Halpin -Tran Paper R12

8/20/2019 Halpin -Tran Paper R12

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2015 ASTM F34 International Symposium on Rolling Element Bearings – An Analytical Model of Four-Point Contact Rolling

Element Ball Bearings – by J. Halpin and A. Tran (submitted for review: 20-Apr-2015)1

Abstract — The purpose of this work is to establish an analytical model and standard way to predict the

performance characteristics of a four-point contact, or gothic arch type, rolling element ball bearing.Classical rolling element bearing theory, as developed by A. B. Jones, has been extended to include theunique aspects of the four-point contact bearing; thereby providing complete element-wise attitude and

internal load distributions of the bearing under operating load conditions. Independent geometrydefinition for the inner and outer raceways, including their associated arching, is provided to allow

assessment of a broad range of manufacturing tolerance conditions. Standard performance parameters,such as element contact stresses, contact angles, inner ring deflections, nonlinear stiffness’s, torque, andL10 life are solved explicitly via standard Newton-Raphson techniques. Race control theory is replacedwith a minimum energy state theory that allows both spin and slip to occur at each contact.

The developed four-point model was programmed within the ORBIS software program. Various test

cases are analyzed and key analytical results are compared with the A. B. Jones Four Point Contact BallBearing Analysis Program, the DG03 design guide and traditional two-point (angular contact) analysis

codes. Model results for the internal distribution of ball loads and contact angles match the Jones program extremely well for all cases considered. Some differences are found with the DG03 analysismethods and it is found that modelling a four-point contact bearing with two opposed angular contact

bearings positioned to a common center can result in gross errors.

Keywords — bearing analysis, four-point contact, gothic arch bearing, double arched bearing, ball bearings, split ring

bearing, rolling element bearings

1J. D. Halpin is the owner of Halpin Engineering, LLC, located in Torrance, CA 90504 USA (phone: 310-650-8982; e-mail:

[email protected]).2 Tran, A. N. is owner of ATEC Corporation, located in Cypress, CA 90630 USA (email: [email protected])

An Analytical Model of Four-Point Contact

Rolling Element Ball Bearings

Jacob D. Halpin1 and Anh N. Tran2

mailto:[email protected]:[email protected]:[email protected]


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NOMENCLATURE

Symbol Description

Ball spin axis angle

Axial distance from outer-right raceway curvature center to inner-left raceway curvaturecenter

Radial distance from outer-right raceway curvature center to inner-left raceway

curvature center Contact angle

+ − 1

L10 fatigue capacity of a single point contact

,,,,, Simplifying expressions, see equations (46) through (51)

Ball diameter

Pitch diameter

Normal approach ball to raceway contact

Young’s modulus of elasticity

Curvature ratio of raceway

F Elastic reaction force of bearing on shaft

Arching dimension

Simplifying expression, see equations (37) and (38)

G, H Generic functions, see equations (34) and (35)

Stiffness, Hertzian contact

M, N Generic functions, see equations (90) through (101)

Azimuth angle (ball station clocking angle)

Radial play of non-arched bearing

Enforced displacement of inner ring at its center

Normal ball load

Raceway radius

, Radius from bearing spin axis to inner or outer raceway curvature center at initial free

contact

Radial play of an arched bearing

Axial play of an arched bearing

Radial height of ball center relative to outer raceway curvature centers

Axial distance to ball center from outer right raceway curvature center

Number of rolling elements (balls) in a bearing row

SUBSCRIPTS

Inner raceway

Outer raceway

Left raceway

Right raceway

1,2,3,4,5 Coordinate directions (1 = x, 2 = y, 3 = z, 4 = yy, 5 = zz)

rest Resting contact angle

free Free contact angle


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Introduction

Gothic arch, or four-point, bearings are a unique class of ball bearings in that they offer capability toreact combined axial, radial and moment loads in a compact single row configuration. This set of

capabilities is desirable for a large array of mechanical designs; perhaps most useful in large scaleapplications, such as turrets for cranes or wind turbine pitch and yaw bearings, both due to the difficulties

in maintaining coaxial alignment requirements of double row angular contact bearings and the obviousweight savings realized from use of a single row.

Despite size and weight advantages offered by gothic arch bearings their rolling kinematics canresult in undesirable performance. In particular, four-point contact bearings are known to exhibit largefrictional efficiency losses which, in turn, cause heat generation within the bearing and accelerate the

lubricant breakdown process. These efficiency losses prohibit four-point contact bearings from beingsuitable for high speed applications.

With the aforementioned pros and cons it becomes necessary for the design engineer to have areliable tool to assess critical performance characteristics of a four-point contact bearing. This work

proposes such a model by solving the complete internal Hertzian contact distribution of a four-point

contact ball bearing. Model outputs include the following key performance characteristics:

Ball normal contact loads

Ball contact angles Frictional losses due to contact spin and slip

Relative displacements of inner rings to outer rings due to external loading

Non-linear bearing stiffness

L10 Fatigue lifeThis model neglects effects from gyroscopic moments and only considers dynamic body forces from

centrifugal effects. A study is presented herein to show removal of gyroscopic moments has aninsignificant effect on accuracy for speeds extending well beyond those advisable for gothic arch type

bearings. Hamrock [1] also neglected gyroscopic moments in his model for an arched outer race bearing.

Nelias and Leblanc [2] provided a model to consider gyroscopic moments yet ultimately caution thatselection of a double arched bearing for high speed applications should be carefully considered.

The problem of ball spin axis determination, which A. B. Jones addressed with his race control

theory [3], is handled herein with a minimum energy theory. This method allows both spin and slip tooccur at a single contact while maintaining a minimum energy state of the power losses. This isimportant for four-point contact bearings as it provides a means to model torque increases due to multiplecontact points. This phenomenon is sometimes referred to as ‘wiping.’

The developed model has been programmed into the ORBIS software program to aid in theevaluation of model results. Multiple test cases are compared against the various references anddiscussed in detail.

Geometry

Geometry of a gothic formed raceway is perhaps best described by splitting a conventional deepgroove bearing down the middle of the raceway, removing equal portions of material from the two halves,

and abutting them back together. FIG. 1 illustrates this concept and a bearing of this type is often referredto as a split ring, or split raceway, bearing. Bearing manufacturers may also use an arched shape tool to

directly cut the gothic profile without requiring ring splitting. Both types are synonymous in the contextherein and no further distinction will be made.


