CN111723423A - Time-frequency mixing prediction method and system for vertical vibration of tunnel and soil body caused by vehicle - Google Patents

Time-frequency mixing prediction method and system for vertical vibration of tunnel and soil body caused by vehicle Download PDF

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CN111723423A
CN111723423A CN202010514190.4A CN202010514190A CN111723423A CN 111723423 A CN111723423 A CN 111723423A CN 202010514190 A CN202010514190 A CN 202010514190A CN 111723423 A CN111723423 A CN 111723423A
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王力东
韩艳
朱志辉
胡朋
李春光
张迅
刘叶
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Central South University
Changsha University of Science and Technology
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Abstract

The invention discloses a time-frequency mixing prediction method and a time-frequency mixing prediction system for vertical vibration of a tunnel and a soil body caused by vehicles, the vertical interaction force of the track and the tunnel is obtained in a time domain by establishing a 2D vehicle-track-tunnel-soil MBS/FEM model (namely a 2D multi-body dynamics-finite element model), the vertical interaction force of the track and the tunnel is converted into a frequency domain and applied to a tunnel-soil body 2.5D FEM-PML model, vibration calculation of the tunnel-soil body in the frequency domain-wave number domain is completed, and finally the vertical vibration level of the tunnel-soil body is calculated, so that the flexibility of time domain solution on track discrete support characteristics and local defect simulation is exerted, the high efficiency of frequency domain solution on soil body vibration calculation is exerted, the final calculation result is more consistent with the actual situation, and meanwhile, the calculation efficiency is improved.

Description

Time-frequency mixing prediction method and system for vertical vibration of tunnel and soil body caused by vehicle
Technical Field
The invention belongs to the technical field of railway engineering application and design, and particularly relates to a time-frequency mixing method and a time-frequency mixing system for predicting vertical vibration of a tunnel and a surrounding soil body caused by a subway vehicle.
Background
Subway and underground traffic systems are common in large urban areas, especially in congested urban centers. Since 1863 the first subway line in london was opened, over 200 cities around the world have adopted this mode of transportation. In recent years, asian countries have contributed most to the rapid development of urban rail transit worldwide, mainly through strong subways. According to the data of the Chinese subway association, as of 2017, 34 cities in China continent have subway systems, which amount to 3884 kilometers, and 6000 kilometers are expected to be reached in 2020. Subway systems provide the maximum capacity and most efficient urban traffic, but trains running in tunnels generate ground vibrations and transmit them to nearby buildings. Ground vibration is a serious problem near public transportation lines and maintenance facilities, and can cause buildings to rock and hear rumble. These perceptible vibrations are typically in the range of 2 to 80 hertz, or transmitted through the ground to the building causing low frequency noise emissions, typically in the range of 30 to 250 hertz. Although such vibrations do not generally reach a level that endangers the safety of the structure, the long lasting exposure of people to vibrations is nowadays considered a public health problem.
Machine for inducing vibration of subway vehicleThe theory generally comprises two parts: firstly, vibration is generated, namely excitation caused by the action of wheel load on a steel rail is added with impact of wheels on the steel rail caused by the irregularity of the surface of the steel rail; and secondly, the propagation of vibration, namely the propagation of vibration waves to the tunnel and the surrounding soil body through the track. In the early days, people do not consider the interaction between vehicles and rails when predicting the vibration caused by the subway, but simplify the vehicle load into a moving constant force or a simple harmonic force applied to the soil body. Obviously, this practice ignores the vehicle-rail dynamic interaction. In fact, the dynamic component in the vehicle load has a significant effect on the vibration of the soil surrounding the tunnel. Then, the vertical vibration of the tunnel and the surrounding soil caused by the ground iron under the action of 1/8, 1/4, 1/2 and the overall vehicle model was studied by assuming that the geometrical characteristics and the material characteristics of the track structure have the property of extending infinitely along the cross section. The researches are carried out based on a frequency domain flexibility matrix method, and dynamic interaction between vehicles, tracks, tunnels and soil bodies can be considered. However, these approaches fail to take into account discrete support characteristics in the track structure, particularly for structures with significant discontinuity characteristics such as steel spring-loaded slab tracks. The Degrande establishes a track-tunnel-soil body model with periodic discrete support by Floquet transformation, and also realizes the coupling vibration with the vehicle by adopting a flexibility matrix method. However, this method is performed by periodically changing the unit of cells, and it is still difficult to simulate the conditions of rail joints, switches, and the like. The above are all frequency domain based analysis methods. In fact, if a time-domain analysis method is used, the simulation of the vehicle-rail interaction will become much more flexible. By establishing a finite element model of the track structure, the conditions of discrete support of sleepers and steel springs, turnouts and the like can be simulated, and the nonlinearity of a vehicle-track system can be considered. On the other hand, at present, a separation iteration method is mostly adopted for solving time-domain vehicle-rail coupling vibration. In fact, due to the high contact rigidity of the wheel and rail, the time step required for solving by the separation iteration method is very small (generally less than 10)-4) Resulting in a large amount of calculation. And the strong coupling method is adopted for solving, so that the limitation of the time integration step length can be avoided.
Various types of vibration wave propagation models are also currently developed, including analytical models, semi-analytical models, and numerical models. Although computationally efficient, analytical/semi-analytical models are limited in their applicability and accuracy by a number of simplifying assumptions. Numerical models including Finite Element Models (FEM), Boundary Element Models (BEM), hybrid finite element-boundary element models (FEM-BEM), etc. are currently common computational models. The numerical model also includes a time domain model and a frequency domain model. The time domain model, although having a strong simulation capability, requires an increase in the size of the model to reduce the reflection of the shock wave at the boundary, resulting in a low computational efficiency. In fact, the strain of the soil body under the action of the vehicle load is very small and is only 10-5Of even smaller order, and therefore the soil can be considered as a string elastomer. Also, in most cases, the geometrical and material characteristics of the tunnel-soil system can be considered to have a constant along-the-track extension. In this case, the 2.5-dimensional (2.5D) analysis method, which is a frequency domain analysis method proposed by the poplar and the inventor, has high computational efficiency. The 2.5D method only needs to simulate the cross section of the model and realize the propagation of the vibration wave in the spatial domain through the dispersion of wave numbers. In the framework of the 2.5D method, various simulation methods of vibration wave propagation models were developed, including finite element-infinite element method (FEM-IEM), FEM-BEM, finite element-best matching layer method (FEM-PML), finite element-fundamental solution (FEM-MFS), and the like.
Disclosure of Invention
Aiming at the defects of the prior art, the invention provides a time-frequency mixing prediction method and a time-frequency mixing prediction system for vertical vibration of a tunnel and a soil body caused by a vehicle, and aims to solve the problems that in the conventional vehicle-track-tunnel-soil body coupled vibration calculation, the local defect of the track is difficult to consider in frequency domain calculation, the efficiency of time domain calculation is low, and the like.
