Environmental vibration and noise pollution caused by the operation of subway is a hot topic of concern at home and abroad. There is a need to research dynamic interaction between trains and track in order to solve environmental vibration and noise pollution caused by the operation of train. As an excitation source of the train vibration, rail surface roughness has significant impact on the wheel-rail interaction. In order to study the effect of rail roughness on the wheel-rail vibration load, the wheel-rail interaction model is established based on excitation of rail surface roughness in this paper. The effect of rail roughness on train vibration load is calculated by this model according to the measured rail roughness data in Beijing subway in the MATLAB program. Wheel-rail force in the most adverse situations is also calculated. The property of wheel-rail force under different conditions such as track forms, driving speed and curve radius is also analyzed in this paper

Linlin Du

Weining Liu

English

Vibroengineering PROCEDIA

  © JVE INTERNATIONAL LTD. VIBROENGINEERING PROCEDIA. SEP 2015, VOLUME 5. ISSN 2345-0533 475 
Study on wheel-rail interaction based on rail roughness 
Linlin Du1, Weining Liu2 
Beijing Jiaotong University, Beijing, China 
1Corresponding author 
E-mail: 114115285@bjtu.edu.cn, 2wnliu@bjtu.edu.cn 
(Accepted 28 July 2015) 
Abstract. Environmental vibration and noise pollution caused by the operation of subway is a hot 
topic of concern at home and abroad. There is a need to research dynamic interaction between 
trains and track in order to solve environmental vibration and noise pollution caused by the 
operation of train. As an excitation source of the train vibration, rail surface roughness has 
significant impact on the wheel-rail interaction. In order to study the effect of rail roughness on 
the wheel-rail vibration load, the wheel-rail interaction model is established based on excitation 
of rail surface roughness in this paper. The effect of rail roughness on train vibration load is 
calculated by this model according to the measured rail roughness data in Beijing subway in the 
MATLAB program. Wheel-rail force in the most adverse situations is also calculated. The 
property of wheel-rail force under different conditions such as track forms, driving speed and 
curve radius is also analyzed in this paper. 
Keywords: rail roughness, rail corrugation, wheel-rail force, property of dynamic load. 
1. Introduction 
Rail transit is the fundamental way to solve traffic congestion in public and is favored by big 
cities all over world. Environmental vibration and noise pollution caused by rail transit draw 
people attention at the same time [1]. For example, the normal life of residents along railways is 
affected greatly by train induced environmental vibration in Beijing. Therefore, various forms of 
damping track are adopted to mitigate environmental vibration. However, the abnormal rail 
corrugation appears in the damping track section also affect the normal operation of Beijing 
subway. Abnormal corrugation intensified wheel-rail interaction and reduced the life of rail 
greatly. The wheel-rail whistle noise induced by corrugation is also very serious [2]. 
Urban Rail Transit Network Planning and construction must face and solve environmental 
vibration and noise caused by the operation of subway trains, and the root of the problem lies in 
interaction between vehicle and rails [3]. As an excitation source of the train vibration, rail surface 
roughness has significantly impact on the wheel-rail interaction. In order to study the effect of rail 
roughness on wheel-rail vibration load, the wheel-rail interaction model is established based on 
excitation of rail surface roughness in this paper. The effect of rail roughness on train vibration 
load is calculated. The property of wheel-rail force under different conditions such as track forms, 
driving speed and curve radius is also analyzed in this paper. 
2. Rail surface roughness and measurement 
Rail surface roughness refers to the small valley irregularities on rail surface. Rail roughness 
can reflect the rail corrugation level. The smaller of rail roughness value, the smoother of rail 
surface. It is defined as follows: 
ܴ௔ =
1
݈ න |ݕ|݀ݔ,
௟
଴
 (1)
where ܴ௔  represents value of rail roughness, ݕ represents amplitude of roughness in each test 
point, ݈ represents length of test section. 
Rail surface roughness measurement method is divided into indirect measurement method and 
direct measurement method. Calculation data used in this paper is continuous data measured by 
STUDY ON WHEEL-RAIL INTERACTION BASED ON RAIL ROUGHNESS.  
LINLIN DU, WEINING LIU 
476 © JVE INTERNATIONAL LTD. VIBROENGINEERING PROCEDIA. SEP 2015, VOLUME 5. ISSN 2345-0533  
the direct measurement method. Continuous data is measured by digital measurement and analysis 
car, as shown in Fig. 1. Sample data measured by continuous measurement are relative value with 
respect to the start point and it is a stochastic process associated with longitudinal direction of rail 
line. The stochastic process is superposition by random irregularity wave with different 
wavelengths, phases and amplitudes. Usually it can be represented by the statistical parameters 
according to stochastic theory. 
 