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go

gi

ro

ri

CL

Pd/4

CL

FIG. 1 — Conventional deep groove bearing with arching region identified and resulting double arched bearing

The removed material, denoted gi for inner race and go for outer race, is called the arching or shimsize. As arching is increased from zero the initial radial play (Pd) decreases and the ball is no longer ableto make contact at the arch point (zero contact angle). To maintain this key distinction between an arched

bearing and conventional bearing Sd shall be used to denote radial play of gothic formed raceways and Pd will denote radial play of the original circular raceway profile.

Raceway arching results in two important initial contact angles: a resting angle and the conventionalfree contact angle. The resting angle, as shown in FIG. 2, is defined as the contact angle made when the

ball is simply resting on the inner or outer raceway (initial unloaded radial contact position). The freecontact angle, which is the standard way to define a conventional angular contact bearing, pertains to theangle made at the initial unloaded axial contact position. These two contact angles should not be mixed

or interchanged as they are often different. For all four-point bearing designs the free contact angles must be greater than or equal to the resting angle. The two are equal when there is no internal free play in the

bearing.


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2βrest, o

2βrest, i

CL

βfree

CL

Sd

FIG. 2 — Resting angle and free contact angle

The arching geometry necessitates definition of additional parameters compared to a conventionaldeep groove or angular contact bearing. In particular, two of the following three parameters must be

known to define an arched bearing: arched internal clearance (Sd), resting angles (αrest) or the arching

dimensions (gi and go). The most practical parameters to measure are the internal clearance and restingangles. The relationship between resting angles and arch dimensions are described by

i ir f d (1)

o or f d (2)

,sin (2 )

i rest i i g r d (3)

,sin (2 )o rest o o g r d (4)

Internal play is perhaps best visualized by considering a normalized circle of radius 2r-d. As shownin FIG. 3 the relationships between internal clearances, contact angles and shim size become readilyapparent when shown on the normalized circle.


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,i o

g

,2

i or d

,i o Pd

,i oSd

,i oSe

free

rest

FIG. 3 — Normalized internal clearance circle depicting geometrical relationship between internal play, resting and free

contact angles, and the shim size.

Jones [3] determined that for a conventional angular contact bearing the relationship between radial

play and the free contact angle is

2 (1 cos )d free P Bd (5)

However, as previously discussed, for a gothic formed raceway the radial play is reduced. To makeuse of equation (5), the original radial play, prior to arching, must be established. Leveille [4] found thereduction to internal play for each gothic formed raceway is expressed by

,(2 )(1 cos )

i i rest i Pd r d (6)

,(2 )(1 cos )

o o rest o Pd r d (7)

Hence, the non-arched radial play is simply the sum of Sd and the reductions from gothic forming ofthe inner and outer raceways. With this relationship the remaining necessary geometry is developed as

follows.

, ,d d d i d o P S P P (8)

1cos 12

d

free

P

Bd

(9)

2 sine free i oS Bd g g (10)


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0.5 ( 0.5) cosi m i free R d f d (11)

0.5 ( 0.5) coso m o free R d f d (12)

Analytical Model

Internal Load Distribution

The general modelling approach expresses each of the four raceway centers of curvature with acircle. The model assumes inner and outer shim sizes be greater than zero (arching is implied) and thuseach center of curvature has a unique position. Outer raceways remain fixed and a moveable coordinateframe is attached at the center of the two inner raceways. Displacements of the moveable coordinate are

analogous to shaft deflections at the center of the bearing, which result in change to the normal approach between inner and outer raceway circles. The normal approach of each contact point is then solved ateach ball station, or azimuth angle, around the bearing as a function of inner ring displacements. The

location of the ball center, relative to the fixed outer raceways, is determined by ensuring the ball is inquasi-static equilibrium.

Given the normal approach of the ball center to each raceway classical Hertzian contact analysis isutilized to determine contact normal loads at each of the four possible points. The vector sum of all ball

loads then provides the resultant force reacting on the shaft for the given set of displacements to the innerrings. Complete bearing equilibrium is achieved when shaft reaction forces are equal to and opposite ofexternally applied loads.

FIG. 4 shows a four-point bearing along with a five degree-of-freedom standard right-handedcoordinate system. As shown, the x-axis is aligned with the bearing spin axis and the y-axis is upward in

the cross-section. The inner ring deflections, q’s, correspond to this coordinate frame. Ball indexingconvention is depicted in FIG. 5; where the first ball is always placed along the y-axis and indexingincreases counter-clockwise in the YZ plane.


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X

Z

Y

q

1

q

2

q

4

q

3

5

FIG. 4 — Coordinate system showing relation to inner ring displacements

Y

Z

φ

n = 1

n = 2

n = z

n = 3

FIG. 5 — Ball position definition relative to global coordinates


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FIG. 6 illustrates the internal kinematics of the ball center and each of the four raceway centers of

curvature for initial and deflected states of the inner rings. Implied here is that when inner ringdeflections are zero (i.e. q1,2,3,4,5 = 0) the bearing is in its free contact angle position. Note that this is notnecessarily the geometric center of the bearing — only in the case where there is zero internal clearancewill this correspond to the geometric center of the bearing.

go

W

V

Ax

Ay

gi

( f i - 0

. 5 ) d

+ δ

i l

βilβir

βolβor

( f o - 0

. 5 ) d

+ δ o l

( f o - 0

. 5 )

d + δ o

r

(

f i - 0 . 5 ) d

+ δ i l

θ

Initial Ball Center

Final Ball Center

Outer-left raceway

curvature centerOuter-right raceway

curvature center

Final inner-right raceway

curvature center

Final inner-left raceway

curvature center

FIG. 6 — Position of ball center and raceway curvature centers at initial and final loaded positions

From FIG. 6, the following relationships are developed.

