Two first order equations are given connecting geometry and internal motions in an angular contact ball bearing. Total speed, kinematic equivalence, basic speed ratio, and modal speed ratio are defined and discussed; charts are given for the speed ratios covering all bearings and all rotational modes. Instances where specific first order assumptions might fail are discussed, and the resulting effects on bearing performance reviewed

Kingbury, E.

English

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NASA Technical Memorandum 83592
First Order Ball Bearing Kinematics
Edward Kingsbury
Lewis Research Center
Cleveland, Ohio
Prepared for the
Annual Meeting of the American Society of Lubrication Engineers
Chicago, Illinois, May 7-10, 1984
NASA
https://ntrs.nasa.gov/search.jsp?R=19840011748 2020-03-21T00:04:40+00:00Z
FIRST ORDER BALL BEARING KINEMATICS
Edward Kingsbury*
National Aeronautics and Space Administration
Lewis Research Center
Cleveland, Ohio 44135
ABSTRACT
Two first order equations are given connecting geometry and internal
motions in an angular contact ball bearing. Total speed, kinematic equiva-
lence, basic speed ratio, and modal speed ratio are defined and discussed;
charts are given for the speed ratios covering all bearings and all rotational
2 modes. Instances where specific first order assumptions might fail are dis-
1—4
uli cussed, and the resulting effects on bearing performance reviewed.
INTRODUCTION
First order kinematic analysis is concerned with the relations between
the sizes and the motions of balls and races in a bearing. The equations
relating race rotations to ball and ball group rotations ("speed ratios") are
usually given separately for the two specific cases were one or the other race
is stationary - four equations for these two special cases (1). There are an
infinite number of other combinations for race rotation where these formulas
fail. A more general treatment has been given (2, 3), but in a rather compli-
cated form which obscures somewhat the simple underlying physical situation.
All these theories are first order, implicitly or explicitly invoking
five assumptions, with the idea that a first order treatment will give a
useful foundation for bearing analysis. Complications may then be added to
* C. S. Draper Laboratory, Cambridge, Massachusetts 02139 and National
Research Council - NASA Research Associate.
the theory one at a time as appears necessary. The assumptions are: (a)
rigid balls and races, (b) zero circumferential ball-race slip, (c) equal
inner and outer contact angles, (d) no rolling friction, and (e) no accelera-
tion of the bearing center of mass.
In this paper it is shown that, under the first order assumptions, two
very general but very simple formulas are adequate to describe all possible
angular contact ball bearings for all possible combinations of race rotation.
Moreover the variables in the equations are natural and intuitive; their phys-
ical meaning makes the equations easy to remember and easy to apply.
With this as background, the application of accurate rate measurement to
first order kinematic analysis is then considered to show how information on
specific dynamical and rheological bearing effects can be obtained.
GEOMETRY
A bearing can be regarded as a group of balls arranged in a circle,
rolling under load between toroidal inner and outer races. The proportions of
the bearing are generalized by D, the ratio of ball to ball group (pitch)
diameter: D = d/E. A bearing has many "small" balls if d « E. D can vary
from zero in a linear bearing (E =<*>) to 1 in a bearing with two large
balls.
Bearing loads are carried from race to race across each ball, along lines
of contact which pass through each ball center. The angle B between a line
of contacts and a normal to the bearing axis is called the contact angle and
determines bearing type. B can vary from zero in a radial bearing to 90
degrees in a thrust bearing (Fig. 1).
D and B form a pair of independent bearing variables which completely
describe first order geometry. Either can take any value over its range,
independent of the other (except that a bearing with D = 1 and B = 0 can
have no inner race). Fig. 2 shows a two-ball bearing for B = 0 and B = 90
degrees.
Some derived geometries are a, the angle around the pitch circle between
two adjacent balls in a full complement bearing:
a = 2 slrT^D) (1)
N, the maximum number of balls in a full complement:
N = int (2n/a) (2)
and G, the angular gap outside a full complement:
G = 2* - Na (3)
Fig. 3 is a plot of N against D. N takes on integral values, jumping
to smaller values as D increases. G = a at each jump and decreases to
zero just before the next jump.
ROTATIONS
A ball bearing has four angular rates:
•
YO outer race
•
YJ inner race
•
6 ball group orbit
6 ball-about-its-own-center (spin)
The inner and outer rates are another pair of independent bearing vari-
ables. Either can take any value over the range *°; each combination is
called a "mode". In practice one or the other rate is usually zero, hence the
terms "inner or outer race rotation," but there is no theoretical requirement
that one race be stationary.
