Study on wheel/rail adhesion force and its control of high-speed trainsconsidering aerodynamic loads and track excitations

Haijun Wang n, Jing Zeng, Ren LuoState Key Laboratory of Traction Power, Southwest Jiaotong University, Chengdu 610031, China

a r t i c l e i n f o

Article history:Received 16 November 2013Accepted 24 November 2013

Keywords:Vehicle/track systemWheel/rail adhesion forceFuzzy controlAdhesion utilization

a b s t r a c t

The wheel/rail vertical loads are strongly affected by the aerodynamic loads and track excitations, andwill further influence the wheel/rail adhesion forces. With the increasing of train speed, the influence ofaerodynamic loads becomes more and more significant. In order to probe into this problem, a simplifiedtrain dynamic model with three vehicles is established in which each vehicle system has 14 degrees offreedom. The track irregularity spectrum of a high speed line in China and steady aerodynamic loads aretaken as the inputs to the vehicle system. Numerical simulation is carried out by using MATLAB. It isfound that the aerodynamic loads obviously affect the wheel/rail contact forces for the high-speed train.Under the same train speed, with the increasing of crosswind velocity, the wheel/rail contact forces forcars at different positions in the train set are reduced gradually. Under the same crosswind velocity, theimpact of aerodynamic loads on the different-position vehicle is different. The model of wheel/railadhesion fuzzy control is built and preliminary analysis is conducted. The strategy of predictive controlbased on the vehicle system is introduced.

& 2013 Elsevier B.V. All rights reserved.

1. Introduction

With the increasing of train speed, the problem of interactionsbetween wheel and rail becomes more and more complicated [1].The traction and braking forces are limited by the wheel/rail adhesion.If the traction or braking force is greater than the wheel/rail adhesionforce, the wheel slip or skidding will occur. In order to improve thetraction and braking performance of the high speed trains in China,the anti-skid control devices are adopted for electric brake and airbrake. The devices of anti-slip and adhesion control are generallyadopted for traction. As one knows, wheel/rail vertical loads aremainly affected by the aerodynamic loads and track excitations, andwill further influence the wheel/rail adhesion forces. Different wheelin the vehicle system may have different vertical load. The purpose ofthe paper is to study the changes of wheel load and adhesion forceunder different aerodynamic loads and track inputs. In order to reducethe wheel/rail wear, the adhesion control strategy is also studied.

2. Dynamic model of vehicle/track system

Because the wheel/rail adhesion force is mainly related to theadhesion coefficient and wheel vertical load, the vertical model ofvehicle/track coupled system is established and shown as Fig. 1.

The vehicle/track coupled system is taken as a multiple rigid-body system with 14 degrees of freedom. They are bouncingand pitching motions of car body and bogie frame, and verticalmotions of each wheelset and the corresponding equivalent trackunit. The mass center of every component is taken as the origin ofcoordinate system. The running direction along the track centerline is taken as the x-axis. The z-axis is normal to the track planeand directed downward. The parameters of the vehicle system arelisted in Table 1.

The wheel/rail vertical contact is described as Hertz nonlinearspring. Steady aero-dynamic lift and pitching torque are exertedon the car body. The track random irregularity is considered. Eq.(1) is the motion equation of vehicle/track coupled system whenrunning on the straight lines.

Mf€zgþCf_zgþKfzg ¼ PfugþF ð1Þ

The state vector {z} in Eq. (1) is expressed as follows:

fzg ¼ fzc; θc; zb1; θb1; zb2; θb2; zw1; zw2; zw3; zw4; zr1; zr2; zr3; zr4gT ð2Þ

where zc, zb, zw and zr represent the bouncing motions of car body,bogie frame, wheelset and equivalent track unit respectively, andθc and θb represent the pitching motions of car body and bogieframe, respectively. M denotes the mass matrix, C the dampingmatrix, and K the stiffness matrix. F is the aerodynamic load actingon the car body and P is the excitation coefficient matrix. The

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n Corresponding author. Tel.: þ868335198339; fax: þ868335198322.E-mail address: whjyxy@aliyun.com (H. Wang).

