Embed Size (px)
Citation preview
This article was downloaded by: [Beijing Jiaotong University]On: 27 September 2011, At: 21:20Publisher: Taylor & FrancisInforma Ltd Registered in England and Wales Registered Number: 1072954 Registeredoffice: Mortimer House, 37-41 Mortimer Street, London W1T 3JH, UK
Vehicle System DynamicsPublication details, including instructions for authors andsubscription information:http://www.tandfonline.com/loi/nvsd20
A new method for determiningwheel––rail multi-point contactZ. Ren a , S. D. Iwnicki b & G. Xie ba Engineering Research Center of Structure Reliability andOperation Measurement Technology of Rail Guided Vehicles,School of Mechanical, Electrical and Engineering, China Ministryof Education, Beijing Jiaotong University, Beijing, 100044,People's Republic of Chinab Rail Technology Unit, Manchester Metropolitan University,Manchester, M1 5GD, UK
Available online: 15 Aug 2011
To cite this article: Z. Ren, S. D. Iwnicki & G. Xie (2011): A new method for determining wheel––railmulti-point contact, Vehicle System Dynamics, 49:10, 1533-1551
To link to this article: http://dx.doi.org/10.1080/00423114.2010.539237
PLEASE SCROLL DOWN FOR ARTICLE
Full terms and conditions of use: http://www.tandfonline.com/page/terms-and-conditions
This article may be used for research, teaching and private study purposes. Anysubstantial or systematic reproduction, re-distribution, re-selling, loan, sub-licensing,systematic supply or distribution in any form to anyone is expressly forbidden.
The publisher does not give any warranty express or implied or make any representationthat the contents will be complete or accurate or up to date. The accuracy of anyinstructions, formulae and drug doses should be independently verified with primarysources. The publisher shall not be liable for any loss, actions, claims, proceedings,demand or costs or damages whatsoever or howsoever caused arising directly orindirectly in connection with or arising out of the use of this material.
Vehicle System DynamicsVol. 49, No. 10, October 2011, 1533–1551
A new method for determining wheel–rail multi-point contact
Z. Rena*, S.D. Iwnickib and G. Xieb
aEngineering Research Center of Structure Reliability and Operation Measurement Technology of RailGuided Vehicles, School of Mechanical, Electrical and Engineering, China Ministry of Education,Beijing Jiaotong University, Beijing 100044, People’s Republic of China; bRail Technology Unit,
Manchester Metropolitan University, Manchester M1 5GD, UK
(Received 21 June 2010; final version received 5 November 2010; first published 15 August 2011 )
A new method for wheel–rail multi-point contact is presented in this paper. In this method, the first- andthe second-order derivatives of the wheel–rail interpolation distance function and the elastic wheel–railvirtual penetration are used to determine multiple contact points. The method takes account of the yawangle of the wheelset and allows the identification of all possible points of contact between wheel andrail surfaces with an arbitrary geometry. Static contact geometry calculations are first carried out usingthe developed method for both new and worn wheel profiles and with a new rail profile. The validity ofthe method is then verified by simulations of a coupled vehicle and track system dynamics over a smallradius curve. The simulation results show that the developed method for multi-point contact is efficientand reliable enough to be implemented online for simulations of vehicle–track system dynamics.
Keywords: multi-point contact method; wheel-rail interpolation distance; extreme point; penetration;vehicle-track system dynamics; curve negotiation; verification
1. Introduction
Wheelsets of railway vehicles are usually constructed with two wheels tightly fixed on anaxle, and in addition to the rolling motion longitudinally along two rails, wheelsets also havemotions in the lateral direction, yaw and roll. The interaction between wheels and rails takesplace in the wheel–rail contact patch, and the motion of wheelset is generally geometricallyconstrained by wheel and rail profiles. Consequently, an essential part of the wheel–rail contactproblem is to solve a three-dimensional geometric constraint problem formed by the wheeland rail spatial surfaces.
The wheel and rail profiles are usually designed with the aim of maintaining a single-pointcontact between the wheel and the rail. In practice, single-point contact is also regarded as oneof the effective measures to reduce the wheel–rail wear and to provide satisfactory dynamicperformance of the vehicle–track system. It has, however, been shown that [1–3] multi-pointcontact frequently occurs, especially when the wheel or rail profiles are worn, the vehiclepasses through a turnout zone [4,5] or negotiates a small radius curve.
*Corresponding author. Email: [email protected]
ISSN 0042-3114 print/ISSN 1744-5159 online© 2011 Taylor & FrancisDOI: 10.1080/00423114.2010.539237http://www.informaworld.com
Dow
nloa
ded
by [
Bei
jing
Jiao
tong
Uni
vers
ity]
at 2
1:20
27
Sept
embe
r 20
11
1534 Z. Ren et al.
Many studies have been carried out in the area of vehicle dynamics simulation dealing withthe wheel–rail multi-point contact problem. Pascal and Sauvage [2], for example, have pre-sented a method in which two separate contact zones were approximated with one equivalentrigid contact patch and a multi-Hertzian contact hypothesis to determine the equivalent conicityfor the case of S1002 wheel profiles and UIC60 rail profiles [3]. They also presented a methodfor the calculation of the wheel–rail forces in non-Hertzian contact patches [6]. In 1991, Elkinspresented a survey of the state-of-the-art in predicting wheel–rail interaction and concludedthat there is a room for further improvement in the calculation of contact geometry [7].
Based on the interpenetration of the two underformed bodies profiles, Ayasse and Chollet[8] proposed a semi-Hertzian method to consider the non-Hertzian wheel–rail contact patch.Piotrowski and Kik [9] presented a simplified model for non-Hertzian wheel–rail contactproblems in which the semi-elliptical normal pressure distribution is assumed in the directionof rolling and the contact area is found by the virtual penetration of wheel and rail. Anextensive survey of wheel–rail contact models for railway vehicle system dynamics includingmulti-point contact was given by Piotrowski and Chollet [1]. Some current methods adopt thetechnique of discretisation of the contact area into a number of longitudinal strips parallel tothe rolling direction with the normal pressure for each strip determined on the assumptionof Hertzian contact. Pombo and Ambrosio [10] also presented a computational method topredict wheel–rail contact including multiple contact points by introducing normal vectorsfor the three-dimensional wheel and rail profiles. All of these methods, however, containconsiderable complexities that inevitably bring higher computation costs as they have to beimplemented at each step of the numerical integration of vehicle dynamic simulations.
