epartment of Mechanical Engineering,TSI-BILBAO,niversity of the Basque Country,8013 Bilbao, Spain
his paper presents a calculation of the general static load-arrying capacity of four-contact-point slewing bearings underxial, radial, and tilting-moment loads. This calculation is basedn a generalization of Sjoväll and Rumbarger’s equations androvides an acceptance surface in the load space. This acceptanceurface provides a solid basis to compute acceptance curves forhe design and selection of bearings of this kind.DOI: 10.1115/1.4001600�
IntroductionSlewing bearings are large-sized bearings with a wide field of
pplications, such as in wind turbine generators, tower cranes, andertical lathe tables. In general, they are used in machines, whichontain large rotational functional elements. Usually, these bear-ngs are driven and so they contain gears in the inner or outer ring.
There are many different types of slewing bearings available onhe market. They differ from each other in the number of rows andn the type of rolling elements. Thus, there are bearings with oner two rows and the rolling elements can be balls, tapered rollers,r cylindrical rollers. Some of these types are shown in Fig. 1.
The loads acting on these bearings usually contain axial andadial forces, as well as tilting moments �see Fig. 2�. In the mostnfavorable load case, the radial force is perpendicular to theesultant of the tilting moments. Several bearing manufacturersrovide acceptance curves that allow one to determine whether orot a bearing is acceptable for a given equivalent load, calculateds a combination of the axial and radial loads. By means of aoment-axial-force diagram, this equivalent load allows a de-
igner to obtain the maximum allowable tilting-moment that theearing can bear. This is illustrated in Fig. 3.
There are some variations in the form and limits of the diagramhown in Fig. 3. These variations are due to the manufacturersaving experimented with or assessed the bearings themselves oraving simply copied other manufacturers’ data. Anyhow, there islways a certain ambiguity and a lack of a clear criterion in the
Contributed by the Design Automation Committee of ASME for publication in theOURNAL OF MECHANICAL DESIGN. Manuscript received November 24, 2009; finalanuscript received March 28, 2010; published online May 25, 2010. Assoc. Editor:
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definition of the equivalent load. The objective of this paper is thedevelopment of a procedure, which defines a surface by the lim-iting values of the loads FA, FR, and M for four-contact-pointslewing bearings. This representation can be used directly to de-termine whether or not a given load combination is acceptable.
There have been some previous publications where conceptsrelevant to the assessment of the static load-carrying capacity offour-contact-point slewing bearings have been examined. Amasor-rain et al. �1� developed a procedure to work out the load distri-bution in bearings of this type subjected to axial and radial forcesand tilting moments. Liao and Lin �2� developed a similar proce-dure in which only axial and radial forces were taken into account.Both of these procedures are similar to the procedure that Zupanand Prebil �3� used to estimate the influence of geometrical andstiffness parameters on the calculation of the load-carrying capac-ity �3�. Other works have also been done on these topics �4,5�. Allof the above papers propose a generalization of the equationsobtained by Jones �6� in which the load distribution is worked outfrom the known external loads, taking account of the variation incontact angle with the loading conditions.
This paper has a different focus, consisting in directly calculat-ing the load combinations that result in static failure �as defined inthe ISO standard �7�� of the most loaded element. This allows oneto obtain a three-dimensional acceptance condition in the form ofa surface inequation. The designer can use this acceptance surfaceas a straightforward way to select a bearing appropriately. Thisapproach is based on the calculations of Sjoväll �8� for combina-tions of axial and radial loads and of Rumbarger �9� for combina-tions of axial and moment loads. These calculations assume zeroclearance in the contact and rigid rings. These assumptions arealso made in the current paper. The axial load-carrying capacity isused to normalize the results and can be obtained from standards�7,10�. Some manufacturers use experimentation to fine-tune thisvalue, taking material quality and geometrical parameters into ac-count.
2 MethodThis section presents the procedure that leads to the three-
dimensional condition of acceptance for the bearing. First, amodel of geometrical-interference is formulated and then equa-tions that reflect the equilibrium of the forces and moments areworked out. Finally, the equilibrium equations are rewritten toprovide an acceptance inequation.
