Bouncing Your Way To Chaos Matt Aggleton Rochester Institute of Technology

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

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

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

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

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

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

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

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)

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)

Hysteresis  (seconds) Gamma increasing

Hysteresis  (seconds) Gamma increasing Gamma decreasing

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

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

Experimental Confirmation of Double Periods  (seconds)

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

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