A Model of the Transient Behavior of Tractive Rolling Contacts

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.

1. Introduction

When¹ 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. Section 2 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 Figure 1(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.

(a)

(b)

Figure 1 

Reduction of the elastic contact between two balls to that of one ball and a rigid flat surface.

When this assumption is used, the equivalent modulus of elasticity, 𝐸_ℎ (Hertzian modulus), and the equivalent radius of the resulting single body of revolution, 𝑅_ℎ, are given by 1𝐸_ℎ=1−𝜈²¹𝐸₁+1−𝜈²²𝐸₂1,𝑅_ℎ=1𝑅₁+1𝑅₂.(1) (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 radius 𝑅 and equivalent modulus of elasticity 𝐸_𝑒=𝐸/(1−𝜈²) is pressed with a load 𝑊 against a rigid plane surface, the following obtaining [17, 19].

The circular contact patch radius 𝑎 is given by𝑎=3𝑊𝑅4𝐸_𝑒^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 𝑝_𝑧 is given by𝑝_𝑧(𝑥,𝑦)=𝑝₀1−(𝑥/𝑎)²−(𝑦/𝑎)²(3)with𝑝₀=3𝑊2𝜋𝑎².(4)

2.3. Tractive Rolling Kinematics

Figure 2 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 𝜓̇, 𝜔, and 𝜙̇, respectively. The creepages, which characterize tractive rolling, are defined as follows:𝛿𝛿𝑉𝑉_𝑥𝑦=−𝜔𝑅+𝑉isthelongitudinalcreepage,=𝜙𝜓̇i̇𝑅sthisetshpeinlacterreaelpcargeee.page,(5)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 velocity 𝐕 (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, where𝐜=𝛿𝑉_𝑥−𝜙̇𝑦,𝛿𝑉_𝑦+𝜙̇𝑥(6) 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 patch 𝐴, 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]: 𝐬=𝐜+𝐷𝐮𝐷𝑡.(7)Writing out the material derivative 𝐷/𝐷𝑡=𝜕/𝜕𝑡+𝑥̇𝜕/𝜕𝑥 and 𝑥̇=−𝑉, one obtains𝐬=𝐜−𝑉𝜕𝐮𝜕𝑥+𝜕𝜕𝐮𝑡.(8) 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 substitution𝑞=^𝑡0𝑉(𝜏)𝑑𝜏.(9)Substituting this into (8) yields𝐬=𝐜−𝑉𝜕𝐮𝜕𝑥+𝑉𝜕𝜕𝐮𝑞.(10)Finally, normalizing creepage and slip by the rolling speed, one obtains𝐒=𝐂−𝜕𝐮𝜕𝑥+𝜕𝜕𝐮𝑞,(11)where 𝐒=𝐬/𝑉=(𝑆_𝑥,𝑆_𝑦) is the relative slip and 𝐂=𝐜/𝑉=(𝜉_𝑥−𝜙𝑦/𝑎,𝜉_𝑦+𝜙𝑥/𝑎) is the relative creepage.

(a)

(b)

Figure 4 

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

(In this formulation, the creepages (𝜉_𝑥=𝛿𝑉_𝑥/𝑉,𝜉_𝑦=𝛿𝑉_𝑦/𝑉,𝜙=𝜙̇𝑎/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 (𝑝_𝑥,𝑝_𝑦) 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,(𝑢,𝑣)=𝐿𝑝_𝑥,𝑝_𝑦,(12)where (𝑝_𝑥,𝑝_𝑦) is the tangential traction field and 𝐿 is the tangential flexibility of the “bristle”.