1 4 50.5 sin 0.5 sin [ sin cos ] x i free o free i A f d f d q R q q (13)


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2 322

544 5

0.5 cos 0.5 cos cos sin

sin ( ) cos ( )2 2

y i free o free

i

A f d f d q q

qq R SGN q SGN q

(14)

0sin .5 x

i iil

l

A W

f d

(15)

5 4

cos( cos sin )sin

0.5

i x

ir

i ir

g Aq

d

q W

f

(16)

sin

0.5or

o or

W

f d

(17)

sin

0.5

o

ol

o ol

g W

f d

(18)

cos

0.5

y

il

i il

A V

f d

(19)

5 4sin( cos sin )

cos0.5

y i

ir

i ir

A V g q q

f d

(20)

cos

0.5or

o or

V

f d

(21)

cos 0.5ol o ol

V

f d (22)

Out of plane inner ring rotations causes a tilting effect of the inner raceway curvature centers(denoted by θ in FIG. 6). This tilting angle will be small and can be sufficiently expressed with a firstterm Taylor Series expansion. From equations (13) through (22), and the Pythagorean Theorem, thenormal approach of each raceway contact is solved. Negative values of normal approach indicate loss ofcontact at the raceway.

2 2

0.5il y x i

A V A W f d (23)

2 2

5 4 5 4sin( cos sin ) 0.5cos( cos sin )ir y i i x iq q A V g q q g A W f d (24)

2 2 0.5or oV W f d (25)

22 0.5ol o oV g W f d (26)

Contact angles are then determined as a function of the normal approach and ball center location.


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1cos

0.5

y

il

i il

A V

f d

(27)

5 41

sin( cos sin )

cos 0.5

y i

ir

i ir

A V g q q

f d

(28)

1cos

0.5or

o or

V

f d

(29)

1cos

0.5ol

o ol

V

f d

(30)

The ball center coordinates, V and W, are now needed to determine both the normal approach andcontact angle at each raceway. Since this model is quasi-static each ball must be in a state of staticequilibrium. Equilibrium force equations in the axial and radial directions become

sin sin sin sin 0il il ol ol ir ir or or Q Q Q Q (31)

cos cos cos cos 0il il ir ir ol ol or or cQ Q Q Q F (32)

The ball normal load can also be defined as

1.5Q K (33)

By substitution the equilibrium equations can be rewritten as follows. These equations are labeled G

and H for further use.

1.5 1.5 1.5

1

4

.5

5cos

0.5 0.5 0.5

0 ( , )0.5

( cos sin )il il x ol ol o ir ir i x

i il o ol i ir

or or

o or

K A W K g W K g A W

f d f d f d

K W G

q q

V W f d

(34)

1.5 1.5 1.55 4

1.5

sin( cos sin )

0.5 0.5 0.5

0 ( , )

0.5

il il y ir ir y i or or

i il i ir o or

ol ol

c

o ol

K A V K A V g q q K V

f d f d f d

K V F H V W

f d

(35)

Fc is the centrifugal force acting at the ball center and is defined by

2Ω

2

b m ec

M d F (36)


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Ball orbit, or separator, speed is taken from Jones [3]. Since these equations are based on a two point contact bearing it becomes necessary to find the inner and outer contacts that provide the

dominating tractive forces. In general this can be determined by identification of the largest normalapproach for each set of inner and outer contacts per equations (23) through (26).

cosi

im

d

d

(37)

coso

o

m

d

d

(38)

1 cosΩ

1 cos 1 cos

i o

e i

o i i o

(for stationary outer ring) (39)

1 cosΩ

1 cos 1 cos

o i

e o

o i i o

(for stationary inner ring) (40)

The normal contact stiffness, which is a function of body curvatures and material properties, of each

ball to race contact is thoroughly described per Jones [5] and will not be repeated herein. With theequilibrium equations now defined the solution to ball center coordinates is developed. Per Hamrock andAnderson [1] and Harris [6] the ball center coordinates can be iteratively solved via Newton-Raphson

method.

1 Λn n V V V (41)

1 Λ

n n W W W (42)

Assigning equilibrium equations (34) & (35) as functions G & H respectively, the operators from

equations (41) and (42) become

1ΛV

H GG H

W W

(43)

1ΛW

G H H G

V V

(44)

G H H G

V W V W

(45)

Partial derivatives of the equilibrium equations are needed to solve ball center coordinates. Usingthe following simplifying constants the complete sets of partial derivatives are solved. Note that

technically the centrifugal force is also a function of the ball center; however, its rate of change is poorlycoupled with radial movements of the ball center and neglecting its derivatives has negligible impact tothe convergence rate of the Newton-Raphson method.

2 2 2 2

1 2 2 y y x xC A A V V A A W W (46)


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2 5 4sin( cos sin ) A y iC A V g q q (47)

2 5 4cos( cos sin )

i B xC q g Aq W (48)

2 2

2 2 2 A BC C C (49)

2

3

2C V W (50)

2 2

4

22o oC V g g W W (51)

The partial derivatives of G(V,W) with respect to V become

31 2 4GG G GG

V V V V V

(52)

1.50.5

1 11

1.5

1 1

0.5 0.5

1.5

i x y i x y

il il

C f d A W V A C f d A W V AG

K K V C C

(53)

1.0.5

2 2 2 2 2 22

1.5

2

5

2

0.5 0.51.5

i A B i A B

ir ir

C f d C C f d C C G K K

V C C

(54)

0.5 1.5

3 33

1.5

3 3

0.5 0.51.5

o o

or or

C f d WV f d WV G K K

V C C

(55)

0.5 1.5

44

1.5

4 4

0.5 4 0.5

1.5

o o o o

ol ol

C f d g W V f d g W V G

K K V C C

(56)

The partial derivatives of G(V,W) with respect to W are

31 2 4GG G GG

W W W W W

(57)

0.5 1.5

1 11

1 1

1.5

1

1.5

1

0.5 0.51.5

0.5

i x x i

il il

i x x

il

C f d A W W A C f d G K K

W C C

C f d A W W A

K C

(58)

0.52

2 2 22

2 2

15

1

2

2 2

1.

.

