![Spikes in the computed pressures p n due to facets in the undeformed distance function. Picture thanks to K. Oldknow [pers.comm].](https://www.researchgate.net/profile/Edwin-Vollebregt/publication/346492412/figure/fig4/AS:964941074939904@1607071288149/Spikes-in-the-computed-pressures-p-n-due-to-facets-in-the-undeformed-distance-function_Q320.jpg)
![Left: track viewed in world-coordinates (not used in CONTACT), with track inclination (elevation) angle φ (adapted from [31]). Right: definition of the track coordinate system, at the centre of the plane resting on the (inclined) rails in initial (design) configuration.](https://www.researchgate.net/profile/Edwin-Vollebregt/publication/346492412/figure/fig5/AS:964941074927618@1607071288187/Left-track-viewed-in-world-coordinates-not-used-in-CONTACT-with-track-inclination_Q320.jpg)
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Vehicle System Dynamics
International Journal of Vehicle Mechanics and Mobility
ISSN: (Print) (Online) Journal homepage: https://www.tandfonline.com/loi/nvsd20
Detailed wheel/rail geometry processing using the
planar contact approach
E. A. H. Vollebregt
To cite this article: E. A. H. Vollebregt (2020): Detailed wheel/rail geometry processing using the
planar contact approach, Vehicle System Dynamics, DOI: 10.1080/00423114.2020.1853180
To link to this article: https://doi.org/10.1080/00423114.2020.1853180
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VEHICLE SYSTEM DYNAMICS
https://doi.org/10.1080/00423114.2020.1853180
Detailed wheel/rail geometry processing using the planar
contact approach
E. A. H. Vollebregt
Vtech CMCC, Rotterdam, The Netherlands
ABSTRACT
The use of detailed wheel/rail contact models has long been frus-
trated by the complicated preparations needed, to analyse the pro-
files for the local geometry and creep situation for the planar con-
tact approach. A new software module is presented for this that
automates the calculations in a generic way. Building on many com-
ponents developed by others, this greatly simplifies the running of
CONTACT for generic wheel/rail contact situations. Fully 3-d con-
tact search algorithms are implemented. This uses the contact locus
approach, that’s simplified for wheel-on-rail situations and extended
to wheel-on-roller contacts. A main characteristic of the new mod-
ule is its extensive use of the multibody formalism, using markers
to represent coordinate systems, using a newly designed, generic,
object oriented-like software foundation. The contact geometry is
analysed twice; first for the location of contact patches, and then
for the local geometry of the contact patches. The contact search
starts from profiles in their actual, overlapping positions. This yields
the extent of interpenetration areas as needed for the potential con-
tact definition. Different strategies may be employed for the tangent
plane needed in the planar contact method. Creepages are formed
automatically using rigid body kinematics, including wheel and track
flexible bending. Numerical results illustrate the viability, generality,
and robustness of the approach. The extension to conformal contacts
is presented in an accompanying paper.
ARTICLE HISTORY
Received 1 May 2020
Revised 4 October 2020
Accepted 9 November 2020
KEYWORDS
Wheel-rail contact; wheel-rail
profile; contact geometry;
contact forces; planar
contact; software design
1. Introduction
Wheel/rail contact plays an essential role in rail vehicle systems, carrying the vehicle load,
guiding the vehicle along the track curve, and allowing for traction and braking. Large
loads are carried on small contact patches, such that high stresses arise, with consequent
challenges for the material and geometrical design. As such, wheel/rail contact is a topic
of continued research, from the perspectives of rail vehicle dynamics [1–3], and wear and
rolling contact fatigue [4,5].
The CONTACT model is currently considered as ‘the golden standard’ for wheel/rail
contact evaluation [6,7]. It was developed in the 1970-ies and 80-ies by professor Kalker
[8–12]. The model is used at many places where detailed contact investigations are wanted,
CONTACT E. A. H. Vollebregt edwin.vollebregt@cmcc.nl
© 2020 The Author(s). Published by Informa UK Limited, trading as Taylor& Francis Group
This is an Open Access article distributed under the terms of the Creative Commons Attribution-NonCommercial-NoDerivatives License
(http://creativecommons.org/licenses/by-nc-nd/4.0/), which permits non-commercial re-use, distribution, and reproduction in anymedium,
provided the original work is properly cited, and is not altered,transformed, or built upon in any way.
2E. A. H. VOLLEBREGT
e.g. [13], and several researchers programmed their own version of Kalker’s equations, e.g.
[14–18]. The original software was acquired by VORtech BV in the year 2000, at Kalker’s
retirement from Delft University of Technology. VORtech extended the programme with
respect to the speed of computations [19–21] and the physics involved, including the
eects of interfacial layers [22–24] and conformal contact situations [25,26]. Free and com-
mercial versions have been provided since 2009 with annual updates. The software was
recently transferred to the company Vtech CMCC that now continues this development
and distribution [27].
The use of the CONTACT software has long been complicated, due to a variety of
compounding factors. A rst one is the complexity of rolling contact mechanics itself,
using concepts such as penetration and creepage that may be dicult to understand and
relatetoone’smainobjectofstudy,andbecauseofalotofmathematicsisinvolvedin
the solution. A second reason comes from the focus of the original CONTACT software,
concentrating on the contact mechanics side of the problem, rather than the embedding
inthecontextofvehiclesystemdynamics.Thisplacedtheburdenontheusertodenea
suited local coordinate system and perform the corresponding preparations, providing the
undeformed distance and rigid slip functions as inputs to CONTACT. This restricted the
possible application of the CONTACT software.
Many works have been published on the incorporation of wheel/rail contact in vehicle
dynamics, e.g. [28–31], and on the subsequent contact geometry problem [32–41]. How-
ever, these works don’t deliver completely workable solutions as needed by CONTACT.
Besides the location of contact patches, this further needs suitable choices for the tangent
plane and coordinate origin [42], estimates for the extent of the contact patches [33], a suit-
able grid denition, and the calculation of the undeformed distance function. No general
solution has been published to date for the automated running of CONTACT for generic
wheel/railcontactsituations.Asaresult,researchersmayputCONTACTasideasbeingtoo
complicated, or waste time on the re-development of prole smoothing and interpolation
by themselves, or settle for short-cuts, ignoring the eects of the yaw angle, for instance,
that may reduce condence in the results.
This paper presents the software module for automated wheel/rail geometry processing
that has been implemented in CONTACT. This greatly simplies the use of CONTACT for
wheel/rail contact investigation. Its design provides for detailed analyses in a generic way,
allows for batch operation, and for the on-line integration in vehicle dynamics simulation.
Thetechniquesthatareusedmostlybuildupontheworksofothersbeforeus:thecited
references [32,34–36,40], the multibody formalism as presented by Shabana [43], and our
fruitful interactions with SIMPACK, RecurDyn, and other providers of vehicle dynamics
software. The main contribution lies in the combination of these ideas, on the basis of an
extended vocabulary for working with markers, coordinates, and coordinate transforma-
tions, and the corresponding software foundation. A further novelty concerns the contact
locus approach, that is simplied for wheel-on-track situations and extended to wheel-on-
roller congurations. A nal benet of the approach that is presented lies in its extension
toconformalcontactsituations.Thisisaddressedinaseparatepaper[44].
This paper is structured as follows. Section 2describes the software foundation, intro-
ducing the main concepts for coordinates and coordinate transformations. Section 3gives
an overview of software design issues and the approach followed for contact geometry pro-
cessing, and elaborates on the main steps that are used. The extension of the contact locus
VEHICLE SYSTEM DYNAMICS 3
approach is presented in Section 4. After this, Section 5shows results that illustrate the
eectiveness of the approach. Conclusions and discussion are presented in Section 6.
2. Coordinates and transformations
A generic solution is constructed for the wheel/rail contact geometry problem. This is
founded on the multibody formalism that may be considered well-known, e.g. [43], but
for which little guidance can be found on its software implementation. Here we present
the notions of markers, grids, splines and interpolations, and the corresponding software
objectsusedinCONTACT.Thisprovidesasoftwarefoundation,uponwhichthefurther
algorithms are built in Sections 3and 4.
The software foundation was constructed in a long and tedious process, were we had to
step back and make changes multiple times. For instance, we rst used dimensions, posi-
tions and orientations at many places in the programme, as seems to be common in the
algorithms for contact mechanics [39–41,45,46], but this created an unmanageable situ-
ation. This is addressed in the nal design using markers, localising the specication of
positions to a single routine, and isolating and shielding the formulas used for rotations.
Adoptingthemultibodyframeworkappearedtobeagamechangerinthisprocess.Another
good choice is to treat proles, surfaces and grids in the same way, facilitating the easy
conversion between them.
2.1. Vectors and coordinate systems
One aspect of multibody formulations that’s important for CONTACT concerns the
various types of coordinates. One and the same material point on a body that occu-
pies a well-dened position in three-dimensional space may be designated by dierent
coordinatevalues.ThisisillustratedbythetipofashingrodTin Figure 1.
Figure 1. Illustration of ‘global’ versus ‘local’ coordinates. Note that the point T is expressed differently
in terms of the blue and black coordinate systems, resulting in different coordinate values.
4E. A. H. VOLLEBREGT
Mathematically, there are vectors from the two origins shown to the point T,expressed
in terms of the base vectors i,j,koftherespectivecoordinatesystems:
pT(g)=pT=[2.5 +√3, 2.5, 0]T=(2.5 +√3)ig+2.5jg+0kg, (1a)
pT(l)=¯
pT=[2,0,0]
T=2il+0jl+0kl. (1b)
Linear algebra allows to speak of abstract vectors, irrespective of the coordinate representa-
tion,butthisisn’tveryhelpfulforthesoftwareimplementation.Thereweconsiderconcrete
vectors that are connected to (‘expressed in’, ‘using’) a coordinate system.
In an object oriented approach, coordinate systems aren’t very lively objects. They are
like sheets of squared paper, providing means for measuring distances and orientations.
•Acoordinate system consists of a name, an origin O=[0,0,0]
T, unit vectors
i=[1,0,0]
T,j=[0,1,0]
T,k=[0,0,1]
T, and a length scale.
Design choices are to restrict attention to 3-dimensional space, using right-handed
coordinate systems and [mm] as the unit of length where possible. Cartesian coordinates,
cylindrical coordinates and (generalised) curvilinear coordinates are used. Left-handed
coordinates are needed sometimes when using mirroring between left and right wheels, to
consider both sides with a single software implementation.
•Avector (3-vector)isatripletv=[v1,v2,v3]Tassociated with the three coordinate axes
of a coordinate system: v=v1i+v2j+v3k.
In the software, the values viare unrestricted (∈R) in cartesian coordinate systems. In
cylindrical coordinates (v=[r,θ,v3]T,v3=x,yor z), r∈R+
0and θ∈[0, 2π).
Vectorscanbesetandread,scaled,andcombinedindierentwayslikeaddingandcom-
puting the dot and cross products. Further, they can be transformed from one coordinate
system to another, using the mechanisms that are described below.
2.2. Notations for the coordinate reference
Thedierentcoordinatesystemsprovidepowerfulmeansforpreparingtheinputsto
CONTACT. They allow to prepare the wheel surface easily in wheel prole coordinates,
transform to track coordinates for the 3-d contact search procedure including yaw, then
transform further to contact local coordinates, describing the contact problems in normal
and tangential directions. However, algorithms will fail if coordinate values are entered in
one form where another form is expected. Therefore it’s of paramount importance to be
clear at all times on the reference system that is used. This is done in dierent ways.
•The coordinate reference is dropped at places where only a single coordinate system
is relevant. For instance, when dealing with the internals of the CONTACT model,
we simply use [x,y,z]Tand [px,py,pz]T, according to the local contact coordinate
system.
•At places where two coordinate systems are used, Shabana’s notation is adopted [43],
distinguishing between ‘global’ and ‘local’ coordinates, writing [¯
x,¯
y,¯
z]Tto indicate a
VEHICLE SYSTEM DYNAMICS 5
vector in local coordinates. This has a clear meaning without introducing confusion
withothersub-orsuperscriptsthatareinuse.Thesameisachievedusing[x,s,n]Tfor
contact local coordinates while using [x,y,z]Tfor the global track system.
