Vibration prediction method of subway superstructure based on null space method and energy method

Abstract

The invention discloses a vibration prediction method for a subway superstructure building based on a null space method and an energy method, comprising the following steps: using the energy method to obtain the energy functional of each shear wall and floor slab; obtaining the connection displacement coordination condition matrix between the shear wall and floor slab of each layer of a single-layer and double-layer subway superstructure building under a consolidation connection condition based on the energy functional; obtaining the constraint condition matrix between the shear wall and floor slab of each layer of the single-layer and double-layer subway superstructure building based on the connection displacement coordination condition matrix, using the null space method to solve the constraint condition matrix, constructing the total energy functional of the subway superstructure building and rewriting it into a weight coefficient column vector, scanning the frequency in the weight coefficient column vector, obtaining the displacement response of the subway superstructure building, and then calculating the vibration transmissibility; based on the framework of the energy method, the invention decouples the displacement shape function from the boundary condition, avoids the boundary dependence and other problems in the shape function construction, and then realizes a simple and fast representation of the displacement field of each component of the superstructure building structure.

Description

Technical Field

[0001] The invention relates to the technical field of transportation engineering, and in particular to a vibration prediction method for a subway superstructure based on a null space method and an energy method. Background Art

[0002] With the acceleration of urbanization, subways have become an indispensable means of transportation in modern cities, and subway-surface buildings have gradually become a new trend in urban development due to their advantages of land saving and comprehensive utilization. The vibration problem caused by subway operation has a significant impact on the structural safety and living comfort of the superstructure. Therefore, the study of subway vibration prediction and calculation methods is particularly important.

[0003] The main methods for predicting and calculating the propagation law of vibration in subway superstructures are empirical formula method, field test method and finite element method. The empirical formula method is mainly used in the preliminary prediction stage. It is just a simple mathematical expression, which is difficult to adapt to various working conditions and environments, and it is difficult to accurately predict the frequency range corresponding to the vibration. The field measurement method uses field measurement data to obtain the transfer function, and then predicts the subway environmental vibration caused by the operation of subway trains. However, due to the late development of subway superstructures in China, there is not enough data support.

[0004] The finite element method has great advantages in calculating complex periodic structures such as subway superstructure vibrations. It is a useful analysis method. However, the calculation accuracy of the finite element method depends on the rationality of mesh division, which means that any slight changes to the model must also require re-meshing, which seriously affects the calculation efficiency during geometric parameter analysis and optimization.

[0005] The energy method has the characteristic of transforming the functional boundary value problem into an extreme value problem, which is conducive to solving the coupling problem and has great advantages in the structural calculation of predicting the vibration of subway superstructures. Summary of the invention

[0006] In view of the deficiencies in the prior art, the present invention provides a subway superstructure vibration prediction method based on a null space method and 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 subway superstructure vibration prediction method based on the null space method and the energy method, comprising the following steps:

[0008] Step S1: construct a two-layer subway superstructure, including an m-th-layer subway superstructure and an m+1-th-layer subway superstructure; decompose the subway superstructure into shear walls and floor slabs, and use the energy method to obtain the energy functional of each shear wall and floor slab;

[0009] Step S2: based on the energy functional of each shear wall and floor slab, the displacement coordination condition matrix of the connection between the shear wall and floor slab of each floor of the m-th subway superstructure under the consolidation connection condition is obtained; based on the displacement coordination condition matrix of the connection between the shear wall and floor slab of each floor of the m-th subway superstructure, the constraint condition matrix of the connection between the shear wall and floor slab of each floor of the m-th subway superstructure is obtained;

[0010] Step S3: Based on the energy functional of each shear wall and floor slab, the displacement coordination condition matrix of the connection between the shear wall and the floor slab of each layer between the m-th floor and the m+1-th floor of the subway superstructure under the consolidation connection condition is obtained; based on the displacement coordination condition matrix of the connection between the shear wall and the floor slab of each layer between the m-th floor and the m+1-th floor of the subway superstructure, the constraint condition matrix of the connection between the shear wall and the floor slab of each layer between the m-th floor and the m+1-th floor of the subway superstructure is obtained;

[0011] Step S4: combining the shear wall and floor connection constraint matrix of each layer of the m-th subway superstructure building and the shear wall and floor connection constraint matrix of each layer between the m-th and m+1-th subway superstructure buildings into a total constraint matrix, and solving the total constraint matrix using the null space method to obtain a set of basic solution systems;

[0012] Step S5: constructing the total energy functional of the subway superstructure based on the set of basic solution systems, and rewriting the total energy functional into a weight coefficient column vector in combination with the Lagrangian equation;

[0013] Step S6: By scanning the frequencies in the weight coefficient column vector, the displacement response of the subway building is obtained, and then the vibration transmissibility is calculated.

