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.
2.72
Elements of
Mechanical Design
Lecture 16: Dynamics
and damping
Schedule and reading assignment
Quiz
�
None
Topics
�
�
�
Vibration physics
Connection to real world
Activity
Reading assignment
�
Skim last gear reading assignment… (gear selection)
© Martin Culpepper, All rights reserved
2
Resonance
Basic Physics
�
�
x
c
Exchange potential-kinetic energy
Energy transfer with loss
k
m
PE
Loss
KE
F(t)
Modeling
�
�
�
2nd order system model
Spring mass damper
Differential equations, Laplace
Transforms
Why Do We Care
�
�
�
�
Critical to understanding motion of
structures- desired and undesired
Generally not steady state
Location error
Large forces, high fatigue
k
ωn =
m
Gain
ωn
© Martin Culpepper, All rights reserved
ω
3
Vibrations - Input
Oscillation of System
Why Categorize
�
�
A
Forced
Different causes
Different solutions
Input Form
�
�
Forced – Steady State
• Command Signal
• Electrical (60 Hz)
Free – Transient
• Impacts
t
A
Free
t
© Martin Culpepper, All rights reserved
4
Vibrations - Source
Source
�
Undriven
Disturbance
Spectrum
10­9
Disturbance
• Driven
10­13
Device Motors
Rotating Components
10­17
10­2 1
102 104
• Undriven
Electrical (60 Hz)
People (2 Hz)
Cars (10 Hz)
Nearby Equipment
�
Driven Disturbance
Command
• Step Response
A
System
t
Command
© Martin Culpepper, All rights reserved
Vibration
Response
5
Vibrations: Command vs. Disturbance
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
6
Vibrations - Example
Example
�
�
A
Chinook
Identify:
• Mode
• Form
• Source
• Response
Forced
A
t
Free
t
10­9
Undriven Disturbance
Spectrum
10­13
10­17
10­2
1
102 104
Driven Disturbance
A
System
t
© Martin Culpepper, All rights reserved
Response
7
Attenuating Vibrations
Change System
�
�
Mass, stiffness, damping
Adjust mode shapes
Change Inputs
�
�
Command: Input Modulation, Feedback
Disturbance, reduce:
• undriven vibrations, e.g. optical table
• driven vibrations – alter device structure (damping on motors, etc.)
A
F
t
© Martin Culpepper, All rights reserved
1
ms2+bs+k
Z
8
Modulating Command Vibrations
Change
m, k, c
© Martin Culpepper, All rights reserved
Input shaping
9
Behavior
Regimes
�
�
�
(1) Low Frequency (ω<ωn)
(2) Resonance (ω≈ωn)
(3) High Frequency (ω>ωn)
Example – Spring/Mass demo
ζ = 0.05
Amplification factor (output/input)
�
10
ζ = 0.1
2
1
3
ζ = 0.2
ζ = 0.3
ζ = 0.4
1
ζ = 0.5
ζ = 0.6
ζ = 0.7
1st order system
ζ = 0.8
ζ = 0.9
ζ = 2.0
ζ = 1.0
-1
-1
ω/ωn
1
10
Figure by MIT OpenCourseWare.
1
1
x
= 2
=
F ms + bs + k k (ω )
© Martin Culpepper, All rights reserved
10
Behavior – Low Frequency
Frequency Response Regimes
�
�
�
(1) Low Frequency (ω<ωn)
(2) Resonance (ω≈ωn)
(3) High Frequency (ω>ωn)
Example – Spring/Mass demo
Low Frequency (ω<ωn)
�
�
�
System tracks commands
Ideal operating range
High disturbance rejection
ζ = 0.05
x 1
≈
F k
Amplification factor (output/input)
�
10
1
ζ = 0.1
2
3
ζ = 0.2
ζ = 0.3
ζ = 0.4
1
ζ = 0.5
ζ = 0.6
ζ = 0.7
1st order system
ζ = 0.8
ζ = 0.9
ζ = 2.0
Resonance Frequency (ω≈ωn)
�
�
�
�
�
System Response >> command
-1
-1
keff ↓
Disturbances will cause very large response
Quality factor = magnitude of peak Figure by MIT OpenCourseWare.