The flexibility parameter 𝐿 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]:𝐿𝐿==ℱ𝐿₁𝐺,𝐶_𝑖𝑗,𝜉_𝑥,𝜉_𝑦,𝜙,(13)|||𝜉_𝑥|||+𝐿₂|||𝜉_𝑦|||+𝐿₃||𝜙||𝜉^2𝑥+𝜉^2𝑦+𝜙².(14)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𝑝^𝑥,𝑝_𝑦=(𝑢𝐿,𝑣).(15)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‖‖𝑝_𝑥,𝑝_𝑦‖‖≤𝜇𝑝^𝑧,(16)where, 𝑝_𝑧 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:𝑝^𝑥,𝑝_𝑦=⎧⎪⎪⎨⎪⎪⎩‖‖(𝑢,𝑣)‖‖𝐿,‖‖𝑝^𝑥𝜇𝑝_𝑧,𝑝_𝑦‖‖≤𝜇𝑝^𝑧,(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 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]:𝑝_𝑥𝑦(𝑟)𝜇𝑝₀=√1−𝑟²−𝑟𝐻𝑟_𝑠−𝑟^𝑟𝑠_𝑟Ψ𝑡,𝑟_𝑠𝑡²√1−𝑡²𝑑𝑡,(18)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, 𝑟_𝑠 denotes the dimensionless radius of the stick region, and Ψ(𝑟,𝑟_𝑠) is given by the following integral expression:Ψ=⎧⎪⎪⎪⎪⎨⎪⎪⎪⎪⎩𝑟,0𝑟,^𝑠if𝑟_𝑠2≤𝑟≤1,𝜋(𝑤𝑟^0𝜋/2𝑑𝜃−𝑟_𝑠1−(1−𝑟²)sin²𝜃)1−𝑟^′2𝑠sin²𝜃)𝐾^′𝑟^𝑠𝑟𝑤𝑟_𝑠−𝑄⁰(𝑤)𝑄₀(𝑟_𝑠)if0≤𝑟≤𝑠,(19)where 𝐾^(𝑟_𝑠) is a complete elliptic integral of the first kind, 𝑟^𝑠2=1−𝑟^2𝑠,𝑄₀(𝑟)=12ln11+−𝑟𝑟,𝑤²=𝑟^2𝑠−𝑟²1−𝑟².(20)In the case of elastic similar bodies, the entire contact zone sticks, thus 𝑟_𝑠=1. This adhesive limit was studied by Goodman [21] and can be expressed in closed form by𝑝_𝑥𝑦(𝑟)𝛽𝑝₀=𝑟𝜋^𝑟′₀ln𝑥𝑑𝑥1−𝑥²²,0<𝑟<1,(21)with 𝛽 is the Dundurs' constant [22, 23] given by𝛽=1−2𝜈¹/2𝐺¹−1−2𝜈²/2𝐺²1−𝜈¹)/𝐺₁+(1−𝜈₂)/𝐺₂.(22)Figure 5 shows the radial shear traction distribution for different values of 𝑟_𝑠. 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−𝜕𝑢𝜕𝑥+𝜕𝜕𝑞𝑢=𝑆^𝑥−𝜉_𝑥+𝜙𝑦−𝜕𝑣𝑎,(23)𝜕𝑥+𝜕𝜕𝑣𝑞=𝑆^𝑦−𝜉_𝑦−𝜙𝑥𝑎.(24)If the slip (𝑆_𝑥,𝑆_𝑦) 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 (𝑆_𝑥,𝑆_𝑦)≡0, we have the following solutions (see the appendix):𝑢(𝑥,𝑦,𝑞)=𝑞⋅−𝜉_𝑥+𝜙𝑦𝑎+𝑓(𝑥+𝑞,𝑦).