Abstract
The objective of this paper is to study the reasonable stiffness of rubber mat layer of vibration reduction slab track and configuration of rubber mat layer of transition section slab tack between vibration reduction section and normal section in highspeed tunnel. Based on achievements of the related studies, a highspeed train, slab track and tunnel finite element coupling dynamic model was established, and corresponding program was developed with MATLAB and verified by in situ measured data. The dynamic responses of slab track under moving highspeed train with different vibration reduction configurations and transition arrangements in Shiziyang tunnel of GuangzhouHong Kong highspeed railway line in China were analyzed. The study shows that: the rubber mat layer under the slab of slab track can greatly reduce the tunnel vibration, but the slab bending moment and the rail vertical displacement will increase, so the stiffness of rubber mat layer under the slab of slab track in tunnel of highspeed railway line should not be too low; the stiffness of rubber mat layer of slab track in tunnel of GuangzhouHong Kong line is controlled by the rail vertical displacement, and the stiffness value of 0.04 N/mm^{3} is reasonable; the vibration and dynamic stress of slab track can be improved greatly by setting transition section between vibration reduction slab track and ordinary slab track, and the design of transition section slab track in tunnel of GuangzhouHong Kong line is reasonable.
1. Introduction
Slab track is a modern railway technology. By using slab track, stability of track can be enhanced, comfort and safety operation of highspeed train can be ensured, and the change rate of track geometry and maintenance work can also be greatly reduced because of the cancellation of the ballast which can cause residual track deformation easily. With the development of slab track technology, slab track has been widely used in highspeed railway in the world, such as German, Japanese and China et al. [1].
Though slab track has many advantages, its stiffness is greater than ballast track, so its influence on environmental vibration is more serious than that of ballast track does in normal conditions. Without vibration reduction measures, it cannot be used directly in some sections of highspeed railway which have special requirements for environmental vibration, for example, the Shiziyang tunnel under the Shiziyang River on GuangzhouHong Kong highspeed railway line. The maximum water depth of Shiziyang River is 26.6 m, the maximum water width of Shiziyang River is 3300 m. The foundation of Shiziyang tunnel under the Shiziyang River consists mainly of sand and soft soil, which is very sensitive to environmental vibration.
At present, lots of research work has been done on the dynamic characteristics of slab track of highspeed railway. However, most of these researches are aimed at ordinary slab track structure in normal line [29]. Few studies have been made on the dynamic characteristics of vibration reduction slab track structure in vibration reduction section of highspeed railway line, as well as the dynamic characteristics of slab track structure of the transition section between the vibration reduction and normal section of highspeed railway line. Furthermore, former studies mainly concentrated on the vibration characteristic of the coupling system, and few studies considered the dynamic stress of the slab track, which is a key factor for the fatigue failure of slab track. Thirdly, only middlelong random track irregularities, whose wavelength is longer than 1m, is taken into account in the vibration reduction simulation studies, while shortwave random track irregularities, whose wave length is less than 1 m and which has significant influence on the dynamic response of track structure, is not taken into account.
In this paper, based on achievements of the related studies, a highspeed train, slab track and tunnel finite element dynamic model, which takes both short wavelength random track irregularities and middlelong wavelength random track irregularities into account, was established. The corresponding program was developed with MATLAB selfdeveloped program and verified by field test results. With the established dynamic model, taking highspeed trains passing vibration reduction section and adjacent transition section of slab track in Shiziyang tunnel of GuangzhouHong Kong line as an example, considering both vibration characteristic of system and dynamic stress of the slab track, the reasonable stiffness value of rubber mat layer under the slab in vibration reduction section of Shiziyang tunnel of GuangzhouHong Kong line is studied. By comparing dynamic characteristic of highspeed trainslab tracktunnel system with or without transition section between vibration reduction slab track section and ordinary slab track section, the reasonability of transition section design is verified.
2. Dynamic model
A schematic describing the dynamic model for a highspeed train traveling on a slab track in a tunnel at a constant speed $V$ along the longitudinal direction is shown in Fig. 1.