Damping ↑ = Q ↓
ζ = 1.0
ω/ωn
1
10
x
1
= 2
F ms + bs + k
High Frequency (ω≈ωn)
�
�
System Response << command
High disturbance rejection
© Martin Culpepper, All rights reserved
11
Behavior - Resonance
10
Frequency Response Regimes
�
�
�
(1) Low Frequency (ω<ωn)
(2) Resonance (ω≈ωn)
(3) High Frequency (ω>ωn)
Example – Spring/Mass demo
Low Frequency (ω<ωn)
�
�
�
System tracks commands
Ideal operating range
High disturbance rejection
x Q
<
≤
Resonance Frequency (ω≈ωn)
k F1 k
� System Response >> command
�
�
�
�
keff ↓
Disturbances will cause very large response
Quality factor = magnitude of peak
Damping ↑ = Q ↓
ζ = 0.05
Amplification factor (output/input)
�
ζ = 0.1
2
1
3
ζ = 0.2
ζ = 0.3
ζ = 0.4
1
ζ = 0.5
ζ = 0.6
ζ = 0.7
1st order system
ζ = 0.8
ζ = 0.9
ζ = 2.0
ζ = 1.0
-1
-1
ω/ωn
1
10
Figure by MIT OpenCourseWare.
x
1
= 2
F ms + bs + k
High Frequency (ω≈ωn)
�
�
System Response << command
High disturbance rejection
© Martin Culpepper, All rights reserved
12
Behavior – High Frequency
Frequency Response Regimes
�
�
�
(1) Low Frequency (ω<ωn)
(2) Resonance (ω≈ωn)
(3) High Frequency (ω>ωn)
Example – Spring/Mass demo
ζ = 0.05
Amplification factor (output/input)
�
10
Low Frequency (ω<ωn)
�
�
�
System tracks commands
Ideal operating range
High disturbance rejection
ζ = 0.1
2
1
ζ = 0.2
ζ = 0.3
ζ = 0.4
1
ζ = 0.5
ζ = 0.6
ζ = 0.7
1st order system
ζ = 0.8
ζ = 0.9
ζ = 2.0
ζ = 1.0
Resonance Frequency (ω≈ωn)
�
�
�
�
�
System Response >> command
keff ↓
Disturbances will cause very large response
Quality factor = magnitude of peak
Damping ↑ = Q ↓
High Frequency (ω≈ωn)
�
�
System Response << command
Poor disturbance rejection
© Martin Culpepper, All rights reserved
3
-1
-1
ω/ωn
1
10
Figure by MIT OpenCourseWare.
x
1
= 2
F ms + bs + k
1
x
≈
F mω 2
13
Constitutive Relations
Relevant equations
�
�
�
�
Damping ratio
ωn
Gain
Quality factor
© Martin Culpepper, All rights reserved
c
ζ =
2 km
k
ωn =
m
GP =
ωd = ωn 1 − ζ
1
2ζ 1− ζ 2
2
1
ζ <
2
1
Q=
2ζ (1− ζ 2 )
14
Application of Theory
Relate Variables to Actual Parameters
�
�
�
�
Vibrational Mode
Mass
Stiffness
Damping
Transfer between Model and Reality
�
�
�
�
Iterative
Start simple (1 mass, 1 spring)
Add complexity
Limits
Example – building (video)
c
x
k
m
F(t)
Iteration
Mode?
k?
m?
c?
© Martin Culpepper, All rights reserved
15
Strategies for damping
Material
�
�
�
Pros and cons of each
Grain boundary
Internal lattice
Viscoelastic (elastomers/goo)
Viscous
�
�
Air
Fluid
Electromagnetic
Friction
Active
Combinations
�
Sponge
© Martin Culpepper, All rights reserved
16
Example: Couette flow relationships
Relevant equations
dx�
τ =µ
dy
dx�
x�
F = τA = Aµ
= Aµ
dy
h
F = cx�
(F, x� )
µ
A
h
Aµ
c=
h
© Martin Culpepper, All rights reserved
17
Exercise (see next page too)
Perform a frequency analysis of the part
Develop & prove (FEA) how to increase nat. freq. via geometry change
� Any constraints you might have? Geometry changes can’t be unbounded
� Explain effect of your change on vibration amplitude (relative to outer
base) at given ω, via sketches & plots
�
Xtra credit, assume:
Flexure is contained between two parallel plates (on top and bottom)
� Viscous air damping in the gaps on both sides
� 1 micron gap between the flexure sides and plates
� Elaborate on how well flexure is damped (don’t just use intuition)
�
Useful equations (c = damping coefficient, k = stiffness, m = mass)
c
ζ =
2 km
Gain at peak amplitude
Frequency at peak with max gain
Damping ratio
ω p = ωn 1 − 2ζ
© Martin Culpepper, All rights reserved
2
Gp=
1
2
4ζ − 4ζ
4
18
Flexure
Flexure design
top
Constrained on 4 sides
Bottom
See this diagram for extra credit:
Top plate
Flexure
© Martin Culpepper, All rights reserved
Bottom plate
19
Multiple Resonances
Gain
10
Center Stage Modes
0
10
10
Measured
Model
Fit
-2
10
Phase Lag (deg)
Z-Axis
2
1
2
10
Frequency (Hz)
10
3
0
-100
-200
-300
10
1
© Martin Culpepper, All rights reserved
2
10
Frequency (Hz)
3
10
20