(25)Similarly, the general solution for 𝑣 is𝑣(𝑥,𝑦,𝑞)=𝑞⋅−𝜉_𝑦−𝜙𝑎𝑥+2𝑞+𝑔(𝑥+𝑞,𝑦),(26)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 have𝑢=𝑣=0∀𝑞at𝑥=𝑎𝑦₀,(27)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 at 𝑞=0 is assumed to be given, for example, from Section 2.4.3, as𝑞=⎧⎪⎨⎪⎩0,𝑢=𝑢⁰0,𝑣=𝑣₀𝑥,𝑦⁰,𝑥,𝑦⁰.(28)Substituting this into (25) and (26), we obtain𝑓𝑔((𝑥𝑥,,𝑦𝑦⁰₀)=𝑢₀𝑥,𝑦⁰,)=𝑣₀𝑥,𝑦⁰.(29)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 that𝑓𝑔𝑥𝑥++𝑞𝑞,,𝑦𝑦₀₀=𝑢⁰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 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) yields𝑢𝑎𝑦₀,𝑦⁰,𝑞==0𝑞⋅−𝜉_𝑥+𝜙𝑦⁰𝑣𝑎𝑦₀𝑎+𝑓𝑎𝑦⁰+𝑞,𝑦⁰,,𝑦⁰,𝑞==0𝑞⋅−𝜉_𝑦−𝜙𝑎𝑎𝑦⁰+2𝑞+𝑔𝑎𝑦⁰+𝑞,𝑦⁰.(31)Replacing 𝑞 by 𝑥+𝑞, on 𝑎(𝑦₀)−𝑞≤𝑥≤𝑎(𝑦₀), and translating 𝑥 by −𝑎(𝑦₀), we have that𝑓𝑥+𝑞,𝑦₀=−𝑥+𝑞−𝑎𝑦⁰−𝜉^𝑥+𝜙𝑦⁰𝑔𝑥+𝑞,𝑦₀𝑎,=−𝑥+𝑞−𝑎𝑦⁰𝑎𝑦−𝜉^𝑦−2𝜙𝑎𝑥+𝑞+𝑎𝑦⁰₀,−𝑞≤𝑥≤𝑎𝑦⁰.(32)Finally, the analytic solution is obtained in closed form by substituting of 𝑓 and 𝑔 from (30) and (32) into (25) and (26):𝑢=𝑥𝑞,⋅𝑦₀,−𝑞𝜉_𝑥+𝜙𝑦+⎧⎪⎨⎪⎩−𝑥+𝑞−𝑎𝑎𝑦₀𝑢₀−𝜉^𝑥+𝜙𝑎𝑦⁰,𝑎𝑦⁰−𝑞≤𝑥≤𝑎𝑦⁰,𝑣𝑥,𝑦₀𝑥+𝑞,𝑦⁰,−𝑎𝑦⁰≤𝑥≤𝑎𝑦⁰−𝑞,=𝑞⋅,−𝑞𝜉_𝑦−𝜙+⎧⎪⎪⎨⎪⎪⎩−𝑥+𝑞−𝑎𝑎𝑦𝑥₀+2𝑞𝑎𝑦−𝜉^𝑦−2𝜙𝑎𝑥+𝑞+𝑎𝑦⁰₀,𝑣₀−𝑞≤𝑥≤𝑎𝑦⁰,𝑥+𝑞,𝑦⁰,−𝑎𝑦⁰≤𝑥≤𝑎𝑦⁰−𝑞.(33)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) as𝑢_∞𝑥,𝑦⁰=−𝑥−𝑎𝑦⁰−𝜉^𝑥+𝜙𝑦⁰𝑣_∞𝑎,𝑥,𝑦⁰=−𝑥−𝑎𝑦⁰−𝜉^𝑦−2𝜙𝑎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 contact problem, the contact region is discretized into a set of nodes. In Figure 7, 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(−𝜉_𝑥+𝜙𝑦/𝑎,−𝜉_𝑦−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𝑝^𝑥,𝑝_𝑦=𝐿(𝑢,𝑣).(35)If ‖(𝑝_𝑥,𝑝_𝑦)‖≤𝜇𝑝_𝑧, no slip occurs (𝑆_𝑥=𝑆_𝑦=0) and the values of (𝑢,𝑣) are retained. If, on the other hand, ‖(𝑝_𝑥,𝑝_𝑦)‖>𝜇𝑝_𝑧, then(𝑢,𝑣)_new=𝜇𝑝^𝑧𝐿‖‖((𝑢𝑢,,𝑣𝑣))‖‖_old.(36) In Figure 8, the flowchart of the algorithm is presented.

