Machine Design

COMPILED BY:
Rolling-Contact Bearings
ENGR. JR ESQUILLO
Rolling-Contact Bearings
• The terms rolling-contact bearing, antifriction bearing,
and rolling bearing are all used to describe that class of
bearing in which the main load is transferred through
elements in rolling contact rather than in sliding contact.
• In a rolling bearing the starting friction is about twice the
running friction, but still it is negligible in comparison with
the starting friction of a sleeve bearing. Load, speed, and
the operating viscosity of the lubricant do affect the
frictional characteristics of a rolling bearing. It is probably
a mistake to describe a rolling bearing as “antifriction,”
but the term is used generally throughout the industry.
Nomenclature of a Ball Bearing

Fig. 11–1
Types of Ball Bearings

Fig. 11–2
 Types of Roller Bearings

Straight Cylindrical Spherical Roller, thrust Tapered roller, thrust

Needle Tapered roller Steep-angle tapered roller

Fig. 11–3
Bearing Life Definitions
• Bearing Failure: Spalling or pitting of
an area of 0.01 in2
• Life: Number of revolutions (or hours
@ given speed) required for failure.
• Rating Life: Life required for 10% of
sample to fail.
• For a group of bearings
• Also called Minimum Life or L10 Life
• Median Life: Average life required for
50% of sample to fail.
• For many groups of bearings
• Also called Average Life or Average
Median Life
• Median Life is typically 4 or 5 times the L10
Life
 Load Rating Definitions
• Catalog Load Rating, C10: Constant radial load that
causes 10% of a group of bearings to fail at the bearing
manufacturer’s rating life.
• Depends on type, geometry, accuracy of fabrication, and
material of bearing
• Also called Basic Dynamic Load Rating, and Basic Dynamic
Capacity
• Basic Load Rating, C: A catalog load rating based on a
rating life of 106 revolutions of the inner ring.
• The radial load that would be necessary to cause failure at such
a low life is unrealistically high.
• The Basic Load Rating is a reference value, not an actual load.
 Load Rating Definitions
• Static Load Rating, Co:
Static radial load which corresponds to a permanent
deformation of rolling element and race at the most
heavily stressed contact of 0.0001d.
• d = diameter of roller
• Used to check for permanent deformation
• Used in combining radial and thrust loads into an equivalent
radial load
• Equivalent Radial Load, Fe:
Constant stationary load applied to bearing with rotating
inner ring which gives the same life as actual load and
rotation conditions.
 Load-Life Relationship
• Nominally identical groups of bearings are tested to the life-
failure criterion at different loads.
• A plot of load vs. life on log-log scale is approximately linear.
 Using a regression equation
to represent the line,

◦ a = 3 for ball bearings

◦ a = 10/3 for roller bearings
(cylindrical and tapered
roller)

Fig. 11–4
 Load-Life Relationship
• Applying Eq. (11–1) to two load-life conditions,

• Denoting condition 1 with R for catalog rating conditions,

and condition 2 with D for the desired design conditions,

• The units of L are revolutions. If life is given in hours at

a given speed n in rev/min, applying a conversion of 60
min/h,
 Load-Life Relationship
• Incorporating Eq. (b) into Eq. (a),
 Load-Life Relationship
• Solving Eq. (a) for FR, which is just another notation for
the catalog load rating,

• The desired design load FD and life LD come from the

problem statement.
• The rated life LR will be stated by the specific bearing
manufacturer. Many catalogs rate at LR = 106 revolutions.
• The catalog load rating C10 is used to find a suitable
bearing in the catalog.
 Load-Life Relationship
• It is often convenient to define a dimensionless multiple
of rating life
 Reliability vs. Life
• At constant load, the life measure distribution is right
skewed.
• The Weibull distribution is a good candidate.
• Defining the life measure in dimensionless form as x =
L/L10, the reliability is expressed with a Weibull
distribution as
 Reliability vs. Life
• An explicit expression for the cumulative distribution
function is
 Relating Load, Life, and Reliability
• Catalog information is at point A, at coordinates C10 and
x10=L10/L10=1, on the 0.90 reliability contour.
• The design information is at point D, at coordinates FD and
xD, on the R=RD reliability contour.
• The designer must move from point D to point A via point B.