•At places where more coordinate systems are needed, these are indicated by additional
sub- or superscripts such as [xc,yc,zc]Tfor contact local coordinates or [xw,yw,zw]Tfor
wheel prole coordinates.
•The reference to the coordinate system is put in parentheses when other subscripts are
used as well, e.g. mcp(tr)for a point mcp expressed in track coordinates (tr).
2.3. Rotation matrices
Objects may have an orientation, such as the shing rod in Figure 1. This orientation is
expressed with respect to a coordinate system. For a person, for instance, the natural ori-
entation may be to look into the direction of the positive x-axis, with the positive y-axis
pointingtotheleftsideandpositivez-axis pointing upwards.
•Arotation matrix isa3×3matrixRthat describes orientation of an object through the
direction cosines [43]. The columns of Rmaybeviewedasthenaturalvectorsi,j,kof
the object, expressed in terms of the coordinate system used.
Rotation matrices are orthonormal, that is, each column has unit norm and is orthogonal
to the other columns, such that RTR=I.TheinverseofarotationRis thus described by
the transpose matrix RT.
Rotation matrices are used as the primary means of storing orientations in CONTACT,
because they allow coordinate transformations to be implemented directly. The formulas
by which this works depend on ones conventions:
•We use column vectors throughout and apply rotations by pre-multiplication,
•using right-handed coordinate systems,
•using the right-hand rule where counter-clockwise rotation is positive when looking
from the positive side of an axis down to the origin.
Rotation matrices can be constructed from basic rotations, multiplied, transposed, and
appliedtoavector.
2.4. Euler angles
Euler angles are used in the input to CONTACT, particularly for the orientation of the
wheelset with respect to the track.
•Euler angles are triplets {φ,ψ,θ}for roll, yaw and pitch, that describe the orientation of
an object (point, coordinate system, body) via three successive rotations starting from
the natural orientation.
Euler angles can be used in dierent ways, e.g. using intrinsic or extrinsic rotations, and
in dierent ordering. One of their complications is the existence of singular congurations.
6E. A. H. VOLLEBREGT
Inourworkweadopttheroll–yaw–pitchconventionthroughout.Thisavoidssingularities,
as long as roll and yaw remain limited. Another complication of Euler angles is that they
cannot be time-integrated, which isn’t relevant to our application.
The basic rotations in R3,withroll,pitchandyawanglesφ,θ,ψ,aboutthex-, y-and
z-axes, are described using the matrices
roll : Rx(φ) def
=⎡
⎣
10 0
0cφ−sφ
0sφcφ
⎤
⎦=⎡
⎣
10 0
0cos(φ) −sin(φ)
0sin(φ) cos(φ)
⎤
⎦, (2a)
pitch : Ry(θ ) def
=⎡
⎣
cθ0sθ
010
−sθ0cθ
⎤
⎦=⎡
⎣
cos(θ) 0sin(θ)
010
−sin(θ) 0cos(θ )
⎤
⎦, (2b)
yaw : Rz(ψ) def
=⎡
⎣
cψ−sψ0
sψcψ0
001
⎤
⎦=⎡
⎣
cos(ψ) −sin(ψ) 0
sin(ψ) cos(ψ) 0
001
⎤
⎦. (2c)
Using the roll–yaw–pitch convention φ–ψ–θ,rotationsarecarriedoutrstaboutthex-
axis, then about the new z-axis, and thirdly about the nal y-axis. Such intrinsic rotations,
about successive intermediate axes, are obtained by multiplying the basic rotations on
original axes in reverse order. The nal rotation matrix is then obtained as
R(φ–ψ–θ) =Ry (θ )Rz(ψ)Rx(φ) =Rx(φ)Rz(ψ )Ry(θ )
=⎡
⎣
cψcθ−sψcψsθ
sφsθ+cφsψcθcφcψ−sφcθ+cφsψsθ
−cφsθ+sφsψcθsφcψcφcθ+sφsψsθ
⎤
⎦.
(3)
The value θ=0isrelevantwhenconsideringacoordinatesystemthatisn’taectedby
pitch rotation:
R(φ–ψ) =⎡
⎣
10 0
0cφ−sφ
0sφcφ
⎤
⎦⎡
⎣
cψ−sψ0
sψcψ0
001
⎤
⎦=⎡
⎣
cψ−sψ0
cφsψcφcψ−sφ
sφsψsφcψcφ
⎤
⎦.(4)
For Wang’s method for the contact locus, a matrix shows up using the yaw–roll–pitch
convention:
R(ψ–φ) =⎡
⎣
cψ−sψ0
sψcψ0
001
⎤
⎦⎡
⎣
10 0
0cφ−sφ
0sφcφ
⎤
⎦=⎡
⎣
cψ−sψcφsψsφ
sψcψcφ−cψsφ
0sφcφ
⎤
⎦.(5)
2.5. Coordinate transformations
The combination of a vector with a rotation matrix is called a marker object. These markers
facilitate transformations between dierent coordinate systems.
•Amarker isa‘pointwithorientation’intermsofabasecoordinatesystem:aposition
vector oplus rotation matrix R.
VEHICLE SYSTEM DYNAMICS 7
In a way, markers dene new, local coordinate systems, within the global base system.
The origin of the local system is described using the vector ol(g),fromOgto Ol,interms
of global coordinates. The orientation is dened using a rotation matrix R, of which each
column expresses one of the local base vectors in terms of global coordinates,
Rl(g)=⎡
⎣
|||
il(g)jl(g)kl(g)
|||
⎤
⎦.(6)
This links the two systems together, allowing for the conversion of vectors between them.
Combining equations (1b) and (6) it follows that
‘to global’ : p(g)=ol(g)+Rl(g)p(l),orp=o+R¯
p(7a)
‘to local’ : p(l)=RT
l(g)(p(g)−ol(g)),or¯
p=RT(p−o)(7b)
In the example of Figure 1,wheretheanglefromigto ilis 30◦,
R=⎡
⎣⎡
⎣
0.87
0.5
0
⎤
⎦⎡
⎣
−0.5
0.87
0
⎤
⎦⎡
⎣
0
0
1
⎤
⎦⎤
⎦,o+R¯
p=⎡
⎣
2.5
1.5
0
⎤
⎦+⎡
⎣
1.73
1
0
⎤
⎦=⎡
⎣
4.23
2.5
0
⎤
⎦.(8)
Markers can be constructed and modied in dierent ways, using translation and basic
rotations. They are used to capture (store) the relative positions of the main objects in the
geometry problem, and to transform vectors, grids, and other markers from one coordinate
system to another.
Expressions that relate the local and global velocities of a point pon a body are obtained
by taking the derivative of equation (7a) with respect to time.
˙
p=˙
o+˙
R¯
p+R˙
¯
p,with
˙
R¯
p=ωl×(R¯
p)=R(¯
ωlׯ
p)and R˙
¯
p=R¯v.(9)
This shows how the total velocity vp=˙
pof a point with respect to the global origin may
be decomposed into three contributions:
(1) Rigid body translation, vl=˙
o, of the local system with respect to the global origin;
(2) Rigid body rotation, ωl, of the local system with respect to the global system, at an arm
R¯
pof the point with respect to the local origin;
(3) Flexible body deformation, motion of the point within the local system, ¯vp=˙
¯
p,atthe
current orientation Rof the local system.
Detailed information on the relations between angular velocities ωl,¯
ωland the deriva-
tive ˙
Risgivenin[43, § 2.4–2.5].
2.6. Grids and grid functions
Primary work horses in our implementation are grids and grid functions.
•A1-dgrid(curve,prole) is an ordered list of points {xi},i=1, ...,n1or {xj},
j=1, ...,n2intermsofagivencoordinatesystem.
8E. A. H. VOLLEBREGT
•A2-dgrid(mesh,surface)isastructuredsetofpointsG={xij },i=1, ...,n1,j=
1, ...,n2in terms of the coordinate system used.
1-d grids are implemented in CONTACT using 2-d grids with either n1=1orn2=1.
Further, all xij =[x1,x2,x3]T
ij are points in 3-d space. Grids can be at but don’t have to be,
at grids can be aligned with coordinate directions but may also be tilted.
Rail proles are implemented as planar curves in the Oyz plane, using n1=1, xj≡0,
with n2points in y-direction. Wheel proles are dened using cylindrical coordinates with
xjˆ=θj≡0, zjˆ=rj−rnom,w. Generic conversion routines are provided to convert to cartesian
coordinates,assumingthewheelisabodyofrevolution.Thecontactlocusonthewheelis
a generic space curve, xj=[xj,yj,zj]T,withxj= 0.
A grid is called ‘uniform’ in CONTACT if the coordinates can be expressed as xij =
x0+iδx(same for all j)andyij =y0+jδy(same for all i), using xed steps δx,δy,with
zij =const. (This is a restricted denition of uniformity; other forms of uniformity could
be recognised also.) Non-uniform grids are called ‘cylindrical’ or ‘curvilinear’. Cylindri-
cal grids can be transformed into cartesian coordinates. Cartesian coordinates can be
transformed from one origin and orientation to another, using the to-global and to-local
operations of equations (7a) and (7b).
One main purpose of uniform grids is to dene the ‘contact grid’, in the Oxy-plane,
alignedwithcontact-localx,y-coordinate axes. The contact grid points are associated
with the centres of rectangular elements, with a secondary grid of corners induced at
xij +[±δx/2, ±δy/2, 0]T. The regular structure allows for quick determination of the
elements in which points are located, as needed in interpolations.
•A1-dgridfunctionis a function f:G→R,assigningscalarvaluesfij to each grid point
xij of a grid G.
•A3-dgridfunctionisafunctionf:G→R3, assigning triplets [f1,f2,f3]T
ij to each grid
point xij of a grid G.
Grid functions are used to store all kinds of information like surface heights hij , surface
tractions pij, elastic displacements uij ,ormaskarraysmij, indicating the status of each
point. 3-d grid functions typically contain vectorial values, relative to the coordinate axes
of an underlying coordinate system.
In the implementation, the two indices ij are collapsed into a single index I=i+(j−
1)n1,I=1, ...,ntot =n1n2.
Using a dierent view point, grids may be considered as grid functions themselves. In this
view, a grid’s essential information consists of the indices, G={ij},makingthecoordinates
agridfunctionx:G→R3. This view relates to the concept of index sets as elaborated
in [47].
2.7. Splines and interpolations
Basic 1-d linear interpolations are provided in CONTACT using a generic routine for
non-equidistant input data that may be given in ascending or descending order. Multiple
interfaces are built on top of this routine, for scalar, array, or grid outputs. Similarly, a single
routine is constructed for the computation of interpolation weights for 2-d curvilinear to
uniform grid interpolation, with multiple wrappers for dierent forms of the actual data.
VEHICLE SYSTEM DYNAMICS 9
Smooth interpolation of curves is provided via cubic splines.
•A1-d simple spline for a 1-d tabulated function {ti,yi}approximates the underlying
function y=y(t)with piecewise cubic sections with parameter t.
Simplesplinesarecalled‘interpolating’whenthetabulatedpointsareretrievedat
appropriate parameter values, or ‘smoothing’ if the input points are not contained in
the resulting curve. Cubic splines are stored using the coecients [a0,a1,a2,a3]ifor all
table segments, of polynomial terms t0,t1,t2,t3,respectively,relatedtotheppform of de
Boor [48].
•A1-d parametric spline for a 1-d grid {[xj,yj,zj]T}approximates the underlying space
curve as (x(s),y(s),z(s)) with sthe arc length parameter.
Parametric splines are composed of three simple splines with the same parametrisation.
Parametric spline objects can be computed, copied, trimmed, shifted, rotated, and
evaluated in direct and indirect ways. Direct evaluation gives the prole positions and
inclinations at specied s-positions. Indirect evaluation takes the desired y-values as input,
locates the corresponding s-positions in the spline function, and delivers x,y,zand their
partial derivatives like ∂y/∂sor ∂z/∂ yas the output values.
