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MIT OpenCourseWare
http://ocw.mit.edu
2.72 Elements of Mechanical Design
Spring 2009
For information about citing these materials or our Terms of Use, visit: http://ocw.mit.edu/terms.
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2.72
Elements of
Mechanical Design
Lecture 03: Shafts
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Schedule and reading assignment
Reading quiz
Hand forward lathe exercise quiz
Topics Finish matrices, errors
Shaft displacements
Stiffness exercise
Reading assignment Shigley/Mischke
Sections 6.16.4: 10ish pages & Sections 6.76.12: 21ish pages
Pay special attention to example 6.12 (modified Goodman portion)
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Deflection within
springs and shafts
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Shafts, axles and rails
Martin Culpepper, All rights reserved
Shafts Rotating, supported by bearings/bushings
Dynamic/fluctuating analysis
Axles Non-rotating, supported by bearings/bushings
Static analysis
Rails
Non-rotating, supports bearings/bushings Static analysis
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Examples drawn from your lathe
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Examples drawn from your lathe
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In practice, we are concerned with
Martin Culpepper, All rights reserved
Deflection Stiffness
Bearings and stiffness of connectivity points
Function of global shaft geometry, sometimes adjacent components
Stress Catastrophic failure: Ductile Brittle Fatigue
Function of local shaft geometry
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What is of concern?
Martin Culpepper, All rights reserved
Deflection and stiffness Beam bending models
Superposition
Load and stress analysis
Bending, shear & principle stresses
Endurance limit
Fatigue strength
Endurance modifiers
Stress concentration
Fluctuating stresses
Failure theories
Von Mises stress Maximum shear stress
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Materials
Steel vs. other materialsAluminum
Brass
Cast iron
Important properties
Modulus Yield stress
Is density important?
Fatigue life CTE
Material treatment Hardening
What does hardening do the material properties
It is expensive
Affects final dimensions
You can usually design without this
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Principles of stiffness: Relationshipsq
q d
4
y=
q
EI dx4
V =
d
3
y VEI dx
3
M =
d
2
y MEI dx
2
=
dy dx
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Modeling: General forms of equations
Lateral bending deflection (middle)F
F L3
F L3
= Const48E I E I
RL RRAxial deflection
F L
=
A E
Lateral bending angles (at ends)
F L2 F L2 M L= = or
6E I Const E I Const E I
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Modeling: Stiffness
These pop up in many places, memorize them
Square cross section
1 3I =
bh12
Circular cross sections
I = [(d )4(d )4]64
outer inner
J =
[(d
)
4(d
)
4
]32 outer inner Martin Culpepper, All rights reserved 13
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Principles of stiffness: RatiosEverything deforms
Impractical to model the stiffness of everything
Mechanical devices modeled as high, medium & low stiffness elements
Stiffness ratios show what to model as high-, medium, or low stiffness
Rk =k1st
Stiffness ratio k2ndAE
3EIAE
l = Rk =
l =4
l2
= klaterall
3kaxiall 3EI h
2
l3
FF
Building intuit ion for stiffness
You cant memorize/calculate everything
Engineers must be reasonable instruments
Car suspension is easy, but flexed muscle vs. bone?
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Principles of stiffness: Sensitivity
CantileverL
FF L3
=3 E I h
mL1
n
h1 3
bI = b
h12
F =
E b
h
3
dF d E b h
3 E b h 3
4 L k =d
=d
4
L
4
L
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Superposition
16 Martin Culpepper, All rights reserved
You must be careful, following assumptions are needed Cause and effect are linearly related
No coupling between loads, they are independent
Geometry of beam does not change too much during loading
Orientation of loads does not change too much during loading
Use your head, when M = 0, what is going on
Superposit ion is not plug and chug
You must visualize You must think
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Types of springs and behaviors
Springs and stiffness
kF =
dF(x)/dx
Constant force spring kF =
dF(x)/dx = 0 Eb-a = F (xb - xa )
Constant stiffness spring kF = Constant
Eb-a = 0.5 kF (xb2 - xa
2 )
Non-linear force spring kF = function of x
Eb-a =
F(x) dx
Force-Displacement Curve
F(x)=F
F
Fa b
x
Force-Displacement Curve F(x)=kFx
F F F 1kF
xa b
Force-Displacement Curve
F
x Martin Culpepper, All rights reserved
a b 17Images from Wikimedia Commons, http://commons.wikimedia.org
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Non-conformal contact ball on flat
Non-conformal contacts often non-linear Example: bearings, belleville washers, structural connections
Is anything ever perfectly conformal?
Specific case: Hertzian contact
F
kn ( )= Constant
1 2
1
F R 3E 3 F 3
In-plane stiffness
0
20
40
60
80
0 250 500 750 1000
Preload [N]
k[N/micron]
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Linearization of non-linear springs
If you can linearize over the appropriate range then
you can use superposition
So how would, and when could, you do this?
R = ball radius
E = modulus of both materials (both steel)
F = contact load
dF
1 2
1
kn ( )= Constant= F R 3E 3 F 3d
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P ti l li ti t th l th bl
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Practical application to the lathe problem
yx F
Case 11 in Appendix A-9 Martin Culpepper, All rights reserved 20
P ti l li ti t th l th bl
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Practical application to the lathe problem
y Fx
( ) =1 F x2(11x 9l )y x
AB 96EI
y( l2)=
96
1
EI F
7
8l3
768
E Ik =
Beam 7 l
3
But, is this really what is going on?
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P ti l li ti t th l th bl
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Practical application to the lathe problemy
Fx
768 E Ik =
Beam 7 l3
Vs. ?F k k
xy
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P ti l li ti t th l th bl
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Practical application to the lathe problem
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F
k k
xy
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Practical application to the lathe problem
F
x
Or is it this?
If so, does it matter?
yk
bearing
k k
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Practical application to the lathe problem
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Practical application to the lathe problem
x F
1 1F F
2F F
2 2
k k Martin Culpepper, All rights reserved 25
Group work
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Group work
Obtain an equation fortotal
in terms of F, k and l
Estimate when k is important / should be considered?
What issue/scenario would cause k not to be infinite?
Look at these causes, if a stiffness is involved, would
linearity, and therefore superposition apply?
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