# Experimental Investigation into the Tractive Prerolling  Behavior of Balls in V-Grooved Tracks

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

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## Abstract

In a rolling element
system, the period of transition between motion
commencement and the attainment of steady state,
gross rolling, and termed prerolling is of common
concern to many engineering applications. This
region is marked by hysteresis friction
behavior, with a characteristic
friction-displacement curve, which is in
particular relevant to motion characterization
and control issues. In a previous paper, the authors carried out a
theoretical analysis of tractive prerolling,
leading to a model for simulating this
phenomenon. The present paper is dedicated to
the experimental investigation of tractive
prerolling friction behavior, including
validation of the theoretical model. Firstly, a
kinematic analysis of the rolling motion in
V-grooved tracks is carried out. Secondly, the
influence of the normal load on the frictional
behavior, in prerolling up to the attainment of
gross rolling, is investigated on a dedicated
test setup. Finally, the newly developed
theoretical model is validated by comparison
with the experimental results. Satisfactory
agreement is obtained between theory and
experiment.


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## Body

## 1. Introduction

The characterization of rolling friction is a pertinent issue in many fields
ranging from rolling stock, aircraft servo systems, and rotary and linear
positioning systems to the control of miniature reading/writing heads, friction
drives, and high precision systems. This topic has therefore been gaining importance over the last decades, owing to the increasing demands of industrial
applications on positioning accuracy, wear, and controllability. A common
concern in all these applications is the rolling element system's behavior
between motion reversal points, in other words, the period between motion
commencement and the attainment of gross rolling. This period is referred to as
prerolling, which is akin to presliding in sliding contacts [1–3]. Prerolling is marked by a
clear hysteresis behavior with nonlocal memory, which lies at the root of the
strongly nonlinear dynamic behavior that characterizes systems comprising
rolling elements [4, 5]. The design parameters of rolling element systems,
such as preload, ball diameter, and track groove angle, evidently, have their
influence on the frictional behavior of the system. Despite this fact, a
coherent, systematic theory that correlates behavior with parameters has not
yet received the attention it deserves in the literature, owing to many
reasons, in particular the complexity and the strong nonlinearity of the
rolling contact problem.Rolling resistance or, as it is often called, rolling
friction arises from two basic mechanisms: internal mechanical losses in the
bulk of rolling objects and external tractions in the rolling contact patch. In
case of elastic rolling objects (such as metallic objects), the former is
generally negligible in magnitude in comparison with the latter. This paper is
concerned with tractive rolling; more specifically that, which is caused by
creepages in the rolling motion, in particular, spin creepage.When 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 steady-state distribution. This distribution, which is independent of
the initial field prior to commencement of rolling, does not vary with further
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 [2].
In this reference, an extensive experimental study of prerolling behavior of a
ball in V-groove application has been reported, where the reader may also find
a short overview of literature around the subject. The conclusions and
correlations presented in that study remained, however, empirical.Recently, the present authors have developed a
theoretical model, which is able to simulate tractive prerolling (and
transition to gross rolling) in an arbitrary rolling contact [6]. The present paper introduces
on the one hand an experimental validation of that model and on the other hand
a check of the validity of Amontons' law as applied to rolling contacts.Bearings can be considered as one of the most
important machine elements in regard to accuracy of motion. However, only few
types of bearings, notably cylindrical and tapered roller bearings, contain
rolling elements which undergo pure rolling, that is, rolling without
creepage, which leads to microslip. In other types, such as pivot bearings,
axially loaded deep groove ball bearings, angular contact bearings, and
recirculating rolling element guideways, the rolling motion of the rolling
elements comprises a significant amount of microslip which in those cases is
responsible for the greatest part of the frictional losses. This microslip
depends on the relative velocity (or creepage) between the contacting surfaces
and on the elasticity.Other applications where rolling with creepage appears
are gears and friction wheel transmissions. The study of rolling contacts
comprising microslip becomes therefore an important research issue.Compared to steady state rolling friction, only a few
experimental studies in the field of prerolling have been conducted. Liu et al.
[7] perform some
experiments on a newly developed test apparatus in this field. They distinguish
three regimes for the coefficient of rolling friction, namely, a gradually
increasing coefficient in the elastic deformation regime, a peak in the
transition zone, and a constant value in the gross rolling regime.
Additionally, the wear behavior of steel is investigated under different
conditions. The breakaway rolling friction (i.e., the peak value prior to
commencement of gross rolling) is studied by Budinski [8] in a case study in which he
presents a modified test procedure for the breakaway coefficient of rolling
friction for rolling element bearings.Endo et al. [9] study the nonlinear spring characteristic of a rolling
element in the transient zone. They link the energy losses in the contact to
the overall damping characteristics of the rolling system and confirm
experimentally that these losses vary when rolling elements with different
diameters are used. Otsuka and Masuda [10] call the prerolling region the “nonlinear spring
area” and try to explain the influence of the nonlinear spring behavior (NSB)
on the frequency response in some practical case studies.Besides dry rolling contacts, lubricated contacts have
also been investigated in the past. Wang et al. [11] investigate the behavior of
EHL films under pure rolling, short stroke reciprocating motion. High-speed
color camera measurements are compared to simulations with multigrid techniques.Xiao et al. [12] have suggested a friction model for dry and
lubricated rolling of rough bodies. They observe a linear increasing
coefficient of rolling friction with increasing mean contact pressure, up to a
maximum limit above which the coefficient of friction stays constant.Load variations in mechanical systems comprising
rolling elements often occur, while their influence on the frictional behavior
is neglected in control models. This may impose a limit on the attainable
positioning accuracy. Since this issue is of importance in machine design and
control, this study deals with the influence of the normal load on the
prerolling behavior. The first objective is to verify the effect of the normal
load on the overall shape and behavior of the hysteresis loop on an existing
test setup. The second objective is the verification of the theory presented in
[6] (which is the
theoretical part of this study) for the case of rolling with spin creepage.The paper is organized as follows. Section2 considers
the kinematics of the rolling element system which is used during the
experiments and estimates the relative order of magnitude of the various types
of possible friction losses. Section 3 gives an overview of the experiments. In
Section 4, the hysteresis identification is discussed. A model equation is
proposed which is used to fit and process the data for further analysis. In
Section 5, the influence of the normal load on the parameters of the model
equation is studied. All these aspects provide the prerequisites for the
validation of the theoretical model, presented in [6], in Section 6. Finally,
appropriate conclusions are drawn and future work is outlined.