2

5

5

0.5 0.51.5

0.5

i B i

ir ir

i B

ir

C f d C C f d G K K

W C C

C f d C K

C

(59)


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0.5 0.5 1.5

2 2

3 3 33

1.5

3 33

0.5 0.5 0.51.5

o o o

or or or

C f d W C f d C f d W G K K K

W C C C

(60)

0.5 1.5

4 44

4 4

1.5

4

1.5

4

0.5 0.51.5

0.5

o o o o

ol ol

o o o

ol

C f d g W W g C f d G K K W C C

C f d g W W g K

C

(61)

The partial derivatives of H(V,W) with respect to V are

31 2 4 H H H H H

V V V V V

(62)

0.5 1.5

1 11

1 1

1.5

1

1.5

1

0.5 0.51.5

0.5

i y y i

il il

i y y

il

C f d A V V A C f d H K K

V C C

C f d A V V A K

C

(63)

0.5 1.5 1.5

2 2

2 2 2 2 22

1.5

2 22

1.5 0.5 0.5 0.5ir i A i ir i Air

K C f d C C f d K C f d C H K

V C C C

(64)

0.5 1.5 1.5

2 2

3 3 3

3 1.5

3 33

0.5 0.5 0.5

1.5o o o

or or or

C f d V C f d C f d V H K K K V C C C

(65)

0.5 11.5

2 2

4 4 44

1.5

4 44

.5

0.5 00.5 .51.5

o o

ol ol o

o

l

C d V C f d C f d V H K K K

V C C C

f

(66)

Finally, the partial derivatives of H(V,W) with respect to W are

31 2 4 H H H H H

W W W W W

(67)

0.5 1.5

1 11

1.5

1 1

[ ( 0.5) ] ( )( ) [ ( 0.5) ] ( )( )1.5

i y x i y x

il il

f d A V W A f d A V W A H K K

W

C C

C C

(68)

0.5 1.5

2 2 2 2 2 22

1.5

2 2

[ ( 0.5) ] [ ( 0.5) ]1.5

i A B i A B

ir ir

f d f d C C C C C C

C

H K

C K

W

(69)


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0.5 1.5

3 o 3 o3

or or 1.5

3 3

f 0.5 d WV f 0.5 d WVH1.5K K

W

C C

C C

(70)

0.5 1.5

4 o o 4 o o4

ol ol 1.5

4 4

f 0.5 d V W g f 0.5 d V W gH1.5K K

W

C C

C C

(71)

With the foregoing solution to the ball center coordinates established the internal load distribution at each

ball contact is obtained and standard Hertzian contact methods can be used to determine contact stressesand sub-surface shear stresses.

Bearing Reaction Forces and Stiffness

Net reaction forces of the bearing on the shaft are obtained by summing the force components foreach inner ring contact. Referring to coordinate definitions shown in FIG. 4 and ball indexing definitionin FIG. 5, the five bearing reaction forces are defined.

1 , , , ,

1

[ sin sin ] z

il n il n ir n ir n

n

F Q Q

(72)

2 , , , ,

1

[ cos cos ]cos z

il n il n ir n ir n

n

F Q Q

(73)

3 , , , ,

1

[ cos cos ]sin z

il n il n ir n ir n

n

F Q Q

(74)

4 , , , ,

1

[ sin sin ]sin z

i il n il n ir n ir n

n

F R Q Q

(75)

5 , , , ,

1

[ sin sin ]cos z

i il n il n ir n ir n

n

F R Q Q

(76)

Stiffness of a four-point contact bearing entails developing the Hertzian contact stiffness of all four

possible load paths. As illustrated in FIG. 7 this results in a parallel pair of springs for the inner-left andinner-right races and another parallel pair of springs for the outer-left and outer-right races. However,since both pairs of springs are tied together at the ball center, the two pairs are connected in series. Recallfrom earlier that the outer raceway centers of curvature are fixed in space while the inner curvaturecenters are allowed to move relative to the fixed outer races.


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K

ilir

K

olor

FIG. 7--Four-Point Contact Stiffness Diagram

The total stiffness of the inner ring pair relative to the fixed outer pair is then expressed by equation(77); which represents the non-linear instantaneous stiffness of the bearing under a given relativedeflection of the inner rings.

1

, , , ,

1 1 j

j j j j j

j il j ir j ol j or

F

F F F F qq q q q

(77)

Let the following notation be used to express the partial derivatives for each ring pair

j j j

inner il ir

F F F

q q q

(78)

j j j

outer ol or

F F F

q q q

(79)

Then the diagonal terms of the full Jacobian become

,sin ,sin1,sin ,sin

1, 1 1 1 1

il ir il il ir ir il il il ir ir ir

inner il ir

M M Q Q F K M N K M N

q q q q q

(80)

,sin ,sin1

,sin ,sin

1, 1 1 1 1

ol or ol ol or or

ol ol ol or or or

outer ol or


q q q q q

(81)

,cos ,cos2

,cos ,cos ,cos

2, 2 2 2 2

cos cosil ir il il ir ir

il il il ir ir ir

inner il ir


q q q q q

(82)

,cos ,cos2

,cos ,cos

2, 2 2 2 2

cos cosol or ol ol or or

ol ol ol or or or

outer ol or


q q q q q

(83)

,cos ,cos3

,cos ,cos ,cos ,cos

3, 3 3 3 3

sin sinil ir il il ir ir

il il il ir ir ir

inner il ir

M M F Q Q K M N K M N

q q q q q

(84)


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,cos ,cos3,cos ,cos

3, 3 3 3 3

sin sinol or ol ol or or

ol ol ol or or or

outer ol or

M M F Q Q K M N K M N

q q q q q

(85)

,sin ,sin4,sin ,sin

4, 4 4 4 4

sin sinil ir il il ir ir

i il il il i ir ir ir

inner il ir

M M Q Q F R K M N R K M N

q q q q q

(86)

,sin ,sin4,sin ,sin

4, 4 4 4 4

sin sinol or ol ol or or

i ol ol ol i or or or

outer ol or

M M Q Q F R K M N R K M N

q q q q q

(87)

,sin ,sin5,sin ,sin ,sin ,sin

5, 5 5 5 5

cos cosil ir il il ir ir

i il il il i ir ir ir

inner il ir

M M F Q Q R K M N R K M N

q q q q q

(88)

,sin ,sin5,sin ,sin

5, 5 5 5 5

cos cosol or ol ol or or

i ol ol ol i or or or

outer ol or

M M F Q Q R K M N R K M N

q q q q q

(89)

Where the functions M and N in equations (80) through (89) are defined as

1.5

( 0.5)

il

il

i il

N f d

(90)

1.5

( 0.5)

ir

ir

i ir

N f d

(91)

1.5

( 0.5)

ol

ol

o ol

N f d

(92)

1.5

( 0.5)

or

or

o or

N f d

(93)

,sinil x M A W (94)

,sin 5 4cos( cos sin )

ir i x M g q q A W (95)

,sinol o M g W (96)

,sinor M V (97)

,cosil y M A V (98)

,cos 5 4sin( cos sin )ir y i M A V g q q (99)

,cosol M V (100)

,cosor M V (101)

Partial derivative of normal load with respect to the normal approach has a general form that is

independent of raceway. By consistent subscript assignment the following expression can be used.


