# Analysis of Antideformation and Antivibration Effect of Different Track Beds under Subway Train Vibration

Authors: Jin Zhang; Chuanhao Xi; Yanhua Sun; Guangsheng Li; Junwei Liu; Mengxue Wang
Journal: Shock and Vibration
 (2022)

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## Abstract

Running metro trains would inevitably influence the vibration deformation in the lower part of the roadbed structure. In order to study the effects of vibration, Qingdao metro line 3 tunnel was taken as the research background. The track coupling model was established using Abaqus-Simpack to analyze the train velocity, train axle load, and trackbed vibration forms for the vertical deformation of roadbed calendar effect to provide the reference in practical engineering design and deformation control. According to the analysis, influenced by the formation of Qingdao Metro Line 3, the time-history curve of the displacement excitation of the trackbed presents the distribution law of “large in the middle and small at both ends.” According to the sensitivity analysis, the deformation of the trackbed is most obviously affected by the form of trackbed; the second was the influence of the axle load of the train. The speed change of the subway train presented relatively little influence on the displacement of the trackbed. Although the deformation control ability of the integral trackbed was better, its vibration reduction performance was poor. The vibration reduction performance of the rubber cushion floating trackbed was the best, followed by the steel spring floating trackbed.


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## Body

## 1. Introduction

With the development of transportation, the vibration influence of rail train operation on the surrounding environment cannot be ignored. Many scholars have done a lot of research on this issue. He and Cui [1] used numerical simulation to research the difference of vibration law between train operation and frozen soil and undisturbed soil. Forrest and Hunt [2] established a three-dimensional numerical model to research the propagation rule of the influence of urban subway operation on tunnel structure and soil layer vibration. Yaseri et al. [3], Gao et al. [4], Zhu et al. [5], and Nejati et al. [6] researched the law of the influence of train operation on ground vibration using numerical simulation. Wang et al. [7] proposed a calculation method to analyze the full-scale three-dimensional random vibration caused by railway traffic, which could effectively solve the train-track-soil coupled vibration problem. Zhu et al. [8] proposed a method to establish a numerical model of sandstone using a two-dimensional particle flow code and a new contact model, providing a reference for the numerical simulation analysis of sandstone tunnel vibration. Yan et al. [9], Connolly et al. [10] Sanayei et al. [11], and Lopes et al. [12] researched the propagation characteristics of vibration influence on existing buildings under train operation. Based on wheel-rail interaction, Cai et al. [13] carried out dynamic simulation calculations for trains, tracks, and bridges through the TTBSIM simulation program. Dapeng et al. [14] established a three-dimensional finite element model to analyze the influence characteristics of train operation on existing foundation pits. Ma et al. [15] used finite element software to simulate and study the vibration characteristics of high-speed train operation on buildings along the line in the soft soil area of Tianjin Binhai New Area. Qiu et al. [16] studied the vertical vibration displacement, dynamic stress, vibration acceleration, and transverse stress distribution of the upper surface layer of the train foundation bed at different speeds using layered system theory and the finite element method. Kaynia et al. [17] studied the influence of different train speeds on the ground vibration response using the finite-infinite element method. Liu et al. [18] and Han et al. [19, 20] established a three-dimensional finite element model and used dynamic implicit analysis to analyze the variation rule of ground vibration intensity under different shield tunnel burial depths, different elastic modulus of tunnel foundation, different subway train speeds, and different soil properties of tunnels. Xie et al. [21] and Liang et al. [22] established a two-dimensional finite element dynamic analysis model and studied the vibration response law of subway operation on adjacent sensitive buildings and the influence of foundation form on vibration by taking speed as the control physical quantity. Actual measurement and experimental research have been done by scholars [19, 23–26]. Schillemans et al. calculated the trainload through the measured transfer function and established a two-dimensional finite element model to study the impact of environmental vibration caused by high-speed railway train operation and analyze the vibration reduction effect of vibration reduction type floating slab track. The following was known from existing research: at present, the research mainly focused on the vibration law of the train operation on the ground, soil layer, and existing building structures. The influence of train operation on the vibration of the trackbed, which was the supporting structure of the track train, cannot be ignored. Based on the existing research, this article comprehensively considered the numerical calculation accuracy and calculation speed. Abaqus-Simpack was used to establish the rigid and flexible coupling numerical model of rail to analyze the law of track vibration deformation under the conditions of train running speed, train axle load, and different modes of trackbed vibration reduction.


## 2. Model Building

### 2.1. Technical Process of Model Building

The Abaqus-Simpack joint simulation was adopted for modeling to guarantee the accuracy and velocity of calculation. The model of the superstructure was built with the multibody dynamics software Simpack, and the model of the substructure composed of rail and trackbed was built with Abaqus. The rail and trackbed generated the flexible volume substructure through finite element calculation [27]; the Abaqus-Simpack joint simulation interface was used to import the Simpack file; the coupling relationships between the two flexible structures and the existing vehicle structures were established. The modeling technical flow is shown in Figure 1.Figure 1
Technical flow route.


#### 2.1.1. Superstructure

The vehicle model was simplified as a system composed of body, hinge, constraint, and force using Simpack [28–31], where parameters of each component of the vehicle model, as shown in Tables 1–4. Qingdao Metro Line 3 adopts the marshaling method of four bullet trains and two trailers [+TC-M-M + M-M-TC+]. As shown in Figure 2, Simpack comes with five forces that both act as coupling connectors and simplify modeling and computation. Therefore, the 5 force element is used instead of the coupler.Table 1
Subway train body parameters.
ComponentValueMass of subway train (t) (single cycle)The locomotive: 36 tThe trailer: 34.6 tSubway train body length (m) (velocity direction)19.5Width of subway train body (m)2.8Height of subway train body (m) (bottom plate from top of rail: 1.1 m)2.7 (the height of the underground train from the rail surface: 3.8 m)Vehicle bogie distance (m), 2Lc12.6Bogie wheelbase fixed distance (m), 2Lt2.2Moment of inertia of nodding mass of subway train (kg·m2), Iyy1328000Moment of inertia of mass side roll of subway train (kg·m2), Ixx32400Moment of inertia of shaking head of a mass of subway train (kg·m2), Izz1317000Table 2
Bogie frame parameters.
ComponentValueQuality of frame (t)A singleThe locomotive: 3.6 tThe trailer: 1.2 tLength of the frame (m) (velocity direction)2.2Width of the frame (m)2.05Moment of inertia of the nodding frame mass (kg·m2), Iyy1760Roll moment of inertia of frame mass (kg·m2), Ixx1430Moment of inertia of shaking head of frame mass (kg·m2), Izz2950Table 3
Parameters of the wheelset.
ComponentValueQuality of the wheelset (t)1.7The radius of the wheelset (m)0.42Moment of inertia of the nodding mass of the wheelset (kg·m2), Iyy104Moment of inertia of the side roll of the wheelset mass (kg·m2), Ixx801Moment of inertia of the wheelset mass (kg·m2), Izz814Table 4
Series I and II suspension parameters.
ComponentValueSeries I longitudinal stiffness (N/m)0.96 × 106Series I lateral stiffness (N/m)0.96 × 106Series I vertical stiffness (N/m)1.2 × 106Series I longitudinal damping (N·s/m)2.5 × 104Series I lateral damping (N·s/m)2.5 × 104Series I vertical damping (N·s/m)5 × 104Series II longitudinal stiffness (N/m)0.21 × 106Series II lateral stiffness (N/m)0.21 × 106Series II vertical stiffness (N/m)0.58 × 106Series II longitudinal damping (N·s/m)3.5 × 104Series II lateral damping (N·s/m)3.5 × 104Figure 2
Subway train marshaling.