The model consists of four submodels, namely, the highspeed train, the slab tracktunnel, wheelrail interaction, and track irregularity submodels. The highspeed train submodel is established based on multirigidbody dynamics theory, and the slab tracktunnel submodel is established based on finite element method. The highspeed train submodel and slab tracktunnel submodel are coupled through wheelrail interaction submodel considering track irregularities. The four submodels are explained as follows.
Fig. 1Schematic of the coupling dynamic model
2.1. Highspeed train submodel
As shown in Fig. 1, the highspeed train submodel consists of a series of identical fourwheel highspeed vehicles. Each vehicle of the train is modeled as a multirigidbody system including a car body, two bogie frames, four wheelsets and the connection components. The connections between the bogie frame and the wheelset are characterized by springdampers, named the primary suspension system. The connection between the car body and the bogie frame are characterized by springdampers, named the secondary suspension system. The springs are all with linear property, and the dampers are all with viscous property. The car body and each of the two bogie frames have vertical and pitch degrees of freedom (DOF), each of the four wheels has vertical and pitch DOF. Each vehicle of the highspeed train submodel has 10 DOF.
2.2. Slab tracktunnel submodel
The 3D slab tracktunnelsoil submodel using volume elements to simulate the slab, tunnel, and surrounding soil can reflect the dynamic characteristics of the slab tracktunnelsoil subsystem. However, the simulation is extremely timeconsuming because long model length, small mesh size and small simulation time step are needed in the simulation of a moving highspeed train along the rail with random irregularity of track in this study.
In order to reduce the calculation time, a 2D slab tracktunnel submodel is used. The submodel is shown in Fig. 2. The rail, slab, and tunnel are simulated by BernoulliEuler beam elements, while fasteners which connect rail and slab, rubber mat layer which connects slab and tunnel, as well as the Winkler foundation of the tunnel are simulated by linear springdamper elements.
Fig. 22D slab tracktunnel dynamic submodel
A sophisticated 3D slab tracktunnelsoil model (Fig. 3) is used to determine the equivalent stiffness of the Winkler foundation of the tunnel. In the 3D slab tracktunnelsoil model, the rail is simulated by BernoulliEuler beam elements; the slab, rubber mat layer, tunnel, and soil are simulated by volume elements; the fasteners are simulated by linear springdamper elements.
Fig. 3Sophisticated 3D slab tracktunnelsoil model
a) Crosssection graph
b) Enlarged graph
Static train loads are respectively applied on the 2D slab tracktunnel submodel shown in Fig. 2 and 3D slab tracktunnelsoil model shown in Fig. 3 to determine the equivalent Winkler foundation stiffness of the tunnel of 2D slab tracktunnel submodel. If the relative error between the maximum vertical displacement of tunnel in the 2D slab tracktunnel submodel and that of the 3D slab tracktunnelsoil model is less than 10^{6}, the equivalent Winkler foundation stiffness has been found. Otherwise, the equivalent stiffness value should be modified until the above requirement can be met.
By different settings of the stiffness of rubber mat layer between slab and tunnel, the slab track of the vibration reduction section at the normal section and the transition section can all be modeled easily in the 2D slab tracktunnel submodel shown in Fig. 2.
2.3. Wheelrail interaction model
The wheelrail interaction model is the same as the model given in reference [10]. The vertical force between wheel and rail is determined by the Hertz nonlinear contact theory considering the track irregularity as follows:
where ${P}_{j}\left(t\right)$ is the wheelrail contact force under the $j$th wheel at time $t$, $G$ is the wheelrail contact constant, ${Z}_{w}(j,t)$ is the displacement of the $j$th wheel at time $t$, ${Z}_{r}(j,t)$ is the rail displacement under the $j$th wheel at time $t$, and ${Z}_{0}\left(t\right)$ is the track irregularity under the $j$th wheel at time $t$.
2.4. Track irregularity model
Because of the lack of track spectrum of highspeed railway in China, low interference spectrum of the German highspeed railway, which is widely used as middlelong wavelength random track irregularity spectrum for highspeed railway all around the world, is used to generate the samples of middlelong wavelength random track irregularity.
