Jtam.dvi


JOURNAL OF THEORETICAL

AND APPLIED MECHANICS

47, 2, pp. 275-293, Warsaw 2009

FRICTION ROLLING WITH LATERAL SLIP IN RAIL
VEHICLES

Robert Konowrocki

Czesław Bajer

Institute of Fundamental Technological Research, Plish Academy of Sciences, Warsaw, Poland

e-mail: rkonow@ippt.gov.pl

The paper deals with the dynamic phenomena accompanying the wheel
rollingovera road(rail, track),with lateral slip effects.Theyoccur in rol-
ling of awheel andwheelset on a straight track in the case of lateral load
and especially on curves.Different curvature radii and rotaryoscillations
of wheelsets result in skew rolling and, in turn, in lateral slip oscillation
in the contact zone between the wheel and rail. It significantly increases
noise and wear in real structures. Double periodicity of motion was de-
tected in anunderground train.Hitherto, this phenomenon has not been
reported in the literature. Experimental investigationwas performed on
a test stand forvariousparameters: theangle of skewrolling,velocityand
contact pressure. Results were related to a two degree-of-freedom the-
oretical system. In the case of steel/polyester and polyamide/polyester
friction pair, qualitatively similar results were obtained.

Key words: rolling, friction, lateral slip, wheel-rail interaction

1. Introduction

Rolling contact has been investigated intensively during the last two decades
(Kalker, 1990; Knothe, 1983). However, full and correct simulation of the rol-
lingprocesswithall phenomenathat occur in the railway transport is complex.
Thegeneration of corrugations (Bajer, 1998;Hempelmann et al., 1991;Kalker,
1994; Sato et al., 2002) and resultingvibrations, the essential problemof rolling
in railway transportation, has not been completely explained yet. Burdensome
vibration transmission to the environment is in focus of researches.
Different hypotheses were assumed as a base of the analysis of wheel rol-

ling. Some of them have minor importance and the postulated phenomena
do not result in the final effect significantly, others are still being intensively



276 R. Konowrocki, C. Bajer

investigated. In the literature, the following cases are pointed out as a source
of generation of vibrations and corrugations: imperfections in rail joints and
wheel geometry (Dżuła, 1995), cone form of wheels which results in different
linear speeds of the left and rightwheel and snaking of trains, periodical struc-
ture of railswith sleepers and the instability ofmotion over periodically placed
supports (Bajer and Bogacz, 2005; Bogacz et al., 1995), contact problems be-
tween wheels and rails, stick and slip zones varyingwith a high frequency and
generating waves which deform elastically or plastically both contact surfa-
ces (Bogacz et al., 1987; Brzozowski et al., 1990), residual stresses caused by
manufacturing andmaintenance of rails andwheels (Bogacz, 1995), non-linear
friction in the stick zone (Bogacz andRyczek, 1997), influence ofmaterial har-
dening (Bajer, 1997), deformation of wheel/axle system as a result of impacts
during rollingmotion, instability of wheelsetmotion (Meinke and Szolc, 1996;
Bogacz andDżuła, 1993), strong hits of a perfectly roundwheel rolling on an
imperfect rail (Kowalska, 2008), wave phenomena in circumferential direction
(Bajer, 1998), etc.

The important contribution to the problemwas published by Bogacz and
Dżuła (1993), Dżuła (1989, 1995). The physical continuous model was tre-
ated analytically. The wheel tire was modeled as an elastic curved Rayleigh’s
beam joined with the axle by means of the continuous elastic Winkler type
foundation. The elastic foundation constituting the wheel disk carries out the
load in three directions: circumferential, radial and perpendicular to the plane
of the wheel. The curved beam theory allows one to assume the real shape
of the cross-section. Frequency response functions for forced vibrations of the
railway wheel prove a significant amplitude increase for the frequency 100Hz
at the velocity 200km/h. There is no doubt that the wave propagation ana-
lysis is essential. In Bajer (1998) it was proved that steady rolling of the disc
results in oscillations of contact forces. Surface waves on the circumference
periodically reflect from the contact zone. The number of oscillations that suit
the circumference decreases with the increase of the travelling speedwhile the
amplitudes of contact forces increase in the same case.