3. Steps used to solve the contact geometry problem
3.1. Scope of the problem
CONTACTaimstosolvecontactproblemsforawiderangeofcongurations,forvehicle
dynamics as well as stand-alone users, without placing much burden on the user. This
appeared challenging in case of the wheel/rail geometry problem, that arises in many
dierent ways.
•Wheel and rail proles are provided in a wide range of forms, from constant radius arcs
on technical drawings, to measured, worn proles in various formats. These alternative
forms may need dierent kinds of preparations (smoothing, alignment, unit conversion,
positioning in the track), and may need or allow for dierent algorithms for the contact
detection.
•Thepositionofawheelwithrespecttoarailmaybespeciedindierentways,using
an ideal design track or from measured data, with dierent conventions for signs and
angles.
•Stand-alone wheel/rail contact investigation has dierent points of departure than vehi-
cle dynamics simulation. Vehicle dynamics typically needs the contact forces at precisely
specied states of wheel and rail, while stand-alone users typically want to specify cer-
taintotalforcesormoments.Thisaddsadditionalbalanceequationstotheproblem,
typically concerning a whole wheelset instead of a single wheel on a rail.
Consideredasasoftwareproblem,thisgivesaveryinformalandunclearrequire-
ments specication: what should or should not be implemented? How can the problem
10 E. A. H. VOLLEBREGT
be specied by a user in a detailed yet convenient way? A solution is implemented in
CONTACT, that relies on the following design decisions for the software implementation:
•Rigorous choices are made regarding the coordinate systems and the use of the multi-
body formalism for their manipulation;
•Wheelandrailprolesarereadfromawiderangeofformats,andbroughtintocanonical
form before starting the actual calculations;
•The geometry problem is analysed twice; rst for the location of contact patches, and
then for the local geometry of the contact patches;
•Surfaces are constructed as grids in local coordinates, transformed, and evaluated using
interpolation, rather than transforming the equations by which the surfaces are dened.
The steps that are used are shown pictorially in Figure 2,andareelaboratedinthe
following sections.
Figure 2. Main steps used for wheel/rail contact geometry processing, using markers and associated
coordinate systems.
VEHICLE SYSTEM DYNAMICS 11
3.2. Wheel and rail proles
Wheel and rail proles are read from various formats, using a wide range of dierent con-
ventions. They are transformed into a canonical form, showing proles for the right side
of the track, using right-handed coordinates with zpositive downwards (Figure 2(a)).
Up/down and left/right mirroring are used as needed. The proles could further be
scaled, shifted, trimmed, and so on, and smoothing may be applied. A marker is placed
at the zero position in the proles, dening the wheel and rail prole coordinate systems,
and the spline representation is built. These preparations simplify the further processing:
providing information in a consistent way to the actual solvers, reducing the number of
optionsthatmustbeconsideredthere.
3.2.1. Requirements for rail profiles
The variety of rail proles is illustrated in Figure 3.
•Rail proles can be provided before (a) or after applying the rail cant angle (b).
•Theprolecoordinateorigincanbechosenatthetopoftherail(a),atthegaugeface
(c), or anywhere else (d, f).
•Theprolescouldbemeasuredfortherailontheleftsideofthetrack(d)orforthe
right side (b).
•Theprolescouldbemeasuredwithzpositive upwards (e) or downwards (a).
•Lateral positions could be increasing monotonically (a), or could be decreasing at some
points (d, e), creating multi-valued functions.
•Rails could consist of multiple sections, particularly at switches and crossings (f).
•For testing purposes, rails could be dened as circular arcs (g) or planar
sections (h).
•Thesidefaceoftherailcouldbetoosmallforthegaugewidthcomputation(h).
•The biggest width needed for the gauge width computation may occur above the gauge
measuring height (d).
•The highest part of the rail need not be the most relevant for contact analysis (i).
•Proles may contain sharp corners (i), that must be dealt with appropriately in opera-
tions like smoothing and interpolation.
•The proles could be measured in [mm], [m] or [in], etc.
Figure 3. Variety of rail profiles that should be supported.
12 E. A. H. VOLLEBREGT
3.2.2. Requirements for wheel profiles
A variety of wheel proles is shown in Figure 4.
•Proles can be given for right (a) or left wheels (b).
•Theprolecoordinateorigincanbeplacedatthetapecircleline(a,b),attheange
back (c), or anywhere else (d).
•Theprolescanbegivenwithzpositive downwards (a, b, c) or upwards (d).
•A wheel may have one (a–d), two (e), or no anges (f–h).
•For testing purposes, wheels may be dened with a circular cross-section (f) or as a
at/conical section (g).
•Idealised two-dimensional contacts require a logarithmical drop at the sides (h), to avoid
pressure concentration [49,50], [51, p.134].
Wheelprolescanbenarrowerorwiderthantherailprole.Incaseofagroovedrail
prole (Figure 3(i)), a narrow circular wheel (Figure 4(f)) may have its lowest z-values
larger than the lowest zon the rail. In that case, the wheel prole may not be extended by
constant extrapolation.
3.2.3. Canonical form, mirroring
Users want to be able to dene a prole just once and then use this denition on both sides
of a rail vehicle. Likewise, it is benecial for programmers to concentrate on a single geo-
metrical conguration, that could either be for the left or right side of the track, yet avoid
switching between sides. This way, the ange will be found at the same position, angles will
beinaconsistentrange,andsoon,allowingtofamiliariseandmemorisethevalues.Positive
values that should be negative are then spotted much quicker, the same for clockwise and
counter-clockwise rotations. Programming errors are detected and corrected more easily,
and prevented by standardisation.
Dierent conventions are possible for the reference position on a prole (on the side or
at the top of the rail), for the orientation (before or after cant rotation), for the mirroring
(left or right side), and ordering of the points (low-to-high, high-to-low).
•We let the reference position in the rail prole be dened freely by the user. We place
amarker at this reference position and use this to move the rail around in the track
system.
Figure 4. Variety of wheel profiles that should be supported.
VEHICLE SYSTEM DYNAMICS 13
•We focus on the wheel/rail pair on the right side of the track. Problems for the left
side are mirrored in the plane ytr =0. This ips the sign on y-components of positions,
velocities, tractions, and forces. Rotations are aected about the x-andz-axes, aecting
angular orientations and velocities and corresponding moments.
Optional mirroring is provided for cases where left side proles are given on input.
3.3. Prole smoothing and spline interpolation
Measured worn proles may exhibit small grooves and ridges that have a strong inuence
on the contact results. The normal contact problem is particularly sensitive to variations
in the undeformed distance, resulting in irregularly shaped contact patches, composed
of multiple disconnected sections, with consequent variations in the contact pressures.
Such irregularities in the results may be desired in some cases, for detailed studies, but
are typically seen as unwanted eects. Dierent results would be obtained for prole mea-
surements that are taken close together on the same wheel or same rail. The uctuations
will turn out dierently with every other instance of the proles. Prole smoothing may be
used to attenuate the most rapid uctuations.
Prole smoothing can be achieved using cubic smoothing splines, as discussed for
instance in [48,52–54]. A particularly convenient description that we used was given by
Pollock [55]. For a given set of data points {si,yi}and weights σi,i=1···n,thisconstructs
thesplinefunctionY(s)that minimises the value of
L=
n
i=1yi−Y(si)
σi2
+λsn
s1Y(s)2ds(10)
Thersttermdescribesthedeviationfromtheinputdata,thelatterthesmoothnessofthe
spline function. The parameter λconcerns the level of smoothing, with no smoothing at
all for λ=0 and using a linear function Y(s)for λ→∞.
Apossibledrawbackisthatsplinesmoothingtendstoexhibitovershootsincurvature
at places where the input curvature jumps from one to another value. This is illustrated
in Figure 5for the UIC60 prole that consists of 300, 80 and 13 mm radius arcs joined
together. This issue requires some special attention for algorithms that use the curvatures,
forinstancetomakeaHertzianapproximation.
Other algorithms for smoothing are the Savitzki-Golay approach [56]orLowessand
Loesslters.Aninterestingideaistouseoptimisationofanobjectivefunctionthatincludes
curvature, as proposed in the method of ‘minimum curvature variation’ [57].
Another way to approach the smoothing problem is by tting of circular arcs to seg-
ments of the prole data. This is used in the Universal Mechanism software [pers.comm].
This tting is intricate for prole segments that have low curvature, where it’s dicult to
decide whether the curve centre should be moved to the left or the right side of the curve.
This ill-posedness can be avoided by reformulation of the problem, using the curvature
instead of the curve radius as optimisation variable [58], by which low curvature segments
approach 0 from positive and negative sides.
A topic related to smoothing concerns prole interpolation. Even if the input data them-
selves are obtained from a smooth prole, a rough prole may result if interpolation is
14 E. A. H. VOLLEBREGT
Figure 5. Curvature of new UIC60 rail profile obtained from smoothing spline interpolation.
Figure 6. Spikes in the computed pressures pndue to facets in the undeformed distance function.
Picture thanks to K. Oldknow [pers.comm].
used in the wrong way. Linear interpolation creates facets that distort the results if the con-
tact grid has higher sampling than the input prole. This is illustrated in Figure 6for a
case where the proles are given with steps of about 1 mm, using 0.2 mm steps in CON-
TACT. This facetting is avoided using higher order interpolation, as provided by cubic
spline interpolation.
VEHICLE SYSTEM DYNAMICS 15
3.4. Track geometry and wheelset denition
A second step in the preparations concerns the positioning of the wheel and rail proles
with respect to each other (Figure 2(b)). Dierent ways of specication may be used that
each have their own benets in dierent circumstances. Absolute positioning of rails can
be used, with respect to a global coordinate system, especially in the case of a design track
layout with track irregularities and/or rail bending. Alternatively, the relative positions of
therailsmaybeused,usingthegaugewidthmeasuredatsomeheightbelowtheplane
resting on the top of the rails. This is especially practical for measured track data, where
theabsolutepositionmaynotbeaccessedeasily.
The input to CONTACT has been designed to strike a balance between the dierent
cases. This is done with the aid of a virtual ‘wheelset’ and ‘track’, to describe the geometry
intermsfamiliartotheuser,usingtwowheelsandtworails.Thesimulationofatestrigis
alsosupported,withrollersreplacingtherails,aswellasacongurationwithasinglewheel
on a rail. Currently, the two sides are considered as separate wheel/rail contact problems.
CONTACT does not yet provide for the interactions between the two wheels, adjusting the
roll angle for instance, to satisfy an equilibrium equation.
3.4.1. Track geometry
CONTACT does not consider gravity and the corresponding vertical direction. The basis of
working is the track plane, dening its own vertical direction. A marker mtr is introduced
as shown in Figure 7(right). In rail vehicle dynamics, the track origin may be placed at
thepointonthetrackcurveclosesttothewheelsetcentreofmass(CM),seee.g.[36]. This
position is denoted here as sws,withs(s1, longitudinal) the parameter used to describe the
track curve. The xtr direction is aligned with the track centre line, ytr is the lateral direction,
positive to the right, by which ztr is positive downwards.
The right rail prole is placed in the track using the gauge computation, dening a
marker mrr(tr).Thisisinitialisedat{0,I}, rotated by the rail cant angle, if not included
in the prole itself, then shifted upwards and to the right to touch precisely the track plane
and gauge stop as shown in Figure 8(left). The left-most point of the prole is used for
zrr(tr)∈[0, gheight ], as suggested by the gauge-stop in the gure. An alternative choice is to
evaluate the prole at zrr(tr)=gheight . This is the convention used in the VAMPIRE soft-
ware. This aects the positioning of rails as shown in Figure 3(d), where the left-most point
liesabovethegaugemeasuringheight.
Next, track irregularities may be dened that displace the rails with respect to their ini-
tial (design) positions and orientations, see Figure 8(right), see [27, § 4.2] for a further
Figure 7. Left: track viewed in world-coordinates (not used in CONTACT), with track inclination (eleva-
tion) angle φ(adapted from [31]). Right: definition of the track coordinate system, at the centre of the
plane resting on the (inclined) rails in initial (design) configuration.