Please cite this article as: H. Wang, et al., Study on wheel/rail adhesion force and its control of high-speed trains consideringaerodynamic loads and track excitations, Wear (2013), http://dx.doi.org/10.1016/j.wear.2013.11.043i

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excitation vector {u} is expressed as

fug ¼ ðz01; z02; z03; z04Þ ð3Þ

The track irregularity inputs corresponding to the second to thefourth wheelset with respect to the first wheelset can be describedby the equations below

z02ðtÞ ¼ z01 t�2bv

� �ð4Þ

z03ðtÞ ¼ z01 t�2lv

� �ð5Þ

z04ðtÞ ¼ z01 t�2lþ2bv

� �ð6Þ

where t is time and v the forward speed.Eq. (7) gives the wheel/rail vertical contact force based on Hertz

nonlinear contact theory [2].

pj ¼ p0þ2 1Gðzwj�zrj�z0jÞ� �3

2 zwj�zrj�z0j40pj ¼ 0 zwj�zrj�z0jr0

8<: ð7Þ

where pj indicates the wheel/rail vertical contact force, p0 is thewheel/rail static vertical force and G denotes the wheel/rail contactconstant.

The wheel/rail contact constant for worn-type wheel profile isas below [3].

G¼ 3:86R�0:116 � 10�8 m=N2=3 ð8Þwhere R is the wheel radius.

In this study, the train systemwith three vehicles is considered.The structure parameters and suspension parameters of eachvehicle are the same. The influence of longitudinal forces betweenvehicles on wheel/rail contact force is ignored.

3. Excitation inputs

3.1. Track random irregularity

The statistical characteristics of track random irregularity canbe obtained by the measurements of the actual track. Themeasured track irregularity of Wuhan–Guangzhou high speedpassenger special line in China is used. The random sample ofthe vertical irregularity of a section of track is shown in Fig. 2.

3.2. Steady aerodynamic loads under crosswind

Under the crosswind environment, the wind induced distribu-tion pressure and shear stress on the carbody surfaces can producea composition force in the vertical direction when the train isrunning. The aerodynamic lift force is the sum of the air pressure-difference lift force and friction lift force on the carbody surface inthe vertical direction. There also has a pitching torque acting on

Fig. 1. Dynamic model of vehicle/track coupled system.

Table 1Parameters of vehicle/track coupled system.

Parameter Symbol Value

Mass of car body mc 26:1 tMass of bogie frame mb 2:6 t

2:1 tMass of equivalent track unit mr 0:33 tPitching motion rotational inertia of car body Jc 1278:90 t=m2

Pitching motion rotational inertia of bogie frame Jb 1:424 t=m2

Spinning motion rotational inertia of wheelset Jw 0:084 t=m2

Length of car body L 25 mHalf of bogie center distance l 8:75 mHalf of wheelbase b 1:25 mRadius of wheel R 0:43 mVertical stiffness of primary Suspension of each wheel set kpz 2352 kN=mVertical stiffness of secondary Suspension of each bogie ksz 378:28 kN=mVertical damping of primary Suspension of each wheel set cpz 39:2 kN=s=mVertical damping of secondary suspension of each bogie csz 40 kN=s=m

0 200 400 600 800 1000-4

-3

-2

-1

0

1

2

3

4

Ver

tical

Tra

ck Ir

regu

larit

y(m

m)

Distance(m)

Fig. 2. Vertical track random irregularity.

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the carbody due to a combined action of the aerodynamicresistance and the aerodynamic lift force [4]. The protuberanceand depression of the carbody surface and the bottom structure ofcarbody are ignored [5]. The train shape is assumed to be smooth.The simplified train model includes a head car, a middle car and arear car. The length of each vehicle is 25 m, height 3.9 m and width3.38 m. Tables 2 and 3 give the aerodynamic lift force and pitchingtorque on the carbody, respectively.