In this paper, we consider a dynamic phenomenon, which occurs in rolling
of thewheel over a rail with lateral slip. Such a case is involved bywind blows
(both on curved and straight tracks), lateral deformation of wheels, wheelsets
and rails, different linear velocity of wheels on curves and rotary oscillations
of wheelsets. The skewwheel plane related to the direction of rolling results in
lateral slip in the rail-wheel contact zone (Fig.1). The rail-wheel system oscil-
lates and generates noise. In the same time, wear considerably increases and
negatively influences the environment and decreases the safety of transporta-



Friction rolling with lateral slip in rail vehicles 277

tion. The noise frequency is lower than in the case of corrugations generated
during straight rolling, and is between 100 and 200Hz. This low frequency
noise is more burdensome for the environment and less damped than a higher
frequency with an evenmore intensive level. Corrugations are a visible exam-
ple of this oscillatory wear. However, the similar phenomenon can be observed
also in rolling of a tire over a road and in the contact of rolls with a guide. In
all cases, friction is themain reason of the describedphenomenon.The friction
law influences it qualitatively and quantitatively.

Fig. 1. Skew rolling

Kowalska in (2004, 2008) andBogacz andKowalska (2001) investigated the
rolling over waved surfaces. Bothwheel and rail irregularities were considered.
The important conclusion states that short periodic irregularities of small
amplitudes (0.01-0.10mm) result in very high instantaneous normal contact
forces.At a speed range of a typical train operation for low-pitch corrugations,
the wheelset-rail system vibrates in the post resonance range. Amplitudes of
thewheelsetmotion are low, while normal contact forces at corrugation peaks
are significantly higher than the static load. During rolling along corrugated
surfaces the high level noise is therefore generated.

Considering up-to-date publications, we can say that there are twomecha-
nisms of wheel and rail wear. In the first one, the surface is temporarily or
permanently deformed by the wave phenomenon or centrifugal forces in the
domain of anisotropy involved by residual or permanent stresses. High speed
rolling results in stress jumps, which in turn are sources of successive stages
of periodic process. Short wave corrugations (2-8cm) occur.

The second scenario concerns long corrugations (20-40cm length). They
appear on curves, when bothwheels run over rails with different curvature. In
sucha case, one orbothwheels roll with lateral slip.This casewill bediscussed



278 R. Konowrocki, C. Bajer

in thepaper.We takehere intoaccount theelasticity of thewheelset, especially
rotary flexibility of the axle and the lateral bending of the wheel disc.

Skew rolling with lateral stick and slip results in another phenomenon
important in practice. Vertical stresses in contact are augmented by a signifi-
cant value of residual and permanent stresses in the circumferential direction.
Lateral motion in the direction of the third coordinate contributes to tangen-
tial stresses. Then the plasticity limit can be locally achieved. The oscillatory
lateral motion can result in consolidation of a waved pattern.

Fig. 2. Time history of vertical accelerationsmeasured on the base of a railroad
between rails on the curved section (left) and straight section (right)

Figure 2 showsvibrationsmeasured in anundergroundrailway tunnel.The
comparison of experimental results exhibits higher amplitudes of accelerations
generated on curves of the track than on its straight segments. The measu-
rements on curves exhibited twenty four characteristic predominant groups
of high amplitude vibrations. These groups correspond with the passage of
successive wheelsets of bogies by the point of measurement. The effect of pre-
dominant groups of vibrations does not occur in the results registered on the
straight section of the track (Fig.2). This important featuremust be emphasi-
sed here. Although vertical accelerations are three times higher than horizon-
tal, vertical displacements are natural in the case of vertical load application.
Horizontal forces are not so intuitive in rolling over a straight track. Howe-
ver, they propagate with significant intensity towards buildings. In Fig.3, we
show vertical displacements of the track basement. The displacements were
obtained by double integration of accelerations and additional smoothing by
the FFT filtering. The FFT filter smoothing stage removed components with
frequencies higher than the cutoff frequency. The measurements showed that
vertical displacements were about two and five times higher than lateral ones
on the curved and straight sections, respectively.



