Lecture5 Bearings

ME 350 Mechanical Design and Manufacturing II ME 350

Transmission Support Bearings （轴承）

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Lectures ME 350

Power/Energy Conversion (Electrical Motors)

Joints (Fasteners, Connectors)

Transmission Support (Bearings)

Power/Energy Transmission (Gears, Belt Drives, Power Screws)

Structural Support (Frames Shafts Axles Spindles)

Tools Stress Analysis, Failure Theories Dynamics, Statics, Etc….

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Why Bearings

ME 350

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 ME 350

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Development of the Bearing ME 350

3500 B.C.

1794 A.D.

1869

700 B.C. 40 A.D.

1995

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Sliding or Rolling Bearings ME 350

Sliding bearings: sliding friction

Rolling bearings: rolling friction

µ µ 7

Principles of Operation

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Rolling Friction (Rolling Bearing 滚动 轴承 )Roller/ball （滚子 / 球） Outer Ring （外圈 ） Lubrication （润滑 剂） Inner Ring （内圈 ）

Sliding Friction (Sleeve Bearing 滑动轴 承) Hydrodynamic lift is generated by fluid being dragged into gap by viscous shear Circumferential pressure profile

Sleeve （轴 瓦） Lubrication

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Types of Antifriction Bearings

内径

外径

Ball Bearings

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内圈 滚道 保持 架 端面

外圈 滚道

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Bearings Components

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Seal Rolling elements Inner ring            Outer ring Cage             Seal

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Types of Antifriction Bearings

Tapered Roller Bearings •Components:

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保持 架

Cone

=Inner ring

Cup

=Outer ring

Tapered rollers Cage =Space

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Types of Antifriction Bearings

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Roller and Ball Contact Area/Form

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Rolling Elements ME 350

Ball

Spherical

Cylindrical

Asymmetrical

Taper Needle

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Types of Ball Bearings

深沟 球

带球面 外衬

有装球 缺 口

双列 球

角接触

双列自 动调 心

带防尘 盖

单向推 力

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带密封 圈

单向推 力带 球面座 圈 15

Types of Ball Bearings

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• 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

ME 350

• 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

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Types of Ball Bearings

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• 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

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Types of Ball Bearings

ME 350

• Self-aligning BB （自动调心球轴 承 ） Spherically ground outer race allows for alignment flexibility  Reduced load bearing capabilities, with minimal thrust loading

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Types of Ball Bearings

ME 350

•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 Bearings A)Cylindrical （圆柱滚子 ）

ME 350

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 )

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Types of Roller Bearings ME 350

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 内圈单挡 内圈单 挡 外圈无挡 内圈单挡 内圈 无挡 rubbing not边 rolling 边并带斜 边并带 平 边 边

挡圈

挡圈

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Types of Roller Bearings Needle RB

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（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

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Types of Roller Bearings • Spherical RB （球面 滚子 轴承）

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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

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Types of Roller Bearings

ME 350

• 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

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Overview of Design

ME 350

• 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 26

How Bearings Fail

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• Static stress Static 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 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

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Life v. Reliability

ME 350

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 L10 life = life at 10% failure (90% Reliability) 28  Described with a Weibull Probability

Life v. Reliability

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• L10 life is usually used as the Rating(or minimum) Life （额定寿命） 29

Load v. Life

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• For 2 groups of identical bearings tested under different loads F1 and F2, the respective lives L1 and aL2 L F   1 2 are related =  by

L2

 F1 

a = 3 for ball bearings a = 10/3 for roller bearings

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Load v. Life 1/ a

FL

=C

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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)

F = rated load of bearing under life L L = life of bearing in millions of revolutions = 90 for 90∗106 revolutions a = 3 for ball bearing a = 10/3 for roller bearings

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Load vs. Life

ME 350

• 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 6 60 min 500 rev L10 = (3000h) × × = 90 ×10 rev h min

• And the basic load rating is 1/ a

C = FR L

= 2140 × 90

3 / 10

= 8263lb

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Load v. Life

ME 350

1/ a

=C a L  FR  =  LR  F  FL

F = actual load L = actual lifetime FR = load rating/catalog load LR = rated lifetime (at test load) a = 3 for ball bearing a = 10/3 for roller bearings

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Load v. Life

ME 350

• Works for FIXED reliability • Rated life standard LR = L10 life • AFBMA uses L10 life of 106 revolutions Have tables of bearing load ratings Actually extrapolated from tests for use in calculations, actually FR ratings are not applied to bearings & would cause early failure a

L  FR  =  LR  F 

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Example

ME 350

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 L10 life of the bearing if it were subjected to a load of 3,500 lb?

