RELATIONSHIP BETWEEN ROLLING AND SLIP RESISTANCE IN ROLLING BEARINGS

Dep. «Applied Mechanics», Dnipropetrovsk National University of Railway Transport named after Academician V. Lazaryan, Lazaryan St., 2, Dnipropetrovsk, Ukraine, 49010, tel. +38 (056) 793 19 19, e-mail admin_diit@inbox.ru, ORCID 0000-0001-6602-2745 Dep. «Transportation Technologies», Lviv branch of Dnipropetrovsk National University of Railway Transport named after Academician V. Lazaryan, I. Blazhkevich St., 12a, Lviv, Ukraine, 79052, tel. +38 (097) 907 50 72, e-mail babjk@mail.ru, ORCID 0000-0001-5125-9133 Dep. «Military Training of Specialists of the State Special Service of Transport», Dnipropetrovsk National University of Railway Transport named after Academician V. Lazaryan, Lazaryan St., 2, Dnipropetrovsk, Ukraine, 49010, tel. +38 (056) 793 19 19, e-mail Sergei_jak@mail.ru, ORCID 0000-0002-6431-4303 Dep. «Military Training of Specialists of the State Special Service of Transport», Dnipropetrovsk National University of Railway Transport named after Academician V. Lazaryan, Lazaryan St., 2, Dnipropetrovsk, Ukraine, 49010, tel. +38 (056) 793 19 19, e-mail admin_diit@inbox.ru, ORCID 0000-0002-8114-8722 Dep. «Military Training of Specialists of the State Special Service of Transport», Dnipropetrovsk National University of Railway Transport named after Academician V. Lazaryan, Lazaryan St., 2, Dnipropetrovsk, Ukraine, 49010, tel. +38 (056) 793 19 19, e-mail admin_diit@inbox.ru, ORCID 0000-0002-9335-7716


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 at the point contact where bis the half-width of the contact area according to Hertz; α -is the coefficient of hysteresis losses.Since the experimental determination of the coefficient α requires considerable time and money, the works [1,2,6,9] proposed the experimentallyanalytical dependence to determine α , which contains only generally accepted dimensions and mechanical contacts.
By analogy with (1) and ( 2) the formulas are obtained in the form ( ) 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 i P of the roller (Fig. 1, a 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 o w (Fig. 1, b), the velocity of the point 1 as the point belonging to the outer cage is 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 1 2 The length of the roller track on the outer cage is o o 2 l r = π , and on the inner one is i i 2 l r = π 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 l ∆ 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 2.9 The force acting on the most loaded ball Ball radius The radius of raceway of the bearing track If the number of balls is 10 z ≥ the load on bearing Q (for example, at 10 z = ) ( ) where γ -is the angle between the balls (here 36 γ = °).
On that basis the load on the side balls is The value of half-widths of the contact areas in the formulas (3) and (4) are determined using the expressions: In formulas ( 4)-( 7) D -is the outer diameter of the bearing; d -is the inner diameter of the bearing; 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.
-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 rol 26.
In this bearing the slipping resistance 3 times exceeds the rolling resistance.
The values in the formulas ( 8) and ( 9) considering the coefficient that takes into account the friction of flanges f 1.2 k = (supporting cranes, central drive, conical wheel rim) if f i 293.9With two bearings no.312 with a total static load 2 98.8 kN the movement of the crane wheel with the diameter w 400 D = mm along the rail KR-70 with a radius of curvature r 400 R = mm is possible [12].
Rolling resistance with this diameter ( 50 D > mm) should be determined taking into account the hysteresis loss coefficient [2,6,7,8] 0,2 0.16 where k R -in meters.Comparison of the formulas ( 4) and (8) shows that that for this class of problems the coefficient α 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 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.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 o 0.9956 η = .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 60 d = mm and 49.4 Q = kN we obtain the moment on the trunnion 1.27 where µ 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 o 1, 27 2 and a one order less than its value with the liquid lubricant.
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: On the inner and outer cages ( ) with the recommended value [12] for the wheels with diameter up to 700 mm 0.020 w = .

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.).

Fig. 1 .
Fig. 1.To the determination of tangential force during rotation of the inner cage [10] (a) and velocity of the points of the outer cage and the ball (b)

,
the radius of the bearing track of the inner ring.With i b for the most loaded ball one should substitute the value 0 P , and for the lateral ball 1 P or 2 P depending on the number of balls.When the ball rolling on the outer ring Наука та прогрес транспорту.Вісник Дніпропетровського національного університету залізничного транспорту, 2016, № 3 (63) РУХОМИЙ СКЛАД І ТЯГА ПОЇЗДІВ doi 10.15802/stp2016/74760 © L. M. Bondarenko, M. O. Babyak, S. O. Yakovlev, S. O. Istin, G. Yu.Moskalev, 2016 where o n is determined as the function b r b o

1 . 1 .
The bearing no.304the outer ring o0 57.77W = N at o0 0,0564 k = mm; two lateral balls on the inner ring i1 18.30 W = N at i1 0.029 k = and o1 23.9 W = N at o1 0.038 k = mm.Let us find the work of forces of rolling friction per revolution of the inner and outer rings: − during rotation of the inner ring in the work[8]  for the diameters of 400 mm, 500, 560 and 630 mm.Rolling resistance of the wheel equal to 365.3 N and the work of the rolling friction force per revolution of the inner cage will be w 458.8A = N/m, and with accounting of the flange friction is wf 550.6A = N.Thus, the work of the friction forces per one revolution of the inner cage of the bearing will be if

Fig. 2 .
Fig. 2. Recommended supports of the blocks with the rotation of the inner ring of the bearings Based on the static carrying capacity of the one bearing 7.94 Q =kN for the scheme «a» we will take max S as equal to this value.The breaking tension of the rope will be taken as