## 2. Analysis of Rolling Friction for  a Ball in a V-Grooved Track

As mentioned in
the previous section, rolling friction may be caused by internal losses and/or
external tractions. Consequently, it may be pertinent to gain some idea of the
mechanisms, parameters, and orders of magnitudes of any possible mechanism
prior to embarkation on a test program. In this section, we first examine the
kinematics of motion of the ball in order to evaluate the creepages associated
with that motion. Secondly, we assess the importance of each rolling friction
mechanism.


### 2.1. Kinematics of a Ball Rolling in V-Grooved Tracks

As mentioned in the
introduction, the tractive rolling subject of this investigation arises mainly
as a consequence of creepages imposed on the otherwise pure rolling motion. For
a formal definition of the principal creepages, the reader is referred to
[6]. In the case of
balls rolling between two V-grooved tracks, the most dominant type of creepage
is spin. Quantification of relative spin, that is, the amount of spin per unit
rolling distance is therefore essential for evaluating tractive friction. To
this end, this section presents a kinematic analysis of the motion of the balls
between the V-grooves.Consider a ball rolling between two V-grooved tracks as
depicted in Figure1. When the lower track moves, with a velocity V normal to the plane of the page, the ball will
roll between the grooves with a main rolling velocity ω. This latter may be resolved into a pure rolling motion ωR and a pure spin motion ωS. In the case of pure rolling, the creepage is assumed zero, thus no microslip
takes place. In other words, for the case of a ball in the V-grooved tracks,
the spin component of the rolling motion is the only creepage responsible for
the frictional losses. As a consequence, the friction resistance will increase
with decreasing groove angle, as the spin per unit rolling increases with a
decreasing groove angle. The other limiting case is the rolling of a sphere in
between two horizontal flat plates, where consequently no creepage occurs. In
this case, the only remaining sources of rolling friction are elastic
hysteresis losses and Heathcote slip (see further below).Figure 1
The roll and spin motion.A linear motion of the lower V-groove corresponds to a
main angular velocityω:(1)ω=VR′,with R′ being the distance depicted in Figure 1. This main
motion can be resolved into a pure rolling motion ωR and a pure spin motion ωS:
(2)ωR=ωcos(δ)=ωsin(α2),ωS=ωsin(δ)=ωcos(α2),thus,(3)R′=Rsin(α2),ωS=Vcos(α/2)Rsin(α/2),ωR=VR.


