# A Model of the Transient Behavior of Tractive Rolling Contacts

Authors: Farid Al-Bender; Kris De Moerlooze
Journal: Advances in Tribology
 (2008)

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## Abstract

When an elastic body of revolution rolls tractively over another,
the period from commencement of rolling until gross rolling ensues
is termed the prerolling regime. The resultant tractions in this regime are
characterized by rate-independent hysteresis behavior with nonlocal memory in
function of the traversed displacement. This paper is dedicated to the theoretical
characterization of traction during prerolling. Firstly, a theory is developed to calculate
the traction field during prerolling in function of the instantaneous rolling displacement,
the imposed longitudinal, lateral and spin creepages, and the elastic contact parameters.
Secondly, the theory is implemented in a numerical scheme to calculate the resulting traction
forces and moments on the tractive rolling of a ball. Thirdly, the basic hysteresis characteristics
are systematically established by means of influence-parameters simulations using dimensionless
forms of the problem parameters. The results obtained are consistent with the limiting cases available
in literature and they confirm experimental prerolling hysteresis observations. Furthermore, in a second
paper, this theory is validated experimentally for the case of V-grooved track.


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## Body

## 1. Introduction

WhenThe authors dedicate this paper to the memory of Prof.  J.  J.
Kalker (1933–2006). an
elastic body of revolution rolls tractively over another, the traction field in
the contact patch changes progressively with the distance traversed, from its
initial distribution, until it reaches a certain constant distribution. This
distribution, which is independent of the initial field prior to commencement
of rolling, does not vary with further (steady-state) rolling. This eventual
rolling regime is termed gross rolling; the period building up to it, from commencement of rolling, is termed
the prerolling regime. The
resultant traction in this regime is characterized by rate-independent
hysteresis behavior with nonlocal memory in function of the traversed
displacement [1].
Although steady-state gross rolling is fairly well understood and theoretically
founded, the situation is different in regard to prerolling. This paper deals
with the theoretical treatment of the prerolling period.The research on tractive rolling contact phenomena
dates back to 1875 when Reynolds [2] describes  the phenomenon of creepage. He
uses creepage measurements between a rubber
cylinder and a metal plate to confirm his proposition that the contact region
of a rolling contact is divided into stick zones and microslip zones,
determined by frictional forces and elastic deformation in the contact. The
findings of Hertz [3]
in 1882 form the necessary basis for the beginning of research on rolling
friction.The treatment of rolling motions starts with Carter
[4] in 1926. He
considered the steady-state tractive rolling of an
elastic cylinder, which transmits a tractive force at the plane on which it is
rolling. Carter presented a solution to this
problem in a two-dimensional form (plane strain). He
defined the relation between creepage and creepage
forces, applied on locomotive wheels, where high tangential forces are
transmitted from the wheel to the rail during accelerating and braking the
vehicle. He proved that from the moment that a
braking or tractive couple is applied to the wheel, creepage occurs. This
two-dimensional theory is extended to the three-dimensional case by Johnson
[5, 6]. He
considered two rolling balls, including the
longitudinal and lateral creepages, however without spin creepage. Vermeulen
and Johnson [7]
subsequently extended this theory to arbitrary
smooth half-space bodies.Kalker [8–10] developed numerical
methods which are able to deal with elastic rolling involving three-dimensional
frictional contacts, with imposed creepage and spin, considered constant
throughout the contact spot. His universal computer algorithm Contact [8] deals with all
contact problems of half-space bodies. This algorithm, which is based on the
exact theory of Kalker [10], is computationally intensive, thus not suitable for
real-time applications. For this reason, Kalker developed
the simplified theory, which is used in his algorithm Fastsim [9]. The reference work of Garg
and Dukkipati [11]
serves as a good summary of the preceding theoretical work.Nielsen [12] considered corrugation by
abrasive wear and the case of a velocity dependent friction coefficient in the
contact of two-dimensional quasi-identical bodies. Two bodies are
quasi-identical when they are geometrically and elastically symmetric, which
means that their elastic constants are equal, considering the homogeneous
isotropic case [10].
Li [13] investigated
the wear of rolling contact in railway applications by developing a simulation
tool to model the evolution of the contacting surfaces, with the objective to
better predict the wear behavior and to optimize the profile geometry of the
wheel and rail.This short literature overview shows that the research
on prerolling hysteresis was little in evidence despite its importance. The
main scope of the research was up to now mainly situated in the field of
locomotive and automotive design. In this paper, the hysteretic frictional
behavior in the prerolling period is treated.The objective of this study is to extend the
understanding of the frictional behavior in the prerolling regime by developing
a theoretical model for (pre-)rolling friction, partly based on existing
theories [10]. In very
precise positioning, this period, which occurs after every velocity reversal
and extends for a distance on the order of magnitude of the contact patch
radius, is of main importance [14].
It is mainly the hysteresis effect in this period that is responsible for the
stiffness and damping characteristics of a rolling element guideway in the
direction of rolling [15, 16].
As rolling element bearings are widely used in machine guidance, knowledge and
theoretical quantification of this hysteretic behavior is important. Moreover,
for the treatment of torsional or rolling vibrations in railway wheels or
rolling elements, a theory of prerolling may be an important prerequisite. To
this end, it is the intent of this paper to extend the simplified theory of
rolling contacts, based on the work of Kalker [10], to the case of prerolling.The paper is structured as follows. Section2 gives an
overview of the contact definition in a nonconforming Hertzian contact, the
tractive rolling kinematics, and the rolling theory. A simplified
traction-displacement relationship is formulated. Moreover, the initial
traction field, formed when the bodies first come into contact, is determined
as an initial value to the prerolling problem. Section 3 introduces the
solution method developed in the scope of this research topic. Here, the
implementation details are introduced and dimensional analysis is applied to
the theory in order to yield tractable results. In Section 4, the model results
are presented. The steady-state results as well as the transient results are
discussed. Afterwards, the evolution of the tractive forces as function of the
relative motion (creepage and/or spin) is given. Furthermore, a parameter study
is applied to obtain better knowledge of the phenomenon and to make it possible
to situate experimental results in a consistent framework. Finally,
appropriate conclusions are drawn and future work is indicated.


## 2. Formulation

### 2.1. Contact Definition and Preliminary
Assumptions

Let us consider
the general elastic, nonconformal contact of two bodies of revolution.
Depending on their elasticity moduli, the two bodies will deform to certain
degrees (see Figure1(a)). For the purpose of our study, we shall assume that
the problem may be reduced to that of a single equivalent elastic body of
revolution with a plain infinitely rigid body (see Figure 1(b)). This
assumption is widely adopted in the theory and application of Hertzian contacts
so that the conversion formulas of geometry and elasticity are well established
[17, 18]. However, this assumption
excludes certain phenomena such as Heathcote slip. Let us note, firstly, that
when the externally imposed creepages result in appreciably larger microslip
levels than Heathcote slip, the assumption will yield good approximations.
Secondly, it would still be possible to adopt this assumption and account for
Heathcote slip by calculating an equivalent creepage field that corresponds to
that case.Reduction of the elastic contact between two balls to that of one ball and a
rigid flat surface.
(a)(b)When this assumption is used, the equivalent modulus
of elasticity,Eh (Hertzian
modulus), and the equivalent radius of the resulting single body of revolution, Rh, are given by
(1)1Eh=1−ν12E1+1−ν22E2,1Rh=1R1+1R2.
(NB. This formula is also
applicable to each of the principle radii of an ellipsoid.)In the rest of this article, we shall consider,
without loss of generality, the contact of an elastic sphere with a smooth,
rigid plane. The case of an elliptical contact may be dealt with in an
analogous manner using appropriate conversion formulas.


### 2.2. Normal Stresses and Contact Patch

When an elastic
sphere of radiusR and equivalent
modulus of elasticity Ee=E/(1−ν2) is pressed with
a load W against a rigid
plane surface, the following  obtaining [17, 19].The circular contact patch radiusa is given
by(2)a=(3WR4Ee)1/3.The contact patch is defined as
the region A in the xy-plane: A={(x,y):x2+y2≤a2}.The normal stresspz is given
by(3)pz(x,y)=p01−(x/a)2−(y/a)2with(4)p0=3W2πa2.