KINEMATICS
For arbitrary race rotations define the total speed of the bearing as
5 = TO - TI (4)
and the average speed as
A = (YQ + Yi)/2 (5)
taking account of the signs of the rotations.
Then, for the first order assumptions
•
6 = S (1/2D) (1 - D2 cos2 B) (6)
and
6 = A + S (D cos B)/2 (7)
These are the general first order kinematic equations connecting geometry
and rotation in any bearing. Ball spin rate depends only on geometry and
total speed; ball orbit rate is the average speed corrected by a geometrical
fraction of total speed. The significance of S and A is seen from a com-
parison of a bearing running in two special modes:
I II
• • • • • •
YO = -YI = Y YO = YI = Y
S = 2Y S = 0
•
A = 0 A = Y
6 = 2 Y (1/2 D) (1 - D2 cos2 B) 6=0
• • • •
3 = Y (D COS B) B = Y
Total speed thus measures the relative rotation inside a bearing; average
speed is a measure of its solid body rotation.
Define the basic speed ratio of any bearing as
P = 6/S = (1/2 D) (1 - D2 cos2 B) (8)
p is called basic because it depends only on geometry, not on mode. It
is a characteristic number for an angular contact ball bearing.
Define modal speed ratio as
PM = B/S = A/S + (D cos B)/2 (9)
PM is the sum of a modal term, A/S, and a geometric correction whose
maximum value could be 1/2. Both p and PM are defined for all modes
except at S = 0 (Mode II). But in II we really have a wheel, not a bearing
at all.
Thus the kinematics of all bearings in all modes can be described with
two diagrams, one for p and one for p,,.
Fig. 4 is such a diagram; p vs B showing lines of constant D. The
number of balls for a full complement at each D is also given (from Fig.
3). Only at large D is there any appreciable effect of contact angle. For
D < 0.1 an approximation good to 1 percent is given by p = 1/(2D). The
chart of p vs B extends to <» as D approaches 0; however most bearings
have p < 10, N < 62, and are included in Fig. 4. In practice p
increases in a general way with bearing size. A typical value for a small
instrument bearing is 2, for an aircraft gas turbine bearing 4, and for a
crane turntable bearing 10. Certain specialty bearings (torque tube) may
have p as large as 50, and contain over 300 balls.
Fig. 5 is a plot of the modal part of PM vs YI/YQ» w^t'1 the P°int
for outer race rotation marked. Both branches are asymptotic to -1/2, one from
above, the other from below. Inner race rotation is at YI/YQ = "
where A/S = -1/2. To find p.. from Fig. 5 it is only necessary to add
the geometric correction (D cos B)/2 to the A/S of interest. The correc-
tion is always positive, approaching zero for small balls or thrust loading,
and 1/2 for large balls or radial loading.
As an example of the use of Fig. 5, consider bearing modes with zero ball
orbit rate (there is a theoretical interest in these modes since ball
centrifugal force is then zero). Eq. (9) shows that A/S must be between 0
* • •
and -1/2 for B =0. In Fig. 5 this region, lying to the left of YJ/YQ
= -1, is shown shaded. Only here could a suitable choice of D and B give
•
B =0. Thus the race rotations must be opposite, giving the name "counter
race rotation" to these modes.
APPLICATIONS
Basic Speed Ratio
Basic speed ratio can be measured extremely accurately in a bearing run
in the counter race rotation mode at zero ball orbit rate. For stationary
ball centers individual spin rates can easily be obtained to 1 part in 10 ,
and p computed from its definition, Eq. (8). Since, to first order p is a
geometric constant, any measured change reflects some deviation from the first
order assumptions. This is useful in determining running conditions where
specific assumptions might break down, and in evaluating numerical models of
bearing performance (4).
In a successful bearing balls and races are separated by an elastohydro-
dynamic lubricant film. Since ball spin is driven by traction forces gener-
ated in this film, p gives a sensitive evaluation of ball-lubricant-race
coupling. Relative changes in film thickness (5) and absolute values for
inner and outer ball-race circumferential slip (6) have been obtained from p
measurements.
To find circumferential slip it is necessary to know the geometry of the
test bearing and its rotation rates to a comparable accuracy. Direct measure-
ment (e.g. with a micrometer) at this accuracy is possible for ball diameter,
but impossible for pitch diameter and contact angle. However it turns out for
no circumferential slip in counter rotation (6)
B = h/2 YO YI| |(YI + YQ)cos /2  YI  (10)
(YI - Yo)_|E = fi/2 YOYI _(YI )_d (ID
Thus E and B can be accurately determined by means of a small slip
experiment (e.g., low total speed, thin film) and then used to find circumfer-
ential slip at high speed and thick film. This particular slip depends among
other things on lubricant rheology and film thickness. Experimental informa-
tion on changes in film configuration can thus easily be obtained in a real
bearing under realistic operating conditions.