A fast and effective method for the prediction of contact conditions in multiple contact mustbe used in vehicle-turnout dynamics studies because of varying profiles of switch and nose rails.Ren et al. [11] have presented a method to determine the two-point contact zone and to modelthe transfer of wheel–rail forces between different rails in a turnout. It should be pointed out thatmost existing methods of wheel–rail contact in vehicle dynamics simulations are limited to twodimensions and the yaw motion of wheelsets is not therefore taken into account. Santamariahas studied the effect of the introduction of two-point contact in three dimensions on wearindexes and derailment risk against traditional two-dimensional analysis and shown that thereare significant differences between the results obtained with the three-dimensional modelsand those with two-dimensional models [12]. Therefore, a fast three-dimensional wheel–rail geometric contact method able to effectively deal with the multi-point contact problemswould appear to be useful, especially for situations encountered in the vehicle-turnout systemdynamics and curving on a small curve radius.
The nature of the multi-point contact problem implies that the solution for it must considera method to determine the initial contact point, which is mainly determined by the geometricrelation between the wheelset and track, and the elastic penetration between wheel and rail,which is influenced by profiles and load conditions. Based on this principle, a new method todetermine multi-point contact in three dimensions is presented in this paper. The geometricrelationship of the wheel–rail system according to Wang [13] is first given in Section 2 asthe mathematical basis and the description of the new method is detailed in Section 3. Theproposed method is then applied to static calculations for new and worn wheel profiles, and theapplicability of the method is further verified in vehicle curving simulations for a full vehicleover a small radius curve.
2. The line tracing method for wheel–rail contact
Kinematic expressions for the wheel–rail geometry problem can be referred to a method called‘tracing line method’ presented by Wang [13]. The principle is based on the fact that, at the
Dow
nloa
ded
by [
Bei
jing
Jiao
tong
Uni
vers
ity]
at 2
1:20
27
Sept
embe
r 20
11
Vehicle System Dynamics 1535
contact point, the wheel surface and the rail surface must have the identical spatial coordinate,the tangent plane and surface normal. The geometry of the wheel can be regarded as a surfaceof revolution about the axle of the wheelset and any contact points with rails must be onthis three-dimensional surface. It is usually not a simple mathematical task to determine thecontact points on the spatial surface. The position of the contact point on the wheel surfaceis determined by parameters including the wheel and rail profiles, the lateral movement, andthe yaw and roll angles of the wheelset. By the ‘line tracing method’, these parameters forma number of nonlinear geometric constraints and a series of possible contact points can befound within a spatial curve. Therefore, the problem to determine the contact points on thespatial surface is transferred to an equivalent one on a spatial curve [13], and this mathematicaltransform greatly simplifies the problems. When the yaw of the wheelset is not present, thespatial curve reduces to a two-dimensional curve in the vertical plane of the track and thiscurve is actually the wheel profile.
Owing to the symmetry of the wheelset structure, a half wheelset is considered here. Inthe wheelset coordinate system reference o − xyz as shown in Figure 1, o is the centre of thewheelset, ox and oy are the longitudinal and lateral axes, respectively, oη is the axis of thewheelset, o1 is a crossing point between the axle oη and the normal line o1C at the contactpoint C of the wheel profile; so, δr is an angle whose tangent value is equal to the slope of thecurvature of the wheel profile at the contact point C when the roll angle θ = 0.
Given ψ is the yaw angle and θ is the roll angle of the wheelset, the coordinate of the rightcontact point C can be determined with Equation (1)
xw = − cos θ sin ψ(ηr − Rrtgδr),
yw = cos θ cos ψa0 + sin θb0,
zw = − sin θa0 + cos θ cos ψb0,
(1)
where Rr = o2C is the rolling radius of the wheel at the contact point C, ηr = oo2 is thelateral distance between the contact point C and the centre of the wheelset, and a0 and b0 arecoefficients given by Equation (2)
a0 = ηr + Rrtgδr cos2 θ sin2 ψ
(cos2 θ cos2 ψ + sin2 θ),
b0 = Rr
√cos2 θ cos2 ψ + sin2 θ − cos2 θ sin2 ψtg2δr
(cos2 θ cos2 ψ + sin2 θ).
(2)
Using direction cosine functions with respect to axis oη
lx = − cos θ sin ψ, ly = cos θ cos ψ, and lz = sin θ. (3)
z
y
x
ψ
C
Rr
o o2 θηω 1
δro
Figure 1. The line tracing method for contact curve calculation.
Dow
nloa
ded
by [
Bei
jing
Jiao
tong
Uni
vers
ity]
at 2
1:20
27
Sept
embe
r 20
11
1536 Z. Ren et al.
Equation (1) may be rewritten as
xw = lx(ηr − Rrtgδr),
yw = lya0 + lzb0,
zw = −lza0 + lyb0,
(4)
and Equation (2) as
a0 = ηr + Rrtgδrl2x
(l2y + l2
z ),
b0 = Rr
√l2y + l2
z − tg2δrl2x
(l2y + l2
z ).
(5)
Equations (4) and (5) define the spatial curve or ‘tracing line’, where all possible contactpoints on the wheel surface are located. The y and z coordinates of this spatial curve andrail profile will then be used to determine the contact points. Additionally, yaw angle ψ andlateral displacement y of wheelset in Equations (3)–(5) are variables, and they are obtainedfrom the motion equation of the wheelset. The roll angle θ is an independent variable whichis determined by the yaw angle and the lateral displacement of the wheelset because of thewheel-rail geometric constraint, and consequently, the roll angle θ is iterated to obtain thecontact geometries. For each discretised point of the wheel profile, parameters such as ηr, Rr,and δr are constants.
The geometric curves with no yaw and a yaw angle of 5π/180 radians are shown in Figure 2.This indicates that the curve is the lowest cross-sectional line of the wheel surface if theyaw angle is zero (ψ = 0) and the curve must be on the two-dimensional curve because thelongitudinal coordinate values of each discretised point are all equal to zero as shown inFigure 2(a). On the other hand, all of the contact points of the wheel must be on a spatial curveif a yaw angle exists (ψ �= 0) as shown in Figure 2(b). Moreover, if the wheelset yaw angleis sufficiently large, the curvature at the flange root becomes smaller and the possibility ofmulti-point contact occurrence increases.