#### 2.1.2. Substructure

The substructure was the key to the model. To meet the stress and strain of rail and trackbed requirements, Abaqus was adopted to establish the finite element model of the substructure and conduct modal analysis for generating a flexible body [32].To simplify the model, a 30 m long Timoshenko beam was built in Abaqus without considering the orbital joints between each segment. One hundred one master nodes were established with one master node every 0.3 m. Besides, regardless of the gap between each trackbed, a plate unit with a width of 2.7 m and a length of 30 m was established, and a total of 102 main nodes was retained. The first 30 modes of rail and trackbed, as the principal modes, were taken for the sake of conservation. The boundary only restricted the two ends of the track with six degrees of freedom. The rail and trackbed’s physical parameters are shown in Table5.Table 5
Orbital physical parameters.
RailValueBulk density (N/m3)7.83 × 1011Mass per unit length (kg/m) (monorail)60.64Modulus of elasticity (Pa)2.10 × 1011Poisson’s ratio (μ)0.3Track bedValueBulk density (N/m3) (cross-sectional area1.08 m2)2300Mass per unit length (kg/m) (velocity direction)2484Modulus of elasticity (Pa)3.25 × 1010Poisson’s ratio (μ)0.25


#### 2.1.3. The Upper and Lower Structures Are Coupled

The track and trackbed in the substructure were imported into the Simpack with the Abaqus-Simpack joint simulation interface, and the track coupler was simplified as a set of springs with three-direction stiffness, damping to coupling the track and trackbed. The bottom of the trackbed was the base cushion, which was simplified as a group of springs with three-direction stiffness, damping to be connected with the base rock of the rigid body. The spring parameters of rail fasteners and different track bed bases are as shown in Table6. The dynamic model was adopted to describe the dynamic wheel-rail interaction relationship. Hertz’s nonlinear contact theory generated the coupling relationship between the wheelset and the track (the distribution law of local stress and strain produced by contact between two objects under compression). Thus, the upper and lower structures were coupled as a whole to form the vehicle-rail coupling model. The connection between every two trains and the coupling mode with the substructure is shown in Figure 3, and the model is shown in Figure 4.Table 6
Parameters of the spring of the substructure.
The name of the springValueVertical dynamic stiffness of rail fasteners (kN/m)4.2 × 103Lateral dynamic stiffness of rail fasteners (kN/m)9.5 × 103Longitudinal dynamic stiffness of rail fasteners (kN/m)4.2 × 103Vertical damping of rail coupler (kN/m)50Rail coupler transverse damping (kN·s/m)50Longitudinal damping of rail fasteners (kN·s/m)50Base stiffness of integral track bed (kN/m/m2)6.4 × 105Base damping of integral track bed (kN·s/m/m2)3.7The base stiffness of rubber cushion floating plate trackbed (kN/m/m2)2 × 104Rubber cushion floating plate track bed base damping (kN·s/m/m2)1 × 103Base stiffness of steel spring floating plate trackbed (kN/m)6.9 × 103Steel spring floating plate track bed base damping (kN·s/m)75Figure 3
Vehicle-rail coupling diagram.Figure 4
Effect diagram of rail coupling.


#### 2.1.4. The Orbital Irregularities Are Stimulated

The train’s vibration is influenced by many factors, such as the wear between the wheel and the track, the distance error of the track, and the deviation degree of the curvature of the track. Therefore, the irregularity of the track, including vertical irregularity, horizontal irregularity, direction irregularity, and track spacing irregularity, is an essential factor affecting the accuracy of the calculation results [33–35].To facilitate the modeling, this simulation uses the irregularity excitation spectrum by US Federal Railway Administration (FRA) in the 1980s to simulate the random vibration in train operation based on a large number of measured data. The wavelength range of the spectrum was 1.524–304.8 m, and the grade of the track was divided into six levels. The maximum velocity of Qingdao Metro Line 3 is 80 km/h, which can be satisfied by the American level-5 track irrationality excitation. Therefore, the American level-5 smoothness spectrum was applied as the random external excitation of the track. The irregularity excitation spectrum relevant parameter is shown in Table7.Table 7
Irregularity excitation spectrum relevant parameter.
ParameterParameter values of each orbitalSymbolUnit654321Avcm2/rad/m0.03390.20950.53760.68161.01811.2107Aacm2/rad/m0.03390.07620.30270.41281.2107303634Ωsrad/m0.43800.82091.13120.85200.93080.6046Ωcrad/m0.82450.82450.82450.82450.82450.8245Maximum running speedkm/h176144128964824


## 2.1. Technical Process of Model Building

The Abaqus-Simpack joint simulation was adopted for modeling to guarantee the accuracy and velocity of calculation. The model of the superstructure was built with the multibody dynamics software Simpack, and the model of the substructure composed of rail and trackbed was built with Abaqus. The rail and trackbed generated the flexible volume substructure through finite element calculation [27]; the Abaqus-Simpack joint simulation interface was used to import the Simpack file; the coupling relationships between the two flexible structures and the existing vehicle structures were established. The modeling technical flow is shown in Figure 1.Figure 1
Technical flow route.


