a model of the transient behavior of tractive rolling...

Hindawi Publishing CorporationAdvances in TribologyVolume 2008, Article ID 214894, 17 pagesdoi:10.1155/2008/214894

Research ArticleA Model of the Transient Behavior of Tractive Rolling Contacts

Farid Al-Bender and Kris De Moerlooze

Mechanical Engineering Department, Division Production Engineering, Machine design and Automation,Katholieke Universiteit Leuven, Celestijnenlaan 300B, B-3001 Heverlee, Belgium

Correspondence should be addressed to Farid Al-Bender, [email protected]

Received 19 September 2007; Accepted 6 February 2008

Recommended by Mihai Arghir

When an elastic body of revolution rolls tractively over another, the period from commencement of rolling until gross rollingensues is termed the prerolling regime. The resultant tractions in this regime are characterized by rate-independent hysteresis be-havior with nonlocal memory in function of the traversed displacement. This paper is dedicated to the theoretical characterizationof traction during prerolling. Firstly, a theory is developed to calculate the traction field during prerolling in function of the instan-taneous 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 aball. Thirdly, the basic hysteresis characteristics are systematically established by means of influence-parameters simulations usingdimensionless forms of the problem parameters. The results obtained are consistent with the limiting cases available in literatureand they confirm experimental prerolling hysteresis observations. Furthermore, in a second paper, this theory is validated experi-mentally for the case of V-grooved track.

Copyright © 2008 F. Al-Bender and K. De Moerlooze. This is an open access article distributed under the Creative CommonsAttribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work isproperly cited.

1. INTRODUCTION

When an elastic body of revolution rolls tractively over an-other, the traction field in the contact patch changes pro-gressively with the distance traversed, from its initial distri-bution, until it reaches a certain constant distribution. Thisdistribution, which is independent of the initial field priorto commencement of rolling, does not vary with further(steady-state) rolling. This eventual rolling regime is termedgross rolling; the period building up to it, from commence-ment of rolling, is termed the prerolling regime. The resultanttraction in this regime is characterized by rate-independenthysteresis behavior with nonlocal memory in function ofthe traversed displacement [1]. Although steady-state grossrolling is fairly well understood and theoretically founded,the situation is different in regard to prerolling. This pa-per deals with the theoretical treatment of the prerollingperiod.

The authors dedicate this paper to the memory of Prof. J. J. Kalker (1933–2006).

The research on tractive rolling contact phenomena datesback to 1875 when Reynolds [2] describes the phenomenonof creepage. He uses creepage measurements between a rub-ber cylinder and a metal plate to confirm his proposition thatthe contact region of a rolling contact is divided into stickzones and microslip zones, determined by frictional forcesand elastic deformation in the contact. The findings of Hertz[3] in 1882 form the necessary basis for the beginning of re-search on rolling friction.

The treatment of rolling motions starts with Carter [4]in 1926. He considered the steady-state tractive rolling ofan elastic cylinder, which transmits a tractive force at theplane on which it is rolling. Carter presented a solution tothis problem in a two-dimensional form (plane strain). Hedefined the relation between creepage and creepage forces,applied on locomotive wheels, where high tangential forcesare transmitted from the wheel to the rail during accelerat-ing and braking the vehicle. He proved that from the mo-ment that a braking or tractive couple is applied to the wheel,creepage occurs. This two-dimensional theory is extendedto the three-dimensional case by Johnson [5, 6]. He consid-ered two rolling balls, including the longitudinal and lateral

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2 Advances in Tribology

creepages, however without spin creepage. Vermeulen andJohnson [7] subsequently extended this theory to arbitrarysmooth half-space bodies.

Kalker [8–10] developed numerical methods which areable to deal with elastic rolling involving three-dimensionalfrictional contacts, with imposed creepage and spin, con-sidered constant throughout the contact spot. His universalcomputer algorithm Contact [8] deals with all contact prob-lems of half-space bodies. This algorithm, which is based onthe exact theory of Kalker [10], is computationally inten-sive, thus not suitable for real-time applications. For this rea-son, Kalker developed the simplified theory, which is used inhis algorithm Fastsim [9]. The reference work of Garg andDukkipati [11] serves as a good summary of the precedingtheoretical work.

Nielsen [12] considered corrugation by abrasive wear andthe case of a velocity dependent friction coefficient in thecontact of two-dimensional quasi-identical bodies. Two bod-ies are quasi-identical when they are geometrically and elasti-cally symmetric, which means that their elastic constants areequal, considering the homogeneous isotropic case [10]. Li[13] investigated the wear of rolling contact in railway appli-cations by developing a simulation tool to model the evolu-tion of the contacting surfaces, with the objective to betterpredict the wear behavior and to optimize the profile geom-etry of the wheel and rail.

This short literature overview shows that the research onprerolling hysteresis was little in evidence despite its impor-tance. The main scope of the research was up to now mainlysituated in the field of locomotive and automotive design. Inthis paper, the hysteretic frictional behavior in the prerollingperiod is treated.

The objective of this study is to extend the understand-ing of the frictional behavior in the prerolling regime by de-veloping a theoretical model for (pre-)rolling friction, partlybased on existing theories [10]. In very precise positioning,this period, which occurs after every velocity reversal and ex-tends for a distance on the order of magnitude of the contactpatch radius, is of main importance [14]. It is mainly the hys-teresis effect in this period that is responsible for the stiffnessand damping characteristics of a rolling element guideway inthe direction of rolling [15, 16]. As rolling element bearingsare widely used in machine guidance, knowledge and theo-retical quantification of this hysteretic behavior is important.Moreover, for the treatment of torsional or rolling vibrationsin railway wheels or rolling elements, a theory of prerollingmay be an important prerequisite. To this end, it is the in-tent of this paper to extend the simplified theory of rollingcontacts, based on the work of Kalker [10], to the case of pre-rolling.

The paper is structured as follows. Section 2 gives anoverview of the contact definition in a nonconformingHertzian contact, the tractive rolling kinematics, and the roll-ing theory. A simplified traction-displacement relationshipis formulated. Moreover, the initial traction field, formedwhen the bodies first come into contact, is determined asan initial value to the prerolling problem. Section 3 intro-duces the solution method developed in the scope of this re-search topic. Here, the implementation details are introduced

(a) (b)

Figure 1: Reduction of the elastic contact between two balls to thatof one ball and a rigid flat surface.

and dimensional analysis is applied to the theory in order toyield tractable results. In Section 4, the model results are pre-sented. The steady-state results as well as the transient resultsare discussed. Afterwards, the evolution of the tractive forcesas function of the relative motion (creepage and/or spin) isgiven. Furthermore, a parameter study is applied to obtainbetter knowledge of the phenomenon and to make it possi-ble to situate experimental results in a consistent framework.Finally, appropriate conclusions are drawn and future workis indicated.

