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ME 350 Mechanical Design and
Manufacturing II
Transmission SupportBearings (轴承)
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Lectures
Power/Energy
Conversion(Electrical Motors)
Power/EnergyConversion
(Electrical Motors) Power/EnergyTransmission
(Gears, Belt Drives,
Power Screws)
TransmissionTransmissionSupportSupport
(Bearings)(Bearings)
Joints(Fasteners,
Connectors)StructuralSupport(FramesShaftsAxles
Spindles)
ToolsStress Analysis, Failure Theories
Dynamics, Statics, Etc….
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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
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History
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Reconstruction of Old Bearing
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Development of the Bearing
700 B.C.
3500 B.C.
40 A.D.
1794 A.D.1869
1995
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Sliding or Rolling Bearings
Sliding bearings:
sliding friction µRolling bearings:
rolling frictionµ
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Principles of OperationRolling 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
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Types of Antifriction BearingsBall Bearings
内径
外径
端面 外圈滚道
内圈滚道
保持架
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Bearings Components
Seal Rolling elements Inner ring Outer ring Cage Seal
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Types of Antifriction Bearings
Tapered Roller Bearings
保持架
•Components:Cone
=Inner ringCup
=Outer ringTapered
rollersCage
=Space retainer
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Types of Antifriction Bearings
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Roller and Ball Contact Area/Form
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Rolling Elements
Cylindrical
Needle
Taper
Ball Spherical Asymmetrical
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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
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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
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Types of Ball Bearings
• Double Row/Deep Groove BB(双列深沟球轴承)Add another row to increase load
capabilitiesGreater load capabilities than SRDGSmaller space requirement than 2 SRDGMore misalignment problems
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• Angular Contact BB (角接触球轴承)One side of race is higher Can accommodate a larger thrustForce resultant preferred between
15º and 40º
Types of Ball Bearings
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• 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
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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
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Types of Roller BearingsA)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)
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Types of Roller Bearings
Cylindrical (Straight)RB (圆柱滚子轴承)Greater radial load capacityTheoretical 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
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Needle RB (滚针轴承)Roller with small diameterSmall d, makes them radially
compact, good for large radial loads at high speeds
Thrust capabilities and misalignment poor
Types of Roller Bearings
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Types of Roller Bearings• Spherical RB (球面滚子轴承)One type of self-aligningIf misaligned - relative rotation of
outer race to rollers and inner raceLoad capability increased
• Thrust RB (推力滚子轴承)Only resist thrustSeveral types: rollers, tapered
rollers
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• 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
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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
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How Bearings Fail
• Static stressStatic Load Rating, Co (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 stressLife, Reliability and Load relationsWill happen due to high contact
stressesMore likely than static failureSpalling or pitting in area of contact
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Life v. ReliabilityLife TestingL = life = # of cycles of revolutionF = applied load, fixed for life tests
After some operating period, t=LReliability = % of surviving bearingsLife is different under different reliabilityL10 life = life at 10% failure (90% Reliability)Described with a Weibull Probability Distribution
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• L10 life is usually used as the Rating(or minimum) Life (额定寿命)
Life v. Reliability
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Load v. Life
•For 2 groups of identical bearings tested under different loads F1 and F2, the respective lives L1 and L2 are related by
a
FF
LL
1
2
2
1
a = 3 for ball bearings
a = 10/3 for roller bearings
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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 106 revolutions of the inner ring (stationary load and stationary outer ring)
CFL a /1
F = rated load of bearing under life LL = life of bearing in millions of revolutions
= 90 for 90106 revolutionsa = 3 for ball bearinga = 10/3 for roller bearings
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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 L10 life is
revrevh
hL 610 1090
min500min60)3000(
•And the basic load rating is
lbLFC aR 8263902140 10/3/1
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Load v. Life
F = actual loadL = actual lifetime FR = load rating/catalog loadLR = rated lifetime (at test load)a = 3 for ball bearinga = 10/3 for roller bearings
LLRFRF
a
CFL a /1
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Load v. Life
LLRFRF
a
• Works for FIXED reliability
• Rated life standard LR = L10 life
• AFBMA uses L10 life of 106 revolutionsHave tables of bearing load ratingsActually extrapolated from tests for use in
calculations, actually FR ratings are not applied to bearings & would cause early failure
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ExampleA 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 L10 life of the bearing if it were subjected to a load of 3,500 lb?
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Example - Solution
ThusL = LR*(FR/F)a = 106*(7050/3500)3
= 8.17*106 revs
LLRFRF
a
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Load v. Life – Different Speeds
aa
RHR
DHDD
R
DDR nL
nLFLLFF
11
•If a bearing is going to be subjected to a load FD for a life LHD*nD
•And the catalog specifies a life of LHR*nR
•The bearing to be selected has to have a radial load rating equal to or greater than FR
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Reliability/Life Equations
If FD=FR, then the reliability at different life is:
• Ball Bearings & Straight RB
4831
4394020
exp 10
.LL
..