4. Wheel/rail vertical contact force

The adhesion force is the product of wheel/rail vertical contactforce and adhesion coefficient, which can be calculated by Eq. (9).The adhesion coefficient is related to various factors, such as thesurface condition of wheel and rail, vehicle speed, wheel load andso on. Eq. (10) generally is taken as the calculation equation of theadhesion coefficient [6]. It is related to dry or wet of rail surfaceand running speed. If rail surface state and running speed areconstant, wheel/rail vertical contact force variations reflect adhe-sion force variations. The adhesion force can be calculated by

Fμ ¼Nμ ð9Þ

where N represents the wheel/rail vertical contact force and μ isthe adhesion coefficient which is expressed by Eq. (10).

μ¼ 0:0624þ 45:6vþ260 Dry Rail Surface

μ¼ 0:0405þ 13:5vþ120 Wet Rail Surface

(ð10Þ

4.1. Wheel/rail vertical contact force

The wheel load variation is affected by the aerodynamic forceand track inputs, while the aerodynamic force is related to thewind velocity and vehicle speed, and the track excitation is alsorelated to the vehicle speed. In this study, the wheel load variationis simulated with MATLAB under the same track and variouscombinations of running speed and crosswind velocity.

Fig. 3 illustrates the variation of wheel/rail vertical contactforces of the four wheelsets in time domain. It is assumed thatonly one controller for adhesion utilization is adopted for thevehicle, and four times of the minimum among the four wheel/railvertical contact forces is used to calculate the vehicle adhesionforce.

4.2. Average minimum wheel/rail contact force under crosswind

Fig. 4 shows the variation of the average minimum wheel/railcontact force of the head car, middle car and rear car underdifferent crosswind velocity. It is seen that the average minimumwheel/rail contact force reduces with the increasing of runningspeed and crosswind velocity, and the influence of the aerody-namic force on the wheel/rail contact force is significant.

4.3. Average minimum wheel/rail contact force of different vehicles

Fig. 5 shows the calculation results of the average minimumwheel/rail contact force in the case of without considering theaerodynamic force. It is seen that the vertical contact forces ofdifferent position vehicles are the same.

From Fig. 6, it is seen that when the aerodynamic loads areconsidered, the average minimum wheel/rail vertical contactforces of different position vehicles are not the same. Whencrosswind velocities are 5 and 10 m/s, the head car has the largestcontact force, and rear car has the smallest contact force. Whenthe crosswind velocity is 15 m/s, the contact force of the head caris still the largest.

Table 2Calculated steady aerodynamic lift forces.

Vehicle speedðkm=hÞ

Wind speedðm=sÞ

Lift force (kN)

Head car Middle car Rear car

200 5 0.51 �4.66 �7.3610 �6.67 �17.50 �15.7715 �21.34 �34.18 �26.70

250 5 1.81 �4.72 �9.6210 �4.57 �18.22 �19.0315 �18.26 �37.88 �31.32

300 5 3.30 �4.77 �12.3010 �2.43 �18.57 �22.4115 �15.07 �39.92 �36.19

350 5 5.00 �4.82 �15.4110 �0.21 �18.75 �26.0415 �11.89 �41.07 �41.06

400 5 6.93 �4.87 �18.9810 2.61 �18.88 �30.0115 �8.68 �41.74 �45.99

Table 3Calculated aerodynamic pitching torques.