### 2.1.1. Superstructure

The vehicle model was simplified as a system composed of body, hinge, constraint, and force using Simpack [28–31], where parameters of each component of the vehicle model, as shown in Tables 1–4. Qingdao Metro Line 3 adopts the marshaling method of four bullet trains and two trailers [+TC-M-M + M-M-TC+]. As shown in Figure 2, Simpack comes with five forces that both act as coupling connectors and simplify modeling and computation. Therefore, the 5 force element is used instead of the coupler.Table 1
Subway train body parameters.
ComponentValueMass of subway train (t) (single cycle)The locomotive: 36 tThe trailer: 34.6 tSubway train body length (m) (velocity direction)19.5Width of subway train body (m)2.8Height of subway train body (m) (bottom plate from top of rail: 1.1 m)2.7 (the height of the underground train from the rail surface: 3.8 m)Vehicle bogie distance (m), 2Lc12.6Bogie wheelbase fixed distance (m), 2Lt2.2Moment of inertia of nodding mass of subway train (kg·m2), Iyy1328000Moment of inertia of mass side roll of subway train (kg·m2), Ixx32400Moment of inertia of shaking head of a mass of subway train (kg·m2), Izz1317000Table 2
Bogie frame parameters.
ComponentValueQuality of frame (t)A singleThe locomotive: 3.6 tThe trailer: 1.2 tLength of the frame (m) (velocity direction)2.2Width of the frame (m)2.05Moment of inertia of the nodding frame mass (kg·m2), Iyy1760Roll moment of inertia of frame mass (kg·m2), Ixx1430Moment of inertia of shaking head of frame mass (kg·m2), Izz2950Table 3
Parameters of the wheelset.
ComponentValueQuality of the wheelset (t)1.7The radius of the wheelset (m)0.42Moment of inertia of the nodding mass of the wheelset (kg·m2), Iyy104Moment of inertia of the side roll of the wheelset mass (kg·m2), Ixx801Moment of inertia of the wheelset mass (kg·m2), Izz814Table 4
Series I and II suspension parameters.
ComponentValueSeries I longitudinal stiffness (N/m)0.96 × 106Series I lateral stiffness (N/m)0.96 × 106Series I vertical stiffness (N/m)1.2 × 106Series I longitudinal damping (N·s/m)2.5 × 104Series I lateral damping (N·s/m)2.5 × 104Series I vertical damping (N·s/m)5 × 104Series II longitudinal stiffness (N/m)0.21 × 106Series II lateral stiffness (N/m)0.21 × 106Series II vertical stiffness (N/m)0.58 × 106Series II longitudinal damping (N·s/m)3.5 × 104Series II lateral damping (N·s/m)3.5 × 104Figure 2
Subway train marshaling.


### 2.1.2. Substructure

The substructure was the key to the model. To meet the stress and strain of rail and trackbed requirements, Abaqus was adopted to establish the finite element model of the substructure and conduct modal analysis for generating a flexible body [32].To simplify the model, a 30 m long Timoshenko beam was built in Abaqus without considering the orbital joints between each segment. One hundred one master nodes were established with one master node every 0.3 m. Besides, regardless of the gap between each trackbed, a plate unit with a width of 2.7 m and a length of 30 m was established, and a total of 102 main nodes was retained. The first 30 modes of rail and trackbed, as the principal modes, were taken for the sake of conservation. The boundary only restricted the two ends of the track with six degrees of freedom. The rail and trackbed’s physical parameters are shown in Table5.Table 5
Orbital physical parameters.
RailValueBulk density (N/m3)7.83 × 1011Mass per unit length (kg/m) (monorail)60.64Modulus of elasticity (Pa)2.10 × 1011Poisson’s ratio (μ)0.3Track bedValueBulk density (N/m3) (cross-sectional area1.08 m2)2300Mass per unit length (kg/m) (velocity direction)2484Modulus of elasticity (Pa)3.25 × 1010Poisson’s ratio (μ)0.25


### 2.1.3. The Upper and Lower Structures Are Coupled

The track and trackbed in the substructure were imported into the Simpack with the Abaqus-Simpack joint simulation interface, and the track coupler was simplified as a set of springs with three-direction stiffness, damping to coupling the track and trackbed. The bottom of the trackbed was the base cushion, which was simplified as a group of springs with three-direction stiffness, damping to be connected with the base rock of the rigid body. The spring parameters of rail fasteners and different track bed bases are as shown in Table6. The dynamic model was adopted to describe the dynamic wheel-rail interaction relationship. Hertz’s nonlinear contact theory generated the coupling relationship between the wheelset and the track (the distribution law of local stress and strain produced by contact between two objects under compression). Thus, the upper and lower structures were coupled as a whole to form the vehicle-rail coupling model. The connection between every two trains and the coupling mode with the substructure is shown in Figure 3, and the model is shown in Figure 4.Table 6
Parameters of the spring of the substructure.
The name of the springValueVertical dynamic stiffness of rail fasteners (kN/m)4.2 × 103Lateral dynamic stiffness of rail fasteners (kN/m)9.5 × 103Longitudinal dynamic stiffness of rail fasteners (kN/m)4.2 × 103Vertical damping of rail coupler (kN/m)50Rail coupler transverse damping (kN·s/m)50Longitudinal damping of rail fasteners (kN·s/m)50Base stiffness of integral track bed (kN/m/m2)6.4 × 105Base damping of integral track bed (kN·s/m/m2)3.7The base stiffness of rubber cushion floating plate trackbed (kN/m/m2)2 × 104Rubber cushion floating plate track bed base damping (kN·s/m/m2)1 × 103Base stiffness of steel spring floating plate trackbed (kN/m)6.9 × 103Steel spring floating plate track bed base damping (kN·s/m)75Figure 3
Vehicle-rail coupling diagram.Figure 4
Effect diagram of rail coupling.


### 2.1.4. The Orbital Irregularities Are Stimulated

The train’s vibration is influenced by many factors, such as the wear between the wheel and the track, the distance error of the track, and the deviation degree of the curvature of the track. Therefore, the irregularity of the track, including vertical irregularity, horizontal irregularity, direction irregularity, and track spacing irregularity, is an essential factor affecting the accuracy of the calculation results [33–35].To facilitate the modeling, this simulation uses the irregularity excitation spectrum by US Federal Railway Administration (FRA) in the 1980s to simulate the random vibration in train operation based on a large number of measured data. The wavelength range of the spectrum was 1.524–304.8 m, and the grade of the track was divided into six levels. The maximum velocity of Qingdao Metro Line 3 is 80 km/h, which can be satisfied by the American level-5 track irrationality excitation. Therefore, the American level-5 smoothness spectrum was applied as the random external excitation of the track. The irregularity excitation spectrum relevant parameter is shown in Table7.Table 7
Irregularity excitation spectrum relevant parameter.
ParameterParameter values of each orbitalSymbolUnit654321Avcm2/rad/m0.03390.20950.53760.68161.01811.2107Aacm2/rad/m0.03390.07620.30270.41281.2107303634Ωsrad/m0.43800.82091.13120.85200.93080.6046Ωcrad/m0.82450.82450.82450.82450.82450.8245Maximum running speedkm/h176144128964824