Friction rolling with lateral slip in rail vehicles 279

Fig. 3. Time history of vertical displacements on the curved section of the base of a
railroad between rails: low frequencymotion (left) and high frequencymotion (right)

Fig. 4. Spectrum of lateral displacements on the base of the curved (left) and
straight (right) section of a track

In Fig.4, spectral analysis of experimental results of lateral displacements
is depicted. Frequencies of the range 1-3Hz on the curved segment can be
noticed, whereas on the straight track the respective frequency range is equal
to 1-2Hz. The predominant frequency on the curved and straight segments of
the track is equal to 1Hz and 2Hz. The spectral analysis showed us a double
periodicity of the base motion in both cases with low contribution of the
third frequency. The measurements demonstrated that lateral displacements
are about 3.5 times higher on the curve than on the straight track. The double
peridicity is demonstrated also in the phase plots in Fig.5. Vibrations of the
lower frequency 1-4Hz coexist with higher frequency oscillations (15-30Hz).
The first case occurs as a result of snaking influenced by the car bodymotion.
The second higher range frequency is caused by the low amplitude interaction
of the wheel/rail system.

The first attempt to the investigation of rolling with lateral slip wasmade
experimentally. The results of tests were compared with real vibrations of the
track and the theoretical model response. Two different friction pairs were



280 R. Konowrocki, C. Bajer

Fig. 5. Phase diagram for lateral low frequencymotion 1-4Hz of the base of the
railroad on the curved section of the track (left) and the higher frequency

component 15-30Hz (right)

considered: polyester/steel and polyester/polyamide. Three different polyester
belts characteristic of different surface patterns were tested, however the re-
sults did not differ qualitatively. The following parameters were examined: the
speed of rolling, the angle between the wheel plane and the rolling direction
(Fig.1) and the contact force. The results strongly depended also on other pa-
rameters. In our case itwas the contact time.For this reason, the experimental
measurements were long lasting in order to stabilise results.

2. Experimental stand

Oscillatory lateral motion was examined in details on the experimental stand
(Fig.6). The driven belt carrier was the main part of the stand. Various belt
materials were tested. Polymer and rubber surfaces were characterised by dif-
ferent elasticity, viscosity and friction. The tested wheels were made of steel,
aluminium and polyamide. The wheel was suspended elastically on the axle
by two springs that enabled its horizontal lateral motion. The springs had the
longitudinal k and torsional kϕ stiffness equal to 90N/m and 0.036Nm/rad,
respectively. They were ended by slide bearings in form of rings immersed
in the grease, giving a low and constant value of the torque. The wheel was
placed on the elastic belt with the adjusted contact force N.

In the axle direction, we consider our physical model as a system of two
viscoelastic spring elements that hold the wheel assumed as a lumped mass.
Damping of the springmaterial is relatively low and can be neglected.Motion
of the wheel along its axle is excited by friction. In the direction of rotation,
the mass M of the inertia Io is supported by the same set of springs with



Friction rolling with lateral slip in rail vehicles 281

Fig. 6. Experimental stand

the stiffness kϕ. A constant low value torque (0.02-0.04Nm) is applied to
the wheel and is balanced by the spring. The wheel is subjected vertically
to an external constant force N and its weight together with the suspension
frame.

The friction force is equal to the force in the spring increased by a com-
ponent perpendicular to the axle, induced by the slide bearings. It increases
until the maximum value Fmaxts is achieved (segment 0-C in Fig.7). We have

Fig. 7. Stages of the oscillation process

the stick state then. In the meantime, the stick limit is achieved (point A) by
contribution of the torque. The lower kinetic friction allows motion towards
the equilibrium state. Low rotary spring forces are partially released. A new
stick point is established and the sameprocess is repeated on the segmentB-C.
In the extremal position, the slip state starts since the friction force is lower
than the elastic spring force (C-E). In the beginning of this stage (C-D), again
the rotational potential energy is released. At the bottom of the diagram (po-



282 R. Konowrocki, C. Bajer

int E) the cycle repeats. Intermediate stages (B-C) can be placed on both the
ascending and descending branch of the diagram, depending on parameters of
the experiment.

Displacements in time in the wheel axle direction were registered by using
laser distance transformers. They permit high resolution measurements (over
0.01mm)andhigh frequency of recording (over 1000 registrations per second).

In the paper. two materials used for the wheel will be presented: steel
and polyamide. The aluminium wheel was tested, however it exhibits similar
properties as polyamide andwill not be discussed in the paper. The diameter
of the wheel was 75mm and the width of the contact path – 9mm.

The friction law for each tested frictionpairwasexamined.Thedependency
of the friction coefficient on the relative speed was determined. The friction
varied with time of sliding. We can say that the local temperature of the
surface in the case of polyester increased friction in the whole speed range.
Here, we do not intend to present the friction law in the test stand since both
the experiment and further numerical simulations proved independency of the
wheel motion of the friction function.