Fig. 1. Continuous measurement of rail roughness 
3. Wheel-rail interaction model  
Based on the number of fixed load model, vehicle balance equation is established and solved 
in the frequency domain. Input the amplitude of measured roughness data in the model, finally, 
the wheel-rail interaction load property is obtained. 
3.1. Calculation model and parameters 
Vertical vehicle vibration is the main consideration when researching environmental influence 
of vibration induced by subway train. Side roll and lateral vibration of the train is ignored in the 
model. The vehicle can be simplified to the combination of a series of secondary suspension 
system. One simplified model of the secondary suspension is shown in Fig. 2. Track is treated as 
rigid in this model [4]. Assuming that vehicle wheel is contact with rail closely and the rail surface 
roughness amplitude equals to the actual displacement of wheel. The wheel-rail coupled model is 
excited by rail roughness measured data in Beijing subway. 
 
Fig. 2. Simplified model of vehicle for its vertical vibration 
Vehicle parameters in the model can be selected in Table 1, as is shown in follows:  
݉ଵ = 1700 N⋅s2/m, ݉ଶ = 1800 N⋅s2/m, ݉ଷ = 10750 N⋅s2/m, ݇ଵ = 4.9×106 N/m,  
݇ଶ = 2.08×106 N/m, ܿଵ = 0.48×106 N⋅s/m, ܿଶ = 0.48×106 N⋅s/m. 
c2
c1k1
z2
P(t)
z1
m1
m2
z0
z3m3
k2
STUDY ON WHEEL-RAIL INTERACTION BASED ON RAIL ROUGHNESS.  
LINLIN DU, WEINING LIU 
 © JVE INTERNATIONAL LTD. VIBROENGINEERING PROCEDIA. SEP 2015, VOLUME 5. ISSN 2345-0533 477 
Table 1. Vehicle parameters in Beijing subway 
Vehicle mass ܯ (t) 43.0 Spring constant of bogie ்݇ (kN/m) 2080 
Inertia moment of vehicle mass ܬ (tm2） 1700 Bogie damping ܥ் (kN⋅S/m) 240 
Bogie mass ܯ (t) 3.60 Spring constant of wheelsets (kN/m) 2450 
Inertia moment of bogie mass ܬ (tm2) 9.62 Wheelsets damping ܥ் (kN⋅S/m) 240 
Length of vehicle ܮ (m) 19.52 distance between bogie centers ݈ (m) 12.66 
Axis spacing ܽ (m) 2.30 Wheelsets mass ݉௪ (t) 1.70 
3.2. Calculation of wheel-rail dynamic load 
According to Fig. 2, vehicle balance equation in vertical direction is established: 
൜݉ଷݖሷଷ + ܿଶ(ݖሶଷ − ݖሶଶ) + ݇ଶ(ݖଷ − ݖଶ) = 0,݉ଶݖሷଶ + ܿଵ(ݖሶଶ − ݖሶଵ) + ݇ଵ(ݖଶ − ݖଵ) − ݇ଶ(ݖଷ − ݖଶ) − ܿଶ(ݖሶଷ − ݖሶଶ) = 0, (2)
where ݉ represents mass, ݇ represents rigidity, ܿ represents damping, ݖ represents displacement. 
Subscripts denote parts can be seen in Fig. 2. Acceleration of wheel, bogie and vehicle is ݖሷଵ, ݖሷଶ, 
ݖሷଷ, respectively. 
After Fourier transforming of Eq. (2), balance equation of vehicle motion in the frequency 
domain is obtained as follows [5]: 
൜݉ଷ(−߱
ଶ̂ݖଷ) + ܿଶ(݅߱̂ݖଷ − ݅߱̂ݖଶ) + ݇ଶ(̂ݖଷ − ̂ݖଶ) = 0,
݉ଶ(−߱ଶ̂ݖଶ) + ܿଵ(݅߱̂ݖଶ − ݅߱̂ݖଵ) + ݇ଵ(̂ݖଶ − ̂ݖଵ) − ݇ଶ(̂ݖଷ − ̂ݖଶ) − ܿଶ(݅߱̂ݖଷ − ݅߱̂ݖଶ) = 0, (3)
where ̂ݖଵ,  ̂ݖଶ,  ̂ݖଷ  represents displacement of wheel, bogie, vehicle in frequency domain, 
respectively. 
Displacement of wheel ̂ݖଵ in the frequency domain can be obtained by Fourier transform on 
the measured rail roughness data. Specific operating method is explained clearly in the following 
section. Interacting load between wheel and rail can be calculated by the following formula: 
ܲ(ݐ) = (݉ଵ + ݉ଶ + ݉ଷ)݃ + ݉ଵݖሷଵ + ݉ଶݖሷଶ + ݉ଷݖሷଷ. (4)
3.3. Input of rail surface roughness 
The wheel-rail interaction model is excited by measured data of rail surface roughness in 
Beijing. Fig. 3 shows rail roughness varies with position, and the horizontal axis represents the 
test point location, the vertical axis represents the value of rail roughness. 
The stochastic process is superposition by random irregularity wave with different 
wavelengths, phases and amplitudes [6]. Since displacement of the wheel is equal to the magnitude 
of rail roughness, it can be written as: 
ݖ1 = ෍ ܣ݆݁−݅ߣ݆ݔ
ܰ
݆=1
, (5)
where ܣ௝  represents amplitude of rail roughness in given frequency, ߣ௝  represents spatial 
frequency, ݔ represents location of test point, it can also be expressed as ݔ = ݒݐ, ݒ represents 
speed of vehicle. 
The test data in Fig. 3 is transformed by Fourier to obtain the relationship between roughness 
amplitude and wave number in spatial frequency domain, as shown in Fig. 4. To calculate the 
worst case of wheel-rail interaction force caused by rail roughness, take the horizontal and vertical 
coordinates at the peak point as spatial frequency and roughness amplitude in the harmonic 
vibration which is used to simulate the motion of vehicle with maximum load. 
STUDY ON WHEEL-RAIL INTERACTION BASED ON RAIL ROUGHNESS.  
LINLIN DU, WEINING LIU 
478 © JVE INTERNATIONAL LTD. VIBROENGINEERING PROCEDIA. SEP 2015, VOLUME 5. ISSN 2345-0533  
 
