kinetic energy of the rail and invert are expressed as: ； ； In the formula, and express strain energy and kinetic energy; and express The stiffness matrix and mass matrix of ; , , , , and They are The elastic modulus, section moment of inertia, shear coefficient, shear modulus, cross-sectional area, and mass density of , The generalized magnitude vector representing the displacement of the rail; The generalized magnitude vector representing the rotation angle of the rail; The generalized magnitude vector representing the displacement of the invert; The generalized magnitude vector representing the rotation angle of the invert; Indicates the symbol of the differential element; express The conjugate transpose of ; express Derivative with respect to time; express Derivative with respect to time; express Derivative with respect to time; express Derivative with respect to time; is the length of the track structure.

6. The method for rapid prediction of urban underground rail structure vibration based on energy method according to claim 5 is characterized by: Considering the strain energy and kinetic energy of the track plate under bending vibration, it can be expressed as: ； ； In the formula, and represent the strain energy and kinetic energy of the track plate respectively; It represents the bending stiffness of the track slab; Indicates the width of the track plate; , , , , and They represent the shear coefficient, shear modulus, cross-sectional area, mass density, Poisson's ratio and thickness of the track slab respectively; and denote the stiffness matrix and mass matrix of the track slab respectively; represents the symbol of partial derivative; express Derivative with respect to time; express Derivative with respect to time; express The derivative with respect to time.

7. The method for rapid prediction of urban underground rail structure vibration based on energy method according to claim 1 is characterized by: Considering the tunnel as a Kirchhoff-Love shell, the strain energy and kinetic energy of 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; express The transpose of is the tunnel strain vector; is the Hookean operator of the tunnel; represents the time derivative of the tunnel normal displacement; represents the time derivative of the tunnel axial displacement; represents the time derivative of the tunnel circumferential displacement; and represent the stiffness matrix and mass matrix of the tunnel respectively.

8. The method for rapid prediction of urban underground rail structure vibration based on energy method according to claim 7 is characterized by: The specific process of step S3 is: The virtual spring is used to simulate the space between the fastener and the track plate. The elastic potential energy of the fastener is obtained through the displacement difference between the fastener and the track plate, which is expressed as: ； In the formula, Indicates that the qth fastener is on the rail and the track plate The coordinates of the direction, Indicates the number of fasteners; Indicates the fastener on the track plate coordinate; Indicates the stiffness of the fastener; represents the stiffness matrix of the fastener; represents the elastic potential energy of the fastener; Indicates Rail displacement at each fastener; Indicates Track plate displacement at each fastener; The support layer between the track plate and the invert is simulated by virtual springs. The elastic potential energy of the support layer between the track plate and the invert is obtained through the displacement difference between the track plate and the invert, which is expressed as: ； In the formula, Indicates that virtual springs are used to simulate the surface stiffness between support layers; represents the stiffness matrix between the track slab and the invert; represents the elastic potential energy of the supporting layer between the track slab and the invert; Indicates the vertical displacement of any position of the track plate; It represents the vertical displacement of any position of the invert; The connection between the tunnel and the invert is simulated by virtual springs. Multiple groups of virtual line springs are used. Each group of virtual line springs is connected along the tangential and normal directions of the tunnel respectively. The elastic potential energy between the tunnel and the invert is obtained by the displacement difference between the tunnel and the invert, which is expressed as: ； In the formula, and represent the virtual spring stiffness in the tangential and normal directions between the tunnel and the invert, respectively; Indicates The angle of the tunnel where the virtual line spring of the group is connected; The stiffness matrix representing the connection between the tunnel and the invert; represents the elastic potential energy of the connection between the tunnel and the invert; Indicates the total number of virtual line springs; represents the circumferential displacement of the tunnel at the virtual line spring connection between the p-th group of tunnels and the invert; represents the tunnel normal displacement at the virtual line spring connection between the p-th group of tunnels and the invert; The soil is considered as a virtual spring outside the tunnel and simulated. The elastic potential energy of the soil is expressed as: ； In the formula, , , They represent the stiffness of the axial, circumferential and normal virtual springs used to simulate the soil outside the tunnel; represents the stiffness matrix of the soil; represents the elastic potential energy of the soil.

9. The method for rapid prediction of urban underground rail structure vibration based on energy method according to claim 8 is characterized by: The specific process of step S4 is: On the rails An excitation force is applied at the position, and the work done by the external force of the excitation force is expressed as: ； In the formula, Indicates the magnitude of the excitation force applied; ; Indicates the incentive point displacement vector; Indicates the incentive point The shape function vector of the rail at ; It represents the work done by the exciting force applied on the rail; represents the time variable; Indicates the incentive point Displacement of the rail at Considering the work done by external force as pseudo potential energy, the total energy functional under excitation is obtained and expressed in 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.

10. The method for rapid prediction of urban underground rail structure vibration based on energy method according to claim 9 is characterized in that: The specific process of step S5 is: according to the Lagrange equation , solve the total energy functional and obtain the characteristic equation of motion of the orbital structure: ； In the formula, represents the circular frequency; By scanning the circular frequency in the characteristic equation of motion of the orbital structure , and we get the column vector ,Will The displacement and rotation angle of the rail and the invert, the displacement and rotation angle of the track plate and the axial displacement, circumferential displacement and normal displacement equations of the tunnel in step S1 are introduced to obtain the vibration displacement of any position of the track structure.

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