Proceedings of PACAM XI
Copyright © 2009 by ABCM
11th Pan-American Congress of Applied Mechanics
January 04-08, 2010, Foz do Iguaçu, PR, Brazil
INTERNAL LOADING DISTRIBUTION IN STATICALLY LOADED BALL
BEARINGS SUBJECTED TO A COMBINED RADIAL, THRUST, AND
MOMENT LOAD, INCLUDING THE EFFECTS OF TEMPERATURE AND
FIT
Mário César Ricci, [email protected]
Brazilian Institute for Space Research/Space Mechanics and Control Division (INPE/DMC)
Abstract. A new, rapidly convergent, numerical procedure for internal loading distribution computation in statically
loaded, single-row, angular-contact ball bearings, subjected to a known combined radial, thrust, and moment load, is
used to find the load distribution differences between a loaded unfitted bearing at room temperature, and the same
loaded bearing with interference fits which might experience radial temperature gradients between inner and outer
rings. For each step of the procedure it is required the iterative solution of Z + 3 simultaneous nonlinear equations –
where Z is the number of the balls – to yield exact solution for axial, radial, and angular deflections, and contact
angles. Numerical results are shown for a 218 angular-contact ball bearing.
Keywords: ball, bearing, static, load, temperature, fit
1. INTRODUCTION
Ball and roller bearings, generically called rolling bearings, are commonly used machine elements. They are
employed to permit rotary motions of, or about, shafts in simple commercial devices and also used in complex
engineering mechanisms.
This work is devoted to study of the internal loading distribution in statically loaded ball bearings. Several
researchers have studied the subject as, for example, Stribeck (1907), Sjoväll (1933), Jones (1946) and Rumbarger
(1962). The methods developed by them to calculate distribution of load among the balls and rollers of rolling bearings
can be used in most bearing applications because rotational speeds are usually slow to moderate. Under these speed
conditions, the effects of rolling element centrifugal forces and gyroscopic moments are negligible. At high speeds of
rotation these body forces become significant, tending to alter contact angles and clearance. Thus, they can affect the
static load distribution to a great extension.
Harris (2001) described methods for internal loading distribution in statically loaded bearings addressing pure
radial; pure thrust (centric and eccentric loads); combined radial and thrust load, which uses radial and thrust integrals
introduced by Sjoväll; and for ball bearings under combined radial, thrust, and moment load, initially due to Jones.
There are many works describing the parameters variation models under static loads but few demonstrate such
variations in practice, even under simple static loadings. The author believes that the lack of practical examples is
mainly due to the inherent difficulties of the numerical procedures that, in general, deal with the resolution of various
non-linear algebraic equations that must to be solved simultaneously.
In an attempt to cover this gap studies are being developed in parallel (Ricci, 2009 to Ricci, 2009e). Particularly in
this work a new, precise numerical procedure, described in Ricci (2009c), for internal load distribution computation in
statically loaded, single-row, angular-contact ball bearings subjected to a known external combined radial, thrust, and
moment load, is used to find the load distribution differences between a loaded bearing with clearance fits at room
temperature, and the same loaded bearing with interference fits, such might experience radial temperature gradients
between inner and outer rings.
In the most usual situation, angular contact bearings would first be fitted, with interference or clearance defined at
room temperature, to their respective shaft and housing; then a defined axial “hard” preload would be applied and
subsequently in operation the bearings might experience radial temperature gradients between inner and outer rings.
Ball bearings and other radial rolling bearings are designed to have a diametral clearance. Due to this radial
clearance the bearing also can experience an axial play. Removal the axial freedom causes the ball-raceway contact line
to assume an oblique angle with respect to the radial plane; hence, a contact angle different from zero will occur. This
angle is called free contact angle and is a function of radial clearance and the raceway groove curvatures.
Press or shrink fitting of the inner ring on the shaft causes the inner ring to expand slightly. Similarly, press fitting
of the outer ring in the housing causes the former member to shrink slightly. Thus, the bearing’s diametral clearance
will tend to decrease. Large amounts of interference in fitting practice can cause bearing clearance to vanish and even
produce negative clearance or interference in the bearing.
Thermal conditions of bearing operation can also affect the diametral clearance. Heat generated by friction causes
internal temperatures to rise. This in turn causes expansion of the shaft, housing, and bearing components. Depending
on the shaft and housing materials and on the magnitude of thermal gradients across the bearing and these supporting
structures, clearance can tend to increase or decrease.
Proceedings of PACAM XI
Copyright © 2009 by ABCM
11th Pan-American Congress of Applied Mechanics
January 04-08, 2010, Foz do Iguaçu, PR, Brazil
2. STATIC LOAD DISTRIBUTION UNDER COMBINED RADIAL, THRUST, AND MOMENT LOAD IN
BALL BEARINGS
Having defined in other works analytical expressions for geometry of bearings and the contact stress and
deformations for a given ball or roller-raceway contact (point or line loading) in terms of load (see, e.g., Harris, 2001) it
is possible to consider how the bearing load is distributed among the rolling elements. In this section a specific load
distribution consisting of a combined radial, thrust, and moment load, which must be applied to the inner ring of a
statically loaded ball bearing, is given.
According Ricci (2009c), let a ball bearing with a number of balls, Z, symmetrically distributed about a pitch circle
according to Fig. 1a, to be subjected to a combined radial, thrust, and moment load. Then, a relative axial displacement,
δa, a relative angular displacement, θ, and a relative radial displacement, δr, between the inner and outer ring raceways
may be expected. Let ψ = 0 to be the angular position of the maximum loaded ball.
Final position, inner raceway groove
curvature center
δa + Riθcosψ
δr
Initial position, inner raceway groove
curvature center
β
βf
s = A + δn
Acosβf
A
Outer raceway groove
curvature center fixed
(a)
(b)
Figure 1. (a) Ball angular positions in the radial plane that is perpendicular to the bearing’s axis of rotation, ∆ψ = 2π/Z,
ψj = 2π/Z(j −1), j = 1…Z; (b) Initial and final curvature centers positions at angular position ψ, with and without applied
load
Figure 1b shows the initial and final groove curvature centers positions at angular position ψ, before and after
loading, considering the centers of curvature of the raceway grooves fixed with respect to the corresponding raceway. If
δa, θ, and δr are known, the contact angle at angular position ψ, after the combined load has been applied, is given by
 A cos β f + δ r cosψ 
,