The expression of low interference spectrum of the German highspeed railway is as follows:
where $S\left(\mathrm{\Omega}\right)$ is power spectral density, $\mathrm{\Omega}$ is spatial frequency, ${A}_{v}$ is the roughness constant and its value is 4.032×10^{7} m rad, ${\mathrm{\Omega}}_{c}$ and ${\mathrm{\Omega}}_{r}$ are cutoff frequency, the value of ${\mathrm{\Omega}}_{c}$ is 0.8246 rad/m, the value of ${\mathrm{\Omega}}_{r}$ is 0.0206 rad/m.
The low interference spectrum of the German highspeed railway is formed according to the measured track irregularities (wavelengths are more than 1 m) which are automatically measured by the track geometry recording car running on the track. However, the short wavelength random track irregularities are not well considered in the German's low interference spectrum. According to author and other researchers work, short wave random track irregularities (wavelengths between 0.01 m and 1 m) have great influence on dynamic characteristics of slab track
To consider the influence of short wave random track irregularities on dynamic characteristics of slab track, not only the middlelong wavelength random track irregularity but also the short wavelength random track irregularity were taken into account.
Sato spectrum [11], which is widely used in the fields such as highfrequency wheelrail contact forces [12], wheelrail rolling noise [1314], and wheelrail wear [15], is used as track irregularity spectrum to generate short wavelength random track irregularities. The expression is as follows:
where $S\left(\mathrm{\Omega}\right)$ is power spectral density, $\mathrm{\Omega}$ is spatial frequency, $A$ is the roughness constant and its value is 4.15×10^{8} m.rad5.0×10^{7} m.rad.
According to measured rolling noise data of QinhuangdaoShenyang line, the value of $A$ is suggested to be 3.15×10^{7} m.rad in reference [16].
3. Vibration equations of the coupling dynamic system
Using the principle of total potential energy with the stationary value in elastic system dynamics presented by Zeng [17], one can derive the vibration equation of the highspeed train, slab track and tunnel coupling dynamic model. The equation can be written in matrix form as follows:
where, ${\ddot{X}}_{v}$, ${\dot{X}}_{v}$, and ${X}_{v}$ denote the acceleration, velocity, and displacement vectors for the DOF of the highspeed train subsystem, respectively; ${\ddot{X}}_{t}$, ${\dot{X}}_{t}$, and ${X}_{t}$ denote the acceleration, velocity, and displacement vectors for the DOF of the slab tracktunnel subsystem, respectively; ${M}_{vv}$, ${C}_{vv}$, and ${K}_{vv}$ denote the mass, damping, and stiffness matrices of the highspeed train subsystem, respectively; ${M}_{tt}$, ${C}_{tt}$, and ${K}_{tt}$ denote the mass, damping, and stiffness matrices of the slab tracktunnel subsystem, respectively; ${F}_{vg}$ denotes the gravity subload vector of the highspeed train subsystem; ${F}_{vi}$ and ${F}_{ti}$ denote the subload vector of the wheelrail interaction forces on highspeed train subsystem and slab tracktunnel subsystem, respectively.
Detailed expressions of the stiffness matrix, mass matrix, damping matrix, and load vector of a single vehicle of the highspeed train submodel can be founded in reference [10]. Likewise, detailed procedures for obtaining the stiffness matrix, mass matrix, damping matrix, and load vector of the slab tracktunnel subsystem can be found in reference [18]. Solution procedures for the vibration equation of the coupled system are as follows.
1) At time $t=$ 0, the static displacement of the coupled system under the combined effect of gravity load of vehicles and track irregularity is calculated and used as the initial value for the dynamic coupled system.
2) When time $t>$ 0, using the wheelrail interaction model considering the track irregularity Eq. (1) and an iteration procedure as described in reference [10], the displacement, velocity, and acceleration of each DOF of the coupled system, as well as the wheelrail interaction forces are calculated. Then, the force of each springdamping element and the bending moment of each beam element can also be calculated. It should be mentioned that the convergence of the wheelrail interaction force must be ensured during each time step.
