Optimization of vibration reduction of the rubber floatingslab tracks
Hao Jin^{1} , Shunhua Zhou^{2} , Weining Liu^{3}
^{1, 2}Key Laboratory of Road and Traffic Engineering of Ministry of Education, Tongji University, Shanghai, China
^{3}School of Civil Engineering, Beijing Jiaotong University, Beijing, China
^{1}Corresponding author
Journal of Vibroengineering, Vol. 19, Issue 2, 2017, p. 12141224.
https://doi.org/10.21595/jve.2016.17234
Received 2 June 2016; received in revised form 17 September 2016; accepted 19 September 2016; published 31 March 2017
JVE Conferences
The floatingslab track is one of the best methods to reduce the vibration from the underground railway. In this paper, the rubber floatingslab track was studied. The traintrack model coupled with the tracktunnelsurrounding soil model was built, which was verified by the results of the insite test. Through this model, the quality of the rubber floatingslab track was optimized by the multiisland genetic algorithm. The optimized Z vibration level was reduced by 4.92 dB with the density of the floating slab 2306 kg/m^{3}, the Young’s modulus of the floating slab 3.21×10^{10} N/m^{2}, the density of the rubber bearing 730 kg/m^{3} and the Young’s modulus of the rubber bearing 9.55×10^{8} N/m^{2}.
Keywords: floatingslab track, vibration reduction, optimization, numerical model.
1. Introduction
Vibration generated by underground railways is one of the most serious engineering problems. The vibration of the subway railway will result in structural vibrations and reradiated noise [1, 2]. There are many measures to control this vibration. The floatingslab track is one of them.
The floating slab can be continuous or discontinuous [3], which mainly can be supported by rubber bearings or steelsprings. In the Singapore Mass Rapid Transit (SMRT) system, the discontinuous floatingslab track supported by the resilient pads was used [4]. In the San Francisco Bay Area Rapid Transit (BART) system, the discontinuous floatingslab track also was used, which was supported by rubber pads [5]. In Xi’an subway line 2, China, the continuous rubber floatingslab track was used. Furthermore, the steelspring floatingslab track was used in Bejing subway line 5, China [6].
Generally, the methods studying the floatingslab track are to build the twodimensional model or the threedimensional model. Using the twodimensional model, both the discontinuous floating slab and the continuous floating slab can be analyzed. Cui and Chew [4] analyzed the effectiveness of the discontinuous floatingslab track to stationary/moving harmonic loads by the receptance method. Hussein and Hunt [79] studied the discontinuous/continuous floatingslab track by the FourierRepeatingunit method, the PeriodicFourier method and the Modifiedphase method. Similarly, Gupta and Degrande [10] build the model of the discontinuous/continuous floatingslab track, which was coupled with the tunnel soil system by a periodic approach. Furthermore, some others did this job as well [1114].
However, the twodimensional model only can analyze the full supporting accurately. If the floating slab rests on the steelsprings, i.e. the point supporting, the twodimensional model cannot be employed to analyze the characteristic of the floatingslab track. The results will not be reasonable. At this time, the threedimensional model must be built. Usually, it is accomplished by the numerical model, e.g. the finite element method. Lei and Jiang [15] build the finite element model of the steelspring floatingslab track to analyze the effectiveness of vibration reduction. Geng et al. [16] constructed a 3D finite element model to analyze the normal models of the rubber floatingslab track. Eigenfrequencies of the floating slab with two densities were analyzed under different stiffness of the rubber bearing and the fastener.
Besides the theoretical analysis, some researchers performed the insite test or the laboratory tests. Saurenman and Phillips [5] tested the rubber floatingslab track used on the current San Francisco Bay Area Rapid Transit (BART) system. Hwang et al. [17] performed the laboratory mockup test to understand the dynamical behaviors of the rubber floatingslab track more accurately. Furthermore, some ones studied the material of the rubber bearings. Wu et al. [18] researched the physical deterioration of the rubber bearings affected by air oxidation, ozone ageing and the random fatigue loads. Montella et al. [19] introduced an innovative floatingslab track supported by the recycled rubber bearings.
In this paper, our focus is the rubber floatingslab track. The present contribution aims to improve the ability of vibration reduction of the rubber floatingslab track through optimizing the properties of the floating slab and the rubber bearings. This paper is organized as follows. Section 2 introduces the mathematical models including the traintrack submodel and the tracktunnelsurrounding soil submodel. To verify the model, the insite test is performed in Section 3. Section 4 gives the optimization objective and parameters of the rubber floatingslab track. Then, the numerical model coupled with the multiisland genetic method is employed to optimize the ability of vibration reduction of the rubber floatingslab track in Section 5. Conclusions are in Section 6.
2. Numerical models
2.1. Traintrack submodel
The numerical model was divided into two submodels. The first submodel was the traintrack submodel, from which the contact force between the train wheels and the rail was obtained. The second submodel was the tracktunnelsurrounding soil submodel, which was a finite element model, built in the software Abaqus.
Fig. 1. Projection of the real train in the 2D traintrack submodel