0=ψ (b) (a) 180/5πψ = rad
–90–60
–300
3060
680700
720740
760780
–460
–450
–440
–430
–420
–410
y (mm)x (m
m)
z (m
m)
z (m
m)
–0.8–0.4
0.00.4
0.8
680700
720740
760780
–460
–450
–440
–430
–420
–410
y (mm)x (
mm)
Figure 2. Wheel–rail contact curve for different yaw angles.
Dow
nloa
ded
by [
Bei
jing
Jiao
tong
Uni
vers
ity]
at 2
1:20
27
Sept
embe
r 20
11
Vehicle System Dynamics 1537
3. Wheel–rail multi-point contact method
Based on the line tracing method, a spatial curve containing all possible contact points hasbeen found and a new approach to determine the multi-point contact in three dimensions isdeveloped. The approach consists of two phases. In the first phase, the well-developed iterationprocedure [12,14] is implemented to determine the first contact point using the spatial curveobtained by the line tracing method. This conventional procedure is purely a geometry problemand usually applied in two dimensions considering the lateral motion of the wheelset. In thepresent approach, the yaw motion is taken into account so that this procedure is implementedin three dimensions. In the second phase, the first- and second-order derivatives of the distancefunction between the wheel and rail and the elastic penetration are introduced to determinethe second contact point or possibly more contact points. For each contact point, the Hertzcontact theory is used to determine the contact shape and pressure. The details of this newapproach are described as follows.
3.1. Determination of the first contact point
The iteration process determining the contact point is well known and extensively used invehicle dynamics to deal with single-point contact problem; this process is briefly describedhere as it is helpful to understand the method to determine multi-point contact later.
Generally, the wheelset has six degrees of freedom (DOFs) with respect to the track. Thesix DOFs include the longitudinal, lateral and vertical displacement and axle rotation, rolland yaw which are indicated here by x, y, z, ω, θ , and ψ , respectively. As the wheelset isconstrained on the rails and moving forwards, only the lateral and yaw motions remain to beactive DOFs, provided that the movement of the rails is not considered, and roll can furtherbe treated as a DOF depending on the lateral motion. When the yaw angle of the wheelset isignored, the wheel–rail contact parameters are associated only with the lateral displacement y
and the information of all possible contact points are often tabulated in a table, which is thenused for the time integration in vehicle dynamics simulations. The zero yaw approximationsimplifies the problem and is effective in most situations but the effect of the yaw angle on thewheel–rail contact needs to be taken into account in some particular situations, for example,in tight curves. The present method therefore includes the effect of wheelset yaw to determinethe first contact point.
As shown in Figure 3, o − xyz is a wheelset coordinate system and o′ − x ′y ′z′ is a trackcoordinate system. The wheel and rail profiles are often described using discrete points inpractice, for instance, from measurements, and spline functions are then used to representprofiles so that the distance between wheel and rail for any position can be obtained byinterpolation. For a given lateral displacement y and yaw angle ψ of the wheelset, the spatialcurve shown in Equation (4) first projects on the rail profile to give the interpolated distances
y
o
z
θ y
'z'
o'
x
x' ddzl zr
Figure 3. Interpolation method for the wheel–rail contact geometry calculation.
Dow
nloa
ded
by [
Bei
jing
Jiao
tong
Uni
vers
ity]
at 2
1:20
27
Sept
embe
r 20
11
1538 Z. Ren et al.
between wheel and rail. The roll angle θ is then iteratively adjusted and the location of thefirst contact point C1 can be determined until the following condition is satisfied:
|dzl − dzr| ≤ εt , (6)
where dzl is the interpolated distance between the left wheel and left rail, dzr is the interpolateddistance between the right wheel and right rail and εt is a specified tolerance. For a 50 kNwheel load, the wheel–rail penetration is in the order of 10−2 mm approximately and thereforea tolerance εt of 10−6 mm is used here to meet the precision requirement for simulations.Additionally, the minimum wheel–rail interpolation distance dmin at the contact point C1isexpressed in a discrete form by
dmin = min(dz1, dz2, . . . , dzn), (7)
where n is the number of the discretised points of the wheel profile and at the contact pointC1, dmin = dzl = dzr.
3.2. Wheel–rail multi-point contact
After the contact point C1 is obtained, in order to determine the other contact points, not onlythe wheel–rail interpolation distances are used but also elastic penetrations associated with theload of wheelset have to be introduced. As shown in Figure 4, zw is the vertical coordinate ofthe spatial curve of the wheel profile as shown in Equation (4) and zr is the vertical coordinateof the rail profile, the interpolation distance dz is then given by Equation (8)
dz = zw − zr. (8)
The interpolation distance dz of the all discretised points forms a function f (dz), and it isdetermined by the wheel and rail profiles, the lateral displacement, and the yaw angle of thewheelset [1] and the lateral and vertical displacements of rails if the rail movements are takeninto account.
The first derivative of the distance function f (dz) can be obtained with Equation (9)
d ′z = f ′(dz) = d(dz)
dy= d(zw − zr)
dy. (9)
If the derivative is zero, Equation (10), one or more turning points or extreme points arepresent in the function f (dz). The number of these points is ne with ne < n
f ′(dz) = 0. (10)
If ne = 1, there is only one wheel–rail contact point and it is the contact point C1 that hasbeen obtained with the iterative procedure given above. If, however, ne > 1, some of the other
y
z
o
dz
zr
zw
Figure 4. Wheel–rail interpolation distance.
Dow
nloa
ded
by [
Bei
jing
Jiao
tong
Uni
vers
ity]
at 2
1:20
27
Sept
embe
r 20
11
Vehicle System Dynamics 1539
extreme points may potentially be in contact. Consequently, from the view of the principleof profile design, there should be only one extreme point in the wheel–rail distance functionf (dz) for any lateral displacement y and yaw angle ψ of the wheelset.
The physical meaning of the first derivative of the wheel–rail distance function is the slopedifference between the wheel profile and rail profile, and at an extreme point, the wheel andrail profiles should have the same normal direction if the slope difference equals zero. This iscoincident with the principle of maintaining the single-point contact condition.
