Introduction
It is considered that ball and roller bearings can replace the slipping friction by rolling friction appearing during the rolling of balls or rollers on the inner and outer beating cage in the rotating pair [4, 10, 13]. However, for some unknown reason it is ignored the fact that in practice in rolling bearings may rotate both the inner ring with a stationary outer one, and vice versa almost in equal relations.
The rolling resistances appearing at this have different values and during the rotation of outer ring the causes are analogous to the problem considered by the ancient Greek mathematician Heron [3] when moving two cylinders of different diameters with a rigid connection. However, without having even the laws of friction and even more the laws of rolling his arguments have philosophical nature.
Without complete rejecting the influence of slipping friction on the resistance in the rolling bearings let us note that the first analytical dependence on its definition obtained O. Reynolds [15]. However, his theory was wrong, because he believed that the reason of rolling resistance lies in the slipping in the contact place. If so (it was not doubted, because of sizable reputation of the author), the rolling bearings were also lubricated as the slipping bearings. Another reason Reynolds could not have imagined as yet there was no theory of Hertzian contact deformation. Only 90 years later, D. Tabor [16] showed by experiments that the role of slipping during rolling is small. The theoretical dependences for the determination of the rolling friction coefficient also belong to him. At the linear contact the rolling friction coefficient is
(1)
at the point contact
(2)
where
– is
the half-width of the contact area according to Hertz;
– is the coefficient of hysteresis losses.
Since
the experimental determination of the coefficient
requires considerable time and money, the works [1, 2, 6, 9]
proposed the experimentally-analytical dependence to determine
,
which contains only generally accepted dimensions and mechanical
contacts.
By analogy with (1) and (2) the formulas are obtained in the form
(3)
(4)
where r – is the radius of the rolling body in meters.
The unresolved parts of the problem should include solution of the two following problems.
One
of the first is the problem related to the Reynolds mistake. Since
the main reason of the rolling resistance is the slip, in the works
[7, 10, 14] the rolling friction coefficients of the roller along
the outer and inner cages are taken as the equal and the tangential
force from the reaction
of the roller (Fig. 1, a)
(5)
|
а |
|
|
b |
|
|
|
Fig.
1. To the determination of tangential force during rotation of
the inner cage [10] (a)
and velocity |
The second problem to be solved is accounting which cage is the rotating one. In practice in rolling bearings may rotate both the inner ring with a stationary outer one, and vice versa. The reference literature does not take into account this fact. For example, the efficiency coefficient of the groove pulley is given equal, although any cage can rotate, especially with fixed blocks.
The peculiarity of the bearing functioning is that the balls (rollers) run different distances per revolution of inner or outer cage.
At
the simplified diagram of the bearing the problem can be solved as
follows. If the outer cage rotates with angular velocity
(Fig. 1, b),
the velocity of the point 1 as the point belonging to the outer cage
is
(6)
where the letters i, o, b – is belonging of the sizes and velocities to the inner, outer cages and the ball; n – is the rotation frequency of both inner and outer cages.
Naturally,
the instantaneous velocity center of this cage will be located at
the point 2
of
the contact with the ball. Assuming that the slippage between the
outer cage and the ball is absent, then
.
The
length of the roller track on the outer cage is,
and on the inner one is
and the difference of distance will be
.
That is, at this distance the roller slipping on the inner cage will
take place.
In
the case of inner cage rotation with the fixed outer cage, the
difference
suggests that on the outer cage the roller will pass a distance
equal to the distance on the inner cage.
Purpose
The article is aimed to find analytically reduced coefficient of friction of the ball and roller bearings taking into account the different values of the rolling friction coefficient on the outer and inner cages and take into account the difference in the rolling distance over them.
Methodology
The solution technique is based on the theory of plane motion of a rigid body, the theory of Hertzian contact deformation and the analytical dependencies for determination of coefficient of rolling friction.