18/31



1.5

2

1.5

( 0.5) [( 0.5) ]

Q

f d f d

(102)

The normal approach equations and the functions expressed in equations (94) through (101) will

have two forms depending on whether the reference is an inner or outer raceway.

yi i x i

x y

A A

q A q A q

(103)

yo o x o

x y

AW A V

q W A q V A q

(104)

yi i x i

x y

A M M A M

q A q A q

(105)

yo o x o

x y

A M M A M W V

q W A q V A q

(106)

( 0.5)

il x

x il i

A W

A f d

(107)

( 0.5)

yil

y il i

A V

A f d

(108)

( 0.5)

ir x i

x ir i

A g W

A f d

(109)

5 4sin cos sin sin

( 0.5)

y i iir

y ir i

A V g q g q

A f d

(110)

( 0.5)ol o

o ol

W g

W f d

(111)

( 0.5)ol

o ol

V

V f d

(112)

( 0.5)

or

o or

W

W f d

(113)

( 0.5)

or

o or

V

V f d

(114)

The remaining partial derivatives needed are shown in Table 1 and Table 2.


19/31



Table 1 — Partial derivatives of Ax and Ay

/ x A y A

1q 1 0

2q 0 cos

3q

0sin

4q sin

i R

4 4sin * ( )

i R q signum q

5q cosi R 5 5cos * ( )i R q signum q

Table 2 — Miscellaneous partial derivatives

/ W V ,sinil M ,sinir M ,sinol M ,sinor M ,cosil M ,cosir M ,cosol M ,cosor M

x A 1 - 1 -1 - - 0 0 - -

y A

- 1 0 0 - - 1 1 - -W - - - - -1 1 - - 0 0V - - - - 0 0 - - 1 1

To make the foregoing analysis useful it is desired to compute the precise bearing deflections due toa known externally applied load. This is achieved by numerical iteration, via the Newton-Raphsontechnique, as follows.

1

1n n applied reaction

n

F q q F F

q

(115)

Frictional Effects

Bearing friction torque, or its internal resistance to rotation, is dependent upon a multitude of

physical phenomena — such as lubricant shearing (viscous effects), retainer or cage drag forces, imperfectgeometry of rolling elements and associated raceways, contact surface asperities, material elasticity and

hysteresis, and rolling/spinning contact slip due to relative surface velocities. For bearings of precisionquality with well-designed mounting conditions, and under loaded conditions, the primary torquecomponents typically come from contact sliding and viscous effects. The focus herein will be to extend

the conventional torque equations for contact slip within an angular contact bearing to a gothic arch bearing. A well-known understanding with balls containing four points of contact is the kinematic over-

constraint prevents pure rolling motion at all contact points. This phenomenon is sometimes referred toas ‘wiping’; which is a loose term that is not clearly defined in an analytical sense but shall be defined

here, qualitatively, to describe a situation where there is an appreciable angle between the surfacevelocities of the ball and the raceway.

Jones [3] developed a general friction model for ball bearings that accounts for interfacial slip due torolling and spinning at the contact. His model includes a ‘race control theory’ that requires one of thecontacts in a two point bearing to contain pure rolling while spin occurs at the other contact. Boness andChapman [7] later developed a novel test apparatus that provided contrary results to the race control

theory. It’s worth mentioning that Jones’ race control theory does result in a very close approximation forwell-behaved operation of angular contact bearings. Chapman and Boness found that the ball spin axis


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angle, which defines the roll-to-spin ratio at the contact, tends to take an angle that minimizes the energystate of the frictional losses.

With gothic formed bearings there can be up to four contact points on a given ball, and race controltheory becomes untenable. The conclusion made by Chapman and Boness — where the ball spin axisattitude resolves to the lowest energy state — shall be applied here with use of general rolling and slidingfriction models developed by Jones. Additionally, to further expand the torque model, rolling and

spinning coefficients of friction will be differentiated to allow separate assignments for each. Per Jones[3] the torque required to rotate a ball bearing under complex loading is

63,025

o i

H

N N

(116)

Where torque has units of in-lbf, H is the horsepower generated from frictional losses, and N i and No are inner and outer race speeds in revolutions per minute. The frictional horsepower, which is now

extended to all four possible contact points, becomes

, , , , , , , ,

1

, , , , , , , ,

1(

6,600)

q q q q q q q q

q q q q q q q q

n

s il s il s ir s ir s or s or s ol s ol

q

R il r il R ir r ir R or r or R ol r ol

HP

(117)

Where τS is the spinning torque and τR is rolling torque. Spinning torque at each contact point is

defined by

3

8 ( )

spin

s

aQ

E e

(118)

Where E(e) is the complete elliptical integral of the second kind with modulus ‘e’ (contact

ellipticity), μspin is the coefficient of sliding friction, ‘a’ is the semi-major dimension of the contact ellipse

and ‘Q’ is the normal ball load. Rolling friction, or Heathcoate slip, is written as follows. Here thetractive braking effects are assumed to be zero, which results in a pure rolling line at 34.37% of the semi-major ellipse dimension.

1 22 2 1 1 1 22 2

1 1 1

3 sin4 2sin4 12sin cos sin cos ( 2 ) 1

4sin 16sin 4sin

roll

R

PR

(119)

1

1 sin

a

R

(120)

1

2

0.3437sin

a

R

(121)

2

2 1

fd R

f

(122)

Ball spin velocity for a stationary inner ring is


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(1 )(1 )

(1 ) cos (1 )cos

m o o i

b

o i i o

d

d

(123)

And for a stationary outer ring becomes

(1 )(1 )

(1 ) cos (1 )cosm i o i

b

o i i o

d d

(124)

Now, the relative surface velocities for spinning and rolling at each contact become

, sin sin( ) s il i il b il (125)

, sin sin( ) s ir i ir b ir (126)

, sin sin( )

s or o or b or (127)

, sin sin( ) s ol o ol b ol (128)

, cos cos( ) R il i il b il (129)

, cos cos( )

R ir i ir b ir (130)

, cos cos( ) R or o or b or (131)

, cos cos( ) R ol o ol b ol (132)

The lowest frictional energy state is achieved when equation (117) is minimized as a function of the

ball spin axis angle (α). Energy minimization can be achieved with a numerical search algorithm thatvaries α between zero and π/2. While this method provides both spin and slip to occur on a given contact

the assumption for tractive braking effects to be zero at each contact is likely invalid for certain operatingconditions. Namely, in cases where a raceway pair (i.e. inner left and outer right) provide dominate

control of the ball spin axis any contact on the opposing raceways must be in a state of slip. Suchinstances could be modelled by modifying equation (121) to adjust the no-slip lines.

