Jtam.dvi


JOURNAL OF THEORETICAL

AND APPLIED MECHANICS

50, 1, pp. 119-129, Warsaw 2012

50th anniversary of JTAM

ON SOME NEW ASPECTS OF CONTACT DYNAMICS WITH

APPLICATION IN RAILWAY ENGINEERING1

Roman Bogacz

Cracow University of Technology, Faculty of Civil Engineering, Kraków, Poland

e-mail: rbogacz@ippt.gov.pl

Kurt Frischmuth

University of Rostock, Institute of Mathematics, Rostock, Germany

e-mail: kurt.frischmuth@uni-rostock.de

The paper is devoted to the study of several dynamical phenomena of
contact problems, where the wheel/rail motion is connected with an
oscillating load caused by corrugation, poligonalisation and other irre-
gularities of rails or wheels. It will be shown that even if the wheel
center moves with constant speed, the load applied by contact forces
moves with variable speed, thus crossing critical resonance boundaries.
Some problems studied by the authors in previous works are revisited.

Key words: dynamics, traveling load, rail-wheel contact, corrugation

1. Introduction

The development of themodern technology of tracked high-speed transporta-
tion systems becomes more and more important. This entrails a strong need
for simplifiedbut reliablemodels of contact andmodels of continuous orhybrid
systems in order to study various dynamical effects of the wheel/rail interac-
tion. There is still no general model valid in the whole range of velocities.
Themodel we are proposing here covers the gap in themedium and high spe-
ed range at the presence of corrugations. The contact loading influences the
durability of infrastructure, the damage to the environment and the comfort
of transportation. The beginning of research in this direction was connected

1
The paper is dedicated to Prof. Dr. Dr. h.c. mult. Oskar Mahrenholtz on the occasion

of his 80th birthday.



120 R. Bogacz, K. Frischmuth

with the study of moving loads acting on beams. Those were supported by
aWinkler foundation (Winkler, 1867) and subjected to a single concentrated
force moving at constant speed. The problem was mentioned by Timoshenko
(1926). The first stationary solution in the case of a Bernoulli-Euler beam on
an elastic foundationwas obtained in a properway byLudwig (1938), but on-
ly for the sub-critical speed range. The case of a moving and oscillating force
was formulated, and partly solved, byMathews (1958). The first fully correct
solution of Mathews’ problem was given by Bogacz and Krzyżyński (1986).
There are various extensions of the classic problem towardsmore complicated
but alsomore realisticmodels of contact loads. A great deal of new effects we-
re discovered by Bogacz et al. (1998) and Bogacz and Kowalska (2001), who
studied problems of oscillating loads, moving along some periodic, i.e. varia-
ble in space, structures. The dynamical effects for two-dimensional problems
with moving loads have important practical engineering applications. Some
problems connected with a system of plates subjected to traveling loads can
be found in a recent paper by Bogacz and Frischmuth (2008). The aim of the
present paper is the systematization of previous results as well as the generali-
zation towards a discussion of some new effects connected with amoving load
along a corrugated rail or rolling surface of thewheel, which lead to oscillation
of the speed of the travelling load.

2. Contact problems

Some investigations were devoted to the analysis of the development of corru-
gations using linear and nonlinear wheel/rail interaction models. A rigid and
an elastic contactmodelwere discussed byFrischmuth (2001), elastic andper-
fect plastic models were studied by Frischmuth and Langemann (2002), and
a discussion of frictional wear as the source of surface pattern development in
rolling contact is given by Bogacz et al. (2005). Similarly as in the approach
made by Kowalska (2008), we will study the two-dimensional problem with
jumps of wheel/rail contact and inelastic impacts.

Typical corrugations, also called slip-waves (Fig.1), have a much smaller
wavelength, and are believed to be related to percussional effects leading to
permanent (plastic) deformations of the surface layer. Analysis of the contact
geometry of rigid bodies with harmonic sinusoidal corrugations shows that
already at comparatively low speeds two effects may be observed. First, the
point of geometrical contact moves forth and back around the actual position
of the wheel. Thus we have to deal with a force traveling at variable speed.



On some new aspects of contact dynamics... 121

Due to large amplitudes of the speed, critical regions of speeds are crossed
(see Bogacz andKrzyżyński, 1986; Bogacz et al., 1998). Second, also the value
of the normal force oscillates considerably around its mean value, and at a
certain velocity the contact is frequently lost and reestablished. At the same
time, we observe the onset of a process of rapid plastic deformation due to the
increase of contact forces (Bogacz and Kowalska, 2008).

