A19_362

13th World Congress in Mechanism and Machine Science, Guanajuato, México, 19-25 June, 2011 A19_362

1

General Static Load Capacity in Slewing Bearings. Unified Theoretical Approach for Crossed Roller Bearings and Four Contact

Point Angular Ball Bearings.

J. Aguirrebeitia* M. Abasolo† R. Aviles‡ I. Fernandez§ ETSIB, UPV/ EHU ETSIB, UPV/ EHU ETSIB, UPV/ EHU ETSIB, UPV/ EHU

Bilbao, Spain Bilbao, Spain Bilbao, Spain Bilbao, Spain

Abstract— this work presents a unified approach to obtain the general static load-carrying capacity for two kinds of slewing bearings: crossed roller bearings and four contact point angular bearings. The bearings are loaded with an axial force, a radial force and a tilting-moment. This approach is based on a generalization of Sjoväll and Rumbarger’s equations and provides an acceptance surface in the load space. This acceptance surface provides a solid basis to compute acceptance curves for the design and selection of these bearings.

Keywords: slewing bearing, ball bearing, roller bearing, load capacity.

I. Introduction

Slewing bearings are large-sized bearings with a wide field of applications, such as in wind turbine generators, tower cranes, tanks, vertical lathe tables… generally, they are used in cases in which large rotational functional elements are involved.

Usually, these bearings are driven, and so they contain gears in the inner or outer ring, and the two rings use to be bolted. There are many different types of slewing bearings which differ from each other in the number of rows and in the type of rolling elements. Thus, there are bearings with one or two rows, and the rolling elements can be balls, tapered rollers or cylindrical rollers. In Fig. 1, the two types of slewing bearings considered in this work appear, as well as the loads acting on them (axial and radial forces, as well as tilting moments). In the most unfavorable load case, the radial force is perpendicular to the resultant of the tilting moments.

Several bearing manufacturers provide acceptance curves that allow one to determine whether or not a bearing is statically acceptable for a given equivalent load, calculated as a combination of the axial and radial loads. By means of a moment–axial-force diagram, this equivalent load allows a designer to obtain the maximum allowable tilting moment that the bearing can bear statically. This is illustrated in Figure 2. 1

There are some variations in the form and limits of the diagram shown in Figure 2. These variations are due to the manufacturers having experimented with or assessed

* [email protected] † [email protected] ‡ [email protected] § [email protected]

the bearings themselves, or having simply copied other manufacturers’ data. Anyhow, there is always a certain ambiguity and a lack of a clear criterion in the definition of the equivalent load.

a) Four contact point bearing b) Crossed roller bearing

c) Setup of a slewing bearing

Fig. 1. Slewing bearings.

Fig. 2. Moment–axial-force diagram.

The objective of this work is to give a common frame to arrange the general static load capacity for crossed roller bearings and four contact point angular bearings, adapting the procedure developed in [1] and reformulated in [2] for the last one. In that work, a procedure defining a surface with the limiting values of the loads FA, FR and MT was developed. This representation can be used directly to determine whether or not a given load combination is acceptable.

There have been some previous publications where concepts relevant to the assessment of the static load-carrying capacity of slewing bearings have been

MT

FA

C0ad/n

Feq

Mmax

C0a

C0a: static axial load rating d: ball centre diameter n = 4.37 (ball bearings)

= 4.06 (roller bearings)

Inner Holes

Outer Holes

Gear Teeth

MT

FA

FR


2

examined, mainly for the case of four-contact-point bearings. Amasorrain et al. [3] developed a procedure to work out the load distribution in bearings of this type subjected to axial and radial forces and tilting moments. Liao and Lin [4] developed a similar procedure, in which only axial and radial forces were taken into account. Both of these procedures are similar to the procedure that Zupan and Prebil [5] used to estimate the influence of geometrical and stiffness parameters on the calculation of the load-carrying capacity. Other work has also been done on these topics [6, 6]. All of the above papers propose a generalization of the equations obtained by Jones [7], in which the load distribution is worked out from the known external loads, taking account of the variation in contact angle with the loading conditions. Our work has a different focus, consisting in directly calculating the load combinations that result in static failure (as defined in the ISO standard [9]) of the most loaded element. This allows one to obtain a three-dimensional acceptance condition in the form of a surface inequation. The designer can use this acceptance surface as a straightforward way to select a bearing appropriately. This approach is based on the calculations of Sjoväll [10] for combinations of axial and radial loads and of Rumbarger [11] for combinations of axial and moment loads. These calculations assume zero clearance in the contact, and rigid rings. These assumptions are also made in the current paper. The axial load-carrying capacity is used to normalize the results and can be obtained from standards [9, 12]. Some manufacturers use experimentation to fine-tune this value, taking material quality and geometrical parameters into account.