Fig. 11–5
Shigley’s Mechanical Engineering Design
 Relating Load, Life, and Reliability
• Along a constant reliability contour (BD), Eq. (11–2)
applies:

Fig. 11–5
Shigley’s Mechanical Engineering Design
 Relating Load, Life, and Reliability
• Along a constant load line (AB), Eq. (11–4) applies:

• Solving for xB,

Fig. 11–5
Shigley’s Mechanical Engineering Design
 Relating Load, Life, and Reliability
• Substituting xB into Eq. (a),

• Noting that FB = C10, and including an application factor

af,

• Note that when RD = 0.90, the denominator equals one

and the equation reduces to Eq. (11–3).
 Weibull Parameters
• The Weibull parameters x0, q, and b are usually provided by
the catalog.
• Typical values of Weibull parameters are given on p. 608 at
the beginning of the end-of-chapter problems and shown
below.
• Manufacturer 1 parameters are common for tapered roller
bearings
• Manufacturer 2 parameters are common for ball and straight
roller bearings
 Relating Load, Life, and Reliability

• Eq. (11–6) can be simplified slightly for calculator entry.

Note that

where pf is the probability for failure

• Thus Eq. (11–6) can be approximated by
Combined Reliability of Multiple
Bearings
• If the combined reliability of multiple bearings on a
shaft, or in a gearbox, is desired, then the total reliability
is equal to the product of the individual reliabilities.
• For two bearings on a shaft, R = RARB
• If the bearings are to be identical, each bearing should
have a reliability equal to the square root of the total
desired reliability.
• If the bearings are not identical, their reliabilities need
not be identical, so long as the total reliability is realized.
 Combined Radial and Thrust Loading
• When ball bearings carry both
an axial thrust load Fa and a
radial load Fr, an equivalent
radial load Fe that does the
same damage is used.
• A plot of Fe/VFr vs. Fa/VFr is
obtained experimentally.
• V is a rotation factor to account
for the difference in ball
rotations for outer ring rotation
vs. inner ring rotation.
• V = 1 for inner ring rotation
• V = 1.2 for outer ring rotation
Fig. 11–6
 Combined Radial and Thrust Loading
• The data can be approximated
by two straight lines

• X is the ordinate intercept and

Y is the slope
• Basically, indicates that Fe
equals Fr for smaller ratios of
Fa/Fr, then begins to rise when
Fa/Fr exceeds some amount e
Fig. 11–6
 Combined Radial and Thrust Loading
• It is common to express the
two equations as a single
equation

where
i = 1 when Fa/VFr ≤ e
i = 2 when Fa/VFr > e

• X and Y factors depend on

geometry and construction of
the specific bearing.
Fig. 11–6
Equivalent Radial Load Factors for Ball Bearings

• X and Y for specific bearing obtained from bearing catalog.

• Table 11–1 gives representative values in a manner common
to many catalogs.
Table 11–1

Shigley’s Mechanical Engineering Design

 Equivalent Radial Load Factors for Ball Bearings

Table 11–1

• X and Y are functions of e, which is a function of Fa/C0.

• C0 is the basic static load rating, which is tabulated in the catalog.
 Dimension-Series Code
 ABMA standardized dimension-series code represents the
relative size of the boundary dimensions of the bearing cross
section for metric bearings.
 Two-digit series number
 First digit designates the width series
 Second digit designates the diameter series
 Specific dimensions are tabulated in catalogs under a specific
series

Shigley’s Mechanical Engineering Design

Fig. 11–7
• Figure 11–8 Shaft and
housing shoulder
diameters ds and dh
should be adequate to
ensure good bearing
support.
Representative Catalog Data for Ball Bearings (Table 11–2)
Representative Catalog Data for Cylindrical Roller Bearings (Table 11–3)
Bearing Life Recommendations
(Table 11–4)
Recommended Load Application
Factors (Table 11–5)
Sample Problem
• A gear-reduction unit uses the countershaft depicted
in the figure. Find the two bearing reactions. The
bearings are to be angular-contact ball bearings,
having a desired life of 50 kh when used at 300
rev/min. Use 1.2 for the application factor and a
reliability goal for the bearing pair of 0.96. Select the
bearings from Table 11–2
Given
• Bearing type: angular−contact ball bearings
• Desired life of 50 kh when used at 300 rev/min.
• Use 1.2 for the application factor
• Reliability goal for the bearing pair of 0.96.
Required
• Find the two bearing reactions
• Select the bearings from Table 11–2
 Solution
• Draw the FBD Diagram

240 lbf sin 20 F sin 25

240 lbf F
Y Gear 3 Y
24 in diameter
20 deg 25 deg
240 lbf cos 20 F cos 25

Z Raz Z Rbz
Gear 4
A B 12 in diameter

Ray Rby
At (y-z plane)

෍ 𝑇𝑂 = 0

24 12
240 𝑙𝑏𝑓 cos 20° 𝑖𝑛 − 𝐹 cos 25° 𝑖𝑛 = 0
2 2

𝐹 = 497.68 𝑙𝑏𝑓
 At (y-z plane)
෍ 𝑓𝑧 = 0
240 lbf sin 20
240 lbf
Y Gear 3 240 𝑙𝑏𝑓 cos 20° − 𝑅𝑎𝑧 = 0
24 in diameter
20 deg 𝑅𝑎𝑧 = 225.53 𝑙𝑏𝑓
240 lbf cos 20