Fig. 1. View of a track segment with the corrugated rail (slip waves)

As an academic example, one can assume a sinusoidal shape of the rail
surface and a constant wheel radius. However, the observed wavelengths on
worn rails are quite different.
Ingeneral,weassumesufficient regularity of thegeometry, to ensure the so-

lvability of dynamic equations ofmotion.As it turns out, however, smoothness
is not sufficient. We need also bounds on the departure from the ideal forms,
straight respectively circular. We assume the domains occupied by the wheel
and rail to intersect in a nonempty set of zero area. The wheel and the rail
touch, but do not yet penetrate. In the classic case, the intersection consists of
just onepoint,which is called thepoint of contact. In this case, there exists also
a unique angle φ, such that the value parametrizes the point of contact on the
wheel circumference. It has tobeadded thatduringmotion the contactmaybe
lost, inwhich case neither position xg nor φg maybedefined (Fig.3).Another
non-classic case occurs for heavily corrugated surfaces, e.g. when the wheel
convexity is lost. Then several different or even a continuum of points may
be in contact. The same is true for models with compliant contact partners.
For a rigid contact model, we introduce a unilateral constraint. In this case,
for relatively great amplitudes of corrugation, a two point contact is possible
(Fig.2).



122 R. Bogacz, K. Frischmuth

Fig. 2. The case of corrugation curvature greater than the wheel curvature

Fig. 3. Profile of the corrugated rail and trajectory of wheel center (left) and the
corresponding contact force (right) for the case of amplitude of corrugation 10µm,
and wave length λ=50mm at speed 50km/h (Bogacz and Kowalska, 2001)

The case when during rollingmotion of awheel the contact is lost is possi-
ble for a corrugation amplitude equal to 10µmand awave length λ=50mm
at the speed V =50km/h. Such a situation is visible on the right-hand side
of Fig.3, where the contact force is equal to zero. It ought to be noticed that
the amplitude 10µm is relatively small, which is why the contact is lost only
in the resonance case. For higher speeds the wheel is rolling without losing
the contact. As for the speed at which the lift-off occurs, andwhere hence the
contact is reestablished by an impact, there are several ways of determination.
The normal force which keeps the wheel at the actual height of the rail enters
the equations ofmotion in the rigidmodel via a Lagrangemultiplier. As such,
it is not determined by thewheel position coordinates x, y. As an alternative,
we may allow the domains of the non-deformed – but rigidly moved – bodies
to penetrate. We stress that, of course, the real bodies to not penetrate –
they avoid this by deforming,mainly in the contact area. The correct solution
to the elastic contact problem, i.e. the form of deformed bodies, is hard to
calculate. The integral equation formulation of this problem turns out to be



On some new aspects of contact dynamics... 123

illposed.For quick simulations,weuseaunilateral nonlinear (Hertzian) spring.
This approach gives a normal contact force depending on the elastic approach
of the bodies. We have to mention that this characteristic depends on the
material, but also on the curvatures of surfaces. Hence, it may vary during
motion. Furthermore, the direction of the contact force is perpendicular to
the surface at the contact point, which is not exactly in the vertical direction.
Themodifications necessary to take this into account are straightforward, we
leave these technicalities out for the sake of readability of the paper. More
crucial is another enhancement of the simple Hertzian contact spring, which
is a damping component.We assume the same form of the dependence, just a
different coefficient connecting the damping force to the speed of penetration.
What remains is the relative velocity in the contact point, which in turn re-
sults in a frictional tangential force. By Coulomb’s law, this force is directed
opposite to the velocity of the tangential motion. The details can be found
by Frischmuth (2002). The computer simulation of slip-wave generation was
discussed by Bogacz and Frischmuth (2001), see also Fig.1.

In the majority of studies devoted to dynamically applied loads in civil
and railway engineering, a simplification was used that a dynamically applied
load can be approximated by addition of 15-30% of the static load value. This
amounts to a big mistake even in the case that only the vertical dynamics is
taken into account. The results shown on the right side of Fig.3 confirm this
statement. Another simplification is connected with the assumption that the
speed of contact loading is constant. This aspect will be discussed in the next
part of the paper.