### 2.3. Tractive Rolling Kinematics

Figure2 gives an overview of the different creepages
which can occur in tractive rolling. Here, one considers a sphere of radius R rolling in the x-direction such
that its center is translating at a velocity V. In addition, the sphere is spinning around its x, y, and z axes with
angular speeds ψ˙, ω, and ϕ˙, respectively. The creepages, which characterize
tractive rolling, are defined as follows:(5)δVx=−ωR+Visthelongitudinalcreepage,δVy=ψ˙Risthelateralcreepage,ϕ˙isthespincreepage.In most cases of interest, the
magnitudes of these creepages are proportional to the magnitude of the rolling
velocity. In that case, they can be expressed in terms of displacements per
traversed rolling distance (as will be apparent further below).Figure 2
Overview of the rolling motion.The problem of an elastic sphere tractively rolling on
a rigid plane can be reduced to that in which the contact patch is stationary
in space and time. This is accomplished by assuming the rigid plane upon which
the sphere is rolling to be moving in the opposite direction with a velocityV (see Figure 2).
Looking through the stationary contact patch in a direction normal to it at
points in the bodies that are sufficiently remote from the interface, one will
then see two surfaces entering it: one moving with velocity V and one with
velocity V+c, where(6)c=(δVx−ϕ˙y,δVy+ϕ˙x)
is the creepage
velocity vector, as depicted in Figure 3.Figure 3
An overview of the relative motion between the two bodies.In order to arrive at an equation to describe the
kinematics of the surface points lying inside the contact patchA, one follows a point on the surface of the sphere,
which enters the contact patch and mates at the entrance with a counterpoint on
the rigid plane. Owing to the creepage, the point on the surface of the sphere
will have to deform in the plane of the contact patch by an amount u=(u,v) being called
the displacement, which is generally function of space and time. This situation
is depicted in Figure 4, which illustrates, in the 2D case, the difference between
pure rolling and tractive rolling. Now, defining the slip s as the relative
velocity between mating points in the interface, and assuming that (u,v) are small as
compared to contact patch dimensions, one obtains the following differential
equation for the plane deformations [10, 19, 20]:
(7)s=c+DuDt.Writing out the material
derivative D/Dt=∂/∂t+x˙∂/∂x and x˙=−V, one obtains(8)s=c−V∂u∂x+∂u∂t. Generally, V=V(t). If, as indicated earlier, slip and creepage are
scaled with V, then one can rewrite the previous equation in terms
of the traversed rolling distance q rather than the
time and rolling velocity by making use of the substitution(9)q=∫0tV(τ)dτ.Substituting this into (8)
yields(10)s=c−V∂u∂x+V∂u∂q.Finally, normalizing creepage
and slip by the rolling speed, one obtains(11)S=C−∂u∂x+∂u∂q,where S=s/V=(Sx,Sy) is the relative
slip and C=c/V=(ξx−ϕy/a,ξy+ϕx/a) is the relative
creepage.The case of rolling without slip (a) and rolling with
slip (b).
(a)(b)(In this formulation, the creepages(ξx=δVx/V,ξy=δVy/V,ϕ=ϕ˙a/V) have units
m/m.) Moreover, when the rolling speed is constant, then C will be a
constant vector. Otherwise, it will be function of the traversed distance. In
the rest of the treatment, the form given by (11) will be adopted.


### 2.4. The Rolling Theory

#### 2.4.1. Traction-Displacement Relationship: Simplified Theory

Following Kalker [10], we consider the simplified “Winkler bedding” model
to determine the relationship between surface displacement (u,v) and the
tangential traction field (px,py) of the rolling
object for the case when no slip occurs. Allowance for slip is considered
subsequently together with normal traction (Hertzian pressure) treatment.In this simplified approach, the surface of the
elastic object is considered to be covered by elastic “bristles”, normal to
it, which have constant stiffness for tangential deformations,L, that is,(12)(u,v)=L(px,py),where (px,py) is the
tangential traction field and L is the
tangential flexibility of the “bristle”.The flexibility parameterL depends not
only on the elasticity characteristics and Poisson's ratio, but also on the
aspect ratio of the contact ellipse and the magnitude of the creepages. The
determination of an appropriate value for L, for a particular case, is carried out by formal
comparison of analytical, no-slip solutions using the simplified theory with
those using exact theory, see Kalker [10]:(13)L=ℱ(G,Cij,ξx,ξy,ϕ),(14)L=L1|ξx|+L2|ξy|+L3|ϕ|ξx2+ξy2+ϕ2.It is shown in Kalker [10] that this approximation
yields solutions which are very close to those given by exact theory.


#### 2.4.2. The Traction Bound and Slip Conditions

When a
tangential displacement field(u,v) is given, (12)
will yield the corresponding traction field(15)(px,py)=(u,v)L.This traction field corresponds
to the no-slip condition. In practice, the local friction coefficient and
normal traction (Hertz pressure) will determine whether no-slip will hold. In
other words, (15) will be valid only if(16)∥(px,py)∥≤μpz,where, pz is given by
(3), (4) and μ is the local
coefficient of friction or adhesion. Generally, μ will depend on
the contact conditions. However, in the present study, we consider only the
case of constant μ.The condition (16) is known as the traction bound,
which leads to the following relationship for determining the tangential
tractions:(17)(px,py)={∥(u,v)∥L,∥(px,py)∥≤μpz,μpz(u,v)∥(u,v)∥,otherwise.


#### 2.4.3. The Initial Traction Field: Hertzian Contacts with
Friction

Before any
prerolling is initiated, a traction field is already present in the contact due
to the normal loading of the contact. The initial value of the traction field,
which is needed to solve the prerolling problem, can be calculated, considering
a Hertzian axisymmetric contact between a ball and a flat surface. Here, we consider
the normal loading of the ball, including the frictional behavior between the
ball and its contacting surface. The shear traction field is given by Hills et
al. [19]:(18)pxy(r)μp0=1−r2−rH(rs−r)∫rrsΨ(t,rs)t21−t2dt,where 0<r<1 is the
dimensionless radial coordinate, μ the local
coefficient of friction, p0 the peak
contact pressure, H(⋅) the Heaviside
step function, rs denotes the
dimensionless radius of the stick region, and Ψ(r,rs) is given by the
following integral expression:(19)Ψ(r,rs)={0,ifrs≤r≤1,2π(wr∫0π/2dθ(1−(1−r2)sin2θ)1−rs'2sin2θ))−rsK'(rs)r(wrs−Q0(w)Q0(rs))if0≤r≤s,where K′(rs) is a complete
elliptic integral of the first kind, rs′2=1−rs2,(20)Q0(r)=12ln(1+r1−r),w2=rs2−r21−r2.In the case of elastic similar
bodies, the entire contact zone sticks, thus rs=1. This adhesive limit was studied by Goodman [21] and can be expressed in
closed form by(21)pxy(r)βp0=rπ∫0r′lnxdx(1−x2)2,0<r<1,with β is the Dundurs'
constant [22, 23] given by(22)β=(1−2ν1)/(2G1)−(1−2ν2)/(2G2)(1−ν1)/G1+(1−ν2)/G2.Figure 5 shows the radial shear
traction distribution for different values of rs. This field serves as an initial value for the
prerolling model. Let us note however that once gross rolling is attained, the
traction field corresponding to it will serve as the initial value for
subsequent rolling.Figure 5
The radial
shear traction distribution for different partial slip cases.


### 3. Solution Procedure

Referring to
the previous section, the problem to be solved may be stated as follows: solve
(11) subject to conditions (12), (15), (16) to determine the tangential
displacements and tractions. First, we show that (11) admits a closed form
general solution for the case of zero slip. Secondly, we apply the stiffness
and slip conditions (12), (15), (16) in a numerical implementation to determine
the resulting traction field. Finally, the traction forces are determined.


#### 3.1. Analytical, Zero-Slip Solution

First, we write
the two members of (11) in the form(23)−∂u∂x+∂u∂q=Sx−ξx+ϕya,(24)−∂v∂x+∂v∂q=Sy−ξy−ϕxa.If the slip (Sx,Sy) is given, (23),
(24) admit general solutions which are derived in detail in the appendix.Since the slip field is not known beforehand, the
solutions of interest for numerical implementation are those corresponding to
the case of zero-slip. Thus, for(Sx,Sy)≡0, we have the following solutions (see the appendix):(25)u(x,y,q)=q⋅(−ξx+ϕya)+f(x+q,y).Similarly, the general solution
for v is(26)v(x,y,q)=q⋅(−ξy−ϕa(x+q2))+g(x+q,y),where f,g are any
arbitrary functions. Note that outside the contact patch A, u≡v≡0.Let us remark first that the problem is not coupled in(x,y), that is, y can be treated
as a parameter in (25) and (26). Consequently, we can solve the problem on any
line y=y0 in the contact
patch. Thus, referring to Figure 6, we consider the solution on the line
segment y=y0 which is
bounded by the leading edge x=a(y0) and the
trailing edge x=−a(y0). Since the surface points enter the contact patch
free of stress, we have(27)u=v=0∀qatx=a(y0),which provides the necessary
boundary condition for the solution.Figure 6
The case of rolling without slip (left) and rolling with slip (right).The initial surface displacement distribution atq=0 is assumed to
be given, for example, from Section 2.4.3, as(28)q={0,u=u0(x,y0),0,v=v0(x,y0).Substituting this into (25) and (26), we obtain(29)f(x,y0)=u0(x,y0),g(x,y0)=v0(x,y0).Equation (29) means
that the initial (u,v) distribution
determines the arbitrary functions f and g on the range of
definition of (u,v). Thus, by replacing x by x+q, on −a≤x+q≤a, we have that(30)f(x+q,y0)=u0(x+q,y0),g(x+q,y0)=v0(x+q,y0),−a(y0)≤x≤a(y0)−q. In order to obtain the form of functions f and g in the interval a(y0)−q≤x≤a(y0), we make use of the initial condition at x=a(y0). Thus, substituting (27) into (25) and (26)
yields(31)u(a(y0),y0,q)=0=q⋅(−ξx+ϕy0a)+f(a(y0))+q,y0),v(a(y0),y0,q)=0=q⋅(−ξy−ϕa(a(y0)+q2))+g(a(y0)+q,y0).Replacing q by x+q, on a(y0)−q≤x≤a(y0), and translating x by −a(y0), we have that(32)f(x+q,y0)=−(x+q−a(y0))(−ξx+ϕy0a),g(x+q,y0)=−(x+q−a(y0))(−ξy−ϕ2a(x+q+a(y0))),a(y0)−q≤x≤a(y0).Finally, the analytic solution
is obtained in closed form by substituting of f and g from (30) and
(32) into (25) and (26):(33)u(x,y0,q)=q⋅(−ξx+ϕya)+{−(x+q−a(y0))(−ξx+ϕy0a),a(y0)−q≤x≤a(y0),u0(x+q,y0),−a(y0)≤x≤a(y0)−q,v(x,y0,q)=q⋅(−ξy−ϕa(x+q2))+{−(x+q−a(y0))(−ξy−ϕ2a(x+q+a(y0))),a(y0)−q≤x≤a(y0),v0(x+q,y0),−a(y0)≤x≤a(y0)−q.This solution shows that the
initial displacement distribution passes as a wave, moving from right to left,
through the contact patch. Gross rolling is thus achieved after a rolling
distance equal to 2a is traversed.
The no-slip displacement field corresponding to it is obtained from the top
members of (33)  as(34)u∞(x,y0)=−(x−a(y0))(−ξx+ϕy0a),v∞(x,y0)=−(x−a(y0))(−ξy−ϕ2a(x+a(y0))),which is evidently independent
of the initial distribution.