### 2.2. Rolling Friction Forces

Beginning with the traction forces, let us note that the forces associated with the kinematics
outlined above may be obtained in an analogous way. Thus, let the motion of the
sphere result in a spinning momentMz, at each contact patch, which gives rise to the frictional force that has to be
overcome by the linear actuator. For one ball, four spin components Mz are formed, as depicted in Figure 2, which
result in a global spin moment Mz,tot, around the rolling axis, that is given by(4)Mz,tot=4Mzcos(α2). The rolling friction force, per ball, associated with
this moment is given by
(5)Ffric,tot=4Mzcos(α/2)2R′=2Mzcos(α/2)Rsin(α/2). Thus, the coefficient of rolling friction owing to
tractive rolling is given by
(6)λtractive=FfricW=Mz,totWRsin(α/2),
with W the externally applied normal load.Figure 2
The local spin
moments result in a global spin momentMz,tot.We turn our attention presently to the other two
phenomena responsible for the measured friction in experiments, namely, bulk
elastic hysteresis and Heathcote slip.Firstly, rolling resistance as a consequence of the
elastic hysteresis losses due to Hertzian compression is given by [13]
(7)λelastic=3ϵa16R,
where ϵ is the coefficient of elastic hysteresis loss
due to Hertzian compression and a is the radius of the contact patch which is given by
(8)a=(34WRE*)1/3. Secondly, Heathcote [14] deduced a formulation for
the frictional moment M for the case of pure rolling by neglecting the
elastic compliance of the contacting bodies [15],(9)MW=0.08μa2R,
with μ being the local coefficient of sliding
friction. From this equation, one can derive an expression for the coefficient
of rolling resistance due to Heathcote slip,
(10)λHC=MRW=0.08μa2R2.


### 2.3. Order of Magnitude Analysis

The three phenomena together (6), (7), and (10) account for the total coefficient of
rolling friction measured in the subsequently presented experiments. It is thus
interesting to gauge the relative importance of these different rolling
friction contributions in the context of our experiments.Thus, let us consider a steel ball of 6 mm diameter, a maximal loadW of 20N and a coefficient of sliding friction μ of 0.5. The contact radius a will be on the order of 10−4 m. Experiments reported in this paper show a
rolling friction value on the order of 10−2. Considering a realistic coefficient of elastic hysteresis loss ϵ of 0.01, the coefficient of rolling friction due to elastic hysteresis losses λelastic is on the order of 10−5. The coefficient of rolling friction due to Heathcote slip λHC is on the order of 10−8.This estimation indicates that the contribution of
traction (in this case, microslip associated with the spin creepage) is by far
the most dominant, compared to the two other energy dissipating phenomena.


## 2.1. Kinematics of a Ball Rolling in V-Grooved Tracks

As mentioned in the
introduction, the tractive rolling subject of this investigation arises mainly
as a consequence of creepages imposed on the otherwise pure rolling motion. For
a formal definition of the principal creepages, the reader is referred to
[6]. In the case of
balls rolling between two V-grooved tracks, the most dominant type of creepage
is spin. Quantification of relative spin, that is, the amount of spin per unit
rolling distance is therefore essential for evaluating tractive friction. To
this end, this section presents a kinematic analysis of the motion of the balls
between the V-grooves.Consider a ball rolling between two V-grooved tracks as
depicted in Figure1. When the lower track moves, with a velocity V normal to the plane of the page, the ball will
roll between the grooves with a main rolling velocity ω. This latter may be resolved into a pure rolling motion ωR and a pure spin motion ωS. In the case of pure rolling, the creepage is assumed zero, thus no microslip
takes place. In other words, for the case of a ball in the V-grooved tracks,
the spin component of the rolling motion is the only creepage responsible for
the frictional losses. As a consequence, the friction resistance will increase
with decreasing groove angle, as the spin per unit rolling increases with a
decreasing groove angle. The other limiting case is the rolling of a sphere in
between two horizontal flat plates, where consequently no creepage occurs. In
this case, the only remaining sources of rolling friction are elastic
hysteresis losses and Heathcote slip (see further below).Figure 1
The roll and spin motion.A linear motion of the lower V-groove corresponds to a
main angular velocityω:(1)ω=VR′,with R′ being the distance depicted in Figure 1. This main
motion can be resolved into a pure rolling motion ωR and a pure spin motion ωS:
(2)ωR=ωcos(δ)=ωsin(α2),ωS=ωsin(δ)=ωcos(α2),thus,(3)R′=Rsin(α2),ωS=Vcos(α/2)Rsin(α/2),ωR=VR.