The invention solves the technical problems through the following technical scheme: a time-frequency mixing prediction method for vertical vibration of a tunnel and a soil body caused by vehicles comprises the following steps:
step 1: establishing a tunnel-soil 2.5D FEM-PML model and a vehicle-track-tunnel-soil 2D multi-body dynamics-finite element model;
step 2: calculating the vertical interaction force of the track-tunnel in the time domain according to the vehicle-track-tunnel-soil 2D multi-body dynamics-finite element model in the step 1;
and step 3: establishing a dynamic equation of the tunnel-soil 2.5D FEM-PML model in the step 1 in a frequency domain-wave number domain according to the vertical interaction force of the track-tunnel in the step 2;
and 4, step 4: solving a dynamic equation of the tunnel-soil 2.5D FEM-PML model in the step 3 to obtain a dynamic response of the tunnel-soil 2.5D FEM-PML model in the frequency domain-wave number domain;
and 5: calculating the vibration response of the tunnel-soil 2.5D FEM-PML model in the time domain-space domain according to the dynamic response of the tunnel-soil 2.5D FEM-PML model in the frequency domain-wave number domain in the step 4;
step 6: and calculating the vertical vibration level corresponding to the tunnel-soil body according to the vibration response of the tunnel-soil body 2.5D FEM-PML model in the step 5.
According to the time-frequency hybrid prediction method of vertical vibration, the vertical interaction force of a track and a tunnel is obtained in a time domain by establishing a 2D vehicle-track-tunnel-soil MBS/FEM model (namely a 2D multi-body dynamics-finite element model), the vertical interaction force of the track and the tunnel is converted into a frequency domain and is applied to a tunnel-soil 2.5D FEM-PML model, vibration calculation of the tunnel-soil in the frequency domain-wave number domain is completed, and finally the vertical vibration level of the tunnel-soil is calculated, so that the flexibility of time domain solution on track discrete support characteristics and local defect simulation is exerted, the high efficiency of the frequency domain solution on soil vibration calculation is exerted, the final calculation result is more in line with the actual situation, and meanwhile, the calculation efficiency is improved.
Further, in the step 1, when a tunnel-soil 2.5D FEM-PML model is established, soil in a research range around the tunnel and the tunnel is dispersed by adopting 4-node or 8-node planar shell units, a longitudinal degree of freedom is added to a node of each planar shell unit, and the degree of freedom coordinate is converted from a spatial domain to a wavenumber domain;
the best matching layer elements are set at the boundaries of the study range and discretized using 4-node or 8-node plane elements.
The best matching layer at the boundary avoids reflections of the vibration wave by increasing the degree of freedom along the track extension to take into account the propagation of the vibration wave along the track extension.
Further, in the step 1, when a vehicle-track-tunnel-soil body 2D multi-body dynamics-finite element model is built, the vehicle adopts multi-body dynamics to simulate, the vehicle comprises a vehicle body, a bogie, a wheel set, a primary suspension and a secondary suspension, the vehicle body is connected with the bogie through the secondary suspension, and the bogie is connected with the wheel set through the primary suspension;
simulating a track-tunnel-soil body by adopting a 2D three-layer elastic foundation beam finite element model, simulating a steel rail, a track plate and a tunnel by adopting an Euler beam model, and dispersing by adopting a finite element method; the fastener, the elastic cushion layer and the elastic support of the soil body are simulated through the linear spring-damper;
and connecting the vehicle model with the track-tunnel-soil body model through a linear Hertz contact spring to obtain the vehicle-track-tunnel-soil body 2D multi-body dynamics-finite element model.
Further, in the step 1, the parameter determination step of the vehicle-track-tunnel-soil body 2D multi-body dynamics-finite element model comprises:
step 1.1: adding a track model on the basis of the tunnel-soil 2.5D FEM-PML model to obtain a track-tunnel-soil 2.5DFEM-PML model;
step 1.2: replacing the 2D three-layer elastic foundation beam finite element model of the track-tunnel-soil body with a 2D three-layer elastic foundation beam analytical model of the track-tunnel-soil body;
step 1.3: respectively calculating the vertical flexibility coefficients of the steel rails of the 2D three-layer elastic foundation beam analytical model and the 2.5D FEM-PML model;
step 1.4: adjusting the rigidity and damping of the fastener, the elastic cushion layer and the elastic support spring-damper unit of the soil body to enable the relative error between the vertical flexibility coefficient of the steel rail calculated based on the three-layer elastic foundation beam analytical model and the vertical flexibility coefficient of the steel rail calculated based on the 2.5D FEM-PML model to be smaller than the set error precision, and determining the relevant parameters of the 2D three-layer elastic foundation beam analytical model of the track-tunnel-soil body;
step 1.5: and determining the rigidity and the damping of a fastener, an elastic cushion layer and a soil body elastic supporting spring-damper in the track-tunnel-soil body 2D three-layer elastic foundation beam finite element model according to the relevant parameters of the track-tunnel-soil body 2D three-layer elastic foundation beam analytical model and based on the principle that the rigidity and the damping are equal in each linear meter.
Because the track-tunnel-soil body 2D three-layer elastic foundation beam finite element model is a plane model, and some parameters cannot be directly obtained, the rigidity coefficients and the damping coefficients of the fasteners, the elastic cushion layers and the soil body elastic supporting spring-damper units in the track-tunnel-soil body 2D three-layer elastic foundation beam finite element model are determined through the tunnel-soil body 2.5D FEM-PML model, and the parameters of the vehicle model can be obtained according to an actual vehicle without determining. Because the track-tunnel-soil 2.5D FEM-PML model assumes longitudinal invariance, the vertical flexibility coefficient of the steel rail calculated by the track-tunnel-soil 2D three-layer elastic foundation beam finite element model cannot be compared with the vertical flexibility coefficient calculated by the 2.5D FEM-PML model. In order to keep consistent with the assumption of longitudinal invariance adopted in the track-tunnel-soil 2.5D FEM-PML model, the track-tunnel-soil 2D three-layer elastic foundation beam finite element model is replaced by a track-tunnel-soil 2D three-layer elastic foundation beam analytical model.