#### 3.2. Numerical Implementation

In
order to apply the procedure numerically to a given contact problem, the
contact region is discretized into a set of nodes. In Figure7, the grid is
depicted, where we have chosen for programming convenience Δx=Δy. Starting from any initial distribution u0(x,y0),v0(x,y0), (33) yields the zero-slip values at any
desired value of q. In the particular case when the increments Δq=Δx, the analytical solution reduces to the following
simple step-wise procedure. After a step Δq, the new (u,v) distribution is
obtained by augmenting the previous value with Δq(−ξx+ϕy/a,−ξy−ϕx/a), shifting each of the u and v vectors one
position to the left, and padding them with the boundary condition u=v=0 at the right.Figure 7
The
discretization of the contact patch.Next, the traction bound needs to be verified at each
step and the values of(u,v) accordingly
corrected. This is carried out as follows.With any new obtained value(u,v), we calculate(35)(px,py)=L(u,v).If ∥(px,py)∥≤μpz, no slip occurs (Sx=Sy=0) and the values
of (u,v) are retained.
If, on the other hand, ∥(px,py)∥>μpz, then(36)(u,v)new=μpzL((u,v)∥(u,v)∥)old. In Figure 8, the flowchart of
the algorithm is presented.Figure 8
The flowchart of the algorithm.


#### 3.3. Calculation of the TractionsFx,Fy,Mz

The total
traction forces and moment are obtained from the following
integrals:(37)Fx=∫−aa∫−a(y)a(y)px(x,y)dxdy,(38)Fy=∫−aa∫−a(y)a(y)py(x,y)dxdy,(39)Mz=∫−aa∫−a(y)a(y)p(x,y)r(x,y)dxdywith r(x,y) the radial
distance from the center of the contact patch to the location of the node.For the no-slip, gross rolling solution, (37) and (38)
yield (see also [10])(40)Fx=−8a3ξx3L,(41)Fy=−8a3ξy3L−πa3ϕ4L.Upon comparing these results
formally with the results obtained from exact theory, the relations (12) and
(13) are obtained.


#### 3.4. Dimensional analysis

In order to
generate, analyze, and present the results systematically, dimensional analysis
is applied to the problem. This is achieved through the application of the Vaschy-Buckingham-Π theorem
[24, 25] as follows. The rolling
problem may be generally expressed as(42)f(W,F,M,a,p0,p,E,R,V,δVx,δVy,ϕ˙,ν,μ,q,x,y)=0.Inspection shows that the
problem posses three independent dimensions
(length, force, and time). Choosing a, p0, and V as the
variables to be eliminated, we obtain the dimensionless form, in which the
number of variables is reduced by
three:(43)ℱ(Wp0a2,Fp0a2,Mp0a3,aa,p0p0,pp0,Ep0,Ra,VV,δVxV,δVyV,ϕ˙aV,ν,μ,qa,xa,ya)=0,ℱ(W*,F*,M*,p*,E*,R*,ξx,ξy,ϕ,ν,μ,q*,x*,y*)=0.Another possibility is to also
eliminate the (dimensionless) friction coefficient μ since it only
scales the traction forces.We refer to the nomenclature for an overview of the
used symbols. Since the model contains no system dynamics of the ball, the
rolling velocityV falls out of
the equations. This dimensional analysis makes it easier to compare the
influence of the different parameters of the problem in a consistent way.
Therefore, in the next section, the results are also given in this
dimensionless form. An arbitrary value of 0.5 has been used for the coefficient
of sliding friction.


### 4. Results

#### 4.1. Steady-State Rolling

Figure9 depicts traction distributions pertaining to
basic cases of steady-state rolling with longitudinal, lateral, and/or spin
creepages of an elastic ball. While rolling with longitudinal and lateral
creepage is of main interest in wheel-rail or tyre-road contact, rolling with
spin is the dominant type in angular contact ball bearings and linear guideways
(e.g., with V-grooved tracks). For the sake of better visibility, a coarse grid
is used. Figure 9(a) gives the example of equal longitudinal and lateral
creepage. The slip zone is symmetric w.r.t. the x-axis while the
traction vectors are oriented at −45 degrees to it.
Figures 9(b) and 9(c) represent the traction fields with spin creepage of
different levels. With increasing spin, the slip zone becomes larger, until the
total contact zone slips. Figure 9(d) shows how the traction field looks like
when combining spin with creepage in both directions.A few results of the steady-state rolling traction
field, the rolling direction is from the left to the right.
(a)(b)(c)(d)


#### 4.2. Transient Rolling

In this
section, the traction behavior during transition to rolling is analyzed. This
behavior is marked by hysteresis of the traction forces in the rolling
displacement. Again, we limit the presentation to the basic cases of
(pre-)rolling with longitudinal, lateral, and spin creepage. The applied
displacement trajectory is provided with reversal points, to ascertain the
nonlocal memory character of the hysteresis curves, as discussed in [1, 26, 27]. The rolling trajectory is
specified as q*=[0→q1→q2→⋯]. For all cases considered, the initial traction field
is a null field since the contacting materials are identical, which results in
a Dundurs' constant β=0.


#### 4.2.1. Pure Longitudinal Creepage

For this case, we choseξx=0.015,ξy=ϕ=0. The rolling trajectory is q*=[0→2.1→−2.1→2.1]. The longitudinal creepage gives rise only to a
traction force Fx* in the rolling
direction. Figure 10(c) plots Fx* against q* to show the
resulting hysteresis loop. Figures 10(a), 10(b), 10(c), 10(d), and 10(e) show the characteristic
traction fields at selected points during the motion. From these, we see also
that all traction components lie in the rolling direction.The results for transient (pre-)rolling with
longitudinal creepage. In the middle, the hysteresis with around it, transient
traction fields at selected points in the trajectory.
(a)(b)(c)(d)(e)


#### 4.2.2. Pure Lateral Creepage

To illustrate this case, we putξx=0,ξy=0.015,ϕ=0. Rolling combined with a lateral creepage component
gives rise only to a traction field in the y-direction,
that is, perpendicular to the rolling direction. Consequently, the hysteresis
loops in Figure 11(a) depict this traction force Fy* versus the
rolling displacement (q*=[0→2.1→0.4→1.6→−2.1→−0.4→−1.6→2.1]). This
trajectory has been so constructed as to show inner hysteresis loops and thus
the nonlocal memory character of hysteresis. The behavior is, otherwise,
similar to that of pure longitudinal creepage. Figure 11(b) depicts the
steady-state traction field.The results for transient (pre-)rolling with lateral
creepage. The left panel depicts the hysteresis curve while the right panel
visualizes the steady-state rolling traction field.
(a)(b)


#### 4.2.3. Pure Spin

Here, we putvx=vy=0,ϕ=0.0255. Rolling with pure spin results in a traction field
having components in the x and y directions. The
resulting traction force in the x direction
equals zero. The traction stress field results in a spin moment Mz*. This moment is plotted as function of the rolling
displacement to yield the hysteresis curve of Figure 12(c). The input is q*=[0→2.04→0.46→1.54→−2.04→−0.46→−1.54→2.04]. Note that the virgin curve, corresponding to
start of motion until first gross slip, overshoots the subsequent hysteresis
curves. The shape of the virgin curve and the amount of overshoot vary
with the assumed initial traction distribution.The results for transient (pre-)rolling with spin. In
the middle, the hysteresis with around it, some transient traction fields at
selected points in the trajectory.
(a)(b)(c)(d)(e)


### 4.3. Evolution of the Steady-State
Tractions in Function of the Creepages

It is
intuitively plausible to assume that the traction force (or moment in the case
of spin) will increase with increasing creepage until gross slip is reached. In
Figure13, the evolution of the steady-state rolling traction force is depicted
as a function of the relative creepage. For rolling with pure longitudinal
creepage, an increasing trend is observed as shown in Figure 13(a). The same
evolution is noticed for rolling with lateral creepage in Figure 13(b).
Considering rolling with pure spin, the spinning moment Mz* increases
towards a saturation value while the lateral force Fy* shows a local
maximum. These trends are depicted in Figure 13(c). As already mentioned, the
longitudinal force component Fx* equals zero for
this case. These results agree with those obtained in [10].The evolution of the steady-state
traction as function of the creepage.
(a)(b)(c)