Fig. 4Arrangement of insitu acceleration sensors
4. Verification of coupling dynamic model
In situ dynamic response measurement was conducted in the vibration reduction slab track section on tunnel of GuangzhouHong Kong line. Acceleration sensors were attached to the end side of the slab as shown in Fig. 4. The measurement is used to obtain the accelerations of rail, slab and tunnel when highspeed train runs through the track at speeds from 110 km/h to 300 km/h. The field measured timehistory of acceleration of rail, slab, and tunnel are shown in Fig. 5Fig. 7 when highspeed train runs through the track at the speed of 300 km/h.
Fig. 5Field measured timehistory of acceleration of rail
Fig. 6Field measured timehistory of acceleration of slab
Fig. 7Field measured timehistory of acceleration of tunnel
Corresponding numerical simulation was conducted for highspeed train passing vibration reduction section of slab track in Shiziyang tunnel of GuangzhouHong Kong highspeed railway line at speed of 300 km/h. The parameters for the highspeed train and those of the slab tracktunnel submodel are listed in Tables 1 and 2, respectively.
The random track irregularity used in this study is plotted in Fig. 8(a). It is the combination of the sample of middlelong wavelength random track irregularity in Fig. 8(b) and short wavelength random track irregularity in Fig. 8(c).
Timehistory curve of maximum acceleration of rail, slab and tunnel calculated by selfdeveloped MATLAB program are shown in Fig. 9Fig. 11.
Table 1Parameters for the highspeed train submodel
Parameter  Unit  Value 
Mass of car body  kg  42400 
Mass of bogie  kg  3400 
Mass of wheelset  kg  2200 
Pitch inertia of car body  kg.m^{2}  1370000 
Pitch inertia of bogie  kg.m^{2}  3600 
Vertical damping of primary suspension  N.s.m^{1}  35000 
Vertical stiffness of primary suspension  N.m^{1}  1040000 
Vertical damping of secondary suspension  N.s.m^{1}  35000 
Vertical stiffness of secondary suspension  N.m^{1}  400000 
Wheelbase  m  2.5 
Distance between center of front bogie and center of rear bogie  m  17.375 
Table 2Parameters for the slab tracktunnel submodel
Parameter  Unit  Value 
Section area of rail  cm^{2}  77.45 
Inertia moment of rail  cm^{4}  3217 
Density of rail  kg.m^{3}  7800 
Elastic modulus of rail  GPa  210 
Spacing of fastener  m  0.625 
Vertical stiffness of fastener  kN·mm^{1}  30 
Damping of fastener  kN.s.m^{1}  20 
Width of slab  m  2.4 
Thickness of slab  m  0.19 
Elastic modulus of slab  GPa  36 
Density of slab  Kg.m^{3}  2500 
Stiffness of rubber mat layer  MPa/m  40 
Section area of tunnel  m^{2}  13 
Inertia moment of tunnel crosssection  m^{4}  250 
Density of tunnel  Kg.m^{3}  2500 
Dynamic modulus of tunnel foundation  MPa  60 
Equivalent Winkler foundation stiffness of tunnel  MPa  12 
Fig. 8a) Samples of the combined random track irregularity, b) middlelong wavelength random track irregularity, c) short wavelength random track irregularity
a)
b)
c)
Comparing Fig. 5Fig. 7 and Fig. 9Fig. 11, the maximum acceleration of rail, slab, tunnel calculated by selfcompiled MATLAB program are 138.245 g, 11.569 g and 48.5 mg, respectively. And the maximum acceleration of rail, slab, and tunnel measured in situ are 155.077 g, 10.133 g and 35 mg, respectively. The results calculated by the selfcompiled MATLAB program are agreed with measured data. Considering that the track irregularities used in this study cannot fully agree with the track irregularities of the field environment and that noise signal has influence on the field test result, the discrepancies between the calculated results and measured data are reasonable, which proves that the dynamics response of trainslab tracktunnels coupling system is correct.
