Lecture5 Bearings

ME 350
1
ME 350
Mechanical Design and
Manufacturing II
Transmission Support
Bearings （轴承）
ME 350
2
Lectures

Power/Energy
Conversion
(Electrical Motors)
Power/Energy
Transmission
(Gears,
Belt Drives,
Power Screws)
Transmission Transmission
Support Support
(Bearings) (Bearings)
Joints
(Fasteners,
Connectors)
Structural
Support
(Frames
Shafts
Axles
Spindles)
Tools
Stress Analysis,
Failure Theories
Dynamics, Statics, Etc….
ME 350
3
Why Bearings
Bearing is defined by Webster’s to be “a 
support or supporting part”
– In machine design, a bearing is a 
component that allows for relative motion 
between two bodies
Bearings are for:

reduce friction

carry load 

guide of moving parts
ME 350
4
History
ME 350
5
Reconstruction of Old Bearing
ME 350
6
Development of the Bearing
700 B.C.
3500 B.C.
40 A.D.
1794 A.D.
1869
1995
ME 350
7
Sliding or Rolling Bearings
Sliding bearings:
sliding friction
µ
Rolling bearings:
rolling friction
µ
ME 350
8
Principles of Operation
Rolling Friction (Rolling Bearing 滚动
轴承 )
Roller/ball
（滚子 / 球）
Lubrication
（润滑剂）
Outer Ring （外圈）
Inner Ring （内圈
）
Sliding Friction (Sleeve Bearing 滑动轴
承 )
Sleeve （轴瓦）
Lubrication
Circumferential
pressure profile
Hydrodynamic lift is
generated by fluid being
dragged into gap by
viscous shear
ME 350
9
Types of Antifriction
Bearings
Ball Bearings
内
径
外
径
端面
外圈
滚道
内圈
滚道
保持
架
ME 350
10
Bearings Components
Seal Rolling elements Inner ring
           Outer ring Cage             Seal
ME 350
11
Types of Antifriction
Bearings
Tapered
Roller
Bearings
保持架
•
Components:

Cone
=
Inner ring

Cup
=
Outer ring

Tapered
rollers

Cage
=
Space
retainer
ME 350
12
Types of Antifriction
Bearings
ME 350
13
Roller and Ball Contact
Area/Form
ME 350
14
Rolling Elements
Cylindrical
Needle
Taper
Ball
Spherical
Asymmetrical
ME 350
15
Types of Ball Bearings
深沟
球
有装球缺
口
角接触
带防尘盖
带密封圈
带球面外衬
圈
双列
球
双列自动调
心
单向推
力
单向推力带
球面座圈
ME 350
16
Types of Ball Bearings
•
Single Row Deep Groove (Conrad) BB
（单列深沟球轴承）

Spherical balls roll in deep groove in both races

Space maintained by separators (retainers/cages)

Ball radius smaller than groove radius

Mostly take radial loads, some thrust load

Theoretical point contact (actually a small circular area), so
high local contact stress

Some permissible misalignment
ME 350
17
Types of Ball Bearings
•
Double Row/Deep Groove BB
（双列深沟球轴 承）

Add another row to increase load
capabilities

Greater load capabilities than SRDG

Smaller space requirement than 2 SRDG

More misalignment problems
ME 350
18
•
Angular Contact BB （角接触球 轴
承）

One side of race is higher

Can accommodate a larger thrust

Force resultant preferred between
15º and 40º
Types of Ball Bearings
ME 350
19
•
Self-aligning BB （自动调心球轴 承
）
Types of Ball Bearings

Spherically ground outer race
allows for alignment flexibility

Reduced load bearing
capabilities, with minimal thrust
loading
ME 350
20
Types of Ball Bearings
•
Thrust BB （推力球 轴承）