3. Experimental results

The following parameters have been investigated: angle α, rolling speed v,
contact pressure σ, type of friction pair. We must emphasise that all experi-
mental results have one common feature: they exhibit double periodicity. This
feature in the case of polyamide/polyester friction pair is depicted in Fig.8.

Fig. 8. Spectral analysis of the displacement function in the case of
polyamide/polyester friction pair (domination of the frequencies 0.8 and 1.6Hz)



Friction rolling with lateral slip in rail vehicles 283

In practice, all experimental signals for such a friction pair exhibit the
same relation. Further, we shall present oscillations of thewheel in the case of
two friction pairs.

In the paper, only selected andmost characteristic diagrams are discussed.
Other relations between the parameters will be given in conclusions.

3.1. Polyamide/polyester friction pair

Experimental results for the polyamide/polyester frictionpair for the angle
α = 5◦ are depicted in Fig.9. The most characteristic sets of diagrams were
selected.Phaseplotsbecomemoresmoothwithan increaseof the rolling speed.
The formof diagrams strongly depends on the vertical contact force (Fig.10).

Fig. 9. Phase diagram for experimental results with the belt speed v=5, 6, 7, 8, 9
and 10cm/s (polyamide/polyester friction pair)

Bychangingparameters of the experiment (i.e. belt speed, angle α andmainly
the contact pressure), we can obtain a phase diagramwith a higher frequency
loop localised in any position around the low frequency loop. In the picture
low, middle and high pressure effects are depicted (3.0, 7.5 and 9.5N of the
vertical force, respectively). Other values of the pressure result in intermediate
shapes.

Another interesting observation was made. The increase of the belt speed
increases the amplitude of the first mode vibration and decreases the second
mode amplitude. The level of the pressure force between the wheel and the



284 R. Konowrocki, C. Bajer

Fig. 10. Experimental phase trajectories, displacements and velocity in time for
different pressure forces: (a) low, (b) middle, (c) high (for α=5◦ and speed

v=9cm/s

belt also influences displacement diagrams. With the increase of the vertical
force, we notice higher amplitudes of displacements related to the first, lower
frequencymode.The secondmodeamplitude remains unchanged.The contact
pressure alteration results in another feature. The phase shift between two
modes strongly depends on it (Fig.10). This effect is visible for small angles.
When α> 10◦, the phase shift is less influenced by the pressure value.

Amplitudes depend on the belt speed but they are independent of the
angle α. In Fig.11, amplitudes vs. belt speed v are plotted for various α. All
lines are almost straight in the range of investigation and coincide.

The next test with the same friction pair was performed. The polyamide
wheelwas loaded by two steel rings attached to its both sides. Themass of the
wheel increased then by four. The final results were not affected qualitatively
by the increased inertia.



Friction rolling with lateral slip in rail vehicles 285

Fig. 11. Amplitudes vs. belt speed for 2◦ ¬α¬ 10◦ (polyamide/polyester pair)

3.2. Steel/polyester friction pair

The steel wheel wasmanufactured as a ring (steel foot) filled insidewith a
polyamide thin disk. Thus the weight of the steel wheel was reduced and was
comparable with the polyamide one. In this case, phase diagrams are similar
to those obtainedwith thepolyamide/polyester friction pair (Fig.12).One can
notice that the basic period is influenced by a higher order oscillation. The
second and third frequency is noticed aside themain one.Mutual contribution
of higher frequencies slightly varies during the experiment and in such a case
the diagrams in time are less uniform than in the case of the polyamidewheel.

Fig. 12. Steel-polyester friction pair: phase plots, displacements and velocities for
α=10◦ (upper) and for α=15◦ (lower) with low contact pressure



286 R. Konowrocki, C. Bajer

4. Theoretical model

Experimental results exhibit double periodicity of vibrations.We propose two
a degree-of-freedom system as a mathematical model of the problem. It can
be described by the following equation

[

m1 0
0 m2

]{

ÿ1
ÿ2

}

+

[

f11(y, ẏ) f12(y, ẏ)
f21(y, ẏ) f22(y, ẏ)

]{

ẏ1
ẏ2

}

+

[

k1 0
0 k2

]{

y1
y2

}

=0

(4.1)
Two degrees of freedom correspond to the lateral and rotary motion of the
wheel, respectively. Both inertia and stiffnessmatrices are constant. The dam-
ping coefficients depend on velocities and displacements of both degrees of
freedom. We have a parametric excitation in a general case. Below we shall
establish the functions fij to fit the experimental results.