A +δn


β (ψ ) = cos −1 
(1)
where A is the distance between raceway groove curvature centers, βf is the free contact angle, and δn is the total ball
normal deflection.
Also,
δ a + Riθ cosψ = ( A + δ n )sinβ − Asinβ f ,
(2)
where
Ri = d e / 2 + ( f i − 0.5)D cos β f
expresses the locus of the centers of the inner ring raceway groove curvature radii, with de being the bearing pitch
diameter, D the ball diameter, and fi the inner race conformity ratio.
From Fig. 1b we can arrive in the transcendental equations for the extend of the loading zone, ψl, that are
cos β l − cos β f =
δr
A
cosψ l
(3)
Proceedings of PACAM XI
Copyright © 2009 by ABCM
11th Pan-American Congress of Applied Mechanics
January 04-08, 2010, Foz do Iguaçu, PR, Brazil
and
sinβ l − sinβ f =
δa
Riθ
cosψ l ,
A
+
A
(4)
where βl is the contact angle regarding the end of loading zone.
From Eq. 1, the total normal approach between two raceways at angular position ψ, after the combined load has
been applied, can be written as
 cos β f
 δ cosψ
− 1 + r
.
cos β

δ n (ψ ) = A
 cos β
(5)
From Fig. 1b and Eq. 5 it can be determined that s, the distance between the centers of the curvature of the inner
and outer ring raceway grooves at any rolling element position ψ, is given by
s (ψ ) = A + δ n = A
cos β f
cos β
+
δ r cosψ
.
cos β
(6)
From Eqs. 2 and 6 yields, for ψ = ψj,
δ a + Riθ cosψ j − δ r tan β j cosψ j − A
(
sin β j − β f
cos β j
)=0,
j = 1,…, Z.
(7)
j = 1,…, Z.
(8)
From load-deflection relationship for ball bearings and Eq. 5 yields, for ψ = ψj,
  cos β f
 δ r cosψ j 

Q j = K nj  A
− 1 +


cos β j 
  cos β j


3/ 2
,
If a thrust load, Fa, a radial load, Fr, and a moment, M, are applied then, for static equilibrium to exist
Z
Fa =
∑ Q sinβ
,
(9)
cos β j cosψ j ,
(10)
j
j
j =1
Z
Fr =
∑Q
j
j =1
and
Z
M=
∑ Q sinβ [(R + δ
j
j
i
r
]
)
cosψ j cosψ j − δ r .
(11)
j =1
Substitution of Eq. 8 into Eq. 9 yields
  cos β f
 δ r cosψ j 