Fig.1 illustrated the relationship between the real train and the mathematical model of the train in the 2D traintrack submodel. The main components of the train are the car body, the bogie and the wheel set. The primary suspension is used to connect the bogie and the wheel set. Similarly, the secondary suspension is employed to connect the car body and the bogie. In this paper, the traintrack submodel is a 2D model. Thus, one car body has two freedom degrees. One bogie has two freedom degrees. Onewheel set has only one freedom degree. Fig. 2 shows the details of the freedom degrees for different components. ${y}_{c1}$ is the vertical displacement of the first car body, ${\mathrm{\phi}}_{\mathrm{c}1}$ is the rotation of the first car body, ${\mathrm{y}}_{\mathrm{b}\mathrm{i}}$ is the vertical displacement of the $i$th bogie, ${\phi}_{bi}$ is the rotation of the $i$th bogie, ${y}_{wj}$ is the vertical displacement of the $j$th wheel set.
Then, we can build the dynamic equations of the train model in matrix as follows:
where $\mathbf{y}$ is the vertical displacement of the train in the $y$ direction as shown in Fig. 2:
$\dot{\mathbf{y}}$ is the vertical velocity of the train:
$\ddot{\mathbf{y}}$ is the vertical acceleration of the train:
$\mathbf{Q}$ is the force loading on the train:
${{m}_{w2}g,{m}_{b1}g,0,{m}_{b2}g,0,{m}_{c1}g,0]}^{T}.$
$\mathbf{M}$ is the mass matrix of the train:
$\mathbf{C}$ is the damping matrix of the train:
$\mathbf{K}$ is the stiffness matrix of the train:
${m}_{c1}$ is the mass of the first car body, ${m}_{bi}$ is the mass of the $i$th bogie, ${m}_{wj}$ is the mass of the $j$th wheel set, ${R}_{wjrail}$ is the interaction force between the $j$th wheel set and the rail, ${k}_{1s}$ is the vertical stiffness of the primary suspension, ${c}_{1s}$ is the vertical damping of the primary suspension, ${k}_{2s}$ is the vertical stiffness of the secondary suspension, ${c}_{2s}$ is the vertical damping of the secondary suspension, ${l}_{b}$ is the distance between two wheel sets, ${\mathrm{l}}_{\mathrm{c}}$ is the distance between two bogies, ${J}_{bi}$ is the moment of inertia of the $i$th bogie, ${J}_{c1}$ is the moment of inertia of the first car body.
In this paper, the rail was simplified to be a simply supported beam [20]. The governing equation of the rail is
where, $y(x,t)$ is the vertical displacement of the rail at the position $x$ when the time is $t$, $E$ is the Young’s modulus of the rail, $I$ is the area moment of inertia of the rail, $\rho $ is the density of the rail per meter, $F(x,t)$ is the force on the rail, i.e. the interaction force between the wheel sets and the rail in this case $\sum _{i=1}^{n}{R}_{wirail}(x,t)$, $n$ is the number of the wheel sets, ${\mathrm{k}}_{\mathrm{f}}$ is the vertical stiffness of the fastener, ${c}_{f}$ is the vertical damping of the fastener.
Fig. 2. Freedom degrees of the train in 2D traintrack submodel
The force of Hertz contact, which connects the train model and the rail model, is:
$=\left\{\begin{array}{l}{\left\{\frac{1}{G}\left[{y}_{wi}\left(x,t\right)y\left(x,t\right){y}_{irr}\left(x,t\right)\right]\right\}}^{\frac{3}{2}},{y}_{wi}(x,t)y(x,t){y}_{irr}(x,t)0,\\ 0,{y}_{wi}(x,t)y(x,t){y}_{irr}(x,t)0.\end{array}\right.$
where, $i$ is the $i$th wheel set, $G$ is the interaction constant, ${y}_{wi}(x,t)$ is the vertical displacement of the $i$th wheel set at the position $x$, $y(x,t)$ is the vertical displacement of the rail at the position $x$, ${y}_{irr}(x,t)$ is the irregularity of the rail at the position $x$.
2.2. Tracktunnelsurrounding soil submodel
The finite element model for the tracktunnelsurround soil submodel was built in the finite element software Abaqus to analyze and optimize the ability of vibration reduction of the rubber floatingslab track loaded by the wheelrail force from the traintrack submodel.
In order to save the calculation time in the optimization process, only the track, the rubber bearing, the tunnel and the surrounding soil were built in Abaqus. Furthermore, the infinite elements were adopted to avoid the wave reflection at the boundaries. The longitudinal size of the model is 100 m. The wheelrail force was loaded at the rail position in the tracktunnelsurrounding soil submodel as shown in Fig. 3.
3. Verification of the calculation results
3.1. Insite test
In order to verify that the results calculated by the above numerical model are reasonable, Xi’an subway line 2, in Shanxi province, China, was tested where the rubber floatingslab track was used. The rubber floatingslab track was constructed in the shielded tunnel. The train was Btype train according to the category of Chinese subway train. The speed of the train was about 40 km/h when it passed through the testing section.
Fig. 3. Tracktunnelsurrounding soil submodel built in Abaqus
Fig. 4. The insite test in Xi’an subway line 2
The vertical acceleration of the tunnel wall, shown in Fig. 5, needs to be measured according to the Chinese code, Technical code for floating slab track [21]. As the requirement of the code, the distance between the rail top and the accelerometer is about 1.25 m. In this case, a 5 g accelerometer was used. The frequency range of the accelerometer is from 0.1 Hz to 2000 Hz.
The data acquisition was fixed firmly to the pipe on the tunnel wall in order not to affect the operation safety, shown in Fig. 6.
Fig. 5. Accelerometer’s position
Fig. 6. The data acquisition system was fixed firmly
3.2. Result comparison between the numerical model and the insite test
The main parameters of the Btype train are in Table 1. The density of the rail is 60 kg/m, the Young’s modulus 2.1×10^{11} Pa, the section area 7.6×10^{3} m^{2}, the area moment of inertia 3.04×10^{5} m^{4}. The stiffness of the fastener is 7.8×10^{7} N/m, the damping of the fastener is 5×10^{4} N∙s/m. The spacing distance between the fasteners is 0.6 m.
The Table 2 shows the density, the Young’s modulus and the Possion’s ratio of the track, the tunnel and the surrounding soil.
Table 1. Parameters of the Btype train
Components

