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2.72 Elements of Mechanical Design
Spring 2009
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2.72
Elements of
Mechanical Design
Lecture 02: Review
Intent
High-level review of undergrad
material as applied to
engineering decision making
NOT an ME “redo” or a “how
to” recitation
Import
Main goal of 2.72 is to teach you how to integrate past knowledge to engineer a system
Given this, how do I engineer a mechanical system?
modular→simple→complex→system
2.001, 2.002 2.003, 2.004 2.005, 2.006 2.007, 2.008
Use of core ME principles that you know…
2.001, 2.002
2.003, 2.004
2.005, 2.006
2.007, 2.008
© Martin Culpepper, All rights reserved
4
Impact
Understand why & how we will
use parts of ME core knowledge
Problem set → Engineering
Future help
We can’t use lecture time to redo the early curriculum
BUT
We are HAPPY to help outside of lecture IF you’ve tried
Schedule and reading assignment
Reading quiz
Changing from sponge to active mode
Lecture
Mechanics
Dynamics
Heat transfer
Matrix math
Hands-on
Mechanics
Dynamics
Heat transfer
Matrix math
Reading assignment
Shigley/Mischke
• Sections 4.1–4.5: 08ish pages & Sections 5.1–5.5: 11ish pages
• Pay special attention to examples 4.1, 4.4, 5.3 , and 5.4
© Martin Culpepper, All rights reserved
7
Mechanics
Free body diagrams
Useful for:
Equilibrium
Stress, deflection, vibration, etc…
v
v
ΣF = 0 = ma
v
ΣM = 0 = Iα
Why do we ALWAYS use free body diagrams?
Communication
Thought process
Documentation
How will we use free body diagrams?
We are dealing with complex systems
We will break problem into modules
We will model, simulate and analyze mechanical behavior
Integrate individual contributions to ascertain system behavior
© Martin Culpepper, All rights reserved
9
Free body diagrams: Bearings/rails
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11
Free body diagrams: Bearings/rails
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12
Static: Head stock deformation
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13
Static: Rail deformation
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14
Example 1: 0 < x < a
y
Cantilever
Shear & normal
Static failure
Fatigue failure
Why do we care?: Stiffness
F
Forces, moments, & torques
y
Why do we care?: Stress
a
Displacement
Rotation
Vibration → (k/m)½
But, ends aren’t all that matters
a
F
M(0)
b
l
b
V(0)
v
ΣF = 0 = − F + V (0 ) → V (0 ) = V ( x ) = F
v
ΣM = 0 = + F ⋅ l + M (0 ) → M (0 ) = − F ⋅ a
a
x
b
F
M(x)
V(x)
v
ΣF = 0 = − F + V (0 ) → V (0 ) = V ( x ) = F
v
ΣM = 0 = + F ⋅ (a − x ) + M ( x ) → M ( x ) = − F ⋅ (a − x )
© Martin Culpepper, All rights reserved
15
Example 1: 0 < x < a
But, ends aren’t all that matters
a
x
b
F
M(x)
Relating V(x) & M(x)
V (x ) = F
M (x ) = F ⋅ (x − a )
V (x ) =
d
M (x )
dx
Shear moment diagrams
V(x)
v
ΣF = 0 = − F + V (0 ) → V (0 ) = V ( x ) = F
v
ΣM = 0 = + F ⋅ (a − x ) + M ( x ) → M ( x ) = −V ( x )⋅ (a − x )
V ( x ) = F
F
V
x
M
x
Solve statics equation
Put point of import on plots
Use V=dM/dx to generate M plot
Master before spindle materials
-F·a
© Martin Culpepper, All rights reserved
M ( x ) = −V ( x )⋅ (a − x ) = −F ⋅ (a − x )
16
Example 1: 0 < x < a
Stress
M(x)
a
x
M(x+dx)
F
M(x)
σ max
h
2
b
V(x)
v
ΣF = 0 = − F + V (0 ) → V (0 ) = V ( x ) = F
y
h
v
ΣM = 0 = +V ( x )⋅ (a − x ) − M ( x ) → M ( x ) = −V ( x )⋅ (a − x )
b
I (x ) =
σ (y) = M
1
b( x ) [h ( x )]3
12
-F·a
y
c
→ σ max = M
I (x )
I (x )
σ max = F ⋅ (a − x )
σ max = 6
M
h(x )
12
2 b( x ) ⋅ [h ( x )]3
F ⋅ (a − x )
b( x ) ⋅ [h ( x )]2
© Martin Culpepper, All rights reserved
x
M ( x ) = −V ( x )⋅ ( x − a ) = − F ⋅ (a − x )
σ =6
6F·l
bh2
F ⋅ (a − x )
bh 2
σ
-6F·l
bh2
x
σ = −6
F ⋅ (a − x )
bh 2
17
Group work: Generate strategy for this…
x
x
z
z
y
y
Vx
z
Vx
z
Vy
z
Vy
z
© Martin Culpepper, All rights reserved
18
Dynamics
Vibration
Vibration principles
Exchange potential-kinetic energy
2nd order system model
c
m
Blocks and squiggles…
What do they really mean?
Why are they important?
How will we apply this?