## 2.1.1. Superstructure

The vehicle model was simplified as a system composed of body, hinge, constraint, and force using Simpack [28–31], where parameters of each component of the vehicle model, as shown in Tables 1–4. Qingdao Metro Line 3 adopts the marshaling method of four bullet trains and two trailers [+TC-M-M + M-M-TC+]. As shown in Figure 2, Simpack comes with five forces that both act as coupling connectors and simplify modeling and computation. Therefore, the 5 force element is used instead of the coupler.Table 1
Subway train body parameters.
ComponentValueMass of subway train (t) (single cycle)The locomotive: 36 tThe trailer: 34.6 tSubway train body length (m) (velocity direction)19.5Width of subway train body (m)2.8Height of subway train body (m) (bottom plate from top of rail: 1.1 m)2.7 (the height of the underground train from the rail surface: 3.8 m)Vehicle bogie distance (m), 2Lc12.6Bogie wheelbase fixed distance (m), 2Lt2.2Moment of inertia of nodding mass of subway train (kg·m2), Iyy1328000Moment of inertia of mass side roll of subway train (kg·m2), Ixx32400Moment of inertia of shaking head of a mass of subway train (kg·m2), Izz1317000Table 2
Bogie frame parameters.
ComponentValueQuality of frame (t)A singleThe locomotive: 3.6 tThe trailer: 1.2 tLength of the frame (m) (velocity direction)2.2Width of the frame (m)2.05Moment of inertia of the nodding frame mass (kg·m2), Iyy1760Roll moment of inertia of frame mass (kg·m2), Ixx1430Moment of inertia of shaking head of frame mass (kg·m2), Izz2950Table 3
Parameters of the wheelset.
ComponentValueQuality of the wheelset (t)1.7The radius of the wheelset (m)0.42Moment of inertia of the nodding mass of the wheelset (kg·m2), Iyy104Moment of inertia of the side roll of the wheelset mass (kg·m2), Ixx801Moment of inertia of the wheelset mass (kg·m2), Izz814Table 4
Series I and II suspension parameters.
ComponentValueSeries I longitudinal stiffness (N/m)0.96 × 106Series I lateral stiffness (N/m)0.96 × 106Series I vertical stiffness (N/m)1.2 × 106Series I longitudinal damping (N·s/m)2.5 × 104Series I lateral damping (N·s/m)2.5 × 104Series I vertical damping (N·s/m)5 × 104Series II longitudinal stiffness (N/m)0.21 × 106Series II lateral stiffness (N/m)0.21 × 106Series II vertical stiffness (N/m)0.58 × 106Series II longitudinal damping (N·s/m)3.5 × 104Series II lateral damping (N·s/m)3.5 × 104Figure 2
Subway train marshaling.


## 2.1.2. Substructure

The substructure was the key to the model. To meet the stress and strain of rail and trackbed requirements, Abaqus was adopted to establish the finite element model of the substructure and conduct modal analysis for generating a flexible body [32].To simplify the model, a 30 m long Timoshenko beam was built in Abaqus without considering the orbital joints between each segment. One hundred one master nodes were established with one master node every 0.3 m. Besides, regardless of the gap between each trackbed, a plate unit with a width of 2.7 m and a length of 30 m was established, and a total of 102 main nodes was retained. The first 30 modes of rail and trackbed, as the principal modes, were taken for the sake of conservation. The boundary only restricted the two ends of the track with six degrees of freedom. The rail and trackbed’s physical parameters are shown in Table5.Table 5
Orbital physical parameters.
RailValueBulk density (N/m3)7.83 × 1011Mass per unit length (kg/m) (monorail)60.64Modulus of elasticity (Pa)2.10 × 1011Poisson’s ratio (μ)0.3Track bedValueBulk density (N/m3) (cross-sectional area1.08 m2)2300Mass per unit length (kg/m) (velocity direction)2484Modulus of elasticity (Pa)3.25 × 1010Poisson’s ratio (μ)0.25


## 2.1.3. The Upper and Lower Structures Are Coupled

The track and trackbed in the substructure were imported into the Simpack with the Abaqus-Simpack joint simulation interface, and the track coupler was simplified as a set of springs with three-direction stiffness, damping to coupling the track and trackbed. The bottom of the trackbed was the base cushion, which was simplified as a group of springs with three-direction stiffness, damping to be connected with the base rock of the rigid body. The spring parameters of rail fasteners and different track bed bases are as shown in Table6. The dynamic model was adopted to describe the dynamic wheel-rail interaction relationship. Hertz’s nonlinear contact theory generated the coupling relationship between the wheelset and the track (the distribution law of local stress and strain produced by contact between two objects under compression). Thus, the upper and lower structures were coupled as a whole to form the vehicle-rail coupling model. The connection between every two trains and the coupling mode with the substructure is shown in Figure 3, and the model is shown in Figure 4.Table 6
Parameters of the spring of the substructure.
The name of the springValueVertical dynamic stiffness of rail fasteners (kN/m)4.2 × 103Lateral dynamic stiffness of rail fasteners (kN/m)9.5 × 103Longitudinal dynamic stiffness of rail fasteners (kN/m)4.2 × 103Vertical damping of rail coupler (kN/m)50Rail coupler transverse damping (kN·s/m)50Longitudinal damping of rail fasteners (kN·s/m)50Base stiffness of integral track bed (kN/m/m2)6.4 × 105Base damping of integral track bed (kN·s/m/m2)3.7The base stiffness of rubber cushion floating plate trackbed (kN/m/m2)2 × 104Rubber cushion floating plate track bed base damping (kN·s/m/m2)1 × 103Base stiffness of steel spring floating plate trackbed (kN/m)6.9 × 103Steel spring floating plate track bed base damping (kN·s/m)75Figure 3
Vehicle-rail coupling diagram.Figure 4
Effect diagram of rail coupling.


