The Mechanical Chaotic Oscillator
-
Upload
brandon-thacker -
Category
Documents
-
view
165 -
download
3
Transcript of The Mechanical Chaotic Oscillator
![Page 1: The Mechanical Chaotic Oscillator](https://reader034.fdocuments.in/reader034/viewer/2022042605/58a26d911a28ab94628b4f95/html5/thumbnails/1.jpg)
The Mechanical
Chaotic OscillatorBY: BRANDON THACKER
![Page 2: The Mechanical Chaotic Oscillator](https://reader034.fdocuments.in/reader034/viewer/2022042605/58a26d911a28ab94628b4f95/html5/thumbnails/2.jpg)
What is Chaos?
![Page 3: The Mechanical Chaotic Oscillator](https://reader034.fdocuments.in/reader034/viewer/2022042605/58a26d911a28ab94628b4f95/html5/thumbnails/3.jpg)
Deterministic vs. Stochastic
Systems
Stochastic Systems
• Random process
• Every possible state of a system is equally probable.
• The state of a system is independent of its past or future states.
![Page 4: The Mechanical Chaotic Oscillator](https://reader034.fdocuments.in/reader034/viewer/2022042605/58a26d911a28ab94628b4f95/html5/thumbnails/4.jpg)
Deterministic Systems
• A deterministic system is allowed
only one unique time evolution of
states for a given set of initial
conditions. Proof: “The
Uniqueness Theorem.”
• If you have complete knowledge of
the state of a system (state
variables), all possible future and
past states can be determined.
• Deterministic systems are “in
theory” completely predictable.
![Page 5: The Mechanical Chaotic Oscillator](https://reader034.fdocuments.in/reader034/viewer/2022042605/58a26d911a28ab94628b4f95/html5/thumbnails/5.jpg)
Characteristics of Chaos
• Chaos manifests in nonlinear dynamical systems.
• A chaotic system’s sensitivity to initial conditions make the system
unpredictable as time evolves.
• The time for which the system can be predicted (within a certain tolerance)
depends on the uncertainty in the initial conditions.
![Page 6: The Mechanical Chaotic Oscillator](https://reader034.fdocuments.in/reader034/viewer/2022042605/58a26d911a28ab94628b4f95/html5/thumbnails/6.jpg)
So What is Chaos?
• The take away: Chaos is a seemingly random
unpredictable behavior present in deterministic
nonlinear dynamical systems. The
unpredictability comes from small differences in
initial conditions leading to vastly different
trajectories due to the nonlinear response of the
system.
![Page 7: The Mechanical Chaotic Oscillator](https://reader034.fdocuments.in/reader034/viewer/2022042605/58a26d911a28ab94628b4f95/html5/thumbnails/7.jpg)
Nonlinear dynamics
• Nonlinear: A nonlinear system is one that gives a response
that is not directly proportional to its stimulus.
• Dynamical: A dynamical system is one that evolves in time.
• The MCO is a nonlinear dynamical system.
![Page 8: The Mechanical Chaotic Oscillator](https://reader034.fdocuments.in/reader034/viewer/2022042605/58a26d911a28ab94628b4f95/html5/thumbnails/8.jpg)
Modeling the MCO
![Page 9: The Mechanical Chaotic Oscillator](https://reader034.fdocuments.in/reader034/viewer/2022042605/58a26d911a28ab94628b4f95/html5/thumbnails/9.jpg)
The Phase Space Portrait
![Page 10: The Mechanical Chaotic Oscillator](https://reader034.fdocuments.in/reader034/viewer/2022042605/58a26d911a28ab94628b4f95/html5/thumbnails/10.jpg)
Key features of the MCO -1.0
Magnetic damping
(micrometer, inertial disc, and magnet)
System tracking
(Optical encoder and disk)
Electronics and communication
(microcontroller, push pull circuit
and serial communication)
Helmholtz chamber
(field coil, drive coil,
shaft and dipole)
Case and materials
![Page 11: The Mechanical Chaotic Oscillator](https://reader034.fdocuments.in/reader034/viewer/2022042605/58a26d911a28ab94628b4f95/html5/thumbnails/11.jpg)
Case and materials
Non magnetic materials
• Brass and nylon fasteners
• Acrylic case and coil spools
• Designed in AutoCad
• Cut using laser
• Brass shaft
![Page 12: The Mechanical Chaotic Oscillator](https://reader034.fdocuments.in/reader034/viewer/2022042605/58a26d911a28ab94628b4f95/html5/thumbnails/12.jpg)
Magnetic Damping
Micrometer adjuster
Magnet
Inertial disk
Lenz’s Law
![Page 13: The Mechanical Chaotic Oscillator](https://reader034.fdocuments.in/reader034/viewer/2022042605/58a26d911a28ab94628b4f95/html5/thumbnails/13.jpg)
Helmholtz Chamber
Field coils
Drive coils
![Page 14: The Mechanical Chaotic Oscillator](https://reader034.fdocuments.in/reader034/viewer/2022042605/58a26d911a28ab94628b4f95/html5/thumbnails/14.jpg)
By Morn - Own work, CC BY-SA 3.0, https://commons.wikimedia.org/w/index.php?curid=32750723
![Page 15: The Mechanical Chaotic Oscillator](https://reader034.fdocuments.in/reader034/viewer/2022042605/58a26d911a28ab94628b4f95/html5/thumbnails/15.jpg)
Magnetic dipole and assembly
3D-Printed assembly
Dipole specs.