Fatigue Life

The ANSI/AFBMA (now named ABMA) standards [8] were specifically developed to allow users toderive bearing fatigue life estimates via hand calculations and lookup tables. According to Jones [3],

these simplifications required assumptions regarding the internal load distribution versus applied externalloading. In particular, the standards make the following assumptions: inner and outer rings remain

parallel (i.e. only axial and radial relative motion), the contact angle is only influenced by thrust loads,

body forces due to element motions are neglected, initial mounted conditions are neglected (i.e. preloading), and combined axial and radial loading can be modelled with an equivalent radial load.Harris [6] further noted the fatigue equations only predict raceway failure and that ball failure apparentlywas not observed in the Lundberg — Palmgren test data. He further postulated that during the Lundberg — Palmgren era the ability to manufacture accurate geometry of balls with good metallurgical properties

exceeded that of the raceways, thus resulting in predominate raceway failure in the test data.


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Another implication, which has not been studied sufficiently to be quantifiable, is that sliding at thecontact is believed to accelerate metallic fatigue of the raceways. The ANSI standard is based on

extensive testing of deep groove radial bearings which can only contain two points of contact on the balland will therefore have minimal sliding at the contact. Since the L10 formulation is all that is available atthis time it will be simply extended to include all four contact points with no additional knockdown toaccount for additional sliding in a four-point contact bearing.

In the case of a ball with four points of contact, versus only two contact points, it is clear the ballstress cycles must increase and hence so will the cumulative fatigue damage; yet the following L10calculations are unable to account for this. To better account for irregular internal load distributionswithin a gothic formed raceway it is not recommended to use ANSI tables with an assumed double rowconfiguration to approximate a four-point contact bearing. Instead, capacities shall be derived at eachcontact point. Per Harris [6], dynamic capacity of a point contact on a raceway is defined by equation

(133) for ball diameters up to 1.0 inch (25.4 mm) and equation (134) for larger ball sizes.

1 1

3 3

0.30.41 1.39 1.812

2 1 1 p p

m

f d d C A

f d z

(133)

1 13 3

0.30.41 1.39 1.412

2 1 1 p p

m

f d d C A

f d z

(134)

The upper sign in (133) and (134) pertains to inner ring contacts and the lower sign is used for

outer ring contacts. Material constant A p tends to vary depending on the source, however Harris showsthis constant to be 7080 for C p in units pounds (71.7 for C p in Newtons). This constant is applicable to air

melted 52100 chromium steel. Note that equations (133) and (134) are the complete general forms fordynamic capacity of a ball-to-raceway contact and are not restricted to a particular conformity, contactangle or load distribution.

Life of each raceway, in total revolutions, is established by summing the ratio of each normal contactforce and the applicable capacity of the raceway contact. The exponent for this ratio has a slight

dependence on the relative fixity of external loading to the rotating ring. Also, since there will be loadcases where some normal contact forces can be zero it is numerically convenient to invert the standardL10 equation such that the capacity term is in the denominator.

L10 life for a raceway that is rotating relative to the external load is

3

6

,

1 ,

10 10 z

n

race rotating

n p n

Q L z

C

(135)

L10 life for a raceway that is stationary relative to the external load is

103

0.9 6

,

1 ,

10 10 z n

race stationary

n p n

Q L z C

(136)

Each raceway’s life capacity is then combined to determine the total life capacity of the bearing. Inthe case of a four-point contact bearing the final L10 life, expressed in bearing revolutions, becomes


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910

10 10 10 109 9 9 9

4

1 1 1 110

10 10 10 10 Pt

il ir or ol

L L L L L

(137)

Discussion

Model Results

To enhance confidence in the model a set of analysis test cases are compared with various

references. The primary parameters needed to validate the core model pertain to the internal loaddistribution — which is comprised of ball normal loads and associated contact angles. Hertzian contact

stresses provide a convenient comparative metric but are essentially a secondary measure of normal loadsfor a given set of contact conformities and material properties. Other performance parameters, namely

friction torque, stiffness and L10 life are essentially all driven by the internal load distributions andtherefore are not discussed. To simplify matters, and maintain focus on the core model, all boundaryconditions will be assumed to be rigid (a.k.a. ‘fixed ring’ analysis).

Each of the analytical references available has certain limitations and therefore a consistent set ofreferences for each test case becomes impractical. The A. B. Jones High Speed Ball and Roller AnalysisProgram

3 and ORBIS bearing analysis program both provide reference for standard angular contact

bearing analysis capabilities. These two references, which are denoted herein as Jones-2PT and ORBIS-2PT, will be utilized to compare various parameters applicable to two-point modelling. They will also be

used to demonstrate that modelling a four-point contact by placing two angular contact bearings with acommon center point and opposing contact angles is not always a valid technique. The only referenceavailable that models a true four-point contact bearing explicitly is A. B. Jones Four Point Ball BearingAnalysis Program4, herein referred to as Jones-4PT. This program provides detailed output for all fourcontact points. Its limitations, however, are that it only performs a static analysis and requires the inner

and outer curvatures be the same value. It therefore does not provide torque losses or accurate results formoderate speed applications. Additionally, the Jones-4PT model is not documented and there appears to

be a discrepancy in the non-linear stiffness output and some minor numerical errors when resolving theelliptical integrals needed for stress results. The Wind Turbine Design Guideline, DG03 [9], is anotherreference and provides hand calculations for estimating maximum ball load, maximum stress, torque and

L10 life. However, it is limited to a fixed curvature assumption, two discrete contact angle options, andzero internal play (resting angle and free contact angle are the same).

Three bearings, as defined in Table 3, will be used to compare the different model predictions.

Bearing ‘A’ is a fairly large cross-section bearing with tight curvature and large internal play. Bearing‘B’ has a large pitch diameter, thin cross-section, and open curvature. Bearing ‘B-DG03’ is a

modification to bearing ‘B’ to accommodate the constraints of the DG03 analysis method.

