UDC 621.86/87(075.8)

Fig. 1. Dependence on the wheel radius for linear a) and point contact b)
(points show the reference values of the rolling friction coefficients):
1 – wheel pressure restraining force; 2 – coefficient of hysteresis losses;
3 – rolling friction coefficient; 4 – rolling resistance coefficient

For this, the load on the wheels with the radii mm and 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. 2

Fig. 2. Dependences of the ratio of the applied forces for linear
(a) and point (b) contacts: 1, 1₁, 1₂ – total value of the pressing force
and the force acting on each wheel, 3₁, 3₂ – rolling friction coefficients;
4, 4₁, 4₂ – total rolling resistance value and rolling resistance of each wheel;
lower position of curves for wheel mm, upper for wheel 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, roller [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 tangential force acting on the ball (Fig. 3, a) is defined as [8]

;

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 (Fig. 3, b), then the speed of point 1 as the point belonging to the outer race will equal:

, (12)

Where o, i, b are the letters of the indices of sizes and speed of outer, inner races and ball; – frequency of rotation of both inner and outer races.

Fig. 3. Elements of bearings: a – scheme for determining the tangential force during
the rotation of inner race [1], b – scheme for determining the speed
of points of outer race and ball; c – contact pattern

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.

The length of the ball rolling track on the outer race , and on the inner race 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 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]:

. (13)

The force acting on the most loaded ball is:

. (14)

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 the load on the bearing (for example, if ) [8]:

, (15)

where is the angle between the balls (here ). Based on this, the load on the side balls

, . (16)

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:

, (17)

where – is the coefficient, depending on the tangency ellipse equation . In formulas (13) – (17) – outer bearing diameter;
– inner bearing diameter; radius of the track of the inner race.

At for the most loaded ball, it is necessary to set optionally the value of P, and for the side balls or depending on the number of balls.

When rolling the ball on the outer race:

, (18)

where is determined as a function; ; is the radius of the outer race track.

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

3.1. Ball bearing of 304 series. Calculation output data: bearing of 304 series, mm, mm, static load kN, average diameter mm, mm, number of balls at , mm; mm; mm.

Half-width contact pattern of the ball, loaded with force N, with the inner race mm for , with the outer race mm for . Correspondingly, the side balls loaded by force :
mm; mm. Resistance to rolling of the most loaded ball: on the inner race N, with the rolling friction coefficient mm, on the outer race N with mm; two side balls on the inner race N with mm and N with mm.

Let us determine the work of the rolling friction forces during one rotation of the inner and outer races.

During rotation of the inner race, Nm:

; (19)

During the rotation of the outer race, Nm:

(20)

Thus, during the rotation of the inner race, the rolling friction force work during one rotation equals Nm, in case of the outer race rotation Nm (1.55 times higher), and taking into account sliding Nm, that is, 6 times higher.

In this case, the value of the conditional coefficient of friction reduced to the shaft is equal to: during the rotation of the inner race , for the recommended value , and during the rotation of the outer race .

3.2. Ball bearing of 2306 series. Calculation output data for the bearing of 2306 series:
mm, mm, static load
kN, roller diameter mm, roller length mm, number of rollers at bearing track radius on the inner race mm, track radius on the outer race mm.

The force acting on the most loaded and side rollers is determined from formulas (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:

, (21)

and on the outer race:

. (22)

According to formula (6), the rolling friction coefficient will be respectively mm, mm. The rolling resistance of rollers: on the outer race Н, N, and on the inner race N.

Work of rolling and sliding friction forces on the inner and outer races, Nm:

(23)

for the friction sliding coefficient of rollers on the inner race .

The motion resistance coefficient is: during the rotation of the inner race , during the rotation of the outer race , for the recommended value [9] for the wheel with up to 700 mm diameter .

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 and for constructive reasons the average radius of the ball centers is taken

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:

(24)

Under the known value of vertical pressure , the pressure on the balloon will be:

Nm, (25)

where is the angle between the reaction of the ball and the vertical line (usually ) (Fig. 4).

Maximum pressures on conditional side balls, Nm:

(26)

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

Nm. (27)

The pressure on the left conditional side balls is found in the same way as for the right ones.