[0014] Furthermore, the specific process of step S1 is as follows:

[0015] The shear wall and floor slab are both represented as Mindlin slabs; the vertical displacement of the Mindlin slab is represented as w; the rotation angle of the Mindlin slab in the x direction is represented as ; The y-direction rotation angle of the mindlin board is expressed as ; The in-plane displacement of the Mindlin plate in the x direction is denoted as u; The in-plane displacement of the Mindlin plate in the y direction is denoted as v; Let the length of the floor be l, the width of the floor be b, the height of the shear wall be h, and the width of the shear wall be k;

[0016] Based on the energy method, the displacement fields of shear wall and floor vibration are expressed in the form of basis functions as follows: as well as and an unknown weight coefficient combination of out-of-plane motion and in-plane motion of shear walls and slabs.

[0017] Furthermore, the out-of-plane motion is expressed as:

[0018] ;

[0019] The in-plane motion is expressed as:

[0020] ;

[0021] In the formula, represents the out-of-plane displacement of the shear wall; represents the x-direction rotation angle of the shear wall; Indicates time; represents the y-direction rotation angle of the shear wall; represents the out-of-plane displacement of the floor slab; Indicates the x-direction rotation angle of the floor; Indicates the y-direction rotation angle of the floor; represents the first unknown weight coefficient; represents the second unknown weight coefficient; represents the third unknown weight coefficient; represents the fourth unknown weight coefficient; represents the fifth unknown weight coefficient; represents the sixth unknown weight coefficient; represents the seventh unknown weight coefficient; represents the eighth unknown weight coefficient; represents the ninth unknown weight coefficient; represents the tenth unknown weight coefficient; The shear wall basis functions, i∈1,2,…,n; Indicates the first basis functions; express indivual A collection of; express indivual A collection of; express The transpose of express The transpose of represents the in-plane displacement of the shear wall in the x direction; represents the in-plane displacement of the shear wall in the y direction; represents the in-plane displacement of the floor in the x direction; represents the in-plane displacement of the floor in the y direction; represents the first unknown coefficient; represents the second unknown coefficient; represents the third unknown coefficient; represents the fourth unknown coefficient; represents the fifth unknown coefficient; represents the sixth unknown coefficient; represents the seventh unknown coefficient; represents the eighth unknown coefficient; represents the ninth unknown coefficient; represents the tenth unknown coefficient; , , , , , , , , , Respectively , , , , , , , , , is the transpose of ; j and b represent shear wall and floor slab respectively; n represents the number of displacement shape functions used to simulate displacement, i.e., the truncation coefficient; T represents the transpose.

[0022] Furthermore, the strain energy and kinetic energy of the out-of-plane motion and in-plane motion of the shear wall and floor slab are expressed as:

[0023] ;

[0024] ;

[0025] ;

[0026] ;

[0027] ;

[0028] ;

[0029] ;

[0030] ;

[0031] In the formula, represents the kinetic energy outside the shear wall; represents the strain energy outside the shear wall; represents the kinetic energy outside the floor surface; represents the strain energy outside the floor surface; represents the kinetic energy within the shear wall; represents the kinetic energy within the floor surface; represents the strain energy in the shear wall; represents the strain energy within the floor surface; represents the density of the shear wall; represents the shear wall thickness; represents the time derivative of the out-of-plane displacement of the shear wall; represents the derivative of the shear wall in the x direction; represents the derivative of the shear wall in the y direction; represents the derivative of the x-direction rotation of the shear wall with respect to time; It represents the derivative of the y-direction rotation angle of the shear wall with respect to time; represents the out-of-plane vibration mass matrix of the shear wall; represents the shear modulus of the shear wall; represents the x direction of the shear wall; represents the y direction of the shear wall; represents the Poisson's ratio of the floor slab; represents the Poisson's ratio of the shear wall; represents the shear coefficient of the shear wall; represents the shear modulus of the shear wall; represents the cross-sectional area of ​​the shear wall; represents the out-of-plane vibration stiffness matrix of the shear wall; Indicates the density of the floor slab; Indicates the thickness of the floor slab; represents the time derivative of the out-of-plane displacement of the floor; represents the derivative of the floor in the x direction; represents the derivative of the floor in the y direction; represents the derivative of the x-direction rotation angle of the floor with respect to time; represents the derivative of the y-direction rotation angle of the floor with respect to time; represents the mass matrix of the floor slab; It represents the flexural rigidity of the floor slab; represents the x direction of the floor; Indicates the y direction of the floor; represents the shear coefficient of the floor slab; represents the shear modulus of the floor slab; Indicates the cross-sectional area of ​​the floor slab; represents the time derivative of the in-plane displacement of the shear wall in the x direction; It represents the time derivative of the in-plane displacement of the shear wall in the y direction; represents the in-plane vibration mass matrix of the shear wall; represents the time derivative of the in-plane displacement of the floor in the x direction; It represents the time derivative of the in-plane displacement of the floor in the y direction; represents the in-plane vibration mass matrix of the floor; represents the shear modulus of the shear wall; represents the in-plane vibration stiffness matrix of the shear wall; represents the in-plane vibration stiffness matrix of the floor; represents the derivative of the unknown coefficient; represents the derivation; represents the x-direction rotation angle of the shear wall; represents the y-direction rotation angle of the shear wall; represents the out-of-plane displacement of the shear wall; represents the unknown coefficient; Represents the unknown coefficient The conjugate transpose of They are , , , , , , , , , The conjugate transpose of ; for Derivative with respect to time; Indicates the x-direction rotation angle of the floor; Indicates the y-direction rotation angle of the floor; represents the out-of-plane displacement of the floor slab; represents the in-plane displacement of the floor in the x-direction.