2. FORMULATION

2.1. Contact definition and preliminary assumptions

Let us consider the general elastic, nonconformal contactof two bodies of revolution. Depending on their elasticitymoduli, the two bodies will deform to certain degrees (seeFigure 1(a)). For the purpose of our study, we shall assumethat the problem may be reduced to that of a single equiv-alent elastic body of revolution with a plain infinitely rigidbody (see Figure 1(b)). This assumption is widely adopted inthe theory and application of Hertzian contacts so that theconversion formulas of geometry and elasticity are well es-tablished [17, 18]. However, this assumption excludes certainphenomena such as Heathcote slip. Let us note, firstly, thatwhen the externally imposed creepages result in appreciablylarger microslip levels than Heathcote slip, the assumptionwill yield good approximations. Secondly, it would still bepossible to adopt this assumption and account for Heath-cote slip by calculating an equivalent creepage field that cor-responds to that case.

When this assumption is used, the equivalent modulus ofelasticity, Eh (Hertzian modulus), and the equivalent radiusof the resulting single body of revolution, Rh, are given by

1Eh= 1− ν 2

1

E1+

1− ν 22

E2,

1Rh= 1R1

+1R2.

(1)

(NB. This formula is also applicable to each of the principleradii of an ellipsoid.)

In the rest of this article, we shall consider, without lossof generality, the contact of an elastic sphere with a smooth,rigid plane. The case of an elliptical contact may be dealt within an analogous manner using appropriate conversion for-mulas.

F. Al-Bender and K. De Moerlooze 3

zy

x

ω

V

φ

ψ

y

x

V

δVy

φ

δVx

Figure 2: Overview of the rolling motion.

2.2. Normal stresses and contact patch

When an elastic sphere of radius R and equivalent modulusof elasticity Ee = E/(1− ν2) is pressed with a load W againsta rigid plane surface, the following obtaining [17, 19].

The circular contact patch radius a is given by

a =(

3WR

4Ee

)1/3

. (2)

The contact patch is defined as the region A in the xy-plane:A = {(x, y) : x2 + y2 ≤ a2}.

The normal stress pz is given by

pz(x, y) = p0

√1− (x/a)2 − (y/a)2 (3)

with

p0 = 3W2πa2

. (4)

2.3. Tractive rolling kinematics

Figure 2 gives an overview of the different creepages whichcan occur in tractive rolling. Here, one considers a sphereof radius R rolling in the x-direction such that its center istranslating at a velocity V . In addition, the sphere is spin-ning around its x, y, and z axes with angular speeds ψ, ω,and φ, respectively. The creepages, which characterize trac-tive rolling, are defined as follows:

δVx = −ωR +V is the longitudinal creepage,

δVy = ψR is the lateral creepage,

φ is the spin creepage.

(5)

In most cases of interest, the magnitudes of these creepagesare proportional to the magnitude of the rolling velocity. Inthat case, they can be expressed in terms of displacements pertraversed rolling distance (as will be apparent further below).

The problem of an elastic sphere tractively rolling on arigid plane can be reduced to that in which the contact patchis stationary in space and time. This is accomplished by as-suming the rigid plane upon which the sphere is rolling tobe moving in the opposite direction with a velocity V (seeFigure 2). Looking through the stationary contact patch in adirection normal to it at points in the bodies that are suffi-ciently remote from the interface, one will then see two sur-faces entering it: one moving with velocity V and one with

−→V

−−→V

−→V

−→C

Figure 3: An overview of the relative motion between the two bod-ies.

velocity V + c, where

c = (δVx − φy, δVy + φx)

(6)

is the creepage velocity vector, as depicted in Figure 3.In order to arrive at an equation to describe the kine-

matics of the surface points lying inside the contact patch A,one follows a point on the surface of the sphere, which entersthe contact patch and mates at the entrance with a counter-point on the rigid plane. Owing to the creepage, the pointon the surface of the sphere will have to deform in the planeof the contact patch by an amount u = (u, v) being calledthe displacement, which is generally function of space andtime. This situation is depicted in Figure 4, which illustrates,in the 2D case, the difference between pure rolling and trac-tive rolling. Now, defining the slip s as the relative velocitybetween 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 planedeformations [10, 19, 20]:

s = c +DuDt

. (7)

Writing out the material derivative D/Dt = ∂/∂t + x∂/∂x andx = −V , one obtains

s = c−V ∂u∂x

+∂u∂t. (8)

Generally,V = V(t). If, as indicated earlier, slip and creepageare scaled withV , then one can rewrite the previous equationin terms of the traversed rolling distance q rather than thetime and rolling velocity by making use of the substitution

q =∫ t

0V(τ)dτ. (9)

Substituting this into (8) yields

s = c−V ∂u∂x

+V∂u∂q. (10)

Finally, normalizing creepage and slip by the rolling speed,one obtains

S = C− ∂u∂x

+∂u∂q

, (11)


−→V

−→c = 0

y z

xO

(a)

−→V

−→c > 0

y z

xO

(b)

Figure 4: The case of rolling without slip (a) and rolling with slip (b).

where S = s/V = (Sx, Sy) is the relative slip and C = c/V =(ξx − φy/a, ξ y + φx/a) is the relative creepage.

(In this formulation, the creepages (ξx = δVx/V , ξy =δVy/V , φ = φa/V) have units m/m.) Moreover, when therolling speed is constant, then C will be a constant vector.Otherwise, it will be function of the traversed distance. In therest 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 “Win-kler bedding” model to determine the relationship betweensurface 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 withnormal traction (Hertzian pressure) treatment.

In this simplified approach, the surface of the elastic ob-ject is considered to be covered by elastic “bristles,” normal toit, which have constant stiffness for tangential deformations,L, that is,

(u, v) = L(px, py

), (12)

where (px, py) is the tangential traction field and L is the tan-gential flexibility of the “bristle.”

