Double Pendulum Undergraduate Research Symposium.pdf
Double Pendulum Undergraduate Research Symposium.pdf
Double Pendulum Undergraduate Research Symposium.pdf
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Measuring Chaos in a <strong>Double</strong><strong>Pendulum</strong>Vasha Dutell, Patrick Freeman, Duncan Luiten,and Professor Eric TorrenceUO <strong>Undergraduate</strong> <strong>Research</strong> <strong>Symposium</strong>UO <strong>Undergraduate</strong> <strong>Research</strong><strong>Symposium</strong>
Motivation & Background- Simple system, chaotic motion- Chaos: small Δ initial ➔ large Δ final- Exponential separation characterized byLyapunov Exponent
Two Modes of AttackSimulationPhysical <strong>Pendulum</strong>• MATLAB generated• Runge-Kutta Method• No Friction• <strong>Double</strong>-bar pendulum• Released from high angle• Circular dots for tracking• Casio EX-F1 at ~5 feet• 600 fps analyzed inMATLAB
Circle Detection & Tracking• Circle Detection – CircularHough• Accumulation Array• Circle Differentiation• Angles ExtractedUO <strong>Undergraduate</strong> <strong>Research</strong><strong>Symposium</strong>
Phase Space Plots• 4 parameters: θ,Φ,δθ, δΦ• Angle 1 vs. itsAngular Velocity• Chaotic ➔ PeriodicUO <strong>Undergraduate</strong> <strong>Research</strong><strong>Symposium</strong>
Lyapunov Exponent• 4 parameters: θ,Φ, δθ, δΦ• |δZ(t)|≈|δZ(t 0 )|e λt• At least 1 positive to showchaotic behavior• Rosenstein Method• Nearest Neighbors• Temporal SeparationUO <strong>Undergraduate</strong> <strong>Research</strong><strong>Symposium</strong>
• Camerao Lighting• Energy FunctionChallenges/Problems• Tracking & InterpolationoCircle Sizeso Could Switch Direction
Future Analysis• Fractal Dimension of attractor• Angle 2 vs. its velocity• Use Box-counting method
Special Thanks to• Professor Eric Torrence• Bryan Boggs• Isaac Hastings Hauss• Professor Richard Taylor• Alexander Elrich (Simulation)• UO Machine Shop Personnel• Ian Pilgrim (Box Counting Analysis)UO <strong>Undergraduate</strong> <strong>Research</strong><strong>Symposium</strong>
Questions?
Lyapunov Exponent Calculation
Attractor Plot
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