Spring 2009 For information about citing these materials ...€¦ · knowledge to engineer a system...
Transcript of Spring 2009 For information about citing these materials ...€¦ · knowledge to engineer a system...
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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 02: Review
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Intent High-level review of undergrad
material as applied to engineering decision making
NOT an ME “redo” or a “how to” recitation
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ImportMain 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
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Use of core ME principles that you know…
2.001, 2.002
2.003, 2.004
2.007, 2.008
2.005, 2.006
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Impact Understand why & how we will use parts of ME core knowledge
Problem set → Engineering
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Future help
We can’t use lecture time to redo the early curriculum
BUTWe are HAPPY to help outside
of lecture IF you’ve tried
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Schedule and reading assignment Reading quiz
Changing from sponge to active mode
Lecture Hands-on Mechanics Mechanics Dynamics Dynamics Heat transfer Heat transfer Matrix math 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
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Mechanics
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Free body diagrams Useful for: v vΣF = 0 = ma
� Equilibrium � Stress, deflection, vibration, etc… ΣM = 0 = Iαv
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
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Free body diagrams: Bearings/rails
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Free body diagrams: Bearings/rails
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Static: Head stock deformation
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Static: Rail deformation
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l
Example 1: 0 < x < a Cantilever
� Forces, moments, & torques
Why do we care?: Stress � Shear & normal � Static failure � Fatigue failure
Why do we care?: Stiffness � Displacement � Rotation � Vibration → (k/m)½
But, ends aren’t all that matters
y
M(0)
y
V(0)
M(x)
F
© Martin Culpepper, All rights reserved 15
ba
F ba
( ) ( ) ( ) FxVVVFF ==→+−==Σ 000 v
( ) ( ) aFMMlFM ⋅−=→+⋅+==Σ 000 v
x
V(x)
F
ba
( ) ( ) ( ) ( )xaFxMxMxaFM −⋅−=→+−⋅+==Σ 0 v
( ) ( ) ( ) FxVVVFF ==→+−==Σ 000 v
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Example 1: 0 < x < a But, ends aren’t all that matters
Relating V(x) & M(x) � V ( ) = Fx
� M ( ) = F ⋅ x − a)x (
( )� V ( )x = d M xdx
Shear moment diagrams � Solve statics equation � Put point of import on plots � Use V=dM/dx to generate M plot � Master before spindle materials
x
V(x)
M(x) F
ba
F =Σ ( )→0VF +−0 v
= V (0) =V (x) = F v
ΣM = 0 = F ⋅ − M+ (a x)+ (x) → M (x) = −V (x)⋅(a − x)
V (x) = F
F
V x
M x -F·a
M (x) = −V (x)⋅(a − x) = −F ⋅(a − x) © Martin Culpepper, All rights reserved 16
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Example 1: 0 < x < a Stress
M(x) M(x+dx)
2
h
h
b
maxσ
y
I ( )x = 1 b( ) ( )x [h x ]3
12
( ) = M y →σmax = M cσ y
I ( )x ( )I x
h x = F ⋅ (a − x) ( ) 12σmax b( ) ( )x [ ]32 ⋅ h x
F ⋅ (a − x)=6σmax ( ) ( )⋅ h x ]2b x [
© Martin Culpepper, All rights reserved
a
FM(x)
V(x)
x b
( ) ( ) ( ) ( ) ( ) ( )xaxVxMxMxaxVM −⋅−=→−−⋅+==Σ 0 v
( ) ( ) ( ) FxVVVFF ==→+−==Σ 000 v
M ( ) ( ) ( ) ( )xaFxxVxM −=
-F·a a −⋅−=−⋅
F ⋅(a − x)σ = 6 bh2
bh2 6F·l
bh2 -6F·l σ x
17
F ⋅(a − x)σ = −6 bh2
x
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Group work: Generate strategy for this…
z
x
y z
x
y
Vx Vxz z
Vy Vyz z
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Dynamics
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Vibration Vibration principles
� Exchange potential-kinetic energy � 2nd order system model
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
Estimate ωn (watch units) for: � A car suspension system
© Martin Culpepper, All rights reserved
x
m
k
c
F(t)
ωn
n =ω k
Gain m
ω
20
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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
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Vibration: MEMS device behavior
Without input shaping With input shaping
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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.
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Vibrations: Reducing amplitude… How to change
m, k, and c?
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So… where find in reality… in lathe… x
m
k
c
F(t)
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Vibration: Lathe structure – 1st mode
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Vibration: Lathe structure – 2nd mode
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Heat transfer
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Thermal growth errors
TL Cm
m oo Δ× −6105.11
For uniform temperature ΔL =α L ΔTo
STEEL :12L14 →ΔL =
× −6106.23 mALUMINUM :6061 T 6 →ΔL = L ΔT m oC o
POLYMER : Delrin →ΔL = × −610100 m Lo ΔT
m oC
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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
© Martin Culpepper, All rights reserved
Conduction: dTq& = k Across dx
Why do we care? � Heat removal from cutting zone � Heat generation in bearings � Thermal growth errors
Common k values to remember � Air 0.026 W /(moC)
� 12L14 51.9 W /(moC)
� 6061 T6 167 W /(moC)
30
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Thermal resistance Thermal resistance
ΔT q& =RT
� Convection
q& = (h ( AT − T∞ )
)−1 a RT =
h A 1
surface surface
� Conduction
k A dx q& = dT cross a R = dx T k Across
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Biot (Bi) number Ratio of convective to conductive heat transfer
q&convection a (h AΔT )convection a Bi =
hLc kq&conduction ⎛
⎜k A ΔT ⎞⎟
⎝ L ⎠conduction
Why do we care?
Low Bi High Bi © Martin Culpepper, All rights reserved 32
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Example of thermal errors For: For:
� h = 0.1 W/(m2oC) � h = 50 W/(m2oC) � Bearing T = 150 oF � Bearing T = 150 oF � Chip T = 180 oF � Chip T = 180 oF
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Example of thermal errors For: For:
� h = 0.1 W/(m2oC) � h = 5000 W/(m2oC) � Bearing T = 150 oF � Bearing T = 150 oF � Chip T = 180 oF � Chip T = 180 oF
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Types of errors
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[ [ ]InputsCOutputs
Machine system perspectiveSystem-level approach ⎡C1 C2 C3⎤
⎢ ⎢ ⎢⎣
⎥ ⎥ ⎥⎦
]=Linking inputs and outputs C C4 5 6
C7 C8 C9Measurement quality
TemperaturePower Material changes Torques
Desired outputs- Motion, location, rotation - Cutting forces- Speeds Perceived
Measured outputs Error - Motion, location, rotation Real- Cutting forces- Speeds error
Actual outputs- Motion, location, rotation - Cutting forces- Speeds
Machine
FabricationForces Vibration Speedserrors
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Errors…. Accuracy
� The ability to tell the “truth”
Repeatability � Ability to do the same thing over & over
Both � 1st make repeatable, then make accurate � Calibrate
Determinism � Machines obey physics! � Model → understand relationships
⎡C1 C2 C3 ⎤ [Outputs]= ⎢
⎢C4 C5 C6 ⎥⎥[Inputs]
⎢C C C ⎥⎣ 7 8 9 ⎦
� Understand sensitivity
[ΔOutputs]= J [ΔInputs]Range
� Furthest extents of motion
Resolution � Smallest, reliable position change
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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 B C D © Martin Culpepper, All rights reserved 38
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Exercise
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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
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Group work: Generate strategy for this…
z
x
y z
x
y
Vx Vxz z
Vy Vyz z
© Martin Culpepper, All rights reserved 41