Total Speed
Total speed measures the relative rotation inside a bearing, giving a
basis for comparing the severity of different operating modes. Since ball
spin rate is the product of p and total speed it is the same for all modes
which have the same total speed. The details of the approach, contact and
retreat of load carrying elements on ball and race are the same for all modes
at the same total speed. To first order "all modes at the same total speed
are kinematically equivalent". Classical calculation of elastohydrodynamic
film thickness depends on these kinematic details.
It can also be shown (7) that the total inner and outer ball-race
pivoting (the components of relative angular velocity normal to each Hertz
area, often loosely referred to as spin), is proportional to total speed for
all positions of the ball spin vector:
PQ + Pj = S sin B (12)
Normalizing with respect to ball spin rate
= sin B/p (13)
a geometric constant independent of mode, speed, or change in ball spin orien-
tation. The ball-race slip associated with pivoting is "kinematic," dictated
solely by bearing geometry and total speed. It can never be zero except in a
pure radial bearing. In particular, pivoting slip is completely independent
of lubricant rheology and film thickness; in a sense just the opposite of cir-
cumferential slip.
Although lubrication does not affect pivoting slip, the reverse is not
true. It has been suggested (8) that lubricant shear associated with pivoting
slip activates polymerization or oxidation reactions in organic lubricants.
In a starved bearing rapid lubricant failure may result if the elastohydro-
dynamic film becomes too thin, even though it is still complete (unpenetrated).
Modal Speed Ratio
Kinematic equivalence means that all modes at the same total speed would
look the same to an observer orbiting with the ball group. This is only true
•
as long as the centrifugal force on a ball due to 0 is small compared to the
service load across the ball. If the total normal force at the ball-outer
contact were significantly larger than that at the inner, the observer would
measure a decreased outer contact angle; the line of contacts would not go
through the ball center, and a first order assumption would be violated. Ball
centrifugal force c depends on ball orbit rate squared; the latter is easily
8
computed from Eq. (9), given a definite mode, total speed, and bearing
geometry.
A reduced outer angle is given by
BQ = tan'1 (sin Bj/fcos Bj + c/n)) (14)
Fig. 6 plots (By - BQ) vs Bj for different values of c/n, cen-
trifugal divided by inner normal ball force. The difference is less than 6
degrees in all bearings if c is no more than 10 percent of the service
load. The effect becomes less important in general as bearings become more
"radial"; for contact angles less than 30° the difference is less than
10 degrees for centrifugal forces as much as half the service load. Ball
•
centrifugal force is proportional to B^, d^, and E. First order
analysis becomes suspect for large bearings and/or high average speeds (Eq. 7),
Another dynamic effect proportional to B? arises in a full complement
bearing with a ball group residual gap (Eq. (3)). Then the ball group center
of mass does not lie on the bearing spin axis and there will be a radial group
unbalance force rotating around the bearing axis at ball orbit rate. A
similar force results from the eccentric whirl motion of the cage in a
conventional bearing. This force cannot be removed by ordinary balancing
since B is never the same as either race rate (except in mode II or in
certain unusual special cases). The forces from two bearings can combine to
produce a variable torque normal to the spin axis of the pair (9); mechanical
"crosstorque" is undesirable in gyroscope and satellite applications.
In a full complement bearing it can be shown (for D small enough so
D = sin" (D), say less than 0.5) that the radius from the bearing axis
to the ball group center of mass is
r = (N + l)e/2 (15)
where e is the difference between the actual ball diameter and the next
larger diameter for zero gap d~Q:
e = dG=Q - d (16)
Thus if a bearing has 10 balls (D = 0.3) it makes no difference if it
has a pitch diameter of one mm or one km: when the ball diameter is reduced
0.001 mm from that required for zero gap the center of mass of the ball group
will be located 0.00175 mm radially from the bearing axis in either case.
CONCLUDING REMARKS
In its first order form an angular contact bearing is a very simple
device. When it is run in the counter rotating mode under a pure axial load
very accurate kinematic measurements can be made easily. Together these fea-
tures make the analysis of real bearings in real operating situations feasi-
ble, interesting, and occasionally surprising. The kinematic, dynamic,
rheologic, and chemical effects reviewed here are not the only facets of bear-
ing operation which can be studied. These techniques thus offer an alterna-
tive to numerical modeling in bearing research.