16 E. A. H. VOLLEBREGT
Figure 8. Left: initial (design) placement of the right rail profile in the track system, with positive
cant angle, using gauge width and gauge measuring height. Right: actual (current) configuration with
rotation φrr and displacements zrr,yrr >0.
specication. These rail irregularities may be static/permanent, but may be due to track
exibility too. In such a case, the corresponding velocities vrr =˙
yrr,...may be specied
also,aectingthecreepcalculation[59,60].
3.4.2. Roller rig configurations
Forthesimulationofrollerrigs,itisassumedthattherolleraxleisxedinaframe,
unable to move except for rotation about its axle. Track coordinates are used largely sim-
ilarly as above for wheelset on track congurations, with slight dierences as indicated in
Figure 9:
•The track plane is aligned with the rollers’ axle, at the nominal roller radius rnom,rabove
the rollers’ centre of mass;
•The prole markers for the rollers are placed directly in the track plane, without
up/down shifting to make the proles touching the track plane. The z-values in the
prole are thus interpreted as variations of the rolling radius with respect to rnom,r.
Figure 9. Definition of track coordinates for the simulation of a roller rig: aligned with the rollers’ axle,
at a distance rnom,rabove the rollers’ centre of mass.
VEHICLE SYSTEM DYNAMICS 17
The gauge computation is used for the lateral positioning of the proles in the initial
(design) conguration. The proles may then be rotated and displaced by deviations for
irregularities and exible bending. This feature may also be used to describe the motion of
the roller rig as a whole, relative to the frame in which it is contained.
3.4.3. Wheelset geometry
The geometry of the wheelset is dened using the ange back distance dng,theyrw location
oftheangebackinwheelprolecoordinates,andthenominalradiusrnom,w(Figure 10,
left). These are used to set the wheel prole marker at xrw =[0, dng /2−ypos,rnom,w]T
with respect to the wheelset marker, in the initial (design) conguration. Flexible wheelset
deviationsmaythenbedenedthatdisplacethewheelprolewithrespecttoitsdesign
position and orientation. Increments may be specied for all six position and orientation
variables to support axle and wheel bending and torsion. The corresponding velocities may
be specied as well.
Next, the wheelset marker is dened, mws, capturing the wheelset position and ori-
entation with respect to the track frame (Figure 10, right). The position of a wheelset is
characterised by the position of its centre of mass along the track curve (sws ), which is
largely irrelevant to CONTACT, and the position and orientation with respect to the track
reference. The orientation is dened with Euler angles in roll–yaw–pitch convention: start-
ingwiththeaxleparalleltotheytr-direction, the wheelset is rolled about its x-axis by φws,
then yawed by ψws about the new z-axis and then pitched by θws about the axle, i.e. the new
y-axis. After this the wheelset is shifted to its position [0, yws,zws −rnom,w]T.
Using the markers mrw and mws with suited local-to-global transformations, these yield
the nal position of the wheel prole in the track frame. The rail prole is positioned sim-
ilarly using the rail prole marker, such that the two proles can be plotted together in a
single coordinate frame. This then provides the basis for the contact geometry problem
(Figure 2(b)).
Figure 10. Left: illustration of wheelset geometry parameters. Right: wheelset (CM) position and orien-
tation with respect to the track system.
18 E. A. H. VOLLEBREGT
3.5. Locating the contact patches
The contact geometry is analysed in track coordinates, using the vertical wheel and rail sur-
face positions, zrr(tr)−zrw(tr)<0. This gives the virtual interpenetration of the proles,
producing the potential extent of the contact patches along the way (Figure 2(c)). The
‘geometrical point of contact’ or ‘initial contact position’ is obtained at the location of max-
imum interpenetration. Multiple contact patches may be detected. Contact patches that lie
close together may be joined, to include interactions between them in the actual contact
solution.
Two dierent methods are implemented for the contact location: a brute-force, grid
based approach, cf. [35], and a rened approach on the basis of the contact locus [28,32].
The grid based approach is easily understood and implemented for generic situations,
includingtheyawofthewheelsetforinstance,andforawheelonaroller.Itcanbegener-
alised further to corrugated rails, wheel ats, and wheels with polygonisation. This method
is discussed further in this section.
The idea of the contact locus approach is to restrict the search to a 1-d curve instead of
a 2-d surface. This is done using some analytical derivations that rely on the smoothness of
the wheel and the rail. This is explained furter in Section 4, where the method is extended
to wheel-on-roller congurations.
Thegridbasedapproachstartsbydeninga‘reasonablearc’onthewheelprolethat
should encompass the contact, typically θ=[−7◦,7
◦] with respect to the lowest point on
the wheel. This arc is divided into a hard-coded, somewhat arbitrary number of n1=231
slices. Considered as a brute-force method, little has been done on optimisation. This is
suggestedasatopicforfurtherresearch.
The wheel prole is revolved around its axle creating a surface in cylindrical coor-
dinates, converted to a cartesian surface in wheel prole coordinates, {xi,yi,zi}(w),and
transformed next to track coordinates. Note that these steps are provided directly by the
software foundation, e.g.
type(t_grid) :: whl_srf ! 1-d grid for wheel profile
type(t_marker) :: m_trk ! marker for wheel reference in track frame
! transform wheel profile from wheel coordinates to track coordinates
call cartgrid_2glob( whl_srf, m_trk )
Auniform‘gapmesh’isthendenedintrackcoordinatesforthewholewidthofthe
rail, using the range x=[min(xi(tr)),max(xi(tr))]obtainedfromthewheelsurface.
The 1-d rail prole is converted to track coordinates and computed at the yjof the gap
mesh using spline evaluation. For wheel–railcontact,thisisthenextrudedinx-direction
to form a 2-d surface; for wheel–roller contact, the prole is revolved around it’s axle and
evaluated at the xjpositions of the gap mesh. The bounding boxes of wheel and rail surfaces
are checked. If there’s no overlap then the user may have specied inappropriate positions
or dimensions, such as the gauge width gwidth or ange-back position ypos,elseoneortwo
proles may need to be mirrored.
The z-values of the wheel surface are then interpolated to the gap mesh using 2-d bilin-
ear interpolation, and the gap is formed as the dierence zr(tr)−zw(tr). All the local minima
of this function are determined and recorded in a list. They are processed one by one, each
time taking the one with largest interpenetration, dening a contact patch, and removing
VEHICLE SYSTEM DYNAMICS 19
all the local minima contained in the patch. Once this list is exhausted, a new list of contact
patches is formed. This may be sorted by lateral position, connected to a previous time-step,
and may further be processed with respect to contact patches that lie close together.
The inputs to these calculations are the wheel and rail at their actual positions, including
yaw, providing a 3-d contact search algorithm. The outputs are zero or more contact patch
objects, and the overall minimum gap and corresponding curvatures that are determined
also. Each contact patch consists of a contact reference marker, and the appropriate sizes for
the potential contact area. The minimum gap value and the curvatures are used if the total
force is prescribed and no contact occurs in the initial conguration. Solving a Hertzian
problem,anappropriatevaluemaythenbeobtainedforthewheelsetverticalposition.
3.6. Contact grid denition
The denition of the contact grid needs information derived from the gap function, and is
therefore implemented together with the location of the contact patches (Figure 2(d)).
The origin of the contact local coordinate system is traditionally placed at the initial
contact position or geometrical point of contact [31,61,62]. Inspired by SIMPACK [38],
we advocated for an alternative choice [42], keeping the origin more centred in asym-
metric geometries. When using the grid based approach, this uses the mid-point of the
interpenetration area, dened in a weighted sense, with the gap values g(x,y)used as
weights.
[xtr,ytr]cwgt1=g<0
[x,y]g(x,y)dxdyg<0
g(x,y)dxdy. (11)
Experimentsreportedin[44] show that quadratic weights work somewhat better, and that
it’s desired to use an averaged surface inclination:
[xtr,ytr,αtr]cwgt2=g<0
[x,y,α]g2(x,y)dxdyg<0
g2(x,y)dxdy. (12)
Here αtr is the surface inclination of the rail prole with respect to the track marker. For
the contact locus approach this averaging is implemented using a quadratic approximation
g(t;y)=c(y)+a(y)t2.Theminimumvaluec(y)is the negative gap value found at the
locus position xlc(y). The coecient a(y)comes from the curvature a(y)=1/2r(y),and
t=x−xlc.Thisisthecurvatureinoverallztr -direction, unaected by the contact angle.
Therailproleisinterpolatedatycwgt2
tr to get the corresponding ztr -value, dening the
contact reference position on the rail. The surface inclination could be obtained from the
spline, and used to set the contact reference angle δcp(tr), the rotation from track zto local
n-direction. A dierent choice is to use αcwgt2
tr , representing the average over the contact,
improving the t to conformal computations in a large set of test-cases [44]. Planar contact
coordinates are then introduced by dening the marker mcp(tr), with origin at the reference
position and orientation using the contact reference angle.
The extent of the interpenetration region is obtained rst in the horizontal (xtr,ytr )
plane using the gap function. The corresponding z-values are obtained from the rail pro-
le.Thedesiredsizeofthepotentialcontactsize[ssta,send] is computed using temporary
markers msta and mend, dened in track coordinates, then transformed into planar con-
tact coordinates. The contact grid is dened in the contact plane, with requested step
20 E. A. H. VOLLEBREGT
sizes δx×δs, with a grid point placed at the contact origin. The number of grid points
in each direction is computed large enough to encompass ssta and send,andsimilarlyfor
the x-direction.
3.7. Combination of contact patches
For two contact patches that lie close together, there are interactions between their respec-
tive surface tractions and elastic deformations. The pressures on one patch typically reduce
the amount of interpenetration found on the other [63], which reduces the pressures
needed to satisfy the contact conditions. These interactions can be included in CONTACT
by solving the patches in one go, using an enlarged potential contact area.
In principle, it’s possible to combine all contact patches together into a single contact
calculation. In the planar contact approach adopted in this paper, this needs a compromise
for the contact reference angle: a single contact plane must be selected that strikes a balance
between the actual contact angles found at dierent locations. This may introduce bigger
approximation errors than when the interactions would be neglected. Therefore, a strategy
is used that combines patches conditionally, on the basis of the mutual distance between
adjacent patches, and on the change in contact angle.
Switching between combination or separation of adjacent patches may cause discon-
tinuities in the numerical results at slight perturbations of input parameters. This is a
concern for the solution of equilibrium equations such as a prescribed vertical force. These
jumps may also introduce spurious, high-frequency oscillations in multibody simulations
thatareunphysical,andthatshouldbepreventedasmuchaspossible[64,65].
A blending approach is introduced to mitigate some of the jumps, using two thresh-
old values dcomb ≤dsep for the distance dbetween their interpenetration areas and a third
threshold αsep for the angle dierence. Two patches are processed separately when |αcp1−
αcp2|≥αsep or when d≥dsep.Else,thepatchesarecombinedfullywhend≤dcomb.In
between, dcomb ≤d≤dsep, the patches are combined with interactions scaled by a factor θ:
θ=dsep −d
dsep −dcomb 0.8
. (13)
This factor θis applied inside CONTACT’s solution algorithms by splitting the pressure
array pninto parts for dierent sub-patches, computing elastic deformations useparately
with inuence coecients A, and combining as follows:
u1,1
u2,1 =Ap1
0,u1,2
u2,2 =A0
p2,u=u1,1 +θu1,2
θu2,1 +u2,2 . (14)
This extends the traditional formula ‘u=Ap’asshowne.g.in[44, § 2.7]. The coecient
0.8 in Equation (13) was found to give a reasonably smooth transition in one case that was
studied in detail. Further research is needed on this, along with suitable choices for dcomb
and dsep in relation to total forces, material parameters and geometry of the problem.
3.8. Undeformed distance calculation
With the contact patches identied in the previous steps, dening markers and grids for
the contact computations, the next step concerns the undeformed distance calculation.
VEHICLE SYSTEM DYNAMICS 21
This is achieved by computing the wheel and rail surfaces in wheel and rail coordinate
systems, then transforming the problem to the local coordinate system (Figure 2(e)). Using
the appropriate software routines, we may sample the wheel prole as a body of revolution,
place the points in the track frame, view the same in terms of contact coordinates, and use
interpolation to get the heights at the contact grid (Figure 2(f)).
