Machine Elements

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Machine Elements

Bearings

GM team

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Outline

• General knowledge  homework + moodle test

• Sizing for bearings with radial contact

• Sizing for bearings with angular contact

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Goals

• Analyze a rotative joint : accuracy, forces transmission, efficiency, fiability

• Acquire technological culture: main technical solutions to design a revolute joint:

contact, plane or friction bearings, ball bearings, etc

• Modeling and sizing a revolute joint by direct contact

• Choosing the most appropriate component between bearings from forces transmission and accuracy expectations.

• Identifying functionnal surfaces with respect to the bearing choice

• Knowing the typical lubrication and seal solutions

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Rolling bearings

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Example: electrical actuator

Bearing 9 : 6304

Bearing 9: « Ball bearing 20/52/15 » in the bill of material

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Example: electrical actuator ^Bearings« Tapered roller bearing 15-42-14,25 »^{6 et 6’ :}

Bearings 6 and 6’ : 30302

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Bearing 21 :

« Needle roller and cage assembly SKF

8-15-12 »

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Simplified representation

Deep grove ball bearing Angular contact ball bearing Self-aligning 2 rows ball bearing

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Sizing

Radial contact rolling bearing

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Rolling bearing sizing

Main failures :

• Permanent deformation at contact points – large static loads, small or oscillating speed

• Bearing fatigue due to fast cycling loadings

Flaking on the raceway

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Static deterioration

Cases: Static + high load / Slow oscillations + load / rotation + high peak loads / low speed (<10r/min) + load + limited life

Sizing criterion:

Contact pressure rolling elements/raceway < max pressure

= 0,01% D_rolling permanent deformation 4600 Mpa for self-aligning ball bearings, 4200 Mpa for other ball bearings, 4000 Mpa for roller bearings

• C₀ : basic static load rating (kN)

• s₀ : static safety factor

• P₀ : equivalent static bearing load :

P₀ = max(X₀.F_r + Y₀.F_a ; F_r)¹

𝑪 _𝟎 > 𝒔 _𝟎 × 𝑷 _𝟎

1From SKF

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Static deterioration: safety factor

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Static deterioration : X0 and Y0 (ISO 76)

Bearing type

1 row 2 rows

X0 Y0 X0 Y0

Ball bearings

Radial contact 0.6 0.5 0.6 0.5

Self aligning 0.5 0.22 cotg α 1 0.4 cotg α

Angular contact

α = 15° 0.5 0.46 1 0.92

α = 20° 0.5 0.42 1 0.84

α = 25° 0.5 0.38 1 0.76

α = 30° 0.5 0.33 1 0.66

α = 35° 0.5 0.29 1 0.58

α = 40° 0.5 0.26 1 0.52

α = 45° 0.5 0.22 1 0.44

Roller bearings

Tappered rollers 0.5 0.22 cotg α 1 0.44 cotg α

Self-aligning 0.5 0.22 cotg α 1 0.44 cotg α

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Electrical actuator

Bearing 9 :

Deep-grove ball bearing 20/52/15 C=15,8 kN

Co=7,8 kN F_rB= 660 N F_aB = 16710 N

P₀ = 0,6 x 660 + 0,5 x 16710 = 8751 N s₀ = 0,5 (normal, low requirements) C_{0 mini} = 4375 N >> OK

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Ruin mechanism in dynamic conditions

Bearings sustain cycling loadings  fatigue

How to take into account fatigue for bearing sizing ? Until middle 1900: based on experimental observations.

