Vibration prediction method for upper cover building of rail transit

Abstract

The invention discloses a vibration prediction method for a rail transit upper cover building. The method comprises the following steps: dividing a building structure model into a plurality of frame units; constructing a vibration simulation model for each frame unit according to a vibration propagation rule; a unit displacement function is constructed based on the vibration simulation model, and the unit displacement function is an improved Fourier series displacement function in which a four-sine function is introduced so as to eliminate potential discontinuity of a boundary; constructing a vibration stress model based on the unit displacement function, wherein the vibration stress model is obtained by functional variation of the Lagrangian quantity of the overall structure; and obtaining a vibration prediction model of the building structure based on the vibration stress model. According to the method, unit discretization does not need to be carried out on a structural component, the introduced virtual spring model can conveniently simulate non-ideal connection and boundary conditions, and the method has the advantages that the displacement shape function construction is flexible, the structural connection and boundary conditions are convenient to set, and the calculation efficiency is high.

Description

Technical Field

[0001] The present application generally relates to the field of rail transit technology, and specifically to a vibration prediction method for rail transit superstructures. Background Art

[0002] With the rapid development of urban rail transit in my country, utilizing the space above subway depots for property development has significantly improved urban land utilization and generated substantial returns on investment. This development has been adopted by major cities such as Beijing, Shanghai, and Guangzhou. With the rapid development of properties above depots, the vibration and noise issues caused by trains have become increasingly prominent. Schools, office buildings, and other structures above subway depots are typically frame structures. Therefore, quickly and accurately predicting the vibrations caused by trains in these structures is a key step in the subsequent vibration and noise reduction design.

[0003] Currently, building vibration prediction methods primarily rely on empirical prediction formulas, numerical analysis, and theoretical models. Empirical models are the mainstream approach used in early feasibility studies. The U.S. Federal Transportation Administration (FTA) applies a 1-2 dB attenuation correction to vibration transmission through buildings. However, existing research indicates that when train vibration is transmitted through the upper floors of a building, the vibration is amplified as the floor increases. Therefore, empirical models have significant limitations in predictive accuracy.

[0004] Numerical models are the primary approach used in the design phase of existing technologies. While they significantly improve computational accuracy, they also suffer from issues like low computational efficiency and uncertain parameters. Current theoretical models offer a relatively balanced balance between predictive accuracy and computational efficiency, but they provide a relatively limited consideration of the connections and boundary conditions of the structures within the models. Research has shown that the connections and boundary conditions of structures subjected to weak vibrations, such as those in subways, differ from those in strong vibration models, significantly reducing the accuracy of existing numerical models. Summary of the Invention

[0005] In view of the above-mentioned defects or deficiencies in the prior art, it is desired to provide a vibration prediction method for rail transit superstructures, which can predict the vibration of rail transit superstructures and improve the prediction accuracy and efficiency.

[0006] The present application provides a vibration prediction method for a rail transit superstructure, the method comprising:

[0007] Divide the building structure model into several frame units;

[0008] Constructing a vibration simulation model for each of the frame units according to vibration propagation rules;

[0009] Constructing a unit displacement function based on the vibration simulation model, wherein the unit displacement function is an improved Fourier series displacement function that introduces four sine functions to eliminate potential discontinuities at the boundary;

[0010] Constructing a vibration force model based on the unit displacement function, wherein the vibration force model is obtained by varying the Lagrangian functional of the overall structure;

[0011] A vibration prediction model of the building structure is obtained based on the vibration force model.

[0012] Optionally, the frame unit includes a first-floor unit, a plurality of basic units, and a top-floor unit sequentially connected from bottom to top, wherein each frame unit includes a transversely arranged beam and a vertically arranged column, the length of the beam is h, and the length of the column is l;

[0013] The first floor unit is a cross frame structure, with a first vertical side length of 3h / 2 and a first horizontal side length of l;

[0014] The basic unit is a regular cross frame structure, and several basic units have the same structure, with a second vertical side length of h and a second horizontal side length of l;

[0015] The top unit is a T-shaped frame structure, the third vertical side length of which is h / 2, and the third horizontal side length is l.