3. Real velocity of moving load

We start with a simple kinematic consideration assuming that the center of
thewheelmoves at a constant prescribed speed.Further, we assume that both
bodies are rigid, and they are in contact all the time in a uniquely defined
point of geometrical contact. Finally, at thepoint of contact there is no relative
motion between the wheel and rail, which amounts to the condition that the
velocity at the contact point on the wheel circumference is zero. The above
conditions define the trajectory of the wheel center and the revolution of the
wheel uniquely.Hence,weobtain thevertical acceleration andangularmoment
and thus thenormal force aswell as the force in the tangential direction.Under
such idealized conditions of rolling contact, wemay calculate the position and
vertical force in the contact pointwithout solving the equations ofmotion.We



124 R. Bogacz, K. Frischmuth

derive from this a new type of the traveling force problem approximated by
the formula

Fy(t;x) = (F0+F1 sin(ω1t))δ[x−V0t+εsin(ω2t)] (3.1)

where F0 is the value of constant force contribution, F1 is the amplitude of the
oscillating force with frequency ω1, V0 is the average speed of motion which
oscillates with amplitude ε and with frequency ω2 = k0V dependent on the
wavelength of the corrugation.

It is new here that the force position oscillates around its mean position
which moves steadily forward. The critical speeds and frequencies have been
derived for the case ε = 0. The oscillating position may exceed the critical
boundaries of speed-frequency regions, cf. Bogacz andKrzyżyński (1986), but
only on small intervals. The effects of this need further considerations.

Let us assume now an ideal wheel and a rail with a wavy surface. We
will try to create a parametrization of the wheel center locus in terms of
the horizontal coordinate of the contact point as the parameter. We have
to perform vector addition of the contact point C(xc;yc) on the rail with
the horizontal coordinate s and the normal vector multiplied by the wheel
radius R0. The resulting curve may be non-smooth or even contain loops.
True rolling – with continuous one-point contact – is only possible as long as
the curve is free of double-points. InBogacz andFrischmuth (2009) we derived
an inequality connecting wavelengths of corrugations and amplitudes on both
contact partners, which ensure the applicability of the procedure.

For a realistic wheel radius andwave amplitude the effect is hardly visible
in Fig.4. Thus we show the same plot for a much smaller wheel of ten times
reduced radius.

Fig. 4. Locus of the wheel center for an ideal wheel [m] versus rail length [m]



On some new aspects of contact dynamics... 125

Fig. 5. Locus of the wheel center for a small ideal disk wheel

Figures 4, 5 and 6 show the consequences of the geometric contact condi-
tion. Ideal rolling at a constant horizontal speed leads to high variation in the
rotational speed, constant revolutions require heavy variation of the horizontal
speed. The vertical position of thewheel center versus time is shown in Fig.6.

Fig. 6. Vertical position of the wheel center [m] versus time [s]

The horizontal position of the contact point versus the position of the
wheel center is shown in Fig.7.

For the case of a constant horizontal speed of moderate value, i.e. 36m/s,
extreme vertical or horizontal accelerations and very high velocities of contact
point motion can be observed. It is interesting and important, particularly
from the point of view of wave generation, that the horizontal speed of the
moving contact point is so high, cf. Fig.8. In Fig.9, we can see the resonance
behavior for the very simple case – Bernoulli-Euler beammodel on an elastic
foundation subjected toaharmonically oscillating loadmovingwitha constant
speed.



126 R. Bogacz, K. Frischmuth

Fig. 7. Horizontal position of the contact point [m] versus the wheel center (steady
forwardmotion)

Fig. 8. Horizontal speed of the contact point [m/s] versus time [s]

Fig. 9. Response of the Bernoulli-Euler beam subjected to a force moving with the
speed V and oscillating with the frequency Ω



On some new aspects of contact dynamics... 127

In the case of load described by expression (3.1), some resonance bounda-
ries will be crossed. The response of a beam subjected to such a periodically
moving load is not easy to determine, even numerically. The difficulties are
connected with non-stationary motion due to varying speed. Some approxi-
mate solutions of this problemwill be given in a forthcoming paper.

The investigated case was devoted to an ideal wheel interacting with the
corrugated rail. But the case of a corrugated wheel or a wheel with poligo-
nalisation is also studied, and the results are also very interesting. The study
gives an explanation of some phenomena existing in real engineering practice.
Part of the investigations can be found in other papers, e.g. in Bogacz and
Frischmuth (2009).

4. Summary

In the paper, an overview of the questions connected with the problem of
contact loading was presented. Some new results of the investigations were
obtained for the case of wheel and rail corrugations.