[0032] Furthermore, the specific process of step S2 is as follows:

[0033] Assume that the boundary between the shear wall and the floor is Department, Indicates the floor direction, Represents shear wall direction, the displacement coordination condition at the connection between the shear wall and the floor slab of the subway cover building on the mth floor is expressed as:

[0034] ;

[0035] In the formula, Indicates that when x is equal to Out-of-plane vibration displacement of the floor at ; Indicates that when x is equal to The x-direction rotation angle of the floor at Indicates that when x is equal to The in-plane vibration displacement of the floor in the x direction; Indicates that when x is equal to The in-plane vibration displacement of the floor in the y direction; Indicates that when x is equal to The in-plane vibration displacement of the shear wall in the x direction; Indicates that when x is equal to The x-direction rotation angle of the shear wall at ; Indicates that when x is equal to Out-of-plane vibration displacement of the shear wall at ; Indicates that when x is equal to The in-plane vibration displacement of the shear wall in the y direction;

[0036] The displacement coordination conditions of the connection between the shear wall and the floor slab of each floor of the m-th subway superstructure are expressed in matrix form, that is, the displacement coordination condition matrix of the connection between the shear wall and the floor slab of each floor of the m-th subway superstructure:

[0037] ;

[0038] in, ;

[0039] In the formula, It represents the constraint matrix of the connection between the shear wall and the floor slab of each floor of the m-th subway building; express At x equals The conjugate transpose of the value at ; express At x equals The conjugate transpose of the value at .

[0040] Furthermore, the specific process of step S3 is as follows:

[0041] Assume that the coordinates of the connection between the shear wall and the floor slab on each floor between the mth floor and the m+1th floor of the subway superstructure are: , Indicates the floor slab of the subway building on the mth floor direction, Indicates the shear wall of the subway building on the m+1 floor direction;

[0042] The displacement coordination condition of the shear wall and floor connection between the mth and m+1th floor subway superstructures is expressed as:

[0043] ;

[0044] In the formula, Indicates that the mth floor of the subway building is equal to x Out-of-plane vibration displacement at ; Indicates the y direction of the m-th subway superstructure building floor; represents the in-plane vibration displacement of the shear wall of the subway building on the m+1th floor in the x direction at the point where x is equal to h; Indicates the y direction of the m+1th floor of the subway superstructure; Indicates that the mth floor of the subway building is equal to x The x-direction angular displacement at It represents the angular displacement of the shear wall of the subway building on the m+1th floor in the x direction at the point where x is equal to h; Indicates that the mth floor of the subway building is equal to x The in-plane vibration displacement in the x direction at ; represents the out-of-plane vibration displacement of the shear wall of the subway building on the m+1th floor at the position where x is equal to h; Indicates that the mth floor of the subway building is equal to x The in-plane vibration displacement in the y direction at ; It represents the in-plane vibration displacement of the shear wall of the subway building on the m+1th floor in the y direction at the point where x is equal to h;

[0045] The constraint matrix of the connection between the shear wall and the floor slab of the m-th subway building is obtained by using The same method is used to obtain the constraint matrix of the connection between the shear wall and the floor slab on each floor between the mth floor and the m+1th floor of the subway superstructure. .

[0046] Furthermore, the specific process of step S4 is: the shear wall and floor connection constraint matrix of each floor of the m-th subway building And the constraint matrix of the connection between the shear wall and the floor slab on each floor between the mth floor and the m+1th floor of the subway superstructure Combined into the total constraint matrix , , and solve it by the null space method to obtain the set of basic solution systems :

[0047] ;

[0048] In the formula, represents the null space for solving the total constraint matrix.

[0049] Furthermore, the specific process of step S5 is: based on the set of basic solution systems Constructing the total energy functional of subway superstructures , expressed as:

[0050] ;

[0051] In the formula, represents the mass matrix of the subway superstructure; represents the stiffness matrix of the subway superstructure; Indicates the external force on the subway superstructure; express The transpose of

[0052] Combining Lagrange equations The total energy functional is rewritten as:

[0053] ;

[0054] In the formula, represents the column vector of weight coefficients; Indicates frequency; express The conjugate transpose of .

[0055] Further, the specific process of step S6 is: by scanning the frequency , the weight coefficient column vector By substituting the out-of-plane motion and in-plane motion into the formula, the displacement response of the subway superstructure is obtained, and then the vibration transmissibility is calculated.

[0056] Compared with the existing technology, the present invention has the following beneficial effects: based on the framework of the energy method, the present invention decouples the displacement shape function from the boundary conditions, avoids problems such as boundary dependence in the shape function construction, and thus realizes a simple and fast representation of the displacement field of each component of the upper cover building structure; the present invention eliminates the need for discretization of structural components, and the introduced null space method can effectively simulate non-ideal connections and boundary conditions, and the structural connections and boundary conditions are easy to set and have high calculation efficiency. BRIEF DESCRIPTION OF THE DRAWINGS

[0057] Figure 1 The present invention is a flow chart of the method.