Large axial loading capabilities

Shaft speeds must be kept low
because of centrifugal forces

ME 350
21
Types of Roller Bearings
A)Cylindrical （圆柱滚子
）
B) Spherical （球面滚子
）
C) Tapered Roller,
Thrust
（推力圆锥滚子）
D) Needle （滚针）
E) Tapered Roller
( 圆锥滚子)
F) Steep-angle
Tapered Roller （大锥角
圆锥滚子）
ME 350
22
Types of Roller Bearings
Cylindrical (Straight)RB （圆柱滚子轴
承）

Greater radial load capacity

Theoretical line contact (actually a
rectangle), so lower contact
stresses

Do not use for thrust - causes
rubbing not rolling
外圈无挡
边
内圈无挡
边
内圈单挡
边
内圈单挡
边并带斜
挡圈
内圈单挡
边并带平
挡圈
ME 350
23
Needle RB （滚针轴承）

Roller with small diameter

Small d, makes them radially
compact, good for large radial loads
at high speeds

Thrust capabilities and misalignment
poor
Types of Roller Bearings
ME 350
24
Types of Roller Bearings
•
Spherical RB （球面滚子轴承）

One type of self-aligning

If misaligned - relative rotation of
outer race to rollers and inner
race

Load capability increased
•
Thrust RB （推力滚子轴承）

Only resist thrust

Several types: rollers, tapered
rollers
ME 350
25
•
Tapered RB( 圆锥滚子轴承 )

Combine advantages of straight roller
and ball type bearings

Can accommodate radial and axial
loading

High load bearing capabilities
Types of Roller Bearings
ME 350
26
Overview of Design
•
Failure Theory for antifriction
bearings is not fully developed
•
Bearing selection is based on life
testing and reliability models
•
Tabulated load ratings for AFBMA
(Anti-Friction Bearing
Manufacturers Association)
•
Design Requirements converted to
required catalog load ratings
ME 350
27
How Bearings Fail
•
Static stress
Static Load Rating, C
o
（额定静载荷）
=
Load that bearing can withstand w/o
permanent deformation

Balls will indent races, cause pitting,
lead to noise, rapid wear
•
Fatigue stress

Life, Reliability and Load relations

Will happen due to high contact
stresses

More likely than static failure

Spalling or pitting in area of contact
ME 350
28
Life v. Reliability
Life Testing

L = life = # of cycles of revolution

F = applied load, fixed for life tests

After some operating period, t=L

Reliability = % of surviving bearings

Life is different under different reliability
L
10
life = life at 10% failure (90% Reliability)

Described with a Weibull Probability
Distribution
ME 350
29
• L
10
life is usually used as the Rating(or minimum)
Life （额定寿命）
Life v. Reliability
ME 350
30
Load v. Life
•
For 2 groups of
identical
bearings tested
under different
loads F
1
and F
2
,
the respective
lives L
1
and L
2

are related by
a
F
F
L
L

,
`

.
|
·
1
2
2
1
a = 3 for ball bearings
a = 10/3 for roller bearings
ME 350
31
Load v. Life
C = basic load rating ( 基本额定动载荷, life = 1 millions of
revolutions )
•
The Basic Load Rating (C) is

The constant radial load which a group of
apparently identical bearings can endure for a
rating life of 10
6
revolutions of the inner ring
(stationary load and stationary outer ring)
C FL
a
·
/ 1
F = rated load of bearing under life L
L = life of bearing in millions of revolutions
= 90 for 90∗10
6
revolutions
a = 3 for ball bearing
a = 10/3 for roller bearings
ME 350
32
Load vs. Life
•
Supposing a company rates its bearings for
3000 hrs at 500 rpm.
•
Suppose one of the bearings has a rated
radial load of 2140 lb.
•Then the corresponding L
10
life is
rev
rev
h
h L
6
10
10 90
min
500 min 60
) 3000 ( × · × × ·
•
And the basic load rating is
lb L F C
a
R
8263 90 2140
10 / 3 / 1
· × · ·
ME 350
33
Load v. Life
F = actual load
L = actual lifetime
F
R
= load rating/catalog load
L
R
= rated lifetime (at test load)
a = 3 for ball bearing
a = 10/3 for roller bearings
L
L
R
·
F
R
F
|
.