The steps used for the undeformed distance look much alike the steps used to determine
the gap function. One dierence is that the undeformed distance is analysed in contact
coordinates, using surface heights nr(cp)−nw(cp), evaluated at the contact grid positions.
A second issue comes from multi-valuedness of the surface heights when the prole is
viewed from a large angle, e.g. at ycp =50 in Figure 2(e). This can happen in the over-
all z-direction as well (Figure 3(d, e)), but can be removed in those cases by restricting the
acceptable proles or discarding points where ‘fold-back’ occurs. When forming the unde-
formed distance, the multi-valuedness is inherent in the problem description. It is resolved
by trimming the proles, using the extent [smin,smax]ofthecontactgrid.
One further dierence with the previous calculation of the gap function, is that the
undeformed distance function is needed with higher precision. This is achieved by evaluat-
ing the wheel and rail prole splines at the sampling density of the contact grid, followed by
bilinear interpolation. Bi-cubic interpolation has been implemented as well, cf. [53, § 3.6],
but the added benets seem low compared to the added computations. Two-dimensional
splines could also be useful, providing bi-cubic interpolation in a dierent way.
The calculation of the wheel surface heights nw(cp)usesaboundingboxasabasisfor
the computation. This is computed using four markers at the corners of the contact grid,
dened in track coordinates, transformed to wheel prole coordinates using to-local oper-
ations. This gives the range of ywvalues of relevance to the contact grid. This range is
mapped to arc-length parameters swon the wheel, divided into steps according to the
contact grid spacing, and used to trim and re-sample the wheel prole as needed for the
undeformed distance calculation. This prole is then revolved around its axle and evalu-
ated at the appropriate range of xwwith sampling δxof the contact grid. From there the
surface is known as {xi,yi,zi}(w)covering the contact grid with sucient resolution, and is
transformed easily into {xi,si,ni}(cp).
3.9. Calculation of the creepages between wheel and rail
A nal step concerns the determination of the creepages between the wheel and the rail.
Thesearedenedasthevelocityofthesurfaceoftheupper(rail)bodyrelativetothatofthe
lower (wheel), evaluated at the origin of the local coordinate system, scaled by the rolling
velocity V. There are three creepages, ξ,η,φ,forthevelocitydierencesinlongitudinaland
lateral directions, x=xc,s=yc, and for the rotation about the normal direction n=zc.
Using Pfor the contact reference point on the rail or roller, and Qfor the corresponding
point on the wheel, the formulas are:
ξ=vx
V=vP
x−vQ
x
V,η=vs
V=vP
s−vQ
s
V,φ=ωn
V=ωP
n−ωQ
n
V. (15)
These creepages approximate the local kinematic motions using a simplied form, w=
[ξ−φs,η+φx]T, with rigid slip function w. This approximation is improved upon in
22 E. A. H. VOLLEBREGT
the conformal approach, by computing velocity dierences at every point of the potential
contact [44].
The velocities vP(cp),ωP(cp)and vQ(cp),ωQ(cp)of Pand Qare obtained using the formulas
that express the velocity of a point on a rigid body, cf. the local to global conversion of
Equation (9). These are worked out for the input velocities of the wheelset and rollers,
including the contributions of wheel and rail exible bending.
3.9.1. Velocity of contact reference point P on a rail
A rail (prole) can move with respect to the track centre due to track exibility. The
corresponding velocities are input as:
vrr(tr)=[0, vy,vz]T,ωrr(tr)=[vφ,0,0]
T. (16)
These are given in track coordinates; the angular velocity ωconcerns a rotation about the
rail marker.
The velocity of the point Pwith respect to the track centre is evaluated with Equation (9),
considering Pasapointonarigidbody.
vP(tr)=vrr(tr)+ωrr(tr)×Rrr(tr)xP(rr),ωP(tr)=ωrr(tr). (17)
These are transformed to the contact local coordinates using the appropriate rotation:
vP(cp)=RT
cp(tr)vP(tr),ωP(cp)=RT
cp(tr)ωP(tr). (18)
The matrix Rcp(tr)comes from the marker mcp(tr),thatdescribesthecontactreferencepoint
relative to the track system. The matrix is transposed here because these are global-to-local
conversions cf. Equation (7b).
3.9.2. Velocity of contact reference point P on a roller
The creep calculation for a roller uses an additional coordinate system mrol ,withaxes
parallel to mtr and located at z=rnom,rbelow the track centre (orol(tr)=[0, 0, rnom,r]T,
Rrol(tr)=I). This is used among others to get the local rolling radius at the contact
reference P:
mP(rol)=to-local(mP(tr),mrol(tr)), i.e. xP(rol)=RT
rol(tr)xP(tr)−orol(tr)(19)
Thevelocitiesfortherollerprolearedenedsimilarlyasforarail,
vrr(rol)=[0, vy,vz]T,ωrr(rol)=[vφ,0,0]
T. (20)
Note that vφis retained even though the marker mrol(tr)is supposed to follow the track
marker. How this should be used properly has not yet been decided. The point Pthat’s
considered is found at the local position xP(rr)in the prole, with local velocity vP(rr)=0.
Equation (9) obtains the velocity with respect to the roller system.
vP(rol)=vrr(rol)+ωrr(rol)×Rrr(rol)xP(rr). (21)
The roller could be moving with respect to the track frame, in principle, upon which the
global velocity would be obtained using Equation (9) a second time,
vP(tr)=vrol(tr)+ωrol(tr)×Rrol(tr)xP(rol)+Rrol(tr)vP(rol). (22)
VEHICLE SYSTEM DYNAMICS 23
As long as the roller marker stands still, with respect to the track marker, this reduces to
vP(tr)=vP(rol),ωP(tr)=ωP(rol). These velocities are transformed to contact local coordi-
nates using Equation (18).
3.9.3. Velocity of contact reference point Q on the wheel
Flexibilities in the wheel and wheelset may be given for all six degrees of freedom, relative
to the wheelset marker,
vrw(ws)=[vx,vy,vz]T,ωrw(ws)=[vφ,vθ,vψ]T. (23)
The angular velocities describe rotations of the wheel prole marker.
As the contact reference point is chosen on the rail prole, the point Qis found at a
height δnbelow the tangent plane.
mQ(cp)=[0, 0, δn]T. (24)
This is transformed to track coordinates using the to-global operation, then to wheel prole
coordinates using to-local twice. The velocity of Qwith respect to the wheelset is then
obtained using Equation (9):
vQ(ws)=vrw(ws)+ωrw(ws)×Rrw(ws)xQ(rw). (25)
The overall velocity of the wheelset is given using Euler angles, i.e.
vws(tr)=[vs,vy,vz]T,˙
νws =[˙
φws,˙
θws,˙
ψws]T. (26)
The latter are rotations with respect to intermediate axes x,y and zcf. Section 2.4.The
angular velocity is obtained using a matrix Gcf. [43, § 2.7]:
ωws(tr)=G˙
νws,withG=⎡
⎣
1−sψ0
0cφcψ−sφ
0sφcψcφ
⎤
⎦. (27)
For small ψthis is approximated as G≈R(φ–ψ).ThevelocityvQ(ws)is then transformed
using Equation (9),
vQ(tr)=vws(tr)+ωws(tr)×Rws(tr)xQ(ws)+Rws(tr)vQ(ws). (28)
This is transformed to contact coordinates using RT
cp(tr), analogously to equation (18).
3.9.4. Obtaining the rolling velocity V
For wheel-on-roller congurations, the rolling velocity Vis obtained using
V=|vP(cp),x+vQ(cp),x|
2. (29)
This doesn’t work for wheel/rail contact, where vQ(cp),xis composed of linear and circum-
ferential velocities, vsand ωwsr,thatalmostcanceleachother.Thisisresolvedcomputing
24 E. A. H. VOLLEBREGT
the eect of pitch rotation separately. The rolling velocity Vis then computed as
V=|vQ(cp),x−vθ(cp),x|+|vθ(cp),x|
2. (30)
The rolling direction is determined from the signs of the contributions. At small rolling
velocity V(|V|<10−6),thecreepagesaresetto±1, using the same signs. The creep-
ages are then computed from Equation (15), completing the inputs for the basic contact
calculation.
The expressions for velocities of points Pand Q, such as Equations (18) and (22),
may be expanded symbolically, deriving analytical formulas for the creepages in terms of
wheelset parameters ˙
θws,vy,ψws ,etc.Seeforinstance[66,67] for wheel-on-track or [68]for
wheel-on-roller congurations. This is viewed as complementary to the approach used in
CONTACT. On the one hand, additional insights are gained from symbolic calculations,
revealing for instance which terms are dominating. On the other hand, this requires careful
derivations, and simplication assuming small angles for instance. This can lead to signif-
icant dierences in the computed creep forces or the predicted L/Vratio [67]. The use
of generic, non-linear equations may therefore be preferred for the nal implementation,
supplemented with analytical calculations for insight and cross-validation.
4. Contact locus
The geometrical point of contact is the starting point for many wheel/rail contact algo-
rithms, e.g. [29,30,36,61]. This actually consists of two separate points, on the rail and wheel
surfaces, that lie opposite to each other and have maximum indentation.
This may be formalised using four algebraic equations using the distance vector between
the two points, and normal and tangent vectors at these locations [61,69,70]:
drw =pr−qw,tr
1·drw =0, tr
2·drw =0, tw
1·nr=0, tw
2·nr=0, (31)
The second and third conditions state that the distance vector drw be parallel to the normal
nr, while the latter two conditions ensure that nris parallel to nw. A further condition is
placed on the curvatures such that the interpenetration is maximum instead of minimum.
Previous authors considered the contact locus as a means for locating the geometrical
point of contact. This is called Wang’s method [32,71]ortracelinemethod[72–74], based
on earlier works by de Pater [28,75]. The contact locus is a curve on the wheel surface where
the gap to the rail is minimum in longitudinal direction, as illustrated in Figure 11.Search-
ing laterally for the minimum gap along this curve then presents the overall minimum, i.e.
the initial contact position.
Thebenetofthecontactlocusisthatitreducesthecontactsearchtoaone-dimensional
problem. Other ways to achieve the same were presented by dierent authors [45,76,77].
Wang’s method was extended further to wheel-on-rollers [72,78]. Involved analyses are
used in all of these cases. Here we present a new derivation for the contact locus for wheel-
on-rail congurations that’s easier to understand, and present its extension to wheel-on-
roller contacts.
VEHICLE SYSTEM DYNAMICS 25
4.1. Contact locus for wheel-on-rail contact
For wheel-on-rail contact, we consider a prismatic rail, where the surface height zrail(r)is
afunctionoflateralpositiononly,zrail(r)(x(r),y(r))=hrail(y(r)).Thismakesforasurface
that’s horizontal in the longitudinal directions of the rail and track systems. A necessary
condition for the gap g=zrail(r)−zwh(r)to be locally minimum is then that the wheel be
horizontal also, when viewed along lines of constant y(r).
∂zwh(r)
∂x(r)=0. (32)
This is illustrated in Figure 11, for a wheel with new S1002 prole, with a nominal radius
rnom,w=120 mm to exaggerate the surface variation, showing the prole height zwh(w)in
the wheel prole system. Light-blue lines indicate the orientation of the rail forward direc-
tion, x(r),forayawangleψws =10◦. Black curves indicate the contour lines of constant
surface height. The contact locus of equation (32) is then obtained as the points where the
contours touch on the light-blue lines of constant y(r), as shown by the magenta curve.
The directional derivative ∂zwh(w)/∂x(r),alongthelight-bluelines,isobtainedmulti-
plying the gradient with the desired direction vector t=[cos(ψ),−sin(ψ )]T.Thisbrings
condition (32) for a horizontal tangent to
cos(ψ ) ∂zwh(w)
∂x(w)−sin(ψ ) ∂zwh(w)
∂y(w)=0. (33)
These principles, illustrated in Figure 11, are formalised using appropriate mathematics
notations.Thewheelsurfaceisdenedimplicitlyinwheelcoordinatesasthesetofpoints
satisfying the equation
x2
wh(w)+(zwh(w)−zmid,w)2=(rwh(ywh(w)))2,zmid,w=−rnom,w. (34)
Figure 11. Surface height zwof a small railway wheel in wheel coordinates (rnom,w=120 mm), with
construction of the contact locus for a yaw angle ψws =10◦.