## 2.2. Rolling Friction Forces

Beginning with the traction forces, let us note that the forces associated with the kinematics
outlined above may be obtained in an analogous way. Thus, let the motion of the
sphere result in a spinning momentMz, at each contact patch, which gives rise to the frictional force that has to be
overcome by the linear actuator. For one ball, four spin components Mz are formed, as depicted in Figure 2, which
result in a global spin moment Mz,tot, around the rolling axis, that is given by(4)Mz,tot=4Mzcos(α2). The rolling friction force, per ball, associated with
this moment is given by
(5)Ffric,tot=4Mzcos(α/2)2R′=2Mzcos(α/2)Rsin(α/2). Thus, the coefficient of rolling friction owing to
tractive rolling is given by
(6)λtractive=FfricW=Mz,totWRsin(α/2),
with W the externally applied normal load.Figure 2
The local spin
moments result in a global spin momentMz,tot.We turn our attention presently to the other two
phenomena responsible for the measured friction in experiments, namely, bulk
elastic hysteresis and Heathcote slip.Firstly, rolling resistance as a consequence of the
elastic hysteresis losses due to Hertzian compression is given by [13]
(7)λelastic=3ϵa16R,
where ϵ is the coefficient of elastic hysteresis loss
due to Hertzian compression and a is the radius of the contact patch which is given by
(8)a=(34WRE*)1/3. Secondly, Heathcote [14] deduced a formulation for
the frictional moment M for the case of pure rolling by neglecting the
elastic compliance of the contacting bodies [15],(9)MW=0.08μa2R,
with μ being the local coefficient of sliding
friction. From this equation, one can derive an expression for the coefficient
of rolling resistance due to Heathcote slip,
(10)λHC=MRW=0.08μa2R2.


## 2.3. Order of Magnitude Analysis

The three phenomena together (6), (7), and (10) account for the total coefficient of
rolling friction measured in the subsequently presented experiments. It is thus
interesting to gauge the relative importance of these different rolling
friction contributions in the context of our experiments.Thus, let us consider a steel ball of 6 mm diameter, a maximal loadW of 20N and a coefficient of sliding friction μ of 0.5. The contact radius a will be on the order of 10−4 m. Experiments reported in this paper show a
rolling friction value on the order of 10−2. Considering a realistic coefficient of elastic hysteresis loss ϵ of 0.01, the coefficient of rolling friction due to elastic hysteresis losses λelastic is on the order of 10−5. The coefficient of rolling friction due to Heathcote slip λHC is on the order of 10−8.This estimation indicates that the contribution of
traction (in this case, microslip associated with the spin creepage) is by far
the most dominant, compared to the two other energy dissipating phenomena.


## 3. Overview of the Experiments

### 3.1. The Test Setup

The experimental investigation is carried out on a previously developed tribometer
(we refer to [3] for
more details) with some minor adaptations. Figure 3 depicts a schematic view of
the setup. The tribometer comprises four parts. (1) the actuator part is fixed
to the base plate. It comprises a Lorenz actuator, which drives the lower
friction member through a flexured rod. (2) The frictional part contains (i)
the lower friction member, which in our case is the lower V-track mounted to a
block that floats on air bearings and is driven by the aforementioned actuator,
and (ii) the upper friction member which is the upper V-track, being connected
to the dynamometer via an elastic hinge; (3) a normal loading part, which
comprises a cantilever to apply the load through an air bearing; and (4) the
sensor part which comprises (i) a Bently eddy current sensor which measures the
relative motion between the two friction members, (ii) the friction force
dynamometer which contains a force cell attached to a fixed frame that is fixed
to the base plate, and (iii) a force cell for measuring the normal load. These different parts are
decoupled as far as possible. The Lorenz actuator imposes a straight motion,
with a periodic velocity, to the lower friction member through the flexured
rod. The Lorenz actuator is current driven and the position signal of the
displacement sensor is used for feedback control to obtain, as best as
possible, the desired motion of the moving part.Figure 3
A schematic overview of the
tribometer.The purpose of the employed elastic joints is to set
off alignment errors of the friction block in the normal, lateral, and
rotational directions.The loading part allows easy control of the normal
load on the frictional part. The load is applied by using a cantilever (see
Figure3) and is transmitted to the frictional part through an air bearing, in
order to avoid affecting the inertia of the frictional part when changing the
normal load (mass). Another advantage of the air bearing is that the load is
applied without direct contact which may introduce adventitious forces. The air
bearing is equipped with a ball joint to allow the bearing to adapt its
orientation with respect to the friction block. In this way, a uniform load is
applied normal to the friction contact. Detailed experimental characterization
of this setup can be found in [16].