The flexibility parameter L depends not only on the elas-ticity characteristics and Poisson’s ratio, but also on the as-pect ratio of the contact ellipse and the magnitude of thecreepages. The determination of an appropriate value for L,for a particular case, is carried out by formal comparison ofanalytical, no-slip solutions using the simplified theory withthose using exact theory, see Kalker [10]:

L = F(G,Cij , ξx, ξy ,φ

), (13)

L = L1∣∣ξx∣∣ + L2

∣∣ξy∣∣ + L3|φ|√ξ2x + ξ2

y + φ2. (14)

It is shown in Kalker [10] that this approximation yields so-lutions 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) willyield the corresponding traction field

(px, py

) = (u, v)L

. (15)

This traction field corresponds to the no-slip condition. Inpractice, the local friction coefficient and normal traction(Hertz pressure) will determine whether no-slip will hold. Inother words, (15) will be valid only if

∥∥(px, py)∥∥ ≤ μpz, (16)

where, pz is given by (3), (4) and μ is the local coefficient offriction or adhesion. Generally, μ will depend on the contactconditions. However, in the present study, we consider onlythe case of constant μ.

The condition (16) is known as the traction bound,which leads to the following relationship for determining thetangential tractions:

(px, py

) =

⎧⎪⎪⎪⎪⎨⎪⎪⎪⎪⎩

∥∥(u, v)∥∥

L,

∥∥(px, py)∥∥ ≤ μpz,

μpz(u, v)∥∥(u, v)

∥∥ , otherwise.

(17)

2.4.3. The initial traction field:Hertzian contacts with friction

Before any prerolling is initiated, a traction field is alreadypresent in the contact due to the normal loading of the con-tact. The initial value of the traction field, which is neededto solve the prerolling problem, can be calculated, consider-ing a Hertzian axisymmetric contact between a ball and a flatsurface. Here, we consider the normal loading of the ball, in-cluding the frictional behavior between the ball and its con-tacting surface. The shear traction field is given by Hills et al.[19]:

pxy(r)

μp0=√

1− r2 − rH(rs − r)∫ rsr

Ψ(t, rs

)t2√

1− t2 dt, (18)

where 0 < r < 1 is the dimensionless radial coordinate, μthe local coefficient of friction, p0 the peak contact pressure,


H(·) the Heaviside step function, rs denotes the dimension-less radius of the stick region, and Ψ(r, rs) is given by the fol-lowing integral expression:

Ψ(r, rs

)

=

⎧⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎨⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎩

0, if rs ≤ r ≤ 1,

2π

(wr∫ π/2

0

dθ(1− (1− r2)sin2θ)

√1− r′2s sin2θ

))

− rsK′(rs)r

(w

rs− Q0(w)Q0(rs)

)if 0 ≤ r ≤ s,

(19)

where K ′(rs) is a complete elliptic integral of the first kind,r′2s = 1− r2

s ,

Q0(r) = 12

ln(

1 + r

1− r)

, w2 = r2s − r2

1− r2. (20)

In the case of elastic similar bodies, the entire contact zonesticks, thus rs = 1. This adhesive limit was studied by Good-man [21] and can be expressed in closed form by

pxy(r)

βp0= r

π

∫ r′0

ln xdx(1− x2

)2 , 0 < r < 1, (21)

with β is the Dundurs’ constant [22, 23] given by

β =(1− 2ν1

)/(2G1

)− (1− 2ν2)/(2G2

)(1− ν1)/G1 + (1− ν2)/G2

. (22)

Figure 5 shows the radial shear traction distribution for dif-ferent values of rs. This field serves as an initial value for theprerolling model. Let us note however that once gross rollingis attained, the traction field corresponding to it will serve asthe initial value for subsequent rolling.

3. SOLUTION PROCEDURE

Referring to the previous section, the problem to be solvedmay be stated as follows: solve (11) subject to conditions(12), (15), (16) to determine the tangential displacementsand tractions. First, we show that (11) admits a closed formgeneral solution for the case of zero slip. Secondly, we applythe stiffness and slip conditions (12), (15), (16) in a numeri-cal 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

−∂u∂x

+∂u

∂q= Sx − ξx +

φy

a, (23)

−∂v∂x

+∂v

∂q= Sy − ξy − φx

a. (24)

If the slip (Sx, Sy) is given, (23), (24) admit general solutionswhich are derived in detail in the appendix.

0

0.2

0.4

0.6

0.8

1

p xy(r

)/(μp 0

)[-]

0 0.2 0.4 0.6 0.8 1

r[-]

rs = 0.2rs = 0.4

rs = 0.6rs = 0.8

Figure 5: The radial shear traction distribution for different partialslip cases.

Since the slip field is not known beforehand, the solu-tions of interest for numerical implementation are those cor-responding to the case of zero-slip. Thus, for (Sx, Sy) ≡ 0, wehave the following solutions (see the appendix):

u(x, y, q) = q ·(− ξx +

φy

a

)+ f (x + q, y). (25)

Similarly, the general solution for v is

v(x, y, q) = q ·(− ξy − φ

a

(x +

q

2

))+ g(x + q, y), (26)

where f , g are any arbitrary functions. Note that outside thecontact 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 liney = y0 in the contact patch. Thus, referring to Figure 6, weconsider the solution on the line segment y = y0 which isbounded by the leading edge x = a(y0) and the trailing edgex = −a(y0). Since the surface points enter the contact patchfree of stress, we have

u = v = 0 ∀q at x = a(y0), (27)

which provides the necessary boundary condition for the so-lution.

The initial surface displacement distribution at q = 0 isassumed to be given, for example, from Section 2.4.3, as

q =⎧⎨⎩

0, u = u0(x, y0

),

0, v = v0(x, y0

).

(28)


y

x

x = a(y)

A

x = −a(y)

y = y0

Rolling

Figure 6: The case of rolling without slip (left) and rolling with slip(right).

Substituting this into (25) and (26), we obtain

f (x, y0) = u0(x, y0

),

g(x, y0) = v0(x, y0

).

(29)

Equation (29) means that the initial (u, v) distribution deter-mines the arbitrary functions f and g on the range of defini-tion of (u, v). Thus, by replacing x by x+q, on−a ≤ x+q ≤ a,we have that

f(x + q, y0

) = u0(x + q, y0

),

g(x + q, y0

) = v0(x + q, y0

),

− a(y0) ≤ x ≤ a

(y0)− q.