## 2.1.4. The Orbital Irregularities Are Stimulated

The train’s vibration is influenced by many factors, such as the wear between the wheel and the track, the distance error of the track, and the deviation degree of the curvature of the track. Therefore, the irregularity of the track, including vertical irregularity, horizontal irregularity, direction irregularity, and track spacing irregularity, is an essential factor affecting the accuracy of the calculation results [33–35].To facilitate the modeling, this simulation uses the irregularity excitation spectrum by US Federal Railway Administration (FRA) in the 1980s to simulate the random vibration in train operation based on a large number of measured data. The wavelength range of the spectrum was 1.524–304.8 m, and the grade of the track was divided into six levels. The maximum velocity of Qingdao Metro Line 3 is 80 km/h, which can be satisfied by the American level-5 track irrationality excitation. Therefore, the American level-5 smoothness spectrum was applied as the random external excitation of the track. The irregularity excitation spectrum relevant parameter is shown in Table7.Table 7
Irregularity excitation spectrum relevant parameter.
ParameterParameter values of each orbitalSymbolUnit654321Avcm2/rad/m0.03390.20950.53760.68161.01811.2107Aacm2/rad/m0.03390.07620.30270.41281.2107303634Ωsrad/m0.43800.82091.13120.85200.93080.6046Ωcrad/m0.82450.82450.82450.82450.82450.8245Maximum running speedkm/h176144128964824


## 3. Results and Discussion

### 3.1. Monitoring Point Selection and Simulation Scheme

Since the main displacement deformation of the substructure during train operation was vertical deformation, the rail-wheel coupling simulation calculation was mainly used to study the time-history curve of vertical displacement excitation of the trackbed and the variation rule of the trackbed. The load of the train was transmitted to the substructure through two tracks, so each vertical section had two contact points at the left side and the right side. The numerical simulation results are shown in Figures5–10, and it is found that compared with other main nodes, the 20th pair of main nodes near the middle of the trackbed has the largest vertical displacement and time-history deformation. Therefore, the 20th pair of main nodes was taken as the object of study in simulation is shown in Figure 11, and the specific simulation scheme is shown in Table 8.Figure 5
Vertical displacement (a) time-history curve and (b) amplitude-frequency cure of left monitoring point of trackbed at different velocities.
(a)(b)Figure 6
Vertical displacement (a) time-history curve and (b) amplitude-frequency curve of right monitoring point of trackbed at different velocities.
(a)(b)Figure 7
Comparison of maximum vertical displacement of track under different train velocities.Figure 8
(a) Time-history and (b) amplitude-frequency curves of vertical displacement of left monitoring point of track bed under different axle loads.
(a)(b)Figure 9
(a) Time-history and (b) amplitude-frequency curves of vertical displacement at the right monitoring point of trackbed under different axle loads.
(a)(b)Figure 10
Comparison of maximum vertical displacement of trackbed under different axle loads.Figure 11
Position diagram of monitoring points.Table 8
Simulation scheme.
Types of variablesVariable valueVelocity (km/h) (full load)406080Axle load (t) (60 km/h)6.23 (no-load)10.28 (complement)11.79 (full load)Form of the trackbedIntegral trackbedRubber cushion floating plate trackbedSteel spring floating plate bed


## 3.1. Monitoring Point Selection and Simulation Scheme

Since the main displacement deformation of the substructure during train operation was vertical deformation, the rail-wheel coupling simulation calculation was mainly used to study the time-history curve of vertical displacement excitation of the trackbed and the variation rule of the trackbed. The load of the train was transmitted to the substructure through two tracks, so each vertical section had two contact points at the left side and the right side. The numerical simulation results are shown in Figures5–10, and it is found that compared with other main nodes, the 20th pair of main nodes near the middle of the trackbed has the largest vertical displacement and time-history deformation. Therefore, the 20th pair of main nodes was taken as the object of study in simulation is shown in Figure 11, and the specific simulation scheme is shown in Table 8.Figure 5
Vertical displacement (a) time-history curve and (b) amplitude-frequency cure of left monitoring point of trackbed at different velocities.
(a)(b)Figure 6
Vertical displacement (a) time-history curve and (b) amplitude-frequency curve of right monitoring point of trackbed at different velocities.
(a)(b)Figure 7
Comparison of maximum vertical displacement of track under different train velocities.Figure 8
(a) Time-history and (b) amplitude-frequency curves of vertical displacement of left monitoring point of track bed under different axle loads.
(a)(b)Figure 9
(a) Time-history and (b) amplitude-frequency curves of vertical displacement at the right monitoring point of trackbed under different axle loads.
(a)(b)Figure 10
Comparison of maximum vertical displacement of trackbed under different axle loads.Figure 11
Position diagram of monitoring points.Table 8
Simulation scheme.
Types of variablesVariable valueVelocity (km/h) (full load)406080Axle load (t) (60 km/h)6.23 (no-load)10.28 (complement)11.79 (full load)Form of the trackbedIntegral trackbedRubber cushion floating plate trackbedSteel spring floating plate bed


## 4. Results Analysis

Most sections of Qingdao metro are integral track beds, which are adopted in this simulation when studying the influence of velocity and axle load on vertical displacement excitation.


### 4.1. Influence of Velocity on Vertical Displacement Excitation Time History

The simulation scheme for the influence of the train velocity on the vertical displacement excitation time history of the trackbed was conducted with the velocities of 40 km/h, 60 km/h, and 80 km/h, respectively, under the full axle load. The solve results are shown in Figures5 and 6.As shown in Figures5(a) and 6(a), the vertical displacement generated by the two trailers at both ends is smaller than that generated by the four-bullet train in the middle. With the increase of velocity, the loading time of the train decreased gradually. As shown in Figure 7, under the condition of the same axle load, the maximum vertical displacement of the tunnel plate shows a nonlinear increasing trend with the increase of velocity. When the train velocity decreases from 80 km/h to 60 km/h and then from 60 km/h to 40 km/h, the vertical displacement attenuation rate of the left monitoring point is 3.7% and 2.3%, respectively, and the vertical displacement attenuation rate of the correct monitoring point is 5.6% and 2.5%, respectively. Under the influence of velocity, the maximum values of vertical displacement excitation on the left and right sides are different, and the greater the velocity, the more pronounced the difference. As the train is affected by the irregular excitation when it is running, the train vibration shows randomness, leading to the asymmetric displacement excitation distribution on the left and right sides of the lower trackbed, which further verifies the reliability of the model. However, the attenuation rate of the left monitoring point from the velocity of 80 km/h to 40 km/h is only 5.9%, and the attenuation rate of the correct monitoring point from the velocity of 80 km/h to 40 km/h is only 7.9%. The results show that the maximum value of vertical displacement excitation is not evenly distributed due to the change of velocity, whose influence, however, is not significant.Figures5(b) and 6(b) show amplitude-frequency curves at different speeds. From the diagram, when the train velocity was 40 km/h, 60 km/h, and 80 km/h, the vertical displacement of the trackbed is mainly concentrated within the vibration frequency range of 0 Hz–2 Hz, 0 Hz–3 Hz, and 0 Hz–4 Hz, respectively, and each of the main frequency segments of vertical displacement amplitudes presents four peaks, all belonging to the low-frequency band. It can be found that the velocity presents an obvious influence on the vertical displacement vibration frequency range of the monitoring points on both sides. With the increase of velocity, the vertical displacement vibration frequency range expands gradually.


