ANALYTICAL DETERMINATION OF THE REDUCED ROTATIONAL RESISTANCE COEFFICIENT OF THE CONSTRUCTION MACHINE SLEWING GEAR

Dep. «Applied Mechanics and Material Science», Dnipro National University of Railway Transport named after Academician V. Lazaryan, Lazaryan St., 2, Dnipro, Ukraine, 49600, tel. +38 (056) 373 15 18, e-mail bondarenko-l-m2015@yandex.ua, ORCID 0000-0002-2212-3058 Dep. «Applied Mechanics and Material Science», Dnipro National University of Railway Transport named after Academician V. Lazaryan, Lazaryan St., 2, Dnipro, Ukraine, 49600, tel. +38 (066) 150 95 00, e-mail AleksandrP@3g.ua, ORCID 0000-0002-9701-3873 3 Dep. «Applied Mechanics and Material Science», Dnipro National University of Railway Transport named after Academician V. Lazaryan, Lazaryan St., 2, Dnipro, Ukraine, 49600, tel. +38 (095) 816 99 90, e-mail kazimir.glavatskij@gmail.com, ORCID 0000-0003-0921-9845


Introduction
There are the following types of slewing gears (SG) of the construction machines: a) with a fixed pillar consisting of an upper support with a thrust and radial bearings; b) with a rotating pillar: consists of a pillar connected to the revolving portion of the construction tower crane; c) with a circular flat or tapered rail consisting of a series of conical or cylindrical rollers, which come in contact with two rails on the revolving and non-revolving portions of the construction crane; g) with a slewing ring: consists of ball or roller single-row or multi-row structures (full-slewing and part-slewing excavators, motor graders).
One of the main causes of rotational resistance is rolling resistance [12,13].There are many studies and suggestions for its definition, but all of them are either inaccurate, like Reynolds's assertion that rolling resistance is the result of sliding friction at the contact point, or require an experimental determination of one or more coefficients.
The analytic dependence of Tabor [3] on determining the rolling friction coefficient, which is based on Hertz contact deformation theorem [6], is quite successful.Tabor obtained the following analytical dependences for determining the rolling friction coefficient, k; for a linear contact: for a point contact where b -half-width of the contact pattern;  -coefficient of hysteresis losses.However, the presence in these formulas of the coefficient  nullifies their practical application.
In [5], there are formulas analogous to (1) and (2) without coefficient  , namely: 0.11 kb  and 0.1 kb  , that essentially differ from those offered by Tabor, and the absence of their coefficient of hysteresis losses testifies to their inaccuracy.
In [4], there are proposed the dependences for determining the rolling friction coefficient with the use of Tabor analytical dependences and the experimental values of the rolling friction coefficient for the wheels of cranes with a flat champignon and bull-headed rails [1,2].
Similarly to formulas (1) and ( 2) they are obtained in the following form: for flat champignon rail: for bull-headed rail: where Rwheel radius, m.
The difference in numerical values from the half-width of the contact pattern is obviously due to the rounding of the coefficient k in experiments to ten millimeters, as well as to the fact that their values are obtained the same for several wheel diameters (400, 500, 560, 630): 0.5 k  mm in the case of a flat champignon rail and 0.6 k  mm for the bull-headed rail).
It should be noted that formulas (3) and ( 4) are obtained independently of ( 1) and (2), and since the coefficients before b for such a class of prob- lems can be considered close by value, we will assume that the general values of k in these for- mulas coincide.Having considered that the coefficients before b in Tabor's formulas are obtained analytically and are exact, the value of  can be found by changing the coefficients before R in the exponents.This equality can be achieved by taking the following values  in formulas ( 1) and ( 2

Purpose
Designing new models of construction machines is closely related to the development of slewing gear, and that, in turn, has a drive whose power and dimensions depend on the rotational resistance and the reduced friction coefficient in the units [14][15][16].The absence of analytical dependencies for determining the reduced coefficient of friction for the rotation of construction machines, first, restricts the designer's ability to select materials, and secondly, does not allow the adoption of optimal design solutions.Therefore, the purpose of the article is to find analytical solutions to determine the rotational resistance in the slewing gear of construction machines, which allows projecting more advanced gears and machines in general.Existing techniques are based on empirical dependencies and experimental coefficients that reduce the accuracy of calculations, increase the size and cost of work.It is proposed to improve the accuracy and simplify the process of determining the rotational resistance and the magnitude of the reduced rotational resistance coefficient of the building tower cranes.More precise definition of the rotational resistance in the slewing gear of construction machines leads to saving the machine manufacturing and operation costs [21], as well as reduction of their harmful impact on the service staff and the environment [17][18][19][20].