Further, in the step 2, the calculation method of the vertical interaction force of the rail-tunnel includes:
step 2.1: establishing a coupling dynamic equation of the vehicle-track-tunnel-soil body 2D multi-body dynamics-finite element model based on a strong coupling method, wherein the coupling dynamic equation specifically comprises the following steps:
Figure BDA0002529559700000041
wherein M isv、Cv、KvRespectively mass, damping and stiffness matrix, M, of the vehicle modelt、Ct、KtRespectively the mass, damping and stiffness matrix, K 'of the track-tunnel-soil model'v、Kt' (t) additional stiffness of wheel-rail contact stiffness in vehicle model and track-tunnel-soil body model, Kvt(t)、Ktv(t) are all wheel-rail coupling interaction matrices, Uv(t)、
Figure BDA0002529559700000042
Respectively displacement, velocity and acceleration vectors, U, of the vehicle modelt(t)、
Figure BDA0002529559700000043
Respectively the displacement, velocity and acceleration vectors of the orbit-tunnel-soil model, Fv(t)、Ft(t) load vectors borne by the vehicle model and the track-tunnel-soil body model respectively;
step 2.2: solving the coupled power equation in the step 2.1 to obtain the vertical vibration displacement and the vertical vibration speed of the track node and the tunnel node;
step 2.3: and (3) calculating the vertical interaction force according to the vertical vibration displacement and the vertical vibration speed in the step 2.3, wherein a specific calculation formula is as follows:
Figure BDA0002529559700000051
wherein k ismj、cmjThe rigidity and the damping of the spring-damper for connecting the jth track and the tunnel are respectively realized; f. ofj(t) is the internal force of the corresponding spring-damper, i.e. the vertical interaction force of the rail-tunnel; u. ofmj(t)、
Figure BDA0002529559700000052
The vertical relative displacement and the vertical relative speed of the corresponding spring-damper are respectively, namely the vertical vibration displacement and the vertical vibration speed of the track node and the tunnel node.
Further, in the step 2.2, the coupled dynamic equation is solved by using an Euler-Gauss method, a Newmark-beta method, a Wilson-theta method or a central difference method.
Further, in the step 3, the dynamic equation of the tunnel-soil 2.5D FEM-PML model is as follows:
Figure BDA0002529559700000053
wherein the content of the first and second substances,
Figure BDA0002529559700000054
global rigidity matrixes of a finite element region and an optimal matching layer region are respectively set;
Figure BDA0002529559700000055
Figure BDA0002529559700000056
respectively corresponding global quality matrixes of a finite element region and an optimal matching layer region; u (k)xω) is a node displacement vector in the frequency-wavenumber domain; f (k)xω) is the node load vector in the frequency-wavenumber domain, F (k)xω) is obtained by performing a fast fourier transformation on the vertical interaction force of the rail-tunnel; k is a radical ofxThe wave number in the track extension direction is given by rad/m, and ω is the vibration circle frequency given by rad/s.
Further, in the step 5, a double inverse fourier transform is adopted to transform the dynamic response in the frequency domain-wave number domain into the vibration response in the time domain-space domain, and the specific transformation expression is as follows:
Figure BDA0002529559700000057
wherein the content of the first and second substances,
Figure BDA0002529559700000058
the dynamic response of the tunnel-soil 2.5D FEM-PML model in the frequency domain-wave number domain is shown, u (x, y, z, t) is the vibration response of the tunnel-soil 2.5D FEM-PML model in the time domain-space domain, and k is shownxThe wave number in the track extension direction is given by rad/m, and ω is the vibration circle frequency given by rad/s.
Further, in the step 6, the vertical vibration level corresponding to the tunnel-soil body includes a vibration speed level, a vibration acceleration level and a Z vibration level.
The invention also provides a time-frequency mixing prediction system for vertical vibration of a tunnel and a soil body caused by vehicles, which comprises the following steps:
the model establishing unit is used for establishing a tunnel-soil 2.5D FEM-PML model and a vehicle-track-tunnel-soil 2D multi-body dynamics-finite element model;
the acting force calculation unit is used for calculating the vertical interaction force of the track-tunnel in the time domain according to the vehicle-track-tunnel-soil body 2D multi-body dynamics-finite element model;
the dynamic equation building unit is used for building a dynamic equation of the tunnel-soil 2.5D FEM-PML model in a frequency domain-wave number domain according to the vertical interaction force of the track-tunnel;
the dynamic response obtaining unit is used for solving a dynamic equation of the tunnel-soil 2.5D FEM-PML model to obtain the dynamic response of the tunnel-soil 2.5D FEM-PML model in the frequency domain-wave number domain;
the vibration response calculation unit is used for calculating the vibration response of the tunnel-soil 2.5D FEM-PML model in the time domain-space domain according to the dynamic response of the tunnel-soil 2.5D FEM-PML model in the frequency domain-wave number domain;
and the vibration level calculating unit is used for calculating the vertical vibration level corresponding to the tunnel-soil body according to the vibration response of the tunnel-soil body 2.5D FEM-PML model.
Advantageous effects
Compared with the prior art, the time-frequency mixing prediction method and the time-frequency mixing prediction system for the vertical vibration of the tunnel and the soil body caused by the vehicle, the vertical interaction force of the track and the tunnel is obtained in the time domain by establishing a 2D vehicle-track-tunnel-soil MBS/FEM model, the vertical interaction force of the track and the tunnel is converted into a frequency domain and applied to a tunnel-soil body 2.5D FEM-PML model, vibration calculation of the tunnel-soil body in the frequency domain-wave number domain is completed, and finally the vertical vibration level of the tunnel-soil body is calculated, so that the flexibility of time domain solution on track discrete support characteristics and local defect simulation is exerted, the high efficiency of frequency domain solution on soil body vibration calculation is exerted, the final calculation result is more consistent with the actual situation, and meanwhile, the calculation efficiency is improved.
Drawings
In order to more clearly illustrate the technical solution of the present invention, the drawings needed to be used in the description of the embodiments are briefly introduced below, and it is obvious that the drawings in the following description are only one embodiment of the present invention, and it is obvious for those skilled in the art to obtain other drawings based on the drawings without creative efforts.
FIG. 1 is a flowchart illustrating a time-frequency mixing prediction method for vertical vibration of a tunnel and a soil caused by a vehicle according to an embodiment of the present invention;
FIG. 2 is a schematic structural diagram of a vehicle-track-tunnel-soil 2D MBS/FEM model in the embodiment of the invention;
FIG. 3 is a schematic diagram of the application of rail-tunnel vertical interaction force on a tunnel-soil 2.5D FEM-PML model in an embodiment of the present invention;
FIG. 4 illustrates the geometry and material properties of a rail-tunnel in an embodiment of the present invention;
FIG. 5 is a schematic diagram of meshing of a tunnel-soil 2.5D FEM-PML model according to an embodiment of the present invention;
FIG. 6 is a graph of the vertical compliance of a rail based on analytical and numerical methods in an embodiment of the present invention;
FIG. 7 is a graph of sample points of track irregularity in accordance with an embodiment of the present invention;
FIG. 8 is a vertical velocity plot of observation point OP1 in an embodiment of the present invention;
FIG. 9 is a vertical velocity plot of observation point OP2 in an embodiment of the present invention;
FIG. 10 is a vertical velocity plot of observation point OP3 in an embodiment of the present invention;
FIG. 11 is a vertical velocity plot of observation point OP4 in an embodiment of the present invention;
FIG. 12 is a frequency doubling distribution plot corresponding to one third of the vertical velocity of observation point OP1 in an embodiment of the present invention;
FIG. 13 is a frequency doubling distribution plot corresponding to one third of the vertical velocity of observation point OP2 in an embodiment of the present invention;
FIG. 14 is a frequency doubling distribution plot corresponding to one third of the vertical velocity of observation point OP3 in an embodiment of the present invention;
FIG. 15 is a frequency doubling distribution plot corresponding to one third of the vertical velocity of observation point OP4 in an embodiment of the present invention;
the device comprises a first-second-system suspension, a first-system suspension, a 3-wheel pair, a 4-Hertz contact spring, a 5-spring-damper, an A-steel rail, a B-fastener, a C-track plate, a D-elastic cushion layer, an E-tunnel and an F-soil body.