• ¼ in separation
of magnets
• ¼ in depth of
magnets
• ¼ in diameter
of the magnets
![Page 16: The Mechanical Chaotic Oscillator](https://reader034.fdocuments.in/reader034/viewer/2022042605/58a26d911a28ab94628b4f95/html5/thumbnails/16.jpg)
Electronics and Microcontroller
Circuit components
• Zeroing DAC signal
• Voltage amplification
• Current amplification
Teensy operations
• Communication
• Resolving the
quadrature signal
from encoder
• Analog drive
signal
![Page 17: The Mechanical Chaotic Oscillator](https://reader034.fdocuments.in/reader034/viewer/2022042605/58a26d911a28ab94628b4f95/html5/thumbnails/17.jpg)
System Tracking
Encoder wheel
Old resolution: 360 steps
New resolution: 1024 steps
Optical encoder
Quadrature output signal
![Page 18: The Mechanical Chaotic Oscillator](https://reader034.fdocuments.in/reader034/viewer/2022042605/58a26d911a28ab94628b4f95/html5/thumbnails/18.jpg)
Quadrature Signal Resolution
![Page 19: The Mechanical Chaotic Oscillator](https://reader034.fdocuments.in/reader034/viewer/2022042605/58a26d911a28ab94628b4f95/html5/thumbnails/19.jpg)
USB-Communication
• Drive Phase, position,
velocity
• Query responses
• Commands (reporting,
frequency, amplitude
and much more)
• Queries
![Page 20: The Mechanical Chaotic Oscillator](https://reader034.fdocuments.in/reader034/viewer/2022042605/58a26d911a28ab94628b4f95/html5/thumbnails/20.jpg)
Table of commands for the MCO
![Page 21: The Mechanical Chaotic Oscillator](https://reader034.fdocuments.in/reader034/viewer/2022042605/58a26d911a28ab94628b4f95/html5/thumbnails/21.jpg)
Digital to analog converter (DAC)
Sine wave
lookup table
(Res. = 256)
Teensy’s on
board DAC
To circuit
![Page 22: The Mechanical Chaotic Oscillator](https://reader034.fdocuments.in/reader034/viewer/2022042605/58a26d911a28ab94628b4f95/html5/thumbnails/22.jpg)
Shifting and amplifying
From TeensyDifference amplifierVDAC = (0 - 3.3)v
Maximum D.C.
VMAX = 3.3v
Class B
Current amplifierVOUT = ±1.65v
Drive coils
VOUT = VDAC – 1/2VMAX
ICOIL = ~5A
VCOIL = ± 10v
![Page 23: The Mechanical Chaotic Oscillator](https://reader034.fdocuments.in/reader034/viewer/2022042605/58a26d911a28ab94628b4f95/html5/thumbnails/23.jpg)
Circuit Diagram
![Page 24: The Mechanical Chaotic Oscillator](https://reader034.fdocuments.in/reader034/viewer/2022042605/58a26d911a28ab94628b4f95/html5/thumbnails/24.jpg)
MCO (1.0) Results
Simple harmonic motion driven at the natural frequency
![Page 25: The Mechanical Chaotic Oscillator](https://reader034.fdocuments.in/reader034/viewer/2022042605/58a26d911a28ab94628b4f95/html5/thumbnails/25.jpg)
Limit cycles and transient behavior
The dying out of transient motion
![Page 26: The Mechanical Chaotic Oscillator](https://reader034.fdocuments.in/reader034/viewer/2022042605/58a26d911a28ab94628b4f95/html5/thumbnails/26.jpg)
From simple motion to chaos
Simple motion at 0.76 (rads/sec) Chaos at 0.82 (rads/sec)
![Page 27: The Mechanical Chaotic Oscillator](https://reader034.fdocuments.in/reader034/viewer/2022042605/58a26d911a28ab94628b4f95/html5/thumbnails/27.jpg)
![Page 28: The Mechanical Chaotic Oscillator](https://reader034.fdocuments.in/reader034/viewer/2022042605/58a26d911a28ab94628b4f95/html5/thumbnails/28.jpg)
![Page 29: The Mechanical Chaotic Oscillator](https://reader034.fdocuments.in/reader034/viewer/2022042605/58a26d911a28ab94628b4f95/html5/thumbnails/29.jpg)
![Page 30: The Mechanical Chaotic Oscillator](https://reader034.fdocuments.in/reader034/viewer/2022042605/58a26d911a28ab94628b4f95/html5/thumbnails/30.jpg)
![Page 31: The Mechanical Chaotic Oscillator](https://reader034.fdocuments.in/reader034/viewer/2022042605/58a26d911a28ab94628b4f95/html5/thumbnails/31.jpg)
Acknowledgements
• Contributors to the Chico State PSRI fund.
• Dr. Eric Ayars, program advisor and mentor
• Tucker Hartland, multifaceted wizardry
• Bill Koperwhats, Laser cutter
• Jadie Lee, Physics machine shop
• Scott Brogden, CSUC Mechanical Engineering dept.