26 E. A. H. VOLLEBREGT
This is embedded in 3D space using the level set fwh(w)(x(w))=0, using a function fwh(w):
fwh(w):R3
(w)→R,
fwh(w)(x(w),y(w),z(w))=x2
(w)+(z(w)−zmid,w)2−(rwh(y(w)))2. (35)
Here rwh =rnom,w+hwh (y(w))is the radius of a circle at y(w), centred at x(w)=0, z(w)=
zmid,w.Thefunctionfwh(w), containing the wheel surface, is transformed to rail coordinates
by successive rotations. The transformation is written as x(w)=x(w)(x(r)):
x(w):R3
(r)→R3
(w),x(w)=orail
(w)+Rrail
(w)x(r),
fwh(r):R3
(r)→R,fwh(r)=fwh(w)◦x(w), i.e. fwh(r)(x(r))=fwh(w)(x(w)(x(r))). (36)
The rotation matrix from rail to wheel coordinates is obtained using the rotation from rail
to track coordinates times the inverse from wheel to track coordinates. Using Equation (4),
for the roll–yaw–pitch convention:
Rrail
(w)=R(φ–ψ)T=⎡
⎣
cψcφsψsφsψ
−sψcφcψsφcψ
0−sφcφ
⎤
⎦(37)
In this case, the angle φmay contain roll of the wheelset as well as the cant of the rail.
The wheel surface is extracted from the function fwh(r)in rail coordinates using
the implicit function theorem. This says that there exists an explicit function zwh(r)
corresponding to the surface fwh(r)=0, except for places with vanishing derivative
∂fwh(r)/∂z(r)=0:
zwh(r):R2
(r)→R,zwh(r)=zwh(r)(x(r),y(r)). (38)
The partial derivatives of zwh(r)are obtained through implicit dierentiation of the level
fwh(r)=0withrespecttox(r):
∂fwh(r)
∂x(r)+∂fwh(r)
∂z(r)
∂zwh(r)
∂x(r)=0→∂zwh(r)
∂x(r)=−∂fwh(r)
∂x(r)∂fwh(r)
∂z(r)
. (39)
Next, we obtain the partial derivatives of fwh(r)using the chain rule and the gradient of
fwh(w):
∇fwh(w)(x(w))=2x(w),−2rwh(y(w))h
wh(y(w)),2(z(w)−zmid,w), (40a)
∇fwh(r)(x(r))=∇fwh(w)(x(w)(x(r))) ·Rrail
(w). (40b)
The contact locus on the wheel follows as
∂zwh(r)
∂x(r)
?
=0→∂fwh(r)
∂x(r)=2x(w)·cψ−2rwh(y(w))h
wh(y(w))·−sψ
?
=0
→xlc(w)=−rwh (y(w))h
wh(y(w))tan(ψ ). (41)
The derivative h
wh takes has a function of ywh, whereas the prole points may be given
with ywh decreasing. This requires a bit of care to get the right sign. Computations for the
VEHICLE SYSTEM DYNAMICS 27
left wheel are done with mirroring, using right-sided proles, such that the derivative h
isn’t aected. Note that mirroring changes the sign of ψws.
The contact locus on the wheel is transformed to track coordinates and used to form
thegapfunction.Alllocalminimaaredeterminedandusedtodenecontactpatches.The
lateral extent follows from the range where negative gap values are found. In longitudinal
direction, the length of interpenetration areas is estimated using the maximum penetration
δnand the eective radius reff =r(yw)/ cos(δcp(tr))[26], using a quadratic approximation.
ThereasonwhyourformulalookssomucheasierthanWang’scontactlocusresidesin
the use of the roll–yaw–pitch convention. A dierent matrix shows up when starting from
the yaw–roll–pitch convention,
Rrail
yaw(w)=R(ψ–φ)T=⎡
⎣
cψsψ0
−sψcφcψcφsφ
sψsφ−cψsφcφ
⎤
⎦. (42)
Therstcolumnmultipliedwiththegradient∇fwh(w)of Equation (40a) gives:
∂fwh(r)
∂x(r)=2x(w)·cψ−2rwh(y(w))h
wh(y(w))·−sψcφ+2(z(w)−zmid,w)·sψsφ
?
=0 (43)
This reduces to (41) when φ=0, dropping the z(w)contribution, but cannot be inverted so
easily when z(w)is retained. The detailed formulas were given by Wang [32]. Equation (41)
can be used as an easy and satisfactory approximation. In this case, the roll angle φshould
considerthewheelsetrollonly.Railcantdoesn’taectthecontactlocusforwheel/rail
contact.
4.2. Contact locus for a wheel on a roller
Our approach for a wheel on a roller works along the same lines, even though it looks
more complicated. The starting point is that a locally minimum value of the gap function
requires equal slopes of the surfaces of wheel and roller:
∂zwh(r)
∂x(r)=∂zrol(r)
∂x(r)
. (44)
Theslopeoftherollersurfaceisdevelopedsimilarlyastheslopeonthewheelsurfaceabove,
except that no coordinate rotation is needed in this case:
frol(r):R3
(r)→R,
frol(r)(x(r),y(r),z(r))=x2
(r)+(z(r)−zmid,r)2−(rrol (y(r)))2, (45a)
∇frol(r)(x(r))=2x(r),2rrol (y(r))h
rol(y(r)),2(z(r)−zmid,r). (45b)
Here rrol =rnom,r−hrol (y(r))is the radius of a circle at y(r), centred at x(r)=0, z(r)=
zmid,r=rnom,r.Afunctionzrol(r)exists to describe the roller’s surface:
zrol(r):R2
(r)→R,zrol(r)=zrol(r)(x(r),y(r)). (46)
28 E. A. H. VOLLEBREGT
The derivative in x(r)-direction is obtained as:
∂zrol(r)
∂x(r)=−∂frol(r)
∂x(r)∂frol(r)
∂z(r)=− x(r)
z(r)−rnom,r
. (47)
The denominator of (39) must now be expanded, which could be ignored in the case of a
wheelonarail.
∂fwh(r)
∂z(r)=2x(w)sφsψ−2rwh(y(w))h
wh(y(w))sφcψ+2(z(w)+rnom,w)cφ. (48)
The contact locus on the wheel follows as
−x(w)cψ+rwh(y(w))h
wh(y(w))sψ
x(w)sφsψ−rwh(y(w))h
wh(y(w))sφcψ+(z(w)+rnom,w)cφ
?
=− x(r)
z(r)−rnom,r
. (49)
This is solved quickly using an iterative procedure, for each position ywwhere the locus is
desired:
xk+1
(w)cψ+rwh(y(w))h
wh(y(w))sψ
xk
(w)sφsψ−rwh(y(w))h
wh(y(w))sφcψ+(zk
(w)+rnom,w)cφ=xk
(r)+xk+1
(w)−xk
(w)
zk
(r)−rnom,r
. (50)
Thisconvergesintwostepsevenwhenstartingfromx0
(w)=0.
When using mirroring of the left side to a right side problem, φws and ψws are both
aected. The resulting method is cross-validated using the brute-force approach.
5. Results
The main capability of the method presented is to analyse wheel/rail contacts using a 3-d
contact search algorithm. Some typical examples will be shown for this using the Manch-
ester contact benchmark [13,79]. Further results are shown for a collection of measured
worn proles, demonstrating the automation and robustness of the calculations.
5.1. Results for the Manchester contact benchmark
TheManchestercontactbenchmarkwasproposedtoassesstheimpactofwheel-railcon-
tact modelling assumptions on the simulation of railway vehicle dynamics [13,79]. Two
simulation cases were considered initially: case A for a single wheelset, to consider contact
methods in isolation, and case B for a railway vehicle, to assess the eect of contact meth-
ods on vehicle dynamics. The paper [13] presents the results for case A for ten codes that
participated. The work seems to have stopped thereafter, with no results published yet for
case B.
Here we focus attention on test-case A-2.2 for the tangential contact problem, for a
wheelset with new S1002 and UIC60 wheel and rail proles with lateral displacement and
yaw angles increasing in proportion to each other. Figure 12 shows results for one such
position, yws =5mm,ψws =12.0 mrad, where two contact patches are found on the right
rail.Thicklinesontherailproleindicatethecurvesx(r)=0andy(r)=0, with the origin
of the rail coordinate system at the crossing of these curves. As the wheelset steers to the
VEHICLE SYSTEM DYNAMICS 29
Figure 12. Wheel/rail contact results for the Manchester benchmark test-case A-2.2: wheelset with
lateral displacement yws =5mm,yawedatψws =12.0 mrad.
Figure 13. Left: results obtained using the Kalker CONTACT add-on to SIMPACK rail [38]. Middle, right:
corresponding results for the new w/r geometry module.
right at a positive yaw angle, contact is found on the left rail at positive x(r),andatx(r)<0
on the right rail. These x-locationsareadirectoutputofthe3-dcontactsearchalgorithm.
A particularly challenging conguration was identied by Kienberger for the contact
ofthewheelangeroottotherailgaugecorner[38], leading to strongly non-elliptic
contact as illustrated in Figure 13 (left). This concerns case A-2.2 of the Manchester
benchmark computed using the Kalker CONTACT add-on to SIMPACK Rail, at lat-
eral shift yws =6.5 mm, yaw angle ψws =15.6 mrad, and a vertical force Fz=80 kN
per wheel. Similar shapes were presented by Baeza et al. [40], at yws =6.3 mm, ψws =
17.5 mrad. Figures 13, middle and right, show the corresponding result obtained from
CONTACT, at yws =4.954 mm, ψws =15.6 mrad. This shows that good agreement is
reached at slightly shifted locations. The discrepancy comes from the dierent conven-
tions used: measuring the lateral shift at the height of the axle or in the track plane.
Further, the contact patch appears shifted by (±2, ±1mm)in the local coordinate sys-
tem. This comes from using slightly dierent weighting in the calculation of the contact
reference position.
The contact positions reported by CONTACT concern the weighted reference cf.
Equation (12). These are shown in Figure 14, plotted on top of the results published for the
Manchester benchmark. In overall sense, the results show good correspondence. Some-
what bigger dierences occur around the centred position, yws ≤0.5 mm, that yields wide,
asymmetric contact patches. These dierences do not much aect the computed forces,
30 E. A. H. VOLLEBREGT
Figure 14. Lateral contact positions on the rail for Manchester contact benchmark case A1.1 new pro-
files 20 kN load, computed by CONTACT, plotted on top of benchmark results presented by Shackleton
and Iwnicki [13,Figure3].
or the integration in a vehicle dynamics simulation, as long as creepages, contact location
and force location are used in a consistent way. This is considered in more detail in the
accompanying paper [44].
The Manchester benchmark concerns a free rolling wheelset, including an equilibrium
equation My(ws)=0 for the net moment on the wheelset in its rolling direction, and a
similar balance to distribute the load over the wheels. Focusing on a single wheel and rail,
thesearenotincludedinCONTACT.Instead,asmalliterationprocedureisprogrammedin
Matlab that tunes the wheelset vertical position, roll angle, and pitch velocity, to satisfy the
balance equations. Figures 15 and 16 show the longitudinal and lateral contact forces com-
puted by CONTACT, plotted on top of the published results. The brute-force method and
the contact locus produce identical values, except for minimal eects of the discretisation.
These new results are generally in good agreement with the earlier ones.
5.2. Results for a thousand freight cars with worn proles
Afurtherapplicationthat’sconsideredhereconcernsaprojectonraillifeextensionper-
formed for CSX, a North American class I railroad. A simulation approach is developed
to assess rail prole evolution considering wear and grinding. This work is performed by
Sentient Science, NRC Canada, and Vtech CMCC, using VAMPIRE and the CONTACT
software.
VEHICLE SYSTEM DYNAMICS 31
Figure 15. Longitudinal contact forces for Manchester contact benchmark case A2.2 new profiles20 kN
load, computed by CONTACT, plotted on top of benchmark results presented by Shackleton and Iwnicki
[13, Figure 16].