Findings
1. Ball bearing (Fig. 1).
The number of balls in the bearing [8] according to the assembly condition
(7)
The force acting on the most loaded ball
(8)
Ball radius
(9)
The radius of raceway of the bearing track
(10)
If
the number of balls is
the load on bearing Q
(for example, at z = 10)
(11)
where
– is the angle between the balls (here
).
On that basis the load on the side balls is
(12)
The value of half-widths of the contact areas in the formulas (5) and (6) are determined using the expressions:
(13)
where
– is the coefficient depending on the equation of contact ellipse
In
formulas (6)–(9) D
–
is the outer diameter of the bearing; d
– is the inner diameter of the bearing;
–
is the
radius of the bearing track of the inner ring. With
for the most loaded ball one should substitute the value
,
and for the lateral ball
depending on the number of balls.
When the ball rolling on the outer ring
(14)
where
is determined as the function
where
– is the radius of the bearing track of the outer ring.
To take into account the influence of the bearing size on the efficiency coefficient of the groove pulley and the resistance coefficient to the motion of the crane wheels let us consider two rolling bearing of one series, but with substantially different sizes.
1.1. The
bearing no. 304:
d=20 mm, D=52, static load Q= 7.94 kN, average diameter
mm,
mm,
number of balls z=7
at
Half-width
of the contact areas of the ball loaded by the force
: with the inner cage
;
with the outer ring
.
Accordingly, loaded by the force
N of the lateral balls:
;
.
The rolling resistance of the most loaded ball: on the inner ring
with a coefficient of rolling friction
,
on the outer ring
;
two lateral balls on the inner ring
and
at
.
Let us find the work of forces of rolling friction per revolution of the inner and outer rings:
− during rotation of the inner ring
– during rotation of the outer ring
The
total work of the rolling friction forces of the balls on the inner
and outer rings, excluding the sliding friction of balls
, taking into account the slipping
,
i.e. the half of the rolling resistance at f=0,1 (grease) accounts for slipping in this ball bearing during rotation of the outer ring.
Using this bearing it is difficult to determine the coefficient values of motion resistance and the friction of bearing reduced to the trunnion, which are used for calculation of the resistances in the running gears of the cranes and groove pooleys because of the small value of Q.
1.2. Bearing
no. 312: d=20 mm, D=130 mm, Q=4,94 kN,
,
,
,
,
;
;
.
By
analogy with the preceding bearing let us write:
,
,
;
,
at
,
;
,
.
The work of forces of the rolling friction: at the rotation of the ring, N/m
of the outer ring taking into account the slipping friction of the balls due to the different diameters of the inner and outer rings, N/m
In this bearing the slipping resistance 3 times exceeds the rolling resistance.
The
values in the formulas (15) and (16) considering the coefficient
that takes into account the friction of flanges
(supporting cranes, central drive, conical wheel rim)
With
two bearings no. 312 with a total static load
the movement of the crane wheel with the diameter
along the rail KR-70 with a radius of curvature
is possible [12].
Rolling
resistance with this diameter should be determined taking into account the hysteresis loss
coefficient [2, 6, 7, 8]
(17)
where
– in meters.
Comparison
of the formulas (6) and (15) shows that that for this class of
problems the coefficient
is quite accurately determined by the exponential.
Half-width of the contact area with the circuit of contact «cylinders with mutually perpendicular axes» [11] is equal to
(18)
where
–
is a coefficient depending on the ratio
and equal to
;
at this
mm,
mm with the value
recommended in the work [8] for the diameters of 400 mm, 500, 560
and 630 mm.
Rolling resistance of the wheel
(19)
is
equal to 365.3 N and the work of the rolling friction force per
revolution of the inner cage will be
,
and with accounting of the flange friction is
.
Thus,
the work of the friction forces per one revolution of the inner cage
of the bearing will be
,
and during rotation of the outer bearing cage
, i.e. it is about 2.7 times more.
On
the basis of these data the motion resistance coefficient is
,
and
at
the recommended value
with
the rolling bearings and wheel diameters from 200 to 400 mm [11].
