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2.72 Elements of Mechanical Design
Spring 2009
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2.72
Elements of Mechanical Design
Lecture 03: Shafts
Schedule and reading assignment
Reading quiz
Hand forward lathe exercise quiz
Topics
Finish matrices, errors
Shaft displacements
Stiffness exercise
Reading assignment
Shigley/Mischke
• Sections 6.1–6.4: 10ish pages & Sections 6.7–6.12: 21ish pages
• Pay special attention to example 6.12 (modified Goodman portion)
© Martin Culpepper, All rights reserved
2
Deflection within springs and shafts
Shafts, axles and rails
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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4
Examples drawn from your lathe
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5
Examples drawn from your lathe
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6
In practice, we are concerned with
Deflection
Stiffness
Bearings and stiffness of connectivity points
Function of global shaft geometry, sometimes adjacent components
Stress
Catastrophic failure:
Ductile
Function of local shaft geometry
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Brittle
Fatigue
7
What is of concern?
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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8
Materials
Steel vs. other materials
Aluminum
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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9
Principles of stiffness: Relationships
q
q d4y
= 4
EI dx
V
d3y
= 3
EI dx
M d2y
= 2
EI dx
θ=
dy
dx
© Martin Culpepper, All rights reserved
q
V
M
θ
10
Modeling: General forms of equations
Lateral bending deflection (middle)
F
F L3
F L3
Const
δ=
48E I
EI
Axial deflection
RL
RR
F L
δ=
AE
Lateral bending angles (at ends)
F L2
F L2
ML
δ=
=
or
6 E I Const E I
Const E I
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11
Modeling: Stiffness
Lateral bending stiffness at middle
kb
(
E I)
= 48
= Const (E I ) Ln
F
L3
Axial stiffness
RL
RR
AE
kA =
L
Torsional stiffness
Stepped shafts?
J G
kθ =
L
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12
Modeling: Stiffness
These pop up in many places, memorize them
Square cross section
1 3
I = bh
12
Circular cross sections
I=
[
(d
64
J=
[
(d
32
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π
4
4
)
(
)
−
d
outer
inner ]
π
)
4
outer
− (d inner )
4
]
13
Principles of stiffness: Ratios
Everything 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
k1st
Rk =
k2nd
Stiffness ratio
½l
F
F
kaxial
AE
=
l
klateral
3EI
= 3
l
AE
2
l
Rk = l = 4 2
3EI
h
l3
Building intuition for stiffness
You can’t memorize/calculate everything
Engineers must be reasonable “instruments”
Car suspension is easy, but flexed muscle vs. bone?
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14
Principles of stiffness: Sensitivity
Cantilever
L
F
F ⋅ L3
δ=
3⋅ E ⋅ I
1
I = ⋅b ⋅ h 3
12
⎛ E ⋅b ⎡ h ⎤ 3 ⎞
⋅ ⎢ ⎥ ⎟ ⋅δ
F =⎜
⎜ 4 ⎣L⎦ ⎟
⎝
⎠
© Martin Culpepper, All rights reserved
h
b
δ
1
Lm
hn
3
3
⎫
⎧
dF
d ⎪ E ⋅b ⎡ h ⎤
E ⋅b ⎡ h ⎤
⎪
→
⋅⎢ ⎥
δ
k=
=
⋅
⋅
⎬
⎨
⎢
⎥
4 ⎣L⎦
dδ dδ ⎪⎩ 4 ⎣ L ⎦
⎪⎭
15
Superposition
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
Superposition is not plug and chug
You must visualize
You must think
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16
Types of springs and behaviors
Springs and stiffness
kF = dF(x)/dx
Force-Displacement Curve
F(x)=F
Constant force spring
= dF(x)/dx = 0
kF
ΔEb-a = F • (xb - xa )
F
a
Constant stiffness spring
F
kF
= Constant
ΔEb-a = 0.5 • kF • (xb2 - xa2 )
Force-Displacement Curve
F
F
b
x
F(x)=kF•x
F
1
kF
a
x
b
a
x
b
Non-linear force spring
kF
= function of x
ΔEb-a = ∫ F(x) • dx
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Force-Displacement Curve
F
Images from Wikimedia Commons, http://commons.wikimedia.org
17
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
80
k [N/micron]
In-plane stiffness
60
40
20
0
0
250
500
750
1000
Preload [N]
⎛ 13 2 3 ⎞ 13
kn (F ) = Constant ⋅ ⎜ R ⋅ E ⎟ ⋅ F
⎝
⎠
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18
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
⎛ 13 2 3 ⎞ 13
= kn (F ) = Constant ⋅ ⎜ R ⋅ E ⎟ ⋅ F
dδ
⎝
⎠
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19
Practical application to the lathe problem
y
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x
F
Case 11 in Appendix A-9
20
Practical application to the lathe problem
y
F
x
y ( x ) A→B
1
2
=
⋅ F ⋅ x (11x − 9l )
96EI
( )
y l
1
7 3
=
⋅F ⋅ l
2 96 EI
8
k Beam
768 E I
=
⋅ 3
7
l
But, is this really what is going on?
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21
Practical application to the lathe problem
y
F
x
k Beam
y
Vs. ?
x
½k
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768 E I
=
⋅ 3
7
l
F
½k
22
Practical application to the lathe problem
y
x
½k
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F
½k
23
Practical application to the lathe problem
Or is it this?
If so, does it matter?
y
x
½k
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F
kbearing
½k
24
Practical application to the lathe problem
δ1
δ2
x
F
δ1
½F
½F
½F
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½F
δ2
½k
δ2
½k
25
Group work
Obtain an equation for δtotal 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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