Bouncing Your Way To Chaos

Matt AggletonRochester Institute of Technology

Experimental Setup

Plastic tube

Watch glass

Vibrator arm

Accelerometer

Mechanicaloscillator

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

Watchglass

Straw

Outertube

Ball

1 inch

1/8 inch

• Forces on ball

• Dimensionless Variables

• Combine into independent dimensionless variables

vmgdt

xdm

2

2

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

Simpledrag term

A

xx ~

tt ~

effg

A 2

'

Wiring and Dataflow

LabVIEW

FunctionGenerator

Amplifier

DAQ Board

MechanicalOscillator

Accelerometer

LabVIEW Program

• Amplitude– Starting point– Step size– Endpoint

• Sampling– Sample Time– Cycle Time– Scan Rate

• DAQ limit 200KHz• 20KHz catches all hits

AmplitudeControls

File Controls

SamplingControls

ProgressIndicator

LabVIEW

• Easy interfacing with equipment– DAQ board

– IEEE 488.2

– Serial & Parallel

• Simple to learn• Wiring diagrams

instead of code

Data

Unfiltered Data Filtered Data

•Fourier Analysis •Subtract 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

0

0.02

0.04

0.06

0.08

0.1

0.12

0.14

0.16

0 0.5 1 1.5 2 2.5 3

Features of Graph• Data collected from

high Γ to low Γ• Single period on left• No obvious double

period region• Sharp transition to

chaosg

A 2

(s

econ

ds)

Hysteresis

0

0.01

0.02

0.03

0.04

0.05

0.06

0.07

0.08

0.09

0.1

0 0.5 1 1.5 2 2.5

g

A 2

(s

econ

ds) • Gamma

increasing

Hysteresis

0

0.01

0.02

0.03

0.04

0.05

0.06

0.07

0.08

0.09

0.1

0 0.5 1 1.5 2 2.5

g

A 2

(s

econ

ds) • Gamma

increasing• Gammadecreasing

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 oscillation

One bounce for two oscillations

Numerical Prediction of Double Periods

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

Experimental Confirmation of Double Periods

0

0.01

0.02

0.03

0.04

0.05

0.06

0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1

g

A 2

(s

econ

ds)

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

vFd

Acknowledgements

• Scott Franklin & research group

• Kevin, Melanie, Jesus, & Ken

• RIT Department of Physics