1. Bouncing Your Way To Chaos Matt Aggleton Rochester Institute of Technology

2. Experimental Setup Plastic tube Watch glass Vibrator arm Accelerometer Mechanical oscillator Sand bucket Packing tape 1 inch

3. Sample Stage Watch glass –Automatic centering –Good approximation to flat –Automatic leveling Ball –Steel –1/8 inch diameter Straw vs. Outer Tube –Viscous drag boundary conditions –Centering Watch glass Straw Outer tube Ball 1 inch 1/8 inch

4. Forces on ball Dimensionless Variables Combine into independent dimensionless variables The Math Once units are gone, only independent parameters left are Γ and μ Γ can be controlled via amplitude and frequency of oscillation μ’ = 6πRυ –R = ball radius –υ = viscosity Simple drag term

5. Wiring and Dataflow LabVIEW Function Generator Amplifier DAQ Board Mechanical Oscillator Accelerometer

6. LabVIEW Program Amplitude –Starting point –Step size –Endpoint Sampling –Sample Time –Cycle Time –Scan Rate DAQ limit 200KHz 20KHz catches all hits Amplitude Controls File Controls Sampling Controls Progress Indicator

7. LabVIEW Easy interfacing with equipment –DAQ board –IEEE 488.2 –Serial & Parallel Simple to learn Wiring diagrams instead of code

8. Data Unfiltered Data Filtered Data Fourier AnalysisSubtract off first 3 dominant sinusoidal terms Set average value to zero

9. Time of Flight vs. Impact Time Time of flight: time ball is in air Impact time: time between ball impacts Accelerometer records impact Low Coefficient of Restitution (impact time fails) High Coefficient of Restitution (impact time succeeds)

10. Typical Data Features of Graph Data collected from high Γ to low Γ Single period on left No obvious double period region Sharp transition to chaos  (seconds)

11. Hysteresis  (seconds) Gamma increasing

12. Hysteresis  (seconds) Gamma increasing Gamma decreasing

13. Single vs. Double Periods At low Γ, ball bounces at each oscillation At high Γ, ball bounces at multiples of oscillations Single oscillation stable to lower Γ than multiple oscillations One bounce for one oscillationOne bounce for two oscillations

14. Numerical Prediction of Double Periods Chaotic dynamics of an air-damped bouncing ball, Naylor, et. al., Phys. Rev. E 66, 057201 (2002)

15. Experimental Confirmation of Double Periods  (seconds)

16. Future Work Submerge ball in viscous fluid (in progress) –Analysis of drag force important –Vary radius, viscosity, buoyant effect –____ is only for laminar flow in infinite fluid –We may be laminar, certainly not infinite

17. Acknowledgements Scott Franklin & research group Kevin, Melanie, Jesus, & Ken RIT Department of Physics