[0016] Optionally, the vibration propagation rule includes a rotation constraint rule, a lateral elastic support rule, and a vertical elastic support rule, and the vibration propagation rule is to apply a boundary constraint spring;

[0017] Wherein, constructing a vibration simulation model for each frame unit according to the vibration propagation rule specifically includes:

[0018] Applying a rotational boundary constraint spring at each end point of the frame element according to a rotational constraint rule;

[0019] Applying lateral boundary constraint springs at each end point of the frame unit according to lateral constraint rules;

[0020] A vertical boundary constraint spring is applied to each end point of the frame unit according to a vertical constraint rule.

[0021] Optionally, the unit displacement function is expressed as:

[0022] Where,

[0023]

[0024] Where m is a positive integer greater than or equal to 5, p = 1, 2, 3, 4; M is a positive integer representing the highest order, am (t), b p (t) is a time weight coefficient.

[0025] Optionally, the unit displacement function comprises an axial vibration displacement function, a bending vibration displacement function and a rotation vibration displacement function, wherein,

[0026] The time weight coefficient corresponding to the axial vibration displacement function u(x, t) is respectively represented as: m (t), U p (t);

[0027] The time weight coefficient corresponding to the bending vibration displacement function ω(x, t) is respectively represented as: m (t), W p (t);

[0028] The time weight coefficient corresponding to the rotation vibration displacement function θ(x, t) is respectively represented as: m (t), Θ p (t).

[0029] Optionally, the vibration force model is represented as:

[0030] L = V - E + W;

[0031] In the formula, L is the Lagrangian of the overall structure, V is the total potential energy of the building structure, E is the total kinetic energy of the building structure, and W is the external force work of the overall structure.

[0032] The total potential energy of the building structure is the accumulation of the potential energy obtained based on the unit displacement function of each frame unit.

[0033] The total kinetic energy of the building structure is the accumulation of the kinetic energy obtained based on the unit displacement function of each frame unit.

[0034] The total potential energy of the building structure is the accumulation of the product of the unit displacement function of the first floor unit and the external force load.

[0035] Optionally, the total potential energy of the building structure is represented as:

[0036] In the formula, β is the label of the basic unit, n is the label of the top floor unit, V (β) is the potential energy of the basic unit, V (n) is the potential energy of the top floor unit, V (1) is the potential energy of the first floor unit, is the connection potential energy between each unit;

[0037] The total kinetic energy of the building structure is represented as:

[0038] Where β is the number of the basic unit, n is the number of the top unit, E (β) is the kinetic energy of the basic unit, E (n) is the kinetic energy of the top unit, E (1) is the kinetic energy of the first-floor unit;

[0039] The total potential energy of the building structure is expressed as:

[0040] Where x f is the coordinate of the load action point, f(t) is the external force load, is the axial vibration displacement function of the first-floor unit, is the bending vibration displacement function of the first-layer unit, is the axial vibration displacement function of the first-layer unit.

[0041] Optionally, the kinetic energy of the frame unit includes axial vibration kinetic energy, bending vibration kinetic energy and rotational kinetic energy, wherein the kinetic energy of the frame unit is expressed as:

[0042]

[0043] Where, l i is the length of the beam or column, ρ i is the density of the beam or column, A i is the cross-sectional area of ​​the beam or column, I i is the section moment of inertia of the beam or column; u i is the axial vibration displacement function of the beam or column, w i is the bending vibration displacement function of the beam or column, θ i is the axial vibration displacement function of the beam or column;

[0044] The potential energy of the frame unit includes the strain energy of the frame unit, the elastic potential energy of the connection between the beam and the column in the frame unit, and the boundary elastic potential energy of the frame unit. The potential energy of the frame unit is expressed as:

[0045] Where, is the frame element strain energy, is the elastic potential energy of the connection between the beam and column in the frame unit, is the boundary elastic potential energy of the frame element;

[0046] in,

[0047]

[0048] Where, κ i , G i are the shear coefficient and shear stiffness of the beam or column, respectively, is the complex elastic modulus of the beam or column; ktx 、k ty , K rz are the axial, lateral and rotational coupling spring stiffness coefficients, respectively.