### 4.2. Influence of Axle Load on Vertical Displacement Time-History Excitation

The axle load of the train varies in different periods under the influence of the loading condition. In the morning and evening rush hours, the train is full, and the corresponding axle load is 11.79 t. At noon, there are relatively few passengers, which can be defined as the scheduled status, and the corresponding axle load is 10.28 t. When the train is about to arrive at the last station, and the passengers have almost got off the train, the axle load of the train is all provided by the components of the train itself, namely, the no-load axle load of 6.23 t. Therefore, the simulation scheme was to solve the vertical displacement excitation time-history of the trackbed caused by train vibration under full-load and no-load axle load with the velocity at 60 km/h. The solution results are shown in Figures8 and 9.As shown in Figures8(a) and 9(a), the vertical displacement generated by the trailers at both ends is slightly smaller than that generated by the middle four bullet trains as a whole. With the increase of axle load, the maximum vertical displacement load shows a monotonically increasing trend. As shown in Figure 5, the maximum attenuation rates of vertical displacement of the left monitoring point of the trackbed from full load to complement load and from complement load to light load are 9.9% and 26.7%, respectively, and the maximum attenuation rates of vertical displacement of the right monitoring point of the trackbed are 10.1% and 28.4% respectively. The variation trends of the monitoring points on both sides are similar, indicating that the axial weight has little influence on the variation of the vertical displacement excitation on both sides of the trackbed at the same velocity. However, the axle load plays a crucial role in the variation of vertical displacement excitation. The attenuation rate of the left side from full-load axle load to light-load axle load is up to 34% and that of the right side from full-load axle load to light-load axle load is up to 35.6%. It can be seen that the vertical displacement excitation of the trackbed is sensitive to the axle load of the train.As shown in Figures8(b) and 9(b), the vertical vibration frequency of vertical displacement of the monitoring points on both sides is mainly between 0 Hz and 4 Hz, belonging to the low-frequency band with the increase of axle load; the vertical displacement of the main vibration frequency range has no obvious change. Within the frequency range of 0 Hz–4 Hz, there are four peak vertical displacement amplitudes.


### 4.3. Vertical Displacement and Vibration Reduction Effect of Track Bed Form

The tunnel section of Qingdao Metro Line 3 is mostly of integral type, and part of the section is made of steel spring float bed and rubber cushion float bed. To study the time-history variation rule of vertical displacement excitation of track bed with different forms, the axial weight was set as full load and the velocity was set as 80 km/h during the simulation. The 20th pair of primary nodes of trackbed was also selected as the research object, as shown in Figure11.Figures12(a) and 13(a) show the comparison diagram of the vertical displacement time history curve of different trackbed types, From the diagram, when the axle load and the velocity are the same, the time-history curve of the trackbed caused by the train vibration with different track bed forms changes significantly. As shown in Figures 13(a) and 14(a), for the trains at 80 km/h velocity, the rubber damping pad of floating plate track on displacement excitation time history is around 8.5 s. Therefore, the monolithic trackbed with steel spring floating plate track on the vertical displacement of the end of the schedule time is roughly the same. For the rubber vibration isolation cushion floating plate track, though the train has been out of monitoring the position, the displacement time history of the track does not immediately end. The attenuation of vertical displacement of the trackbed is slow because the material damping of the rubber cushion floating plate trackbed is larger than that of the other two types of trackbed, and the resistance to deformation rebound is larger.Figure 12
(a) Time-history and (b) frequency-history curves of vertical displacement of left monitoring point under different trackbed forms.
(a)(b)Figure 13
The vertical displacement (a) time-history and (b) amplitude-frequency curve of the right monitoring point under different trackbed forms.
(a)(b)Figure 14
Acceleration (a) time-history curve and (b) amplitude-frequency curve under different trackbed forms.
(a)(b)As shown in Figure15, the change of the maximum vertical displacement produces an order of magnitude under different vibration forms. For left monitoring, steel spring floating plate trackbed on the vertical displacement of the maximum value compared to the rubber cushion floating plate trackbed on the vertical displacement of the maximum value attenuation is 61.6%; rubber cushion floating plate trackbed on the vertical displacement of the maximum value compared to the integrated track on the vertical displacement of the maximum value attenuation is 93.6%. For the right monitoring, steel spring floating plate trackbed on the vertical displacement of the maximum value compared to rubber cushion floating plate trackbed on the vertical displacement of the maximum value attenuation is 61.7%; rubber floating plate track on the vertical displacement of the maximum value compared to integrated track on the vertical displacement of the maximum value attenuation is 93.5%. This phenomenon is because the base material of the integral trackbed was more rigid than that of the other two kinds of the trackbed and the resistance to deformation of the integral trackbed is stronger.Figure 15
Maximum vertical displacement under different trackbed types.Figures12(b) and 13(b) show the comparison diagram of the vertical displacement amplitude-frequency curve of different trackbed typs. From the diagram, the vertical displacement vibration frequencies of different track beds are mainly concentrated within 0 Hz–4 Hz, a low-frequency band, where the peak value of the vertical displacement amplitude of rubber damper trackbed and steel spring float trackbed appears three times, and the peak value of vertical displacement amplitude of integral track bed appear fours times.The right monitoring points were taken as the research object when analyzing the acceleration time-history curves of different trackbed forms. As shown in Figure14(a), the acceleration of the rubber damping cushion float bed and the steel spring float bed is attenuated compared with that of the integral track bed, indicating that the rubber cushion float bed and the steel spring float bed have a more significant effect on vibration reduction. As shown in Figure 14(b), the main frequency of acceleration of the integral trackbed is concentrated within the range of 100 Hz–200 Hz. The main frequency of acceleration of the rubber cushion floating trackbed and steel spring floating trackbed are roughly the same, both concentrating within the range of 0 Hz–100 Hz.