`
,

a
C FL
a
·
/ 1
ME 350
34
Load v. Life
L
L
R
·
F
R
F
|
.

`
,

a
• Works for FIXED reliability
•
Rated life standard L
R
= L
10
life
•
AFBMA uses L
10
life of 10
6
revolutions

Have tables of bearing load ratings

Actually extrapolated from tests for use in
calculations, actually F
R
ratings are not
applied to bearings & would cause early
failure
ME 350
35
Example
A catalog lists the basic dynamic
load rating for a BB to be 7,050 lb
for a rated life of 1 million rev.
What would be the expected L
10
life
of the bearing if it were subjected
to a load of 3,500 lb?
ME 350
36
Example - Solution
Thus
L = L
R
*(F
R
/F)
a
= 10
6
*(7050/3500)
3
= 8.17*10
6
revs
L
L
R
·
F
R
F
|
.

`
,

a
ME 350
37
Load v. Life – Different
Speeds
a a
R HR
D HD
D
R
D
D R
n L
n L
F
L
L
F F
1 1

,
`

.
|
·

,
`

.
|
·
•If a bearing is going to be subjected to a load F
D

for a life L
HD
*n
D
•And the catalog specifies a life of L
HR
*n
R
•
The bearing to be selected has to have a radial
load rating equal to or greater than F
R
ME 350
38
Reliability/Life Equations
If F
D
=F
R
, then the reliability at
different life is:
•
Ball Bearings & Straight RB
]
]
]
]





,
`

.
|
−
− ·
483 1
439 4
02 0
exp
10
.
L
L
.
.
R
•
Tapered RB
]
]
]
]






,
`


.
|
− ·
5 . 1
48 . 4
exp
10
L
L
R
ME 350
39
If F
D
≠F
R
(or C
10
), the

Extant Reliability:
Reliability/Life Equations
R gives a predicted reliability for
bearings that are more than adequately
sized
ME 350
40
If we want a different reliability:
Reliability/Life Equations
R
D
= Desired
reliability
ME 350
41
Reliability/Life Equations
Combining life terms and Load/Life
equation gives
1
1/1.483
1
0.02 4.439(ln( ))
a
D
HD D
HR R
R D
R
L n
L n
F F
]
| `
÷
]
. ,
·
]
+
]
]
L
R
= L
Rh
n
R
= “3000 hrs @ 500 rpm”
= 90 * 10
6
revs
ME 350
42
Reliability/Life Equations
1
1/1.483
1
0.02 4.439(ln( ))
a
D
HD D
HR R
R D
R
L n
L n
F F
]
| `
÷
]
. ,
·
]
+
]
]
F
R
(C
10
) = catalog radial load rating
corresponding to L
HR
hours of life at the rated
speed of n
R
rpm.
F
D
= design radial load corresponding to the
required life of L
HD
hours at a design speed of
n
D
rpm and a reliability of R
D
ME 350
43
Example
A ball bearing is to be selected to
withstand a radial load of 4 kN and
have an L
10
life of 1200 h at a speed of
600 rev/min. The bearing maker's
catalog rating sheets are based on an
L
10
life of 3800 h at 500 rev/min.
a. What load should be used to enter
the catalog?
b. What is the reliability of this
application if the catalog rating is 3.8
kN?
ME 350
44
Example – Solution (a)
a
R HR
D HD
D R
n L
n L
F F
1

,
`

.
|
·
( )
3
1
500 * 60 * 3800
600 * 60 * 1200
* 4 ·
kN 89 . 2 ·
ME 350
45
Example – Solution (b)
379 . 0
500 60 3800
600 60 1200
10
·
× ×
× ×
·
L
L
D
97 . 0
4
8 . 3
439 . 4
4
8 . 3
02 . 0 379 . 0
exp
439 . 4
02 . 0 /
exp
483 . 1
3
3
483 . 1
10
10
10
·
¹
¹
¹
¹
¹
'
¹
¹
¹
¹
¹
¹
'
¹
]
]
]
]
]
]







