ТЕKA. COMMISSION OF MOTORIZATION AND ENERGETICS IN AGRICULTURE – 2014, Vol. 14, No.2, 53-58
Modelling of rolling friction with sliding
Irina Kirichenko, Alexandr Kashura, Marina Morneva, Sergey Popov
Volodymyr Dahl East-Ukrainian National University
Molodzhny bl., 20a, Lugansk, 91034, Ukraine,
e-mail: [email protected], [email protected], [email protected]
Received May 22.2014: accepted June 09.2014
S u m m a r y . The article proposes a mathematical model of
the process of rolling friction with slip at the point of contact
with the rail wheels of the locomotive in the implementation
of thrust.
K e y w o r d s . The friction force, the force of adhesion,
the contact, the contact area, rolling friction, sliding
friction.
INTRODUCTION
A sliding friction and friction of rolling
with sliding are the most widespread types of
friction in a technique. So, for example a
friction of rolling with sliding is in toothed and
toothed-spiral transmissions or between
wheels and rails [16].
An experimental investigation on the
process of friction interaction between the
wheels of a railway vehicle and rails was
carried out by using many methods for
different locomotives and conditions [11]. The
results obtained allow the possibility to put
forth common laws for the adhesion process as
follows:
• the basis of adhesion is the force of
external friction;
• realization of adhesion force is
impossible without a slip of the wheel relative
to the rail.
RESULTS
The problem of interaction of a rolling
stock and railway way concerns to number of
the major in a transport science [2, 9, 18, 22,
23].
The urgency of the numerous researches
devoted given problem, is caused by what
from the processes occurring in contact of
interaction of a wheel and a rail, safety and
technical and economic parameters of a
traction rolling stock of railways (speed of
movement, the losses connected with
overcoming of resistance to movement,
deterioration of wheels, rails, etc.) depend [3,
27].
Long-term operation of a rolling stock
shows, that the resource of bandages of wheel
pairs is defined by hire and, in a greater
measure, deterioration of crests. The numerous
publications testify to it, devoted to above
permitted standard wear process of crests of
wheel pairs and a lateral surface of a head of a
rail. So, for example in [1] diagrams of
structure of turnings of wheel pairs on
operational park of locomotives on a network
of railways of the Russian Federation 2012
which Analysis are presented shows, that
made turnings of bandages of wheel pairs on
admissible size of hire make 4 % while on
54
IRINA KIRICHENKO, ALEXANDR KASHURA, MARINA MORNEVA, SERGEY POPOV
deterioration of a crest their value reaches
65%.
With the purpose of decrease in intensity
of wear process of wheel pairs and rails up to
comprehensible values last years a number of
actions of technical and organizationaltechnological character [8] (lubrication,
improvement of a design of a way and
locomotive servicing of a rolling stock,
perfection of geometry of a structure of a
surface of driving of wheels and rails,
monitoring in system a wheel – a rail, etc.) is
spent. From all listed directions of works most
quickly introduction in a zone of contact of the
third body with the set characteristics is sold.
The choice of the materials applied to
cooling and greasing of contacting surfaces of
crests of wheel pairs and rails, should be
carried out in view of temperature conditions
of interaction of the considered pair friction.
How the temperature developing at friction,
causes heating thin superficial layers of the
interfaced bodies and a layer of greasing
dividing them, it is one of the most important
factors influencing all complex of service
properties of lubricants, defining their
antifriction properties.
During friction of a lateral surface of a
crest of a wheel about a rail depending on
loading, high-speed and temperature modes
various modes of friction are realized: dry,
boundary and полужидкого friction.
Change of modes of friction including
formation of the mixed greasing, can be
considered on diagram Hersy (fig. 1), showing
dependence of factor of friction on the
characteristic of a mode of greasing [25]. The
dimensionless size is defined under the
equation:
λ=
μ ⋅V
μ ⋅ω
or λ =
,
р
P
(1)
key parameters enter into it defining a
mode of friction (viscosity of oil, speed,
loading).