Vehicle speed(km=h)

Wind speed(m=s)

Pitching torque (kN=m)

Head car Middle car Tripper car

200 5 50.89 �1.33 24.9110 79.57 �5.67 15.4515 114.40 �19.78 9.66

250 5 14.57 �2.24 38.6710 104.42 �3.89 30.7515 147.08 �15.71 19.23

300 5 101.80 �2.98 61.1910 133.59 �3.14 50.6415 180.53 �11.56 35.56

350 5 135.03 �4.24 84.8010 167.51 �3.16 74.6015 217.02 �8.55 57.68

400 5 173.46 �5.76 112.0210 206.32 �3.75 102.3715 257.61 �6.63 84.68

0.0 0.5 1.0 1.5 2.040

60

80

100

120

140

160

Whe

el/R

ail F

orce

(kN

)

Time(s)

1st Wheelset2nd Wheelset3rd Wheelset4th Wheelset

Fig. 3. Wheel/rail vertical contact force.

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5. Wheel/rail adhesion force control

The PID controller and fuzzy controller were used for adhesionforce control in [7–9]. Seiji Yasunobu and Hiroyasu Oshimaestablished fuzzy rules through summarizing operation experi-ences of train drivers, and built a fuzzy predictive train controlsystem. Their study shows that the predictive train control systemis superior to PID control system [10,11]. In this study, the wheel/rail adhesion control model is established as described in Fig. 7.The input of the control system is the adhesion force, and theoutput is the traction force/braking force. The control objective isthat the traction force/braking force should be close to themaximum adhesion force and the difference between them shouldbe as small as possible.

5.1. Equations of adhesion control

Eq. (11) is the motion equation of the wheelset.

M¼ Mcþ2Mf

4R2þ Jw

� �€θwþFR

Rð11Þ

where M is the torque acting on wheelset, θw denotes therotational angle of wheelset, and FR the running resistance.

The vehicle running velocity is given by Eq. (12).

vt ¼ vt�1þ12€θw � R� t2 ð12Þ

where t is running time and vt is running velocity at time t.

5.2. Adhesion force predictive fuzzy control system

If a great deal of basic data can be gained, it is possible to establishthe database of track irregularity and aerodynamic loads. Thecalculation model of the vehicle/track coupled system is establishedaccording to the real conditions. Wind velocity and wind directionare detected. Rail surface state is also detected. The direction ofstraight line is conformed. Running speed range can be predicted byEq. (11) and Eq. (12). After inputting all this information, the wheel/rail adhesion forces corresponding to line position can be gained.

According to Eq. (10), adhesive coefficient decreases with theincrease of running speed. In addition, the average minimumwheel/rail contact force decreases with the increase of runningspeed as shown in Figs.4–6. Thus, adhesion force decreases withthe increase of running speed according to Eq. (9). Therefore, foradhesion utilization security, adhesion force corresponding to themaximum of running speed range is taken as the maximum ofcontrol target. Because it needs some time to adjust the torqueacting on the wheelset, the torque adjustment should be taken inadvance. To accomplish this, the adhesion force predictive fuzzy

4 6 8 10 12 14 1660

64

68

72

76

80

84

88

92

96

Ave

rage

Min

imum

Whe

el/R

ail F

orce

(kN

)

Ave

rage

Min

imum

Whe

el/R

ail F

orce

(kN

)

Ave

rage

Min

imum

Whe

el/R

ail F

orce

(kN

)Crosswind Velocity(m/s)

V=200km/h V=250km/h V=300km/h V=350km/h V=400km/h

4 6 8 10 12 14 1660

64

68

72

76

80

84

88

92

96

Crosswind Velocity(m/s)

V=200km/h V=250km/h V=300km/h V=350km/h V=400km/h

4 6 8 10 12 14 1660

64

68

72

76

80

84

88

92

96 V=200km/h V=250km/h V=300km/h V=350km/h V=400km/h

Crosswind Velocity(m/s)

Fig. 4. Average minimum wheel/rail contact force. (a) head car; (b) middle car; (c) rear car.

180 220 260 300 340 380 42072

74

76

78

80

82

84

86

88

90

92

Head CarMiddle CarRear Car

Running Speed(km/h)

Ave

rage

Min

imum

Whe

el/R

ail F

orce

(kN

)

Fig. 5. Average minimum wheel/rail contact force without considering aerody-namic loads.