[0049] Optionally, the Lagrangian of the overall structure is expressed as:

[0050]

[0051] Where K is the total stiffness matrix of the whole structure, M is the total mass matrix of the whole structure, {γ (1,n)} is the displacement matrix of the overall structure.

[0052] Optionally, the method of obtaining a vibration prediction model of a building structure based on the vibration force model includes:

[0053] The Lagrangian of the overall structure is solved using the Euler-Lagrange equation to obtain the solution of the external force load; wherein the solution of the external force load is expressed as:

[0054]

[0055] Construct a harmonic load model, and obtain the vibration prediction model based on the solution results; the harmonic load model includes an external force response submodel and a displacement response submodel, wherein,

[0056] The external force response sub-model is expressed as:

[0057] The displacement response submodel is expressed as:

[0058] The vibration prediction model is expressed as:

[0059] Where i is the imaginary unit; ω is the frequency; and t is the time.

[0060] The technical solutions provided by the embodiments of the present application may have the following beneficial effects:

[0061] The vibration prediction method for rail transit superstructures provided in the embodiment of the present application does not require unit discretization of structural components. The introduced virtual spring model can facilitate the simulation of non-ideal connections and boundary conditions. It has the advantages of flexible displacement shape function construction, convenient setting of structural connections and boundary conditions, and high computational efficiency. BRIEF DESCRIPTION OF THE DRAWINGS

[0062] Other features, objects and advantages of the present application will become more apparent upon reading the detailed description of non-limiting embodiments made with reference to the following drawings:

[0063] Figure 1 A schematic diagram showing a building structure model provided in an embodiment of the present application divided into a plurality of frame units;

[0064] Figure 2 A schematic diagram of constructing a vibration simulation model for each frame unit provided in an embodiment of the present application;

[0065] Figure 3 A diagram of a vibration prediction model for a school building on a train depot provided in an embodiment of the present application;

[0066] Figure 4 Comparison of the first 50 modal frequencies of the vibration prediction model and the finite element model provided in the embodiments of the present application;

[0067] Figure 5 Comparison between the vibration prediction results and measured results of the school building provided in the embodiments of this application. DETAILED DESCRIPTION

[0068] The present application will be further described in detail below with reference to the accompanying drawings and examples. It should be understood that the specific embodiments described herein are merely for the purpose of explaining the relevant invention and are not intended to limit the invention. It should also be noted that, for ease of description, only portions relevant to the invention are shown in the accompanying drawings.

[0069] It should be noted that, in the absence of conflict, the embodiments and features of the embodiments in this application can be combined with each other. The present application will be described in detail below with reference to the accompanying drawings and in combination with the embodiments.

[0070] This paper provides an energy-based method for predicting train-induced vibrations in subway depot roof frame structures. This method is illustrated using a planar frame structure as an example. Research has shown that train vibrations propagate through the vertical load-bearing structure to the upper structure. Since medium- and high-frequency vibration waves decay rapidly in the floor slab, they are unlikely to propagate far from the nodes between columns and floor slabs. The influence of vibration propagation between individual columns is minimal. Therefore, vibration propagation in a building can be considered to occur along a single vertical load-bearing structure, traveling to the upper floors.

[0071] The present application provides a vibration prediction method for a rail transit superstructure, the method comprising:

[0072] S100, dividing the building structure model into a plurality of frame units;

[0073] S200, constructing a vibration simulation model for each frame unit according to vibration propagation rules;

[0074] S300, constructing a unit displacement function based on the vibration simulation model, wherein the unit displacement function is an improved Fourier series displacement function that introduces four sine functions to eliminate potential discontinuities at boundaries;

[0075] S400, constructing a vibration force model based on the unit displacement function, wherein the vibration force model is obtained by varying the Lagrangian functional of the overall structure;

[0076] S500: Obtain a vibration prediction model of the building structure based on the vibration force model.

[0077] Based on this, the vibration transmission model of the frame structure in the embodiment of the present application can be simplified to a single vertical load-bearing structure and a connected beam model. The vertical load-bearing structure and the connected beam model can be divided into a basic unit, a top unit and a first-floor unit according to geometric and material parameters, as shown in FIG. Figure 1 shown.