SKF 6002-2RSL

D = 4.762 mm dm = 23.5 mm Z = 9

Ni = 1500 tr /mn Ne = 0

Rolling element diameter: 4.7 mm Average bearing diameter: 23.5 mm Number of rolling elements: 9 Contact angle: 0°

Outer ring rotating speed: 0 r/min Inner ring rotating speed: 1500 r/min

Rolling elements rotation frequency: 9.967 Hz

Frequency of chocks on the inner ring due to rolling elements: 135.3 Hz

Frequency of chocks on the outerr ring due to rolling elements: 89.7 Hz

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Bearing ruin

1947: Basic Rating Life Model, to obtain basic dimensions of bearing based on load with simplified equation:

𝑳 _𝟏𝟎 = 𝑪 𝑷

𝒏

𝑳 _𝟏𝟎𝒉 = 𝑪 𝑷

𝒏 𝟏𝟎 ^𝟔

𝟔𝟎. 𝑵

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Basic Rating Life Model : experimentations

Destroy 100 bearings for a given radial load at different levels Fr1>Fr2>Fr3.

Life before ruin (Million revolutions)

Nb of bearings

L₁₀₍₁₎

Radial load Fr1

90 % 90 %

Fr2

17 bearings

15 Mr

Fr1 > Fr2 > Fr3

L₁₀₍₂₎ L₁₀₍₃₎

Fr3

90 %

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Same experiment for each bearing from the catalogue

0,00 1000,00 2000,00 3000,00 4000,00 5000,00 6000,00 7000,00

0,00 200,00 400,00 600,00 800,00 1000,00 1200,00

Radial load (N)

Life before ruin (Mr)

5 6 7 8

d (N))

ln

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Basic Rating Life Model : experimentations Logarithmic relationship between R and L

Ln(Fr (DaN))

Ln (life before ruin (Mr))

1 0 10 100

1 10 100

1 C

C: basic dynamic load rating ↔ 1 Mr

= purely radial load (or purely axial for thrust bearings) leading to L₁₀ = 1 Mr.

L Fr

ln 𝐶 − ln 𝐹𝑟

ln 1 − ln 𝐿 = 𝑐𝑠𝑡

ln 𝐶

𝐹𝑟 = 𝑐𝑠𝑡 ∗ ln 1 𝐿

𝐿 = 𝐶 𝐹𝑟

−1/𝑐𝑠𝑡

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Fr

In practice loading is not only radial  What happens if we add an axial load ?

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Equi-rating life curve approximated by two straight lines defining P = equivalent dynamic bearing load

Fa Fr

tg  = e

If Fa/Fr > e : P = X Fr + Y Fa If Fa/Fr  e : P = Fr



P = hypothetical load, constant in magnitude and direction, that acts radially on radial bearings and axially and centrically on thrust bearings.

This hypothetical load, when applied, would have the same influence on bearing rating life as the actual loads to which the bearing is subjected.

X, Y and e  to be determined from Fa/Co

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Attention: X, Y and e are bearing characteristics that depend on Fa/C₀

Fa/C0 e X Y

0,014 0,19 0,56 2,3

0,028 0,22 0,56 1,99

0,056 0,26 0,56 1,71

0,084 0,28 0,56 1,55

0,11 0,3 0,56 1,45

0,17 0,34 0,56 1,31

0,28 0,38 0,56 1,15

0,42 0,42 0,56 1,04

0,56 0,44 0,56 1

X = 0,56

Y = 0,8647(Fa/C0)^-0,232

1 1,5

2 e

X Y

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Basic Rating Life Model

To sum up:

• C = Basic dynamic load rating (manufacturer data)

• P = Equivalent dynamic bearing load (to calculate)

• n = Exponent of life equation

• Roller bearings : n = 10/3

• Ball bearings : n = 3

𝑳 _𝟏𝟎 = 𝑪 𝑷

𝒏

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Calculation method

L₁₀, ØD mini, Fa, Fr are known

Assumption: F_a/F_r < e P = Fr

C_mini = L₁₀^1/n.P

Find a bearing that satisfies

C > C_mini Check if F_a/F_r < e

Modify P:

P = 0,56.F_r + Y.F_a Starting with

Y = 0,8647.(Fa/C0)^-0,232 OK !

