Civil Engineering
Vitaliy Statsenko
Scientific supervisor DSc. Leonid Bondarenko
English supervisor DSc. Leonid Khmara
ANALITICALY FINDING OF COEFFICIENT OF FRICTION FOR FLANGES
АНАЛІТИЧНЕ ЗНАХОДЖЕННЯ КОЕФІЦІЕНТУ ТЕРТЯ РЕБОРД
With the proven value of the coefficient of friction of flanges selected component,
which falls on the slip resistance of the wheel rim surface of the head rail because of the
discrepancy between longitudinal axes of rails and wheels. Proved that component, which
accounted for slip depends on the mode of contact between the wheel rim its head rail and is
in linear contact about 10%, and at a point - about 12% of the total coefficient of conflict
between axes 3°.
Known [1] that the movement resistance of the mechanism of movement due with
friction on a straight line segment path defined by the formula:
𝑊тр = (𝐺 + 𝐺к) ∙
𝜇𝑑 + 2𝑓
∙ 𝐾𝑝
𝐷
where G and Gk - the weight of cargo by weight and the crane or trolley; D and d - diameters of
the wheel and his neck; f - coefficient of rolling friction; Kp - coefficient which takes into
account the friction wheel hubs and flanges; μ - coefficient of friction bearings , brought to the
pin wheel.
The analysis of Kp ([1], tab. 6.5.3) shows that even for similar nomination arrangements,
it differs more than a third. For example, Kp = 1,5 for crane supported with cylindrical rim
running wheels and a central drive and Kp = 2.5 for crane trolley with the same rim and drive.
No doubt, that one of the components of the resistance movement, which takes into
account the coefficient Kp is friction between flanges and head side rails. Its value depends on
the degree of lozenging of wheels on one side over another. Identify this value of resistance, in
our view, reveals significant difficult.
Definitely, second element to be considered as sliding friction wheel rim, on the surface
of the rails, which appears due to distortion. These two components will be the key, and the
possible appearance of additional components is unlikely brings significant value to specify the
first two.
We believe that the second component can be determined analytically, and the first - by
reducing the experimental value Wтр its size, which depends on friction with the surface of the
rim track.
As a result of a distortion vector wheel speed will not match the longitudinal axis of the
rail and also there will be a rolling and sliding.
Consider the first point contact when the contact spot looks like an ellipse with major axis
b and small axis a (Fig. 1).
3"
0,35
3'
V
0,25
0,20
C
Ê,Êêçà ,Êêçâ ,Êêç ,ì ì
0,30
a
â
3
0,15
2
0,10
1
0,05
0
0
1
2
3
4
5
C ,8
Figure 1. Dependence of deviation angle ψ longitudinal axis of the wheel from the axis of the
rails:
1.
adjusted coefficient of rolling friction KK3a axis a;
2.
the same respect in KK3b 3 - the total value of rolling friction coefficient is given
KPC, which falls on the sliding friction rim surface of the rails; 3’, 3” - normative coefficient of
friction bearings for the flat and rounded surface.
Suppose that the b axis coincides with the axis of the rails, and the axis and perpendicular to it.
Then the b axis will skate with speed
𝑉𝑏 = 𝑉 ∙ 𝑠𝑖𝑛𝜓
(І)
where ψ - angle from the axis of the rails; V = ωR = πnR - Circular speed wheel rim; n rotational speed; R - radius of the wheel.
Obviously, the pressure wheels on the rail on the section contact b is variable in its length
and change in Parabolic law [2]. We find such a wide area in which b parabola diagram of
pressure will be equal to the square of standard rectangle height σmax corresponding to the
maximum value of contact stress.
2
𝑏΄ = 3 𝑏
(2)
Similarly, progress and in determining the size of a
2
𝑎΄ = 3 𝑎
(3)
Assume that the coefficient of sliding friction does not depend on the size and
distribution of pressure force P rail lines to make a and b.
𝑃
3𝑃
𝑞 = 𝑎΄+𝑏΄ = 2(𝑎+𝑏)
(4)
Efforts PKB which falls on the lines of bK that glides
𝑃𝑏
PKB = (𝑎+𝑏) sin 𝜓
(5)
Power of friction at the site
𝑁𝐾𝐵 =
𝑃𝑏𝑉𝜂
𝑎+𝑏
sin2 𝜓
(6)
where η - coefficient of friction between wheel and rail.