[0078] Specifically, the frame unit includes a first-floor unit, several basic units and a top-floor unit connected in sequence from bottom to top, wherein each of the frame units includes a horizontally arranged beam (i.e., along the X direction) and a vertically arranged column (i.e., along the Y direction), the length of the beam is h, and the length of the column is l.

[0079] In the embodiment of the present application, the structures of the first-floor unit, the basic unit, and the top-floor unit are different. The first-floor unit is a cross-frame structure with a first vertical side length of 3h / 2 and a first horizontal side length of l; the basic unit is a regular cross-frame structure, and several of the basic units have the same structure, with a second vertical side length of h and a second horizontal side length of l; the top-floor unit is a T-frame structure with a third vertical side length of h / 2 and a third horizontal side length of l.

[0080] In S200 of the embodiment of the present application, the beams and columns in each unit are considered to be two-dimensional beam units that can propagate axial waves and bending waves. Artificial spring simulation is introduced to simulate the coupling connection between the beams and columns, and the boundary support conditions of the beams and columns are simulated using linear and rotational constraint springs.

[0081] Specifically, the vibration propagation rule includes a rotation constraint rule, a lateral elastic support rule, and a vertical elastic support rule, and the vibration propagation rule is to apply a boundary constraint spring.

[0082] Among them, a vibration simulation model is constructed for each of the frame units according to the vibration propagation rules, specifically including: applying rotational boundary constraint springs on each end point of the frame unit according to the rotational constraint rules; applying lateral boundary constraint springs on each end point of the frame unit according to the lateral constraint rules; and applying vertical boundary constraint springs on each end point of the frame unit according to the vertical constraint rules.

[0083] like Figure 2 (a) shows the vibration propagation rule applied to the basic unit, such as Figure 2 (b) shows the vibration propagation rule applied to the top unit, such as Figure 2 (c) shows that the vibration propagation rule is applied to the first-floor unit, and the boundary support conditions of the beams and columns are simulated using linear and rotational constraint springs. In the embodiment of this application, the basic unit is used as an example for illustration.

[0084] In an embodiment of the present application, different connection and boundary conditions can be simulated by setting different connection and boundary spring stiffness coefficients. For example, setting the boundary constraint spring stiffness coefficient to infinity or zero can simulate fixed and free boundary conditions respectively.

[0085] In S300 of the embodiment of the present application, the unit displacement function is expressed as:

[0086]

[0087] Where,

[0088]

[0089] Where m is a positive integer greater than or equal to 5, p = 1, 2, 3, 4; M is a positive integer representing the highest order, a m (t), b p (t) is the time weight coefficient.

[0090] According to the present application, a boundary constraint spring is applied to the frame unit, and the unit displacement function includes an axial vibration displacement function, a bending vibration displacement function, and a rotational vibration displacement function, wherein the time weight coefficients corresponding to the axial vibration displacement function u(x, t) are respectively expressed as: U m (t), U p (t); The time weight coefficients corresponding to the bending vibration displacement function ω(x,t) are expressed as: W m (t), W p (t); The time weight coefficients corresponding to the rotational vibration displacement function θ(x,t) are expressed as: Θ m (t), Θ p (t).

[0091] In the embodiment of this application, Figure 2Taking the basic unit of as an example, the beam and column are named beam unit 1 and beam unit 2, respectively. The displacement expression of the beam unit is shown in Equation (1-1). Its displacement function is constructed using the improved Fourier series expansion method, as shown in Equation (1-2). Compared with the traditional Fourier series method, the improved Fourier series expansion method introduces four sinusoidal functions (corresponding to m = 1, 2, 3, 4; p = 1, 2, 3, 4 in the formula). The introduced sinusoidal functions can eliminate the potential discontinuity problem of the displacement function itself and its spatial derivatives at the boundary when performing periodic expansion, and transfer the discontinuity problem of the first-order and third-order partial derivatives of the displacement function to the added four sinusoidal functions. In addition, the sinusoidal function can also improve the continuity of the series expansion within the entire solution domain (including the boundary).

[0092] The function of each framework unit in the embodiment of the present application can be expressed as:

[0093]

[0094] Where u(x, t), w(x, t), and θ(x, t) represent the axial vibration displacement, bending vibration displacement, and rotational vibration displacement of the beam element, respectively. u1 and u2 are the axial displacements of beam element 1 and beam element 2, respectively. U, W, and θ represent time weight coefficients, and ψ is the displacement function of the beam element.