Detailed Description
The technical solutions in the present invention are clearly and completely described below with reference to the drawings in the embodiments of the present invention, and it is obvious that the described embodiments are only a part of the embodiments of the present invention, and not all of the embodiments. All other embodiments, which can be derived by a person skilled in the art from the embodiments given herein without making any creative effort, shall fall within the protection scope of the present invention.
Example 1
As shown in fig. 1, the time-frequency mixing prediction method for vertical vibration of a tunnel and a soil body caused by a vehicle provided by the invention comprises the following steps:
1. and establishing a tunnel-soil 2.5D FEM-PML model and a vehicle-track-tunnel-soil 2D multi-body dynamics-finite element model.
When a tunnel-soil 2.5D FEM-PML model is built, assuming that the geometric properties and material characteristics of a track, a tunnel and soil are kept unchanged along the track extension direction, dispersing the soil in a research range around the tunnel and the tunnel by adopting 4-node or 8-node plane shell units, adding a longitudinal degree of freedom on a node of each plane shell unit in order to consider the propagation of vibration waves along the track extension direction, and converting the degree of freedom coordinate from a space domain to a wave number domain; the best matching layer unit is arranged at the boundary of the research range to avoid the reflection of the vibration wave, and the 4-node or 8-node plane unit is adopted for dispersion, but the best matching layer of the 2.5DFEM-PML model formed based on the stretching function has the function of absorbing the wave field, which is different from the finite element region. The specific modeling process of the tunnel-Soil 2.5D FEM-PML model can be referred to Lopes Patri cia, costap's, Ferrarz M, et al.
In the process of establishing a vehicle-track-tunnel-soil 2D multi-body dynamics-finite element model (MBS/FEM model), the vehicle adopts multi-body dynamics to simulate, the vehicle comprises a vehicle body, a bogie, a wheel set 3, a primary suspension 2 and a secondary suspension 1, the vehicle body is connected with the bogie through the secondary suspension 1, and the bogie is connected with the wheel set 3 through the primary suspension 2; the rail-tunnel-soil body is simulated by adopting a 2D three-layer elastic foundation beam finite element model, the steel rail A, the rail plate C and the tunnel E are simulated by adopting an Euler beam model, and the finite element method is adopted for dispersion; the elastic support of the fasteners B, the elastic cushion D and the soil body F is simulated by linear spring-dampers 5 uniformly distributed along the track direction. The vehicle model and the track-tunnel-soil body model are connected through a linear Hertz contact spring 4 to obtain the vehicle-track-tunnel-soil body 2D MBS/FEM model, as shown in figure 2.
Each car body and bogie contains 2 degrees of freedom, i.e. heave and nod; each wheel pair only contains 1 degree of freedom, namely sinking and floating; the total degree of freedom of the single-section vehicle body is 10. According to multi-rigid-body assumption and linear suspension system assumption, parameters of each section of vehicle body can be simplified into parameters such as primary suspension rigidity and damping, secondary suspension rigidity and damping, mass and nodding moment of inertia of the vehicle body, mass and nodding moment of inertia of a framework and mass of a wheel pair, and the like, and specific reference can be made to the influence of a train-track-bridge coupling system dynamic equation solving method on calculation precision and efficiency, which is proposed by the Chinese railway science, 2016,37(05):17-26.
Because the track-tunnel-soil body 2D three-layer elastic foundation beam finite element model is a plane model, and some parameters cannot be directly obtained, the rigidity coefficients and the damping coefficients of the fasteners, the elastic cushion layers and the soil body elastic supporting spring-damper units in the track-tunnel-soil body 2D three-layer elastic foundation beam finite element model are determined through the tunnel-soil body 2.5D FEM-PML model, and the parameters of the vehicle model can be obtained according to an actual vehicle without determining. Because the track-tunnel-soil 2.5D FEM-PML model assumes longitudinal invariance, the vertical flexibility coefficient of the steel rail calculated by the track-tunnel-soil 2D three-layer elastic foundation beam finite element model cannot be compared with the vertical flexibility coefficient calculated by the 2.5D FEM-PML model. In order to keep consistent with the assumption of longitudinal invariance adopted in the track-tunnel-soil 2.5D FEM-PML model, the track-tunnel-soil 2D three-layer elastic foundation beam finite element model is replaced by a track-tunnel-soil 2D three-layer elastic foundation beam analytical model, and the specific determination steps are as follows:
1.1 adding a track model on the basis of the tunnel-soil 2.5D FEM-PML model to obtain a track-tunnel-soil 2.5DFEM-PML model;
1.2 replacing a 2D three-layer elastic foundation beam finite element model of the track-tunnel-soil body with a 2D three-layer elastic foundation beam analytical model of the track-tunnel-soil body;
1.3, respectively calculating the vertical flexibility coefficients of the steel rail in the range of 0-80 Hz of the 2D three-layer elastic foundation beam analytical model and the 2.5D FEM-PML model;
1.4, adjusting the rigidity and the damping of elastic supporting spring-damper units of the fastener, the elastic cushion layer and the soil body, so that the relative error between the vertical flexibility coefficient of the steel rail calculated based on the three-layer elastic foundation beam analytical model and the vertical flexibility coefficient of the steel rail calculated based on the 2.5D FEM-PML model within the range of 0-80 Hz is smaller than a set error precision, and the rigidity and the damping of the elastic supporting spring-damper units of the fastener, the elastic cushion layer and the soil body in the 2D three-layer elastic foundation beam analytical model of the track-tunnel-soil body are determined, wherein the set error precision is 5% of the engineering precision;
1.5 determining the rigidity and the damping of the fasteners, the elastic cushion layer and the elastic support spring-damper of the soil body in the track-tunnel-soil body 2D three-layer elastic foundation beam finite element model according to the rigidity and the damping of the fasteners, the elastic cushion layer and the elastic support spring-damper of the soil body in the track-tunnel-soil body 2D three-layer elastic foundation beam analytical model and based on the principle that the rigidity and the damping are equal in each linear meter.