The target for a simulation is to evaluate the life of rails at one segment of track for
a given operational scenario. An example considered in the project concerns a 5◦curve,
radius 349 m, near Cartersville, GA. Trac data are collected for the segment: the number
of trains, types of wagons, loads, speeds, wheel proles, as well as track data, e.g. geometry,
friction management practices. These data are analysed and classied, e.g. distinguishing
dierent quality levels [80]. Loading ensembles are then dened that represent the actual
situation, maintaining the ranges and variability of input parameters, however, with a much
reduced number of loadings. Simulations with VAMPIRE are used to assess the vehicle
dynamics in steady curving, using VAMPIRE’s internal contact algorithms. The contact
situationisthenanalysedindetailusingCONTACT,usingthewheelsetkinematicsas
obtained from VAMPIRE, maintaining the vertical force Fz(tr)that governs the level of
the contact stresses. These stresses are then fed into Sentient’s wear and grinding calcula-
tions. This is repeated multiple times, until the rail prole is worn down to a predened
threshold and the life is exhausted.
The challenge for CONTACT in this application lies in the robust and automatic calcu-
lation of a wide range of contact situations. This is demonstrated considering the so-called
‘group 5 data’, the 20% of wheels with the lowest prole quality index. A thousand freight
carsaresimulated,amixtureofat,hopperandtankcars.Speedsvarybetween7and
40 mph (11–64 km/h), and weights between 50,000 and 300,000 lbs (222–1334 kN) for
32 E. A. H. VOLLEBREGT
Figure 16. Lateral contact forces for Manchester contact benchmark case A2.2 new profiles 20 kN load,
computed by CONTACT, plotted on top of benchmark results presented by Shackleton and Iwnicki [13,
Figure 17].
empty and full wagons. Two axles are considered for each car, for the leading bogie. Four
coecients of friction are set for each axle, two for each wheel, taken from normal distribu-
tions with mean ¯μ=0.35 and standard deviation σμ=0.16 at the top of the rail (surface
inclination α≤20◦)and ¯μ=0.20, σμ=0.0167 at the gauge face (α≥30◦). All wheels
encountered in this collection are heavily worn, while some proles have missing data:
gaps, or ending prematurely.
This test-set of four thousand wheels appeared to present scenarios that had not yet
been encountered. Additional work was needed to detect and handle all edge cases appro-
priately. For instance for the arc length along the proles, used in the parametric spline
representation, it appears tricky sometimes to dene the reference s=0attheproleref-
erence marker. This happened particularly for wheels where the wheel marker lies at the
backoftheange,withmultiplecrossingsoftheprolewiththeabcissayw=0. Another
example concerns the contact locus with xlc tending to innity at vertical slopes in the pro-
le. Next, the computation of the weighted centre (Equation (12)), must be prepared for
cases with only minimal interpenetration (e.g. O(10−9)). Such cases aren’t relevant from
a practical perspective, still, the programme should take care not to divide by zero.
Figure 17 shows a situation where the combination of contact patches is activated. Ini-
tially, using a combination distance dcomb =dsep =2 mm, combination of patches occurs
at vertical positions zws ≥0.1125. The computed vertical force then jumps from 90 to
76 kN, hampering convergence for a solution procedure that tries to solve zws for Fz=
82 kN. This may be resolved using dcomb =dsep =5mm,butthatmerelymovesthetrouble
VEHICLE SYSTEM DYNAMICS 33
Figure 17. Combination of contact patches for wheel 2291. Thever tical force jumps suddenly at increas-
ing vertical positions when using a single threshold, and changes gradually when using the blending
approach.
Figure 18. Computed pressures pnfor one case of the group 5 data with varying levels of profile
smoothing.
to other cases. A more durable resolution is provided by the blending approach, smooth-
ing the function Fz=Fz(zws). The solution algorithm used for this balance equation is also
extended such that it proceeds with sensible values when a jump is encountered.
Further issues are related to prole smoothing. We currently don’t have good guidelines
for the parameter λof Equation (10). This depends on the spacing of the prole points and
on the inherent curvature contained in the prole. λis currently tuned by trial and error,
on the basis of Figure 18 for instance. Parameters λ=0.1–1 could be used for these proles
if there’s good condence in the measured values and if one is interested in the eects of
pressure variations. Bigger values like λ=5–10 could be appropriate if one is interested
more in the general behaviour.
Smoothing appears to have subtle interactions with the positioning of wheel and rail
proles. This occurs when certain characteristics of proles are used to put them in place,
like raising or lowering the rail to touch the track plane precisely. Another case where this
occurs is in the gauge point computation. This is shown in Figure 19 for a rail that bulges
out due to plastic deformation. Using λ=10, reduces the bump by 0.24mm, shifting the
whole prole by the same amount. These are quite large displacements considering the
strong sensitivity of contact problems to the amount of indentation. As a result, position
values should be treated with care, with an eye on the context in which they were computed,
and should usually be obtained using additional balance equations.
34 E. A. H. VOLLEBREGT
Figure 19. Strong profile smoothing (λ=10) changes the gauge point by 0.24 mm on this profile with
localised features.
6. Conclusions and discussion
This paper presented the computational methods and software design for the new module
for wheel/rail contact geometry analysis that is implemented in CONTACT. Building on
many components developed by others, this module automates and greatly simplies the
running of CONTACT for generic wheel/rail contact situations. Fully 3-d contact search
algorithms are implemented. This uses the contact locus approach, that’s simplied for
wheel-on-rail situations and extended to wheel-on-roller contacts.
A main characteristic of the new module is its extensive use of the multibody formal-
ism, using markers to represent coordinate systems, and the design using a generic, object
oriented-like software foundation. This simplies the development of new algorithms. We
no longer have to analyse the 3-d contact geometry using pen and paper, making transfor-
mations ourselves, yet can build the geometry in one coordinate system and transform to
another. This complements analytical approaches that provide more insight but are more
dicult to develop and generalise. 1-d parametric spline curves and bi-linear interpolation
areusedandgivesatisfactoryresults;2-dsplinesurfacescouldprovideadditionalexibility
with respect to the needed operations.
The contact geometry is analysed twice; rst for the location of contact patches, and then
for the local geometry of the contact patches. The contact search works in overall vertical
direction and starts from the wheel and rail proles in ther actual, overlapping positions.
Besides the contact location, dened using weighted averaging, this identies the extent
of interpenetration areas also. Potential contact areas are dened accordingly, discretised,
and used to compute the undeformed distance in normal direction. Creepages are formed
automatically using rigid body kinematics, including wheel and track exible bending. This
again complements analytical approximations, allowing for cross-validation.
Numerical results illustrate the viability, generality, and robustness of the approach,
dealing with strongly asymmetric contact patches and situations where multiple contact
patches occur. Results are shown for the Manchester contact benchmark that correspond
well to the published results. Further results concern a collection of heavily worn wheel
proles, illustrating the level of automation of the approach. The dierent strategies used
in the planar contact approach, for the contact reference point and the reference angle, are
discussed in a separate paper [44].
VEHICLE SYSTEM DYNAMICS 35
A challenge for the planar contact approach comes from the combination of nearby
contact patches. This is desired to account for their interaction but unwanted because of
the compromise introduced for the contact reference angle. To strike a balance between
these dierent wishes, a criterion is introduced on the basis of the distance between adja-
cent patches. However, such switches may cause discontinuities in the numerical results at
slight perturbations of input parameters, such as a prescribed vertical force. This hampers
the integration in multi-body dynamics simulation, or the solution of additional balance
equations. A blending approach is introduced to mitigate some of the jumps, more research
is needed to reduce the jumps further.
Further work is needed on the extension of the approach to more general geometri-
cal congurations. For instance, the overall vertical direction may not be appropriate in
the computation of contacts with a check rail. This may be addressed using a prior analy-
sis to nd a reasonably ‘normal’ orientation. In rolling direction, geometrical deviations
are introduced by wheel ats, out-of-roundness, and corrugation. It’s unclear whether
the contact locus approach can be extended for this. Otherwise, the grid-based approach
should be improved with respect to its performance. Finally, short wavelength variations
in rolling direction may call for transient instead of steady rolling computations, address-
ing uctuations in the creepage and creep forces. This calls for connecting the grids and
contact patches between consecutive time steps, and for further integration of the contact
calculations in multi-body dynamics simulation.
Acknowledgments
Thanks to many colleagues for valuable discussions that contributed to this work indirectly. Thanks
to Julio Blanco Lorenzo, Javier Santamaria, and Ernesto Garcia Vadillo for their direct contributions
and reviewing the draft version of this paper. Thanks to CSX (Dan Hampton) for permitting use
of the test-cases discussed in Section 5.2, and thanks to the colleagues at LORAM, NRC and Sen-
tient Science that contributed to the collection and preparation of these data. Thanks nally to the
anonymous reviewers for their valuable feedback on this paper.
Disclosure statement
No potential conict of interest was reported by the author(s).
ORCID
E. A. H. Vollebregt http://orcid.org/0000-0003-2752-1589
References
[1] Iwnicki S, editor. Proceedings of the 22nd International Symposium on Dynamics of Vehicles
on Roads and Tracks. Manchester: IAVSD; 2011.
[2] Rosenberger M, editor. Proceedings of the 24th International Symposium on Dynamics of
Vehicles on Roads and Tracks. Graz, Austria: IAVSD; 2015.
[3] Spir yagin M, Gordon T, Cole C, et al., editor. Proceedings of the 25th International Symposium
on Dynamics of Vehicles on Roads and Tracks. Rockhampton, Queensland, Australia: IAVSD;
2017.
[4] Lunden R, Ekberg A, Berg M, et al. Special edition of “Wear” to contain CM2015 proceedings.
Wear. 2016;366–367:1–2.
36 E. A. H. VOLLEBREGT
[5] Li Z, Núñez A, editors. Proceedings of the 11th International Conference on Contact Mechan-
ics and Wear of Rail/Wheel Systems. The Netherlands: Delft University of Technology;
2018.
[6] Meymand S, Keylin A, Ahmadian M. A survey of wheel-rail contact models for rail vehicles.
Veh Syst D y n . 2016;54:386–428.
[7] Zhai W, Han Z, Chen Z, et al. Train-track-bridge dynamic interaction: a state-of-the-art review.
Veh Syst D y n . 2019;57:984–1027.
[8] Kalker J. The computation of three-dimensional rolling contact with dry friction. Int J Numer
Methods Eng. 1979;14:1293–1307.
[9] KalkerJ.Twoalgorithmsforthecontactprobleminelastostatics.Delft(TheNetherlands):Delft
University of Technology, report TWI 82-26; 1982.
[10] Kalker J. Numerical calculation of the elastic eld in a half-space. Comm Appl Num Meth.
1986;2:401–410. Reprinted as Appendix C in Kalker J. Three-dimensional elastic bodies in
rolling contact. Dordrecht: Kluwer Academic Publishers; 1990. (Solid mechanics and its
applications; vol. 2).
[11] Kalker J. Mathematical models of friction for contact problems in elasticity. Wear.
1986;113:61–77.
[12] Kalker J. Three-dimensional elastic bodies in rolling contact. Dordrecht: Kluwer Academic
Publishers; 1990. (Solid mechanics and its applications; vol. 2).
[13] Shackleton P, Iwnicki S. Comparison of wheel-rail contact codes for railway vehicle simula-
tion: an introduction to the Manchester Contact Benchmark and initial results. Veh Syst Dyn.
2008;46(1–2):129–149.
[14] Baeza L, Vila P, Roda A, et al. Prediction of corrugation in rails using a non-stationary contact
model. Wear. 2008;265:1156–1162.
[15] Willner K. Fully coupled frictional contact using elastic halfspace theory. ASME J Tribol.
2008;130:031405-1–031405-8.
[16] Pieringer A, Kropp W, Thompson D. Investigation of the dynamic contact lter eect in
vertical wheel/rail interaction using a 2D and a 3D non-Hertzian contact model. Wear.
2011;271(1–2):328–338.
[17] Kaiser I. Rening the modelling of vehicle-track interaction. In: Iwnicki S, editor. Proceed-
ings of the 22nd International Symposium on Dynamics of Vehicles on Roads and Tracks.
Manchester: IAVSD; 2011. p. 1–6.