[0058] Figure 2 This is a schematic diagram of the two-story subway superstructure structure.

[0059] Figure 3 It is a comparison diagram of characteristic frequencies of the present invention and the finite element method.

[0060] Figure 4 It is a comparison diagram of the transmissibility between the present invention and the finite element method. DETAILED DESCRIPTION

[0061] This embodiment is based on the condition of two-story subway superstructure building, the m-th-story subway superstructure building and the m+1-th-story subway superstructure building.

[0062] like Figure 1 As shown, the present invention provides a technical solution: a subway superstructure vibration prediction method based on a null space method and an energy method, comprising the following steps:

[0063] Step S1: construct a double-layer subway superstructure, including an m-th-layer subway superstructure and an m+1-th-layer subway superstructure; decompose the subway superstructure into shear walls and floor slabs, and use the energy method to obtain the energy functional of each shear wall and floor slab.

[0064] The specific process of step S1 is:

[0065] The shear wall and floor slab are both represented as Mindlin slabs; the vertical displacement of the Mindlin slab is represented as w; the rotation angle of the Mindlin slab in the x direction is represented as ; The y-direction rotation angle of the mindlin board is expressed as ; The in-plane displacement of the Mindlin plate in the x direction is represented by u; the in-plane displacement of the Mindlin plate in the y direction is represented by v; let the length of the floor be l, the width of the floor be b, the height of the shear wall be h, and the width of the shear wall be k.

[0066] The boundary and connection problems between shear walls and floors are solved by the null space method; Figure 2 As shown, the dotted line frame is the calculation unit of the m-th floor subway superstructure.

[0067] Based on the energy method, the displacement fields of shear wall and floor vibration are expressed in the form of basis functions as follows: as well as and an unknown weight coefficient A combination of:

[0068] Out-of-plane motion:

[0069] ;

[0070] In-plane motion:

[0071] ;

[0072] In the formula, represents the out-of-plane displacement of the shear wall; represents the x-direction rotation angle of the shear wall; Indicates time; represents the y-direction rotation angle of the shear wall; represents the out-of-plane displacement of the floor slab; Indicates the x-direction rotation angle of the floor; Indicates the y-direction rotation angle of the floor; represents the first unknown weight coefficient; represents the second unknown weight coefficient; represents the third unknown weight coefficient; represents the fourth unknown weight coefficient; represents the fifth unknown weight coefficient; represents the sixth unknown weight coefficient; represents the seventh unknown weight coefficient; represents the eighth unknown weight coefficient; represents the ninth unknown weight coefficient; represents the tenth unknown weight coefficient; The shear wall basis functions, i∈1,2,…,n; Indicates the first basis functions; express indivual A collection of; express indivual A collection of; express The transpose of express The transpose of represents the in-plane displacement of the shear wall in the x direction; represents the in-plane displacement of the shear wall in the y direction; represents the in-plane displacement of the floor in the x direction; represents the in-plane displacement of the floor in the y direction; represents the first unknown coefficient; represents the second unknown coefficient; represents the third unknown coefficient; represents the fourth unknown coefficient; represents the fifth unknown coefficient; represents the sixth unknown coefficient; represents the seventh unknown coefficient; represents the eighth unknown coefficient; represents the ninth unknown coefficient; represents the tenth unknown coefficient; , , , , , , , , , Respectively , , , , , , , , , is the transpose of ; j and b represent shear wall and floor slab respectively; n represents the number of displacement shape functions used to simulate displacement, i.e., the truncation coefficient; T represents the transpose.

[0073] The strain energy and kinetic energy of the out-of-plane motion and in-plane motion of the shear wall and floor slab are expressed as:

[0074] ;

[0075] ;

[0076] ;

[0077] ;

[0078] ;

[0079] ;

[0080] ;

[0081] ;