3 A. B. JONES HIGH SPEED BALL AND ROLLER BEARING ANALYSIS PROGRAM, JONES ENGINEERING COMPANY AND

DAVID A. JONES, LICENSED FOR USE BY ATEC/ANH N.TRAN, CYPRESS CA, LICENSE 060907, VERSION 5.5Q, SERIAL NO.

06140755Q1385, ISSUE DATE JULY 3, 20074 A.B.JONES FOUR POINT CONTACT BALL BEARING ANALYSIS PROGRAM (4PT), JONES ENGINEERING COMPANY AND

DAVID A. JONES, LICENSED FOR USE BY ATEC, CYPRESS, CA, UNDER LICENSE NO. 060907, VERSION 2.0B, SERIAL NO.06090720B0005, ISSUE DATE JUNE 16, 2007


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Table 3 — Bearing Parameters

Parameter Bearing 'A' Bearing 'B' Bearing ‘B-DG03’

Pitch Diameter, in. (mm) 7.3838 (187.549) 30.5 (774.7) 30.5 (774.7)Ball Diameter, in. (mm) 0.875 (22.225) 0.25 (6.35) 0.25 (6.35)No. Balls 24 189 189Radial Play, in. (mm) 0.0098 (0.249) 0.0041 (0.104) 0.0 (0.0)

βrest 26.83° 30° 45°βfree 41.12° 39.6° 45°f i .52 .543 .53f o .52 .543 .53Young's Modulus, Balls & Rings, psi. (N/m ) 2.9E+07 (2.0E+11) 2.9E+07 (2.0E+11) 2.9E+07 (2.0E+11)Poisson's Ratio of Balls & Rings 0.25 0.25 0.25

Static Load Capacity — Static load capacity for conventional bearing steel (52100) is defined perISO 76 [10] to be the external load that results in 406 ksi (2.64 GPa) mean stress on one or more of the

ball contacts. Since stress is proportional to contact normal load, and normal loads are a function ofinternal load distributions and element contact angles, the static load capacity is a convenient parameter

for comparative purposes.

Table 4 shows predicted thrust, radial and moment load capacities from the programmed model(ORBIS-4PT) and reference programs for each of the three bearings listed in Table 3. ORBIS-4PT resultsshow strong correlation with Jones-4PT results for most load directions and bearing configurations. Sincethe Jones-4PT model is expected to contain a rigorous four-point contact model, similar to the developed

model herein, good correlation between these two models is expected. With the exception of bearing ‘A’thrust and moment cases the model results correlate with Jones-4PT to within about 1%. Furtherinvestigation into the thrust and moment cases for bearing ‘A’, which consequently have about 6%difference, showed both ORBIS-4PT and Jones-4PT will predict identical ball loads and contact anglesfor a given input load but resolve the corresponding Hertzian contact stresses with some difference. Since

this study fixed the stress to 406 ksi, and stress is proportional to load to the third power, smalldifferences in resolved Hertzian contact stress account for the 6% differences in predicted static load

capacities. It was further found that these particular cases result in computation of elliptic integrals where

the modulus is very close to 1.0 and, knowing the integrals become tougher to resolve to high accuracy inthis vicinity, it is believed that numerical precision errors likely explain these differences.

The model also shows good correlation with the DG03 results for radial and moment loading.However, the predicted thrust capacity from the DG03 model appears to disagree (~10% difference) with

all other models. This difference can be explained by the fact that the DG03 analysis only uses the initialcontact angle in the equations and provides no means to adjust it due to contact deflection. For this thrustcase the contact angle increases from 45° to 52.6°; thus resulting in about 10% change to the normal ball

loads.As expected, the 2PT results correlate well with thrust loading results from the 4PT models.

However, f or bearings ‘A’ and ‘B’, which contain internal clearances that produce differences betweenthe resting angle and free contact angle, the 2PT modelling technique consistently diverges from the 4PTresults for both radial and moment load cases. In the case of bearing ‘B’ moment capacities the 2PT

model diverges by ~27%.


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Table 4 — Thrust, Radial and Moment Static Load Capacity Results

Bearing Model Thrust, lbf Radial, lbf Moment, ft-lbf

A

ORBIS-2PT 193,600 91,900 26,400

Jones-4PT 206,700 88,100 23,350

ORBIS-4PT 193,600 87,500 22,000

B

ORBIS-2PT 81,100 41,800 46,100

Jones-4PT 81,600 37,200 36,350

ORBIS-4PT 81,100 36,800 36,200

B-DG03

ORBIS-2PT 112,400 50,300 63,900

DG03 100,600 50,300 63,900

Jones-4PT 113,200 50,560 63,900

ORBIS-4PT 112,400 50,300 63,900

Four-Point Analysis Overview

Since most bearing performance parameters — such as stress, stiffness, torque, and life — are all

related to the internal load distribution it is important to compute distributions with good accuracy. FIG.8 and FIG. 9 illustrate the effects to internal load distribution and friction torque when varying the

arching, or shim size, on a gothic arch bearing that is under pure radial loading. At low resting angles(less arching) there are fewer balls available to react the applied load and thus the peak ball loads areamplified. As arching, or resting angle, is increased the distribution spreads out the applied load over

more balls. For the example shown in FIG. 8 and FIG. 9 the free contact angle is 39.6° and once theresting angle reaches this value all internal play within the bearing is taken up (Sd = 0). Further increases

to the resting angle result in internal preloading (interference fitting on the balls) and eventually the loaddistribution transitions from a subset of balls carrying the load to all balls carrying the load. Once all

balls have been loaded the load distribution takes a fixed profile and any additional shimming shifts thedistribution curve up the ordinate axis (increase in ball loads). While achieving minimum peak ball loadsis generally desirable, as shown in FIG. 9, there is a rapid torque increase once all balls are engaged(above 39.6° resting angle for the example shown).


26/31



FIG. 8 — Internal load distributions for a bearing under pure radial loading with various arching (expressed by resting

angle).

0

50

100

150

200

250

0 20 40 60 80 100 120 140 160 180 200

B a l l L o a d

Ball No.