All of the extreme points ne as shown in Figure 5 such as C1, A, Ci , B, and Cj satisfyEquation (10), but some extreme points such as A and B in Figure 5, having local maximalinterpolation distance, are not possible contact points and they can be excluded from ne easilybecause the second derivative f ′′(dz) of the wheel–rail distance function does not satisfyEquation (11):
d ′′z = f ′′(dz) ≥ 0. (11)
To finally determine the contact points among the remained points such as C1, Ci , and Cj asshown in Figure 5, the real contact points must satisfy the following constraint condition too
dzi − dmin ≤ εd, (12)
where dzi is the wheel–rail interpolation distance at the extreme point Ci , εd is the elasticpenetration difference between at the extreme point Ci , and the contact point C1 which hasthe minimum interpolation distance dmin.
The principle to determine the multiple contact points can be summarised by⎧⎪⎨⎪⎩
f ′(dzi) = 0
f ′′(dzi) ≥ 0
dzi − dmin ≤ εd
(i = 1, 2, . . . , nc), (13)
where nc is the wheel–rail contact number and satisfies nc ≤ ne < n. If nc > 1, multi-pointcontact occurs and if nc = 1, only one contact point C1 exists.
The elastic penetration difference εd between the contact point Ci and C1 is determined by
εd = δ1 − δi, (14)
where δ1 and δi are the elastic penetrations at contact points C1 and Ci , respectively, and theyare determined by Zhai [14]
δ1 = [zw + dz1 − dz0 + ηr1 sin(θe − θi)] cos δr1 − Rr1 sin(θe − θi) sin δr1 cos ψ,
δi = [zw + dzi − dz0 + ηri sin(θe − θi)] cos δri − Rri sin(θe − θi) sin δri cos ψ,(15)
where zw is the vertical displacement of the wheelset, dz1 and dzi are the interpolated distancesat contact points C1 and Ci , dz0 is the initial minimal interpolation distance between the wheel
o y
zC i j
A B
C C
x
)( zdf
Figure 5. Extreme points of interpolation distance function.
Dow
nloa
ded
by [
Bei
jing
Jiao
tong
Uni
vers
ity]
at 2
1:20
27
Sept
embe
r 20
11
1540 Z. Ren et al.
and rail which is introduced in the interpolation method to calculate the contact geometries inwhich the wheel is above the rails for some distance as shown in Figure 4. θe and θi are the rollangles of the wheelset obtained from the movement equation (21d) which is presented in thenext section and from the iterative process of the contact geometries calculation, respectively.ηr1 is the distance between the wheelset centre and the point C1, ηri is the distance betweenthe wheelset centre and the point Ci . Rr1 and Rri are the rolling radii at the points C1 and Ci
respectively, δr1 and δri are the contact angles at the points C1 and Ci , respectively, as shownin Equation (1). ψ is the yaw angle of the wheelset.
In fact, the penetrations δi and δ1 in Equation (14) are related to the wheel–rail contactnormal force, so the force balance equation of the wheelset must be used to solve them in thevehicle system dynamics study.
The interpolation distance difference εd can be approximated by the elastic penetration δ1
which is given by Equation (16) if only the static load is taken account to check the staticwheel–rail contact geometry.
δ1 = G · (N1)2/3, (16)
where N1 is the static load of the wheel and G is an elastic coefficient, and it is different for aconical or concave profile as shown in Equation (17) [14]
Cone-shaped profile: G = 4.57R−0.149 × 10−8(m/N2/3),
Concave profile: G = 3.68R−0.115 × 10−8(m/N2/3),
(17)
where R is the rolling radius of the wheel.It is worthy to point out that, using the half-axle load as an approximated load condition, this
method can be conveniently and effectively used in the profile design to inspect the wheel–railstatic contact condition.
Another necessary condition for multi-point contact is that each independent contact pointmust be sufficiently distant from each other and, as shown in Figure 6, at least one extremepoint B must exist between the contact points Ci and Cj . Thus, this condition for multiplecontact points can be satisfied if
dzb > dzi + δi,
dzb > dzj + δj ,(18)
where dzi , dzj , and dzb are the interpolation distances at the two contact points Ci and Cj andextreme point B, δj is the wheel–rail penetration at contact point Cj .
If Equation (18) is not satisfied, it means that the contact points are coplanar or collinearas shown in Figure 7. In this situation, profile slopes at the contact points are introduced to
jδ
Rail profile
Wheel profile
δi
zid dzj
z
y
x
B
dzb
o
Ci jC
Figure 6. Wheel–rail penetrations.
Dow
nloa
ded
by [
Bei
jing
Jiao
tong
Uni
vers
ity]
at 2
1:20
27
Sept
embe
r 20
11
Vehicle System Dynamics 1541
jδ
Rail profile
Wheel profile
δi
zid dzj
z
y
x
B
dzb
o
Ci jC
Figure 7. Two coplanar contact points.
determine whether the contact should be treated as one point or multiple points in simulationand the criteria used for multi-point contact is then given by
|si − sj | ≥ εs, (19)
where si is the slope of the wheel profile at the contact point Ci , sj is the slope of the wheelprofile at the contact point Cj and εs is a specified slope difference between the two contactpoints. The minimum slope of the wheel profile at the tread of the high-speed train is generally0.025 and εs = 0.05 used here.
If Equation (19) is not satisfied, the condition of contact should be regarded as one pointcontact and there is very little difference of the simulation results for single-point contact andtwo-point contact in which it is demonstrated in the following section.
4. Validation with static contact geometry calculations
Initially, a static analysis is carried out to validate the new method.
4.1. CHN-LMA wheel profiles
The CHN-LMA wheel profile shown in Figure 8 is currently used by one type of high-speedelectrical multiple units (EMU) in China. With the method presented above, static contact
–75 –60 –45 –30 –15 0 15 30 45 60–10
0
10
20
30
40
Hei
ght (
mm
)
Width (mm)
New Worn
Figure 8. LMA profile in new and worn state.
