Nonlinear Dynamics Laboratory Equipment
Andrew Dick Joe Houtz
Jeremy Redlecki Ashley Rice
James Streeter
Client: Mechanical Engineering Dept. at R•I•T Faculty Mentor: Dr. J.S. Török
Presentation Outline
• Nonlinear Dynamics and Chaos• Project Goals• Design Process• Preliminary Design• Expenses and Budget• Spring Quarter Schedule• Conclusion
Nonlinear Dynamics and Chaos
• Nonlinear Dynamics• Physical systems will nonlinear electrical/mechanical
components• State Equations contain nonlinear terms• Closed form solutions are unlikely• Variety of dynamic behavior
• Chaos• Deterministic systems, not stochastic• Long-term apparent erratic behavior• Dynamics governed by ‘strange’ attractor• Sensitive Dependence on initial conditions
Project Goals
• Design, build, and test nonlinear dynamics laboratory equipment to aid in the teaching of nonlinear dynamics
• Develop a set of procedures and manuals for experiments and demonstrations
Description
• Dynamic systems governed by nonlinear equations
• Ability to demonstrate nonlinear and chaotic behavior
• Equipped with data acquisition equipment to collect data for analysis
• Compatible with three tier system of experimentation1. Demonstration
2. Experimentation
3. Design oriented experimentation
• Extensive testing to validate equipment and gather data required for the completion of the project
Equipment Set
• Chua’s Circuit• Nonlinear Electric Circuit
• Multi-well Oscillator• Magneto-Elastic System
• Chaotic Waterwheel• Mechanical-Fluid System
Design Facets
• Recognize & quantify the need
• Concept development
• Feasibility assessment
• Establishing design objectives & criteria
• Analysis of problems & synthesis into the design
• Preliminary design
Facet One: Needs Assessment
• Laboratory equipment usage• One, two and three tier experiments
• Design constraints
• Financial parameters• $2000 budget
• Performance parameters
Facet Two: Concept Development
• Brainstorming
• Research existing systems
• Group consensus of preliminary concepts
• Develop drawings & descriptions of each preliminary concept• Group exercises
• Concept drawing
• Empathy method
• Preliminary BOM
Facet Three: Feasibility Assessment
• Technical aspects
• Economical aspects
• Market
• Schedule
• Performance
Facet Four: Design Objectives & Performance Specifications
• Safety codes and standards
• List of design objectives • Yes/no question format
• Evaluation checklist
• Performance specification table
Chua’s Circuit• Kirchoff’s laws
)( 1121
1 VgR
VV
dt
dVC
LIR
VV
dt
dVC
212
2
2Vdt
dIL L
-3 -2 -1 0 1 2 3-2
-1.5
-1
-0.5
0
0.5
1
1.5
2x 10
-3 Chua's Circuit I-V Characteristic
V (volts)
I (a
mps
)
xydt
dx
xz y rxdt
dy
bzxydt
dz
Lorenz Equations:
Chua’s Circuit:Simulation and Analysis
• PSpice simulation
• Matlab• Frequency Spectrum
Time
0s 5ms 10ms 15ms 20ms 25ms 30ms 35ms 40ms 45ms 50msV(V1)
-3.0V
-2.0V
-1.0V
-0.0V
1.0V
2.0V
3.0V
V(V1)
-3 -2 -1 0 1 2 3-0.8
-0.6
-0.4
-0.2
0
0.2
0.4
0.6Chua's Circuit V1 vs. V2
V1 (volts)
V2
(vol
ts)
Chua’s Circuit Apparatus Design
• Op-amps represent Chua’s Diode.• Output voltages can be connected to an
oscilloscope, LabView, or speakers.• Setup:
• A base is to be fabricated • Transparent cast acrylic cover• Two speakers
Multi-Well Oscillator
• Research• Magneto-elastic system
• Theory
)cos(32
2
tFxxdt
dx
dt
xd
Multi-well Oscillator:Simulation and Analysis
• Control parameters:• , Damping Factor
• F, Force Amplitude
Multi-well Oscillator Apparatus Design
• Analyses• Stress
• Fatigue
• Frequency
• Magnetic
Chaotic Waterwheel
• Willem Malkus and Lou Howard developed the first chaotic waterwheel at MIT in the 1970’s.
Chaotic Waterwheel:Simulation and Analysis
111 Kaba
1111 qKbab
I
grav 1
Waterwheel Equations:
Chaotic Waterwheel Apparatus Design
• General operation
• Analyses
• Flow analysis
• Moment of inertia analysis
• Stress analysis
• Brake
• Encoder Bearing
Encoder bearing
Brake
In flow
Overflow
Needle valve
Out flow
Expenses and Budget
• Bill of material created for each device• Raw material, purchased components, electrical components, sensors,
nuts and bolts.
• Cost per device and complete set• Chua’s Circuit Apparatus 141 parts $267.71
• Multi-well Oscillator Apparatus 153 parts $597.01
• Chaotic Waterwheel Apparatus 447 parts $1135.81
• Complete set of three Apparatuses 741 parts $1895.29
• Available Balance$104.71
Spring Quarter Schedule
• Phase One: Construction 4 weeks• Quality/performance test components• Fabricate components• Assemble equipment
• Phase Two: Testing 4 weeks• Confirm/optimize design• Determine critical parameters• Develop experimentation procedures
• Phase Three: Documentation 2 weeks• Complete user manual• Final Design Report• Critical Design Presentation
Conclusion
• Successfully completed design facets 1 through 6
• Conducted many different analyses on each of the devices
• Completion of a preliminary design
• Expenses and Budget
• Spring Quarter Schedule