10° 20° 30° 39.6° 41.63° 43.6°


27/31



FIG. 9 — Torque due to slip and spin bearing ‘B’ under pure radial loading with various arching (expressed by resting

angle)

Gyroscopic Moment Assessment — To investigate any modelling errors attributed to omission ofgyroscopic moments a study of constant thrust loading versus speed is performed and compared against

the 2PT model (accounts for gyroscopic moment effects). FIG. 10 and FIG. 11 show this comparison for bearing ‘A’ and bearing ‘B’. Ball normal loads, contact angles and mean stresses for a constant thrustload are plotted against shaft speed. As shown, the model shows negligible difference for all three

parameters at low speeds. Not until around 3,000 RPM is there a perceptible difference in results andeven up to 8,000 RPM these differences are not large. Maximum errors, at max speeds shown in the

figures, are: ~3% error for normal ball load, ~14% error for contact angles, and ~ 1% error for meanHertzian stresses. Bearing ‘A’ exhibits a larger divergence than bearing ‘B’, which is expected since

bearing ‘A’ will ex perience greater gyroscopic moments due to its larger ball diameter (maximum % errorin contact angle for bearing ‘B’ is ~8% at 10,000 RPM compared with 14% error on bearing ‘A’ at 8,000RPM).

0

100

200

300

400

500

600

0 10 20 30 40 50

R o l l & S

p i n T o r q u e ( i n - l b f )

Resting Angle (deg)

Free Contact Angle


28/31



(a) (b)

(c)

FIG. 10 —Comparison of 4PT model to 2PT analysis with gyroscopic moment effects, bearing ‘A’, 5,000 lbf thrust load:

(a) normal ball loads, (b) contact angles, and (c) mean Hertzian stress


29/31



(a) (b)

(c)

FIG. 11 —Comparison of 4PT model to 2PT analysis with gyroscopic moment effects, bearing ‘B’, 10,000 lbf thrust load:

(a) normal ball loads, (b) contact angles, and (c) mean Hertzian stress


30/31



Conclusion

Classical rolling element theory, as originally developed by A. B. Jones, has been extended todevelop an analytical model of four-point contact ball bearings. The model includes independent

geometry definition for both inner and outer raceways, including their associated arching, therebyallowing assessment of a broad range of manufactured bearing conditions. All necessary equations,

including detailed partial derivatives, are completely derived and provided. Key performance parameters,such as element contact stresses, contact angles, deflections, nonlinear stiffness’s, torque and L10 life are

solved. The complete model has been programmed within the ORBIS software program and various testcases covering all loading directions and different internal bearing geometry were analyzed. Modelresults were verified with various references and thoroughly discussed. A study was presented to show

neglecting gyroscopic moments has negligible impact to accuracy for slow to moderate bearing spinspeeds that extend well beyond bearing manufacturer recommendations. The model has replaced Jones’

race control theory with a minimum energy criterion; thereby allowing interfacial slip and spin to occursimultaneously within a given contact.

Analytical modelling of a four-point contact bearing by overlay of two angular contact bearings, with

opposing contact angle orientation and common center location, is shown to produce gross internal loaddistribution errors for radial and moment loading. For this reason use of a rigorous four-point model,

such as presented herein, is highly recommended.

Further Research Suggestions

The model provided herein establishes a core foundation for the analysis of gothic formed bearings.However, it is expected the model will be expanded upon to include many of the common challenges with

real-world bearing applications. For instance, items such as press fitting, thermal expansions, andclamping effects present real-world application challenges that can have significant influence on bearing

performance. Many techniques to model these conditions have developed and used on angular contact bearings. Most of these techniques and are expected to be directly applicable, or useable, in this model by

simply adjusting the initial bearing geometry accordingly. One area that was not covered herein is performance effects due to raceway runouts. This is likely of concern to large diameter, thin section,

four-point bearings. Higher aspect ratios (ratio of diameter to cross section) inherently lose their hoopstiffness and pose challenges for manufacturers to produce precise circular geometry. Raceway runoutswill invariably cause ball contact load fluctuations that will result in increased rotational resistance of the

bearing. Such effects can be further magnified when internal clearances are small or zero. One methodfor modelling this, which is showing promising results, is to superimpose a sinusoidal error function onthe circle describing the raceway center of curvature.

Another area requiring further study is the frictional losses due to wiping. The model providedherein used assumptions for no-slip bands on sliding friction. It is highly likely these bands do not exist

on contacts where a pair of raceways defines the ball spin axis yet the opposing pair of raceways is incontact and must be wiping (the entire elliptical contact is in a state of slip). Preliminary investigationsshow dramatic torque increase can be predicted by extending the slip region to include the full major

dimension of the ellipse. A fairly simple test setup could be constructed to incrementally introduce

multiple ball contact points, perhaps by application of radial load increments, and would provide helpfuldata for further model refinement.


31/31



References

[1] B. J. Hamrock and W. J. Anderson, "Arched-Outer-Race Ball-Bearing Analysis Considering

Centrifugal Forces," NASA, Cleveland, OH, 1972.

[2] D. Nelias and A. Leblanc, "Ball Motion and Sliding Friction in a Four-Contact-Point Ball Bearing," Journal of Tribology, ASME, vol. 129, no. Oct 2007, pp. 801-808, 2007.

[3] A. B. Jones, "The Mathematical Theory of Rolling-Element Bearings," in Mechanical Design and

Systems Handbook , New York, McGraw-Hill, 1964.

[4] A. R. Leveille, "The Non-Reversible Nature of Ball Bearing Internal Geometry," in REBG International Bearing Symposium, Orlando, FL, 1997.

[5] A. B. Jones, Analysis of Stresses and Deflections, Vol I, Bristol, Conn: General Motors Corp, 1946.

[6] T. A. Harris, Rolling Bearing Analysis, 4th Ed., New York: John Wiley & Sons, 2001.

[7] J. J. Chapman and R. J. Boness, "The Measurement and Analysis of Ball Motion in High SpeedDeep Groove Ball Bearings," Journal of Lubrication Technology, ASME, no. July 1975, pp. 341-348,1975.

[8] S.-9. ANSI, "Load Ratings and Fatigue Life for Ball Bearings," American National StandardsInstitute, 1990.

[9] T. A. Harris, J. H. Rumbarger and C. P. Butterfield, "Wind Turbine Design Guideline, DG03: Yawand Pitch Rolling Bearing Life," National Renewable Energy Laboratory, Golden, CO, 2009.

[10] ISO-76, "Static Load Ratings, International Organization forr Standardization," Geneva,

Switzerland, 2006.

[11] ISO-281, "Rolling bearings - Dynamic load ratings and rating life," Technical Committee ISO/TC 4,2007.

Documents

Halpin -Tran Paper R12