## 4.1. Influence of Velocity on Vertical Displacement Excitation Time History

The simulation scheme for the influence of the train velocity on the vertical displacement excitation time history of the trackbed was conducted with the velocities of 40 km/h, 60 km/h, and 80 km/h, respectively, under the full axle load. The solve results are shown in Figures5 and 6.As shown in Figures5(a) and 6(a), the vertical displacement generated by the two trailers at both ends is smaller than that generated by the four-bullet train in the middle. With the increase of velocity, the loading time of the train decreased gradually. As shown in Figure 7, under the condition of the same axle load, the maximum vertical displacement of the tunnel plate shows a nonlinear increasing trend with the increase of velocity. When the train velocity decreases from 80 km/h to 60 km/h and then from 60 km/h to 40 km/h, the vertical displacement attenuation rate of the left monitoring point is 3.7% and 2.3%, respectively, and the vertical displacement attenuation rate of the correct monitoring point is 5.6% and 2.5%, respectively. Under the influence of velocity, the maximum values of vertical displacement excitation on the left and right sides are different, and the greater the velocity, the more pronounced the difference. As the train is affected by the irregular excitation when it is running, the train vibration shows randomness, leading to the asymmetric displacement excitation distribution on the left and right sides of the lower trackbed, which further verifies the reliability of the model. However, the attenuation rate of the left monitoring point from the velocity of 80 km/h to 40 km/h is only 5.9%, and the attenuation rate of the correct monitoring point from the velocity of 80 km/h to 40 km/h is only 7.9%. The results show that the maximum value of vertical displacement excitation is not evenly distributed due to the change of velocity, whose influence, however, is not significant.Figures5(b) and 6(b) show amplitude-frequency curves at different speeds. From the diagram, when the train velocity was 40 km/h, 60 km/h, and 80 km/h, the vertical displacement of the trackbed is mainly concentrated within the vibration frequency range of 0 Hz–2 Hz, 0 Hz–3 Hz, and 0 Hz–4 Hz, respectively, and each of the main frequency segments of vertical displacement amplitudes presents four peaks, all belonging to the low-frequency band. It can be found that the velocity presents an obvious influence on the vertical displacement vibration frequency range of the monitoring points on both sides. With the increase of velocity, the vertical displacement vibration frequency range expands gradually.


## 4.2. Influence of Axle Load on Vertical Displacement Time-History Excitation

The axle load of the train varies in different periods under the influence of the loading condition. In the morning and evening rush hours, the train is full, and the corresponding axle load is 11.79 t. At noon, there are relatively few passengers, which can be defined as the scheduled status, and the corresponding axle load is 10.28 t. When the train is about to arrive at the last station, and the passengers have almost got off the train, the axle load of the train is all provided by the components of the train itself, namely, the no-load axle load of 6.23 t. Therefore, the simulation scheme was to solve the vertical displacement excitation time-history of the trackbed caused by train vibration under full-load and no-load axle load with the velocity at 60 km/h. The solution results are shown in Figures8 and 9.As shown in Figures8(a) and 9(a), the vertical displacement generated by the trailers at both ends is slightly smaller than that generated by the middle four bullet trains as a whole. With the increase of axle load, the maximum vertical displacement load shows a monotonically increasing trend. As shown in Figure 5, the maximum attenuation rates of vertical displacement of the left monitoring point of the trackbed from full load to complement load and from complement load to light load are 9.9% and 26.7%, respectively, and the maximum attenuation rates of vertical displacement of the right monitoring point of the trackbed are 10.1% and 28.4% respectively. The variation trends of the monitoring points on both sides are similar, indicating that the axial weight has little influence on the variation of the vertical displacement excitation on both sides of the trackbed at the same velocity. However, the axle load plays a crucial role in the variation of vertical displacement excitation. The attenuation rate of the left side from full-load axle load to light-load axle load is up to 34% and that of the right side from full-load axle load to light-load axle load is up to 35.6%. It can be seen that the vertical displacement excitation of the trackbed is sensitive to the axle load of the train.As shown in Figures8(b) and 9(b), the vertical vibration frequency of vertical displacement of the monitoring points on both sides is mainly between 0 Hz and 4 Hz, belonging to the low-frequency band with the increase of axle load; the vertical displacement of the main vibration frequency range has no obvious change. Within the frequency range of 0 Hz–4 Hz, there are four peak vertical displacement amplitudes.


## 4.3. Vertical Displacement and Vibration Reduction Effect of Track Bed Form

The tunnel section of Qingdao Metro Line 3 is mostly of integral type, and part of the section is made of steel spring float bed and rubber cushion float bed. To study the time-history variation rule of vertical displacement excitation of track bed with different forms, the axial weight was set as full load and the velocity was set as 80 km/h during the simulation. The 20th pair of primary nodes of trackbed was also selected as the research object, as shown in Figure11.Figures12(a) and 13(a) show the comparison diagram of the vertical displacement time history curve of different trackbed types, From the diagram, when the axle load and the velocity are the same, the time-history curve of the trackbed caused by the train vibration with different track bed forms changes significantly. As shown in Figures 13(a) and 14(a), for the trains at 80 km/h velocity, the rubber damping pad of floating plate track on displacement excitation time history is around 8.5 s. Therefore, the monolithic trackbed with steel spring floating plate track on the vertical displacement of the end of the schedule time is roughly the same. For the rubber vibration isolation cushion floating plate track, though the train has been out of monitoring the position, the displacement time history of the track does not immediately end. The attenuation of vertical displacement of the trackbed is slow because the material damping of the rubber cushion floating plate trackbed is larger than that of the other two types of trackbed, and the resistance to deformation rebound is larger.Figure 12
(a) Time-history and (b) frequency-history curves of vertical displacement of left monitoring point under different trackbed forms.
(a)(b)Figure 13
The vertical displacement (a) time-history and (b) amplitude-frequency curve of the right monitoring point under different trackbed forms.
(a)(b)Figure 14
Acceleration (a) time-history curve and (b) amplitude-frequency curve under different trackbed forms.
(a)(b)As shown in Figure15, the change of the maximum vertical displacement produces an order of magnitude under different vibration forms. For left monitoring, steel spring floating plate trackbed on the vertical displacement of the maximum value compared to the rubber cushion floating plate trackbed on the vertical displacement of the maximum value attenuation is 61.6%; rubber cushion floating plate trackbed on the vertical displacement of the maximum value compared to the integrated track on the vertical displacement of the maximum value attenuation is 93.6%. For the right monitoring, steel spring floating plate trackbed on the vertical displacement of the maximum value compared to rubber cushion floating plate trackbed on the vertical displacement of the maximum value attenuation is 61.7%; rubber floating plate track on the vertical displacement of the maximum value compared to integrated track on the vertical displacement of the maximum value attenuation is 93.5%. This phenomenon is because the base material of the integral trackbed was more rigid than that of the other two kinds of the trackbed and the resistance to deformation of the integral trackbed is stronger.Figure 15
Maximum vertical displacement under different trackbed types.Figures12(b) and 13(b) show the comparison diagram of the vertical displacement amplitude-frequency curve of different trackbed typs. From the diagram, the vertical displacement vibration frequencies of different track beds are mainly concentrated within 0 Hz–4 Hz, a low-frequency band, where the peak value of the vertical displacement amplitude of rubber damper trackbed and steel spring float trackbed appears three times, and the peak value of vertical displacement amplitude of integral track bed appear fours times.The right monitoring points were taken as the research object when analyzing the acceleration time-history curves of different trackbed forms. As shown in Figure14(a), the acceleration of the rubber damping cushion float bed and the steel spring float bed is attenuated compared with that of the integral track bed, indicating that the rubber cushion float bed and the steel spring float bed have a more significant effect on vibration reduction. As shown in Figure 14(b), the main frequency of acceleration of the integral trackbed is concentrated within the range of 100 Hz–200 Hz. The main frequency of acceleration of the rubber cushion floating trackbed and steel spring floating trackbed are roughly the same, both concentrating within the range of 0 Hz–100 Hz.