Fig. 9Timehistory of maximum acceleration of rail calculated by MATLAB
Fig. 10Timehistory of maximum acceleration of slab calculated by MATLAB
Fig. 11Timehistory of maximum vibration acceleration of tunnel calculated by MATLAB
5. Case studies
With the verified highspeed trainslab tracktunnel finite element coupling dynamic model, taking highspeed train passing vibration reduction slab track with different rubber mat stiffness and ordinary slab track in Shiziyang tunnel at speed of 300 km/h as cases, the dynamic characteristic of highspeed trainslab tracktunnel system for 7 cases was theoretical studied and compared. The stiffness value of rubber mat layer for vibration reduction slab track and the stiffness value of cement asphalt (CA) mortar for ordinary slab track for cases 17 are listed in Table 3.
With the verified coupling dynamic model, taking highspeed train passing slab track with different configurations of mat layer at speed of 300 km/h as cases, the dynamic characteristic of highspeed trainslab tracktunnel system with or without transition section between vibration reduction slab track and ordinary slab track are studied and compared. The configurations of length and stiffness of mat layer for cases 89 are shown in Table 4.
It should be mentioned that except the stiffness value of rubber mat layer for vibration reduction slab track and the stiffness value of CA mortar for ordinary slab track, which are listed in Table 34 for different cases, the other calculation parameters are the same as those in Section 4.
The dynamic response of the car body, bogie, wheelset, rail, fastening, slab, mat layer, tunnel of the coupling system for Cases 19 are listed in Table 5.
Table 3The stiffness of rubber mat layer or CA mortar for case 17
Case  Type of slab track  Stiffness of rubber mat layer for vibration reduction slab track or CA mortar for ordinary slab track (N/mm^{3}) 
1  Vibration reduction slab track  0.01 
2  Vibration reduction slab track  0.02 
3  Vibration reduction slab track  0.04 
4  Vibration reduction slab track  0.06 
5  Vibration reduction slab track  0.08 
6  Vibration reduction slab track  0.1 
7  Ordinary slab track  4 
Table 4Configurations of mat layer for cases 89
Case  Type of slab track  Layout of length of rubber mat layer or CA mortar (m)  Layout of stiffness of rubber mat layer or CA mortar (N/mm^{3}) 
8  With transition section  505050505050505050  40.10.080.060.040.060.080.14 
9  Without transition section  150150150  40.044 
From the simulation of the case study, the following points can be obtained:
1) The vibration acceleration of tunnel can be reduced greatly with rubber mat layer used under the slab. The lower the stiffness of rubber mat layer is, the smaller the vibration acceleration of tunnel will be.
2) The rail vertical displacement and the slab bending moment increases significantly with rubber mat layer used under the slab. The lower the stiffness of rubber mat layer is, the larger the rail vertical displacement and the slab bending moment will be.
3) According to the relevant codes and regulations, the vertical displacement of rail of highspeed railway above 300 km/h should not be more than 2 mm. Considering the limited value of rail vertical displacement, the stiffness of rubber mat layer of vibration reduction section of Shiziyang tunnel of GuangzhouShenzhenHong Kong line should not be smaller than 0.04 N/mm^{3}.
4) Comparing with the ordinary slab track, the acceleration of tunnel decreased 16 dB by adopting vibration reduction slab track with 0.04 N/mm^{3} stiffness of rubber mat layer. The vibration reduction effect of rubber mat layer is excellent.
5) The bending moment of slab increases substantially with rubber mat layer used under the slab, so the strength of slab should be carefully checked to avoid fatigue failure of slab.
6) Without transition section between vibration reduction slab track and ordinary slab track, the pressure force of rubber mat layer, bending moment of slab, pressure and tension force of fastener, and acceleration of slab and tunnel increase substantially. With transition section between vibration reduction slab track and ordinary slab track, dynamic characteristic of slab track and tunnel are excellent, design of transition section between vibration reduction slab track and ordinary slab track in tunnel of GuangzhouHong Kong line is reasonable.