Methodology
Now the formulas of Tabor ( 1) and ( 2) can be written as follows: for a linear contact: for a point contact: With formulas ( 6) and ( 7), we can solve the set problems analytically.
In [7] it is indicated that the value of hysteresis losses  in Tabor formulas is small.We can use formula (5) for its determination and ( 6), (7) for determination of the resistance.

Findings
1. Wheel rolling resistance.For a linear contact, we can take   800


MPa (steel 65G, crane operating mode 4M [11]), the elastic modulus  2 0.418 the half-width of the contact pattern will be 1.526 where Вwheel width, m; while the rolling friction coefficient can be determined by the formula (6).
For the point contact we can take   1040


MPa, the radius of the bull-head rail r 300 R  mm.Similarly to the formulas ( 8) and ( 9) we can determine the values for the point contact where b ncoefficient depending on the tangent ellipse equation coefficient Depending on the wheel radius of the pressure restraining force, the coefficient of hysteresis losses, the coefficient of rolling friction and resistance are shown in Fig. 1.
Since the rolling friction coefficient for the wheels of the construction cranes corresponds to their certain radius, it can be assumed that the relationship between the force of rolling resistance and the load on the wheel is linear.But the rolling friction coefficient is determined by the half-width of the contact pattern, depending on several parameters not linearly, therefore, it is necessary to establish the dependence of the wheel rolling resistance on the load.For this, the load P on the wheels with the radii 1 500 R  mm and 2 100 R  mm can be divided on two wheels in the ratio Dependences of the coefficients of rolling friction, loading and rolling resistance of the wheel and the total resistance of the wheels are shown in Fig.  1, 11, 12total value of the pressing force and the force acting on each wheel, 31, 32rolling friction coefficients; 4, 41, 42total rolling resistance value and rolling resistance of each wheel; lower position of curves for wheel 100 R  mm, upper for wheel 500 R  mm Analytic dependencies ( 6) and ( 7) are used to determine the coefficients rolling friction, so it is possible to restore one lacuna in the reference literature.Losses in roller bearings are found by the coefficient of friction reduced to the shaft (ball 0.01...0.015  , roller 0.015...0.02  [2]).However, this does not take into account, which race is rotating, inner or outer one.
Assuming that the deviation in the coefficient is negligible, it should be borne in mind that the number of locally positioned bearings may be significant (conveyors, vehicles), as well as an increase in the efficiency from 0.99 to 0.995 per ten bearings gives it an increase in more than 5%.
2. Ball bearings.The tasks to be clarified when calculating resistance: 1) To take into account the difference in the coefficients of rolling friction during rolling of the ball on the inner and outer races, since for calculating their size we take them equal, and the tangen-tial force acting on the ball (Fig. 3, a) is defined as [8] к 2) To take into account the rotation of the race, since the special feature of the roller bearings design is that the balls (rollers) pass different lines during one revolution of the inner or outer race.
Under the simplified scheme of the bearing, the problem is solved as follows.If the outer race rotates at an angular velocity o  (Fig. 3, b), then the speed of point 1 as the point belonging to the outer race will equal: Where o, i, b are the letters of the indices of sizes and speed of outer, inner races and ball; n - frequency of rotation of both inner and outer races.Naturally, that the instantaneous velocity center of this race is located at point 2 of the ball touch.Assuming that there is no slip between the outer race and the ball, then 12 vv  .The length of the ball rolling track on the outer race oo 2 lr , and on the inner race ii 2 lr  and the length difference will be that is, on this track there will be ball sliding on the inner race.
In case of rotation of the inner race with the fixed outer race the difference l  is evident that the ball will pass the outer race track that equals the inner race track.
We find the load on the balls based on their number [8]: The force acting on the most loaded ball is: For further calculations, the radius of the ball (without rounding to the standard one) and of the rolling bearing track will be equal [8,9]: For the number of balls 10 z  the load on the bearing where  is the angle between the balls (here