[0082] In the formula, represents the kinetic energy outside the shear wall; represents the strain energy outside the shear wall; represents the kinetic energy outside the floor surface; represents the strain energy outside the floor surface; represents the kinetic energy within the shear wall; represents the kinetic energy within the floor surface; represents the strain energy in the shear wall; represents the strain energy within the floor surface; represents the density of the shear wall; represents the shear wall thickness; represents the time derivative of the out-of-plane displacement of the shear wall; represents the derivative of the shear wall in the x direction; represents the derivative of the shear wall in the y direction; represents the derivative of the x-direction rotation of the shear wall with respect to time; It represents the derivative of the y-direction rotation angle of the shear wall with respect to time; represents the out-of-plane vibration mass matrix of the shear wall; represents the shear modulus of the shear wall; represents the x direction of the shear wall; represents the y direction of the shear wall; represents the Poisson's ratio of the floor slab; represents the Poisson's ratio of the shear wall; represents the shear coefficient of the shear wall; represents the shear modulus of the shear wall; represents the cross-sectional area of ​​the shear wall; represents the out-of-plane vibration stiffness matrix of the shear wall; Indicates the density of the floor slab; Indicates the thickness of the floor slab; represents the time derivative of the out-of-plane displacement of the floor; represents the derivative of the floor in the x direction; represents the derivative of the floor in the y direction; represents the derivative of the x-direction rotation angle of the floor with respect to time; represents the derivative of the y-direction rotation angle of the floor with respect to time; represents the mass matrix of the floor slab; It represents the flexural rigidity of the floor slab; represents the x direction of the floor; Indicates the y direction of the floor; represents the shear coefficient of the floor slab; represents the shear modulus of the floor slab; Indicates the cross-sectional area of ​​the floor slab; represents the time derivative of the in-plane displacement of the shear wall in the x direction; It represents the time derivative of the in-plane displacement of the shear wall in the y direction; represents the in-plane vibration mass matrix of the shear wall; represents the time derivative of the in-plane displacement of the floor in the x direction; It represents the time derivative of the in-plane displacement of the floor in the y direction; represents the in-plane vibration mass matrix of the floor; represents the shear modulus of the shear wall; represents the in-plane vibration stiffness matrix of the shear wall; represents the in-plane vibration stiffness matrix of the floor; represents the derivative of the unknown coefficient; represents the derivation; represents the x-direction rotation angle of the shear wall; represents the y-direction rotation angle of the shear wall; represents the out-of-plane displacement of the shear wall; represents the unknown coefficient; Represents the unknown coefficient The conjugate transpose of They are , , , , , , , , , The conjugate transpose of ; for Derivative with respect to time; Indicates the x-direction rotation angle of the floor; Indicates the y-direction rotation angle of the floor; represents the out-of-plane displacement of the floor slab; represents the in-plane displacement of the floor in the x-direction.

[0083] Step S2: Based on the energy functional of each shear wall and floor slab, the displacement coordination condition matrix of the connection between the shear wall and floor slab of each floor of the m-th subway building is obtained under the condition of consolidation connection; Based on the displacement coordination condition matrix of the connection between the shear wall and floor slab of each floor of the m-th subway building, the constraint condition matrix of the connection between the shear wall and floor slab of each floor of the m-th subway building is obtained. .

[0084] The specific process of step S2 is:

[0085] Assume that the boundary between the shear wall and the floor is Department, Indicates the floor direction, Represents shear wall direction, so the displacement coordination condition at the connection between the shear wall and the floor slab of the m-th floor subway cover building can be expressed as:

[0086] ;

[0087] In the formula, Indicates that when x is equal to Out-of-plane vibration displacement of the floor at ; Indicates that when x is equal to The x-direction rotation angle of the floor at Indicates that when x is equal to The in-plane vibration displacement of the floor in the x direction; Indicates that when x is equal to The in-plane vibration displacement of the floor in the y direction; Indicates that when x is equal to The in-plane vibration displacement of the shear wall in the x direction; Indicates that when x is equal to The x-direction rotation angle of the shear wall at ; Indicates that when x is equal to Out-of-plane vibration displacement of the shear wall at ; Indicates that when x is equal to The in-plane vibration displacement of the shear wall in the y direction.

[0088] The displacement coordination conditions of the connection between the shear wall and the floor slab of each floor of the m-th subway superstructure are expressed in matrix form, that is, the displacement coordination condition matrix of the connection between the shear wall and the floor slab of each floor of the m-th subway superstructure:

[0089] ;

[0090] in, ;

[0091] In the formula, It represents the constraint matrix of the connection between the shear wall and the floor slab of each floor of the m-th subway building; express At x equals The conjugate transpose of the value at ; express At x equals The conjugate transpose of the value at .

[0092] Step S3: Based on the energy functional of each shear wall and floor slab, the displacement coordination condition matrix of the connection between the shear wall and the floor slab of each layer between the mth floor and the m+1th floor of the subway superstructure under the consolidation connection condition is obtained; Based on the displacement coordination condition matrix of the connection between the shear wall and the floor slab of each layer between the mth floor and the m+1th floor of the subway superstructure, the constraint condition matrix of the connection between the shear wall and the floor slab of each layer between the mth floor and the m+1th floor of the subway superstructure is obtained. .

[0093] Assume that the coordinates of the connection between the shear wall and the floor slab on each floor between the mth floor and the m+1th floor of the subway superstructure are: , Indicates the floor slab of the subway building on the mth floor direction, Indicates the shear wall of the subway building on the m+1 floor direction.

[0094] The displacement coordination condition of the shear wall and floor connection between the mth and m+1th floor subway superstructures is expressed as:

[0095] ;

[0096] In the formula, Indicates that the mth floor of the subway building is equal to x Out-of-plane vibration displacement at ; Indicates the y direction of the m-th subway superstructure building floor; represents the in-plane vibration displacement of the shear wall of the subway building on the m+1th floor in the x direction at the point where x is equal to h; Indicates the y direction of the m+1th floor of the subway superstructure; Indicates that the mth floor of the subway building is equal to x The x-direction angular displacement at It represents the angular displacement of the shear wall of the subway building on the m+1th floor in the x direction at the point where x is equal to h; Indicates that the mth floor of the subway building is equal to x The in-plane vibration displacement in the x direction at ; represents the out-of-plane vibration displacement of the shear wall of the subway building on the m+1th floor at the position where x is equal to h; Indicates that the mth floor of the subway building is equal to x The in-plane vibration displacement in the y direction at ; It represents the in-plane vibration displacement of the shear wall of the subway building on the m+1th floor in the y direction at the point where x is equal to h.