Table 5Calculation results for cases 19
Case  1  2  3  4  5  6  7  8  9 
Maximum vertical acceleration of wheelset (m/s^{2})  58.37  61.09  57.64  57.01  56.91  57.05  57.22  0.48  1.07 
Maximum vertical acceleration of bogie (m/s^{2})  4.46  5.29  5.31  5.98  6.34  6.59  6.35  0.09  0.22 
Maximum vertical acceleration of car body (m/s^{2})  0.43  0.41  0.42  0.42  0.42  0.42  0.42  0.00  0.13 
Maximum force of wheelset (kN)  145.09  147.51  145.19  143.18  144.45  143.93  143.25  83.94  90.76 
Maximum displacement of rail (mm)  4.37  2.97  1.94  1.77  1.65  1.57  1.24  1.76  1.85 
Maximum acceleration of rail (m/s^{2})  1354.02  1354.03  1356.18  1349.25  1346.97  1346.06  1354.55  29.84  35.59 
Maximum acceleration of slab (m/s^{2})  113.88  116.89  113.49  106.76  103.82  103.58  166.75  17.47  23.04 
Maximum acceleration of tunnel (m/s^{2})  0.21  0.28  0.48  0.71  0.81  0.86  3.11  0.10  0.15 
Vibration level of tunnel (dB)  106.30  108.97  113.56  117.00  118.12  118.70  129.85  100.11  103.68 
Effects on vibration suppression (dB)  23.55  20.88  16.29  12.85  11.72  11.14  0.00  –  – 
Maximum positive moment of rail (kN.m)  27.90  25.52  23.35  23.49  23.51  23.69  23.46  18.86  20.26 
Maximum negative moment of rail (kN.m)  14.17  10.66  11.06  11.56  11.62  11.53  12.10  8.70  10.09 
Maximum pressure force of fastener (kN)  47.86  47.89  47.49  48.66  49.09  48.71  49.53  31.70  37.67 
Maximum tension force of fastener (kN)  11.84  9.28  7.28  7.32  7.40  7.42  7.88  2.70  3.40 
Maximum positive moment of slab (kN.m/m)  16.69  11.89  8.98  8.82  8.21  7.52  3.11  7.42  8.53 
Maximum negative moment of slab (kN.m/m)  19.69  14.43  9.35  7.28  6.08  5.17  1.47  7.21  7.33 
Maximum compressive stress of rubber mat layer (MPa)  0.04  0.05  0.05  0.05  0.05  0.05  0.11  0.05  0.06 
Maximum compressive displacement of rubber mat layer (mm)  3.90  2.35  1.20  0.83  0.64  0.52  0.03  0.99  1.02 
6. Conclusions
The highspeed trainslab tracktunnel finite element coupling dynamic model was established, and corresponding program was developed with MATLAB program language and verified by in situ measured data. Taking highspeed train passing slab track of vibration reduction section and adjacent transition section on tunnel of GuangzhouHong Kong line as an example, dynamic characteristics of highspeed train, slab track and tunnel system was theoretical studied and compared. The main conclusions include:
1) The lower the stiffness of rubber mat layer is, the larger the vertical displacement of rail and the bending moment of slab will be. The stiffness of rubber mat layer of slab track in tunnel of GuangzhouHong Kong line is controlled by the rail vertical displacement, and the 0.04 N/mm^{3} stiffness value for rubber mat layer is reasonable. By using rubber mat layer under the slab, vibration of tunnel can decrease 16 dB, and the effect of vibration reduction of rubber mat layer is excellent.
2) With transition section between vibration reduction slab track and ordinary slab track, vibration and dynamic stress of slab track are excellent. The design of transition section between vibration reduction slab track and ordinary slab track in tunnel of GuangzhouHong Kong line is reasonable.
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About this article
The works described in this paper are supported by National Natural Science Foundation of China (No. 51178469), the National Science Joint Highspeed Railway Foundation of China (No. U1334203) and China Postdoctoral Science Foundation (2014M552158), as well as the State Scholarship Fund of China Scholarship Council (No. 201208430112).