[0095] In the embodiment of the present application, the four corresponding sinusoidal functions in each function can be expressed as:

[0096]

[0097] Where l and h represent the lengths of beams and columns in the basic unit, respectively; m and p are the numbers of trigonometric functions of the displacement functions of beams and columns in the basic unit.

[0098] In S400 of the embodiment of the present application, the vibration force model is expressed as:

[0099] L=V-E+W; (2-1)

[0100] Where L is the Lagrangian of the overall structure, V is the total potential energy of the building structure, E is the total kinetic energy of the building structure, and W is the external work of the overall structure;

[0101] In the embodiment of the present application, the basic unit is used as an example for explanation, and the formula is also applicable to the first-level unit and the top-level unit.

[0102] The total kinetic energy of the building structure is the accumulation of kinetic energy obtained based on the unit displacement function of each frame unit; the total kinetic energy of the building structure is expressed as:

[0103]

[0104] Where θ is the number of the basic unit, n is the number of the top unit, and E (β) is the kinetic energy of the basic unit, E (n) is the kinetic energy of the top unit, E (1) is the kinetic energy of the first-floor unit.

[0105] Specifically, in formula (1-3), each kinetic energy parameter is expressed by the following formula. The kinetic energy of the frame unit includes axial vibration kinetic energy, bending vibration kinetic energy and rotational kinetic energy, wherein the kinetic energy of the frame unit is expressed as:

[0106]

[0107] Where, l i is the length of the beam or column, ρ i is the density of the beam or column, A i is the cross-sectional area of ​​the beam or column, I i is the section moment of inertia of the beam or column; u i is the axial vibration displacement function of the beam or column, w i is the bending vibration displacement function of the beam or column, θ i is the axial vibration displacement function of the beam or column.

[0108] It should be noted that, in the embodiments of the present application, i as a subscript represents a serial number, and i as a superscript represents an imaginary unit. In formula (1-4), i = 1, 2, where the corresponding u i 、w i ,θ i These are formulas (1-1) and (1-2).

[0109] The potential energy of the frame unit includes the strain energy of the frame unit, the elastic potential energy of the connection between the beam and the column in the frame unit, and the boundary elastic potential energy of the frame unit. The potential energy of the frame unit is expressed as:

[0110]

[0111] Where, is the frame element strain energy, is the elastic potential energy of the connection between the beam and column in the frame unit, is the boundary elastic potential energy of the frame element.

[0112] in,

[0113]

[0114] In formula (1-6), i = 1, 2, where the corresponding u i 、w i ,θ iFor formulas (1-1) and (1-2), l i is the length of the beam or column, κ i , G i are the shear coefficient and shear stiffness of the beam or column, respectively, is the complex elastic modulus of the beam or column.

[0115]

[0116] In formulas (1-7) and (1-8), k tx 、k ty , K rz are the axial, lateral and rotational coupling spring stiffness coefficients respectively; u1, w1 and θ1 are the formulas corresponding to beam unit 1 (i.e., the lateral beam) in formulas (1-1) and (1-2); u2, w2 and θ2 are the formulas corresponding to beam unit 2 (i.e., the longitudinal column) in formulas (1-1) and (1-2). Formula (1-1) Formula (1-1) Where l is the length of the beam and h is the length of the column. is the square of formula (1-1) x=0, It is the square of formula (1-1) x=l.

[0117] The total potential energy of the building structure is the accumulation of the product of the unit displacement function of the first-floor unit and the external force load; the total potential energy of the building structure is expressed as:

[0118]

[0119] Where x f is the coordinate of the load action point, f(t) is the external force load, is the axial vibration displacement function of the first-floor unit, is the bending vibration displacement function of the first-layer unit, is the axial vibration displacement function of the first-floor unit. The corresponding u in each formula of the vibration force model is i 、w i ,θ i These are formulas (1-1) and (1-2).