4.4. After the value , we roughly take the diameters of the ball .

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

4.6. Maximum pressure on one ball of the right sector and the ball radius, based on Hertz contact pressure for the track radius mm:

Nm, (28)

where is the value depending on the ratio of tangible ellipse equation factors ;
– boundary contact stresses depending on the steel grade, contact type and Brinell-hardness; for mm, (; [4]).

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 balls and total 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 kNm, the largest vertical reaction kN, the average diameter of the ball centers mm ( mm).

In this case, the vertical pressure from the moment, taking into account the side balls (24) will be kN, and pressure on the right and left conditional balls (25) and (27) kN, kN. Having taken the ball diameter mm and the maximum pressure on one ball, we will have:

kN,

where is determined from geometric conditions . Let us check the taken ball radius

Fig. 4. Design diagram of the ball-bearing SG

by the theory of contact stresses provided that :

Nm, (29)

where mm for MPa (surface hardened steel 45 [2]).

For taking into account the number of balls in the sector and pressure on the conditional central and side balls (26), we find the half-width of the contact pattern:

. (30)

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 kN by kN, we obtained the value of the reduced rotational resistance of the crane that significantly exceeds the recommended value.

The reasons for this discrepancy may be: a) irrelevance of the value adopted here to the valid one; b) understated value during the experiment.

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].

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. However, in the case of ball contact with both the plane and the bearing track, the contact pattern is not a point, but the ellipse with the axes and the length of which is determined from the Hertz contact deformation formulas.

The average pressure per ball during its rotation by 360⁰ is kN. Herewith, the minor axis of the ellipse is mm.

Concentrating the pressure at 4 points, we find that the pressure at the points v and n (see Figure 3) is kN. In this case, the vertical axis of the ellipse [4]:

mm. (31)

The distance from these points to is, i.e. mm.

The difference in the distance travelled by one rotation of the ring is mm. For kN, (steel on steel, no lubrication), the work of sliding friction forces will be Nm. Expressing the work of normal forces through the reduced coefficient, we can obtain , wherefrom

(32)

and is about 0.01 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 , includes the average radius of the ball centres 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. Distribution of pressures per one ball and resistance to its rolling along the ring

5. 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 mm; diameter of the bearing ring mm; horizontal reaction t, where – the resulting moment of the rotary part in the vertical plane; m –the distance between the line of application of reactions and the journal; vertical reaction t.

Fig. 6. Design diagram of the slewing gear of a construction 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 kN, the force acting on each of the two side rollers is the same and is equal to kN.

For the roller width mm and MPa (steel 75, mode of operation 5M), its radius is determined from the Hertz contact forces formula:

, (33)

it is 130 mm.

Half-width of the contact pattern

, (34)

and it is 1.75 mm.

Rolling coefficient of the side roller (6) mm, and that located on the horizontal axis mm.

Rolling resistance of three rollers:

N.

Due to the 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 coefficient and with the journal diameter mm.

Friction resistance in roller journals:

kN. (35)

For the top journal diameter mm the sliding resistance in it

kN. (36)

5.2. Resistance in thrust bearing. According to the value of static load on the bearing kN we take the bearing of 8216 series with mm, mm, ball diameter mm, number of balls , track radius
mm.

When loading one ball kN, the half-width of the contact pattern (11) mm .

The rolling friction coefficient according to formula (7) is mm.

The rolling resistance of 20 balls N. The ball sliding resistance coefficient in accordance with (32) for (thick lubrication) and for its rolling resistance value is 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 .

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, mm, the horizontal reaction is equal to kN, the boundary contact stress is MPa, the roller width .

The radius of the roller for the diagram a (Fig. 7) can be found from formula (33), by replacing with . According to the taken values mm, the journal radius is taken to be equal to mm for cases a and b (Fig. 7).

The radius of the roller for the diagram b (Fig. 7) can be found by the same formula (33) in the case of change of the sign under the radical to the inverse, and it will be equal to mm.

Half-width of the contact patterns is found by the formula (34) with the change of the sign to the inverse, according to the diagram b before (see Figure 7).