2. According to the vehicle-track-tunnel-soil 2D MBS/FEM model in the step 1, the vertical interaction force of the track-tunnel is calculated in the time domain, and the calculation method of the vertical interaction force of the track-tunnel comprises the following steps:
2.1, establishing a coupling dynamic equation of a vehicle-track-tunnel-soil 2D MBS/FEM model based on a strong coupling method, wherein the coupling dynamic equation specifically comprises the following steps:
Figure BDA0002529559700000091
wherein M isv、Cv、KvRespectively mass, damping and stiffness matrix, M, of the vehicle modelt、Ct、KtRespectively the mass, damping and stiffness matrix, K 'of the track-tunnel-soil model'v、Kt' (t) additional stiffness of wheel-rail contact stiffness in vehicle model and track-tunnel-soil body model, Kvt(t)、Ktv(t) are all wheel-rail coupling interaction matrices, Uv(t)、
Figure BDA0002529559700000092
Respectively displacement, velocity and acceleration vectors, U, of the vehicle modelt(t)、
Figure BDA0002529559700000093
Respectively the displacement, velocity and acceleration vectors of the orbit-tunnel-soil model, Fv(t)、FtAnd (t) load vectors borne by the vehicle model and the track-tunnel-soil body model respectively.
In the time domain, the time domain [ t ] is dividedmin,tmax]Discrete as m1Points are formed to form parameters such as initial mass, initial damping and initial stiffness matrix of the vehicle-track-tunnel-soil 2D MBS/FEM model, specific expressions and calculation methods of each parameter matrix can refer to the influence of a train-track-bridge coupling system dynamic equation solution method on calculation precision and efficiency, which is proposed by Zhushihui and the like [ J]China railway science,2016,37(05):17-26.。
As shown in fig. 1, according to the current calculated time or current running time t (t) of the vehiclemin≤t≤tmax，tminTo calculate the time, t, to a minimummax△ t is a time integral step length for maximum calculation time), determining the position of the steel rail where each wheel pair of the vehicle is positioned, calculating the track irregularity at the contact position of each wheel pair and the steel rail and the parameters of the mass, damping and rigidity matrix of the vehicle-track-tunnel-soil 2D MBS/FEM model according to the position, and updating the parameters in real time until the current calculation time t is more than tmaxThe influence of the train-track-bridge coupled system dynamic equation solving method on the calculation precision and efficiency can be referred to in detail [ J]2016 (05):17-26. in China railway science. Minimum calculation time tminMay be 0, maximum computation time tmaxEqual to the ratio of the sum of the length of the steel rail and the length of the vehicle to the speed of the vehicle.
2.2, solving the coupled dynamic equation in the step 2.1 by adopting an Euler-Gauss method, a Newmark-beta method, a Wilson-theta method or a center difference method to obtain the vertical vibration displacement and the vertical vibration speed of the track node and the tunnel node.
In the embodiment, a Newmark-beta method is selected to solve the coupling dynamic equation of the formula (1), and the specific solving process can refer to Liu Jing Bo et al, structural dynamics [ M ] Beijing, mechanical industry Press, 2005.
2.3 calculating the vertical interaction force according to the vertical vibration displacement and the vertical vibration speed in the step 2.3, wherein the specific calculation formula is as follows:
Figure BDA0002529559700000094
wherein k ismj、cmjThe rigidity and the damping of a spring-damper for connecting the jth track and the tunnel are respectively set, j is 1-M, and M is the number of connecting springs between the track and the tunnel; f. ofj(t) is the internal force of the corresponding spring-damper (i.e. the vertical interaction force of the rail-tunnel); u. ofmj(t)、
Figure BDA0002529559700000101
The vertical relative displacement and the vertical relative velocity (namely the vertical vibration displacement and the vertical vibration velocity of the track node and the tunnel node) of the corresponding spring-damper are respectively.
Because the dynamic response and the vibration response of the subsequent tunnel-soil 2.5D FEM-PML model are solved in the frequency domain and the wave number domain, the current calculation time t is more than tmaxIn the process, the finally calculated track-tunnel vertical interaction force needs to be converted from a time domain to a frequency domain through fast Fourier transform and then applied to a tunnel-soil body 2.5D FEM-PML model, as shown in figure 3, fj(ωi) To be f isj(t) the result of the transformation from time domain to frequency domain, j is 1 to M, M is the number of connecting springs between the track and the tunnel, ω isiDiscrete points in the frequency domain.
3. And (3) establishing a dynamic equation of the tunnel-soil body 2.5DFEM-PML model in the step 1 in the frequency domain-wave number domain according to the vertical interaction force of the track-tunnel in the step 2.
The dynamic equation of the tunnel-soil 2.5D FEM-PML model is as follows:
Figure BDA0002529559700000102
wherein the content of the first and second substances,
Figure BDA0002529559700000103
global rigidity matrixes of a finite element region and an optimal matching layer region are respectively set;
Figure BDA0002529559700000104
Figure BDA0002529559700000105
respectively corresponding global quality matrixes of a finite element region and an optimal matching layer region; u (k)xω) is a node displacement vector in the frequency-wavenumber domain; f (k)xω) is the node load vector in the frequency-wavenumber domain, F (k)xω) is obtained by performing a fast fourier transformation on the vertical interaction force of the rail-tunnel; k is a radical ofxThe unit rad/m is the wave number in the orbital extension direction, ω is the vibration circle frequency, the unit rad/s, the wave number, circle frequency are variables in the fourier transform, the wave number corresponds to the spatial coordinate x, and the circle frequency corresponds to the time t.
4. And (4) solving the dynamic equation of the tunnel-soil 2.5D FEM-PML model in the step (3) to obtain the dynamic response of the tunnel-soil 2.5DFEM-PML model in the frequency domain-wave number domain.
Wave number domain [ k ]x-min,kx-max]Discrete as m2And (4) forming parameters such as initial mass and initial rigidity matrix of the tunnel-soil 2.5D FEM-PML model. As shown in fig. 1, according to the current wave number kx(kx-min≤kx≤kx-max，kx-minIs the minimum wave number, kx-maxAt maximum wavenumber, △ kxWave number integration step length) updating the global rigidity and the global quality matrix of a finite element region and the global rigidity and the global quality matrix of an optimal matching layer region of the tunnel-soil 2.5D FEM-PML model, and forming the overall rigidity and the quality matrix of the model; calculating a tunnel-soil load vector corresponding to the current wave number, applying the tunnel-soil load vector to the tunnel-soil 2.5D FEM-PML model, and obtaining the dynamic response of the tunnel-soil 2.5D FEM-PML model through solving the formula (3) until the current wave number kx＞kx-max. Reference may be made specifically to Lopes patri cia, costap.
Figure BDA0002529559700000106
R.，Cardoso，A.Silva.Numerical modeling of vibrations induced by railwaytraffic in tunnels:From the source to the nearby buildings[J].Soil Dynamicsand Earthquake Engineering,2014,61-62:269-85.。
The range of the frequency domain and the discrete point number respectively correspond to the time integral step length and the integral time step number selected during the calculation of the vertical interaction force of the track-tunnel. And the range and the discrete point number of the wave number domain are determined according to the precision requirement of the calculation result.
5. And 4, calculating the vibration response of the tunnel-soil 2.5D FEM-PML model in the time domain-space domain according to the dynamic response of the tunnel-soil 2.5D FEM-PML model in the frequency domain-wave number domain in the step 4.