REFERENCES
1. Jones, A. B., Analysis of Stresses and Deflections, New Departure Divi-
sion of General Motors, Bristol,CT. (1946).
2. Bisson, E. E. and Anderson, W. J., "Advanced Bearing Technology," NASA
SP-38 (1964).
3. Hamrock, B. J. and Dowson, D., Ball Bearing Lubrication: The
Elastohydrodynamics of Elliptical Contacts, John Wiley, New York (1981).
4. Kingsbury, E. P., "Basic Speed Ratio of an Angular Contact Bearing," J.
Lubr. Techno!., 102. 3, pp 391-394 (1980).
5. Kingsbury, E. P., "Lubricant-Breakdown in Instrument Ball Bearings," J.
Lubr. Techno!.. 100. 3, pp 386-394 (1978).
6. Kingsbury, E. P., "Slip Measurement in an Angular Contact Ball Bearing,"
J. Lubr. Techno!., 105. 2, pp 162-165 (1983).
10
7. Kingsbury, E., "Pivoting and Slip in an Angular Contact Bearing," NASA
TM-83408 (1983).
8. Kingsbury, E., "Transient EHL Effects in Starved Ball Bearings,"
Tribology in the Eighties, NASA CP- (1983), pp 641-649.
9. Kingsbury, E., "Gyroscope Bearing Cross-Torque," J. Lubr. Techno!., 92,
pp 303-309 (1970).
11
RADIAL
LINE OF
CONTACT
Fig. 1. - Angular contact ball bearing.
ROTATION
AXIS
LINES OF CONTACT
ROTATION
AXIS
(a) Radial bearing.
ir TT
B = 90°
(b) Thrust bearing.
Fig. 2. - Radial and thrust bearings with the largest (D = 1) and fewest
(N = 2) balls.
50
40
30
20
10
0 .1 .2 .3 .4 .5 .6 .7 .8 .9 1.0
D
Fig. 3. -Maximum ball complement versus ball diameter to
pitch diameter ratio.
10
8
6
4
2
0
(RAC
6EAI
D
0.05
a 075
~~ 0.1
— 0.15
0.2
^ 1 0.5
N
62
41
31
20
15
6 2
|— . 1
30
HAL
RING)
60 90
(THRUST
BEARING)
B, deg
Fig. 4. - Basic speed ratio versus contact angle.
2 i—
-1+1
-1/2
-2
OUTER RACE RTN
REGION WHERE p=0
OPERATION IS POSSIBLE
INNER RACE RTN
-25 25
Fig. 5. - Modal part of pM (Eq. (9)) versus [inner/outer] rate.
6
CO
CQ
RADIAL
BEARING
r, deg
90
THRUST
BEARING
Fig. 6. - Contact angle difference versus inner contact angle.
1. Report No
NASA TM-83592
2. Government Accession No 3 Recipient s Catalog No
4 Title and Subtitle
First Order Ball Bearing Kinematics
5 Report Date
6 Performing Organization Code
505-33-1B
7 Authorts)
Edward Kingsbury
8 Performing Organization Report No
E-1918
10 Work Unit No
9 Performing Organization Name and Address
National Aeronautics and Space Administration
Lewis Research Center
Cleveland, Ohio 44135
11 Contract or Grant No
12 Sponsoring Agency Name and Address
National Aeronautics and Space Administration
Washington, D.C. 20546
13 Type Of Report and Period Covered
Technical Memorandum
14 Sponsoring Agency Code
15 Supplementary Notes
E. Kingsbury, C. S. Draper Lab., Cambridge, Massachusetts 02139 and National
Research Council-NASA Research Associate. Prepared for the Annual Meeting of the
American Society of Lubrication Engineers, Chicago, Illinois, May 7-10, 1984.
16 ADstract
Two first order equations are given connecting geometry and internal motions in
an angular contact ball bearing. Total speed, kinematic equivalence, basic speed
ratio, and modal speed ratio are defined and discussed; charts are given for the
speed ratios covering all bearings and all rotational modes. Instances where
specific first order assumptions might fail are discussed, and the resulting
effects on bearing performance reviewed.
7 Kay Words (Suggested by Authorts))
Ball bearing
Kinematics
General first order theory
18 Distribution Statement
Unclassified - unlimited
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20 Security Cluslt (of this page)
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First order ball bearing kinematics

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