In previous publications [1,2] the authors developed the theoretical procedure to establish a three dimensional acceptance condition for the selection of four contact point slewing bearings. In this work that approach is generalized and unified also for crossed roller slewing bearings.

II. Common geometrical features

Both types of bearings share some geometrical properties. In Fig. 3 one can observe that in both cases a contact angle α is defined, and if rollers are considered pairwise, four contact areas are identified in both bearings. Besides, the inner ring contact diameter d is considered for both. So, from now on, the only difference between these two types of bearings will be the power of the hertzian contact relationship between applied force and deformation if rollers are considered pairwise.

III. Theoretical approach

This section summarizes the procedure that leads to the three-dimensional condition of acceptance for the bearings. First, a model of geometrical interference is

formulated, and then equations that reflect the equilibrium of the forces and moments are worked out.

a) Four contact point angular slewing bearing

b) Crossed roller slewing bearing

Fig. 3. Geometrical interference model.

Finally, the equilibrium equations are rewritten to provide an acceptance inequation. Fig. 3 shows the geometrical interference between the rolling elements and the raceways 1 and 2 in a four-contact-point slewing bearing, which can be analytically expressed as in (1) appears.

Regarding with force and moment equilibrium, adapting from expressions developed by Sjoväll [10] and Rumbarger [11], as appear in [13-14], force and moment equilibrium equations can be written as shown in (2), where the contributions of raceways 1 and 2 are added.

ψ

α

raceway 1

raceway 2

d

δa

δr θ

αraceway 1

raceway 2

ψ

d

δa

δr θ


3

sin 2

cossin and

where

2

cos1

2

cos1

2

2 0

1

1 0

2

1

2 0

1 0

2

1

dMRA

MRA

MRA

MRA

MRA

Tra

(1)

2

1

21

21

21

sin

cos

sin 1

Q

Q

MM

RR

AA

R

A

MAX F

F

JJ

JJ

JJ

dM

F

F

ZQ

(2)

Table 1 shows the five different interference cases that can happen for each raceway, along with the reformulation of the interference fields over their maximum values.

INTERFERENCE CASES

CONDITIONS AND EQUATIONS

1

0

0

0

0

0 MAX l

cos1

2

11

2

1

000

20

0

0

0

MAX l

cos12

1 1

2

1 00

3 0

00

0 MAX

0

01cos

l

cos1

2

11

2

1

000

4 0

00

MAX

0

01cos

l

cos12

1 1

2

1 00

5 0

00

0 MAX 0l

0

TABLE 1. Five cases of geometrical interference.

where:

1

,max

,max then

,,,max

or

,,,max

if

,max

,max

1

then

,,,max

or

,,,max

if

2

22 0

11 01

22 0

11 0

2

22 0

11 0

2 0

11 0

22 02

1

22 0

11 0

1

22 0

11 0

1 0

Q

q

Q

q

Q

Q

F

F

F

F

(3)

and

dJJ

dJJ

dJJ

q

MAX

iiii

Mi M

q

MAX

iiii

Ri R

q

MAX

iiii

Ai A

cos4

1,

cos2

1,

2

1,

0

0

0

(4)

where q=3/2 for ball bearings and q=29/27 for roller bearings. Parameter q is called here the bearing type parameter. The maximum load can be expressed as a function of the axial load-carrying capacity. This is done in order to represent graphically the values of FA, FR and MT that cause permanent deformation in the most loaded ball, as detailed in [9]. We have:

sin0

Z

CQ a

MAX (5)

When we substitute equation (5) into equation (2), we obtain:

a

T

MAX

T

a

R

MAX

R

a

A

MAX

A

C

dM

ZQ

dM

C

F

ZQ

F

C

F

ZQ

F

0

0

0

sin

tan cos

sin

(6)

which can be seen as the coordinates of a point in a three-dimensional diagram, with axes FA/C0a, (FR/C0a)tanα and (MT/d)/C0a. When we study different interference combinations defined by parameters (A, R, M) according to equation (1) and solve equations (2) for each one, the final result is a cloud of points that define the acceptance surface.