,
`

.
|

,
`

.
|
−
− ·
¹
¹
¹
¹
¹
¹
¹
'
¹
¹
¹
¹
¹
¹
¹
¹
'
¹
]
]
]
]
]
]
]









,
`


.
|


,
`


.
|
−
− ·
a
D
a
D
D
F
C
F
C
L L
R
ME 350
46
Bearing Selection Process
•
Select the type of bearing
•
Find the equivalent radial load ( 当量径向
载荷 ), F
e


Accounts for any thrust/axial load
•
Apply a load factor, K
a
, such that
F
D
= K
a
F
e
•
Determine the minimum acceptable
shaft diameter that limits the bore size
•
Determine the design life
L
D
= (hours)(rpm)(60min/hr)
ME 350
47
Bearing Selection Process
• Compute the dynamic load rating, F
R
(C
10
)
•
Identify candidate bearings with
required rating
•
Select bearing with most convenient
geometry, also considering cost and
availability
•
Determine mounting conditions
ME 350
48
Bearing Type Selection
ME 350
49
Bearing Type Selection
Criteria:
•
Type of load: radial, thrust,
combination of both, steady or
shock
•
Magnitude of load
•
Rotation speed
•
Shaft misalignment
•
Diameter of both shaft and housing
•
Packaging constraints
•
Desired life
•
Maintenance requirements
ME 350
50
Bearing Type Selection
ME 350
51
Axial, radial and combine
loads
Axial load
radial load
Combine load
ME 350
52
Thrust bearings
Cylindrical 
thrust:
Spherical roller thrust bearing
Thrust ball
Thrust:
Thrust ball Thrust ball:
ME 350
53
Recap of Bearing Types
ME 350
54
Speed rates
Oil lub.
Grease lub
Max rotating 
speed
r/min
ME 350
55
Equivalent Loads
Radial bearings with thrust (axial) loads
must have the load transformed into
equivalent radial load (F
e
) for bearing
design
F
r
= applied radial load
F
a
= applied thrust load
V = rotation factor = 1.0 when inner race
rotates,
=1.2 when outer race
rotates
ME 350
56
Equivalent Loads
X = radial factor （径向载荷系数） (Table 11-1)
Y = thrust factor （轴向载荷系数） (Table 11-
1)
These depend on the geometry of the
bearing.
Determined by ratio of:
F
a
– thrust component
C
0
– basic static load rating (Table 11-
2,3)
e – variable reference value
Note:
This requires iteration, since the basic
static load rating C
0
is not known until
bearing is selected
ME 350
57
Equivalent Loads
ME 350
58
Equivalent Loads
ME 350
59
Determine the Thrust Load of
Tapered Roller Bearings
•Thrust component, F
a(180)
, due to
pure radial load, F
r
, is given by
K
F
F
r
a
47 . 0
) 180 (
·
K =0.389cotα, ratio of radial rating of
bearing to thrust rating( Figure 11-17)
Can be approximated in the preliminary
selection process as:
= 1.5 for radial bearings
= 0.75 for steep-angle bearings
α
N
ME 350
60
Determine the Thrust Load of
Tapered Roller Bearings
F
ae
ME 350
61
F
re
F
ae F
rA F
rB
F
a180A
F
a180B
F
ae
F
re
F
a180A
F
rA
F
rB
F
a180B
Indirect mounting (m=-1)
Direct mounting (m=1)
Determine the Thrust Load of
Tapered Roller Bearings
ME 350
62


,
`


.
|
− + ·
ae
B
rB
A rA eA
mF
K
F
K F F
47 . 0
4 . 0
•
Equivalent radial load on bearing A and B is
•F
ae
= external thrust load
rB eB
F F ·
ae
B
rB
A
rA
mF
K
F
K
F
− ≤
47 . 0 47 . 0
B
rB
aB ae
B
rB
aA
K
F
F mF
K
F
F
47 . 0
,
47 . 0
· − ·
So
Then
If
Determine the Thrust Load of
Tapered Roller Bearings
ME 350
63