[18] Blanco-Lorenzo J, Santamaria J, Vadillo E, et al. On the inuence of conformity on wheel-rail
rolling contact mechanics. Tribol Int. 2016;103:647–667.
[19] Vollebregt E. Improving the speed and accuracy of the frictional rolling contact model “CON-
TACT”. In: Topping B, Adam J, Pallarés F, et al., editors. Proceedings of the 10th Interna-
tional Conference on Computational Structures Technology; Stirlingshire, United Kingdom.
Civil-Comp Press; 2010. p. 1–15. DOI:10.4203/ccp.93.17.
[20] Vollebregt E. A new solver for the elastic normal contact problem using conjugate gradients,
deation, and an FFT-based preconditioner. J Comput Phys. 2014;257, Part A:333–351.
[21] Zhao J, Vollebregt E, Oosterlee C. A fast nonlinear conjugate gradient based method for 3D
concentrated frictional contact problems. J Comput Phys. 2015;288:86–100.
[22] Vollebregt E. Numerical modeling of measured railway creep versus creep-force curves with
CONTACT. Wear. 2014;314:87–95.
[23] Wekken Cvd, Vollebregt E. Local plasticity modelling and its inuence on wheel-rail fric-
tion. In: Li Z, Núñez A, editors. Proceedings of the 11th International Conference on Contact
Mechanics and Wear of Rail/Wheel Systems. The Netherlands: Delft University of Technology;
2018. p. 1013–1018.
[24] Vollebregt E, van der Wekken C. Advanced modeling of wheel-rail friction phenomena.
VORtech; 2019. TR19-11. FRA project.
[25] Vollebregt E, Segal A. Solving conformal wheel-rail rolling contact problems. Veh Syst Dyn.
2014;52(suppl. 1):455–468. DOI:10.1080/00423114.2014.906634
[26] Vollebregt E. Conformal contact: corrections and new results. Veh Syst Dyn. 2018;56(10):
1622–1632. DOI:10.1080/00423114.2018.1424917
VEHICLE SYSTEM DYNAMICS 37
[27] Vollebregt E. User guide for CONTACT, Rolling and sliding contact with friction. Vtech
CMCC; 2020. TR20-01, version 20.2. Available from: www.cmcc.nl/documentation.
[28] Pater Ad. The geometrical contact between track and wheelset. Veh Syst Dyn. 1988;17(3):
127–140.
[29] Shabana A, Zaazaa K, Escalona J, et al. Development of elastic force model for wheel/rail
contact problems. J Sound Vib. 2004;269:295–325.
[30] Pombo J, Ambrósio J, Silva M. A new wheel-rail contact model for railway dynamics. Veh Syst
Dyn. 2007;45:165–189.
[31] Shabana A, Zaazaa K, Sugiyama H. Railroad vehicle dynamics: a computational approach. Boca
Raton: CRC Press; 2008.
[32] Wang K. The track of wheel contact points and the calculation of wheel/rail geometric contact
parameters. J Southwest Jiaotong Univ. 1984;1:89–99.
[33] Li Z. Wheel-rail rolling contact and its application to wear simulation [dissertation]. Delft
University of Technology; 2002.
[34] Jin X, Wen Z, Zhang W, et al. Numerical simulation of rail corrugation on a curved track.
Comput Struct. 2005;83:2052–2065.
[35] Santamaría J, Vadillo E, Gómez J. A comprehensive method for the elastic calculation of the
two-point wheel-rail contact. Veh Syst Dyn. 2006;44, suppl:240–250.
[36] Malvezzi M, Meli E, Falomi S, et al. Determination of wheel-rail contact points with semiana-
lytic methods. Multibody Syst Dyn. 2008;20:327–358.
[37] Sugiyama H, Suda Y. On the contact search algorithms for wheel/rail contact problems.
J Comput Nonlinear Dyn Trans ASME. 2009;4:041001.
[38] Vollebregt E, Weidemann C, Kienberger A. Use of “CONTACT” in multi-body vehicle dynam-
ics and prole wear simulation: initial results. In: Iwnicki S, editor. Proceedings of the 22nd
International Symposium on Dynamics of Vehicles on Roads and Tracks. Manchester: IAVSD;
2011. p. 1–6.
[39] Yang X, Gu S, Zhou S, et al. A method for improved accuracy in three dimensions for
determining wheel/rail contact points. Veh Syst Dyn. 2015;53:1620–1640.
[40] Baeza L, Thompson D, Squicciarini G, et al. Method for obtaining the wheel-rail contact loca-
tion and its application to the normal problem calculation through “CONTACT”. Veh Syst
Dyn. 2018;56:1734–1746.
[41] An B, Wen J, Wang P, et al. Numerical investigation into the eect of geometric gap ideal-
isationonwheel-railrollingcontactinpresenceofyawangle.MathProblEng.2019;2019:
9895267.
[42] Vollebregt E. Comments on “the Kalker book of tables for non-Hertzian contact of wheel and
rail”. Veh Syst Dyn. 2018;56(9):1451–1459. DOI:10.1080/00423114.2017.1421767
[43] Shabana A. Dynamics of multibody systems. 5th ed. New York: Cambridge University Press;
2020.
[44] Vollebregt E. Detailed wheel/rail geometry processing using the conformal contact approach.
Multibody Syst Dyn. 2020.DOI:10.1007/s11044-020-09762
[45] Alonso A, Giménez J, García M. Analytical methodology to solve the geometric wheel-rail
contact problem taking into account the wheelset yaw angle. In: Rosenberger M, editor. Pro-
ceedings of the 24th International Symposium on Dynamics of Vehicles on Roads and Tracks.
Graz, Austria: IAVSD; 2015. p. 1–9.
[46] Liu B, Bruni S, Vollebregt E. A non-Hertzian method for solving wheel-rail normal contact
problem taking into account the eect of yaw. Veh Syst Dyn. 2016;54(9):1226–1246.
[47] Vollebregt E. Abstract level parallelization of nite dierence methods. Sci Program.
1997;6:331–344.
[48] Boor CD. Practical guide to splines. New York: Springer-Verlag; 1978.
[49] Lundberg G. Elastische Berührung zweier Halbräume. Forschung ad Gebeite des Ingenieur-
wesens. 1939;10:201.
[50] Fujiwara H, Kobayashi T, Kawase T, et al. Optimized logarithmic roller crowning design of
cylindrical roller bearings and its experimental demonstration. Tribol Trans. 2010;53:909–916.
[51] Johnson K. Contact mechanics. Cambridge: Cambridge University Press; 1985.
38 E. A. H. VOLLEBREGT
[52] Dierckx P. Algorithms for smoothing data with periodic and parametric splines. Comput Gr
Image Process. 1982;20:171–184.
[53] Press W, Teukolsky S, Vetterling W, et al. Numerical recipes in Fortran 77: the art of scientic
computing. 2nd ed. Cambridge: Cambridge University Press; 1992.
[54] Weinert H. A fast compact algorithm for cubic spline smoothing. Comput Statist Data Anal.
2009;53:932–940.
[55] PollockD.Smoothingwithcubicsplines.London:QueenMaryandWesteldCollege,The
University of London; 1999.
[56] Savitzky A, Golay M. Smoothing and dierentiation of data by simplied least squares proce-
dures. Anal Chem. 1964;36:1627–1639.
[57] Moreton H. Minimum curvature variation curves, networks, and surfaces for fair free-form
shape design [dissertation]. University of California at Berkeley; 1992.
[58] Chernov N, Lesort C. Least squares tting of circles. J Math Imaging Vis. 2005;23:239–252.
[59] Escalona J, Sugiyama H, Shabana A. Modelling of structural exibility in multibody railroad
vehicle systems. Veh Syst Dyn. 2013;51(7):1027–1058.
[60] Mikheev G, Pogorelov D, Rodikov A. Methods of simulation of railway wheelset dynamics
taking into account elasticity. In: Proceedings of the First International Conference on Rail
Transportation; Chengdu, China; 2017. p. 1–11.
[61] Zaazaa K, Schwab A. Review of Joost Kalker’s wheel-rail contact theories and their implemen-
tation in multibody codes. In: Proceedingsof the ASME 2009 International Design Engineering
Technical Conferences; September; 2009. p. 1–12.
[62] PiotrowskiJ,BruniS,LiuB.Replytocommentson“theKalkerbookoftablesfornon-
Hertzian contact of wheel and rail” by E.A.H. Vollebregt. Veh Syst Dyn. 2018;56:1460–1469.
DOI:10.1080/00423114.2018.1437274
[63] Piotrowski J, Kal ker J. The elastic cross-inuence between two quasi-Hertzian contact zones.
Veh Syst D y n . 1988;17:337–355.
[64] Aceituno J, Urda P, Briales E, et al. Analysis of the two-point wheel-rail contact scenario
using the knife-edge-equivalent contact constraint method. Mech Mach Theory. 2020;148:
103803.
[65] Piotrowski J, Chollet H. Wheel-rail contact models for vehicle system dynamics inlcuding
multi-point contact. Veh Syst Dyn. 2005;43:455–483.
[66] Wickens A. Fundamentals of rail vehicle dynamics: guidance and stability. Lisse: Swets and
Zetlinger; 2003.
[67] Shabana A, Tobaa M, Marquis B, et al. Eect of the linearization of the kinematic equations in
railroad vehicle system dynamics. ASME J Comput Nonlinear Dyn. 2006;1:25–34.
[68] Liu B, Bruni S. Analysis of wheel-roller contact and comparison with the wheel-rail case. Urban
Rail Transit. 2015;1(4):215–226.
[69] Pombo J, Ambrósio J. A wheel/rail contact model for rail guided vehicles dynamics. In:
Ambrósio J, editor. Proceedings of the ECCOMAS Thematic Conference on Advances in
Computational Multibody Dynamics; Lisbon, Portugal; 2003. p. 47–56.
[70] Shabana A, Tobaa M, Sugiyama H, et al. On the computer formulations of the wheel/rail contact
problem. Nonlinear Dyn. 2005;40:169–193.
[71] Burgelman N, Li Z, Dollevoet R. Eect of the longitudinal contact location on vehicle dynamics
simulation. Math Probl Eng. 2016;2016:1901089.
[72] Zhang W. Calculation method of wheel/roller (rail) spatial contact point. China Railway Sci.
2006;27:76–79. In Chinese.
[73] Zhang W, Dai H, Shen Z, et al. Roller rigs. In: Iwnicki S, editor. Handbook of railway vehicle
dynamics. Chapter 14. Boca Raton: CRC Press; 2006. p. 457–506.
[74] Zhu B, Zeng J, Zhang D, et al. A non-Hertzian wheel-rail contact model considering wheelset
yaw and its application in wheel wear prediction. Wear. 2019;432-433:202958.
[75] Pater Ad. The general theory of the motion of a single wheelset moving through a curve with
constant radius and cant. Z Angew Math Mech. 1981;61(7):277–292.
[76] Auciello J, Meli E, Falomi S, et al. Dynamic simulation of railway vehicles: wheel/rail contact
analysis. Veh Syst Dyn. 2009;47:867–899.
VEHICLE SYSTEM DYNAMICS 39
[77] Marques F, Magalhães H, Pombo J, et al. A three-dimensional approach for contact detection
between realistic wheel and rail surfaces for improved railway dynamics. Mech Mach Theory.
2020;149:103825.
[78] Zeng Y, Shu X, Wang C, et al. Study on three-dimensional wheel/rail contact geometry using
generalized projection contour method. In: Zhang W, editor. Proceedings of the 23rd Interna-
tionalSymposiumonDynamicsofVehiclesonRoadsandTracks;August;Qingdao,P.R.China.
IAVSD; 2013. p. 1–10.
[79] Shackleton P, Iwnicki S. Wheel-rail contact benchmark, version 3.0 [Rail technology unit,
Manchester metropolitan university]; 2006.
[80] Magel E, Oldknow K. Quality indices for managing rail through grinding. In: Li Z, Núñez A,
editors. Proceedings of the 11th International Conference on Contact Mechanics and Wear of
Rail/Wheel Systems. The Netherlands: Delft University of Technology; 2018. p. 658–667.