And converting the finally calculated dynamic response into the vibration response in a time domain-space domain by adopting double Fourier inversion conversion, wherein the specific conversion expression is as follows:
Figure BDA0002529559700000111
wherein the content of the first and second substances,
Figure BDA0002529559700000112
the dynamic response of the tunnel-soil 2.5D FEM-PML model in the frequency domain-wave number domain is shown, u (x, y, z, t) is the vibration response of the tunnel-soil 2.5D FEM-PML model in the time domain-space domain, and k is shownxThe wave number in the track extension direction is given by rad/m, and ω is the vibration circle frequency given by rad/s.
The vibrational response includes a vibrational velocity and a vibrational acceleration, from which a vibrational velocity level can be calculated, from which a vibrational acceleration level, a fractional maximum vibration level, etc. can be calculated.
6. And (5) calculating the vertical vibration level corresponding to the tunnel-soil body in the time domain according to the vibration response of the tunnel-soil body 2.5D FEM-PML model in the step 5. The vertical vibration level corresponding to the tunnel-soil body comprises a vibration speed level, a vibration acceleration level and a Z vibration level.
(1) Vibration velocity level
The U.S. department of railway transportation adopts a vibration speed level L when evaluating environmental vibration caused by trafficvTo express the intensity of the vibration, the specific calculation formula is:
Figure BDA0002529559700000113
in the formula: l isvVibration speed level, unit: dB; v. ofRMSRoot mean square vibration velocity, unit: m/s; v. ofrefIs the reference velocity.
(2) Vibration acceleration level LaComprises the following steps:
Figure BDA0002529559700000114
in the formula: a isrmsIs effective value of vibration acceleration, m/s2；a0Is the reference acceleration.
(3) Z vibration level VL
According to ISO2631-1, the vibration acceleration level obtained after the whole body vibration is corrected according to weighting factors of different frequencies, is called Z vibration level for short, and is recorded as VL, and the unit is dB. The calculation formula of the Z vibration level VL is as follows:
Figure BDA0002529559700000121
in the formula, a0Is a 'reference acceleration'rmsFor the corrected effective value (m/s) of the vibration acceleration2)。
(4) Frequency division maximum vibration level
The frequency division maximum vibration level VLmax is the maximum vibration acceleration level (obtained after correction according to the weighting factor Z) corresponding to the center frequency of one third octave. 1/3 octave center frequencies and their frequency ranges are shown in Table 1.
TABLE 11/3 octave center frequency and frequency Range (Hz)
Figure BDA0002529559700000122
The invention also provides a time-frequency mixing prediction system for vertical vibration of a tunnel and a soil body caused by vehicles, which comprises the following steps:
the model establishing unit is used for establishing a tunnel-soil 2.5D FEM-PML model and a vehicle-track-tunnel-soil 2D multi-body dynamics-finite element model;
the acting force calculation unit is used for calculating the vertical interaction force of the track-tunnel in the time domain according to the vehicle-track-tunnel-soil body 2D multi-body dynamics-finite element model;
the dynamic equation building unit is used for building a dynamic equation of the tunnel-soil 2.5D FEM-PML model in a frequency domain-wave number domain according to the vertical interaction force of the track-tunnel;
the dynamic response obtaining unit is used for solving a dynamic equation of the tunnel-soil 2.5D FEM-PML model to obtain the dynamic response of the tunnel-soil 2.5D FEM-PML model in the frequency domain-wave number domain;
the vibration response calculation unit is used for calculating the vibration response of the tunnel-soil 2.5D FEM-PML model in the time domain-space domain according to the dynamic response of the tunnel-soil 2.5D FEM-PML model in the frequency domain-wave number domain;
and the vibration level calculating unit is used for calculating the vertical vibration level corresponding to the tunnel-soil body according to the vibration response of the tunnel-soil body 2.5D FEM-PML model.
Example 2
As shown in fig. 4, taking a single-line tunnel as an example, the center of mass of the tunnel is located at a depth of 13.0m from the ground, the inner diameter of the tunnel is 6.0m, and the thickness e of the lining is 0.4 m. The tunnel being made of concrete and having a Young's modulus Ed32.5GPa, density pd＝2500kg/m3Poisson ratio vdDamping ratio ζ of 0.2d0.02. The guide rail consists of two R60 steel rails with a gauge of 1.5m and is supported by fasteners with a spacing of 0.625m along the extension direction of the rails. The track structure adopts a continuous slab ballastless track, and the height of the track slab is 0.4m and the width of the track slab is 2.5 m. The steel rail is connected with a track plate, and the track plate is fixed on the tunnel. The tunnel is surrounded by homogeneous half-space soil body, and the transverse wave velocity C of the soil bodys170m/s, longitudinal wave velocity CpDensity p of soil mass 318m/sc＝2000kg/m3Poisson ratio vcDamping ratio ζ of 0.3c0.05. OP1, OP2, OP3 and OP4 are observation points, wherein OP1 is located on the upper surface of the tunnel right below the right steel rail, OP2 is located at the height position of 1/2, OP3 is located on the ground surface right above the center line of the tunnel, and OP4 is located on the ground surface 20m away from the OP3 point by a perpendicular line.
As shown in fig. 5, a mesh partition diagram of a tunnel-soil 2.5D FEM-PML model is shown, and due to the symmetry of the tunnel-soil system, the present embodiment only establishes a finite element model of a right region with a vertical line passing through the center of the tunnel as a symmetry axis. Fig. 5 shows the finite element-optimal matching layer grid division of the tunnel-soil 2.5DFEM-PML model, wherein the finite element area FEM is 30m × 30m, the peripheral area is the optimal matching layer area PML with the width of 1m, both areas are simulated by 8-node plane shell elements, and the element size is not more than 0.6 m.
Because the vehicle-track-tunnel-soil 2D multi-body dynamics-finite element model is a plane model, and some parameters cannot be directly obtained, the rigidity coefficients and the damping coefficients of the fasteners, the elastic cushion layers and the soil elastic supporting spring-damper units in the track-tunnel-soil 2D three-layer elastic foundation beam finite element model are determined through the tunnel-soil 2.5D FEM-PML model. As shown in fig. 6, the parameters of the 2D three-layer elastic foundation beam finite element model of the track-tunnel-soil are determined by using the vertical flexibility coefficient of the steel rail obtained by comparing the 2D three-layer elastic foundation beam analytical model with the track-tunnel-soil 2.5D FEM-PML model. The specific implementation method comprises the following steps: adjusting parameters (rigidity and damping) of a fastener, an elastic cushion layer (steel spring) and a soil body elastic supporting spring-damper of the three-layer elastic foundation beam analytical model, and enabling a relative error between a steel rail vertical flexibility coefficient calculated by the three-layer elastic foundation beam analytical model and a steel rail vertical flexibility coefficient calculated by the track-tunnel-soil body 2.5D FEM-PML model to meet an engineering precision requirement of less than 5%, considering that the rigidity and damping parameters corresponding to the three-layer elastic foundation beam analytical model at the moment are reasonable, and finally determining the rigidity and damping of the fastener, the elastic cushion layer and the soil body elastic supporting spring-damper in the track-tunnel-soil body 2D three-layer elastic foundation beam finite element model based on a principle that the rigidity and the damping are equal every linear meter. In fig. 6, the analytical solution corresponds to the vertical compliance of the rail calculated based on the three-layer elastic foundation beam analytical model, and the numerical solution corresponds to the vertical compliance coefficient of the rail calculated by the track-tunnel-soil 2.5D FEM-PML model.
