Vibrator arm. Accelerometer. Mechanical. oscillator. San

1
Bouncing Your Way To Chaos

• Matt Aggleton
• Rochester Institute of Technology

2
Experimental Setup
Packing tape
Plastic tube
Accelerometer
Watch glass
Mechanical oscillator
Vibrator arm
Sand bucket
3
Sample Stage
Outer tube
1 inch

• 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

Straw

Watch glass
Ball
1/8 inch
4
The Math

• Forces on ball
• Dimensionless Variables
• Combine into independent dimensionless variables

• Once units are gone, only independent parameters
  left are G and µ
• G can be controlled via amplitude and frequency
  of oscillation
• µ 6pR?
• R ball radius
• ? viscosity

Simple drag term
5
Wiring and Dataflow
Mechanical Oscillator
Function Generator
LabVIEW
DAQ Board
Accelerometer
Amplifier
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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
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LabVIEW

• Easy interfacing with equipment
• DAQ board
• IEEE 488.2
• Serial Parallel
• Simple to learn
• Wiring diagrams instead of code

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Data

• Fourier Analysis

• Subtract off first 3 dominant sinusoidal terms

• Set average value to zero

• Unfiltered Data Filtered Data

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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)
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Typical Data

• Features of Graph
• Data collected from high G to low G
• Single period on left
• No obvious double period region
• Sharp transition to chaos

? (seconds)
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Hysteresis

• Gammaincreasing

? (seconds)

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Hysteresis

• Gammaincreasing

? (seconds)

• Gammadecreasing

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Single vs. Double Periods

• At low G, ball bounces at each oscillation
• At high G, ball bounces at multiples of
  oscillations
• Single oscillation stable to lower G than
  multiple oscillations

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

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Acknowledgements

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