### 3.2. The Normal Load Limitations

In order to guarantee the applicability of Hertz's theory and the elastic behavior of the
contacting bodies, the maximum allowable load is calculated. From the work of
Tabor [17], plasticity
first occurs when the Hertzian contact pressure reaches the limit of
approximately 0.6H, where H is the hardness. Thus for a circular contact,
(11)p0=1π(6WE*2R2)1/3<0.6H.
In Figure 4, the evolution of
the contact pressure is depicted as function of the normal load on the
V-grooved track for different opening angles of the V grooves and the case of
two steel balls of diameter 6.35 mm. For the used V groove of 60°, the maximum applicable load approximates to 200N. For higher loads, the brinelling effect occurs which means that plastic
damage of the contacting surfaces takes place.Figure 4
Determination of the maximum allowable normal load (6.35 mm
steel balls).


### 3.3. Measurement Procedure

For the purpose of
this study, the displacement of the moving block and the friction force are
measured by the sensors described above and captured using a
dSPACE system.
To improve the quality of the measured data, a lowpass filter with a bandwidth
of 30 Hz is used to exclude most of the measurement
noise. The filter is applied to all channels in the same way in order to avoid
phase shift effects. Afterwards, the data samples, obtained from repeated test
runs for any single test condition, are time averaged to remove random noise
and to improve their statistical relevance.


## 3.1. The Test Setup

The experimental investigation is carried out on a previously developed tribometer
(we refer to [3] for
more details) with some minor adaptations. Figure 3 depicts a schematic view of
the setup. The tribometer comprises four parts. (1) the actuator part is fixed
to the base plate. It comprises a Lorenz actuator, which drives the lower
friction member through a flexured rod. (2) The frictional part contains (i)
the lower friction member, which in our case is the lower V-track mounted to a
block that floats on air bearings and is driven by the aforementioned actuator,
and (ii) the upper friction member which is the upper V-track, being connected
to the dynamometer via an elastic hinge; (3) a normal loading part, which
comprises a cantilever to apply the load through an air bearing; and (4) the
sensor part which comprises (i) a Bently eddy current sensor which measures the
relative motion between the two friction members, (ii) the friction force
dynamometer which contains a force cell attached to a fixed frame that is fixed
to the base plate, and (iii) a force cell for measuring the normal load. These different parts are
decoupled as far as possible. The Lorenz actuator imposes a straight motion,
with a periodic velocity, to the lower friction member through the flexured
rod. The Lorenz actuator is current driven and the position signal of the
displacement sensor is used for feedback control to obtain, as best as
possible, the desired motion of the moving part.Figure 3
A schematic overview of the
tribometer.The purpose of the employed elastic joints is to set
off alignment errors of the friction block in the normal, lateral, and
rotational directions.The loading part allows easy control of the normal
load on the frictional part. The load is applied by using a cantilever (see
Figure3) and is transmitted to the frictional part through an air bearing, in
order to avoid affecting the inertia of the frictional part when changing the
normal load (mass). Another advantage of the air bearing is that the load is
applied without direct contact which may introduce adventitious forces. The air
bearing is equipped with a ball joint to allow the bearing to adapt its
orientation with respect to the friction block. In this way, a uniform load is
applied normal to the friction contact. Detailed experimental characterization
of this setup can be found in [16].


## 3.2. The Normal Load Limitations

In order to guarantee the applicability of Hertz's theory and the elastic behavior of the
contacting bodies, the maximum allowable load is calculated. From the work of
Tabor [17], plasticity
first occurs when the Hertzian contact pressure reaches the limit of
approximately 0.6H, where H is the hardness. Thus for a circular contact,
(11)p0=1π(6WE*2R2)1/3<0.6H.
In Figure 4, the evolution of
the contact pressure is depicted as function of the normal load on the
V-grooved track for different opening angles of the V grooves and the case of
two steel balls of diameter 6.35 mm. For the used V groove of 60°, the maximum applicable load approximates to 200N. For higher loads, the brinelling effect occurs which means that plastic
damage of the contacting surfaces takes place.Figure 4
Determination of the maximum allowable normal load (6.35 mm
steel balls).


## 3.3. Measurement Procedure

For the purpose of
this study, the displacement of the moving block and the friction force are
measured by the sensors described above and captured using a
dSPACE system.
To improve the quality of the measured data, a lowpass filter with a bandwidth
of 30 Hz is used to exclude most of the measurement
noise. The filter is applied to all channels in the same way in order to avoid
phase shift effects. Afterwards, the data samples, obtained from repeated test
runs for any single test condition, are time averaged to remove random noise
and to improve their statistical relevance.