− 1 +
Fa −
K nj sinβ j  A

cos β j 
  cos β j
j =1


Z
∑
3/ 2
=0.
(12)
Similarly,
  cos β f
 δ r cosψ j 

Fr −
K nj cosψ j cos β j  A
− 1 +

cos β j 
  cos β j
j =1


Z
∑
3/ 2
= 0,
(13)
and
  cos β f
 δ r cosψ j 

K nj  A
− 1 +

cos β j 
  cos β j
j =1


Z
M−
∑
3/ 2
[(
)
]
sinβ j Ri + δ r cosψ j cosψ j − δ r = 0 .
(14)
Proceedings of PACAM XI
Copyright © 2009 by ABCM
11th Pan-American Congress of Applied Mechanics
January 04-08, 2010, Foz do Iguaçu, PR, Brazil
Equations 7, 12, 13 and 14 are Z + 3 simultaneous nonlinear equations with unknowns δa, δr, θ, and βj, j = 1,…, Z. Since
Knj are functions of final contact angle, βj, the equations must be solved iteratively to yield an exact solution for δa, δr, θ
and βj.
3. EFFECTS OF INTERFERENCE FITTING, THERMAL GRADIENTS, AND SURFACE FINISH ON
CLEARANCE
In this section, the principal relationships between interference fittings, thermal gradients, surface finish and
changes in diametral clearance are summarized. As described in Harris (2001), the increase in the inner raceway
diameter, di, due a press fitting between a bearing inner ring and a shaft of hole diameter d2, is given by
∆s =
[(d / d ) − 1] ((dd // dd ))
i
b
2

i
b
i
b
2 Id i / d b
2
+1
2
−1
+ν b +
(15)
 
Eb  (d b / d 2 ) + 1
−ν s 

E s  (d b / d 2 )2 − 1
 
2
,
where I, db, Eb, Es, υb and υs are diametral interference, bearing inner diameter, modulus of elasticity for inner ring and
shaft, and poisson’s ratio for inner ring and shaft, respectively.
If the bearing inner ring and shaft are both fabricated from the same material, then
∆s = I
di
db
 (d b / d 2 )2 − 1  .


2
 (d i / d 2 ) − 1 
(16)
For a bearing inner ring mounted on a solid shaft of the same material, diameter d2 is zero and
∆ s = Id b / d i .
(17)
Similarly, the decrease in the outer raceway diameter, do, due a press fitting between a bearing outer ring and a
housing of outside diameter d1, is given by
∆h =
2 Id a / d o
[(d
a
 (d / d )2 + 1
 
E  (d / d )2 + 1
/ d o )2 − 1  a o 2
−ν b + b  1 a 2
+ ν h 
E h  (d1 / d a ) − 1
 
 (d a / d o ) − 1
]
(18)
,
where da is the bearing outer diameter.
If the bearing outer ring and housing are both fabricated from the same material, then
∆h = I
d a  (d1 / d a )2 − 1  .