Dow
nloa
ded
by [
Bei
jing
Jiao
tong
Uni
vers
ity]
at 2
1:20
27
Sept
embe
r 20
11
1542 Z. Ren et al.
calculations for a new and a worn CHN-LMA wheel profiles on a new CHN60 rail profile arecarried out. The worn profile was measured before it was reprofiled for the first time, and itwas observed that the tread was heavily worn and the flange was worn slightly.
The contact geometry for the new wheel profile is shown in Figure 9. It can be seen thatonly a single wheel–rail contact point occurs if the yaw angle ψ of the wheelset is less than1.5π/180 radians while two-point contact occurs if the yaw angle is larger than 1.5π/180radians and the lateral displacement is 9.2 mm approximately.
4.2. Contact geometry for the worn wheel profile
In the track coordinate system o′ − x ′y ′z′, the left wheel–rail interpolation distance functionf (dz) and its first differentials f ′(dz) for the new and worn profiles are shown in Figures 10and 11, respectively, for the lateral displacement of the wheelset of 1.0 mm. It is seen that thereis only one contact point for the new wheel profile because there is only one extreme point
–10 0 10 20 30 40 50–60
–40
–20
0
20
40
60
Lateral displacement of wheelset (mm)
Con
tact
poi
nt lo
catio
n on
whe
el p
rofi
le (
mm
)
Wid
th (
mm
)
Height (mm)
Worn profile
New profile
–3 0 3 6 9 12 15
–60
–40
–20
0
20
40
60
New profile location: y =0New profile location: y =1.5p/180Worn profile location: y =0
Figure 9. Contact geometry of the new LAM profile.
–790 –780 –770 –760 –750 –740 –730 –7200
1
2
3
4
d z (
mm
)
y (mm)
A
B
C
New
Worn )( zdf
Figure 10. Interpolated distance function for worn profile.
Dow
nloa
ded
by [
Bei
jing
Jiao
tong
Uni
vers
ity]
at 2
1:20
27
Sept
embe
r 20
11
Vehicle System Dynamics 1543
(dz)f ′
–790 –780 –770 –760 –750 –740 –730 –720–0.4
–0.2
0.0
0.2
0.4
d(d z
)/dy
y (mm)
A BC
New
Worn
Figure 11. Derivative of the interpolated distance function.
in its interpolation distance function. But for the worn profile, three extreme points satisfyEquation (10) and two extreme points A and C satisfy Equation (11).
Elastic penetration is introduced to find out whether two-point contact occurs or not. The axleload for this vehicle is 14,000 kg and the nominal rolling radius is 0.43 m so that the penetrationδ1 is therefore given as 6.89 × 10−2 mm by Equation (16) and this is taken as the approximateelastic penetration.
The results show that two-point contact occurs when the wheelset has a lateral displacementyw of 1.0 mm as shown in Figure 12 in the track coordinate system o − x ′y ′z′. In the wheelcoordinate system as shown in Figure 8, the two contact points are approximately located at−7.1 and 20.2 mm of the wheel tread. In the rail coordinate system, i.e. the track coordinatesystem, as shown in Figure 12, the two contact points approximately locate at −14.0 and13.5 mm of the rail profile. The occurrence of the two-point contact at these locations is due tothe fact that the wheel tread was worn to a concave shape in the region of the nominal rollingradius after a period of service and contact points occur on each side of the worn concavezone. If the lateral displacement of the wheelset is between 9.1 and 9.2 mm, two-point contactoccurs again with one point located at the flange and the other point at the flange root of thewheel, which is verified in the next section.
–820 –800 –780 –760 –740 –720 –700 –680460
450
440
430
420
410
z (m
m)
y (mm)
–754 mm
–746.5 mm
Figure 12. Locations of two contact points.
Dow
nloa
ded
by [
Bei
jing
Jiao
tong
Uni
vers
ity]
at 2
1:20
27
Sept
embe
r 20
11
1544 Z. Ren et al.
5. Validations with dynamic simulations
A coupled vehicle and track dynamics model including the new multi-point contact modelwas set up to simulate curving through a small radius curve.
5.1. Track system model
As shown in Figure 13, the track system consists of two Euler–Bernoulli beams representingtwo rails, rail pads and ballast which are modelled by lumped masses. The stiffness kp anddamping cp of the rail pad are taken into account and the ballast is regarded as lumped mass[14,15].All of the vertical and lateral elastic deformations of the rails and sleepers are includedin the system model.
The equation of motion for transverse vibration ur(x, t) of the rails in the vertical and lateralplanes is:
ErIr∂4ur(x, t)
∂x4+ mr
∂2ur(x, t)
∂t2= −
Ns∑i=1
Frsi (t)δ(x − xi) +4∑
j=1
Fwrj (t)δ(x − xwj ), (20)
where Ns is the number of sleeper support points, δ(x) the Dirac delta function, ErIr theflexural moment of inertia of the rail, mr the rail mass per unit length, Fwrj the wheel–railforce, and Frsi the rail–sleeper force. Each Fwrj includes the forces of every multi-point contactfor a wheel. Similarly, the sleeper is treated as a free–free Euler–Bernoulli beam on an elasticfoundation of ballast and its vibration equation is not presented here.
Additionally, Equation (20) is a partial differential equation, and it can be transferred to anordinary differential equation if a mode function is introduced and the orthogonality of themode function is applied on it [16].
5.2. Vehicle system model
The vehicle is the CRH2 high-speed EMU manufactured by Sifang Rolling Stock Companyand in service in China. The overall layout of the vehicle model is shown in Figure 14 andconsists of four wheelsets, two bogie frames, and one carbody. All of the six DOF of thecarbody and bogie frame and the five DOFs of the wheelset except rotation ω are included inthe model. The primary and secondary suspension forces of the vehicle are also included inthe model. The parameters of the vehicle model are listed in Table 1.
Considering two-point contact on each wheel with wheel–rail forces illustrated in Figure 15,equations of motion for the five DOFs of the wheelset can be described as
Longitudinal movement : mwxw = Fxr1 + Fxr2 + Fxl1 + Fxl2 + Fpxl + Fpxr, (21a)
l
F
F
xo
u
V
FFF
FFF
r
r
wr4 wr3 wr2 wr1
rs1 rs2 rsnrsi
Figure 13. Track model.