Figure 8 

The flowchart of the algorithm.

3.3. Calculation of the Tractions 𝐹_𝑥,𝐹_𝑦,𝑀_𝑧

The total traction forces and moment are obtained from the following integrals:𝐹_𝑥=^𝑎−𝑎^{𝑎(𝑦)−𝑎(𝑦)}𝐹𝑝_𝑥(𝑥,𝑦)𝑑𝑥𝑑𝑦,(37)_𝑦=^𝑎−𝑎^{𝑎(𝑦)−𝑎(𝑦)}𝑀𝑝_𝑦(𝑥,𝑦)𝑑𝑥𝑑𝑦,(38)_𝑧=^𝑎−𝑎^{𝑎(𝑦)−𝑎(𝑦)}𝑝(𝑥,𝑦)𝑟(𝑥,𝑦)𝑑𝑥𝑑𝑦(39)with 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])𝐹_𝑥=−8𝑎³𝜉_𝑥𝐹3𝐿,(40)_𝑦=−8𝑎³𝜉_𝑦3𝐿−𝜋𝑎³𝜙4𝐿.(41)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 as𝑓𝑊,𝐹,𝑀,𝑎,𝑝₀,𝑝,𝐸,𝑅,𝑉,𝛿𝑉_𝑥,𝛿𝑉_𝑦,𝜙̇,𝜈,𝜇,𝑞,𝑥,𝑦=0.(42)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:ℱ𝑊𝑝₀𝑎²,𝐹𝑝₀𝑎²,𝑀𝑝₀𝑎³,𝑎𝑎,𝑝𝑝⁰₀,𝑝𝑝₀𝐸,𝑝₀,𝑅ℱ𝑊𝑎,𝑉𝑉,𝛿𝑉𝑉^𝑥,𝛿𝑉𝑉^𝑦,𝜙𝑉̇𝑎,𝜈,𝜇,𝑎𝑞,𝑥𝑎,𝑎𝑦=0,_∗,𝐹^∗,𝑀^∗,𝑝^∗,𝐸^∗,𝑅^∗,𝜉_𝑥,𝜉_𝑦,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 used symbols. Since the model contains no system dynamics of the ball, the rolling velocity 𝑉 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

Figure 9 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)

(b)

(c)

(d)

Figure 9 

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

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 chose 𝜉_𝑥=0.015,𝜉_𝑦=𝜙=0. The rolling trajectory is 𝑞^∗=[0→2.1→−2.1→2.1]. The longitudinal creepage gives rise only to a traction force 𝐹^∗𝑥 in the rolling direction. Figure 10(c) plots 𝐹^∗𝑥 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.

(a)

(b)

(c)

(d)

(e)

Figure 10 

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.

4.2.2. Pure Lateral Creepage

To illustrate this case, we put 𝜉_𝑥=0,𝜉_𝑦=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 𝐹^∗𝑦 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.

(a)

(b)

Figure 11 

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.

4.2.3. Pure Spin

Here, we put 𝑣_𝑥=𝑣_𝑦=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 𝑀^∗𝑧. 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.

(a)

(b)

(c)

(d)

(e)

Figure 12 

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.

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 Figure 13, 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 𝑀^∗𝑧 increases towards a saturation value while the lateral force 𝐹^∗𝑦 shows a local maximum. These trends are depicted in Figure 13(c). As already mentioned, the longitudinal force component 𝐹^∗𝑥 equals zero for this case. These results agree with those obtained in [10].

(a)

(b)

(c)

Figure 13 

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

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) becomes1=3𝑊^∗𝑅^∗4𝐸^∗ℎ^1/3⟺𝑅^∗=4𝐸^∗ℎ3𝑊^∗.(44)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 of 𝑅^∗, 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 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 dimensionless ball radius in Figure 14 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 𝐸^∗ℎ. A certain limit of applicability, 𝑅^∗_lim 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 𝑅^∗_lim (marked by the box “theoretical region”) are only given for the sake of mathematical completeness.

(a)

(b)

(c)

(d)

Figure 14 

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

(a)

(b)

(c)

(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).

From Figures 14(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 theory𝑎∼(𝑊𝑅)^(1/3).(46)From Hills et al. [19], we have𝑀𝑊𝑀𝑅∼𝜃∼𝑅𝑞,(47)𝑞∼𝑊𝑅𝑅∼𝑎𝑅³,(48) since𝑀^∗=𝑀𝑝₀𝑎³.(49)This leads to𝑀^∗∼𝑞𝑘𝑅∼𝑎𝑞.𝑅𝑎,(50)_∗𝑖=𝜕𝑀^∗𝜕𝑞^∗∼1𝑅^∗since𝑞𝑎∼𝑞^∗,𝑅𝑎=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^∗, 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.