The data obtained for the bearing no. 304 are used for determination of the efficiency coefficient of the running and stationary blocks with rotation of the inner (Fig. 2, a, b, c) and outer (conventional design) rings.
|
а |
|
|
b |
|
|
|
Fig.
2. Recommended supports of the blocks |
Based
on the static carrying capacity of the one bearing
for the scheme «a»
we will take
as
equal to this value. The breaking tension of the rope will be taken
as
the
rope with diameter
corresponds to this, diameter of the block
.
Effective work when rotating the block for one revolution
The work of friction forces in the bearing during rotation of the inner ring
Efficiency coefficient of the block scheme «а»
During
rotation of the outer ring
N/m and
i.e. the difference in the efficiency coefficient of the block is 1.8%.
At the scheme «b» (the running block) the value Q with the same bearings is 15.88 kN and the work of effective force per revolution will be the same value as in the previous scheme, the value of the efficiency coefficient will be the same.
It
should be noted that the recommended value of efficiency coefficient
for the rolling bearing is 0.97−0.98,
which is close to the obtained value
In spite of slight difference in the values of efficiency coefficient when rotating the inner and outer ring of the bearings (1.8%), it should be noted that even at the five bearings, this difference is about 9.6%, which, obviously, should be taken into account when calculating and designing.
Here, during the calculation of the friction works the work for the rope bend on the block is not taken into account. However, as shown in [12], a decrease in the rope contact angle of block does not lead to decrease in its efficiency coefficient, which is clearly associated with a decrease in pressure on the balls, and naturally a decrease of friction forces in the same degree.
It
should also be noted that the diameter of the worn-in rolling
bearing sleeve equal to the inner diameter of the rolling bearing
no. 312 with
and
we obtain the moment on the trunnion
,
where
is
a coefficient of sliding friction.
With
the known work of the frictional forces during rotation of the outer
ring in one revolution the required value of the coefficient is
and a one order less than its value with the liquid lubricant.
2. Roller
bearing. Let us consider the bearing of the medium narrow series no.
2306 with the following dimensions:
,
, the diameter roller
,
the number of rollers
,
the radius of the bearing track of the inner ring
, and the same of the outer ring
.
The force acting on the most loaded roller [10]
(22)
on the lateral rollers
(23)
The works [5, 6] proved that if the load is applied to a group of bodies according to the cosine law, to determine the resistance to their rolling the entire load can be applied to a single body, i.e. the rolling resistance of all five rollers on the inner ring at the linear contact is determined using the expression:
(24)
on the outer ring
(25)
The
rolling friction coefficient is determined from the formula (1)
with
and will be
,
accordingly.
The
rolling resistance of the rollers: on the outer cage is
,
on the inner one is
.
The work of friction forces of the rolling and slipping
On the inner and outer cages
with
the coefficient of friction rolling of the rollers on the inner cage
.
Coefficient of the motion resistance is:
– at
the rolling of the inner cage
;
– at
the rolling of the outer cage
with the recommended value [12] for the wheels with diameter up to
700 mm
.
Originality and practical value
Analytical dependences for determining the reduced coefficient of friction for steel wheels and pulleys efficiency coefficient of the groove pulleys were obtained.
These formulas make it possible for the designer to operate not only the design, but also the materials of units at the design stage of rolling units.
Conclusions
Analysis of the obtained formulas and calculation results makes it possible to make the following conclusions and recommendations:
– because of the different diameters of the bearing tracks of the inner and outer rings of the rolling bearings and, consequently, because of the different path of the balls or rollers during rotation of the outer ring (with fixed inner one) occurs balls or rollers slipping on the inner ring;
– value of sliding friction in rolling bearings is approximately 50% from the total in ball bearings and about 30% in roller bearings (as a result of different diameter of balls and rollers); consequently, the efficiency coefficient of the groove pulley decreases by about 2%, and the resistance coefficient of the crane wheels by about 15%;
– when constructing the rolling units of rolling bearings the preference should be given to the rotation of the inner cage, especially for machines with their serial connection (railway trains, belt conveyors, etc.).
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