IV. Results of the theoretical approach

In Fig. 4 the point clouds are represented for q=3/2 and q=29/27, in the {FA/C0a, FRtanα/C0a} plane for some positive intervals of MT/d/C0a and in Figure 5 shows them in the {FRtanα/C0a, MT/d/C0a } plane for some positive intervals of FA/C0a.

δ0

δπ

δ0

δπ

ψl

-ψl

δ0

δπ

ψl-ψl

δ0

δπ

δ0

δπ


4

a) q=3/2. Point cloud in the load space of a four contact point angular slewing bearing

b) q=29/27. Point cloud in the load space of a crossed roller slewing bearing

Fig. 4. Cloud of points forming the acceptance surface. {FA/C0a, FRtanα/C0a} plane


5

a) q=3/2. Point cloud in the load space of a four contact point angular slewing bearing a)

b) q=29/27. Point cloud in the load space of a crossed roller slewing bearing

Fig. 5. Cloud of points forming the acceptance surface. {FRtanα/C0a, MT/d/C0a } plane


6

Figures 4 and 5 show that the structure of both point clouds is quite similar.

V. Finite element results for q=3/2

In order to check the influence of the flexibility of the rings in four contact point angular slewing bearings, the authors presented in previous work some preliminary [15] and detailed [16] finite element calculations of a medium-sized bearing. In the present work a finite element calculation is to be presented for a smaller four contact point bearing, just for testing the match between theoretical results and those given by the finite element model (It must be noted that the smaller the bearing the stiffer it will be, if cross sectional data remains constant and only diameters change). Figure 6 shows the finite element model of the bearing along with its dimensions:

Fig. 6. Geometrical and Finite element model of a small four contact

point angular slewing bearing.

The rolling elements were modeled with the experimentally validated procedure developed in [17] by

the “Laboratoire de Génie Mécanique de Toulouse”, based in the adaptation of the Hertz formulas by Houpert [18].

Fig. 7. FEM results and Theoretical results in {FRtanα/C0a, MT/d/C0a } plane for various values of FA/C0a

Fig. 8. FEM results and Theoretical results in {FA/C0a, MT/d/C0a } and {FA/C0a, FRtanα/C0a}

FA/C0a=0

FA/C0a=0,2

FA/C0a=0,4

FA/C0a=0,6

FA/C0a=0,8

212,5

168142Φ14

10

46

46

215,5260

286

Φ22 ?

Φ14

FRtanα/C0a=0

MT/d/C0a=0


7

In figures 7 and 8, a comparison is made between the theoretical solution and the finite element solution. It should be noted that all the actions are normalized with respect to the axial load capacity. This load capacity might be calculated with the finite element model rather than with standards, due to the high variation of the contact angle with the load, which increases it greatly. However, even if the theoretical model does not consider any change in contact angle, if the axes are normalized with respect to the axial load capacity given by the finite element analysis (which takes into account this variation), figures 7 and 8 show an amazing correspondence between theoretical and finite element calculations. From this fact we can conclude that the contact angle variation has not a decisive influence in the load distribution.

VI. Future work. Finite element calculations for q=29/27

The authors are working now in a parametric finite element model of crossed roller slewing bearings in order to check the deviations between the theoretical model and the finite element model. Each rolling element-raceway contact has been modeled via a parallel set of 6 nonlinear compression-only springs in order to check partiality in the line contact. Some preliminary results have shown that the theoretical model fits extraordinarily well the finite element results. In this sense, it must be considered that in this case the contact angle does not vary with the applied load and therefore the load distribution predicted by the theoretical model for a roller bearing is more accurate than in the case of a ball bearing, though in some situations and for some rolling elements only partial line contact arises.