(30)

In order to obtain the form of functions f and g in the inter-val a(y0)−q ≤ x ≤ a(y0), we make use of the initial conditionat x = a(y0). Thus, substituting (27) into (25) and (26) yields

u(a(y0), y0, q

)=0

=q·(−ξx+

φy0

a

)+ f(a(y0)

+q, y0),

v(a(y0), y0, q

)=0

=q·(− ξy − φ

a

(a(y0)

+q

2

))+g(a(y0)

+q, y0).

(31)

Replacing q by x+q, on a(y0)−q ≤ x ≤ a(y0), and translatingx by −a(y0), we have that

f(x + q, y0

)= −(x + q − a(y0))(− ξx +

φy0

a

),

g(x + q,y0

)=−(x + q− a(y0))(−ξy − φ

2a

(x + q + a

(y0)))

,

a(y0)− q ≤ x ≤ a

(y0).

(32)

Finally, the analytic solution is obtained in closed form bysubstituting of f and g from (30) and (32) into (25) and (26):

u(x, y0, q

)

= q ·(− ξx +

φy

a

)

+

⎧⎪⎨⎪⎩−(x + q− a(y0

))(− ξx +φy0

a

), a

(y0)−q≤x≤a(y0

),

u0(x+q,y0

), −a(y0

)≤x≤a(y0)−q,

v(x, y0, q

)

= q ·(− ξy − φ

a

(x +

q

2

))

+

⎧⎪⎪⎪⎪⎪⎨⎪⎪⎪⎪⎪⎩

−(x + q − a(y0))(− ξy − φ

2a

(x + q + a

(y0)))

,

a(y0)− q ≤ x ≤ a

(y0),

v0(x + q, y0

), −a(y0

) ≤ x ≤ a(y0)− q.

(33)

This solution shows that the initial displacement distributionpasses as a wave, moving from right to left, through the con-tact patch. Gross rolling is thus achieved after a rolling dis-tance equal to 2a is traversed. The no-slip displacement fieldcorresponding to it is obtained from the top members of (33)as

u∞(x, y0

) = −(x − a(y0))(− ξx +

φy0

a

),

v∞(x, y0

) = −(x − a(y0))(− ξy − φ

2a

(x + a

(y0)))

,

(34)

which is evidently independent of the initial distribution.

3.2. Numerical implementation

In order to apply the procedure numerically to a given con-tact problem, the contact region is discretized into a set ofnodes. In Figure 7, the grid is depicted, where we have chosenfor programming convenience Δx = Δy. Starting from anyinitial distribution u0(x, y0), v0(x, y0), (33) yields the zero-slip values at any desired value of q. In the particular casewhen the increments Δq = Δx, the analytical solution re-duces to the following simple step-wise procedure. After astep Δq, the new (u, v) distribution is obtained by augment-ing the previous value withΔq(−ξx+φy/a,−ξy−φx/a), shift-ing each of the u and v vectors one position to the left, andpadding them with the boundary condition u = v = 0 at theright.

Next, the traction bound needs to be verified at each stepand the values of (u, v) accordingly corrected. This is carriedout as follows.

With any new obtained value (u, v), we calculate

(px, py

) = L(u, v). (35)


y

x

Rolling

Figure 7: The discretization of the contact patch.

If ‖(px, py)‖ ≤ μpz, no slip occurs (Sx = Sy = 0) and the val-ues of (u, v) are retained. If, on the other hand, ‖(px, py)‖ >μpz, then

(u, v)new =μpzL

((u, v)∥∥(u, v)

∥∥)

old

. (36)

In Figure 8, the flowchart of the algorithm is presented.

3.3. Calculation of the tractions Fx,Fy ,Mz

The total traction forces and moment are obtained from thefollowing integrals:

Fx =∫ a−a

∫ a(y)

−a(y)px(x, y)dx dy, (37)

Fy =∫ a−a

∫ a(y)

−a(y)py(x, y)dx dy, (38)

Mz =∫ a−a

∫ a(y)

−a(y)p(x, y)r(x, y)dx dy (39)

with r(x, y) the radial distance from the center of the contactpatch to the location of the node.

For the no-slip, gross rolling solution, (37) and (38) yield(see also [10])

Fx = −8a3ξx3L

, (40)

Fy =−8a3ξy

3L− πa3φ

4L. (41)

Upon comparing these results formally with the results ob-tained from exact theory, the relations (12) and (13) are ob-tained.

3.4. Dimensional analysis

In order to generate, analyze, and present the results sys-tematically, dimensional analysis is applied to the problem.

Start

- Determine initialtraction field

- Prescribe q

Increase q byincrement Δq

Calculate newpressure distribution

Verify traction boundand correct pressure

distribution if necessary

Displacement trajectorycompleted?

No

Yes

Calculate tractions

Done

Figure 8: The flowchart of the algorithm.

This is achieved through the application of the Vaschy-Buckingham-Π theorem [24, 25] as follows. The rollingproblem may be generally expressed as

f(W ,F,M, a, p0, p,E,R,V , δVx, δVy , φ, ν,μ, q, x, y

) = 0.(42)

Inspection shows that the problem posses three independentdimensions (length, force, and time). Choosing a, p0, and Vas the variables to be eliminated, we obtain the dimensionlessform, in which the number of variables is reduced by three:

F(W

p0a2,F

p0a2,M

p0a3,a

a,p0

p0,p

p0,

E

p0,R

a,V

V,δVx

V,δVy

V,φa

V, ν,μ,

q

a,x

a,y

a

)= 0,

F(W∗,F∗,M∗, p∗,E∗,R∗, ξx, ξy ,φ, ν,μ, q∗, x∗, y∗

) = 0.(43)

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 usedsymbols. Since the model contains no system dynamics ofthe ball, the rolling velocity V falls out of the equations. This


dimensional analysis makes it easier to compare the influ-ence of the different parameters of the problem in a consis-tent way. Therefore, in the next section, the results are alsogiven in this dimensionless form. An arbitrary value of 0.5has been used for the coefficient of sliding friction.

4. RESULTS

4.1. Steady-state rolling

Figure 9 depicts traction distributions pertaining to basiccases of steady-state rolling with longitudinal, lateral, and/orspin creepages of an elastic ball. While rolling with longitu-dinal and lateral creepage is of main interest in wheel-railor tyre-road contact, rolling with spin is the dominant typein angular contact ball bearings and linear guideways (e.g.,with V-grooved tracks). For the sake of better visibility, acoarse grid is used. Figure 9(a) gives the example of equallongitudinal and lateral creepage. The slip zone is symmet-ric 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 trac-tion fields with spin creepage of different levels. With increas-ing spin, the slip zone becomes larger, until the total contactzone slips. Figure 9(d) shows how the traction field looks likewhen combining spin with creepage in both directions.