### 4.4. Parameter Analysis

In the previous
section, the general behavior of the steady-state traction force in function of
the creepage value is discussed. In this section, the influence of the other
model parameters is discussed. For brevity, we confine the treatment to the
case of pure spin creepage.Consider the dimensionless model of (43). Because the
model parameters are reduced by normalizing with respect to the radius of the
contact patch a and the maximum
normal pressure p0, these parameters fall out of the model equations, so
that (2) becomes(44)1=(3W*R*4Eh*)1/3⟺R*=4Eh*3W*.In other words, R* increases if E* increases or W* decreases. In
this way, R* contains (or
coalesces) all geometry, elasticity, and load information.To study the influence ofR*, this parameter is varied between reasonable bounds.
To determine its influence on the prerolling behavior, three main parameters
are chosen for analysis.(1)
The prerolling
distancexpr*.(2)
The initial
stiffnesski* of the
hysteresis system, that is, the initial slope of the hysteresis
curve,(45)ki*=∂Mz*∂q*|q*=0.(3)
The
steady-state frictional momentMz,ss*.These variables
are depicted as functions of the dimensionless ball radius in Figure14 for
different values of the spin creepage ϕ and in Figure
15 for different values of the coefficient of friction μ. One can notice from (44) that an increase in R* corresponds to
a decrease in W* for a constant
value of Eh*. A certain limit of applicability, Rlim* applies, owing
to geometrical limitations: a cannot be
larger than about R/4 without
violating basic Hertzian assumptions. The plotted values in Figures 14 and 15
below this value of Rlim* (marked by the
box “theoretical region”) are only given for the sake of mathematical
completeness.The behavior of the prerolling region on a variation
of the ball radius for different spin creepages (μ = 0.5).
(a)(b)(c)(d)The behavior of the prerolling region on a variation
of the ball radius for different coefficients of frictionμ(ϕ=0.05).
(a)(b)(c)(d)From Figures14(a) and 15(a), one can see
that the prerolling distance stays constant for low and moderate values of R* (i.e., for
highly loaded contacts), while for higher values (lightly loaded case), the
prerolling distance decreases. Thus, the maximum possible value for the
prerolling distance is twice the contact patch radius. The initial stiffness of
the hysteresis at the beginning of the prerolling region is depicted in Figures
14(b) and 15(b). For low values of R*, the stiffness is constant; for medium values, a
maximal stiffness is observed, while for higher values of R*, the stiffness decreases with R*. In the fully elastic region, we have from the Hertz
theory(46)a∼(WR)(1/3).From Hills et al. [19], we have(47)MWR∼θ∼qR,(48)Mq∼WRR∼a3R, since(49)M*=Mp0a3.This leads to(50)M*∼qR∼qa.aR,(51)ki*=∂M*∂q*∼1R*sinceqa∼q*,Ra=R*.The steady-state tractive moment
is depicted in Figures 14(c) and 15(c). After a constant behavior for low
values of R*, a steep decrease is noticed to end with a quasizero
value for very high values of R*. To get a value for the coefficient of rolling
friction λ*, this steady-state moment is divided by the
dimensionless normal load W*. These results are depicted in Figures 14(d) and
15(d). We notice a steep increase in the region of moderate values for R* to saturate towards
a constant value for higher R*, which results in a similar behavior as compared to
“Amontons' law” [28]
for sliding friction.


## 2.1. Contact Definition and Preliminary
Assumptions

Let us consider
the general elastic, nonconformal contact of two bodies of revolution.
Depending on their elasticity moduli, the two bodies will deform to certain
degrees (see Figure1(a)). For the purpose of our study, we shall assume that
the problem may be reduced to that of a single equivalent elastic body of
revolution with a plain infinitely rigid body (see Figure 1(b)). This
assumption is widely adopted in the theory and application of Hertzian contacts
so that the conversion formulas of geometry and elasticity are well established
[17, 18]. However, this assumption
excludes certain phenomena such as Heathcote slip. Let us note, firstly, that
when the externally imposed creepages result in appreciably larger microslip
levels than Heathcote slip, the assumption will yield good approximations.
Secondly, it would still be possible to adopt this assumption and account for
Heathcote slip by calculating an equivalent creepage field that corresponds to
that case.Reduction of the elastic contact between two balls to that of one ball and a
rigid flat surface.
(a)(b)When this assumption is used, the equivalent modulus
of elasticity,Eh (Hertzian
modulus), and the equivalent radius of the resulting single body of revolution, Rh, are given by
(1)1Eh=1−ν12E1+1−ν22E2,1Rh=1R1+1R2.
(NB. This formula is also
applicable to each of the principle radii of an ellipsoid.)In the rest of this article, we shall consider,
without loss of generality, the contact of an elastic sphere with a smooth,
rigid plane. The case of an elliptical contact may be dealt with in an
analogous manner using appropriate conversion formulas.


## 2.2. Normal Stresses and Contact Patch

When an elastic
sphere of radiusR and equivalent
modulus of elasticity Ee=E/(1−ν2) is pressed with
a load W against a rigid
plane surface, the following  obtaining [17, 19].The circular contact patch radiusa is given
by(2)a=(3WR4Ee)1/3.The contact patch is defined as
the region A in the xy-plane: A={(x,y):x2+y2≤a2}.The normal stresspz is given
by(3)pz(x,y)=p01−(x/a)2−(y/a)2with(4)p0=3W2πa2.


## 2.3. Tractive Rolling Kinematics

Figure2 gives an overview of the different creepages
which can occur in tractive rolling. Here, one considers a sphere of radius R rolling in the x-direction such
that its center is translating at a velocity V. In addition, the sphere is spinning around its x, y, and z axes with
angular speeds ψ˙, ω, and ϕ˙, respectively. The creepages, which characterize
tractive rolling, are defined as follows:(5)δVx=−ωR+Visthelongitudinalcreepage,δVy=ψ˙Risthelateralcreepage,ϕ˙isthespincreepage.In most cases of interest, the
magnitudes of these creepages are proportional to the magnitude of the rolling
velocity. In that case, they can be expressed in terms of displacements per
traversed rolling distance (as will be apparent further below).Figure 2
Overview of the rolling motion.The problem of an elastic sphere tractively rolling on
a rigid plane can be reduced to that in which the contact patch is stationary
in space and time. This is accomplished by assuming the rigid plane upon which
the sphere is rolling to be moving in the opposite direction with a velocityV (see Figure 2).
Looking through the stationary contact patch in a direction normal to it at
points in the bodies that are sufficiently remote from the interface, one will
then see two surfaces entering it: one moving with velocity V and one with
velocity V+c, where(6)c=(δVx−ϕ˙y,δVy+ϕ˙x)
is the creepage
velocity vector, as depicted in Figure 3.Figure 3
An overview of the relative motion between the two bodies.In order to arrive at an equation to describe the
kinematics of the surface points lying inside the contact patchA, one follows a point on the surface of the sphere,
which enters the contact patch and mates at the entrance with a counterpoint on
the rigid plane. Owing to the creepage, the point on the surface of the sphere
will have to deform in the plane of the contact patch by an amount u=(u,v) being called
the displacement, which is generally function of space and time. This situation
is depicted in Figure 4, which illustrates, in the 2D case, the difference between
pure rolling and tractive rolling. Now, defining the slip s as the relative
velocity between mating points in the interface, and assuming that (u,v) are small as
compared to contact patch dimensions, one obtains the following differential
equation for the plane deformations [10, 19, 20]:
(7)s=c+DuDt.Writing out the material
derivative D/Dt=∂/∂t+x˙∂/∂x and x˙=−V, one obtains(8)s=c−V∂u∂x+∂u∂t. Generally, V=V(t). If, as indicated earlier, slip and creepage are
scaled with V, then one can rewrite the previous equation in terms
of the traversed rolling distance q rather than the
time and rolling velocity by making use of the substitution(9)q=∫0tV(τ)dτ.Substituting this into (8)
yields(10)s=c−V∂u∂x+V∂u∂q.Finally, normalizing creepage
and slip by the rolling speed, one obtains(11)S=C−∂u∂x+∂u∂q,where S=s/V=(Sx,Sy) is the relative
slip and C=c/V=(ξx−ϕy/a,ξy+ϕx/a) is the relative
creepage.The case of rolling without slip (a) and rolling with
slip (b).
(a)(b)(In this formulation, the creepages(ξx=δVx/V,ξy=δVy/V,ϕ=ϕ˙a/V) have units
m/m.) Moreover, when the rolling speed is constant, then C will be a
constant vector. Otherwise, it will be function of the traversed distance. In
the rest of the treatment, the form given by (11) will be adopted.


## 2.4. The Rolling Theory

### 2.4.1. Traction-Displacement Relationship: Simplified Theory

Following Kalker [10], we consider the simplified “Winkler bedding” model
to determine the relationship between surface displacement (u,v) and the
tangential traction field (px,py) of the rolling
object for the case when no slip occurs. Allowance for slip is considered
subsequently together with normal traction (Hertzian pressure) treatment.In this simplified approach, the surface of the
elastic object is considered to be covered by elastic “bristles”, normal to
it, which have constant stiffness for tangential deformations,L, that is,(12)(u,v)=L(px,py),where (px,py) is the
tangential traction field and L is the
tangential flexibility of the “bristle”.The flexibility parameterL depends not
only on the elasticity characteristics and Poisson's ratio, but also on the
aspect ratio of the contact ellipse and the magnitude of the creepages. The
determination of an appropriate value for L, for a particular case, is carried out by formal
comparison of analytical, no-slip solutions using the simplified theory with
those using exact theory, see Kalker [10]:(13)L=ℱ(G,Cij,ξx,ξy,ϕ),(14)L=L1|ξx|+L2|ξy|+L3|ϕ|ξx2+ξy2+ϕ2.It is shown in Kalker [10] that this approximation
yields solutions which are very close to those given by exact theory.