Half-width of the contact patterns (Fig. 7):

according to the diagram a mm, according to the diagram b – mm;

rolling resistances of the two rollers are respectively N and N.

The resistance in the journals of the two rollers according to the first and second diagrams is kN, that is more than two orders higher than the rolling resistance of rollers.

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.

Fig. 7. Design diagram of the slewing gear of cranes:
a – with stationary rollers; b – with moving rollers

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...1.5 times 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 higher – in 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%).

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Л. М. Бондаренко¹, О. П. ПосмітюхА^2*, К. Ц. Главацький³.

¹Каф. «Прикладна механіка та матеріалознавство», Дніпровський національний університет залізничного
транспорту імені академіка В. Лазаряна, вул. Лазаряна, 2, Дніпро, Україна, 49010, тел. +38(056)3731518,
ел. пошта bondarenko-l-m2015@yandex.ua, ORCID 0000-0002-2212-3058
^2*Каф. «Прикладна механіка та матеріалознавство», Дніпровський національний університет залізничного
транспорту імені академіка В. Лазаряна, вул. Лазаряна, 2, Дніпро, Україна, 49010, тел. +38 (066) 150 95 00,
ел. пошта AleksandrP@3g.ua, ORCID 0000-0002-9701-3873
³Каф. «Прикладна механіка та матеріалознавство», Дніпровський національний університет залізничного
транспорту імені академіка В. Лазаряна, вул. Лазаряна, 2, Дніпро, Україна, 49010, тел. +38 (095) 816 99 90,
ел. пошта kazimir.glavatskij@gmail.com , ORCID 0000-0003-0921-9845

АНАЛІТИЧНЕ ВИЗНАЧЕННЯ ПРИВЕДЕНОГО КОЕФІЦІЄНТА ОПОРУ

ОБЕРТАННЮ МЕХАНІЗМІВ ПОВОРОТУ БУДІВЕЛЬНИХ МАШИН

Мета. Проектування нових зразків будівельних машин тісно пов’язане з розробкою механізмів повороту, а ті, в свою чергу, мають привід, потужність та габарити якого залежать від опору повороту та приведеного коефіцієнта тертя у вузлах. Відсутність аналітичних залежностей для визначення приведеного коефіцієнта тертя обертанню будівельних машин, по-перше, обмежує можливості конструктора у виборі матеріалів, а по-друге, не дає можливості приймати оптимальні конструктивні рішення. Тому мета статті – знайти аналітичні рішення для визначення опорів обертанню в механізмах повороту будівельних машин, що дозволяє проектувати досконаліші механізми й машини у цілому. Існуючі методики спираються на емпіричні залежності та експериментальні коефіцієнти, що зменшують точність підрахунків, збільшують габарити та вартість робіт. Пропонується підвищити точність та спростити процес визначення опору повороту й величину приведеного коефіцієнта опору обертанню будівельних баштових кранів. Методика. Досягти поставленої мети можна за допомогою аналітичних залежностей для визначення коефіцієнтів тертя кочення за лінійного й точкового контактів. Це дозволить точніше знайти величину коефіцієнта опору, а конструктору під час розрахунків вжити цілеспрямованих заходів щодо його зменшення, використовуючи механічні константи матеріалів вузлів кочення і їх геометричні параметри. Розрахунок ґрунтується на теорії контактних деформацій Герца й теорії плоского руху точок тіла. Результати. Отримані залежності дозволять аналітично знайти опір кочення роликів у будівельних машинах із нерухомою й обертовою колонами, із круговими поворотними пристроями, а також у кулькових і роликових опорно-поворотних кругах. З’ясовані значення коефіцієнтів опору обертанню для деяких їх типів механізмів дають близькі значення з рекомендованими, а для деяких – істотно відрізняються і вимагають їх уточнення у довідкових величинах. Наукова новизна роботи полягає у використанні аналітичних залежностей для визначення приведеного коефіцієнта опору обертанню за лінійного і точкового контактів із використанням теорії контактних деформацій Герца та частково аналітичних залежностей Табора. Практична значимість. Отримані залежності дозволять проектувати нові типи вузлів обертання механізмів повороту будівельних машин і виявляти додаткові опори обертанню.