This curve has two characteristic
branches: left, falling for area of boundary
greasing and right, increasing, for area of
liquid greasing. Between them there is the
transitive site corresponding area semi fluid
greasing (area). At transition in area unstable
semi fluid greasing change of any parameter
promoting decrease λ (reduction of viscosity,
increase in loading, increase in speed of
sliding), leads to increase of factor of friction
and working temperature tribosystem. Growth
of factor of friction in the given area occurs
due to increase of a share of boundary greasing
down to formation cleanly boundary greasing
(area).
Fig. 1. Diagram Hersy
Modeling of thermal processes in pair
friction a crest of a wheel-rail definition of
power
and
geometrical
conditions
контактирования is carried out according to
with described in [4] sequence.
At interaction of a crest of a wheel with a
lateral surface of a head of a rail at presence
between them a lubricant the factor of
boundary friction can be calculated under the
formula resulted in work [8]:
f тг = f тр − k
ηсм ⋅ vск
,
Pпог
(2)
where: f тр - factor of friction of not
greased surfaces, η см -dynamic viscosity of oil,
vск -speed of sliding, Рпог - running loading,
k -factor of proportionality.
MODELLING OF ROLLING FRICTION WITH SLIDING
Modeling modes of friction considered
tribosystem it is necessary to mean, that at dry
friction the second member of the equation (1)
can not be considered. The second part of the
equation (1) generally is function of the
characteristic of a mode of greasing. From the
equation (1) it is visible, that three factors
influencing a mode of greasing are speed,
loading and viscosity of oil.
The factor of friction f тг depends, from
actual (working) viscosity of greasing at
temperature Т on a surface of contact, instead
of from the rating value defined in
вискозиметре at conditional temperature and
atmospheric pressure.
For the calculation of temperature on the
surfaces of wheel and rail, it is necessary to
define a form and size of spot of contact. We
will represent the touch of wheel and rail, as a
contact of cylinder with toroid with
perpendicular axes. Sizes of spot of contact of
wheel and rail is possible to define on the
equations of theory of elasticity [24]:
a = 1.397na ⋅
3
Pк
1
⋅
,
1
1
1
1
E
+
+
−
R1 R2 R3 R4
(3)
where:
conductivity,
λcp -
55
coefficient of heat
heat capacity of materials
by volume, q - thermal thread through a
surface, T - temperature; t - time.
Normal loading P(x, y ) we will define
as work of pressure in every cell of spot of
contact on the area of cells:
P( x,y ) = σ( x,y ) ⋅ F .
(6)
For determination of pressure σ(x, y) in
every point of contact ground the equation of
cycle per a second was used:
2
2
⎛ x⎞ ⎛ y⎞
σ(x,y) = σ
max 1− ⎜ ⎟ − ⎜ ⎟ ,
⎝a⎠ ⎝ b⎠
(7)
where: σmax - maximal value of pressure
in the center of spot of contact. At the accepted
chart of contact of wheel flange with the
lateral surface of head of rail, value of
maximal pressure, it is possible to define on
the equation (8):
2
P
1
b = 1.397nb ⋅ к ⋅
,
1
1
1
1
3 E
+
+
−
R1 R2 R3 R4
(4)
where: na, nb - coefficients values of
which are determined on tables; R1, R2, R3,
R4 – radiuses of curvature of contacting
bodies, E – the elastic module.
Change of temperature on the surface of
friction of wheel flange with the head of rail in
time, it is possible to define, deciding the
unstationary task of heat conductivity which,
in the case of independence of thermo physical
properties of materials of contacting bodies
from a temperature, is described by differential
equalization of heat conductivity, which has a
kind [31]:
⎛ ∂ 2T ∂ 2T ∂ 2T ⎞
∂T
, (5)
λ⎜ 2 + 2 + 2 ⎟ ± q = c p
⎟
⎜ ∂i
∂t
∂j
∂k ⎠
⎝
⎛1 1 1 1⎞
σmax = 0.245⋅ n p ⋅ 3 Pк ⋅ E 2 ⋅ ⎜⎜ + + − ⎟⎟ ,
⎝ R1 R2 R3 R4 ⎠
(8)
where: np - coefficient, the value of
which is determined on tables [26].