෍ 𝑓𝑦 = 0
Z Raz
A 𝑅𝑎𝑦 − 240 𝑙𝑏𝑓 sin 20° = 0

Ray 𝑅𝑎𝑦 = 82.08 𝑙𝑏𝑓

 At (y-z plane)
෍ 𝑓𝑧 = 0

𝑅𝑏𝑧 − F cos 25° = 0

F sin 25 F
Y 𝑅𝑏𝑧 − 497.68 𝑙𝑏𝑓 cos 25° = 0

𝑅𝑏𝑧 = 451.05 𝑙𝑏𝑓

25 deg
F cos 25 ෍ 𝑓𝑦 = 0

𝑅𝑏𝑦 − 𝐹 sin 25° = 0

Z Rbz
Gear 4
B 12 in diameter 𝑅𝑏𝑦 − 497.68 𝑙𝑏𝑓 sin 25° = 0

𝑅𝑏𝑦 = 210.33 𝑙𝑏𝑓

Rby
 • Draw the FBD Diagram (Shaft only)
y
Roz
• To balance the downward
Ray
force of Ray and Rby, Roy and
Rcy will be in UPWARD
O
Rby direction.
z
Roy A Rbz
• To balance the force of Raz
Raz and Rbz, assume the
B direction of Roz and Rcz. If
16 “
the reaction of Roz and Rcz is
C negative, change the
x
14 “ assumed direction
Rcz
Rcy
12 “
 • Draw the FBD Diagram (Shaft only)
y
Roz
• To balance the downward
Ray
force of Ray and Rby, Roy and
Rcy will be in UPWARD
O
Rby direction.
z
Roy A Rbz
• To balance the force of Raz
Raz and Rbz, assume the
B direction of Roz and Rcz. If
16 “
the reaction of Roz and Rcz is
C negative, change the
x
14 “ assumed direction
Rcz
Rcy
12 “
 At X-Z Plane
• Reactions
Roz Rbz ෍ 𝑀𝑜𝑦 = 0

x
−𝑅𝑎𝑧 16" + 𝑅𝑏𝑧 16" + 14"
O A B C
− 𝑅𝑐𝑧 16+14 + 12" = 0
Raz Rcz
−225.53 𝑙𝑏𝑓 16"
+ 451.05 𝑙𝑏𝑓 30" − 𝑅𝑐𝑧 42"
16 “ 14 “ 12 “
=0
𝑅𝑐𝑧 = 236.26 𝑙𝑏𝑓
z
 At X-Z Plane
• Reactions
Roz Rbz ෍ 𝑓𝑧 = 0

O A B C x 𝑅𝑎𝑧 + 𝑅𝑐𝑧 − 𝑅𝑏𝑧 − 𝑅𝑜𝑧

=0
Raz Rcz
225.53 𝑙𝑏𝑓 + 236.26 𝑙𝑏𝑓
16 “ 14 “ 12 “ − 451.05 𝑙𝑏𝑓 − 𝑅𝑜𝑧 = 0

z 𝑅𝑜𝑧 = 10.74 𝑙𝑏𝑓

 At X-Y Plane
• Reactions
y Ray Rby ෍ 𝑀𝑜𝑧 = 0

𝑅𝑎𝑦 16" + 𝑅𝑏𝑦 16" + 14"

x
O A B C − 𝑅𝑐𝑦 16+14 + 12" = 0
Roy Rcy
82.08 𝑙𝑏𝑓 16"
+ 210.33 𝑙𝑏𝑓 30" − 𝑅𝑐𝑦 42"
16 “ 14 “ 12 “ =0

𝑅𝑐𝑦 = 181.50 𝑙𝑏𝑓

 At X-Y Plane
• Reactions
y Ray Rby ෍ 𝑓𝑦 = 0

x
𝑅𝑜𝑦 + 𝑅𝑐𝑦 − 𝑅𝑎𝑦 − 𝑅𝑏𝑦 = 0
O A B C

Roy Rcy 𝑅𝑜𝑦 + 181.50 𝑙𝑏𝑓 − 82.08 𝑙𝑏𝑓

− 210.33 𝑙𝑏𝑓 = 0
16 “ 14 “ 12 “
𝑅𝑜𝑦 = 110.91 𝑙𝑏𝑓
• For bearing reaction at O

𝑅𝑜 = 𝑅𝑜𝑧 2 + 𝑅𝑜𝑦 2

𝑅𝑜 = 10.74 𝑙𝑏𝑓 2 + 110.91 𝑙𝑏𝑓 2

𝑅𝑜 = 111.43 𝑙𝑏𝑓
• For bearing reaction at C

𝑅𝑐 = 𝑅𝑐𝑧 2 + 𝑅𝑐𝑦 2

𝑅𝑐 = 236.26 𝑙𝑏𝑓 2 + 181.50 𝑙𝑏𝑓 2

𝑅𝑐 = 297.93 𝑙𝑏𝑓
• For bearing selection
• Determine the multiple of rating life, 𝑥𝐷
𝐿𝐷
𝑥𝐷 =
𝐿𝑅