### 2.4.2. The Traction Bound and Slip Conditions

When a
tangential displacement field(u,v) is given, (12)
will yield the corresponding traction field(15)(px,py)=(u,v)L.This traction field corresponds
to the no-slip condition. In practice, the local friction coefficient and
normal traction (Hertz pressure) will determine whether no-slip will hold. In
other words, (15) will be valid only if(16)∥(px,py)∥≤μpz,where, pz is given by
(3), (4) and μ is the local
coefficient of friction or adhesion. Generally, μ will depend on
the contact conditions. However, in the present study, we consider only the
case of constant μ.The condition (16) is known as the traction bound,
which leads to the following relationship for determining the tangential
tractions:(17)(px,py)={∥(u,v)∥L,∥(px,py)∥≤μpz,μpz(u,v)∥(u,v)∥,otherwise.


### 2.4.3. The Initial Traction Field: Hertzian Contacts with
Friction

Before any
prerolling is initiated, a traction field is already present in the contact due
to the normal loading of the contact. The initial value of the traction field,
which is needed to solve the prerolling problem, can be calculated, considering
a Hertzian axisymmetric contact between a ball and a flat surface. Here, we consider
the normal loading of the ball, including the frictional behavior between the
ball and its contacting surface. The shear traction field is given by Hills et
al. [19]:(18)pxy(r)μp0=1−r2−rH(rs−r)∫rrsΨ(t,rs)t21−t2dt,where 0<r<1 is the
dimensionless radial coordinate, μ the local
coefficient of friction, p0 the peak
contact pressure, H(⋅) the Heaviside
step function, rs denotes the
dimensionless radius of the stick region, and Ψ(r,rs) is given by the
following integral expression:(19)Ψ(r,rs)={0,ifrs≤r≤1,2π(wr∫0π/2dθ(1−(1−r2)sin2θ)1−rs'2sin2θ))−rsK'(rs)r(wrs−Q0(w)Q0(rs))if0≤r≤s,where K′(rs) is a complete
elliptic integral of the first kind, rs′2=1−rs2,(20)Q0(r)=12ln(1+r1−r),w2=rs2−r21−r2.In the case of elastic similar
bodies, the entire contact zone sticks, thus rs=1. This adhesive limit was studied by Goodman [21] and can be expressed in
closed form by(21)pxy(r)βp0=rπ∫0r′lnxdx(1−x2)2,0<r<1,with β is the Dundurs'
constant [22, 23] given by(22)β=(1−2ν1)/(2G1)−(1−2ν2)/(2G2)(1−ν1)/G1+(1−ν2)/G2.Figure 5 shows the radial shear
traction distribution for different values of rs. This field serves as an initial value for the
prerolling model. Let us note however that once gross rolling is attained, the
traction field corresponding to it will serve as the initial value for
subsequent rolling.Figure 5
The radial
shear traction distribution for different partial slip cases.


## 2.4.1. Traction-Displacement Relationship: Simplified Theory

Following Kalker [10], we consider the simplified “Winkler bedding” model
to determine the relationship between surface displacement (u,v) and the
tangential traction field (px,py) of the rolling
object for the case when no slip occurs. Allowance for slip is considered
subsequently together with normal traction (Hertzian pressure) treatment.In this simplified approach, the surface of the
elastic object is considered to be covered by elastic “bristles”, normal to
it, which have constant stiffness for tangential deformations,L, that is,(12)(u,v)=L(px,py),where (px,py) is the
tangential traction field and L is the
tangential flexibility of the “bristle”.The flexibility parameterL depends not
only on the elasticity characteristics and Poisson's ratio, but also on the
aspect ratio of the contact ellipse and the magnitude of the creepages. The
determination of an appropriate value for L, for a particular case, is carried out by formal
comparison of analytical, no-slip solutions using the simplified theory with
those using exact theory, see Kalker [10]:(13)L=ℱ(G,Cij,ξx,ξy,ϕ),(14)L=L1|ξx|+L2|ξy|+L3|ϕ|ξx2+ξy2+ϕ2.It is shown in Kalker [10] that this approximation
yields solutions which are very close to those given by exact theory.


## 2.4.2. The Traction Bound and Slip Conditions

When a
tangential displacement field(u,v) is given, (12)
will yield the corresponding traction field(15)(px,py)=(u,v)L.This traction field corresponds
to the no-slip condition. In practice, the local friction coefficient and
normal traction (Hertz pressure) will determine whether no-slip will hold. In
other words, (15) will be valid only if(16)∥(px,py)∥≤μpz,where, pz is given by
(3), (4) and μ is the local
coefficient of friction or adhesion. Generally, μ will depend on
the contact conditions. However, in the present study, we consider only the
case of constant μ.The condition (16) is known as the traction bound,
which leads to the following relationship for determining the tangential
tractions:(17)(px,py)={∥(u,v)∥L,∥(px,py)∥≤μpz,μpz(u,v)∥(u,v)∥,otherwise.


## 2.4.3. The Initial Traction Field: Hertzian Contacts with
Friction

Before any
prerolling is initiated, a traction field is already present in the contact due
to the normal loading of the contact. The initial value of the traction field,
which is needed to solve the prerolling problem, can be calculated, considering
a Hertzian axisymmetric contact between a ball and a flat surface. Here, we consider
the normal loading of the ball, including the frictional behavior between the
ball and its contacting surface. The shear traction field is given by Hills et
al. [19]:(18)pxy(r)μp0=1−r2−rH(rs−r)∫rrsΨ(t,rs)t21−t2dt,where 0<r<1 is the
dimensionless radial coordinate, μ the local
coefficient of friction, p0 the peak
contact pressure, H(⋅) the Heaviside
step function, rs denotes the
dimensionless radius of the stick region, and Ψ(r,rs) is given by the
following integral expression:(19)Ψ(r,rs)={0,ifrs≤r≤1,2π(wr∫0π/2dθ(1−(1−r2)sin2θ)1−rs'2sin2θ))−rsK'(rs)r(wrs−Q0(w)Q0(rs))if0≤r≤s,where K′(rs) is a complete
elliptic integral of the first kind, rs′2=1−rs2,(20)Q0(r)=12ln(1+r1−r),w2=rs2−r21−r2.In the case of elastic similar
bodies, the entire contact zone sticks, thus rs=1. This adhesive limit was studied by Goodman [21] and can be expressed in
closed form by(21)pxy(r)βp0=rπ∫0r′lnxdx(1−x2)2,0<r<1,with β is the Dundurs'
constant [22, 23] given by(22)β=(1−2ν1)/(2G1)−(1−2ν2)/(2G2)(1−ν1)/G1+(1−ν2)/G2.Figure 5 shows the radial shear
traction distribution for different values of rs. This field serves as an initial value for the
prerolling model. Let us note however that once gross rolling is attained, the
traction field corresponding to it will serve as the initial value for
subsequent rolling.Figure 5
The radial
shear traction distribution for different partial slip cases.


## 3. Solution Procedure

Referring to
the previous section, the problem to be solved may be stated as follows: solve
(11) subject to conditions (12), (15), (16) to determine the tangential
displacements and tractions. First, we show that (11) admits a closed form
general solution for the case of zero slip. Secondly, we apply the stiffness
and slip conditions (12), (15), (16) in a numerical implementation to determine
the resulting traction field. Finally, the traction forces are determined.