d o  (d1 / d o )2 − 1 
(19)
For a bearing outer ring mounted inside a solid housing of the same material, diameter d1 approaches infinity and
∆ h = Id o / d a .
(20)
A reduction in I due to surface finish must be taking into account (Harris, 2001).
Now, considering bearing outer and inner rings at temperatures To – Ta and Ti – Ta above ambient, respectively, the
approximate increases in do and di are Γbdo(To – Ta) and Γbdi(Ti – Ta), respectively. Thus the diametral clearance
increase due to thermal expansion is
∆ T = Γb d o (To − Ta ) + Γb d i (Ti − Ta ) .
(21)
When the housing and shaft are not fabricated from the same material (usually steel) as the bearing, an increase in
the interference can be wait, that are given by (Γb – Γh)da(To – Ta) and (Γs – Γb)db(Ti – Ta), respectively.
Considering a bearing having a clearance Pd prior to mounting at room temperature, the change in clearance, after
mounting with bearing outer and inner rings at temperatures To and Ti, respectively, is given by
∆Pd = ∆ T − ∆ s − ∆ h .
(22)
Proceedings of PACAM XI
Copyright © 2009 by ABCM
11th Pan-American Congress of Applied Mechanics
January 04-08, 2010, Foz do Iguaçu, PR, Brazil
4. NUMERICAL RESULTS
To show an application of the theory developed in this work a numerical example is presented, which uses the
Newton-Rhapson method to solve the simultaneous nonlinear equations 7, 12, 13 and 14.
I have chosen the 218 angular-contact ball bearing as example, which was also used by Harris. The 218 angularcontact ball bearing has a 0.09 m bore, a 0.16 m o.d. and is manufactured to ABEC 7 tolerance limits. The bearing is
mounted on a hollow steel shaft of 0.0635 m bore with a k6 fit and in a titanium housing having a effective o.d. of
0.2032 m with an M6 fit. Considers that the inner ring operates at a mean temperature of 148.9oC, that the outer ring is
at 121.1oC and that the bearing was assembled at 21.1oC.
There are three steps in the numerical procedure. The first, considering the bearing unfitted at assembling
temperature; the second, considering the fits above at assembling temperature; and the third, considering the fits above
at operational temperatures for the inner and outer rings. Before each step the geometry of the bearing is obtained from
which, the nonlinear equations are solved simultaneously to obtain radial and axial deflections and contact angles.
Figures 2 to 4 show some parameters, as functions of the applied thrust load, for the three steps of the procedure,
for some values of the applied radial load and for moment values of 0 Nm and 100 Nm.
Figures 2a and 2b show the normal ball loads for the maximum loaded ball and for the loaded ball located at ψ =
180o, respectively. For higher applied radial loads and for the range of thrust and moment loads adopted, the normal ball
load of the maximum loaded ball is nearly constant. The normal ball load of the loaded ball located at ψ = 180o increase
monotonically with the thrust load for all values of the applied radial and moment loads considered.
10000
M = 0 Nm
4
1.8x10 N
8000
6000
4
1.4x10 N
4
4000
1.0x10 N
4
2000
0.6x10 N
4
0
0.2
0.6
0.8
1
___ unfitted-T = 21.1°C
a
_ _ fitted-T = 21.1°C
a
.....
fitted-Ti = 148.9°C, To = 121.1°C
3000
4
1.8x10 N
4
1.4x10 N
2500
M = 0 Nm
4
0.6x10 N
2000
4
1.0x10 N
1500
4
Fr = 0.2x10 N
1000
M = 100 Nm
500
M = 100 Nm
Fr = 0.2x10 N
0.4
Radial, Thrust, and Moment Load - 218 Angular-contact Ball Bearing
3500
Normal ball load, Q(ψ = 180 ° ) [N]
Normal ball load for the maximum
loaded ball, Q(ψ = 0) [N]
Radial, Thrust, and Moment Load - 218 Angular-contact Ball Bearing
___ unfitted-T = 21.1°C
a
_ _ fitted-T = 21.1°C
a
.....
fitted-Ti = 148.9°C, To = 121.1°C
1.2
1.4
1.6
1.8
0
0.2
2
Thrust load, Fa [N]
0.4
0.6
0.8
1
1.2
1.4
1.6
1.8
Thrust load, Fa [N]
4
x 10
(a)
2
4
x 10
(b)
Figure 2. Normal ball load as a function of the thrust load, Fa. (a) for the maximum loaded ball, Q(ψ = 0); (b) for the
loaded ball located at ψ = 180o, Q(ψ = 180o)
Figures 3a and 3b show the contact angle for the maximum loaded ball and for the loaded ball located at ψ = 180o,
respectively. The straight lines represent the free contact angles for the three steps of the procedure. For higher applied
radial loads and for the range of thrust and moment loads adopted, the contact angle of the maximum loaded ball (of the
ball located at ψ = 180o) is always less than (greater than) the free contact angles.
Radial, Thrust, and Moment Load - 218 Angular-contact Ball Bearing
Contact angle for the maximum
loaded ball, β (ψ = 0) [° ]
45
M = 0 Nm
βf
M = 100 Nm
65
4
0.6x10 N
30
25
20
15
4
1.4x10 N
10
5
0.2
4
1.8x10 N
4
1.0x10 N
0.4
0.6
0.8
___ unfitted-T = 21.1°C
a