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control system may be built and used alongside the anti-skid andanti-slip devices. The anti-skid and anti-slip systems take pre-cedence over the adhesion force predictive fuzzy control system toavoid skidding and slipping.

5.3. Control simulation example

In this example, the traction adhesion control of the middle caris simulated under constant crosswind velocity when the runningspeed changes from 200 km/h to 400 km/h.

The running resistance is not considered here. Eq. (9) is used tocalculate wheel/rail adhesion force. Adhesion coefficient of dry railsurface in Eq. (10) is adopted. Under crosswind velocity of 15 m/s,the average minimum wheel/rail contact force of the middle car isadopted. The wheelset adhesion forces for different runningspeeds in Table 4 are calculated with Eq. (9), Eq. (10) and theaverage minimum wheel/rail contact force shown in Fig. 6(c).

Fig. 8 is gained with adhesion force predictive fuzzy control.In Fig. 8, the adhesion force curve is fitted with the data in Table 4.Its adhesion utilization ratio is 92.04%. The figure shows that the

180 220 260 300 340 380 42060

64

68

72

76

80

84

88

92

Running Speed(km/h)

Head CarMiddle CarRear Car

180 220 260 300 340 380 42060

64

68

72

76

80

84

88

92

Running Speed(km/h)

Head CarMiddle CarRear Car

180 220 260 300 340 380 42060

64

68

72

76

80

84

88

92

Running Speed(km/h)

Head CarMiddle CarRear Car

Ave

rage

Min

imum

Whe

el/R

ail F

orce

(kN

)

Ave

rage

Min

imum

Whe

el/R

ail F

orce

(kN

)

Ave

rage

Min

imum

Whe

el/R

ail F

orce

(kN

)

Fig. 6. Average minimum wheel/rail contact force of different position vehicle. (a) Crosswind velocity 5 m/s; (b) crosswind velocity 10 m/s; (c) crosswind velocity 15 m/s.

Crosswind Velocity

Running SpeedVehicle/Track CoupledDynamic System

Wheel/Rail Adhesion Force

Difference The Change of Difference

Adhesion Coefficient

Traction Force /Braking Force

Fuzzy Controller

Fig. 7. Wheel/rail adhesion control.

Table 4Wheelset adhesion force.

Running speed (km/h) 200 250 300 350 400

Wheelset adhesion force (kN) 13.12 11.79 10.62 9.48 8.36

200 250 300 350 4007

8

9

10

11

12

13

14

Trac

tion

Forc

e an

d A

dhes

ion

Forc

e (K

N)

Running Speed (km/h)

Traction ForceAdhesion Force

Fig. 8. Adhesion force control curve.

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traction force is always less than adhesion, and it has enough securityredundancy.

6. Conclusions

The vertical contact forces between the rail and four wheelsetsare not equal at the same moment. The minimum wheel set/railcontact force is available for adhesion force utilization.

The influence of aerodynamic loads on wheel/rail verticalcontact force is significant. Thus, the aerodynamic loads must beconsidered as a factor affecting adhesion force variation.

When aerodynamic loads are considered, the average mini-mum wheelset/rail vertical contact forces of different positionvehicles are not equal. The influences of aerodynamic loads onwheelset/rail vertical contact force of different position vehiclesare also different. To study this it is necessary to build the vehicle/track coupled system model for the high speed train.

The adhesion force predictive fuzzy control system may beestablished based on a great deal of basic data. Once it is usedtogether with anti-skid and anti-slip devices, the high adhesionutilization ratio and low wheel/rail wear can be obtained for high-speed trains.

Acknowledgements

This work is supported by the National 973 Program (2011CB-711106), the Open Project of State Key Laboratory of Traction Power

(TPL1104), and the Fundamental Research Funds for the CentralUniversities (SWJTU2011BR055EM).

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