[0120] In S300 of the embodiment of the present application, the Lagrangian of the overall structure is expressed as:

[0121]

[0122] Where K is the total stiffness matrix of the whole structure, M is the total mass matrix of the whole structure, {γ (1,n)} is the displacement matrix of the overall structure, represents the first-order derivative of the displacement matrix, {γ (1,n)} T Represents the transpose of the displacement matrix.

[0123] In the embodiment of the present application, an equation relationship between formulas (2-1) and (2-2) can be established.

[0124] In S500 of the embodiment of the present application, the method for obtaining a vibration prediction model of the building structure based on the vibration force model includes:

[0125] S510: solving the Lagrangian of the overall structure using the Euler-Lagrange equation to obtain a solution result of the external force load;

[0126] Using the Euler-Lagrange equation from the Lagrangian formula (2-2) of the overall structure, we can obtain:

[0127]

[0128] Where, To find the partial derivative, {γ (1,n)} is the displacement matrix of the overall structure, represents the first derivative of the displacement matrix.

[0129] Substituting formula (2-2) into formula (1-10) yields the solution for the external load, expressed as:

[0130]

[0131] {γ (1,n)} is the displacement matrix of the overall structure, represents the second-order derivative of the displacement matrix.

[0132] S520: Construct a harmonic load model, and obtain the vibration prediction model based on the solution result; the harmonic load model includes an external force response sub-model and a displacement response sub-model, wherein:

[0133] The external force response sub-model is expressed as:

[0134]

[0135] The displacement response submodel is expressed as:

[0136]

[0137] Substituting equations (1-12) and (1-13) into (1-11), the vibration prediction model is obtained, which is expressed as:

[0138]

[0139] Where i is the imaginary unit; ω is the frequency; and t is the time.

[0140] The present invention designs a method for predicting vehicle-induced vibration of the frame structure of the subway depot roof based on the energy principle. It is mainly based on the virtual spring model and the energy functional variation principle. A vibration prediction method is established for the frame structure commonly found in subway depot roof buildings. This method does not require unit discretization of structural components. The introduced virtual spring model can facilitate the effective simulation of non-ideal connections and boundary conditions. Therefore, it has the advantages of convenient setting of structural connections and boundary conditions and high computational efficiency. Therefore, the beneficial effects of the present invention will be further illustrated in combination with simulation analysis and field measurements below.

[0141] This method is used to establish a vibration prediction model for the school building above the vehicle depot, and the commercial finite element software ANSYS is used to build its finite element model. The calculation results of this model are compared and analyzed with the finite element simulation results and the field measurement results.

[0142] Figure 3 Schematic diagram of the vibration prediction model established for the vertical load-bearing structure of the selected vehicle depot under-cover and superstructure and the connected beams. The structural parameters are shown in Tables 1 and 2.

[0143] Table 1 Vertical load-bearing structure parameters

[0144]

[0145]

[0146] Table 2 Beam structure parameters

[0147]

[0148] Table 3 compares the calculation results of the modal frequencies of the first 20 orders (i.e., M=20 in the unit displacement function) of the method established in this application and the finite element model. Figure 4 Comparison of the calculation results of the first 50 modal frequencies between this method and the finite element model. Figure 4 (a) is the comparison of the first 50 modal frequencies between the vibration prediction model and the finite element model. Figure 4 (b) is the error analysis of the vibration prediction model and the finite element model. Figure 4 It can be found that the relative error between the calculation results of this method and the finite element calculation results is within 5%, and the calculation results of this method are in good agreement with the finite element simulation results.

[0149] Table 3 Comparison of the first 20 modal frequencies

[0150]

[0151]

[0152] A unit harmonic load is applied to the bottom of the unit column on the first floor of the building vibration prediction model, and the vibration displacement response of the columns and beams on each floor is calculated. The vibration acceleration response can be calculated by multiplying the vibration displacement response by the circular frequency. Then, the vibration transfer function along each floor can be calculated. Based on the measured vibration acceleration of the column on the first floor of the superstructure school building, the vibration response of the remaining floors of the superstructure school building on the vehicle depot is obtained. Figure 5 (a) and Figure 5 (b) Comparative analysis of the vibration prediction model calculation values ​​and on-site measured values ​​for the 2nd and 3rd floors of the school building above the vehicle depot. It can be seen from the figure that the vibration prediction model calculation values ​​and measured values ​​for the 2nd and 3rd floors of the school building are relatively close as a whole, and the law of change with frequency is basically consistent. The main vibration frequency of the 2nd floor is between 20 and 30 Hz, and the main vibration frequency of the 3rd floor is between 12 and 40 Hz. Therefore, it can be considered that the vibration prediction values ​​of the frame structure vibration prediction model established in this application are relatively consistent with the measured values.