2.1 Geometrical-Interference Model. Figure 4 shows thegeometrical-interference between the rolling elements and therings in a four-contact-point slewing bearing. To simplify the fig-ure, the raceways have been supposed to be conical instead oftoroidal and it has been assumed that only the inner ring can bedisplaced, while the outer one remains fixed.
The geometrical-interference fields ��1 and ��
2 can be expressedas follows, measured as displacements perpendicular to raceways1 and 2:
��1 = �a sin � + �r cos � cos � +
�d
2sin � cos � �1�
��2 = − �a sin � + �r cos � cos � −
�d
2sin � cos � �2�
We define parameters A, R, and M as follows:
A = �a sin �
R = �r cos � �3�
M =�d
2sin �
Using Eq. �3�, Eqs. �1� and �2� can be written as
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Tc
Ttftim
Ficr
Fs
0
Downloaded Fr
��1 = A + �R + M�cos � �4�
��2 = − A + �R − M�cos � �5�
he maximum and minimum values of the interference fields oc-ur when �=0 and �=� so that
�01 = A + R + M
��1 = A − R − M
�6��0
2 = − A + R − M
��2 = − A − R + M
he assessment of the load-carrying capacity is conditioned bywo factors. On the one hand, the point where the greatest inter-erence appears must be detected �at �=0 or �=�� so that we canhen calculate the field of interference for the maximum load �thiss critical since the load-carrying capacity is calculated for the
ost loaded ball�. On the other hand, obviously, a load exists only
ig. 1 Different types of slewing bearings „from Iraundi Bear-ngs…: „a… four-contact-point single row ball bearing, „b…rossed cylindrical roller bearing, „c… four-contact-point doubleow ball bearing, and „d… three-row cylindrical roller bearing
Fig. 2 Loads acting on a slewing bearing
ig. 3 Moment–axial-force diagram for a four-contact-point
lewing bearing
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for those values of � for which the interference is positive. Theinterference field must be formulated based on these ideas. Thisfield will depend on both the maximum value of the interferenceand the zones of positive interference. In accordance with this, werewrite Eqs. �1� and �2� in terms of the values of the interferencefor �=0 and �=�.
��1 =
1
2���0
1 + ��1 � + ��0
1 − ��1 �cos �� �7�
��2 =
1
2���0
2 + ��2 � + ��0
2 − ��2 �cos �� �8�
Table 1 shows the five possible interference fields together withthe conditions and equations to be used. It also shows, which ofthe extreme values of the interference is the maximum one ��0 or��� and which angle limits the positive interference zone in eachcase.
2.2 Force and Moment Equilibrium. Using the expressionsdeveloped by Sjoväll �8� and Rumbarger �9�, force and momentequilibrium equations can be written as shown below, where thecontributions of raceways 1 and 2 are added.
FA
Z sin �= QMAX
1 JA��01,��
1 � − QMAX2 JA��0
2,��2 � �9�
FR
Z cos �= QMAX
1 JR��01,��
1 � + QMAX2 JR��0
2,��2 � �10�
M/dZ sin �
= QMAX1 JM��0
1,��1 � − QMAX
2 JM��02,��
2 � �11�
Here, JA, JR, and JM are the following integrals:
JA��0,��� =1
2�� � ��
�MAX�3/2
d� �12�
Fig. 4 Geometrical-interference model
�+
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Tam
w
J
Downloaded Fr
JR��0,��� =1
2��
�+
� ��
�MAX�3/2
cos �d� �13�
JM��0,��� =1
4��
�+
� ��
�MAX�3/2
cos �d� �14�
hese integrals must be evaluated for every case in Table 1 inccordance with the values given in Eq. �6�. Also, Eqs. �9�–�11�ust be rewritten as a function of the maximum load.