Ключові слова: будівельна машина; опір; обертання; поворот; поворотний круг; рейка; тертя кочення

Л. Н. Бондаренко¹, А. П. ПосмИтюхА^2*, К. Ц. Главацкий³

¹Каф. «Прикладная механика и материаловедение», Днипровский национальный университет железнодорожного
транспорта имени академика В. Лазаряна, ул. Лазаряна, 2, Днипро, Украина, 49010, тел. +38 (056) 373-15-18,
эл. почта bondarenko-l-m2015@yandex.ua, ORCID 0000-0002-2212-3058
^2*Каф. «Прикладная механика и материаловедение», Днипровский национальный университет железнодорожного
транспорта имени академика В. Лазаряна, ул. Лазаряна, 2, Днипро, Украина, 49010, тел. +38 (066) 150 95 00,
эл. почта AleksandrP@3g.ua, ORCID 0000-0002-9701-3873
³Каф. «Прикладная механика и материаловедение», Днипровский национальный университет железнодорожного
транспорта имени академика В. Лазаряна, ул. Лазаряна, 2, Днипро, Украина, 49010, тел. +38 (095) 816 99 90,
эл. почта kazimir.glavatskij@gmail.com, ORCID 0000-0003-0921-9845

АНАЛИТИЧЕСКОЕ ОПРЕДЕЛЕНИЕ приведенного коэффициента

сопротивления вращению механизмов поворота

СТРОИТЕЛЬНЫХ МАШИН

Цель. Проектирование новых образцов строительных машин тесно связано с разработкой механизмов поворота, а те, в свою очередь, имеют привод, мощность и габариты которого зависят от сопротивления поворота и приведенного коэффициент трения в узлах. Отсутствие аналитических зависимостей для определения приведенного коэффициента трения вращению строительных машин, во-первых, ограничивает возможности конструктора в выборе материалов, а во-вторых, не дает возможности принимать оптимальные конструктивные решения. Поэтому цель статьи – найти аналитические решения для определения сопротивлений вращению в механизмах поворота строительных машин, которые позволят проектировать более совершенные механизмы и машины в целом. Существующие методики опираются на эмпирические зависимости и экспериментальные коэффициенты, уменьшающие точность подсчетов, увеличивающие габариты и стоимость работ. Предлагается повысить точность и упростить процесс определения сопротивления поворота и величину приведенного коэффициента сопротивления вращению строительных башенных кранов. Методика. Достичь поставленной цели можно с помощью аналитических зависимостей для определения коэффициентов трения качения при линейном и точечном контактах. Это позволит точнее найти величину коэффициента сопротивления, а конструктору при расчетах принять целенаправленные меры по его уменьшению, используя механические константы материалов узлов качения и их геометрические параметры. Расчет основывается на теории контактных деформаций Герца и теории плоского движения точек тела. Результаты. Полученные зависимости позволят аналитически найти сопротивление качению роликов в строительных машинах с неподвижной и вращающейся колоннами, с круговыми поворотными устройствами, а также в шариковых и роликовых опорно-поворотных кругах. Найденные значения коэффициентов сопротивления вращению для некоторых типов механизмов дают близкие значения с рекомендованными, а для некоторых – существенно отличаются и требуют их уточнения в справочных величинах. Научная новизна работы заключается в использовании аналитических зависимостей для определения приведенного коэффициента сопротивления вращению для линейного и точечного контактов с использованием теории контактных деформаций Герца и частично аналитических зависимостей Табора. Практическая значимость. Полученные зависимости позволят проектировать новые типы узлов вращения механизмов поворота строительных машин и выявлять дополнительные опоры вращению.

Ключевые слова: строительная машина; сопротивление; вращения; поворот; поворотный круг; рельс; трение качения

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Received: Sep. 06, 2018

Accepted: Jan. 16, 2019

Creative Commons Attribution 4.0 International

doi: 10.15802/stp2019/ © L. M. BondarenkO, O. P. Posmityukha, K. T. Hlavatskyi, 2019