The critical speed – from the point of view of the theory of traveling
loads on an elastically supported beam –may be exceeded for very moderate
speeds of the wheelset motion on a corrugated track. Above certain levels of
expression of the corrugation patterns the lift-offmay occur, which is followed
byan impact. Ingeneral, there is also aconsiderable influenceof the suspension
and control motion on the resulting trajectories. This problemwill be studied
in next papers.

References

1. Bogacz R., Chudzikiewicz A., Frischmuth K., 2005, Friction, wear and
heat, [In:] Simulation inR&D.,L.BobrowskiandA.Tylikowski (Eds.),Warsaw,
67-76

2. Bogacz R., Frischmuth K., 2001, Computer simulation of slip-wave gene-
ration, [In:] Simulation in R&D, J. Rybicki andA. Tylikowski (Eds.), Gdansk,
42-47

3. Bogacz R., Frischmuth K., 2009, Vibration in sets of beams and plates
induced by traveling loads,Archive of Applied Mechanics, 79, 509-516



128 R. Bogacz, K. Frischmuth

4. Bogacz R., Frischmuth K., 2009, Analysis of contact forces between corru-
gated wheels and rails,Machine Dynamics Problems, 33, 2, 19-28

5. Bogacz R., Frischmuth K., 2010, On some corrugation related dynamical
problems of wheel/rail interaction,The Archives of Transport,XXII, 1, 27-41

6. Bogacz R., Kowalska Z., 2001, Computer simulation of the interaction be-
tween a wheel and a corrugated rail,Eur. J. Mech. A/Solids, 20, 673-684

7. Bogacz R., Krzyżyński T., 1986,On the Bernoulli-Euler Beam on a Visco-
Elastic Foundation under Moving Oscillating Load, Inst. Fund. Technol. Rese-
arch, Report No. 38

8. Bogacz R., Krzyżyński T., Popp K., 1998, Wave propagation in two dy-
namically coupled periodic systems, [In:]Dynamics of Continua: International
Symposium, D. Besdo, R. Bogacz (Eds.), Bad Honnef, Aachen: Shaker Verlag
ISBN3-8265-3834X, 55-64

9. Frischmuth K., 2001, Contact, motion andwear in railwaymechanics, Jour-
nal of Theoretical and Applied Mechanics, 39, 3, 507-522

10. Frischmuth K., 2002, Numerical methods for evolving manifolds, and appli-
cations in rail-wheel contact mechanics, [In:] Selected Topics in Geometry and
Mathematical Physics, 1, E. Barletta (Ed.), 161-187

11. FrischmuthK., LangemannD., 2002,Distributed numerical calculations of
wear in thewheel-rail contact, [In:]SystemDynamics and Long-TermBehavior
of RailwayVehicles, Track and Sub-grade, PoppandSchiehlen (Eds.), Springer,
85-100

12. Kowalska Z., 2008, Vibro-impacts induced by irregular rolling surfaces of
railway rails and wheels, Journal of Theoretical and Applied Mechanics, 46, 1,
205-221

13. Ludwig K., 1938, Deformation of rail elastically supported of infinite length
by loadsmoving at a constanthorizontal velocity,Proc. 5th Int. Congress Appl.
Mech., 650-655

14. Mathews P.M., 1958, Vibration of a beam on elastic foundation, Z. Angew.
Math. und Mech., 38, 105-115

15. Timoshenko S., 1926,Method of analysis of static and dynamical stresses in
rail,Proc. 2nd Int. Congr. Appl. Mech., Zurich, 407-418

16. Winkler E., 1867,Die Lehre von der Elasticitaet undFestigkeitmit besonde-
rer Ruecksicht auf ihre Anwendungen in der Technik, 1. Theil. H. Dominicus,
Prag



On some new aspects of contact dynamics... 129

O pewnych nowych aspektach dynamiki kontaktu z zastosowaniem

w inżynierii kolejowej

Streszczenie

Praca została poświęcona przeglądowemustudium zjawisk dynamicznego kontak-
tu tocznego, w którym ruch układu koło-szyna jest związany z oscylującym obcią-
żeniem spowodowanymkorugacją, poligonalizacją i innymi nierównościami szyny lub
koła kolejowego.Wykazano, że nawet w przypadku toczenia się koła z umiarkowaną
i stałą prędkością obciążenie kontaktowe szyny jest zmienne i przekracza krytyczne
granice rezonansów. Przytoczone zostały niektóre wyniki wcześniejszych badań.

Manuscript received November 10, 2010; accepted for print May 4, 2011