### 3.1. Analytical, Zero-Slip Solution

First, we write
the two members of (11) in the form(23)−∂u∂x+∂u∂q=Sx−ξx+ϕya,(24)−∂v∂x+∂v∂q=Sy−ξy−ϕxa.If the slip (Sx,Sy) is given, (23),
(24) admit general solutions which are derived in detail in the appendix.Since the slip field is not known beforehand, the
solutions of interest for numerical implementation are those corresponding to
the case of zero-slip. Thus, for(Sx,Sy)≡0, we have the following solutions (see the appendix):(25)u(x,y,q)=q⋅(−ξx+ϕya)+f(x+q,y).Similarly, the general solution
for v is(26)v(x,y,q)=q⋅(−ξy−ϕa(x+q2))+g(x+q,y),where f,g are any
arbitrary functions. Note that outside the contact patch A, u≡v≡0.Let us remark first that the problem is not coupled in(x,y), that is, y can be treated
as a parameter in (25) and (26). Consequently, we can solve the problem on any
line y=y0 in the contact
patch. Thus, referring to Figure 6, we consider the solution on the line
segment y=y0 which is
bounded by the leading edge x=a(y0) and the
trailing edge x=−a(y0). Since the surface points enter the contact patch
free of stress, we have(27)u=v=0∀qatx=a(y0),which provides the necessary
boundary condition for the solution.Figure 6
The case of rolling without slip (left) and rolling with slip (right).The initial surface displacement distribution atq=0 is assumed to
be given, for example, from Section 2.4.3, as(28)q={0,u=u0(x,y0),0,v=v0(x,y0).Substituting this into (25) and (26), we obtain(29)f(x,y0)=u0(x,y0),g(x,y0)=v0(x,y0).Equation (29) means
that the initial (u,v) distribution
determines the arbitrary functions f and g on the range of
definition of (u,v). Thus, by replacing x by x+q, on −a≤x+q≤a, we have that(30)f(x+q,y0)=u0(x+q,y0),g(x+q,y0)=v0(x+q,y0),−a(y0)≤x≤a(y0)−q. In order to obtain the form of functions f and g in the interval a(y0)−q≤x≤a(y0), we make use of the initial condition at x=a(y0). Thus, substituting (27) into (25) and (26)
yields(31)u(a(y0),y0,q)=0=q⋅(−ξx+ϕy0a)+f(a(y0))+q,y0),v(a(y0),y0,q)=0=q⋅(−ξy−ϕa(a(y0)+q2))+g(a(y0)+q,y0).Replacing q by x+q, on a(y0)−q≤x≤a(y0), and translating x by −a(y0), we have that(32)f(x+q,y0)=−(x+q−a(y0))(−ξx+ϕy0a),g(x+q,y0)=−(x+q−a(y0))(−ξy−ϕ2a(x+q+a(y0))),a(y0)−q≤x≤a(y0).Finally, the analytic solution
is obtained in closed form by substituting of f and g from (30) and
(32) into (25) and (26):(33)u(x,y0,q)=q⋅(−ξx+ϕya)+{−(x+q−a(y0))(−ξx+ϕy0a),a(y0)−q≤x≤a(y0),u0(x+q,y0),−a(y0)≤x≤a(y0)−q,v(x,y0,q)=q⋅(−ξy−ϕa(x+q2))+{−(x+q−a(y0))(−ξy−ϕ2a(x+q+a(y0))),a(y0)−q≤x≤a(y0),v0(x+q,y0),−a(y0)≤x≤a(y0)−q.This solution shows that the
initial displacement distribution passes as a wave, moving from right to left,
through the contact patch. Gross rolling is thus achieved after a rolling
distance equal to 2a is traversed.
The no-slip displacement field corresponding to it is obtained from the top
members of (33)  as(34)u∞(x,y0)=−(x−a(y0))(−ξx+ϕy0a),v∞(x,y0)=−(x−a(y0))(−ξy−ϕ2a(x+a(y0))),which is evidently independent
of the initial distribution.


### 3.2. Numerical Implementation

In
order to apply the procedure numerically to a given contact problem, the
contact region is discretized into a set of nodes. In Figure7, the grid is
depicted, where we have chosen for programming convenience Δx=Δy. Starting from any initial distribution u0(x,y0),v0(x,y0), (33) yields the zero-slip values at any
desired value of q. In the particular case when the increments Δq=Δx, the analytical solution reduces to the following
simple step-wise procedure. After a step Δq, the new (u,v) distribution is
obtained by augmenting the previous value with Δq(−ξx+ϕy/a,−ξy−ϕx/a), shifting each of the u and v vectors one
position to the left, and padding them with the boundary condition u=v=0 at the right.Figure 7
The
discretization of the contact patch.Next, the traction bound needs to be verified at each
step and the values of(u,v) accordingly
corrected. This is carried out as follows.With any new obtained value(u,v), we calculate(35)(px,py)=L(u,v).If ∥(px,py)∥≤μpz, no slip occurs (Sx=Sy=0) and the values
of (u,v) are retained.
If, on the other hand, ∥(px,py)∥>μpz, then(36)(u,v)new=μpzL((u,v)∥(u,v)∥)old. In Figure 8, the flowchart of
the algorithm is presented.Figure 8
The flowchart of the algorithm.


### 3.3. Calculation of the TractionsFx,Fy,Mz

The total
traction forces and moment are obtained from the following
integrals:(37)Fx=∫−aa∫−a(y)a(y)px(x,y)dxdy,(38)Fy=∫−aa∫−a(y)a(y)py(x,y)dxdy,(39)Mz=∫−aa∫−a(y)a(y)p(x,y)r(x,y)dxdywith r(x,y) the radial
distance from the center of the contact patch to the location of the node.For the no-slip, gross rolling solution, (37) and (38)
yield (see also [10])(40)Fx=−8a3ξx3L,(41)Fy=−8a3ξy3L−πa3ϕ4L.Upon comparing these results
formally with the results obtained from exact theory, the relations (12) and
(13) are obtained.


### 3.4. Dimensional analysis

In order to
generate, analyze, and present the results systematically, dimensional analysis
is applied to the problem. This is achieved through the application of the Vaschy-Buckingham-Π theorem
[24, 25] as follows. The rolling
problem may be generally expressed as(42)f(W,F,M,a,p0,p,E,R,V,δVx,δVy,ϕ˙,ν,μ,q,x,y)=0.Inspection shows that the
problem posses three independent dimensions
(length, force, and time). Choosing a, p0, and V as the
variables to be eliminated, we obtain the dimensionless form, in which the
number of variables is reduced by
three:(43)ℱ(Wp0a2,Fp0a2,Mp0a3,aa,p0p0,pp0,Ep0,Ra,VV,δVxV,δVyV,ϕ˙aV,ν,μ,qa,xa,ya)=0,ℱ(W*,F*,M*,p*,E*,R*,ξx,ξy,ϕ,ν,μ,q*,x*,y*)=0.Another possibility is to also
eliminate the (dimensionless) friction coefficient μ since it only
scales the traction forces.We refer to the nomenclature for an overview of the
used symbols. Since the model contains no system dynamics of the ball, the
rolling velocityV falls out of
the equations. This dimensional analysis makes it easier to compare the
influence of the different parameters of the problem in a consistent way.
Therefore, in the next section, the results are also given in this
dimensionless form. An arbitrary value of 0.5 has been used for the coefficient
of sliding friction.


## 3.1. Analytical, Zero-Slip Solution

First, we write
the two members of (11) in the form(23)−∂u∂x+∂u∂q=Sx−ξx+ϕya,(24)−∂v∂x+∂v∂q=Sy−ξy−ϕxa.If the slip (Sx,Sy) is given, (23),
(24) admit general solutions which are derived in detail in the appendix.Since the slip field is not known beforehand, the
solutions of interest for numerical implementation are those corresponding to
the case of zero-slip. Thus, for(Sx,Sy)≡0, we have the following solutions (see the appendix):(25)u(x,y,q)=q⋅(−ξx+ϕya)+f(x+q,y).Similarly, the general solution
for v is(26)v(x,y,q)=q⋅(−ξy−ϕa(x+q2))+g(x+q,y),where f,g are any
arbitrary functions. Note that outside the contact patch A, u≡v≡0.Let us remark first that the problem is not coupled in(x,y), that is, y can be treated
as a parameter in (25) and (26). Consequently, we can solve the problem on any
line y=y0 in the contact
patch. Thus, referring to Figure 6, we consider the solution on the line
segment y=y0 which is
bounded by the leading edge x=a(y0) and the
trailing edge x=−a(y0). Since the surface points enter the contact patch
free of stress, we have(27)u=v=0∀qatx=a(y0),which provides the necessary
boundary condition for the solution.Figure 6
The case of rolling without slip (left) and rolling with slip (right).The initial surface displacement distribution atq=0 is assumed to
be given, for example, from Section 2.4.3, as(28)q={0,u=u0(x,y0),0,v=v0(x,y0).Substituting this into (25) and (26), we obtain(29)f(x,y0)=u0(x,y0),g(x,y0)=v0(x,y0).Equation (29) means
that the initial (u,v) distribution
determines the arbitrary functions f and g on the range of
definition of (u,v). Thus, by replacing x by x+q, on −a≤x+q≤a, we have that(30)f(x+q,y0)=u0(x+q,y0),g(x+q,y0)=v0(x+q,y0),−a(y0)≤x≤a(y0)−q. In order to obtain the form of functions f and g in the interval a(y0)−q≤x≤a(y0), we make use of the initial condition at x=a(y0). Thus, substituting (27) into (25) and (26)
yields(31)u(a(y0),y0,q)=0=q⋅(−ξx+ϕy0a)+f(a(y0))+q,y0),v(a(y0),y0,q)=0=q⋅(−ξy−ϕa(a(y0)+q2))+g(a(y0)+q,y0).Replacing q by x+q, on a(y0)−q≤x≤a(y0), and translating x by −a(y0), we have that(32)f(x+q,y0)=−(x+q−a(y0))(−ξx+ϕy0a),g(x+q,y0)=−(x+q−a(y0))(−ξy−ϕ2a(x+q+a(y0))),a(y0)−q≤x≤a(y0).Finally, the analytic solution
is obtained in closed form by substituting of f and g from (30) and
(32) into (25) and (26):(33)u(x,y0,q)=q⋅(−ξx+ϕya)+{−(x+q−a(y0))(−ξx+ϕy0a),a(y0)−q≤x≤a(y0),u0(x+q,y0),−a(y0)≤x≤a(y0)−q,v(x,y0,q)=q⋅(−ξy−ϕa(x+q2))+{−(x+q−a(y0))(−ξy−ϕ2a(x+q+a(y0))),a(y0)−q≤x≤a(y0),v0(x+q,y0),−a(y0)≤x≤a(y0)−q.This solution shows that the
initial displacement distribution passes as a wave, moving from right to left,
through the contact patch. Gross rolling is thus achieved after a rolling
distance equal to 2a is traversed.
The no-slip displacement field corresponding to it is obtained from the top
members of (33)  as(34)u∞(x,y0)=−(x−a(y0))(−ξx+ϕy0a),v∞(x,y0)=−(x−a(y0))(−ξy−ϕ2a(x+a(y0))),which is evidently independent
of the initial distribution.