The friction influence is the fundamental question in our problem. It affects
the higher frequency vibration. We assume it as a nonlinear one depending
on the relative velocity of the belt and wheel and the position of the wheel.
We neglect the dependency of friction on the adhesion time and the force
rate (Bogacz and Ryczek, 1997). The periodic motion depends on the belt
velocity v, angle α and,mainly, physical (frictional) properties of the friction
pair (wheel-belt).We simplify our theoretical model. Thewheel is assumed to
be a rigid body, the springs exhibit linear elasticity (axial and rotational) and
negligible internal viscous damping, and the belt is indeformable in the plane.
Its moving velocity is constant.

The principal mode of vibrations is related to lateral motion of the wheel
and the highermode – to the rotarymotion. In our discretemodel (4.1), these
both degrees of freedom are denoted by y1 and y2.

4.1. Polyamide/polyester friction pair

Below we present our experimental results registered for the polyami-
de/polyester friction pair in comparison with theoretical functions. We must
emphasise that diagrams obtained experimentally were repeatable with good
accuracy. Double periodicity was observed in all cases. The following equation
generally describes the wheel motion

x=sint+asin(2t+ b) v=cost+2acos(2t+ b) (4.2)



Friction rolling with lateral slip in rail vehicles 287

Fig. 13. Theoretical phase plots for parameters a and b collected in Table 1

Table 1.Parameters a and b for different belt speed

v [cm/s] 5 6 7 8 9 10

a 0.80 0.70 0.65 0.60 0.55 0.52

b 1.90 1.90 1.90 1.90 1.90 1.90

Fig. 14. Theoretical and experimental phase trajectories (u – speed,
x – displacement, angle α=5◦ and belt speed v=5-10cm/s)



288 R. Konowrocki, C. Bajer

Phase diagrams allowed us to select characteristic features of motion for
various rolling parameters (Fig.14). For various belt speeds from the range
v =5-10cm/s, the parameters a and b have values presented in Table 1 and
plotted in Fig.15.

Fig. 15. Dependency of the parameters a and b on the belt speed

f12 is a linear combination of the velocity and displacement of the second
degree of freedom.Unilateral coupling of both equationsmust be pointed out.
In such a case, the second degree of freedom serves as a source of a harmonic
signal for the first degree of freedom. In practice both ways of coupling exist,
however, with minor influence of the way opposite to the dominating one.

For example, we can obtain phase plots identical as in Fig.10 – the first
one for the initial conditions: q1 = q2 = 0, v1 =−0.1, v2 =1, and excitation
F1 = 0.2v2 +0.15q2, the second one for q1 = q2 = 0, v1 = −0.1, v2 = 1,
F1 =−0.2v2 +0.20q2, and the third one for q1 = q2 = 0, v1 = 0.1, v2 =−1,
F1 =0.2v2+0.15q2.

4.2. Steel/polyester friction pair

Thediscrete system is composedof twodegrees of freedom,as in theprevio-
us case.Aproper selection of functions fij allowsus toobtainnumerical results
coinciding the experimental ones. One can select several pairs of parameters
a, b and initial conditions that give simulation results coincidingwith the expe-
riment. We propose the following initial conditions: q1 = q2 = 0, q̇1 = −0.2,
q̇2 =1.0.The coincidencewith experiments is achieved for f11 = f21 = f22 =0
and f12 as a function of the velocity and the displacement

f12 = c1(|v2|+c2)
2+ c3q2 (4.3)

As an example we consider m11 = 16, m22 = 1, k11 = k22 = 1 and c1 = 24,
c2 = −0.5, c3 = −18 in function (4.3). The velocity v2 contributes to the



Friction rolling with lateral slip in rail vehicles 289

friction term. The displacement q2 shifts the second branch on the diagram
(Fig.16d). f12 introduces a parametric excitation (comparewithFig.12). The

Fig. 16. Displacement in time, velocity in time, phase plane and friction coefficient
(m1/m2 =1 and parameters p1 =0, p2 =0, p3 =24, v1 =3.00 and v2 =5.6)

analytical function which describes lateral motion of the steel wheel can be
written as

y=−sint+asin(2t)+ bsin
t

4
(4.4)

ẏ=−cost+2acos(2t)+
b

4
cos
t

4

We can notice that the multiplier of the second frequency is four instead of
two as in the case of the polyamidewheel.We can explain this as the influence
of the U-shaped friction function for the steel/polyester pair.