[0153] It should be understood that the terms "length", "width", "up", "down", "front", "back", "left", "right", "vertical", "horizontal", "top", "bottom", "inside", "outside", etc., indicating the orientation or position relationship, are based on the orientation or position relationship shown in the accompanying drawings, and are only for the convenience of describing the present invention and simplifying the description, and do not indicate or imply that the device or element referred to must have a specific orientation, be constructed and operate in a specific orientation, and therefore cannot be understood as limiting the present invention.

[0154] Furthermore, the terms "first" and "second" are used for descriptive purposes only and should not be construed as indicating or implying relative importance or implicitly specifying the number of the technical features being referred to. Thus, a feature identified as "first" or "second" may explicitly or implicitly include one or more of the features. In the description of the present invention, "plurality" means two or more, unless otherwise specifically defined.

[0155] Unless otherwise defined, the technical and scientific terms used in this application have the same meaning as commonly understood by those skilled in the art in the technical field of the present invention. The terms used in this application are only for describing specific implementation purposes and are not intended to limit the present invention. Terms such as "setting" appearing in this application can mean that one component is directly attached to another component, or that one component is attached to another component through an intermediate component. Features described in one embodiment of this application can be applied to another embodiment alone or in combination with other features, unless the feature is not applicable in the other embodiment or otherwise stated.

[0156] The present invention has been described through the above embodiments, but it should be understood that the above embodiments are for illustrative and illustrative purposes only and are not intended to limit the present invention to the described embodiments. Those skilled in the art will appreciate that various variations and modifications may be made based on the teachings of the present invention, and such variations and modifications fall within the scope of protection claimed in the present invention.

Claims

1. A vibration prediction method for a rail transit superstructure, characterized in that: The method comprises: Divide the building structure model into several frame units; Constructing a vibration simulation model for each of the frame units according to vibration propagation rules; Constructing a unit displacement function based on the vibration simulation model, wherein the unit displacement function is an improved Fourier series displacement function that introduces four sine functions to eliminate potential discontinuities at the boundary; Constructing a vibration force model based on the unit displacement function, wherein the vibration force model is obtained by varying the Lagrangian functional of the overall structure; A vibration prediction model of the building structure is obtained based on the vibration force model.

2. The vibration prediction method for rail transit superstructure according to claim 1, characterized in that: The frame unit includes a first-floor unit, a plurality of basic units, and a top-floor unit sequentially connected from bottom to top, wherein each frame unit includes a horizontally arranged beam and a vertically arranged column, the length of the beam is h, and the length of the column is l; The first floor unit is a cross frame structure, with a first vertical side length of 3h / 2 and a first horizontal side length of l; The basic unit is a regular cross frame structure, and several basic units have the same structure, with a second vertical side length of h and a second horizontal side length of l; The top unit is a T-shaped frame structure, the third vertical side length of which is h / 2, and the third horizontal side length is l.

3. The vibration prediction method for rail transit superstructure according to claim 1, characterized in that: The vibration propagation rule includes a rotation constraint rule, a lateral elastic support rule, and a vertical elastic support rule, and the vibration propagation rule is to apply a boundary constraint spring; Wherein, constructing a vibration simulation model for each frame unit according to the vibration propagation rule specifically includes: Applying a rotational boundary constraint spring at each end point of the frame element according to a rotational constraint rule; Applying lateral boundary constraint springs at each end point of the frame unit according to lateral constraint rules; A vertical boundary constraint spring is applied to each end point of the frame unit according to a vertical constraint rule.

4. The vibration prediction method for rail transit superstructure according to claim 1, characterized in that: The unit displacement function is expressed as: Where, Where m is a positive integer greater than or equal to 5, p = 1, 2, 3, 4; M is a positive integer representing the highest order, a m (t), b p (t) is the time weight coefficient.

