INFLUENCE OF THE BALLS KINEMATICS AND BALL/RACE CONTACT MODELS
ON QUASI-STATIC APPROACHES FOR BALL BEARING
TRACK OR CATEGORY
Rolling element bearings
AUTHORS AND INSTITUTIONS
C. Servais, J.-L. Bozet, Faculty of Applied Science, University of Liège, Liège, Belgium
INTRODUCTION
The frictional power dissipated within dry lubricated ball bearings is a prime concern, especially for
high speed applications. Nevertheless, the exact balls behavior is still a source of interrogations. Firstly, a
comparison of several ball/race contact models has been performed by prescribing the kinematical
variables. This has been done by using a newly developed quasi-static approach to reach the ball bearing
equilibrium. Secondly, a parametric study has been carried out on the ball kinematics. This time the
contact model remained fixed.
BALL BEARING KINEMATICS AND QUASI-STATIC EQUILIBRIUM
Despite the fact that ball bearing geometry looks simple at first, the description of the ball bearing
behavior is not straightforward. More precisely, the kinematics appears to be one of the keys of the
problem regarding the description of the bearing dynamics, the ball/race contact tangential pressure or
the dissipated power.
The kinematics of a bearing ball is depicted in Fig. 1. Assuming that the inner race rotates at a speed
ω I and the outer race remains fixed, then the ball has a spinning speed ω S . Generally speaking, the
loading and the rotational speed of the bearing imply that the ball cannot exhibit a pure rolling motion on
both inner and outer races (contacts I and E in Fig. 1, respectively) at the same time. As a consequence,
the ball spin ω S makes an angle χ with the bearing axis. This position induces a pivoting component
between the ball and the races unless the angle χ takes one of the specific values χIRC or χORC . If
χ = χIRC , the pivoting component at contact I vanishes, leading to a pure rolling motion. This is the socalled inner race control. Conversely, if χ = χORC , the pure rolling motion only occurs at contact E . This
configuration is known as the outer race control. As detailed in [1], the angle χ is always comprises
between the two extreme values χORC and χIRC, assuming that the balls do not skid. The combination of
rolling and pivoting motions gives the slip speed field depicted in Fig. 1. x and y are respectively the
rolling and the transverse directions.
Concerning quasi-static approaches for studying the ball bearing equilibrium, let us mention that
recent studies, based on the description of Jones/Harris [2], still use inner or outer race control
hypotheses to access the kinematics. Indeed, once the kinematics is known, the gyroscopic torques and
the centrifugal forces applied on the bearing balls are available. However, as the present paper will show,
using race control hypotheses leads to huge errors. This is especially true regarding the dissipated power
and the tangential pressure located within ball/race contacts. The actual angle χ is calculated by using
the method developed in [1], method which removes race control hypotheses.
INFLUENCE OF THE BALL/RACE CONTACT MODEL
Normal pressure and surface areas of ball/race contacts are given by the well known Hertzian theory.
However, the Hertzian model does not give any information about the friction phenomena within contacts.
In this section, a comparison of three ball/race contact models is performed. Each of them necessitates a
discretization of the surface in small rectangular elements. The model Nº1 takes into account the
tangential elastic strains in the rolling direction but neglects the transverse slip; the model Nº2 considers
the transverse slip in both direction (x and y) but neglects elastic strains and the model Nº3 only
considers the slip along the x-axis without elastic strains. The friction is Coulombic in each case.
The influence of the contact model on the computation of the dissipated power density is depicted in
Fig. 2 for a ball/outer race contact. These examples are calculated for a dry lubricated 40 mm bore
diameter ball bearing. A thrust load of 300 daN is applied to the bearing with a DN value of 2 × 106. In Fig.
2, Pf is the dissipated power density, a the semi-major axis and b the semi-minor axis. As depicted in Fig.
2, the influence of the transverse slip is negligible (results produced by models Nº2 and Nº3 are similar).
Nevertheless, elastic strain effects are quite significant (compare model Nº1 with Nº2 and Nº3).
The method described in [1], giving access to the kinematics, requires a specific ball/race contact
model. Fig. 3 represents the influence of the model selected on the kinematics computation. The outer
race control percentage is plotted as a function of the inner ring rotational speed (100% = full outer race
control, 0% = full inner race control). As expected, the race control sharing is always located between the
inner and outer race controls. Fig. 3 also shows that the choice of the contact model is not crucial to
compute the kinematics. Transverse effects are negligible once again. Fig. 4 shows that the number of
elements chosen for the discretization process has not to be large to give satisfying results. This means
that the ball bearing equilibrium can be calculated with a model having a low accuracy, requiring low
computational resources. It is then possible to refine results obtained for the dissipated power or the
tangential pressure within contact by using more surface elements once the kinematics and the
equilibrium of the bearing are known.