Yes

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Electrical actuator

Bearing 9 :

Deep-grove ball bearing 20/52/15 C=15.8KN

Co=7.8KN F_rB= 660 N F_aB = 6710 N N₁₉ = 168 r/min L₁₀= 1000 h mini

Fa/Fr = 10

P = 0.56 x 660 + 6710 = 7080 N L₁₀ = (15800/7080)³ = 11.1 Mr

L_10h = L₁₀/(60 N₁₉) = 1102 h

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Variable load

• All previous calculation are done for constant load.

• In practice: mechanical systems subjected to variable load

• Ex: gear box

• Load modeling: divide variable load into different constant

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Variable load

Load (N)

Number of rotations

(rounds)

Speed level (r/min)

% of total time

Portion of total rounds

P1 n1 N1 α1 α^tr1

P2 n2 N2 α2 α^tr2

P3 n3 N3 α3 α^tr3

𝛼

_𝑖

= 𝑥 % 𝛼

_𝑖^𝑡𝑟

= 𝛼

_𝑖

𝑁

_𝑖

σ 𝛼

_𝑗

𝑁

_𝑗

To evaluate the equivalent load P:

𝑃

_𝑒𝑞

=

^𝑛

෍

𝑖

𝛼

_𝑖^𝑡𝑟

𝑃

_𝑖^𝑛

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Variable load

Load (N)

Number of rotations

(rounds)

Speed level

(r/min) % of total time Portion of total rounds

P1 = 1000N N1 = 2000 r/min α1 = 10%  α^tr1 = 3%

P2 = 5000N N2 = 6000 r/min α2 = 60%  α^tr2 = 58%

P3 = 2000N N3 = 8000 r/min α3 = 30%  α^tr3 = 39%

𝑡𝑟

𝛼

_𝑖

𝑁

_𝑖

Equivalent load P:

෍ 𝛼_𝑗𝑁_𝑗 = 0.1 ∗ 2000 + 0.6 ∗ 6000 + 0.3 ∗ 8000 = 6200 𝑟/𝑚𝑖𝑛

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Arrangements

Radial contact bearings

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Rolling on free rings

Outer ring rolling Inner ring rolling

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Interference fit for the INNER ring

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Example : reducer

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Example : bench grinder

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Example : campervan wheel

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Radial fit (from SNR)

INT. FIT SHAFT HOUSING

Inner ring on shaft

Load case FIT Load case FIT

Normal: P<C/5 j6/k6 General H7/J7

High: P> C/5 m6/p6

Non-located ring G7/H7 Roller bearing M7/P7

Outer ring in housing

General g6/h6 Normal: P<C/5 M7/N7

Non-located ring f6/g6 Hugh: P> C/5 N7/P7

Other Purely axial h6/g6

Purely axial G7/H7

Adapter sleeve h9

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Axial locating

Principle lock nut snap ring shaft shoulder spacer

adapter sleeve

principle housing shoulder spacer snap ring housing cover

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Typical arrangements: interference fit on the shaft, long shaft

a

_cover

Ja

a

_bearing

a

_housing

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Typical arrangements: interference fit in the housing, long shaft

Locating should be close to the loading point – the other bearing is not located Dilatation :

l= 17.10^-6°C^-1 If Ød 20mm and L=100 mm and

Dq=50°C, DL = 0,085 mm

Corresponding to: F = S.E.e = 56 kN !

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Typical arrangements: economical

Easier to mount, less secure

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Lubrication and sealing

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Lubrication modes

Oil Grease

Advantages

• Good penetration

• Good stability

• Cooling

• Easy control

• Cleanliness

• Easier to seal

• Good protection from outside

• Easy to mount and manipulate

• Pre-greased bearings

Drawbacks

• Sealing during mounting

• Bad protection against oxydation and humidity in case of long stop

• Delay if autonom circulation

necessary

• Higher friction

• Worse heat

evacuation

• Replacement

requires bearing dismounting and cleaning

• Control less easy

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Lubrication and sealing methods

See SKF documentation

Machine Elements