FA
QMAXZ sin �= FQ
1 JA��01,��
1 � − FQ2 JA��0
2,��2 � �15�
FR
QMAXZ cos �= FQ
1 JR��01,��
1 � + FQ2 JR��0
2,��2 � �16�
M/dQMAXZ sin �
= FQ1 JM��0
1,��1 � − FQ
2 JM��02,��
2 � �17�
Table 1 The five different ca
here
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if��01 = max�0
1,��1 ,�0
2,��2
or
��1 = max�0
1,��1 ,�0
2,��2 �then� FQ
1 = 1
FQ2 = �max�0
2,��2
max�01,��
1 �3/2 �
if��02 = max�0
1,��1 ,�0
2,��2
or
��2 = max�0
1,��1 ,�0
2,��2 �then�FQ
1 = �max�01,��
1 max�0
2,��2 �3/2
FQ2 = 1
��18�
3 Acceptance Surface of the BearingThe maximum load is expressed as a function of the axial load-
carrying capacity. This is done in order to represent graphicallythe values of FA, FR, and M that cause permanent deformation in
of geometrical-interference
ses
the most loaded ball, as detailed in Ref. �7�. We have
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W
wd�bea
4
s�i
e p
0
Downloaded Fr
QMAX =C0a
Z sin ��19�
hen we substitute Eq. �19� into Eqs. �15�–�17�, we obtain
FA
QMAXZ sin �=
FA
C0a
FR
QMAXZ cos �=
FR
C0atan � �20�
M/dQMAXZ sin �
=M/dC0a
hich can be seen as the coordinates of a point in a three-imensional diagram with axes FA /C0a, �FR /C0a�tan �, andM /d� /C0a. When we study different interference cases definedy �A ,R ,M� according to Eq. �6� and solve Eqs. �15�–�17� forach case, the final result is a cloud of points that define thecceptance surface.
Results and DiscussionIt is difficult to represent the acceptance surface graphically
ince it must be mapped with the parametric coordinatesA ,R ,M�; each set of parametric coordinates results in another setn the coordinate system �FA /C0a, FR tan � /C0a, �M /d� /C0a�.
Fig. 5 Acceptanc
Fig. 6 Rendered triangulation
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However, the surface can be represented, for instance, by meansof a Non Uniform Rational B-Splines �NURBS� approximation tothe resulting cloud of points.
A MAPLE �R� application has been developed to assess thecloud of points with various levels of accuracy N in order to mapthe surface, as follows:
� A�i� = i
R�i� = i
M�i� = i� i = − N . . . N �21�
This is done in such a way that all possible combinations arechosen, resulting in �2N+1�3 points. The cloud of points is repre-sented in Fig. 5 for N=20 �68,921 points� and Fig. 6 shows arendered triangulation of the resulting surface. The points near thecoordinate planes are also represented in Figs. 7–9.
In Eq. �21�, the parameters A, R, and M are real values varyingfrom −� to �. Each value has a direct correspondence in the�FA /C0a , FR tan � /C0a , M /d /C0a� coordinate system. Ac-cording with Eq. �3�, the variation range of A, R, and M would besimilar to the variation range of �a, �r, and �, and one can thinkthat very large values for these displacements would not be real-istic. However, the integrals �12�–�14� reveal that the key variableis �� /�MAX, i.e., a relation among the values of A, R, and M andnot their absolute values. Therefore the adoption of very large
oint cloud. N=20
of the acceptance surface
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vma
cItab
Fig. 9 Acceptance curve in the FA−FR plane
J
Downloaded Fr
alues for these parameters results irrelevant for the calculationethod, although one could think that those deformation values
re physically unattainable.The validity of the method has been verified by, for instance,
hecking the intersection of the surface with the coordinate axes.n fact, as can be found using the results in Refs. �11,12� appliedo a four-contact-point slewing bearing, the limiting values of thexial and radial forces and tilting-moment should be determinedy
FA,max = C0a
FR,max =2
4.37C0a = 0.4577C0a �22�
Fig. 7 Acceptance curve in the FA−M plane
Fig. 8 Acceptance curve in the FR−M plane
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Fig. 10 Acceptance curves in the FA−FR plane for various val-
ues of the normalized moment
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TooItsRwbei
5
dsptaf
6
lbrveatfstvim
0
Downloaded Fr
Mmax =d
4.37C0a = 0.2288C0ad
hese values match up with the ones given by the method devel-ped in this paper �see Figs. 7–9�. Nevertheless, it must be pointedut that the curves are slightly different away from these values.n this sense, the method developed in this paper is more completehan those presented in Refs. �8,9� and it should not be seen as auperposition of the results of those methods. The reason is that inef. �8�, only axial- and radial force equilibrium is assumed,hereas the moment generated by axial nonuniform loading is notalanced, and in Ref. �9�, only axial-force and tilting-momentquilibrium is assumed, whereas the axially generated radial forces not balanced.