Viscosity of all drop liquids and their
mixes with rise in temperature decreases. For
recalculation of viscosity from one
temperature on another the equation (9) is
recommended:
⎛t
ηT = ηи ⋅ ⎜⎜ и
⎝ tT
n
⎞
⎟⎟ ,
⎠
(9)
where: ηT – required viscosity of oil at
temperature tT , η и – Resulted in tables of
GOST viscosity (in Nsm/m2) at temperature of
test of oil, t и – Temperature of test of oil on
definition of its viscosity, tT – temperature for
56
IRINA KIRICHENKO, ALEXANDR KASHURA, MARINA MORNEVA, SERGEY POPOV
which count viscosity, n – factor – an
exponent for each grade of oil. Value of factor
can be accepted under tables [23].
Results of calculation of parameters
considered tribosystem at presence between
them a lubricant are presented on fig. 2-5.
Fig. 3. Dependences maximal (a curve 1) and average (a
curve 2) relative temperature in contact of a crest of a
wheel to a lateral surface of a rail from size of lateral
force (Тsm.max/Тmax - curve 1, Тm.cm./Тm. - curve 2)
Fig. 2. Dependence of the attitude
f тг / f тр
from size
of lateral force in contact of a crest of a wheel to a
lateral surface of a rail (
f тр
and
f тг
– factor of
friction of not greased surfaces and at boundary friction)
Proceeding from data of calculation of
temperature on a spot of contact at presence of
a lubricant layer the processes proceeding on a
surface of friction, it is possible to present in a
following kind. So, at constant speed of sliding
and accruing loading there is a gradual
increase in intensity of allocation of heat of
friction fig. 3 and, hence, rise in temperature
of a layer of the oil dividing rubbing surfaces.
Apparently from fig. 2, in a zone of force of
interaction of contacting bodies 3 … 25 kN
Fig. 4. A field of temperatures (t, °C) on a surface of contact of a crest of a wheel with a rail at presence between
them greasings MC-20, (speed of movement Vline=60km/h, sliding έ=10%, P=65kN)
MODELLING OF ROLLING FRICTION WITH SLIDING
there is an excess. It is possible to explain
presence of an excess change of a physical and
chemical condition of greasing due to a
thermal emission in contact. To the further
growth of temperature leads to that the oil
layer loses protective properties and factor of
friction becomes same, as for not greased
surfaces.
57
between them greasing MC-20, (speed of
movement Vline=60km/h, sliding έ=10%,
P=65kN are shown on fig. 5.
CONCLUSIONS
1. The analysis of the received results
shows, that the effect from drawing greasing
on rubbing surfaces is observed at rather small
values of directing energies up to 10kN. In
these
conditions
rather
insignificant
deterioration is observed.
2. Decrease in intensity of deterioration
and a temperature mode with growth of value
of directing energies can be reached at the
viscosity of oil corresponding to temperatures
considerably
smaller
than
settlement
temperature in a zone of friction.
REFERENCES
Fig. 5. Dependence of the attitude capacities of friction
in contact of a crest of a wheel to a lateral surface of a
rail at presence of greasing and without it from size of
directing energies.
The similar result is received in works
[20] where the spasmodic increase in factor of
a sliding friction was observed at rise in
temperature in a zone of contact due to relative
friction.
By means of the given calculation it is
possible to estimate possible temperature in
contact of interface under the heaviest
conditions of its work and to pick up a
lubricant
with
demanded
operational
properties.
For an estimation of influence applied at
lubrication
materials
on
conditions
контактирования in considered a feather of
friction we shall take advantage of power
criterion of deterioration [9, 10]. Quality
standard of deterioration we shall make
calculation of capacity of forces of friction in a
zone of contact for a series of oils, according to
the described calculation.
Results of calculation of capacity of
force of friction on a surface of contact of a
crest of a wheel with a rail at presence
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МОДЕЛИРОВАНИЕ ПРОЦЕССА ТРЕНИЯ
КАЧЕНИЯ СО СКОЛЬЖЕНИЕМ
Ирина Кириченко, Александр Кашура,
Марина Морнева, Сергей Попов
А н нот а ц ия . В работе предложена математическая
модель
процесса
трения
качения
с
проскальзыванием в точке контакта колеса
локомотива с рельсом при реализации силы тяги.
К л юч ев ы е с ло ва : сила трения, сила сцепления,
контакт, площадь контакта, трение качения, трение
скольжения.