## 4. Identification of Prerolling Hysteresis

The aim of the measurements is to evaluate the prerolling behavior of the friction force for
different normal loads. The prerolling friction force is function of the
displacement which can be described by a rate-independent hysteresis function.
The hysteresis  curve is fully determined by itsvirgin curve (see
[18] for detail). This
theoretical curve starts at the origin (with the very first commencement of
motion) has positive first and second derivatives for all displacements, and
saturates for large displacements. Therefore, it is interesting to use the
virgin curve as reference for comparing the hysteresis curves for the different
experiments, since this curve incorporates all information of the hysteresis
behavior. Once a hysteresis loop is identified, one can determine the virgin
curve by scaling the identified curve, by factor 2 in both x and y directions, and shifting the turning points to
the origin [18].A measured hysteresis curve is identified by fitting
it to a parametric model using nonlinear regression, in order to capture the
curve in a mathematical form. In this study, a two parameter model is chosen
for simplicity which is of the form(12)F(x)=h0(1−e−a(x−x0)).
The parameter h0 gives the saturation value of the curve, while
the parameter a is a measure of the curvature. Finally, the
parameter x0 is necessary for the translation along the
x-axis. For other suitable model equations, we refer to [2].Figure5 shows an example of the model equation fitted
to measured hysteresis data. Both the external loop and the derived virgin
curve are depicted.Figure 5
Identification of the hysteresis loop, including the determination of the
virgin curves. (Steel ballsϕ6.35mm, groove angle: 60°, W=88.9 N).


## 5. Influence of the Normal Load

Examples of the fitted virgin curves for the different
normal loads are visualized in Figure6. The determining parameters for these
curves are h0 and a which are depicted in Figures 7(a) and 7(b).
One can observe an increasing trend for h0 as function of the external normal load W, while the curvature a as function of the external normal load W shows a decreasing trend. Both trends can be
fitted by a linear equation, as shown in the figures. The range of performed
experiments result in a spread of h0 of approximately factor 3 (between minimum and maximum values), while
the results for parameter a give a spread of approximately a factor of 1.1. For the sake of gaining an idea about the shape deviations, let us take a to be fixed to a value equal to its mean value
of 0.0325. In that case, the error on a approximates to ±5%. In this study however, the parameter a is considered to have a linear decreasing trend as function of the normal load W which reduces the error further to
approximately 3%. In other words, the effect of the normal load on the shape of the hysteresis
curve is not appreciable.Figure 6
The identified virgin curves for the different
normal load cases (46.5N≤W≤123.2N).The evolution ofh0 (a) and a (b) with the normal load.
(a)(b)In Figure8, the virgin curves are scaled by
the applied normal load to obtain the evolution of λ=F/W, the coefficient of rolling friction. One can observe a slightly increasing
trend for λ in function of the external normal load.Figure 8
The evolution ofλ with the normal load.In order to ascertain the influence of sliding
friction characteristics (of the ball/guideway surfaces) on the rolling
friction values, experiments with other ball types are carried out on the same
setup. Balls of diameter of 6 mm, coated with a thin layer of diamond-like
carbon, are compared to the former uncoated steel balls. The results are
depicted in Figure9. While the same general trends are visualized for these
DLC coated balls, one can notice the appreciably lower rolling friction values
(by factor 0.3) for the coated balls. This confirms our previous conclusion
(see Section 2.2) that the greatest part of the rolling resistance, for this
bearing configuration, is owing to microslip tractions and not to elastic
hysteresis in the ball/groove material. Furthermore, for the DLC-coated balls,
the spread of the parameter h0 approximates to a factor of 2.8.
As for the parameter a, the error committed by taking it as a constant (equal to 0.125), is ±12%, while considering the linear decreasing trend reduces the error to
approximately 1%.Results for the diamond coated balls (ϕ6mm, 60° opening angle).
(a)(b)(c)(d)


## 6. Comparison between Model and Experiment

The second aim of this article is the validation of a theoretical model presented in [6]. The model is able to
predict creepage induced traction during and after prerolling.For the sake of completeness, a short overview of the
model is given in the next section. Afterwards, the model is validated by means
of the previously presented experimental results for the case of rolling with spin.