Dow
nloa
ded
by [
Bei
jing
Jiao
tong
Uni
vers
ity]
at 2
1:20
27
Sept
embe
r 20
11
Vehicle System Dynamics 1545
1920
35 2731 2336322428 26 22 30 34 2129 25331718
13,1415,16 01,921,11
7,8 5,6 3,4 1,2
ballaster
Roadbed
kp pc
bk cb
kr rc
1–8 Wheel/rail force 9–16 Primary suspension force17–20 Secondary spring force 21–24 Lateral damper force25–28 Lateral damper force 29–32 Lateral stoper force33–36 Tract force
cw
kw
sleeper
Figure 14. Vehicle–track model.
Table 1. Parameters of the high-speed EMU.
Item Notation Value
Mass of the carbody Mc 31,600 kgMass moment of inertia of the carbody around the x-axis Icx 102,384 kg m2
Mass moment of inertia of the carbody around the y-axis Icy 1,548,400 kg m2
Mass moment of inertia of the carbody around the z-axis Icz 1,335,100 kg m2
Mass of bogie frame Mb 3200 kgMass moment of inertia of bogie frame around the x-axis Ibx 2592 kg m2
Mass moment of inertia of bogie frame around the y-axis Iby 1752 kg m2
Mass moment of inertia of bogie frame around the z-axis Ibz 3204 kg m2
Mass of the wheelset Mw 1900 kgMass moment of inertia of the wheelset around the x-axis Iwx 798 kg m2
Mass moment of inertia of the wheelset around the y-axis Iwy 160 kg m2
Mass moment of inertia of the wheelset around the z-axis Iwz 980 kg m2
Half of the longitudinal distance between the centre ofgravity of carbody and rear bogie
lc 8.75 m
Half of the wheelbase lw 1.25 mHalf of the transverse distance between the axle springs lp 1.0 mHalf of the transverse distance between air springs ls 1.23 mTransverse distance of the anti-hunting damper lh 2.7 mHeight of the centre of gravity of the carbody hc 1.52 mHeight of the centre of gravity of bogie frame hb 0.51 mStiffness of the vertical primary suspension spring Kpz 1.18 MN/mDamping of the vertical primary damper Cpz 196 kN s/mStiffness of lateral primary suspension Kpy 6.5 MN/mStiffness of longitudinal primary suspension Kpx 14.7 MN/mStiffness of vertical second suspension Ksz 1.15 MN/mStiffness of lateral second suspension Ksy 0.174 MN/mDamping of the lateral damper Csy 294 kN s/mDamping of the anti-hunting damper Csx 2450 kN s/mNominal radius of the wheel R0 0.43 m
Lateral movement : mwyw = Fyr1 + Fyr2 + Fyl1 + Fyl2 + Fpyl + Fpyr + Mwg(θd − θw).
(21b)
Vertical movement : mwzw = Fpzl + Fpzr − Fzl1 − Fzr2 − Fzl1 − Fzr2 + Mwg. (21c)
Dow
nloa
ded
by [
Bei
jing
Jiao
tong
Uni
vers
ity]
at 2
1:20
27
Sept
embe
r 20
11
1546 Z. Ren et al.
Roll movement:
Iwx
(θw + d2θse
dt2
)= Iwyω
(ψw − V
Rc
)− Fzr1lr1 − Fzr2lr2 + Fzl1ll1 + Fzl2ll2
− (Fpzr − Fpzl)lp − Fyr1rr1 − Fyr2rr2 − Fyl1rl1 − Fyl2rl2
+ Mxl1 + Mxl2 + Mxr1 + Mxr2. (21d)
Yaw movement:
Iwz
[ψw − V
(1
Rc
)]= −Iwyω
(θw + dθse
dt
)+ (Fpxl − Fpxr)lp + Mzl1 + Mzl2
+ Mzr1 + Mzr2 + Fxl1ll1 + Fxl2ll2 − Fxr1lr1 − Fxr2lr2
+ (Fyl1ll1 + Fyl2ll2 − Fyr1lr1 − Fyr2lr2)ψw, (21e)
where Mxl1,2, Mxr1,2, Mzl1,2 and Mzr1,2 are the longitudinal and vertical creep moment compo-nents at each contact patch determined by Kalker’s nonlinear creep theory [17], ω the nominalrotation speed of the wheelset, θse the track superelevation angle, Rc the radius of the track,θd the track cant superelevation angle when the vehicle passes the curve with speed V , ll1, ll2,lr1, and ll2 the lateral distances between the contact points and the centre of the wheelset, andrl1, rl2, rr1, and rr2 the rolling radii at the contact points as shown in Figure 15.
The components of contact forces on each contact patch are determined by the normal force,creep forces, and the other parameters. For example, the component forces of the first contactpatch on the right wheel can be described as Equation (22)
(Fxr1, Fyr1, Fzr1) = f (Nr1, δr1, θ, ψ, σ, E, μ, ξ1, ξ2, ϕ), (22)
where f denotes a function, Nr1 the normal force, δr1 the wheel–rail contact angle, θ the rollangle, ψ the yaw angle, σ Poisson’s ratio, E Young’s modulus, μ the friction coefficient, andξ1, ξ2 and φ the longitudinal, lateral and spin creepage, respectively.
5.3. Calculation of wheel–rail forces
In Equation (22), there are nine unknown variables: four normal forces, Nl1, Nl2, Nr1, and Nr2,and five accelerations, xw, yw, zw, θw, and ψw. The normal force for each wheel–rail contactpatch is calculated by introducing the wheel–rail elastic penetration that is determined by the
l r1l
rl1 r1r rr2
r2l
xr1FFyr1
Fxr2
Fzr1 zr2F
yr2F
Fxl1 xl2F
yl1FFyl2
Fzl1zl2F
pxrF
Fpyr
pzrFpylF
Fpxl
Fpzl
l2r
lp2l
Mwg
l1
l2
Figure 15. Wheelset vibration model.
Dow
nloa
ded
by [
Bei
jing
Jiao
tong
Uni
vers
ity]
at 2
1:20
27
Sept
embe
r 20
11
Vehicle System Dynamics 1547
vertical and lateral displacements, roll and yaw angles of the wheelset, the wheel–rail contactangles, the lateral and vertical displacement of rails, and the interpolation distances betweenwheel and rail profiles.