5. The vibration prediction method for rail transit superstructure according to claim 4, characterized in that: The unit displacement function includes an axial vibration displacement function, a bending vibration displacement function and a rotational vibration displacement function, wherein: The time weight coefficients corresponding to the axial vibration displacement function u(x,t) are expressed as: U m (t), U p (t); The time weight coefficients corresponding to the bending vibration displacement function ω(x,t) are expressed as: W m (t), W p (t); The time weight coefficients corresponding to the rotational vibration displacement function θ(x,t) are expressed as: Θ m (t), Θ p (t).

6. The vibration prediction method for rail transit superstructure according to claim 2, characterized in that: The vibration force model is expressed as: L = V - E + W; Where L is the Lagrangian of the overall structure, V is the total potential energy of the building structure, E is the total kinetic energy of the building structure, and W is the external work of the overall structure; The total potential energy of the building structure is the accumulation of potential energies obtained based on the unit displacement functions of each of the frame units; The total kinetic energy of the building structure is the accumulation of kinetic energy obtained based on the unit displacement function of each frame unit; The total potential energy of the building structure is the accumulation of the product of the unit displacement function of the first-floor unit and the external force load.

7. The vibration prediction method for rail transit superstructure according to claim 1, characterized in that: The total potential energy of the building structure is expressed as: Where β is the number of the basic unit, n is the number of the top unit, V (β) is the potential energy of the basic unit, V (n) is the potential energy of the top unit, V (1) is the potential energy of the first-floor unit, is the connection potential energy between each unit; The total kinetic energy of the building structure is expressed as: Where β is the number of the basic unit, n is the number of the top unit, E (β) is the kinetic energy of the basic unit, E (n) is the kinetic energy of the top unit, E (1) is the kinetic energy of the first-floor unit; The total potential energy of the building structure is expressed as: Where x f is the coordinate of the load action point, f(t) is the external force load, is the axial vibration displacement function of the first-floor unit, is the bending vibration displacement function of the first-layer unit, is the axial vibration displacement function of the first-layer unit.

8. The vibration prediction method for rail transit superstructure according to claim 1, characterized in that: The kinetic energy of the frame unit includes axial vibration kinetic energy, bending vibration kinetic energy and rotational kinetic energy, wherein the kinetic energy of the frame unit is expressed as: Where, l i is the length of the beam or column, ρ i is the density of the beam or column, A i is the cross-sectional area of ​​the beam or column, I i is the section moment of inertia of the beam or column; u i is the axial vibration displacement function of the beam or column, w i is the bending vibration displacement function of the beam or column, θ i is the axial vibration displacement function of the beam or column; The potential energy of the frame unit includes the strain energy of the frame unit, the elastic potential energy of the connection between the beam and the column in the frame unit, and the boundary elastic potential energy of the frame unit. The potential energy of the frame unit is expressed as: Where, is the frame element strain energy, is the elastic potential energy of the connection between the beam and column in the frame unit, is the boundary elastic potential energy of the frame element; in, Where, κ i , G i are the shear coefficient and shear stiffness of the beam or column, respectively, is the complex elastic modulus of the beam or column; k tx 、k ty , K rz are the axial, lateral and rotational coupling spring stiffness coefficients, respectively.

9. The vibration prediction method for rail transit superstructure according to claim 1, characterized in that: The Lagrangian of the overall structure is expressed as: Where K is the total stiffness matrix of the whole structure, M is the total mass matrix of the whole structure, {γ (1,n) } is the displacement matrix of the overall structure.

10. The vibration prediction method for rail transit superstructure according to claim 9, characterized in that: The method for obtaining a vibration prediction model of a building structure based on the vibration force model includes: The Lagrangian of the overall structure is solved using the Euler-Lagrange equation to obtain the solution of the external force load; wherein the solution of the external force load is expressed as: Construct a harmonic load model, and obtain the vibration prediction model based on the solution results; the harmonic load model includes an external force response submodel and a displacement response submodel, wherein, The external force response sub-model is expressed as: The displacement response submodel is expressed as: The vibration prediction model is expressed as: Where i is the imaginary unit; ω is the frequency; and t is the time.

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