Parameters

Mass of the car body

35000 (kg)

Mass of the bogie

4600 (kg)

Mass of the wheel set

1420 (kg)

Moment of inertia of the car body

1700000 (kg∙m^{2})

Moment of inertia of the bogie

9620 (kg∙m^{2})

Vertical stiffness of the primary suspension

2.45 (MN/m)

Vertical damping of the primary suspension

0.24 (MN∙s/m)

Vertical stiffness of the secondary suspension

2.08 (MN/m)

Vertical damping of the secondary suspension

0.24 (MN∙s/m)

Distance between two bogies

12.6 (m)

Distance between two Wheel sets

2.2 (m)

Table 2. Properties of the track, the tunnel and the surrounding soil
Components

Density (kg/m^{3})

Young’s modulus (N/m^{2})

Possion’s ratio

Floating slab

2500

3.1×10^{10}

0.167

Rubber bearing

1000

8×10^{8}

0.47

Invert

2200

3×10^{10}

0.2

Shielded tunnel

2350

3.6×10^{10}

0.2

Surrounding soil

1990

3.1×10^{8}

0.33

The vertical acceleration of the tunnel wall in the frequency domain, ranging from 1 Hz to 200 Hz, was calculated by the numerical model, which was compared with the tested result as shown in Fig. 7. Obviously, the tested result was more complicated than the calculated result. Because of the simplification of the train, the track, the tunnel and the surrounding soil, the numerical model lost some components. Thus, the values at some frequencies were very small. Although the numerical model was not accurate, the tendency of the values calculated by the numerical model was close to the insite test. Therefore, the numerical model was adopted to analyze the following contents.
Fig. 7. Frequency spectrum of the vertical acceleration of the tunnel wall from the numerical model and the insite test
4. Optimization objective and parameters
According to the code, Technical code for floating slab track, the $Z$ vibration level of the tunnel wall was adopted to assess the ability of vibration reduction of the floatingslab track [21]. The definition of the $Z$ vibration level $V{L}_{z}$ as follow:
where, $VL\left(i\right)$ is the vibration level at the 1/3 octave frequency for the number $i$, $w\left(i\right)$ is the weighted factor for the number $i$. Table 3 shows the details for the different numbers [22].
Table 3. Weighted factor in 1/3 octave frequency
Number

Frequency (Hz)

Weighted factor (dB)

Number

Frequency (Hz)

Weighted factor (dB)