𝐿𝐷 = 60ℒ𝐷 𝑛𝐷
60ℒ𝐷 𝑛𝐷
𝑥𝐷 =
𝐿𝑅

• From the given

• Desired life, ℒ𝐷 = 50 𝑘ℎ 𝑜𝑟 50,000 ℎ𝑟

𝑟𝑒𝑣
• Desired speed, 𝑛𝐷 = 300
𝑚𝑖𝑛
• For rating life, 𝐿𝑅
• Manufacturer 2 parameters are common for ball and straight
roller bearings

• use rating life, 𝐿𝑅 = 106

• For bearing selection
• Determine the multiple of rating life, 𝑥𝐷

60ℒ𝐷 𝑛𝐷
𝑥𝐷 =
𝐿𝑅
𝑚𝑖𝑛 𝑟𝑒𝑣
60 50,000 ℎ𝑟 300
ℎ𝑟 𝑚𝑖𝑛
𝑥𝐷 =
106

𝑥𝐷 = 900
• For bearing at O and C
1
𝑥𝐷 𝑎
𝐶10 = 𝑎𝑓 𝐹𝐷 1
1 𝑏
𝑥0 + 𝜃 − 𝑥0 ln
𝑅𝐷

• Where,
• Application factor, 𝑎𝑓 = 1.2 ⇒ 𝑓𝑟𝑜𝑚 𝑡ℎ𝑒 𝑔𝑖𝑣𝑒𝑛
• 𝐹𝐷 = 𝑑𝑒𝑠𝑖𝑔𝑛 𝑙𝑜𝑎𝑑
• 𝑥𝐷 = multiple of rating life
• 𝑎 = 𝑓𝑢𝑛𝑐𝑡𝑖𝑜𝑛 𝑜𝑓 𝑟𝑜𝑙𝑙 𝑣𝑠 𝑏𝑎𝑙𝑙 𝑏𝑒𝑎𝑟𝑖𝑛𝑔
• 𝑥0 , 𝜃, 𝑏 = 𝑤𝑒𝑖𝑏𝑢𝑙𝑙 𝑝𝑎𝑟𝑎𝑚𝑒𝑡𝑒𝑟𝑠
• 𝑅𝐷 = 𝑑𝑒𝑠𝑖𝑔𝑛 𝑟𝑒𝑙𝑖𝑎𝑏𝑖𝑙𝑖𝑡𝑦
• Function of roll vs ball bearing
• a = 3 for ball bearings
• a = 10/3 for roller bearings (cylindrical and tapered roller)

• Weibull parameters
• Design Reliability, 𝑅𝐷
• If the bearings are to be identical, each bearing should have
a reliability equal to the square root of the total desired
reliability.

𝑅𝐷 = 0.96

𝑅𝐷 = 0.98
• For bearing at O
1
𝑥𝐷 𝑎
𝐶10 = 𝑎𝑓 𝐹𝐷 1
1 𝑏
𝑥0 + 𝜃 − 𝑥0 ln
𝑅𝐷
1
900 3
𝐶10 = 1.2 111.43 𝑙𝑏𝑓 1
1 1.483
0.02 + 4.59 − 0.02 ln
0.980

𝐶10 = 1833.62 𝑙𝑏𝑓

4.448 𝑁 1 𝑘𝑁
𝐶10 = 1833.62 𝑙𝑏𝑓 𝑥 𝑥
1 𝑙𝑏𝑓 1000 𝑁

𝐶10 = 8.16 𝑘𝑁
• From Table 11-
2
• For bearing at
O, use an
angular contact
02-17 mm.
• For bearing at C
1
𝑥𝐷 𝑎
𝐶10 = 𝑎𝑓 𝐹𝐷 1
1 𝑏
𝑥0 + 𝜃 − 𝑥0 ln
𝑅𝐷
1
900 3
𝐶10 = 1.2 297.93 𝑙𝑏𝑓 1
1 1.483
0.02 + 4.59 − 0.02 ln
0.980

𝐶10 = 4902.54 𝑙𝑏𝑓

4.448 𝑁 1 𝑘𝑁
𝐶10 = 4902.54 𝑙𝑏𝑓 𝑥 𝑥
1 𝑙𝑏𝑓 1000 𝑁

𝐶10 = 21.81 𝑘𝑁
• From Table 11-
2
• For bearing at
C, use an
angular contact
02-35 mm.
Thank you