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