4.2. Transient rolling

In this section, the traction behavior during transition torolling is analyzed. This behavior is marked by hysteresisof the traction forces in the rolling displacement. Again, welimit the presentation to the basic cases of (pre-)rolling withlongitudinal, lateral, and spin creepage. The applied displace-ment trajectory is provided with reversal points, to ascertainthe nonlocal memory character of the hysteresis curves, asdiscussed in [1, 26, 27]. The rolling trajectory is specified asq∗ = [0 → q1 → q2 → ·· · ]. For all cases considered, the ini-tial traction field is a null field since the contacting materialsare identical, which results in a Dundurs’ constant β = 0.

4.2.1. Pure longitudinal creepage

For this case, we chose ξx = 0.015, ξy = φ = 0. The rollingtrajectory is q∗ = [0 → 2.1 → −2.1 → 2.1]. The longitudinalcreepage gives rise only to a traction force F∗x in the rollingdirection. Figure 10(c) plots F∗x against q∗ to show the re-sulting hysteresis loop. Figures 10(a), 10(b), 10(c), 10(d), and10(e) show the characteristic traction fields at selected pointsduring the motion. From these, we see also that all tractioncomponents lie in the rolling direction.

4.2.2. Pure lateral creepage

To illustrate this case, we put ξx = 0, ξy = 0.015, φ = 0.Rolling combined with a lateral creepage component givesrise only to a traction field in the y-direction, that is, perpen-dicular to the rolling direction. Consequently, the hystere-sis loops in Figure 11(a) depict this traction force F∗y versusthe rolling displacement (q∗ = [0 → 2.1 → 0.4 → 1.6 →

−2.1 → −0.4 → −1.6 → 2.1]). This trajectory has beenso constructed as to show inner hysteresis loops and thusthe nonlocal memory character of hysteresis. The behavioris, otherwise, similar to that of pure longitudinal creepage.Figure 11(b) depicts the steady-state traction field.

4.2.3. Pure spin

Here, we put vx = vy = 0, φ = 0.0255. Rolling with pure spinresults in a traction field having components in the x andy directions. The resulting traction force in the x directionequals zero. The traction stress field results in a spin momentM∗

z . This moment is plotted as function of the rolling dis-placement to yield the hysteresis curve of Figure 12(c). Theinput is q∗ = [0 → 2.04 → 0.46 → 1.54 → −2.04 → −0.46 →−1.54 → 2.04]. Note that the virgin curve, corresponding tostart of motion until first gross slip, overshoots the subse-quent hysteresis curves. The shape of the virgin curve and theamount of overshoot vary with the assumed initial tractiondistribution.

4.3. Evolution of the steady-state tractions infunction of the creepages

It is intuitively plausible to assume that the traction force(or moment in the case of spin) will increase with increasingcreepage until gross slip is reached. In Figure 13, the evolu-tion of the steady-state rolling traction force is depicted as afunction of the relative creepage. For rolling with pure longi-tudinal creepage, an increasing trend is observed as shown inFigure 13(a). The same evolution is noticed for rolling withlateral creepage in Figure 13(b). Considering rolling withpure spin, the spinning moment M∗

z increases towards a sat-uration value while the lateral force F∗y shows a local max-imum. These trends are depicted in Figure 13(c). As alreadymentioned, the longitudinal force component F∗x equals zerofor this case. These results agree with those obtained in [10].

4.4. Parameter analysis

In the previous section, the general behavior of the steady-state traction force in function of the creepage value is dis-cussed. In this section, the influence of the other model pa-rameters is discussed. For brevity, we confine the treatmentto the case of pure spin creepage.

Consider the dimensionless model of (43). Because themodel parameters are reduced by normalizing with respect tothe radius of the contact patch a and the maximum normalpressure p0, these parameters fall out of the model equations,so that (2) becomes

1 =(

3W∗R∗

4E∗h

)1/3

⇐⇒ R∗ = 4E∗h3W∗ . (44)

In other words, R∗ increases if E∗ increases orW∗ decreases.In this way, R∗ contains (or coalesces) all geometry, elasticity,and load information.


−1

−0.8

−0.6

−0.4

−0.2

0

0.2

0.4

0.6

0.8

1

y∗[-

]

−1 −0.5 0 0.5 1

x∗[-]

ξx /= 0, ξy /= 0, φ = 0

Contact contourSlip region

Stick regionTraction stress

(a)

−1

−0.8

−0.6

−0.4

−0.2

0

0.2

0.4

0.6

0.8

1

y∗[-

]

−1 −0.5 0 0.5 1

x∗[-]

ξx = ξy = 0, φ small



(b)

−1

−0.8

−0.6

−0.4

−0.2

0

0.2

0.4

0.6

0.8

1

y∗[-

]

−1 −0.5 0 0.5 1

x∗[-]

ξx = ξy = 0, φ large



(c)

−1

−0.8

−0.6

−0.4

−0.2

0

0.2

0.4

0.6

0.8

1

y∗[-

]

−1 −0.5 0 0.5 1

x∗[-]

ξx /= 0, ξy /= 0, φ /= 0



(d)

Figure 9: A few results of the steady-state rolling traction field, the rolling direction is from the left to the right.

To study the influence of R∗, this parameter is varied be-tween reasonable bounds. To determine its influence on theprerolling behavior, three main parameters are chosen foranalysis.

(1) The prerolling distance x∗pr.(2) The initial stiffness k∗i of the hysteresis system, that is,

the initial slope of the hysteresis curve,

k∗i =∂M∗

z

∂q∗

∣∣∣∣q∗=0

. (45)

(3) The steady-state frictional moment M∗z,ss.

These variables are depicted as functions of the dimen-sionless ball radius in Figure 14 for different values of thespin creepage φ and in Figure 15 for different values of thecoefficient of friction μ. One can notice from (44) that anincrease in R∗ corresponds to a decrease in W∗ for a con-stant value of E∗h . A certain limit of applicability, R∗lim applies,owing to geometrical limitations: a cannot be larger thanabout R/4 without violating basic Hertzian assumptions. Theplotted values in Figures 14 and 15 below this value of R∗lim(marked by the box “theoretical region”) are only given forthe sake of mathematical completeness.


