2.72 Elements of Mechanical Design Lecture 07: Rolling contact bearings
Dec 20, 2015
2.72Elements of
Mechanical Design
Lecture 07: Rolling contact bearings
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Schedule and reading assignmentQuiz
Constraints
Topics Bearing types & failure modes Experimental results to modeling Bearing load-life-reliability Start spindle exercise
Reading assignment None, work on getting bearings done
Quiz Tuesday Bearings (conceptual)
www.renault4.co.uk/
Besides the children’s toy (sorry bout that), what do you notice about the bearing that should be of import?
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Loose endsThermal stability
Function of separation distance and contact geometry - cancellation Will you always have bearing death if it is not thermally stable?
Mounting of races: Inside rotating vs. outside rotating Friction torques Look at the tapered roller example from lab
TR
Tq
crossCondT Ak
tR
surfaceConvT Ah
R1
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Rough design steps – Inherently iterativeStep 1: Functional requirements
DOF Stiffness Lifetime/reliability Etc…
Step 2: Bearing type/layout
Step 3: System design & mfg issues Housing geometry Shaft geometry Shaft deflection Preload Thermal Tolerances
Step 4: Assembly specifications
Iteration
Types of rolling contact bearings
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Examples of rolling element bearings
static.howstuffworks.com
www.machinedesign.com
www.gm-exports.com/
www.bearingsworld.com
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Inner Ring Race
Outer Ring Race
Diameters Outer Bore
Ball/roller
Cage/separator
Face
Width
Elements: Rotary rolling contact bearing
Bearing failure:Causes and failure mode
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Brinelling plus cracks
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Subsurface face fatigue crack that has progressed to surface. Magnification 100X
Cracks in bearings elements
Brinelling of surface
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Failure mode: Spalling
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http://www.tsb.gc.ca
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Causes of spallingSpalling
Surface fatigue that occurs as a result of contact
Seeds of failure Crack growth Inclusions Impact Cyclic high stress Degradation of the lubricant
Once it starts, what happens? Minor spalling + correct problem may slow/stop Typically increases in size with continued service
http://www.tsb.gc.ca
http://www.tsb.gc.ca
Steel quality key to making long-lasting bearings
Preload
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Model ball-groove contacts as six balls on flats
Important relationships for ball-flat contact
Ball-flat elastomechanics: Hertz contact
1 12 22n ek R E
F
Hertzian contactsact as non-linearsprings
Hertz1857-1894
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Location, magnitude of max shear stress
Max Max shear stress occurs below surface, in the member with largest R if ball and flat of same material
3
1
2
2
48.0max
6131.0
e
econtactofcenter
radiuscontactdepth R
EF
ormajorormajor
e
RRRR
R
min22min11
11111
2
22
1
21 11
1
EE
Ee
Equivalent modulus
Equivalent radius
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Preload increases stiffness
A little preload goes a long way
Classes of preload, as % of static load capacity Heavy 5% Medium 3% Light 2%
Sensitivity of contact stiffness to F
31
32
31
oK neen FERk
Contact stiffness
0
05
10
20
30
0 250 500 750 1000
Fn [N]
k [N
/mic
ron
] Large k for small Preload
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How do you preload a bearing?Direct
Nuts pressing directly on races Uses compliance of contacts
Internal Oversized balls Uses compliance of ball-race contacts
Nuts-springs Spring in series with bearing Primarily uses compliance of spring
Think in terms of relative stiffness because… Sensitivity of force to the displacement…
Many bearings comepreloaded “out of the box”
Check to make sure so that you do not add preload that will act to overload
Bearing life and reliability
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Bearing life depends on: Load and revolutions
From experimentation: For a given R, linear log behavior Reliability = % chance that the part will function as expected
Where a = 3 for ball bearings a = 3.333 for roller bearings
Bearing life at rated reliability
aLF1log(F)
log(L)
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Regression fit to experimental dataManufacturers provide a match of:
Cycle rating: Revolutions For example 106 or 906 revs
Load rating: Force Anything…
that defines bearing failure for a given reliability
Given these two numbers, and:
This may be used to extrapolate behavior at different loads and revs
designa
designia
i LFLC11
aLF1
Units of force
Constant
Units of revolutions
Why C vs. F?
This is for common reliability
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Design life in terms of hours or revolutions?We can think in terms of life as time if:
Lhr
revhrt
min
60min
adesigndesignadesigna tFLFLC
111
1010 60
adesigndesigndesignaratedrated tFtC11
10 6060
aratedrated
adesigndesigndesign
t
tFC 1
1
10
60
60
Dynamicload
ratingRevs at failure Be careful with units
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Reliability vs. lifeReliability often well-predicted via Weibull distribution
xo = minimum guaranteed value of x
θ = corresponds to 63.2 percentile of the variate (stochastic variable) b = a shape parameter (controls skew, large = right)
For bearings, we use this as:
10L
Lx
b
o
o
x
xxxR
exp
r(x)
x
xo
Commonly used to fit experimental data; b & come from fit
This is for common load on bearings
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BUT the catalogue never tells me what happens for…. My exact desired load My exact desired life Situations when I want a reliability that is different than R = 0.90
C10
x
Design
R = 0.90
Relationship between load, life and reliability
PFL
Lx
log(x)
log(F)
A
D
B 2
1
aDDaBB xFxF11
:yreliabilitConstant
1
b
o
oB
x
xxxR
exp
loadConstant
2
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Relationship between load, life and reliabilityLoad as a function of reliability & vice versa for given:
C10
L10
R = 0.90; x = 1 when L = L10
a
b
DOO
DD
Rxx
xFC
1
110
1ln
C10
x
Design
R = 0.90
PFL
Lx
log(x)
log(F)
A
D
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Reliability of multi-bearing setsWhat is the reliability of:
One bearing? A spindle with two bearings? With N bearings?
For first order design, how should individual bearing reliability scale as a function of N?
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How to handle combined loadingSo far we have only considered radial loading…
What about combined radial, Fr, and axial loading , Fa? Use an equivalent load, Fe, that does the same amount of damage.
Where V = 1.2 for outer ring rotation and 1 for inner ring This has to do with the fact that outer ring fails more often
Xi and Yi are a function of the Axial load, Fa Static load rating, Co
airie FYFVXF
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Other issues, Shigley/Mischke covers wellLife recommendations (hrs)
Aircraft engines 0 500 – 2 000 24-hour critical service 100 000 – 200 000
Application factors General commercial 1.1 – 1.3 Moderate impact 1.5 – 3.0
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Group exerciseWork on your spindle housing-shaft-bearing design
Constraint layout Loads Preload Thermal stability Cost