FIG. 7 is a plot of sample points of track irregularity. The sample point curve is artificially simulated by a trigonometric series method, wherein the wavelength component comprises two parts, the first part is 1-100 m wavelength and is simulated by a 6-level spectrum of the U.S. orbit irregularity; the second part is 0.2 m-1 m wavelength, which is modeled using the short-wave irregularity spectrum proposed by Sato.
A vibration response curve of the tunnel and the surrounding soil body can be obtained according to the steps 2-5, and a third frequency doubling distribution graph corresponding to the vibration response curve can be obtained through frequency spectrum analysis, as shown in fig. 8-15. Fig. 8 is a vertical velocity diagram of observation point OP 1; fig. 9 is a vertical velocity diagram of observation point OP 2; fig. 10 is a vertical velocity diagram of observation point OP 3; fig. 11 is a vertical velocity diagram of observation point OP 4; fig. 12 is a one-third frequency multiplication distribution diagram corresponding to the vertical velocity of observation point OP 1; fig. 13 is a one-third frequency multiplication distribution diagram corresponding to the vertical velocity of observation point OP 2; fig. 14 is a one-third frequency multiplication distribution diagram corresponding to the vertical velocity of observation point OP 3; fig. 15 is a one-third frequency multiplication distribution diagram corresponding to the vertical velocity of observation point OP 4. As can be seen from fig. 8 and 9, observation points OP1 and OP2 located in the tunnel can better reflect the impact effect caused when the axle passes through the observation points; as can be seen from fig. 10 and 11, the impact effect of observation points OP3 and OP4 located on the ground due to the distance from the tunnel is not obvious; as can be seen from FIGS. 12-15, the vertical vibration response of the tunnel and the soil body is mainly low-frequency (0-6 Hz) and high-frequency (30-80 Hz), but the medium-frequency (6-30H) vibration is not significant.
The time-frequency mixing prediction method is superior to a 3D finite element method in computational efficiency, and can also consider the vertical vibration of the tunnel and the surrounding soil body caused by discrete support of the track and local defects (such as rail joints, turnouts and track damage); the soil body is simulated by adopting a 2.5D finite element, so that the number of model degrees of freedom is obviously reduced, and the requirement on computing equipment is lower; the method for calculating the tunnel-soil body vibration response based on the time domain 3D finite element method generally needs to increase the size of a model to reduce the influence of the boundary vibration wave reflection on the calculation result, so that the calculation efficiency of the 3D finite element method is low, but in the method, the tunnel-soil body vibration response is solved in a frequency domain-wave number domain by establishing a 2.5D FEM-PML model, and the influence of the boundary vibration wave reflection on the calculation result can be avoided on the premise of not increasing the size of the model.
The above disclosure is only for the specific embodiments of the present invention, but the scope of the present invention is not limited thereto, and any person skilled in the art can easily conceive of changes or modifications within the technical scope of the present invention, and shall be covered by the scope of the present invention.

Claims (10)

1. A time-frequency mixing prediction method for vertical vibration of a tunnel and a soil body caused by vehicles is characterized by comprising the following steps:
step 1: establishing a tunnel-soil 2.5D FEM-PML model and a vehicle-track-tunnel-soil 2D multi-body dynamics-finite element model;
step 2: calculating the vertical interaction force of the track-tunnel in the time domain according to the vehicle-track-tunnel-soil 2D multi-body dynamics-finite element model in the step 1;
and step 3: establishing a dynamic equation of the tunnel-soil 2.5D FEM-PML model in the step 1 in a frequency domain-wave number domain according to the vertical interaction force of the track-tunnel in the step 2;
and 4, step 4: solving a dynamic equation of the tunnel-soil 2.5D FEM-PML model in the step 3 to obtain a dynamic response of the tunnel-soil 2.5D FEM-PML model in the frequency domain-wave number domain;
and 5: calculating the vibration response of the tunnel-soil 2.5D FEM-PML model in the time domain-space domain according to the dynamic response of the tunnel-soil 2.5D FEM-PML model in the frequency domain-wave number domain in the step 4;
step 6: and calculating the vertical vibration level corresponding to the tunnel-soil body according to the vibration response of the tunnel-soil body 2.5D FEM-PML model in the step 5.
2. The time-frequency mixing prediction method for the vertical vibration of the tunnel and the soil body caused by the vehicle as claimed in claim 1, characterized in that: in the step 1, when a tunnel-soil 2.5D FEM-PML model is built, soil in a research range around a tunnel and the tunnel is dispersed by adopting 4-node or 8-node plane shell units, a longitudinal degree of freedom is added to a node of each plane shell unit, and the degree of freedom coordinate is converted from a spatial domain to a wavenumber domain;
the best matching layer elements are set at the boundaries of the study range and discretized using 4-node or 8-node plane elements.
3. The time-frequency mixing prediction method for the vertical vibration of the tunnel and the soil body caused by the vehicle as claimed in claim 1, characterized in that: in the step 1, when a vehicle-track-tunnel-soil body 2D multi-body dynamics-finite element model is built, the vehicle adopts multi-body dynamics to simulate, the vehicle comprises a vehicle body, a bogie, a wheel set, a primary suspension and a secondary suspension, the vehicle body is connected with the bogie through the secondary suspension, and the bogie is connected with the wheel set through the primary suspension;
simulating a track-tunnel-soil body by adopting a 2D three-layer elastic foundation beam finite element model, simulating a steel rail, a track plate and a tunnel by adopting an Euler beam model, and dispersing by adopting a finite element method; the fastener, the elastic cushion layer and the elastic support of the soil body are simulated through the linear spring-damper;
and connecting the vehicle model with the track-tunnel-soil body model through a linear Hertz contact spring to obtain the vehicle-track-tunnel-soil body 2D multi-body dynamics-finite element model.