INFLUENCE OF THE KINEMATICS
The previous section compared the dissipated power density for the three studied contact models in
the case of a specific bearing (see Fig. 2). In that example, the comparison has been performed for the
kinematics computed with the method available in [1]. Fig. 5 shows what the dissipated power density
becomes if the kinematics is not correctly calculated, viz. if inner or outer race controls are considered
instead of the actual kinematics. The results have been obtained with the contact model Nº2 for the ball
bearing presented in the previous section. As depicted in Fig. 5, the dissipated power density changes a
lot with the sharing control of the rings. This means that, no matter the contact model selected, the
kinematics must be properly computed before determining the dissipated power or tangential pressure
within contacts. This is especially true for cryogenic applications, like rocket engine turbopump [3].
Outer ring
x
E
B
×
y
Slip speed field
I
ωS
χ
χIRC
Bearing axis
Figure 1 — Kinematics of a bearing ball
!
'""
!$"
ï"#$
!""
Model Nº1
"
%""
!$"
ï"#$
!""
"#$
%$"
"
Model Nº2
"
%""
!$"
ï"#$
Pf = dissipated power
a = ellipse major-axis
b = ellipse minor-axis
!""
$"
$"
ï!
ï! " !
x/b
&$"
&""
y/a
%$"
"
$"
ï!
ï! " !
x/b
&$"
&""
y/a
%""
Pf [W/mm2 ]
y/a
%$"
"
"#$
'""
Pf [W/mm2 ]
&$"
&""
Inner ring
!
!
'""
"#$
ωI
Pf [W/mm2 ]
χORC
ï!
ï! " !
x/b
Model Nº3
"
Figure 2 — Example of dissipated power density, comparison of the three contact models
90
85
85
75
70
Model 1
Model 2
Model 3
65
4
6
8
Inner ring rotational speed [rpm]
ï"#$
"#$
!$"
!""
$"
4
6
8
Inner ring rotational speed [rpm]
!
!%""
&$"
&""
%$"
"
%""
!$"
ï"#$
!""
"#$
Outer race control
ï!
ï! " !
x/b
"
Real kinematics
!"""
(""
"
ï"#$
'""
Pf = dissipated power d
a = ellipse major-axis
b = ellipse minor-axis
&""
%""
$"
ï!
ï! " !
x/b
10
4
x 10
Figure 4 — Evolution of the outer race control as a
function of the bearing rotational speed, comparison
of the contact discretization (model Nº2)
'""
y/a
"
2
!
%""
Pf [W/mm2 ]
y/a
"#$
234 surface elements
494 surface elements
5174 surface elements
65
10
4
x 10
Figure 3 — Evolution of the outer race control as a
function of the bearing rotational speed, comparison
of the three contact models
!
70
y/a
2
75
Pf [W/mm2 ]
60
80
Pf [W/mm2 ]
80
Outer race control [%]
Outer race control [%]
90
ï!
ï! " !
x/b
"
Inner race control
Figure 5 — Influence of kinematics on dissipated power density (model Nº2)
CONCLUSIONS
The link between ball/race contact models and kinematics of dry lubricated ball bearing has been
investigated in this paper. The results demonstrate the predominance of the kinematics on the ball
bearing behavior with an emphasis on the dissipated power within ball/race contacts. This shows that the
kinematics must be mandatorily rightly computed. On the other hand, the ball/race contact model doesn’t
necessitate a high refinement in order to evaluate the ball bearing equilibrium. Conversely, this refinement
is essential for a precise evaluation of the power losses or tangential pressure within bearing contacts.
REFERENCES
[1] Bozet, J.-L., and Servais, C., 2016. “Influence of the balls kinematics of axially loaded ball bearings on
Coulombic frictional dissipations”. ASME Journal of Tribology (In press).
[2] Harris, T. A., 1991. Rolling bearing analysis, third ed. John Wiley & Sons.
[3] Servais, C., Bozet, J.-L., Kreit, P., and Guingo, S., 2014. “Experimental validation of a thermal model of
a LOx flooded ball bearing”. Tribology International, 80, pp. 71–75.
KEYWORDS
Ball Bearings, Contact Mechanics, Rolling Friction