### 6.1. Overview of the Model Structure [6]

For rolling objects, the relative rigid motion of the counter surfaces termed the creepage,C, which consists of longitudinal and lateral creepages, and spin. The (elastic)
surface displacement field, u, is related to the slip field, S, by the equation
(13)S=C−∂u∂x+∂u∂q, Analysis for the case of no slip results in closed
form expression for the displacement of the surface particles:(14)u=f(C,x,y,q),
where x,y are the coordinates of the points in the
contact patch, and q is the traversed rolling distance.To take slip into consideration, the newly obtained
valuesu=(u,v) are used to calculate the pressure
distribution in the contact patch:
(15)(px,py)=L(u,v). If ∥(px,py)∥≤μpz, no slip occurs (Sx=Sy=0) and the values of (u,v) are retained. In the other case, ∥(px,py)∥>μpz, the no slip deformation is replaced by
(16)(u,v)new=μpzL((u,v)∥(u,v)∥)old.
Finally, the traction forces and spin moment are calculated as follows:
(17)(Fx,Fy)=∫−aa∫−a(y)a(y)(px,py)dxdy,Mz=∫−aa∫−a(y)a(y)p(x,y)r(x,y)dxdy. In order to calculate the tractions using this model,
one needs to know (i) the geometry and elastic properties of the rolling
objects, (ii) the normal load, (iii) the creepages, and (iv) the local sliding
friction coefficient in the contact. From the kinematics of a V-grooved setup,
it appears that only spin creepage is present. Furthermore, only the spin
moments are of interest, since the force in the rolling direction Fx equals zero for every contact patch, and the
force, Fy, perpendicular to the rolling direction will have a resultant that is equal to
zero when taking all four contact patches into consideration.


### 6.2. Experimental Validation of the Model

To validate the model, presented in [6], the experimental results are compared to the model
predictions. The model is implemented for the kinematics of the V-grooved
track, using reasonable values for the elastic parameters. The local
coefficient of sliding friction, necessary for the model, is determined by an
additional experiment. In this experiment, the coefficient of sliding friction
is determined for the system of a ball on the track as sliding counter surface.
This is carried out by gluing two balls on a plate, and using this as counter
surface for one of the V-grooves. Periodic sliding motion is imposed for
different normal loads, while the friction force is measured. The coefficient
of sliding friction is determined by dividing the measured friction force by
the applied normal load. Although this rudimentary experiment ignores some
basic rules as, for example, the velocity dependence of the coefficient of
friction; these estimates are accurate enough for a first evaluation of the
model and are a good starting point for further analysis. Note that, Lampaert
et al. [3] show that
the velocity dependence of the coefficient of friction for metallic contacts is
small. The sliding friction results of this experiment are depicted in Figure 10 for different normal loads. The results of this short experiment are
used as input for the model.Figure 10
The evolution of the coefficient of
sliding friction as function of the normal load.The experimentally obtained virgin curves of Section4
are now compared with the model results in Figure 11. We note a good
resemblance between model and experimental results. The initial slope as well
as the saturation value behave in ways similar to theoretical predictions.Figure 11
A comparison between
experimental results and model results of [6] (6.35 mm steel balls, 60° groove angle, 46.5N≤W≤123.2N).In Figure12, the error, according to (18) below,
between model and experiment is presented for the different virgin curves. For
clarity, only a few characteristic results are presented. The error is defined
as follows:
(18)error=|(Fmeasured−Fmodel)|Fmean,where Fmean=(Fmeasured+Fmodel)/2, calculated as function of the displacement.Figure 12
The error between model and experimental results.In Figure13, for every virgin curve, the error is
averaged and divided by the maximum friction force. One can distinguish two
regions. Firstly, experiments with high normal loads show an error lying within 5%.
Secondly, experiments with small normal loads show an error lying in the range
of 6–16%.
An obvious explanation to this is the smaller signals and thus the lower
signal-to-noise ratio of those experiments.Figure 13
The
maximum error between model and experimental results as function of the normal
load.


## 6.1. Overview of the Model Structure [6]

For rolling objects, the relative rigid motion of the counter surfaces termed the creepage,C, which consists of longitudinal and lateral creepages, and spin. The (elastic)
surface displacement field, u, is related to the slip field, S, by the equation
(13)S=C−∂u∂x+∂u∂q, Analysis for the case of no slip results in closed
form expression for the displacement of the surface particles:(14)u=f(C,x,y,q),
where x,y are the coordinates of the points in the
contact patch, and q is the traversed rolling distance.To take slip into consideration, the newly obtained
valuesu=(u,v) are used to calculate the pressure
distribution in the contact patch:
(15)(px,py)=L(u,v). If ∥(px,py)∥≤μpz, no slip occurs (Sx=Sy=0) and the values of (u,v) are retained. In the other case, ∥(px,py)∥>μpz, the no slip deformation is replaced by
(16)(u,v)new=μpzL((u,v)∥(u,v)∥)old.
Finally, the traction forces and spin moment are calculated as follows:
(17)(Fx,Fy)=∫−aa∫−a(y)a(y)(px,py)dxdy,Mz=∫−aa∫−a(y)a(y)p(x,y)r(x,y)dxdy. In order to calculate the tractions using this model,
one needs to know (i) the geometry and elastic properties of the rolling
objects, (ii) the normal load, (iii) the creepages, and (iv) the local sliding
friction coefficient in the contact. From the kinematics of a V-grooved setup,
it appears that only spin creepage is present. Furthermore, only the spin
moments are of interest, since the force in the rolling direction Fx equals zero for every contact patch, and the
force, Fy, perpendicular to the rolling direction will have a resultant that is equal to
zero when taking all four contact patches into consideration.