Fig. 3. Measured rail roughness 
The Eq. (5) can be simplified as: 
ݖ1 = ܣ݇݁−݅ߣ݇ݒݐ, (6)
where ߣ௞, ܣ௞ is the horizontal and vertical coordinates at the peak point at test point in the spatial 
frequency domain, respectively. 
Therefore, the wheel acceleration is: 
ݖ1 = ܣ݇(ߣ݇ݒ)2݁−݅ߣ݇ݒݐ. (7)
 
Fig. 4. Rail roughness in spatial frequency 
4. Calculation and analysis of wheel-rail force 
Based on the wheel-rail interaction model described above, the influence of speed, curve radius 
and type of track on the vehicle vibration is calculated and analyzed in this section. 
4.1. The influence of vehicle speed 
The measured roughness data in straight line with common track type with vehicle velocity 
60 km/h, 80 km/h and 95 km/h are selected to analysis the influence of train speed. Roughness 
data with different speed is shown in Fig. 5. Relationship between amplitude of wheel-rail force 
and vehicle speed is shown in Fig. 6. 
As it can be seen from Fig. 6 that wheel-rail force has a positive correlation with train speed, 
the faster the vehicle speed, the greater the wheel-rail forces is. 
0 20 40 60 80 100
-1000
-500
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500
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m)
distance(m)
1 10 100 1000
0.1
1
10
100
1000
rai
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es
s (
μm
)
spatial frequency
STUDY ON WHEEL-RAIL INTERACTION BASED ON RAIL ROUGHNESS.  
LINLIN DU, WEINING LIU 
 © JVE INTERNATIONAL LTD. VIBROENGINEERING PROCEDIA. SEP 2015, VOLUME 5. ISSN 2345-0533 479 
 