## 3.2. Numerical Implementation

In
order to apply the procedure numerically to a given contact problem, the
contact region is discretized into a set of nodes. In Figure7, the grid is
depicted, where we have chosen for programming convenience Δx=Δy. Starting from any initial distribution u0(x,y0),v0(x,y0), (33) yields the zero-slip values at any
desired value of q. In the particular case when the increments Δq=Δx, the analytical solution reduces to the following
simple step-wise procedure. After a step Δq, the new (u,v) distribution is
obtained by augmenting the previous value with Δq(−ξx+ϕy/a,−ξy−ϕx/a), shifting each of the u and v vectors one
position to the left, and padding them with the boundary condition u=v=0 at the right.Figure 7
The
discretization of the contact patch.Next, the traction bound needs to be verified at each
step and the values of(u,v) accordingly
corrected. This is carried out as follows.With any new obtained value(u,v), we calculate(35)(px,py)=L(u,v).If ∥(px,py)∥≤μpz, no slip occurs (Sx=Sy=0) and the values
of (u,v) are retained.
If, on the other hand, ∥(px,py)∥>μpz, then(36)(u,v)new=μpzL((u,v)∥(u,v)∥)old. In Figure 8, the flowchart of
the algorithm is presented.Figure 8
The flowchart of the algorithm.


## 3.3. Calculation of the TractionsFx,Fy,Mz

The total
traction forces and moment are obtained from the following
integrals:(37)Fx=∫−aa∫−a(y)a(y)px(x,y)dxdy,(38)Fy=∫−aa∫−a(y)a(y)py(x,y)dxdy,(39)Mz=∫−aa∫−a(y)a(y)p(x,y)r(x,y)dxdywith r(x,y) the radial
distance from the center of the contact patch to the location of the node.For the no-slip, gross rolling solution, (37) and (38)
yield (see also [10])(40)Fx=−8a3ξx3L,(41)Fy=−8a3ξy3L−πa3ϕ4L.Upon comparing these results
formally with the results obtained from exact theory, the relations (12) and
(13) are obtained.


## 3.4. Dimensional analysis

In order to
generate, analyze, and present the results systematically, dimensional analysis
is applied to the problem. This is achieved through the application of the Vaschy-Buckingham-Π theorem
[24, 25] as follows. The rolling
problem may be generally expressed as(42)f(W,F,M,a,p0,p,E,R,V,δVx,δVy,ϕ˙,ν,μ,q,x,y)=0.Inspection shows that the
problem posses three independent dimensions
(length, force, and time). Choosing a, p0, and V as the
variables to be eliminated, we obtain the dimensionless form, in which the
number of variables is reduced by
three:(43)ℱ(Wp0a2,Fp0a2,Mp0a3,aa,p0p0,pp0,Ep0,Ra,VV,δVxV,δVyV,ϕ˙aV,ν,μ,qa,xa,ya)=0,ℱ(W*,F*,M*,p*,E*,R*,ξx,ξy,ϕ,ν,μ,q*,x*,y*)=0.Another possibility is to also
eliminate the (dimensionless) friction coefficient μ since it only
scales the traction forces.We refer to the nomenclature for an overview of the
used symbols. Since the model contains no system dynamics of the ball, the
rolling velocityV falls out of
the equations. This dimensional analysis makes it easier to compare the
influence of the different parameters of the problem in a consistent way.
Therefore, in the next section, the results are also given in this
dimensionless form. An arbitrary value of 0.5 has been used for the coefficient
of sliding friction.


## 4. Results

### 4.1. Steady-State Rolling

Figure9 depicts traction distributions pertaining to
basic cases of steady-state rolling with longitudinal, lateral, and/or spin
creepages of an elastic ball. While rolling with longitudinal and lateral
creepage is of main interest in wheel-rail or tyre-road contact, rolling with
spin is the dominant type in angular contact ball bearings and linear guideways
(e.g., with V-grooved tracks). For the sake of better visibility, a coarse grid
is used. Figure 9(a) gives the example of equal longitudinal and lateral
creepage. The slip zone is symmetric w.r.t. the x-axis while the
traction vectors are oriented at −45 degrees to it.
Figures 9(b) and 9(c) represent the traction fields with spin creepage of
different levels. With increasing spin, the slip zone becomes larger, until the
total contact zone slips. Figure 9(d) shows how the traction field looks like
when combining spin with creepage in both directions.A few results of the steady-state rolling traction
field, the rolling direction is from the left to the right.
(a)(b)(c)(d)


### 4.2. Transient Rolling

In this
section, the traction behavior during transition to rolling is analyzed. This
behavior is marked by hysteresis of the traction forces in the rolling
displacement. Again, we limit the presentation to the basic cases of
(pre-)rolling with longitudinal, lateral, and spin creepage. The applied
displacement trajectory is provided with reversal points, to ascertain the
nonlocal memory character of the hysteresis curves, as discussed in [1, 26, 27]. The rolling trajectory is
specified as q*=[0→q1→q2→⋯]. For all cases considered, the initial traction field
is a null field since the contacting materials are identical, which results in
a Dundurs' constant β=0.


### 4.2.1. Pure Longitudinal Creepage

For this case, we choseξx=0.015,ξy=ϕ=0. The rolling trajectory is q*=[0→2.1→−2.1→2.1]. The longitudinal creepage gives rise only to a
traction force Fx* in the rolling
direction. Figure 10(c) plots Fx* against q* to show the
resulting hysteresis loop. Figures 10(a), 10(b), 10(c), 10(d), and 10(e) show the characteristic
traction fields at selected points during the motion. From these, we see also
that all traction components lie in the rolling direction.The results for transient (pre-)rolling with
longitudinal creepage. In the middle, the hysteresis with around it, transient
traction fields at selected points in the trajectory.
(a)(b)(c)(d)(e)


### 4.2.2. Pure Lateral Creepage

To illustrate this case, we putξx=0,ξy=0.015,ϕ=0. Rolling combined with a lateral creepage component
gives rise only to a traction field in the y-direction,
that is, perpendicular to the rolling direction. Consequently, the hysteresis
loops in Figure 11(a) depict this traction force Fy* versus the
rolling displacement (q*=[0→2.1→0.4→1.6→−2.1→−0.4→−1.6→2.1]). This
trajectory has been so constructed as to show inner hysteresis loops and thus
the nonlocal memory character of hysteresis. The behavior is, otherwise,
similar to that of pure longitudinal creepage. Figure 11(b) depicts the
steady-state traction field.The results for transient (pre-)rolling with lateral
creepage. The left panel depicts the hysteresis curve while the right panel
visualizes the steady-state rolling traction field.
(a)(b)


### 4.2.3. Pure Spin

Here, we putvx=vy=0,ϕ=0.0255. Rolling with pure spin results in a traction field
having components in the x and y directions. The
resulting traction force in the x direction
equals zero. The traction stress field results in a spin moment Mz*. This moment is plotted as function of the rolling
displacement to yield the hysteresis curve of Figure 12(c). The input is q*=[0→2.04→0.46→1.54→−2.04→−0.46→−1.54→2.04]. Note that the virgin curve, corresponding to
start of motion until first gross slip, overshoots the subsequent hysteresis
curves. The shape of the virgin curve and the amount of overshoot vary
with the assumed initial traction distribution.The results for transient (pre-)rolling with spin. In
the middle, the hysteresis with around it, some transient traction fields at
selected points in the trajectory.
(a)(b)(c)(d)(e)


## 4.1. Steady-State Rolling

Figure9 depicts traction distributions pertaining to
basic cases of steady-state rolling with longitudinal, lateral, and/or spin
creepages of an elastic ball. While rolling with longitudinal and lateral
creepage is of main interest in wheel-rail or tyre-road contact, rolling with
spin is the dominant type in angular contact ball bearings and linear guideways
(e.g., with V-grooved tracks). For the sake of better visibility, a coarse grid
is used. Figure 9(a) gives the example of equal longitudinal and lateral
creepage. The slip zone is symmetric w.r.t. the x-axis while the
traction vectors are oriented at −45 degrees to it.
Figures 9(b) and 9(c) represent the traction fields with spin creepage of
different levels. With increasing spin, the slip zone becomes larger, until the
total contact zone slips. Figure 9(d) shows how the traction field looks like
when combining spin with creepage in both directions.A few results of the steady-state rolling traction
field, the rolling direction is from the left to the right.
(a)(b)(c)(d)


## 4.2. Transient Rolling

In this
section, the traction behavior during transition to rolling is analyzed. This
behavior is marked by hysteresis of the traction forces in the rolling
displacement. Again, we limit the presentation to the basic cases of
(pre-)rolling with longitudinal, lateral, and spin creepage. The applied
displacement trajectory is provided with reversal points, to ascertain the
nonlocal memory character of the hysteresis curves, as discussed in [1, 26, 27]. The rolling trajectory is
specified as q*=[0→q1→q2→⋯]. For all cases considered, the initial traction field
is a null field since the contacting materials are identical, which results in
a Dundurs' constant β=0.