,
`


.
|
+ + ·
ae
A
rA
B rB eB
mF
K
F
K F F
47 . 0
4 . 0
•
Equivalent radial load on bearing A and B is
rA eA
F F ·
ae
B
rB
A
rA
mF
K
F
K
F
− >
47 . 0 47 . 0
ae
A
rA
aB
A
rA
aA
mF
K
F
F
K
F
F + · ·
47 . 0
,
47 . 0
So
Then
If
Determine the Thrust Load of
Tapered Roller Bearings
ME 350
64
Load Factors
Modify the design load to account
for the type of application before
looking up in catalog:
e a D
F K F ·
K
a
: Load Application Factor, Table
11-5
F
D
: Design load
ME 350
65
Load Factors
5
ME 350
66
Dimension-series
Dimension-series code （尺寸系列代号 ）
ME 350
67
Dimension-series
ME 350
68
Bearing-Life
Recommendations(Table 11-4)
ME 350
69
Rated load
a a
R HR
D HD
D
R
D
D Rre
n L
n L
F
L
L
F F
1 1


,
`


.
|
·


,
`


.
|
·
F
R
: Catalog rated load
(sometimes C
0
in
catalogs)
Calculate the required rated load
of the application:
Rre
F
Rre R
F F >
ME 350
70
Lubrication
The most common lubricants have
traditionally been grease and oil.
Newer lubricants can be used to
withstand higher temperatures, decrease
the friction coefficient, etc. However,
this is not typical in roller bearings.
ME 350
71
Example
A countershaft is supported by roller
bearings using indirect mounting.
The radial bearing loads are 1120 lb
for the left-hand bearing and 2190 lb
for the right-hand bearing. The shaft
rotates at 400 rev/min and is to have
an L
10
life of 40 kh and an application
factor of 1.4.
Assume K = 1.5, and find the
required radial rating for each
bearing, the rating life is 3 kh at 500
rev/min.
ME 350
72
Example – Solution
No external thrust. Thus

,
`

.
|
+ ·
B
rB
A rA eA
K
F
K F F
47 . 0
4 . 0
( ) lb 1477
5 . 1
2190 * 47 . 0
5 . 1 1120 * 4 . 0 · + ·
lb F F
rB eB
2190 · ·
ME 350
73
Example – Solution
Thus
lb F
DA
2068 1477 4 . 1 · ∗ ·
lb F
DB
3066 2190 4 . 1 · ∗ ·
( ) lb F
RA
4207
500 * 3000
400 * 10 * 40
* 2068
10
3
3
· ·
lb F
RB
6237
500 * 3000
400 * 10 * 40
* 3066
10
3
3
·


,
`


.
|
·
ME 350
74
Summary
•
Nomenclature of Ball bearing and Roller
Bearing
•
Performance of Various Types of Bearings
•
Life/Reliability Trade-off at Constant Load
•
Life/Load Trade-off at Constant Reliability
•
Load-Life-Reliability Trade-off
•
Bearing Selection Criteria
•
Equivalent Radial Load
ME 350
75
Mounting
ME 350
76
11-13
A gear-reduction unit uses the
countershaft shown in the figure. Find
the two bearing reactions.
These bearings are to be plain radial ball
bearings, selected for an L
10
life of 40 kh
corresponding to a shaft speed of 400
rev/min. Use 1.2 for the application factor
and specify the bearings selected.
ME 350
77
11-14
The worm shaft shown in part a of the figure
transmits 1.35 hp at 600 rev/min. A static force
analysis gave the results shown in part b of the
figure. Bearing A is to be an angular-contact ball
bearing mounted to take the 555lb thrust load. The
bearing at B is to take only the raidal load and so a
straight bearing will be employed. Use an application
factor of 1.3 and a life of 25kh corresponding to a
reliability of 99% and specify each bearing.