R
• Tapered RB
5.1
48.4exp 10L
L
R
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If FD≠FR(or C10), the Extant Reliability:
Reliability/Life Equations
R gives a predicted reliability for bearings that are more than adequately sized
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If we want a different reliability:
Reliability/Life Equations
RD = Desired reliability
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Reliability/Life Equations
Combining life terms and Load/Life equation gives
1
1/1.48310.02 4.439(ln( ))
a
D
HD D
HR RR D
R
L nL n
F F
LR = LRhnR = “3000 hrs @ 500 rpm”
= 90 * 106 revs
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Reliability/Life Equations1
1/1.48310.02 4.439(ln( ))
a
D
HD D
HR RR D
R
L nL n
F F
FR (C10) = catalog radial load rating corresponding to LHR hours of life at the rated speed of nR rpm.
FD = design radial load corresponding to the required life of LHD hours at a design speed of nD rpm and a reliability of RD
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ExampleA ball bearing is to be selected to withstand a radial load of 4 kN and have an L10 life of 1200 h at a speed of 600 rev/min. The bearing maker's catalog rating sheets are based on an L10 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?
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Example – Solution (a)
a
RHR
DHDDR nL
nLFF1
31
500*60*3800600*60*1200*4
kN 89.2
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Example – Solution (b)
379.0500603800
600601200
10
L
LD
97.0
48.3
439.4
48.3
02.0379.0exp
439.4
02.0/
exp
483.1
3
3
483.1
10
1010
a
D
a
DD
FC
FC
LL
R
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Bearing Selection Process• Select the type of bearing• Find the equivalent radial load ( 当量径向载荷 ),
Fe Accounts for any thrust/axial load
• Apply a load factor, Ka , such that
FD = Ka Fe
• Determine the minimum acceptable shaft diameter that limits the bore size
• Determine the design life
LD = (hours)(rpm)(60min/hr)
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Bearing Selection Process• Compute the dynamic load rating, FR
(C10)
• Identify candidate bearings with required rating
• Select bearing with most convenient geometry, also considering cost and availability
• Determine mounting conditions
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Bearing Type Selection
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Bearing Type SelectionCriteria:• 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
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Bearing Type Selection
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Axial, radial and combine loads
Axial load
radial loadCombine load
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Thrust bearings
Cylindrical thrust:
Spherical roller thrust bearingThrust ball
Thrust:Thrust ballThrust ball:
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Recap of Bearing Types
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Speed rates
Oil lub.
Grease lubMax rotating speed
r/min
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Equivalent LoadsRadial bearings with thrust (axial) loads must have the load transformed into equivalent radial load (Fe) for bearing design
Fr = applied radial load
Fa = applied thrust load
V = rotation factor = 1.0 when inner race rotates,
=1.2 when outer race rotates
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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:
Fa – thrust componentC0 – basic static load rating (Table 11-
2,3)e – variable reference value
Note: This requires iteration, since the basic
static load rating C0 is not known until bearing is selected
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Equivalent Loads
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Equivalent Loads
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Determine the Thrust Load of Tapered Roller Bearings
•Thrust component, Fa(180), due to pure radial load, Fr, is given by
K
FF ra
47.0)180(
K =0.389cot, 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
N
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Determine the Thrust Load of Tapered Roller Bearings
Fae
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Fre
FaeFrA FrB
Fa180A Fa180B
Fae
FreFa180A
FrA FrB
Fa180B
Indirect mounting (m=-1)
Direct mounting (m=1)
Determine the Thrust Load of Tapered Roller Bearings
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ae
B
rBArAeA mF
K
FKFF
47.04.0
•Equivalent radial load on bearing A and B is
•Fae = external thrust loadrBeB FF
aeB
rB
A
rA mFK
F
K
F
47.047.0
B
rBaBae
B
rBaA K
FFmF
K
FF
47.0,
47.0
So
Then
If
Determine the Thrust Load of Tapered Roller Bearings
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ae
A
rABrBeB mF
K
FKFF
47.04.0
•Equivalent radial load on bearing A and B is
rAeA FF
aeB
rB
A
rA mFK
F
K
F
47.047.0
aeA
rAaB
A
rAaA mF
K
FF
K
FF
47.0,
47.0
So
Then
If
Determine the Thrust Load of Tapered Roller Bearings
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Load FactorsModify the design load to account for the type of application before looking up in catalog:
eaD FKF Ka : Load Application Factor, Table 11-5FD: Design load
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Load Factors
5
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Dimension-series
Dimension-series code (尺寸系列代号)
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Dimension-series
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Bearing-Life Recommendations(Table 11-4)
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Rated load
aa
RHR
DHDD
R
DDRre nL
nLF
L
LFF
11
FR : Catalog rated load
(sometimes C0 in catalogs)
Calculate the required rated load of the application:
RreF
RreR FF
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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.
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ExampleA 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 L10 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.
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Example – Solution
No external thrust. Thus
B
rBArAeA K
FKFF 47.04.0
lb14775.12190*47.05.11120*4.0
lbFF rBeB 2190
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Example – Solution
Thus
lbFDA 206814774.1
lbFDB 306621904.1
lbFRA 4207500*3000
400*10*40*206810
33
lbFRB 6237500*3000
400*10*40*3066
103
3
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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
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Mounting
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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 L10 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.
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11-14The 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.