Conclusions

In this work a unified theoretical approach for crossed roller slewing bearings and four contact point angular slewing ball bearings has been presented, in order to arrange their general static load capacity. This general static load capacity is represented by a surface in the load space defined by an axial load, a radial load and a tilting moment. This supposes a generalization of previous work done by the authors in the field of four contact point ball bearings.

In order to prove the adequation of the theoretical model, some finite element calculations have been presented for a small-sized and very stiff four contact point ball bearing. The theoretic-experimental correlation has been successful. Some work remains to be done with

finite element models of crossed roller slewing bearings to fully complete the aim of the research.

Acknowledgments

This work is a result of the close collaboration that the authors maintain with company IRAUNDI S.A. The authors are grateful for the dedication and generosity with which Iraundi has provided its know-how for this work.

References

[1] Aguirrebeitia, J., Abasolo, M., Avilés, R. Fernandez de Bustos, I. General Static Load Capacity in Four Contact Point Slewing Bearings. Third International Conference on Tribology and Design, Algarve, Portugal. May 2010.

[2] Aguirrebeitia, J., Abasolo, M., Avilés, R. Fernandez de Bustos, I. Calculation of General Static Load-Carrying Capacity for the Design of Four Contact-Point Slewing Bearings. ASME Journal of Mechanical Design, 132, 064501, 2010.

[3] Amasorrain, J.I., Sagartzazu, X., Damián, J., Load distribution in a four contact-point slewing bearing. Mechanism and Machine Theory, 38, 479–496, 2003.

[4] Liao, N.T., Lin, J.F., A new method for the analysis of deformation and load in a ball bearing with variable contact angle. Journal of Mechanical Design, 123, 304–312, 2001.

[5] Zupan, S., Prebil, I., Carrying angle and carrying capacity of a large single row ball bearing as a function of geometry parameters of the rolling contact and the supporting structure stiffness. Mechanism and Machine Theory, 36, 1087–1103, 2001.

[6] Antoine, J.F., Abba, G., Molinari, A., A new proposal for explicit angle calculation in angular contact ball bearing. Journal of Mechanical Design, 128, 468–478, 2006.

[7] Hernot, X., Sartor, M., Guillot, J., Calculation of the stiffness matrix of angular contact ball bearings by using the analytical approach. Journal of Mechanical Design, 122, 83–90, 2000.

[8] Jones, A., Analysis of Stresses and Deflections. New Departure Engineering Data, Bristol, CT, 1946.

[9] International Organization for Standardization, Rolling Bearings – Static Load Ratings, ISO 76:2006, 3rd edition, 2006.

[10] Sjoväll, H. The load distribution within ball and roller bearings under given external radial and axial load. Teknisk Tidskrift, Mek., h.9, 1933.

[11] Rumbarger, J., Thrust bearing with eccentric loads. Mach. Des., February, 1962.

[12] International Organization for Standardization, Explanatory Notes on ISO 76, ISO/TR 10657:1991, 1st edition, 1991.

[13] Harris, T.A., Kotzalas, M.N., Rolling Bearing Analysis: Essential Concepts of Bearing Technology. Taylor & Francis/CRC Press, 2007.

[14] Harris, T.A., Kotzalas, M.N., Rolling Bearing Analysis: Advanced Concepts of Bearing Technology. Taylor & Francis/CRC Press, 2007.

[15] Aguirrebeitia, J., Abasolo, M, Avilés, R., Ansola, R., Fernandez de Bustos, I., Static load carrying capacity if four contact point slewing bearings. Theoretical and preliminary finite element calculations. IMECE2010, Vancouver, Canada, Nov 2010.

[16] Aguirrebeitia, J., Abasolo, M, Ansola, R, Vallejo, J, Fernandez de Bustos, I., Modelo detallado para análisis estático de conjuntos de rodamiento de vuelco, XVIII Congreso Nacional de Ingeniería Mecánica, Ciudad Real, Spain ,Nov 2010

[17] Daidié, A., Chaib, Z., Ghosn, A., 3D Simplified Finite Elements Analysis of Load and Contact Angle in a Slewing Ball Bearing, Journal of Mechanical Design, 130, 082601, 2008.

[18] Houpert, L., An Engineering Approach to Hertzian Contact Elasticity—Part I. J. Tribol., 123(3), 582 (7pg), 2001.

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