−1

−0.8

−0.6

−0.4

−0.2

0

0.2

0.4

0.6

0.8

1

y∗[-

]

−1 −0.5 0 0.5 1

x∗[-]

Stick and slip regions


Stick regionTraction arrows

(a)

−1

−0.8

−0.6

−0.4

−0.2

0

0.2

0.4

0.6

0.8

1

y∗[-

]

−1 −0.5 0 0.5 1

x∗[-]




(b)

−1

−0.8

−0.6

−0.4

−0.2

0

0.2

0.4

0.6

0.8

1

F∗ x

[-]

−3 −2 −1 0 1 2 3

q∗[-]

Hysteresis for longitudinal creepage

A

B

C

D

(c)

−1

−0.8

−0.6

−0.4

−0.2

0

0.2

0.4

0.6

0.8

1

y∗[-

]

−1 −0.5 0 0.5 1

x∗[-]




(d)

−1

−0.8

−0.6

−0.4

−0.2

0

0.2

0.4

0.6

0.8

1

y∗[-

]

−1 −0.5 0 0.5 1

x∗[-]




(e)

Figure 10: The results for transient (pre-)rolling with longitudinal creepage. In the middle, the hysteresis with around it, transient tractionfields at selected points in the trajectory.

From Figures 14(a) and 15(a), one can see that the pre-rolling distance stays constant for low and moderate values ofR∗ (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 istwice the contact patch radius. The initial stiffness of the hys-teresis at the beginning of the prerolling region is depicted inFigures 14(b) and 15(b). For low values of R∗, the stiffness is


−1

−0.8

−0.6

−0.4

−0.2

0

0.2

0.4

0.6

0.8

1

F∗ y[

-]

−3 −2 −1 0 1 2 3

q∗[-]

Hysteresis for lateral creepage

(a)

−1

−0.8

−0.6

−0.4

−0.2

0

0.2

0.4

0.6

0.8

1

y∗[-

]

−1 −0.5 0 0.5 1

x∗[-]




(b)

Figure 11: The results for transient (pre-)rolling with lateral creepage. The left panel depicts the hysteresis curve while the right panelvisualizes the steady-state rolling traction field.

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

a∼(WR)(1/3). (46)

From Hills et al. [19], we have

M

WR∼θ∼ q

R, (47)

M

q∼WR

R∼a

3

R, (48)

since

M∗ = M

p0a3. (49)

This leads to

M∗∼ qR∼qa.a

R, (50)

k∗i =∂M∗

∂q∗∼ 1R∗

sinceq

a∼q∗,

R

a= R∗. (51)

The steady-state tractive moment is depicted in Figures 14(c)and 15(c). After a constant behavior for low values of R∗, asteep decrease is noticed to end with a quasizero value forvery high values of R∗. To get a value for the coefficient ofrolling friction λ∗, this steady-state moment is divided by thedimensionless normal load W∗. These results are depictedin Figures 14(d) and 15(d). We notice a steep increase in theregion of moderate values for R∗ to saturate towards a con-stant value for higher R∗, which results in a similar behavioras compared to “Amontons’ law” [28] for sliding friction.

5. DISCUSSION AND CONCLUSIONS

In the foregoing, a theory is developed to characterize thetraction behavior during the transition to rolling. The fol-lowing remarks are in order. Firstly, although based on the“Winkler bedding” simplification, it is shown in [10] that thisapproximation, with appropriate choice of the stiffness pa-rameter L, yields solutions which are very close to those givenby exact theory. The advantage gained is the transparencyand easy application of this theory. Secondly, although onlythe case of point contact has been treated, this theory canbe directly extended to the general cases of elliptical and linecontacts. Thirdly, the cases of variable (e.g., pressure depen-dent) local coefficient of friction in the contact patch, vari-able (e.g., time or position dependent) creepages, variablenormal load, and variable rolling velocity can all be directlytreated by this theory. In that way, dynamical contact phe-nomena, such as those obtaining during motion reversals,acceleration, deceleration, oscillation, and so forth, can beaccommodated by this theory. Other cases not covered im-mediately by this theory are discussed subsequently.

Rough contacts

The replacement of a smooth contact by a rough one gen-erally requires a higher computational effort, which oftenmortgages the development of a simplified version suitablefor real-time execution. Bucher et al. [29], for instance,outline that in the case of rough contact, the stresses anddeformations can only be calculated using special bound-ary element methods, while transient three-dimensionalrolling contacts comprising rough surfaces are not possi-ble at present. However, this “bristle” model offers the ad-vantage that it can be directly extended to rough contacts.


−1

−0.8

−0.6

−0.4

−0.2

0

0.2

0.4

0.6

0.8

1

y∗[-

]

−1 −0.5 0 0.5 1

x∗[-]




(a)

−1

−0.8

−0.6

−0.4

−0.2

0

0.2

0.4

0.6

0.8

1

y∗[-

]

−1 −0.5 0 0.5 1

x∗[-]




(b)

−1

−0.8

−0.6

−0.4

−0.2

0

0.2

0.4

0.6

0.8

1

M∗ z

[-]

−3 −2 −1 0 1 2 3

q∗[-]

Hysteresis for rolling with spin

A

B

C

D

(c)

−1

−0.8

−0.6

−0.4

−0.2

0

0.2

0.4

0.6

0.8

1

y∗[-

]

−1 −0.5 0 0.5 1

x∗[-]




(d)

−1

−0.8

−0.6

−0.4

−0.2

0

0.2

0.4

0.6

0.8

1

y∗[-

]

−1 −0.5 0 0.5 1

x∗[-]




(e)

Figure 12: The results for transient (pre-)rolling with spin. In the middle, the hysteresis with around it, some transient traction fields atselected points in the trajectory.

The challenge is to translate an actual rough surface into anequivalent bristle set. Alternatively, recent numerical meth-ods offer an affordable solution for calculating the rough sur-face contact by using dedicated contact solvers [30, 31].