Given δ(N1) and δ(N2) are the penetrations for two contact patches of a wheel, the normalforce N1 or N2 on each patch is calculated by
N1 =(
δ(N1)
δ(1)
)3/2
, N2 =(
δ(N2)
δ(1)
)3/2
, (23)
where δ(1) is the penetration caused by unit force, and it is determined by [6]
δ(1) = 3(1 − σ)2
πEa
∫ π/2
0
dϕ√1 − (1 − λ2) sin2 ϕ
, λ = a
b, (24)
where a and b are the semi-axes of the contact ellipse which is determined with the Hertzcontact theory here.
The creep forces on each contact patch are determined with the Johnson–Vermeulen andShen’s creep theory [17] as given by
FR =
⎧⎪⎨⎪⎩
μN
[F ′
R
μN− 1
3
(F ′
R
μN
)2
+ 1
27
(F ′
R
μN
)3]
for F ′R ≤ 3μN,
μN for F ′R > 3μN,
(25)
where F ′R =
√F 2
kx + F 2ky , μ is the coefficient of friction, N the wheel–rail normal force, and
Fkx and Fky the creep forces determined with linear Kalker rolling contact theory.The different components of the resultant creep force are then calculated as
Fx = FR
F ′R
Fkx,
Fy = FR
F ′R
Fky.
(26)
Finally, the different contact force components of each contact patches are projected alongthe common local track coordinate system and summed over the different contacts.
5.4. Validation of curving simulation results
Simulations were carried out for a small radius curve (R = 400 m) and vehicle speeds of 80 and100 km/h. The length of the track was 300 m and each curving simulation took no more than12 min on an Intel 2.8 GHz CPU computer to complete time integration with online wheel–rail contact calculation. It was found that the two-point contact occurs for a vehicle speedof 100 km/h with one contact on the flange and the other on the flange root. For the case ofvehicle speed of 80 km/h, flange contact did not occur. The results are compared with thoseobtained with the single-point contact method.
For a speed of 100 km/h, the results for normal forces of the contact points are shown inFigure 16 and locations of the contact points on the wheel profile are shown in Figure 17. Itis seen that the two-point contact occurs on the left wheel tread when the lateral displacementof the wheelset is 1.0 mm approximately and this agrees with the static contact geometriccalculation. When the wheelset lateral displacement is 10.2 mm approximately, the two-point
Dow
nloa
ded
by [
Bei
jing
Jiao
tong
Uni
vers
ity]
at 2
1:20
27
Sept
embe
r 20
11
1548 Z. Ren et al.
0 50 100 150 200
0
20
40
60
80
100
Second contact force on left
First contact force on the left
Whe
else
t lat
eral
dis
plac
emen
t (m
m)
Whe
el c
onta
ct n
orm
al f
orce
(kN
)
Track length (m)
First contact force on the right
Second contact force on the right0
3
6
9
12
15
Wheelset lateral displacement
Figure 16. Normal forces on two contact points.
0 20 40 60 80 100 120–70
–35
0
35
70
Con
tact
poi
nt lo
catio
n on
whe
el p
rofi
le (
mm
)
Track length (m)
Wid
th (
mm
)
Height (mm)
Wheel profile
0 50 100 150 200
-70
-35
0
35
70
First contact on the left Second contact on the left
Second contact on the right
First contact on the right
Figure 17. Two contact points on wheel profile.
contact also occurs on the right wheel profile: one is on the flange root and the other is on theflange. Additionally, the force on the flange is seen to be smaller than that on the flange root.This means that the load on the right wheel is mainly supported by the flange root contactpatch in the curve negotiation.
With the single-point contact method, results of the normal forces and the locations of thecontact points on the wheel profile are shown in Figures 18 and 19, respectively. For a lateraldisplacement of the wheelset of approximately 1.0 mm, the contact point jumps across the wornzone and the normal force fluctuates slightly because the contact angle at this zone is very small.However, the normal force at the right wheel fluctuates severely when the lateral displacementof the wheelset is over 10.2 mm because the location of the contact point jumps betweenthe flange root and the flange. As a result, the wheel–rail impact occurs and unrealistic highnormal forces are obtained. Compared with the results from the multi-point contact method,the results obtained from the single-point contact method are inaccurate when flange contactoccurs. Therefore, a multi-point contact method should be used for curving simulations fortight curves.
Dow
nloa
ded
by [
Bei
jing
Jiao
tong
Uni
vers
ity]
at 2
1:20
27
Sept
embe
r 20
11
Vehicle System Dynamics 1549
0 50 100 150 200
40
80
120
160
Whe
else
t lat
eral
dis
plac
emen
t (m
m)
Con
tact
nor
mal
for
ce (
kN)
Track length (m)
Left force Right force
–6
0
6
12
18
Lateral displacement
Figure 18. Normal forces on contact points (single-point contact method).
0 20 40 60 80 100 120–70
–35
0
35
70
Track length (mm)
Con
tact
poi
nt lo
catio
n (m
m)
Wid
th (
mm
)
Height (mm)
0 50 100 150 200
–70
35
0
35
70
Left Right
Wheel profile
Figure 19. Contact point on wheel profile.
Figure 20 shows the simulations of normal forces at the contact patch obtained with themulti-point contact method for the speed of 80 km/h. In this case, a two-point contact isseen again on the tread of the left wheel profile when the wheelset lateral displacement isapproximately 1.0 mm but there is only one contact on the right wheel profile when thevehicle runs over the curve. This is probably due to the lower running speed compared withthe case discussed above. The comparison between the total normal forces of the two contactpoints from the multi-point contact method and the normal force from the single-point contactmethod is shown in Figure 21 and the results from the two methods are almost same atthis speed.
From the results above, it has been demonstrated that the multi-point contact method pro-duced more reliable results than those obtained with the single-point contact method if flangecontact occurs and equivalent results if flange contact does not occur. The new method istherefore efficient enough to be used online in vehicle dynamics simulations where multi-pointcontact problems are important.
Dow
nloa
ded
by [
Bei
jing
Jiao
tong
Uni
vers
ity]
at 2
1:20
27
Sept
embe
r 20
11
1550 Z. Ren et al.
0 50 100 150 200
0
20
40
60
80
Second contact force on left
First contact force on left
Whe
else
t lat
eral
dis
plac
emen
t (m
m)
Nor
mal
for
ce (
kN)
Track length (m)
0
3
6
9
12First contact force on right
Second contact force on right
Wheelset lateral displacement
Figure 20. Normal force on two contact points (80 km/h).