Find the power of conventional friction coefficient of sliding friction KK3b.
2 𝑃𝑏𝑉
𝑁𝐾𝐵 = 𝐷 𝑎+𝑏 sin 𝜓 ККЗВ
(7)
Equating the right sides of expressions (6) and (7) gives the conditional coefficient of friction
bearings, which accounted for sliding along an axis that coincides with the longitudinal axis of
the rails.
After similar calculations for the axis a, perpendicular to the axis of b implies an
𝑃𝑎
Pka = 𝑎+𝑏 (1 − cos 𝜓)
𝑁𝐾𝐵 =
𝑃𝑏𝑉𝜂
𝑎+𝑏
sin 𝜓 (1 − cos 𝜓)
(9)
(10)
The formula for determining the coefficient of friction bearings, which accounted for sliding
along an axis that coincides with the transverse axis of the rails
(11)
The total coefficient of friction bearings, which accounted for as a result of twisting the wheel
slip at an angle ψ
(12)
Experimental value of the resistance movement of friction flange determined from the
expression (1). At Kp = 1 and the current value of Kr, ie
𝑊𝑒р = (𝐺 + 𝐺к) ∙
𝜇𝑑+2𝑓
𝐷𝐾
∙ (𝐾𝑝 − 1)
(13)
But given the value of coefficient of friction bearings
2𝑊𝑒р
𝐾𝑒𝑝 = (𝐺+𝐺к)𝐷
(14)
In the case of linear contact classes remain preliminary, but a change to B - the width of the head
rail. Size a and b in the case of equal values of elastic modules and Poisson’s ratio node wheels
and rails are determined by the expressions [3]
3
𝑃𝑅
3
𝑃𝑅
𝑎 = 1.397𝑛𝑎 √ 𝐸 ;
𝑏 = 1.397𝑛𝑏 √ 𝐸 ;
2,0
2000
1
1,8
1'
1600
1400
1,6
1,4
4
1200
1,2
1000
1,0
2'
800
600
0,8
Nêâ ,Nêà , êÂò
1800
Wò ð ,Wåð ,Wêç , H
(15)
0,6
2
0,4
400
5
200
5'
4'
0,2
0
0
0
1
2
3
4
5
C,8.
Fig.2. Dependence of deviation angle ψ longitudinal axis of the wheel axis rails:
1 - total experimental values of resistance movement WTP, which includes the friction surface
sliding on rails, 2 - component, which accounted for only friction flange WEP; 3 - theoretical
component of sliding at the contact point; 1̕, 2̕ - size, similar 1 and 2 with linear contact; 4,5 power that is on a sliding WK3 at contact lines and point; 4’, 5’ - same with linear contact
We find values of resistance movement from the friction of flanges in according with
normative method WTP and that component, which falls on the sliding friction wheels on rails
WK3 surface due to a distortion angle. Calculations was done for wheels with
wheel diameter = 250 mm with a static load P = 40 kN of linear and point contacts for crane
trolley connected by flexible cable (Kp = 2) and d = 50mm; f = 0,4, f = 0, 3 - rounded and flat
(B = 45 mm) head rails, if η = 0,1; μ = 0,15.
Fig. 1 and Fig. 2 shows that dependency on the angle of deviation from the
longitudinal axis of the wheel axis rails.
Analysis of the above expressions and graphs in Fig. 1 and Fig. 2 allows the following
conclusions:
1. Factor that takes into account the friction of wheels on rail flange consists of two
main components: direct friction of flanges and component of sliding friction wheel rim on
the back rails due to mismatch longitudinal axis with the axis of the wheel rails.
2. A component of sliding friction on the rim rail depends on the angle discrepancy
axis rails and wheels and such at ψ = 3 ° is about 12% of the total value of point contacts about 10% linear.
REFERENCES:
1.Справочник по кранам: В 2 т. Т. 2 / Александров М.П., Гохберг М.М., Ковин А.А.
и др.:-Л.: Машиностроение, 1988.-559 с.
2.Тимошенко С.П. курс теории упругости:-Киев:Наук. Думка, 1972.-501 с.
3.Справочник по сопротивлению материалов / Писаренко Г.С., Яковлев А.П.,
Матвеев В.В.; -Киев:Наук. Думка,1988.-736 с.