Wear and Heathcote slip

The influence of wear and running-in of the surfaces is ob-viously very important in many applications. In the former


0

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1F∗ x

[-]

0 0.01 0.02 0.03 0.04 0.05

ξx[-]

Evolution of the steady state traction forceas function of the longitudinal creepage

(a)

0

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1

F∗ y[

-]

0 0.01 0.02 0.03 0.04 0.05

ξy[-]

Evolution of the steady state tractionforce as function of the lateral creepage

(b)

0

0.1

0.2

0.3

0.4

0.5

0.6

0.7

M∗ z

[-],F∗ y[

-]

0 0.1 0.2 0.3 0.4 0.5

φ[-]

Evolution of the steady state tractionas function of the spin creepage

Mz

Fy

(c)

Figure 13: The evolution of the steady-state traction as function of the creepage.

algorithm, the creepage was kept constant over the contactpatch. Taking wear in consideration, the need for a variablecreepage becomes important. When the flat surface in whichthe sphere is rolling, wears, a groove develops in which thecontact between both bodies, is conformal. This situationis similar to that of a ball in a groove, for example, a deepgroove ball bearing, where the contact is conformal. In thatcase, the traction field for pure rolling, that is, with zero rel-ative creepage motion in a conformal groove, is described byHeathcote [32] and is qualitatively depicted in Figure 16. Thecontact area is no longer plane. When rolling freely, there isno net tangential force, which explains that the contact spotis subdivided into three zones: the central zone contains pos-itive slip vectors, while in the outer zones, the slip is negative.

In order to be able to deal with this problem using the al-gorithm presented in this paper, the hitherto constant creep-age over the whole contact patch should be replaced by a vari-

able creepage field. The latter may be obtained from geomet-rical and kinematical considerations of the contact with theborder line between positive and negative creepage fields be-ing the only unknown.

In conclusion, this paper considers the hysteretic behav-ior of the prerolling friction between a ball and a flat surface.Extending the existing steady-state gross rolling model, dueto J. J. Kalker, to a transient prerolling model, the evolutionof the traction field in the rolling displacement is determinedand hysteresis curves are generated. The traction field and itsbehavior in the presence of rolling with creepage and spinare systematically investigated. Finally, a parameter study iscarried out to gain more insight into this phenomenon. In aforthcoming paper, an experimental validation of the modelis carried out for the case of rolling with spin creepage, us-ing a configuration consisting of two V-grooved tracks with2 balls in between.


0

0.5

1

1.5

2

Pre

-rol

ling

dist

ancex∗ pr

10−5 100 105

R∗[-]

φ = 0.001φ = 0.035φ = 0.1

φ = 0.35φ = 0.5

Theoreticalregion

(a)

10−2

100

102

Init

ials

tiff

nes

sk∗ i

10−5 100 105

R∗[-]

φ = 0.001φ = 0.035φ = 0.1

φ = 0.35φ = 0.5

Theoreticalregion

(b)

10−4

10−2

100

102

Stea

dyst

ate

mom

entM∗ z,s

s

10−5 100 105

R∗[-]

φ = 0.001φ = 0.035φ = 0.1

φ = 0.35φ = 0.5

Theoreticalregion

(c)

10−10

10−5

M∗ z,s

s/W∗

10−5 100 105

R∗[-]

φ = 0.001φ = 0.035φ = 0.1

φ = 0.35φ = 0.5

Theoreticalregion

(d)

Figure 14: The behavior of the prerolling region on a variation of the ball radius for different spin creepages (μ = 0.5).

APPENDIX

Consider the p.d.e.

−∂u∂x

+∂u

∂q= F (x, y, q). (A.1)

Equation (A.1) corresponds to the Lagrangrian system [33]:

dx

−1= dy

0= dq

1= du

F (x, y, q). (A.2)

The integrals of the system (A.2) are

x + q = α,

y = β,

u− G(x, y, q) = γ,

(A.3)

where α, β, γ are arbitrary constants and G is determined by

G(x, y, q) =∫

F (x, y, q)dq

=∫

F (α− q,β, q)dq

=∫dg

dq(α,β, q)dq

= g(α,β, q) = g(x + q, y, q).

(A.4)

Note that the last integration step is carried out on the as-sumption that the function F (α − q,β, q) posses an an-tiderivative g. This is always true if F can be expressed asa polynomial in q.


0

0.5

1

1.5

2

Pre

-rol

ling

dist

ancex∗ pr

10−5 100 105

R∗[-]

μ = 0.01μ = 0.25μ = 0.5

μ = 0.75μ = 1

Theoreticalregion

(a)

10−4

10−2

100

102

Init

ials

tiff

nes

sk∗ i

10−5 100 105

R∗[-]

μ = 0.01μ = 0.25μ = 0.5

μ = 0.75μ = 1

Theoreticalregion

(b)

10−10

10−5

100

Stea

dyst

ate

mom

entM∗ z,s

s

10−5 100 105

R∗[-]

μ = 0.01μ = 0.25μ = 0.5

μ = 0.75μ = 1

Theoreticalregion

(c)

10−10

10−5

100

M∗ z,s

s/W∗

10−5 100 105

R∗[-]

μ = 0.01μ = 0.25μ = 0.5

μ = 0.75μ = 1

Theoreticalregion

(d)

Figure 15: The behavior of the prerolling region on a variation of the ball radius for different coefficients of friction μ (φ = 0.05).

Figure 16: The traction distribution of pure rolling: Heathcote slip.

The general solution to (A.1) is then [33]

F (α,β, γ) = 0 (A.5)

or

F(x + q, y,u− G(x, y, q)

) = 0, (A.6)

where F is an arbitrary function. Assuming that (A.6) ad-mits an explicit solution for u, this solution will have theform

u = G(x, y, q)− f (x + q, y), (A.7)

where f is an arbitrary function. As an example, consider-ing F (x, y, q) = −ξx + φy/a and assuming ξx and φ to beconstant, then

u(x, y, q) = q(− ξx +

φy

a

)+ f (x + q, y). (A.8)


NOMENCLATURE

a: Footprint radiusA: Contact patchc: CreepageC: Relative creepageEh: Hertzian modulus of elasticityF: Traction forceg: Traction boundG: Modulus of rigidityH(·): Heaviside step functionki: Initial stiffnessL: FlexibilityMz: Spin momentp: Pressureq: Traversed rolling distancer: Distance from the centerR: Contact radiusRh: Equivalent radius of contacts: SlipS: Relative slipt: Timeu, v: Particle displacementV : Rolling velocityW : Normal loadx: Displacementxpr: Prerolling distanceβ: Dundurs’ constantδ: Normal elastic deformationδVx: Dimensional longitudinal creepageδVy : Dimensional lateral creepageλ: Coefficient of rolling frictionμ: Coefficient of frictionν: Poisson’s coefficientξx: Nondimensional longitudinal creepageξy : Nondimensional lateral creepageφ: Dimensional spin creepageφ: Nondimensional spin creepageψ: Angular lateral creepageω: Angular longitudinal creepage

ACKNOWLEDGEMENTS

This research is sponsored by the Fund for Scientific Re-search, Flanders (F.W.O.), under Grant no. FWO4283. Thescientific responsibility is assumed by its authors.