## 6.2. Experimental Validation of the Model

To validate the model, presented in [6], the experimental results are compared to the model
predictions. The model is implemented for the kinematics of the V-grooved
track, using reasonable values for the elastic parameters. The local
coefficient of sliding friction, necessary for the model, is determined by an
additional experiment. In this experiment, the coefficient of sliding friction
is determined for the system of a ball on the track as sliding counter surface.
This is carried out by gluing two balls on a plate, and using this as counter
surface for one of the V-grooves. Periodic sliding motion is imposed for
different normal loads, while the friction force is measured. The coefficient
of sliding friction is determined by dividing the measured friction force by
the applied normal load. Although this rudimentary experiment ignores some
basic rules as, for example, the velocity dependence of the coefficient of
friction; these estimates are accurate enough for a first evaluation of the
model and are a good starting point for further analysis. Note that, Lampaert
et al. [3] show that
the velocity dependence of the coefficient of friction for metallic contacts is
small. The sliding friction results of this experiment are depicted in Figure 10 for different normal loads. The results of this short experiment are
used as input for the model.Figure 10
The evolution of the coefficient of
sliding friction as function of the normal load.The experimentally obtained virgin curves of Section4
are now compared with the model results in Figure 11. We note a good
resemblance between model and experimental results. The initial slope as well
as the saturation value behave in ways similar to theoretical predictions.Figure 11
A comparison between
experimental results and model results of [6] (6.35 mm steel balls, 60° groove angle, 46.5N≤W≤123.2N).In Figure12, the error, according to (18) below,
between model and experiment is presented for the different virgin curves. For
clarity, only a few characteristic results are presented. The error is defined
as follows:
(18)error=|(Fmeasured−Fmodel)|Fmean,where Fmean=(Fmeasured+Fmodel)/2, calculated as function of the displacement.Figure 12
The error between model and experimental results.In Figure13, for every virgin curve, the error is
averaged and divided by the maximum friction force. One can distinguish two
regions. Firstly, experiments with high normal loads show an error lying within 5%.
Secondly, experiments with small normal loads show an error lying in the range
of 6–16%.
An obvious explanation to this is the smaller signals and thus the lower
signal-to-noise ratio of those experiments.Figure 13
The
maximum error between model and experimental results as function of the normal
load.


## 7. Discussion and Conclusions

In this paper, an experimental investigation of prerolling is presented. More specifically,
the influence of the normal load on the hysteresis behavior is studied on an
existing setup, in which the frictional element comprises two balls rolling
between a pair of V-grooved tracks. The frictional hysteresis is fitted by an
exponential model form, in which two parameters characterize the shape, denoted
by the curvature, and the saturation value of the friction force, respectively.
The evolution of these parameters as function of the normal load is presented.
We observe a slight increase in the coefficient of rolling frictionλ with an increasing normal load and a very
small change in the curvature.Another part of the investigation carries out a
validation of an existing theoretical model for tractive rolling by use of the
obtained experimental data. Kinematic analysis of a rolling configuration shows
that the only creepage present is spin. Using the experimentally determined sliding
friction coefficient and spin values as input to the model, a comparison of
model prediction and experiment is carried out. Agreement between model and
experiment falls within an error margin of less than 16% for low normal loads and 5% for high normal loads.


## 8. Future Work

The global aim of this experimental research is the determination of the relation between the
hysteretic friction force and the different system parameters. In [2], Al-Bender and Symens
investigate other system parameters as groove angle, ball diameter, and so
forth. A global synthesis of these results, together with the results of this
study, can improve the understanding of the frictional behavior of rolling
elements. A more correct estimation of the hysteresis shape can find its
applications in feed forward position control to avoid more complicated model
structures which are computationally intensive and less appropriate for online application.



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*Source: 561280-2008-05-18.xml*