Let us compare the experimental results (Fig.12) with analytical plots of
functions (4.4) (Fig.17). Both results in Fig.17 correspond to the low contact
pressure case at angle 10◦ and 15◦. They are plotted for a=−0.35, b=1.4
and a=−0.15, b=1.4 (Eqs. (4.4)), respectively.One cannotice that the third
term in (4.4) with the scaling coefficient b produces long period oscillations
(four times longer comparingwith the base period, see for exampleFig.16b,c).



290 R. Konowrocki, C. Bajer

Characteristic oscillations for the steel wheel are generated by the first two
terms and are qualitatively identical as in the case of the polyamide wheel.

Fig. 17. Displacement in time, velocity in time, phase plane and friction coefficient
(m1/m2 =4 and parameters p1 =0, p2 =0, p3 =24, v1 =0.70 and v2 =3.6)

5. Conclusions

Experimental investigations of the problemof skew rolling allowed us to prove
double periodicity of lateral motion of the wheel. In the case of strongly non-
linear friction law (steel), the third frequencyofminor contribution isobserved.
This frequency is double according to the fundamental period.

First of all, the frequencies of oscillations donotdependon theangle αwhi-
le the amplitudes increase with the increase of this angle. In the case of small
angles α, the phase shift of twomodes strongly depends on the contact pres-
sure. For higher values of N this dependency is not so direct. Stick-slip type of
vibrations can benoticed for small N. For higher values, the stick phasedisap-
pears. Further research will enable us to define the mechanical system which
produces displacement/velocity time characteristics identical with the expe-
rimental ones. A single degree of freedom system with a strongly non-linear



Friction rolling with lateral slip in rail vehicles 291

friction law applied to the vibratingmass does not allow obtaining satisfactory
results, since it requires a harmonic excitation. In our experiment, the rotary
degree of freedom supplies this mode. That is why the two degree-of-freedom
system seems to be appropriate to result in double periodic vibrations. This
also concerns the rolling of the steel wheel which exhibits an increase of the
friction coefficient with the velocity. Theoretical results coincide with the re-
sponse of our physicalmodel with the first degree of freedom related to lateral
motion of the wheel and the second one corresponding to wheel rotations. A
soft coupling is promising in investigation of the self-excited response.

Intuitively, we expect faster lateral motion and spring compression in the
case of higher angles. Thus, one oscillation could be performed in a shorter
time. The experiment proves the independency of the main frequency of the
angle. In turn, the angle influences the amplitude level.

The experimental investigation will be continued for other friction pairs.
The dependence of the friction coefficient on the rolling speed, the angle and
contact pressure in a full formwill be established. The experiment preformed
in a real scale should be the final stage of the research.

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Tarcie toczne z poślizgiem bocznym w pojazdach szynowych

Streszczenie

Wpracyomówionozjawiskadynamiczne związanez toczniemsiękołazpoślizgiem
bocznym po drodze (szynie, torze).Mają onemiejsce przy toczeniu się kół lub zesta-
wów kołowych po torze prostym, w przypadku bocznego obciążenia oraz na łukach.
Różne promienie krzywizn oraz drgania skrętne zestawówkołowychpowodują ukośne



Friction rolling with lateral slip in rail vehicles 293

toczenie i boczne poślizgiw strefie kontaktukoła z szyną. Zjawisko to znacząco zwięk-
sza hałas oraz zużycie elementów konstrukcji.W kolei podziemnej wykryto dwuokre-
sowość drgań. Zjawisko to dotąd nie było opisywane w literaturze. Przeprowadzono
stanowiskowebadania eksperymantalne zuwzględnieniemnastępującychparametrów:
kąta ukośnego toczenia, prędkości oraz siły naciskuw strefie kontaktu.Wyniki odnie-
siono do ruchu teoretycznego układu o dwóch stopniach swobody. W przypadku par
ciernych stal-poliester i poliamid-poliester uzyskano zbliżone jakościowo rezultaty.

Manuscript received July 11, 2008; accepted for print November 12, 2008