).Based on this, the load on the side balls The values of the half-width of contact patterns in formulas ( 9) and ( 11) are determined from expressions ( 17) and (18).When rolling the ball on the inner ring: where  During rotation of the inner race, Nm: During the rotation of the outer race, Nm: 14.99 5.83 59.98 20.82 59.98 80.8.15) and (16).
It was proved in [9] that if a load is applied to a group of bodies according to the cosine law, then to determine the resistance to their rolling, all loads can be applied to one body, that is, the rolling resistance of all five rollers on the inner race for the linear contact is determined from the expression: and on the outer race: According to formula (6), the rolling friction coefficient will be respectively the recommended value [9] for the wheel with up to 700 mm diameter 0.02  .4. Ball-bearing slewing gear (SG).The formula for determining the greatest pressure on the ball, given in [11], contains two unknowns: the average diameter of the rolling circle and the number of balls.
If the first unknown can be set on the basis of constructive considerations, then the number of balls can be set after finding their diameters.In addition, this formula is acceptable only if the reaction from the moment does not go beyond the support contour.
We propose finding the moment of the friction forces in the following sequence.
4.1.The slewing ring is broken, for example, into 10 sectors with a central angle 1 36  and for constructive reasons the average radius of the ball centers is taken .av R 4.2.We apply the load to one conditional ball in the sector, similar to the ball bearing (15), we find the maximum vertical pressure on it from the moment, Nm: Under the known value of vertical pressure V , the pressure on the balloon will be: where  is the angle between the reaction of the ball and the vertical line (usually 45  ) (Fig. 4).Maximum pressures on conditional side balls, Nm: The pressure on the left conditional side balls is found in the same way as for the right ones.4.7.We find the final diameter of the ball, while proceeding from conditions 4.4 and 4.6 and determine the number of balls.
4.8.Based on the equations (25), ( 26), ( 27) and the number of balls, we determine the pressure on one ball per sector, the rolling resistance by the formula (7) of one of the balls of 10 sectors.
We find the rolling resistance of z balls and to- tal pressure as the sum of values obtained by the formulas (25) and (27).
We find the rotational resistance coefficient as the ratio of the total rotational resistance to the total pressure.
Calculations are carried out according to the following data: the greatest moment acting on the slewing ring 427 M  kNm, the largest vertical reaction 178 V  kN, the average diameter of the The rolling friction coefficient is determined by the formula (7), and the rolling resistance subject to two rolling surfaces, i.e.The distribution of pressure per ball on the ring length and the rolling resistance of each ball in the form of graphs are shown in Fig. 5 [5,[12][13].
When adding all the pressures on the balls and their resistance to rolling and disision of It can be emphasized that in the examples of SG calculations, given in [11,12], the coefficient  is taken in relation to these quantities, and in [13] 0.04  .Let us find the value of the rotational resistance coefficient, which falls on sliding during rolling along the ring.Usually it is taken into account only when moving along a cylinder ring. in the case of ball contact both the plane and the bearing track, the contact pattern is not a point, but the ellipse with the axes 2а and 2, b the length of which is determined from the Hertz contact deformation formulas.
The average pressure per ball during its rotation by 360 0 is

Roller slewing gear.
For the calculation example, we considered the slewing gear of the construction tower crane with a fixed pillar and fixed rollers (Fig. 6).
Calculation output data: construction tower crane; average diameter of the thrust ball bearing and it is 1.75 mm.Rolling coefficient of the side roller (6) N, which is about half of the rolling resistance and requires consideration when calculating the units with thrust bearings.
The total moment of frictional forces during the turning of a construction crane with a fixed tower consists of the resistances: rolling of support rollers 771 N (20% of the total); in the roller journals 17 N (0.4% of the total); in the top journal 21 N (0.5% of the total); in the support roller from the rolling of balls 2 160 N (56% of the total) and their sliding 900 N (23% of the total) for a total rotational resistance value of 3 870 N.
6. Rotational resistance of SG rollers with stationary and fixed rollers.In some construction cranes, the support rollers are stationary (Fig. 7, a) or movable (Fig. 7, b).Load per a roller 2 cos It is obviously that for cylindrical rollers, the values of the maximum contact stresses will be different, and the diameters of the rollers and their rolling support on the slewing ring will have different values as well.
For calculations we take the same radius, as in the previous example,  Originality and practical value.The paper proposes to use analytical dependences to determine the reduced rotational resistance coefficient for linear and point contacts using Hertz contact deformations theory and Tabor partial analytic dependencies.The obtained dependencies will allow to design new types of slewing gear assemblies of the construction machines and to find additional rotational supports, which depend on the overall dimensions, shape and type of material from which the components of the assembly are made and do not contain any empirical data.