## 4.2.1. Pure Longitudinal Creepage

For this case, we choseξx=0.015,ξy=ϕ=0. The rolling trajectory is q*=[0→2.1→−2.1→2.1]. The longitudinal creepage gives rise only to a
traction force Fx* in the rolling
direction. Figure 10(c) plots Fx* against q* to show the
resulting hysteresis loop. Figures 10(a), 10(b), 10(c), 10(d), and 10(e) show the characteristic
traction fields at selected points during the motion. From these, we see also
that all traction components lie in the rolling direction.The results for transient (pre-)rolling with
longitudinal creepage. In the middle, the hysteresis with around it, transient
traction fields at selected points in the trajectory.
(a)(b)(c)(d)(e)


## 4.2.2. Pure Lateral Creepage

To illustrate this case, we putξx=0,ξy=0.015,ϕ=0. Rolling combined with a lateral creepage component
gives rise only to a traction field in the y-direction,
that is, perpendicular to the rolling direction. Consequently, the hysteresis
loops in Figure 11(a) depict this traction force Fy* versus the
rolling displacement (q*=[0→2.1→0.4→1.6→−2.1→−0.4→−1.6→2.1]). This
trajectory has been so constructed as to show inner hysteresis loops and thus
the nonlocal memory character of hysteresis. The behavior is, otherwise,
similar to that of pure longitudinal creepage. Figure 11(b) depicts the
steady-state traction field.The results for transient (pre-)rolling with lateral
creepage. The left panel depicts the hysteresis curve while the right panel
visualizes the steady-state rolling traction field.
(a)(b)


## 4.2.3. Pure Spin

Here, we putvx=vy=0,ϕ=0.0255. Rolling with pure spin results in a traction field
having components in the x and y directions. The
resulting traction force in the x direction
equals zero. The traction stress field results in a spin moment Mz*. This moment is plotted as function of the rolling
displacement to yield the hysteresis curve of Figure 12(c). The input is q*=[0→2.04→0.46→1.54→−2.04→−0.46→−1.54→2.04]. Note that the virgin curve, corresponding to
start of motion until first gross slip, overshoots the subsequent hysteresis
curves. The shape of the virgin curve and the amount of overshoot vary
with the assumed initial traction distribution.The results for transient (pre-)rolling with spin. In
the middle, the hysteresis with around it, some transient traction fields at
selected points in the trajectory.
(a)(b)(c)(d)(e)


## 4.3. Evolution of the Steady-State
Tractions in Function of the Creepages

It is
intuitively plausible to assume that the traction force (or moment in the case
of spin) will increase with increasing creepage until gross slip is reached. In
Figure13, the evolution of the steady-state rolling traction force is depicted
as a function of the relative creepage. For rolling with pure longitudinal
creepage, an increasing trend is observed as shown in Figure 13(a). The same
evolution is noticed for rolling with lateral creepage in Figure 13(b).
Considering rolling with pure spin, the spinning moment Mz* increases
towards a saturation value while the lateral force Fy* shows a local
maximum. These trends are depicted in Figure 13(c). As already mentioned, the
longitudinal force component Fx* equals zero for
this case. These results agree with those obtained in [10].The evolution of the steady-state
traction as function of the creepage.
(a)(b)(c)


## 4.4. Parameter Analysis

In the previous
section, the general behavior of the steady-state traction force in function of
the creepage value is discussed. In this section, the influence of the other
model parameters is discussed. For brevity, we confine the treatment to the
case of pure spin creepage.Consider the dimensionless model of (43). Because the
model parameters are reduced by normalizing with respect to the radius of the
contact patch a and the maximum
normal pressure p0, these parameters fall out of the model equations, so
that (2) becomes(44)1=(3W*R*4Eh*)1/3⟺R*=4Eh*3W*.In other words, R* increases if E* increases or W* decreases. In
this way, R* contains (or
coalesces) all geometry, elasticity, and load information.To study the influence ofR*, this parameter is varied between reasonable bounds.
To determine its influence on the prerolling behavior, three main parameters
are chosen for analysis.(1)
The prerolling
distancexpr*.(2)
The initial
stiffnesski* of the
hysteresis system, that is, the initial slope of the hysteresis
curve,(45)ki*=∂Mz*∂q*|q*=0.(3)
The
steady-state frictional momentMz,ss*.These variables
are depicted as functions of the dimensionless ball radius in Figure14 for
different values of the spin creepage ϕ and in Figure
15 for different values of the coefficient of friction μ. One can notice from (44) that an increase in R* corresponds to
a decrease in W* for a constant
value of Eh*. A certain limit of applicability, Rlim* applies, owing
to geometrical limitations: a cannot be
larger than about R/4 without
violating basic Hertzian assumptions. The plotted values in Figures 14 and 15
below this value of Rlim* (marked by the
box “theoretical region”) are only given for the sake of mathematical
completeness.The behavior of the prerolling region on a variation
of the ball radius for different spin creepages (μ = 0.5).
(a)(b)(c)(d)The behavior of the prerolling region on a variation
of the ball radius for different coefficients of frictionμ(ϕ=0.05).
(a)(b)(c)(d)From Figures14(a) and 15(a), one can see
that the prerolling distance stays constant for low and moderate values of R* (i.e., for
highly loaded contacts), while for higher values (lightly loaded case), the
prerolling distance decreases. Thus, the maximum possible value for the
prerolling distance is twice the contact patch radius. The initial stiffness of
the hysteresis at the beginning of the prerolling region is depicted in Figures
14(b) and 15(b). For low values of R*, the stiffness is constant; for medium values, a
maximal stiffness is observed, while for higher values of R*, the stiffness decreases with R*. In the fully elastic region, we have from the Hertz
theory(46)a∼(WR)(1/3).From Hills et al. [19], we have(47)MWR∼θ∼qR,(48)Mq∼WRR∼a3R, since(49)M*=Mp0a3.This leads to(50)M*∼qR∼qa.aR,(51)ki*=∂M*∂q*∼1R*sinceqa∼q*,Ra=R*.The steady-state tractive moment
is depicted in Figures 14(c) and 15(c). After a constant behavior for low
values of R*, a steep decrease is noticed to end with a quasizero
value for very high values of R*. To get a value for the coefficient of rolling
friction λ*, this steady-state moment is divided by the
dimensionless normal load W*. These results are depicted in Figures 14(d) and
15(d). We notice a steep increase in the region of moderate values for R* to saturate towards
a constant value for higher R*, which results in a similar behavior as compared to
“Amontons' law” [28]
for sliding friction.


## 5. Discussion and Conclusions

In the foregoing, a theory is developed to
characterize the traction behavior during the transition to rolling. The
following remarks are in order. Firstly, although based on the “Winkler
bedding” simplification, it is shown in [10] that this approximation, with appropriate choice of
the stiffness parameter L, yields solutions which are very close to those given
by exact theory. The advantage gained is the transparency and easy application
of this theory. Secondly, although only the case of point contact has been
treated, this theory can be directly extended to the general cases of
elliptical and line contacts. Thirdly, the cases of variable (e.g., pressure
dependent) local coefficient of friction in the contact patch, variable (e.g.,
time or position dependent) creepages, variable normal load, and variable
rolling velocity can all be directly treated by this theory. In that way,
dynamical contact phenomena, such as those obtaining
during motion reversals, acceleration, deceleration, oscillation, and so forth,
can be accommodated by this theory. Other cases not covered immediately by this
theory are discussed subsequently.Rough Contacts
The replacement of a smooth contact by a rough one generally requires a
higher computational effort, which often mortgages the development of a
simplified version suitable for real-time execution. Bucher et al. [29], for instance, outline that
in the case of rough contact, the stresses and deformations can only be
calculated using special boundary element methods, while transient
three-dimensional rolling contacts comprising rough surfaces are not possible
at present. However, this “bristle” model offers the advantage that it can be
directly extended to rough contacts. The challenge is to translate an actual
rough surface into an equivalent bristle set. Alternatively, recent numerical
methods offer an affordable solution for calculating the rough surface contact
by using dedicated contact solvers [30, 31].Wear and Heathcote Slip
The influence of wear and running-in of the surfaces
is obviously very important in many applications. In the former algorithm, the
creepage was kept constant over the contact patch. Taking wear in
consideration, the need for a variable creepage becomes important. When the
flat surface in which the sphere is rolling, wears, a groove develops in which
the contact between both bodies, is conformal. This
situation is similar to that of a ball in a groove, for example, a deep groove
ball bearing, where the contact is conformal. In that case, the traction field
for pure rolling, that is, with zero relative creepage motion in a conformal
groove, is described by Heathcote [32] and is qualitatively depicted in Figure 16. The
contact area is no longer plane. When rolling freely, there is no net tangential
force, which explains that the contact spot is subdivided into three zones: the
central zone contains positive slip vectors, while in the outer zones, the slip
is negative.Figure 16
The traction
distribution of pure rolling: Heathcote slip.In order to be able to deal with this problem using
the algorithm presented in this paper, the hitherto constant creepage over the
whole contact patch should be replaced by a variable creepage field. The latter
may be obtained from geometrical and kinematical considerations of the contact
with the border line between positive and negative creepage fields being the
only unknown.In conclusion,
this paper considers the hysteretic behavior of the prerolling friction between
a ball and a flat surface. Extending the existing steady-state gross rolling
model, due to J. J. Kalker, to a transient prerolling model, the evolution of
the traction field in the rolling displacement is determined and hysteresis
curves are generated. The traction field and its behavior in the presence of
rolling with creepage and spin are systematically
investigated. Finally, a parameter study is carried out to gain more insight
into this phenomenon. In a forthcoming paper, an experimental validation of the
model is carried out for the case of rolling with spin creepage, using a
configuration consisting of two V-grooved tracks with 2 balls in between.



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*Source: 214894-2008-04-13.xml*