Fig. 5. Rail roughness with different speed 
 
 
Fig. 6. Relationship between amplitude of  
vibration load and speed 
4.2. The influence of curve radius 
Inuence of curve radius on the train vibration load is very complex. Wheel-rail force is varying 
with the running speed of train because it is different under different curve radius. Since the 
superelevation in curve segment, rail roughness and wheel-rail forces in inner and outer rail of the 
track will be different. Ignoring wheel-rail force difference between inner and outer rail caused by 
superelevation setting, the inner rail roughness data in curve segment is adopted for calculating. 
The selected roughness data is shown in Fig. 7. As it can be seen from Fig. 7 that the smaller the 
radius, the larger roughness of rail. 
Fig. 7. Rail roughness with different curve radius 
 
Fig. 8. Relationship between amplitude of  
vibration load and curve radius 
Relationship between amplitude of vibration load and curve radius is shown in Fig. 8. As it 
can be seen from Fig. 8 that maximum wheel-rail force occurred in segment with radius of 1000 m. 
Speed of train is different in different curve radius and the greater the radius of curve, the greater 
of train speed. Wheel-rail interacting force in curve segment is affect by velocity and rail 
roughness. Since the rail roughness in the section with radius of 450 m is bigger, the amplitude of 
wheel-rail force is larger than the section with radius of 800 m. 
4.3. The influence of track type 
Three types of tracks are analyzed in this section, which is normal track with DTVI2 fasteners, 
track with longitudinal sleeper and track with elastic short sleeper. Rail roughness data for 
calculation of three track types is shown in Fig. 9, as it is shown that difference of rail roughness 
0.1 1 10 100 1000
1E-3
0.01
0.1
1
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rai
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 60km/h
 80km/h
 95km/h
55 60 65 70 75 80 85 90 95 100
300
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 R=450m
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STUDY ON WHEEL-RAIL INTERACTION BASED ON RAIL ROUGHNESS.  
LINLIN DU, WEINING LIU 
480 © JVE INTERNATIONAL LTD. VIBROENGINEERING PROCEDIA. SEP 2015, VOLUME 5. ISSN 2345-0533  
amplitude of track is very small. Rail roughness amplitude in normal track with DTVI2 fasteners 
is the smallest and the track with elastic short sleeper is the biggest. 
Calculating results shows that wheel-rail force in track with longitudinal sleeper is about 
1000 kN, which is the largest, wheel-rail force in normal track is about 730 kN, which is the 
smallest. 
 
Fig. 9. Rail roughness with different track types 
5. Conclusions 
Wheel-rail dynamic interaction model is established based on excitation of rail surface 
roughness and the interacting force under different conditions is calculated and analyzed in this 
paper. Conclusions are found as follows: 
1. Rail roughness has significant influence on vehicle dynamic load and the wheel-rail force is 
a positive correlation with the rail roughness, that is the bigger the rail roughness amplitude, the 
greater the wheel-rail forces. 
2. The increase of vehicle speed will lead to the increase of wheel-rail forces. 
3. Influence of curve radius on the train vibration load is very complex. As the radius of curve 
is small, the influence of rail roughness on interaction load is obviously. 
References 
[1] Ding Deyun Study on Low Frequency Characteristics of Environmental Vibrations due to Metro 
Trains. Beijing Jiaotong University, Beijing, 2009. 
[2] Liu Weining, Ren Jing, et al. In-situ tests and analysis on rail corrugation of Beijing metro. Urban 
Rapid Rail Transit, Vol. 24, Issue 3, 2011, p. 6-9. 
[3] Li Kefei, Liu Weining, et al. In-situ test and analysis on the vibration mitigation measures of the 
elevated line in Beijing metro line 5. China Railway Science, Vol. 30, Issue 4, 2009, p. 25-29. 
[4] Sun Xiaojing Prediction of Environment Vibrations Induced by Metro Trains and Mitigation 
Measures Analysis. Beijing Jiaotong University, Beijing, 2008. 
[5] Jia Yingxuan Study on Analytical Model of Coupled Vehicle and Track and Effect to Environment 
by Metro Train-Induced Vibrations. Beijing Jiaotong University, Beijing, 2009. 
[6] Xia He Dynamic Interaction between Vehicles and Structures. Science Press, Beijing, 2005. 
0.1 1 10 100 1000
1E-3
0.01
0.1
1
10
100
1000
rai
l ro
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hn
es
s (
μm
)
spatial frequency
 longitudinal sleepers 
 normal track with DTVI2
 track with elastic short sleeper


Study on wheel-rail interaction based on rail roughness

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Study on wheel-rail interaction based on rail roughness

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