Curve-Fitting ApproximationAlthough we intend to construct surface approximations in or-
er to systematically assess equivalent loadings in future work, wehow in Fig. 10 some curve fits in the FA /C0a , FR tan � /C0alane for some positive values of �M /d� /C0a in order to clarifyhe structure of the points calculated with Eq. �20�. These curvesre the limits of the axial and radial normalized loads when dif-erent values of the normalized moment are imposed.
Concluding Remarks and Future WorkThis paper has presented a procedure for the assessment of the
imiting values of the loads acting on a four-contact-point slewingearing. The loads are an axial-force, a tilting-moment, and aadial force perpendicular to that moment �this is the most unfa-orable load case�. These limiting values are obtained by consid-ring the equilibrium of the forces and moments in the inner ring,nd then equating the maximum load to the value obtained fromhe axial load-carrying capacity. This procedure results in a sur-ace formed by a cloud of points in a three-dimensional coordinateystem whose axes are the axial load, the radial load and theilting-moment. A designer can use this acceptance surface to pro-ide a straightforward way to select a bearing. When this methods applied to simpler models found in the literature, the results
atch up.
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Further work remains to be done, using multiparametricNURBS surfaces to assess equivalent loads in order to systemati-cally obtain curves for the selection of bearings. We also believethat the methodology will be applicable to other types of bearings,such as crossed roller bearings and three-row cylindrical rollerbearings.
AcknowledgmentThis paper is a result of the close collaboration that the authors
maintain with the companies Iraundi Bearings and LKS Engineer-ing. The authors are grateful for the dedication and generositywith which Iraundi has provided its know-how for this work andfor the professional support of LKS.
References�1� Amasorrain, J. I., Sagartzazu, X., and Damián, J., 2003, “Load Distribution in
a Four Contact-Point Slewing Bearing,” Mech. Mach. Theory, 38, pp. 479–496.
�2� Liao, N. T., and Lin, J. F., 2001, “A New Method for the Analysis of Defor-mation and Load in a Ball Bearing With Variable Contact Angle,” ASME J.Mech. Des., 123, pp. 304–312.
�3� Zupan, S., and Prebil, I., 2001, “Carrying Angle and Carrying Capacity of aLarge Single Row Ball Bearing as a Function of Geometry Parameters of theRolling Contact and the Supporting Structure Stiffness,” Mech. Mach. Theory,36, pp. 1087–1103.
�4� Antoine, J. F., Abba, G., and Molinari, A., 2006, “A New Proposal for ExplicitAngle Calculation in Angular Contact Ball Bearing,” ASME J. Mech. Des.,128, pp. 468–478.
�5� Hernot, X., Sartor, M., and Guillot, J., 2000, “Calculation of the StiffnessMatrix of Angular Contact Ball Bearings by Using the Analytical Approach,”ASME J. Mech. Des., 122, pp. 83–90.
�6� Jones, A., 1946, “Analysis of Stresses and Deflections,” New Departure Engi-neering Data, Bristol, CT.
�7� International Organization for Standardization, 2006, “Rolling Bearings—Static Load Ratings,” 3rd ed., ISO 76:2006.
�8� Sjoväll, H., 1933, “The Load Distribution Within Ball and Roller BearingsUnder Given External Radial and Axial Load,” Teknisk Tidskrift, Mek., h.9.