[0097] The constraint matrix of the connection between the shear wall and the floor slab of the m-th subway building is obtained by using The same method is used to obtain the constraint matrix of the connection between the shear wall and the floor slab on each floor between the mth floor and the m+1th floor of the subway superstructure. .

[0098] Step S4: The constraint matrix of the connection between the shear wall and the floor slab of each floor of the m-th subway building And the constraint matrix of the connection between the shear wall and the floor slab on each floor between the mth floor and the m+1th floor of the subway superstructure Combined into a total constraint matrix, and the null space method is used to solve the total constraint matrix to obtain a set of basic solution systems .

[0099] The specific process of step S4 is:

[0100] The constraint matrix of the connection between the shear wall and the floor slab of each floor of the m-th subway building is calculated. And the constraint matrix of the connection between the shear wall and the floor slab on each floor between the mth floor and the m+1th floor of the subway superstructure Combined into the total constraint matrix , , and solve it by the null space method to obtain the set of basic solution systems :

[0101] ;

[0102] In the formula, represents the null space for solving the total constraint matrix.

[0103] The key to solving the null space method is to find a set of admissible functions that satisfy all constraints, and to use conventional methods such as penalty functions or Lagrange multiplier method to deal with the problem of result convergence. This method does not require the original admissible function to satisfy the constraints.

[0104] Step S5: Based on the set of basic solutions Constructing the total energy functional of subway superstructures , combined with the Lagrange equation, the total energy functional is rewritten as a column vector of weight coefficients.

[0105] The specific process of step S5 is:

[0106] Based on the set of basic solutions Constructing the total energy functional of subway superstructures , expressed as:

[0107] ;

[0108] In the formula, represents the mass matrix of the subway superstructure; represents the stiffness matrix of the subway superstructure; Indicates the external force on the subway superstructure; express The transpose of .

[0109] Combined with the Lagrange equation (Eular-Lagrange) The total energy functional is rewritten as:

[0110] ;

[0111] In the formula, represents the column vector of weight coefficients; Indicates frequency; express The conjugate transpose of .

[0112] Step S6: By scanning the frequencies in the weight coefficient column vector, the displacement response of the subway building is obtained, and then the vibration transmissibility is calculated; specifically, by scanning the frequencies , the weight coefficient column vector By substituting the out-of-plane motion and in-plane motion formulas, the displacement response of the subway superstructure can be obtained, and the vibration transmissibility can be calculated.

[0113] To verify the accuracy of this embodiment, the same material parameters were taken and the method of the present invention and the finite element software COMSOL (hereinafter referred to as the finite element method) were used for calculation to obtain the following results: Figure 3-Figure 4 As shown, Figure 3 The characteristic frequencies obtained by the two methods are compared. Figure 4 The transmissibility obtained by the two methods was compared, and the characteristic frequency and transmissibility results of the method of the present invention were in good agreement.

[0114] 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 subway building vibration prediction method based on the null space method and the energy method, characterized in that: The steps include: Step S1: construct a two-layer subway superstructure, including an m-th-layer subway superstructure and an m+1-th-layer subway superstructure; decompose the subway superstructure into shear walls and floor slabs, and use the energy method to obtain the energy functional of each shear wall and floor slab; Step S2: based on the energy functional of each shear wall and floor slab, the displacement coordination condition matrix of the connection between the shear wall and floor slab of each floor of the m-th floor subway building under the condition of consolidation connection is obtained; Based on the displacement coordination condition matrix of the shear wall and floor slab connection of each floor of the m-th floor subway building, the constraint condition matrix of the shear wall and floor slab connection of each floor of the m-th floor subway building is obtained; Step S3: based on the energy functional of each shear wall and floor slab, the displacement coordination condition matrix of the connection between the shear wall and floor slab of each floor between the mth floor and the m+1th floor of the subway superstructure under the consolidation connection condition is obtained; Based on the displacement coordination condition matrix of the shear wall and floor connection between the mth and m+1th floors of the subway superstructure, the constraint condition matrix of the shear wall and floor connection between the mth and m+1th floors of the subway superstructure is obtained; Step S4: combining the shear wall and floor connection constraint matrix of each layer of the m-th subway superstructure building and the shear wall and floor connection constraint matrix of each layer between the m-th and m+1-th subway superstructure buildings into a total constraint matrix, and solving the total constraint matrix using the null space method to obtain a set of basic solution systems; Step S5: constructing the total energy functional of the subway superstructure based on the set of basic solution systems, and rewriting the total energy functional into a weight coefficient column vector in combination with the Lagrangian equation; Step S6: By scanning the frequencies in the weight coefficient column vector, the displacement response of the subway building is obtained, and then the vibration transmissibility is calculated.