Notes

¹The authors dedicate this paper to the memory of Prof. J. J. Kalker (1933–2006).

Appendix

Consider the p.d.e.−𝜕𝑢𝜕𝑥+𝜕𝜕𝑞𝑢=ℱ(𝑥,𝑦,𝑞).(A.1) Equation A.1 corresponds to the Lagrangrian system [33]:𝑑𝑥−1=𝑑0𝑦=𝑑1𝑞=ℱ(𝑥𝑑,𝑢𝑦,𝑞).(A.2) The integrals of the system A.2 are𝑢−𝑥𝒢+(𝑦𝑥=,𝑞𝑦=𝛽,𝑞,𝛼),=𝛾,(A.3)where 𝛼,𝛽,𝛾 are arbitrary constants and 𝒢 is determined by𝒢(𝑥,𝑦,𝑞)===ℱℱ𝑑𝑔((𝑥𝛼,−𝑦,𝑞𝑞,)𝛽𝑑,𝑞𝑞)𝑑𝑞=𝑔(𝛼𝑑𝑞,𝛽(𝛼,𝑞,)𝛽=,𝑞)𝑔𝑑(𝑞𝑥+𝑞,𝑦,𝑞).(A.4) Note that the last integration step is carried out on the assumption that the function ℱ(𝛼−𝑞,𝛽,𝑞) posses an antiderivative 𝑔. This is always true if ℱ can be expressed as a polynomial in 𝑞.

The general solution to A.1 is then [33]ℱ(𝛼,𝛽,𝛾)=0(A.5)orℱ𝑥+𝑞,𝑦,𝑢−𝒢(𝑥,𝑦,𝑞)=0,(A.6)where ℱ is an arbitrary function. Assuming that (A.6) admits an explicit solution for 𝑢, this solution will have the form𝑢=𝒢(𝑥,𝑦,𝑞)−𝑓(𝑥+𝑞,𝑦),(A.7)where 𝑓 is an arbitrary function. As an example, considering ℱ(𝑥,𝑦,𝑞)=−𝜉_𝑥+𝜙𝑦/𝑎 and assuming 𝜉_𝑥 and 𝜙 to be constant, then𝑢(𝑥,𝑦,𝑞)=𝑞−𝜉_𝑥+𝜙𝑦𝑎+𝑓(𝑥+𝑞,𝑦).(A.8)

Nomenclature

𝑎:	Footprint radius
𝐴:	Contact patch
𝐜:	creepage
𝐂:	Relative creepage
𝐸ℎ:	Hertzian modulus of elasticity
𝐹:	Traction force
𝑔:	Traction bound
𝐺:	Modulus of rigidity
𝐻(⋅)∶	Heaviside step function
𝑘𝑖:	Initial stiffness
𝐿:	Flexibility
𝑀𝑧:	Spin moment
𝑝:	Pressure
𝑞:	Traversed rolling distance
𝑟:	Distance from the center
𝑅:	Contact radius
𝑅ℎ:	Equivalent radius of contact
𝐬:	Slip
𝐒:	Relative slip
𝑡:	Time
𝑢,𝑣:	Particle displacement
𝑉:	Rolling velocity
𝑊:	Normal load
𝑥:	Displacement
𝑥pr:	Prerolling distance
𝛽:	Dundurs' constant
𝛿:	Normal elastic deformation
𝛿𝑉𝑥:	Dimensional longitudinal creepage
𝛿𝑉𝑦:	Dimensional lateral creepage
𝜆:	Coefficient of rolling friction
𝜇:	Coefficient of friction
𝜈:	Poisson's coefficient
𝜉𝑥:	Nondimensional longitudinal creepage
𝜉𝑦:	Nondimensional lateral creepage
𝜙̇:	Dimensional spin creepage
𝜙:	Nondimensional spin creepage
𝜓:	Angular lateral creepage
𝜔:	Angular longitudinal creepage

Acknowledgements

This research is sponsored by the Fund for Scientific Research, Flanders (F.W.O.), under Grant no. FWO4283. The scientific responsibility is assumed by its authors.