4. The time-frequency mixing prediction method for vertical vibration of a tunnel and a soil body caused by a vehicle as claimed in claim 1 or 3, characterized in that: in the step 1, the parameter determination step of the vehicle-track-tunnel-soil 2D multi-body dynamics-finite element model comprises the following steps:
step 1.1: adding an orbit model on the basis of the tunnel-soil 2.5D FEM-PML model to obtain an orbit-tunnel-soil 2.5D FEM-PML model;
step 1.2: replacing the 2D three-layer elastic foundation beam finite element model of the track-tunnel-soil body with a 2D three-layer elastic foundation beam analytical model of the track-tunnel-soil body;
step 1.3: respectively calculating the vertical flexibility coefficients of the steel rails of the 2D three-layer elastic foundation beam analytical model and the 2.5D FEM-PML model;
step 1.4: adjusting the rigidity and damping of the fastener, the elastic cushion layer and the elastic support spring-damper unit of the soil body to enable the relative error between the vertical flexibility coefficient of the steel rail calculated based on the three-layer elastic foundation beam analytical model and the vertical flexibility coefficient of the steel rail calculated based on the 2.5D FEM-PML model to be smaller than the set error precision, and determining the relevant parameters of the 2D three-layer elastic foundation beam analytical model of the track-tunnel-soil body;
step 1.5: and determining the rigidity and the damping of a fastener, an elastic cushion layer and a soil body elastic supporting spring-damper in the track-tunnel-soil body 2D three-layer elastic foundation beam finite element model according to the relevant parameters of the track-tunnel-soil body 2D three-layer elastic foundation beam analytical model and based on the principle that the rigidity and the damping are equal in each linear meter.
5. The time-frequency hybrid prediction method for vertical vibration of tunnel and soil caused by vehicles according to any one of claims 1-3, characterized in that: in the step 2, the calculation method of the vertical interaction force of the track-tunnel comprises the following steps:
step 2.1: establishing a coupling dynamic equation of the vehicle-track-tunnel-soil body 2D multi-body dynamics-finite element model based on a strong coupling method, wherein the coupling dynamic equation specifically comprises the following steps:
Figure FDA0002529559690000021
wherein M isv、Cv、KvRespectively mass, damping and stiffness matrix, M, of the vehicle modelt、Ct、KtRespectively the mass, damping and stiffness matrix, K 'of the track-tunnel-soil model'v、K′t(t) additional stiffness of the wheel-rail contact stiffness in the vehicle model and the track-tunnel-soil model, Kvt(t)、Ktv(t) are all wheel-rail coupling interaction matrices, Uv(t)、
Figure FDA0002529559690000022
Respectively displacement, velocity and acceleration vectors, U, of the vehicle modelt(t)、
Figure FDA0002529559690000023
Respectively the displacement, velocity and acceleration vectors of the orbit-tunnel-soil model, Fv(t)、Ft(t) load vectors borne by the vehicle model and the track-tunnel-soil body model respectively;
step 2.2: solving the coupled power equation in the step 2.1 to obtain the vertical vibration displacement and the vertical vibration speed of the track node and the tunnel node;
step 2.3: and (3) calculating the vertical interaction force according to the vertical vibration displacement and the vertical vibration speed in the step 2.3, wherein a specific calculation formula is as follows:
Figure FDA0002529559690000024
wherein k ismj、cmjThe rigidity and the damping of the spring-damper for connecting the jth track and the tunnel are respectively realized; f. ofj(t) is the internal force of the corresponding spring-damper, i.e. the vertical interaction force of the rail-tunnel; u. ofmj(t)、
Figure FDA0002529559690000031
The vertical relative displacement and the vertical relative speed of the corresponding spring-damper are respectively, namely the vertical vibration displacement and the vertical vibration speed of the track node and the tunnel node.
6. The time-frequency mixing prediction method for the vertical vibration of the tunnel and the soil body caused by the vehicle as claimed in claim 5, characterized in that: in the step 2.2, the coupled dynamic equation is solved by adopting an Euler-Gauss method, a Newmark-beta method, a Wilson-theta method or a central difference method.
7. The time-frequency hybrid prediction method for vertical vibration of tunnel and soil caused by vehicles according to any one of claims 1-3, characterized in that: in the step 3, the dynamic equation of the tunnel-soil 2.5D FEM-PML model is as follows:
Figure FDA0002529559690000032
wherein the content of the first and second substances,
Figure FDA0002529559690000033
global rigidity matrixes of a finite element region and an optimal matching layer region are respectively set;
Figure FDA0002529559690000034
Figure FDA0002529559690000035
respectively corresponding global quality matrixes of a finite element region and an optimal matching layer region; u (k)xω) is a node displacement vector in the frequency-wavenumber domain; f (k)xω) is the node load vector in the frequency-wavenumber domain, F (k)xω) is obtained by performing a fast fourier transformation on the vertical interaction force of the rail-tunnel; k is a radical ofxThe wave number in the track extension direction is given by rad/m, and ω is the vibration circle frequency given by rad/s.
8. The time-frequency hybrid prediction method for vertical vibration of tunnel and soil caused by vehicles according to any one of claims 1-3, characterized in that: in the step 5, a double inverse fourier transform is adopted to transform the dynamic response in the frequency domain-wave number domain into the vibration response in the time domain-space domain, and the specific transformation expression is as follows:
Figure FDA0002529559690000036
wherein the content of the first and second substances,
Figure FDA0002529559690000037
the dynamic response of the tunnel-soil 2.5D FEM-PML model in the frequency domain-wave number domain is shown, u (x, y, z, t) is the vibration response of the tunnel-soil 2.5D FEM-PML model in the time domain-space domain, and k is shownxThe wave number in the track extension direction is given by rad/m, and ω is the vibration circle frequency given by rad/s.
9. The time-frequency hybrid prediction method for vertical vibration of tunnel and soil caused by vehicles according to any one of claims 1-3, characterized in that: in the step 6, the vertical vibration level corresponding to the tunnel-soil body comprises a vibration speed level, a vibration acceleration level and a Z vibration level.
10. The utility model provides a time frequency mixing prediction system that vehicle arouses tunnel and soil body vertical vibration which characterized in that includes:
the model establishing unit is used for establishing a tunnel-soil 2.5D FEM-PML model and a vehicle-track-tunnel-soil 2D multi-body dynamics-finite element model;
the acting force calculation unit is used for calculating the vertical interaction force of the track-tunnel in the time domain according to the vehicle-track-tunnel-soil body 2D multi-body dynamics-finite element model;
the dynamic equation building unit is used for building a dynamic equation of the tunnel-soil 2.5D FEM-PML model in a frequency domain-wave number domain according to the vertical interaction force of the track-tunnel;
the dynamic response obtaining unit is used for solving a dynamic equation of the tunnel-soil 2.5D FEM-PML model to obtain the dynamic response of the tunnel-soil 2.5D FEM-PML model in the frequency domain-wave number domain;
the vibration response calculation unit is used for calculating the vibration response of the tunnel-soil 2.5D FEM-PML model in the time domain-space domain according to the dynamic response of the tunnel-soil 2.5D FEM-PML model in the frequency domain-wave number domain;
and the vibration level calculating unit is used for calculating the vertical vibration level corresponding to the tunnel-soil body according to the vibration response of the tunnel-soil body 2.5D FEM-PML model.
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