## 5. Field Measurement Data Analysis

To preliminary verify the correctness of the model, the actual data of the Qingdao metro were used to verify the model. LMS SCADAS Mobile SCM01 information acquisition system, PCB 393A03 acceleration sensor, Lenovo T460P as the terminal, and the schematic diagram of the test equipment connection are shown in Figure16(a). According to the requirements of Technical Specification for Floating Plate Track (CJJ/T191-2012), a vertical acceleration sensor is arranged on the tunnel wall 1.25 m away from the rail surface, as shown in Figures 16(c) and (d).Figure 16
(a) LMS signal acquisition analyzer. (b) Vertical acceleration sensor. (c) Schematic diagram of measuring points of floating plate trackbed section. (d) Schematic diagram of measuring points of the integral trackbed section.The curves of vertical velocity amplitude and acceleration amplitude with horizontal distance for different roadbed forms are shown in Figures17(a) and 17(b). The maximum vertical vibration response appears directly above the tunnel from the diagram. Moreover, there is a large area between 40 and 60 m. In other distances, the vertical vibration response generally decreases with the increase of the horizontal distance. Rubber cushion floating plate trackbed and steel spring floating plate bed structure on the vertical vibration response has pronounced attenuation effect. The two largest vertical vibration velocity amplitude attenuation rates are 92% and 83%, respectively; the largest vertical vibration acceleration amplitude attenuation rates are 93% and 84%, respectively; both numerical simulations to calculate the maximum vertical vibration velocity amplitude attenuation rate are 90% and 79%. The top amplitude attenuation rates of vertical vibration acceleration are 91% and 80%, respectively. The measured results are in good agreement with the numerical simulation results, and the damping effect of the rubber cushion floating plate trackbed on vertical vibration is slightly more significant than that of the steel spring floating plate bed.Figure 17
Vertical velocity amplitude (a) and vertical acceleration amplitude (b) varying with horizontal distance.
(a)(b)As shown in Figure18, with the increase of horizontal distance, the Z vibration level generally indicates a trend of gradual attenuation. Still, it is in the small amplification area between 40 and 60 m and around 90 m. The ambient vibration value caused by the integral track bed structure exceeds the standard daytime ambient vibration value by 30 m. The range of influence on the expected value of ambient vibration at night is about 60 m. The environmental vibration caused by the rubber cushion floating plate trackbed and steel spring floating plate bed structure and the integral track bed structure is below the standard value.Figure 18
The variation of Z vibration level with horizontal distance.Figures19(a) and 20(a) show the comparison diagram of damping capacity of rubber cushion floating plate trackbed and steel spring floating plate bed with horizontal distance. From the diagram, the maximum vibration reduction of the rubber cushion floating plate trackbed is 24.3 dB, and the ultimate vibration reduction occurs at a horizontal distance of 50 m. The minimum vibration reduction is 22.7 dB. In addition, the vibration reduction within 10 m and 40–50 m distance is significant, and the vibration reduction near 30 m is slight. The maximum vibration reduction of the steel spring floating plate bed is 19.5 dB, and the ultimate vibration reduction occurs near the horizontal distance of 60 m. The minimum vibration reduction is 15.7 dB. Additionally, the vibration reduction is more significant at the space of 50–65 m and near 100 m and smaller at the distance of 30 m and 80 m. Rubber cushion floating plate trackbed can eliminate the load generated by train operation in a short distance. As shown in Figures 19(b) and 20(b), the maximum vibration reduction rate of the rubber cushion floating plate trackbed is 37%, and the minimum vibration reduction rate is 31.5%. The maximum vibration reduction rate of the steel spring floating plate bed is 29%, and the minimum vibration reduction rate is 23%. The damping capacity and damping rate of the rubber cushion floating plate trackbed at the tunnel center are 23.76 dB and 31.56%, respectively. The damping capacity and damping rate of the steel spring floating plate bed at the center of the tunnel are 17.5 dB and 23.3%, respectively. Therefore, the measured results show that the damping effect of the rubber cushion floating plate trackbed is better than that of the steel spring floating plate bed. The numerical simulation results are identical to the measured results.Figure 19
The variation law of damping capacity (a) and damping rate (b) of rubber cushion floating plate trackbed with horizontal distance.
(a)(b)Figure 20
The variation law of damping capacity (a) and damping rate (b) of steel spring floating plate bed with horizontal distance.
(a)(b)


## 6. Conclusion

(1)
Under the influence of train marshaling, the displacement time history of trackbed presents the distribution law of large in the middle and small at both ends.(2)
The displacement excitation size of the ballast bed is most obviously affected by the damping form of trackbed. The maximum attenuation rate of vertical displacement from steel spring floating plate ballast bed to integral ballast bed is 97.5%. The second is the influence of axle load. When the axle load of the train changes from full load to light load, the maximum attenuation rate of vertical displacement of the trackbed is 35.6%.And finally, the influence of velocity is the smallest influence. When the train speed drops from 80 km/h to 40 km/h, the maximum attenuation rate of vertical displacement of the trackbed is only 7.9%.(3)
Although the integral trackbed had better deformation control ability, its antivibration performance is poor, and the rubber cushion ballast bed has the best antivibration performance, followed by the steel spring floating plate trackbed. The rubber floating cushion trackbed is the ideal ballast bed based on its deformation resistance and vibration reduction effect.



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*Source: 4814039-2022-02-04.xml*