2. The subway superstructure vibration prediction method based on the null space method and the energy method according to claim 1 is characterized by: The specific process of step S1 is: The shear wall and floor slab are both represented as Mindlin slabs; the vertical displacement of the Mindlin slab is represented as w; the rotation angle of the Mindlin slab in the x direction is represented as ; The rotation angle of the Mindlin board in the y direction is expressed as ; The in-plane displacement of the Mindlin plate in the x direction is denoted as u; The in-plane displacement of the Mindlin plate in the y direction is denoted as v; Let the length of the floor be l, the width of the floor be b, the height of the shear wall be h, and the width of the shear wall be k; Based on the energy method, the displacement fields of shear wall and floor vibration are expressed in the form of basis functions as follows: as well as and an unknown weight coefficient combination of out-of-plane motion and in-plane motion of shear walls and slabs.

3. The subway superstructure vibration prediction method based on the null space method and the energy method according to claim 2 is characterized by: The out-of-plane motion is expressed as: ； The in-plane motion is expressed as: ； In the formula, represents the out-of-plane displacement of the shear wall; represents the x-direction rotation angle of the shear wall; Indicates time; represents the y-direction rotation angle of the shear wall; represents the out-of-plane displacement of the floor slab; Indicates the x-direction rotation angle of the floor; Indicates the y-direction rotation angle of the floor; represents the first unknown weight coefficient; represents the second unknown weight coefficient; represents the third unknown weight coefficient; represents the fourth unknown weight coefficient; represents the fifth unknown weight coefficient; represents the sixth unknown weight coefficient; represents the seventh unknown weight coefficient; represents the eighth unknown weight coefficient; represents the ninth unknown weight coefficient; represents the tenth unknown weight coefficient; The shear wall basis functions, i∈1,2,…,n; Indicates the first basis functions; express indivual A collection of; express indivual A collection of; express The transpose of express The transpose of represents the in-plane displacement of the shear wall in the x direction; represents the in-plane displacement of the shear wall in the y direction; represents the in-plane displacement of the floor in the x direction; represents the in-plane displacement of the floor in the y direction; represents the first unknown coefficient; represents the second unknown coefficient; represents the third unknown coefficient; represents the fourth unknown coefficient; represents the fifth unknown coefficient; represents the sixth unknown coefficient; represents the seventh unknown coefficient; represents the eighth unknown coefficient; represents the ninth unknown coefficient; represents the tenth unknown coefficient; , , , , , , , , , Respectively , , , , , , , , , is the transpose of ; j and b represent shear wall and floor slab respectively; n represents the number of displacement shape functions used to simulate displacement, i.e., the truncation coefficient; T represents the transpose.

4. The subway superstructure vibration prediction method based on the null space method and the energy method according to claim 3 is characterized by: The strain energy and kinetic energy of the out-of-plane motion and in-plane motion of the shear wall and floor are expressed as: ； ； ； ； ； ； ； ； In the formula, represents the kinetic energy outside the shear wall; represents the strain energy outside the shear wall; represents the kinetic energy outside the floor surface; represents the strain energy outside the floor surface; represents the kinetic energy within the shear wall; represents the kinetic energy within the floor surface; represents the strain energy in the shear wall; represents the strain energy within the floor surface; represents the density of the shear wall; represents the shear wall thickness; represents the time derivative of the out-of-plane displacement of the shear wall; represents the derivative of the shear wall in the x direction; represents the derivative of the shear wall in the y direction; represents the derivative of the x-direction rotation of the shear wall with respect to time; It represents the derivative of the y-direction rotation angle of the shear wall with respect to time; represents the out-of-plane vibration mass matrix of the shear wall; represents the shear modulus of the shear wall; represents the x direction of the shear wall; represents the y direction of the shear wall; represents the Poisson's ratio of the floor slab; represents the Poisson's ratio of the shear wall; represents the shear coefficient of the shear wall; represents the shear modulus of the shear wall; represents the cross-sectional area of ​​the shear wall; represents the out-of-plane vibration stiffness matrix of the shear wall; Indicates the density of the floor slab; Indicates the thickness of the floor slab; represents the time derivative of the out-of-plane displacement of the floor; represents the derivative of the floor in the x direction; represents the derivative of the floor in the y direction; represents the derivative of the x-direction rotation angle of the floor with respect to time; represents the derivative of the y-direction rotation angle of the floor with respect to time; represents the mass matrix of the floor slab; It represents the flexural rigidity of the floor slab; represents the x direction of the floor; Indicates the y direction of the floor; represents the shear coefficient of the floor slab; represents the shear modulus of the floor slab; Indicates the cross-sectional area of ​​the floor slab; represents the time derivative of the in-plane displacement of the shear wall in the x direction; It represents the time derivative of the in-plane displacement of the shear wall in the y direction; represents the in-plane vibration mass matrix of the shear wall; represents the time derivative of the in-plane displacement of the floor in the x direction; It represents the time derivative of the in-plane displacement of the floor in the y direction; represents the in-plane vibration mass matrix of the floor; represents the shear modulus of the shear wall; represents the in-plane vibration stiffness matrix of the shear wall; represents the in-plane vibration stiffness matrix of the floor; represents the derivative of the unknown coefficient; represents the derivation; represents the x-direction rotation angle of the shear wall; represents the y-direction rotation angle of the shear wall; represents the out-of-plane displacement of the shear wall; represents the unknown coefficient; Represents the unknown coefficient The conjugate transpose of They are , , , , , , , , , The conjugate transpose of ; for Derivative with respect to time; Indicates the x-direction rotation angle of the floor; Indicates the y-direction rotation angle of the floor; represents the out-of-plane displacement of the floor slab; represents the in-plane displacement of the floor in the x-direction.