1

1

–6.33

13

16

–2.28

2

1.25

–6.29

14

20

–3.93

3

1.6

–6.12

15

25

–5.8

4

2

–5.49

16

31.5

–7.86

5

2.5

–4.01

17

40

–10.05

6

3.15

–1.9

18

50

–12.19

7

4

–0.29

19

63

–14.61

8

5

0.33

20

80

–17.56

9

6.3

0.46

21

100

–21.04

10

8

0.31

22

125

–25.35

11

10

–0.1

23

160

–30.91

12

12.5

–0.89

24

200

–36.38

In order to improve the ability of vibration reduction of the rubber floatingslab track, four parameters, i.e. the density of the floating slab ${\rho}_{slab}$, the Young’s modulus of the floating slab ${E}_{slab}$, the density of the rubber bearing ${\rho}_{rubber}$ and the Young’s modulus of the rubber bearing ${E}_{rubber}$, were analyzed. According to the engineering suggestions, the ranges of these four parameters were 2200 kg/m^{3}$\le {\rho}_{slab}\le $ 2400 kg/m^{3}, 2.2×10^{10}·N/m^{2}$\le {E}_{slab}\le $ 3.8×10^{10} N/m^{2}, 500 kg/m^{3}$\le {\rho}_{rubber}\le $ 1500 kg/m^{3}, 5×10^{8} N/m^{2}$\le {E}_{rubber}\le $ 10×10^{8} N/m^{2}.
5. Optimization by the numerical model coupled with the multiisland genetic algorithm
5.1. Optimization procedure
Three softwares were employed to accomplish this procedure. Firstly, the multiisland genetic algorithm was used in the software Isight. Through Isight, the software Abaqus was activated. The vertical vibration acceleration of the tunnel wall in the time domain was calculated in the Abaqus. Then, the software Matlab was employed to analyze the data from the Abaqus, i.e. the $Z$ vibration level of the tunnel wall, i.e. the optimization objective, was calculated. According to the calculated $Z$ vibration level, Isight adjusted the values of four parameters, the density of the floating slab, the Young’s modulus of the floating slab, the density of the rubber bearing and the Young’s modulus of the rubber bearing in the inp document, which was the input document of the Abaqus. Fig. 8 shows the workflow of the optimization procedure.
Fig. 8. Workflow of the program
The parameters of the multiisland genetic algorithm were subpopulation size 10, number of island 3, number of generation 10, rate of crossover 0.8, rate of mutation 0.03, rate of migration 0.25, interval of migration 5.
5.2. Optimization results
The program described above was operated for the minimum $Z$ vibration level. Fig. 9 shows the value of the density of the floating slab in the optimization process. The value of the density of the floating slab is 2306 kg/m^{3}^{}for the minimum $Z$ vibration level.
Fig. 10 shows the value of the Young’s modulus of the floating slab in the optimization process. The value of the Young’s modulus of the floating slab is 3.21×10^{10} N/m^{2} for the minimum Z vibration level finally.
Fig. 9. Density of the floating slab with calculation numbers
Fig. 10. Young’s modulus of the floating slab with calculation numbers
Fig. 11 shows the value of the density of the rubber bearing with the calculation numbers. The value of the density of the rubber bearing is 730 kg/m^{3} for the minimum $Z$ vibration level.
Fig. 12 shows the value of the Young’s modulus of the rubber bearing with the calculation numbers. The value of the Young’s modulus of the rubber bearing is 9.55×10^{8} N/m^{2} for the minimum $Z$ vibration level.
Fig. 13 shows the optimization process of the $Z$ vibration level. Obviously, the $Z$ vibration level decreases with the optimization numbers, especially at the start of the optimization. Eventually, the $Z$ vibration level stops at 57.46 dB, which is the minimum value in the total optimization process.
Fig. 11. Density of the rubber bearing with calculation numbers
Fig. 12. Young’s modulus of the rubber bearing with calculation numbers
Fig. 13. $Z$ vibration level of the tunnel wall with calculation numbers
The Z vibration levels were compared between the present rubber floatingslab track and the optimized rubber floatingslab track. The parameters of the present rubber floating slab track were the density of the floating slab 2500 kg/m^{3}, the Young’s modulus of the floating slab 3.1×10^{10} N/m^{2}, the density of the rubber bearing 1000 kg/m^{3} and the Young’s modulus of the rubber bearing 8×10^{8} N/m^{2}. The $Z$ vibration of the present rubber floatingslab track is 62.38 dB. Therefore, the $Z$ vibration level was improved by 4.92 dB.
6. Conclusions
In order to improve the ability of vibration reduction of the rubber floatingslab track, the numerical model coupled with the multiisland genetic algorithm was built. The numerical model consists of the traintrack submodel and the tracktunnelsurrounding soil submodel, which was verified by the result of the insite test in Xi’an subway line 2, Shanxi province, China. Through the numerical model coupled with the multiisland genetic algorithm, the $Z$ vibration level was improved by 4.92 dB with the density of the floating slab 2306 kg/m^{3}, the Young’s modulus of the floating slab 3.21×10^{10} N/m^{2}, the density of the rubber bearing 730 kg/m^{3} and the Young’s modulus of the rubber bearing 9.55×10^{8} N/m^{2} compared with the present rubber floatingslab track.
Acknowledgements
This project was supported by National Natural Science Foundation of China 51478353.
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