Multi-degree-of-freedom system
Mode shape
Resonant frequency
x
F(t)
k
k
m
ωn =
Gain
Estimate ωn (watch units) for:
A car suspension system
ωn
© Martin Culpepper, All rights reserved
ω
20
Images removed due to copyright restrictions. Please see:
http://www.hpiracing.com/graphics/kits/547/_MG_1962e.jpg
http://www.societyofrobots.com/images/mechanics_suspension_honda.gif
http://www.bose.com/images/learning/lc_susp_frontmodule.jpg
© Martin Culpepper, All rights reserved
21
Vibration: MEMS device behavior
Without input shaping
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With input shaping
22
Vibrations: Meso-scale device behavior
Underdamped response
Noise
Free vibration
Golda, D. S., “Design of High-Speed, Meso-Scale Nanopositioners
Driven by Electromagnetic Actuators,” Ph.D. Thesis, Massachusetts
Institute of Technology, 2008.
© Martin Culpepper, All rights reserved
23
Vibrations: Reducing amplitude…
How to change
m, k, and c?
© Martin Culpepper, All rights reserved
24
So… where find in reality… in lathe…
x
c
m
F(t)
k
© Martin Culpepper, All rights reserved
25
Vibration: Lathe structure – 1st mode
© Martin Culpepper, All rights reserved
26
Vibration: Lathe structure – 2nd mode
© Martin Culpepper, All rights reserved
27
Heat transfer
Thermal growth errors
For uniform temperature
ΔL = α Lo ΔT
STEEL :12L14 → ΔL = 11.5 × 10
−6
m
Lo ΔT
o
m C
−6
m
Lo ΔT
o
m C
−6
m
Lo ΔT
o
m C
ALUMINUM :6061 T 6 → ΔL = 23.6 × 10
POLYMER : Delrin → ΔL = 100 × 10
© Martin Culpepper, All rights reserved
29
Convection and conduction
Convection:
q& = h Asurface (T − T∞ )
Why do we care?
Heat removal from cutting zone
Heat generation in bearings
Thermal growth errors
Conduction:
q& = k Across
dT
dx
Why do we care?
Heat removal from cutting zone
Heat generation in bearings
Thermal growth errors
Common k values to remember
© Martin Culpepper, All rights reserved
Air
0.026 W /(moC)
12L14
51.9 W /(moC)
6061 T6
167 W /(moC)
30
Thermal resistance
Thermal resistance
ΔT
q& =
RT
Convection
q& =
(T − T∞ )
(h Asurface )
−1
a RT =
1
h Asurface
Conduction
k Across
dx
q& = dT
a RT =
dx
k Across
© Martin Culpepper, All rights reserved
31
Biot (Bi) number
Ratio of convective to conductive heat transfer
(h AΔT ) convection
q&convection
hLc
a
a Bi =
q&conduction
k
ΔT ⎞
⎛
⎜k A
⎟
L ⎠
conduction
⎝
Why do we care?
Low Bi
© Martin Culpepper, All rights reserved
High Bi
32
Example of thermal errors
For:
For:
h=
Bearing T =
Chip T =
© Martin Culpepper, All rights reserved
0.1 W/(m2oC)
150 oF
180 oF
h=
Bearing T =
Chip T =
50 W/(m2oC)
150 oF
180 oF
33
Example of thermal errors
For:
For:
h=
Bearing T =
Chip T =
© Martin Culpepper, All rights reserved
0.1 W/(m2oC)
150 oF
180 oF
h=
Bearing T =
Chip T =
5000 W/(m2oC)
150 oF
180 oF
34
Types of errors
Machine system perspective
System-level approach
Linking inputs and outputs
Measurement quality
Power
Material
Temperature
changes
⎡
C1 C2 C3
⎤
[Outputs]
=
⎢⎢C4 C5 C6
⎥⎥[Inputs]
⎢⎣
C7 C8 C9
⎥⎦
Torques
Desired outputs
- Motion, location, rotation
- Cutting forces
- Speeds
Measured outputs
- Motion, location, rotation
- Cutting forces
- Speeds
Machine
Perceived
Error
Real
error
Actual outputs
- Motion, location, rotation
- Cutting forces
- Speeds
Forces
Vibration
Fabrication
errors
© Martin Culpepper, All rights reserved
Speeds
36
Errors….
Accuracy
The ability to
tell the “truth”
Repeatability
Ability to do the
same thing over &
over
Determinism
Machines obey physics!
Model → understand relationships
⎡ C1 C2 C3 ⎤
[Outputs] = ⎢⎢C4 C5 C6 ⎥⎥[Inputs]
⎢C
⎣ 7 C8 C9 ⎥⎦
Understand sensitivity
[ΔOutputs] = J [ΔInputs ]
Range
Both
1st make repeatable,
then make accurate
Calibrate
Furthest extents of motion
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Resolution
Smallest, reliable position change
37
Categorizing error types
Systematic errors
Inherent to the system, repeatable and may be calibrated out.
Non-systematic errors
Errors that are perceived and/or modeled to have a statistical nature
Machines are not “random,” there is no such thing as a random error
Consider the error for each set below
Link behavior with systematic and non-systematic errors.
A
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B
C
D
38
Exercise
Exercise
Due Tuesday, start of class:
Lathe components
Rough sketch(es) of lathe
Annotate main components
1 page bullet point summary of where need to use:
2.001, 2.002
2.003, 2.004
2.005, 2.006
2.007, 2.008
Rules:
You may not re-use examples from lecture!
You are encouraged to ask any question!
You may work in groups, but must submit your own work
© Martin Culpepper, All rights reserved
40
Group work: Generate strategy for this…
x
x
z
z
y
y
Vx
z
Vx
z
Vy
z
Vy
z
© Martin Culpepper, All rights reserved
41