Conclusions
The analysis of the dependencies and graphs obtained makes it possible to draw the following conclusions and suggestions: rolling friction coefficient and rolling resistance of the crane type wheels practically linearly depends on the wheel radius, and the coefficient of hysteresis losses linearly decreases from 0.9 to 0.6 per linear contact and linearly increases from 1.01 to 1.06 per point contact; the friction coefficient value of rolling bearings, reduced to the shaft, depends on whether inner or outer race rotates; during the outer race rotation on a different path, passed by a ball or a roller on the tracks of the outer and inner races, the friction coefficient reduced to the shaft, which falls on pure rolling, is 1.3...1.5times higher than its value during the inner race rotation, and taking into account the sliding of balls or rollers on the inner race, it is 4...6 times higher than its size in ball bearings and 3 ... 4 times higherin roller bearings; due to the high value of friction in case of outer race rotation, during the design of rolling assemblies, it is necessary to avoid such solutions, and, if it is impossible, to take into account this fact both for the determination of resistance and for the lubrication of the assembly; the given value of the resistance of the construction crane with the ball-bearing sewing gear is obtained analytically, is 70% higher than the one recommended by supplier; in case of construction cranes with a turning tower, the greatest resistance to rotation falls on support rollers (about 80%).

Fig. 1 .
Fig. 1.Dependence on the wheel radius for linear a) and point contact b) (points show the reference values of the rolling friction coefficients):

Fig. 2 .
Fig. 2. Dependences of the ratio of the applied forces for linear (a) and point (b) contacts:

Fig. 3 .
Fig. 3. Elements of bearings: ascheme for determining the tangential force during the rotation of inner race [1], bscheme for determining the speed of points of outer race and ball; ccontact pattern

1 P or 2 P
the track of the inner race.At i b for the most loaded ball, it is necessary to set optionally the value of P, and for the side balls depending on the number of balls.When rolling the ball on the outer race: of rolling and sliding friction forces on the inner and outer races, Nm:

3 .
Maximum pressure on the opposite (left) conditional ball:

5 .
We find the number of balls in one sector with geometric conditions:

Fig. 4 .
Fig. 4. Design diagram of the ball-bearing SG by the theory of contact stresses provided that tr b 1.2 rr  : this discrepancy may be: a) irrelevance of the value adopted here   to the valid one; b) understated value М during the experiment.

R
steel, no lubrication), the work of sliding friction forces will be 01 of the recommended value of the reduced rotational resistance coefficient of the building cranes.However, it should be borne in mind that the denominator of the formula defining с  , includes the average radius of the ball centres av .The distribution of pressure per ball on the ring length and the rolling resistance of each ball in the form of graphs are shown in Fig.5

Fig. 5 .
Fig. 5. Distribution of pressures per one ball and resistance to its rolling along the ring

4 h 18 V
the rotary part in the vertical plane;  m -the distance between the line of applica- tion of reactions H and the journal; vertical reac- tion  t.

Fig. 6 . 5 . 1 .
Fig. 6.Design diagram of the slewing gear of a tower crane with fixed rollers on roller bearings, thrust bearing and top slide journal 5.1.Calculation of support rollers.Load on the roller, located on the line of force H 0 88.65 1 2cos H P   kN, the force acting on

5 . 2 .
high pressure on the rollers and the impossibility of selecting the appropriate roller bearing, it is possible to apply in the construction tower cranes the bearings with friction sliding co-Resistance in thrust bearing.According to the value of static load on the bearing 180 V  kN we take the bearing of 8216 series with 80 Figure 7).
Наука та прогрес транспорту.Вісник Дніпропетровського національного університету залізничного транспорту, 2019, № 1 (79) МАШИНОБУДУВАННЯ Creative Commons Attribution 4.0 International doi: 10.15802/stp2019/159499 © L. M. BondarenkO, O. P. Posmityukha, K. T. Hlavatskyi, 2019Half-width of the contact patterns (Fig.7):according to the diagram a 0.11 в  mm, according to the diagram b -0.12 в  mm; rolling resistances of the two rollers are respectively 17resistance in the journals of the two rollers according to the first and second diagrams 0is more than two orders higher than the rolling resistance of rollers.

Influence of resistance in bearings on wheel rolling resistance.
Let us consider two rolling bearings of one series, but of essentially different sizes.