REFERENCES

[1] F. Al-Bender and W. Symens, “Characterisation of frictionalhysteresis in ball-bearing guideways,” Wear, vol. 258, no. 11-12, pp. 1630–1642, 2005.

[2] O. Reynolds, “On rolling friction,” Philosophical Transactionsof the Royal Society of London, vol. 166, pp. 155–174, 1876.

[3] H. Hertz, “Uber die Beruhrung fester elastischer Korper,” Jour-nal fur die reine und angewandte Mathematik, vol. 92, pp. 156–171, 1882.

[4] F. W. Carter, “On the action of a locomotive driving wheel,” inProceedings of the Royal Society of London, vol. 112, pp. 151–157, 1926.

[5] K. L. Johnson, “The effect of spin upon the rolling motion ofan elastic sphere upon a plane,” Journal of Applied Mechanics,vol. 25, pp. 332–338, 1958.

[6] K. L. Johnson, “The effect of a tangential force upon the rollingmotion of an elastic sphere upon a plane,” Journal of AppliedMechanics, vol. 25, pp. 339–346, 1958.

[7] P. J. Vermeulen and K. L. Johnson, “Contact of non-sphericalbodies transmitting tangential forces,” Journal of Applied Me-chanics, vol. 31, pp. 338–340, 1964.

[8] J. J. Kalker, “On the rolling contact of two elastic bodies in thepresence of dry friction,” Ph.D. thesis, Delft University of Tech-nology, Delft, The Netherlands, 1967.

[9] J. J. Kalker, “A fast algorithm for the simplified theory of rollingcontact,” Vehicle System Dynamics, vol. 11, no. 1, pp. 1–13,1982.

[10] J. J. Kalker, Three-Dimensional Elastic Bodies in Rolling Con-tact, Kluwer Academic Publishers, Dordrecht, The Nether-lands, 1990.

[11] V. K. Garg and R. V. Dukkipati, Dynamics of Railway VehicleSystems, Academic Press, Waterloo, Ontario, Canada, 1984.

[12] J. B. Nielsen, “New developments in the theory of wheel/railcontact mechanics,” Ph.D. thesis, Technical University of Den-mark, Lyngby, Denmark, 1998.

[13] Z. Li., “Wheel-rail rolling contact and its applications towear simulation,” Ph.D. thesis, Delft University of Technology,Delft, The Netherlands, 2002.

[14] T. Prajogo, “Experimental study of pre-rolling frictionfor motion-reversal error compensation on machine tooldrive systems,” Ph.D. thesis, Department of Werktuigkunde,Katholieke Universiteit Leuven, Leuven, Belgium, 1999.

[15] F. Al-Bender and W. Symens, “Dynamic characterisation ofhysteresis elements in mechanical systems. I. Theoretical anal-ysis,” Chaos, vol. 15, no. 1, Article ID 013105, 11 pages, 2005.

[16] F. Al-Bender and W. Symens, “Dynamic characterisation ofhysteresis elements in mechanical systems. II. Experimentalvalidation,” Chaos, vol. 15, no. 1, Article ID 013106, 9 pages,2005.

[17] R. D. Arnell, P. B. Davies, J. Halling, and T. L. Whomes, Tribol-ogy Principles and Design Applications, MacMillan, New York,NY, USA, 1991.

[18] J. Halling, Principles of Tribology, MacMillan, New York, NY,USA, 1979.

[19] D. A. Hills, D. Nowell, and A. Sackfield, Mechanics of ElasticContacts, Butterworth-Heinemann, Oxford, UK, 1993.

[20] K. L. Johnson, Contact Mechanics, Cambridge UniversityPress, Cambridge, UK, 1985.

[21] L. E. Goodman, “Contact stress analysis of normally loadedrough spheres,” Journal of Applied Mechanics, vol. 29, no. 3,pp. 515–522, 1962.

[22] J. Dundurs and M. S. Lee, “Stress concentration at a sharpedge in contact problems,” Journal of Elasticity, vol. 2, no. 2,pp. 109–112, 1972.

[23] D. A. Spence, “The Hertz contact problem with finite friction,”Journal of Elasticity, vol. 5, no. 3-4, pp. 297–319, 1975.

[24] E. Buckingham, “On physically similar systems; illustrationsof the use of dimensional equations,” Physical Review, vol. 4,no. 4, pp. 345–376, 1914.

[25] M. D. Hersey, Theory and Research in Lubrication: Foundationsfor Future Developments, John Wiley & Sons, New York, NY,USA, 1966.

[26] W. Symens, “Motion and vibration control of mechatronicsystems with variable configuration and local non-linearities,”


Ph.D. thesis, Katholieke Universiteit Leuven, Division ofProduction Engineering, Machine design and Automation,Leuven, Belgium, 2004.

[27] I. Mayergoyz, Mathematical Models of Hysteresis, Springer,New York, NY, USA, 1991.

[28] G. Amontons, “On the resistance originating in machines,” inProceedings of the French Royal Academy of Sciences, pp. 206–222, 1699.

[29] F. Bucher, A. I. Dmitriev, M. Ertz, et al., “Multiscale simulationof dry friction in wheel/rail contact,” Wear, vol. 261, no. 7-8,pp. 874–884, 2006.

[30] I. A. Polonsky and L. M. Keer, “Fast methods for solving roughcontact problems: a comparative study,” Journal of Tribology,vol. 122, no. 1, pp. 36–41, 2000.

[31] I. A. Polonsky and L. M. Keer, “A numerical method forsolving rough contact problems based on the multi-levelmulti-summation and conjugate gradient techniques,” Wear,vol. 231, no. 2, pp. 206–219, 1999.

[32] H. L. Heathcote, “The ball bearing: in the making, under testand on service,” Proceedings of the Institute of Automotive En-gineers, vol. 15, pp. 569–702, 1921.

[33] A. R. Forsyth, Theory of Differential Equations Part IV: Par-tial Differential Equations, vol. 5, University Press, Cambridge,UK, 1906.

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