_ _ fitted-T = 21.1°C
a
.....
fitted-Ti = 148.9°C, To = 121.1°C
4
Fr = 0.2x10 N
40
35
Radial, Thrust, and Moment Load - 218 Angular-contact Ball Bearing
70
___ unfitted-T = 21.1°C
a
_ _ fitted-T = 21.1°C
a
.....
fitted-Ti = 148.9°C, To = 121.1°C
Contact angle, β ( ψ = 180 ° ) [° ]
50
1
1.2
Thrust load, Fa [N]
4
1.4x10 N
4
1.8x10 N
60
55
50
M = 100 Nm
45
4
Fr = 0.2x10 N 0.6x10 4 N 1.0x10 4 N
βf
M = 0 Nm
40
1.4
1.6
1.8
2
0.2
0.4
4
x 10
(a)
0.6
0.8
1
1.2
Thrust load, Fa [N]
1.4
1.6
1.8
2
4
x 10
(b)
Figure 3. Contact angle as a function of the thrust load, Fa. (a) for the maximum loaded ball, β(ψ = 0); (b) for the loaded
ball located at ψ = 180o, β(ψ = 180o)
Figures 4a and 4b show the axial and radial deflections, respectively.
Proceedings of PACAM XI
Copyright © 2009 by ABCM
5
Axial deflection, δ a [m]
Radial, Thrust, and Moment Load - 218 Angular-contact Ball Bearing
-4
4
___ unfitted-T = 21.1°C
a
_ _ fitted-T = 21.1 C
°
a
.....
fitted-Ti = 148.9°C, To = 121.1°C
4
Fr = 0.2x10 N
0
4
1.8x10 N
-5
4
1.4x10 N
4
0.6x10 N
-10
x 10
Radial, Thrust, and Moment Load - 218 Angular-contact Ball Bearing
___ unfitted-T = 21.1°C
a
_ _ fitted-T = 21.1°C
a
.....
fitted-Ti = 148.9°C, To = 121.1°C
Radial deflection, δ r [m]
-5
x 10
11th Pan-American Congress of Applied Mechanics
January 04-08, 2010, Foz do Iguaçu, PR, Brazil
4
1.0x10 N
4
1.0x10 N
3
2
4
0.6x10 N
1
4
Fr = 0.2x10 N
M = 0 Nm
M = 100 Nm
M = 100 Nm
0.4
0.6
0.8
4
1.8x10 N
M = 0 Nm
-15
0.2
4
1.4x10 N
1
1.2
Thrust load, Fa [N]
1.4
1.6
1.8
2
0
0.2
0.4
0.6
0.8
4
x 10
(a)
1
1.2
1.4
1.6
1.8
Thrust load, Fa [N]
2
4
x 10
(b)
Figure 4. Deflections as a function of the thrust load, Fa. (a) axial deflection, δa; (b) radial deflection, δr
5. CONCLUSIONS
The importance of this work lies in the fact that it uses a new procedure for get numerically, accurately and quickly,
the static load distribution of a ball bearing under radial, thrust, and moment loading, taking into account the influence
of fits and thermal gradients. Precise applications, as for example, space applications, require a precise determination of
the static loading. Models available in literature are approximate and often are not compatible with the desired degree of
accuracy. This work can be extended to determine the loading on high-speed bearings where centrifugal and gyroscopic
forces do not be discarded. The results of this work can be used in the accurate determination of the friction torque of
the ball bearings, under any operating condition of temperature and speed.
6. REFERENCES
Harris, T., 2001. “Rolling Bearing Analysis”, 4th ed., John Wiley & Sons Inc., New York.
Jones, A., 1946. “Analysis of Stresses and Deflections”, New Departure Engineering Data, Bristol, Conn.
Ricci, M. C., 2009. “Ball bearings subjected to a variable eccentric thrust load”, DINCON’09 Proceedings of the 8th
Brazilian Conference on Dynamics, Control and Applications, May, 18-22, Bauru, Brazil. ISBN: 978-85-86883-453.
Ricci, M. C., 2009a. “Internal loading distribution in statically loaded ball bearings”, ICCCM09 1st International
Conference on Computational Contact Mechanics, Program and Abstracts, p. 21-22, Sept. 16-18, Lecce, Italy.
Ricci, M. C., 2009b. “Internal loading distribution in statically loaded ball bearings subjected to a combined radial and
thrust load, including the effects of temperature and fit”, Proceedings of World Academy of Science, Engineering
and Technology, Volume 57, September 2009, WCSET 2009, Amsterdam, Sept. 23-25. ISSN: 2070-3724.
Ricci, M. C., 2009c. “Internal loading distribution in statically loaded ball bearings subjected to a combined radial and
thrust load”, 6th ICCSM Proceedings of the 6th International Congress of Croatian Society of Mechanics, Sept. 30
to Oct. 2, Dubrovnik, Croatia. ISBN 978-953-7539-11-5.
Ricci, M. C., 2009d. “Internal loading distribution in statically loaded ball bearings subjected to a combined radial,
thrust, and moment load”, Proceedings of the 60th International Astronautical Congress, October, 12-16, Daejeon,
South Korea. ISSN 1995-6258.
Ricci, M. C., 2009e. “Internal loading distribution in statically loaded ball bearings subjected to an eccentric thrust
load”, accepted to publication in Mathematical Problems in Engineering.
Rumbarger, J., 1962. “Thrust Bearings with Eccentric Loads”, Mach. Des., Feb. 15.
Sjoväll, H., 1933. “The Load Distribution within Ball and Roller Bearings under Given External Radial and Axial
Load”, Teknisk Tidskrift, Mek., h.9.
Stribeck, R., 1907. “Ball Bearings for Various Loads”, Trans. ASME 29, 420-463.
7. RESPONSIBILITY NOTICE
The author is the only responsible for the printed material included in this paper.