5. The subway superstructure vibration prediction method based on the null space method and the energy method according to claim 4 is characterized by: The specific process of step S2 is: Assume that the boundary between the shear wall and the floor is Department, Indicates the floor direction, Represents shear wall Direction, the displacement coordination condition at the connection between the shear wall and the floor slab of the subway cover building on the mth floor is expressed as: ； In the formula, Indicates that when x is equal to Out-of-plane vibration displacement of the floor at ; Indicates that when x is equal to The x-direction rotation angle of the floor at Indicates that when x is equal to The in-plane vibration displacement of the floor in the x direction; Indicates that when x is equal to The in-plane vibration displacement of the floor in the y direction; Indicates that when x is equal to The in-plane vibration displacement of the shear wall in the x direction; Indicates that when x is equal to The x-direction rotation angle of the shear wall at ; Indicates that when x is equal to Out-of-plane vibration displacement of the shear wall at ; Indicates that when x is equal to The in-plane vibration displacement of the shear wall in the y direction; The displacement coordination conditions of the connection between the shear wall and the floor slab of each floor of the m-th subway superstructure are expressed in matrix form, that is, the displacement coordination condition matrix of the connection between the shear wall and the floor slab of each floor of the m-th subway superstructure: ； in, ; In the formula, It represents the constraint matrix of the connection between the shear wall and the floor slab of each floor of the m-th subway building; express At x equals The conjugate transpose of the value at ; express At x equals The conjugate transpose of the value at .

6. The subway superstructure vibration prediction method based on the null space method and the energy method according to claim 5 is characterized by: The specific process of step S3 is: Assume that the coordinates of the connection between the shear wall and the floor slab on each floor between the mth floor and the m+1th floor of the subway superstructure are: , Indicates the floor slab of the subway building on the mth floor direction, Indicates the shear wall of the subway building on the m+1 floor direction; The displacement coordination condition of the shear wall and floor slab connection between the mth and m+1th floor subway superstructures is expressed as: ； In the formula, Indicates that the mth floor of the subway building is equal to x Out-of-plane vibration displacement at ; Indicates the y direction of the m-th subway superstructure floor; represents the in-plane vibration displacement of the shear wall of the subway building on the m+1th floor in the x direction at the point where x is equal to h; Indicates the y direction of the m+1th floor of the subway superstructure; Indicates that the mth floor of the subway building is equal to x The x-direction angular displacement at It represents the angular displacement of the shear wall of the subway building on the m+1th floor in the x direction at the point where x is equal to h; Indicates that the mth floor of the subway building is equal to x The in-plane vibration displacement in the x direction at ; represents the out-of-plane vibration displacement of the shear wall of the subway building on the m+1th floor at the position where x is equal to h; Indicates that the mth floor of the subway building is equal to x The in-plane vibration displacement in the y direction at ; It represents the in-plane vibration displacement of the shear wall of the subway building on the m+1th floor in the y direction at the point where x is equal to h; The constraint matrix of the connection between the shear wall and the floor slab of the m-th subway building is obtained by using The same method is used to obtain the constraint matrix of the connection between the shear wall and the floor slab on each floor between the mth floor and the m+1th floor of the subway superstructure. .

7. The subway superstructure vibration prediction method based on the null space method and the energy method according to claim 6 is characterized by: The specific process of step S4 is: the shear wall and floor connection constraint matrix of each floor of the m-th subway building And the constraint matrix of the connection between the shear wall and the floor slab on each floor between the mth floor and the m+1th floor of the subway superstructure Combined into the total constraint matrix , , and solve it by the null space method to obtain the set of basic solution systems : ； In the formula, represents the null space for solving the total constraint matrix.

8. The method for predicting vibration of subway superstructures based on the null space method and the energy method according to claim 7 is characterized in that: The specific process of step S5 is: based on the set of basic solution systems Constructing the total energy functional of subway superstructures , expressed as: ； In the formula, represents the mass matrix of the subway superstructure; represents the stiffness matrix of the subway superstructure; Indicates the external force on the subway superstructure; express The transpose of Combining Lagrange equations The total energy functional is rewritten as: ； In the formula, represents the weight coefficient column vector; Indicates frequency; express The conjugate transpose of .

9. The method for predicting vibration of subway superstructures based on the null space method and the energy method according to claim 8 is characterized in that: The specific process of step S6 is: by scanning the frequency , the weight coefficient column vector By substituting the out-of-plane motion and in-plane motion into the formula, the displacement response of the subway superstructure is obtained, and then the vibration transmissibility is calculated.

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