0 50 100 150 20052
56
60
64
68
72Right-multi-contact method
Left-multi-contact methodLeft-single contact method
Nor
mal
for
ce (
kN)
Track length (m)
Right-single contact method
Figure 21. Comparison of normal forces with two methods.
6. Conclusions
A new method to determine the multi-point contact based on wheel–rail geometric constraintand elastic penetration has been presented in this paper. The main advantage of this method isthat it is implemented in three dimensions including the effect of the yaw of the wheelset andis fast to be used in vehicle dynamics simulations. The method has been validated with staticcontact geometry calculations and online calculations in a coupled vehicle–track dynamicmodel for curving simulations in tight curves.
In the static contact geometry calculations, it was found that the two-point contact occurs forthe new wheel profile only when the yaw angle is sufficiently large and, for worn profiles, thetwo-point contact may occur more frequently depending on the lateral displacement and thewheel load conditions of the wheelset. The new method can therefore be used in the processof the wheel profile design to check the wheel–rail contact condition.
Dynamic simulations have demonstrated that the new method is fast enough to be imple-mented online in a time-stepping integration of vehicle–track system dynamics and is effectiveat handling the multi-point contact. Comparisons have been made between the method ofthe single-point contact and multi-point contact for the curving simulations and the results
Dow
nloa
ded
by [
Bei
jing
Jiao
tong
Uni
vers
ity]
at 2
1:20
27
Sept
embe
r 20
11
Vehicle System Dynamics 1551
have shown that a single-point contact model may predict unrealistically high wheel-rail impactforces due to jumps of the contact point where a more reliable result with smooth contact forcesbetween two contact points may be obtained with the multi-point contact model.
Finally, the multi-point contact method developed in this paper can be further extended forthe use in simulations of turnout systems to deal with situations with more than two contactpoints [4–5,16,18].
Acknowledgements
This work was carried out during a visit by the first author to Manchester Metropolitan University supported by theNational Natural Science Foundation of China (50875019).
References
[1] J. Piotrowski and H. Chollet, Wheel–rail contact models for vehicle system dynamics including multi-pointcontact, Veh. Syst. Dyn. 43(6–7) (2005), pp. 455–483.
[2] J.P. Pascal and G. Sauvage, New Method for Reducing the Multi-contact Wheel–Rail Problem to One EquivalentRigid Contact Patch, Proceeding of 12th IAVSD Symposium, Lyon, 26–30 August 1991.
[3] J.P. Pascal, About multi Hertzian contact hypothesis and equivalent conicity in the case of S1002 and UIC60analytical wheel-rail profile, Veh. Syst. Dyn. 22(2) (1993), pp. 57–78.
[4] R. Schmid, K. Endlicher, and P. Lugner, Computer-simulation of the dynamical behavior of a railway-bogiepassing a switch, Veh. Syst. Dyn. 23 (1994), pp. 481–499.
[5] G. Schupp, C. Weidemann, and L. Mauer, Modelling the contact between wheel and rail within multi-bodysystem simulation, Veh. Syst. Dyn. 41(5) (2004), pp. 349–364.
[6] J.P. Pascal and G. Sauvage, The available methods to calculate the wheel–rail forces in non-Hertzian contactpatches and rail damaging, Veh. Syst. Dyn. 22 (1993), pp. 263–275.
[7] J.A. Elkins, Prediction of wheel–rail interaction: The state of the art. The dynamics of vehicle system on roads andon tracks, Proceedings of the 12th IAVSD Symposium, Lyon, France, August 1991, Veh. Syst. Dyn. 20(Suppl.)(1992), pp. 1–27.
[8] J.B. Ayasse and H. Chollet, Determination of the wheel rail contact patch for semi-Hertzian conditions, Veh.Syst. Dyn. 43(3) (2005), pp. 157–170.
[9] J. Piotrowski and W. Kik, A simplified model of wheel-rail contact mechanics for non-Hertzian problems andits application in rail vehicle dynamic simulations, Veh. Syst. Dyn. 46(1–2) (2008), pp. 27–48.
[10] J. Pombo and J. Ambrosio, A computational efficient general wheel–rail contact detection method, J. Mech. Sci.Technol. 19(1) (2005), pp. 411–421.
[11] Z. Ren, S. Sun, and G. Xie, A method to determine the two point contact zone and transfer of wheel–rail forcesin a turnout, Veh. Syst. Dyn. 48(10) (2010), pp. 1115–1133.
[12] J. Santamaria, E.G. Vadillo and J. Gomez, A comprehensive method for the elastic calculation of the two-pointwheel–rail contact, Veh. Syst. Dyn. 44(Suppl.) (2006), pp. 240–250.
[13] K. Wang, Calculation of wheel–rail contact parameters [J], J. Southwest Jiaotong Univ. 19(1) (1984), P88–99(in Chinese).
[14] W. Zhai, Vehicle-Railway Coupling Dynamics and Application [M], China Science Publishing House, Beijing,2007 (in Chinese).
[15] W. Zhai, K. Wang and C. Cai, Fundamentals of vehicle–track coupled dynamics, Veh. Syst. Dyn. 47(11) (2009),pp. 1349–1376.
[16] Z. Ren, S. Sun, and W. Zhai, Study on lateral dynamics characteristics of vehicle-turnout system, Veh. Syst.Dyn. 43(4) (2005), pp. 285–303.
[17] Z.Y. Shen, J.K. Hedrick, and J.A. Elkins, A Comparison of Alternative Creep Force Models for Rail VehicleDynamics Analysis, Proceedings of the 8th IAVSD Symposium, Cambridge, MA, Swets & Zeitlinger, Lisse,August, 1983, pp. 591–605.
[18] E. Kassa, C. Andersson, and J. Nielson, Simulation of dynamics interaction between train and railway turnout,Veh. Syst. Dyn. 44(3) (2006), pp. 247–258.
Dow
nloa
ded
by [
Bei
jing
Jiao
tong
Uni
vers
ity]
at 2
1:20
27
Sept
embe
r 20
11