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Example - Solution

L  FR  =  LR  F 

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a

Thus L = LR*(FR/F)a = 106*(7050/3500)3 = 8.17*106 revs

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Load v. Life – Different Speeds 1

L L   D  HD nD  FR = FD   = FD    LR   LHR nR  a

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1

a

•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 37

Reliability/Life Equations

ME 350

If FD=FR, then the reliability at different life is: • Ball Bearings & Straight RB   L L10 − 0.02 1.483  R = exp −      4.439   • Tapered RB

  L L 1.5  10   R = exp −    4.48  

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Reliability/Life Equations

ME 350

If FD≠FR(or C10), the Extant Reliability:

R gives a predicted reliability for bearings that are more than adequately 39

Reliability/Life Equations

ME 350

If we want a different reliability:

RD = Desired reliability

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Reliability/Life Equations

ME 350

LR = LRhnR = “3000 hrs @ 500 rpm” = 90 * 106 revs Combining life terms and Load/Life equation gives







  LHR nR   FR  FD  1/1.483  1  0.02  4.439(ln( RD ))    



 LHD nD

1

a

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Reliability/Life Equations 





1

a

  LHR nR   FR  FD  1/1.483  1  0.02  4.439(ln( RD ))    



 LHD nD

ME 350

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 42

Example

ME 350

A 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 43 kN?

Example – Solution (a)

L  HD nD  FR = FD    LHR nR 

(

1

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a

= 4 * 1200 * 60 * 600 3800 * 60 * 500

)

1

3

= 2.89 kN 44

Example – Solution (b) LD 1200 × 60 × 600 = = 0.379 L10 3800 × 60 × 500 a 1.483       C   L / L − 0.02 10    F    D 10  D    R = exp−  a    C10        4.439      F  D     3 1.483     3 . 8     0.379 − 0.02      4    = 0.97 = exp−   3    3.8    4.439       4     

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Bearing Selection Process

ME 350

• 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

L = (hours)(rpm)(60min/hr)

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Bearing Selection Process

ME 350

• 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 ME 350

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Bearing Type Selection

ME 350

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

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Bearing Type Selection

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Axial, radial and combine loads radial load

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Combine load

Axial load

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Thrust bearings Thrust ball:

Thrust ball

Thrust ball

Cylindrical  thrust:

Thrust:

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Spherical roller thrust bearing

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Recap of Bearing Types

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Speed rates ME 350

r/min

Oil lub.

Grease lub

Max rotating  speed

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Equivalent Loads

ME 350

Radial 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 55 rotates,

Equivalent Loads

ME 350

X = radial factor （径向载 荷系 数） (Table 11-1) Y = thrust factor （轴向 载荷 系数） (Table 111) These depend on the geometry of the bearing. Determined by ratio of: Fa – thrust component C0 – basic static load rating (Table 112,3) e – variable reference value Note: 56

Equivalent Loads

ME 350

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Equivalent Loads

ME 350

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Determine the Thrust Load of Tapered Roller Bearings

ME 350

•Thrust component, Fa(180), due to pure radial load, Fr, is given by

Fa (180)

0.47 Fr = K

K =0.389cotα, ratio of radial rating of α bearing to thrust rating( Figure 11-17) Can be approximated in the preliminary selection process as: = 1.5 for radial bearings N = 0.75 for steep-angle bearings

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Determine the Thrust Load of Tapered Roller Bearings

ME 350

Fae

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Determine the Thrust Load of Tapered Roller Bearings

ME 350

Indirect mounting (m=-1)

Fre

Fa180A

Fa180B

Fae

FrA

FrB

Direct mounting (m=1)

Fre

Fa180A FrA

Fae

Fa180B FrB 61

Determine the Thrust Load of Tapered Roller Bearings

ME 350

•Equivalent radial load on bearing A and B is

0.47 FrA 0.47 FrB ≤ − mFae KA KB 0.47 FrB 0.47 FrB − mFae , FaB = Then FaA = KB KB  0.47 FrB  − mFae  So FeA = 0.4 FrA + K A   KB  If

FeB = FrB •Fae = external thrust load

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Determine the Thrust Load of Tapered Roller Bearings

ME 350

•Equivalent radial load on bearing A and B is

0.47 FrA 0.47 FrB > − mFae KA KB 0.47 FrA 0.47 FrA , FaB = + mFae Then FaA = KA KA If

So

FeA = FrA

 0.47 FrA  FeB = 0.4 FrB + K B  + mFae   KA 

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Load Factors

ME 350

Modify the design load to account for the type of application before looking up in catalog:

FD = K a Fe Ka : Load Application Factor, Table 11-5 FD: Design load 64

Load Factors

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5

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Dimension-series

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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

ME 350

Calculate the required rated load FRre of the application: 1

FRre

 LD   LHD nD   = FD   = FD   LR   LHR nR  a

1

a

FR > FRre FR : Catalog rated load (sometimes C0 in

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Lubrication

ME 350

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. 70

Example

ME 350

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 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

ME 350

No external thrust. Thus

0 . 47 F  rB  FeA = 0.4 FrA + K A    KB 

(

)

= 0.4 *1120 + 1.5 0.47 * 2190 = 1477lb 1. 5 FeB = FrB = 2190lb 72

Example – Solution

ME 350

FDA = 1.4 ∗1477 = 2068lb

FDB = 1.4 ∗ 2190 = 3066lb Thus

(

3 * 400 40 * 10 FRA = 2068 * 3000 * 500

FRB

)

3 10

 40 *103 * 400   = 3066 *   3000 * 500 

= 4207lb

3 10

= 6237lb 73

Summary ME 350

• 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 ME 350

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11-13

ME 350

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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11-14

ME 350

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.

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