Chaos in a Pendulum

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Chaos in a Pendulum

• Simulation,
• Experimental Characterization,
• and Prediction of
• Deterministic Periodic and Chaotic States

Eugene Donev
27th March 2002
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Galileos Apocryphal Chandelier
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The Plane Pendulum
Newtons Second Law
Torque
Important -- The restoring force is nonlinear
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The Damped and Driven Pendulum

• Swinging under the influence of
• gravity (restoring force, -)
• velocity-proportional damping (resisting force,
  -)
• sinusoidal driver ()

The equation of motion (relating the above
torques)
After some intense algebra
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The Damped and Driven Pendulum (continued)
Natural units for the pendulum
Then fiddling with derivatives and constants
turns this
into that
where
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The Real (Physical) Pendulum

• Getting physical
• pendulum length 2 cm eddy-current damping
  (controlled by a micrometer) optical encoding of
  angular position brushless slotless linear AC/DC
  motor
• driver sinusoidal torque frequency 0.3 - 3.0
  Hz, /- 0.001 Hz output voltage 0 - 7 V
• PC card and software

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The Physical Pendulum (continued)
Seen that already
The good old dimensionless parameters
But what about ?0, and b, and I, and T, and
Tcritical?
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Parameterization of the Pendulum
Determining the Natural Frequency, ?0 ?
collecting time series of angular displacement,
?, at minimum damping ? performing nonlinear
fitting of damping equation to part of the time
series ? calculating ?0 from the obtained ?
and ?, damping parameter
Now the driving frequency parameter can be
calculated and varied ?
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Parameterization of the Pendulum (continued)
b/I as a function of Micrometer Setting, S ?
collecting time series of angular displacement,
?, at various values of S ? performing
nonlinear fitting of damping equation to part of
the time series for each S ? obtaining
calibration curve for various micrometer settings
Now the damping parameter can be calculated and
varied ?
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Parameterization of the Pendulum (continued)
Normalized Torque as a function of Driving
Voltage (DC) ? pendulum turned on-edge to
eliminate gravitational restoring force ?
measuring terminal angular velocity, ?term , at
constant damping and varying drive voltage ?
obtaining calibration curve for normalized
torque, T/Tcritical , as a function of voltage
(DC)
Now the driving amplitude parameter can be
calculated and varied ?
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Tweaking the Parameters
Driving frequency (kept constant, ?d 2/3)
Micrometer damping (two different settings, q 2
and q 4)
Driving amplitude (a range of settings)
g 0.92 Period 1
g 1.44 Period 2
g 1.61 Period 3
g 1.75 Chaos
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g 0.92
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g 1.44
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g 1.61
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g 1.75
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CHAOS Aperiodic long-term behavior in
a deterministic system that exhibits sensitivity
to initial conditions.
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Phase Portrait
How do we read the Phase Portrait ?
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How to read the Phase Plane diagram
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g 0.92
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g 1.44
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g 1.61
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Pendulum Bifurcation Diagrams
Solving the differential equation of motion
numerically
becomes
I have used the fourth-order Runge-Kutta
method, with an adjustable time increment.
Solving numerically for x and for v, at small
increments of g.
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g 1.44
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g 1.75
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g 1.61
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g 1.61
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g 1.75
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g 3.49
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Prerequisites for Chaotic Behavior in a Physical
System
1. at least three independent dynamical variables
2. nonlinearity in the equations of motion
3. appropriate values for the parameters
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The Chaotic Pendulum (again)
1. at least three independent dynamical variables
2. nonlinearity in the equations of motion
3. appropriate values for the parameters
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Angular Displacement time series at minimum
damping
Angle (rad)
Time (s)
?
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Data and nonlinear fit for minimum damping
Angle (rad)
Time (s)
?
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Micrometer setting 8.00 By t 4 s, the
oscillations have almost completely died
away, i.e., much damping.
Micrometer setting 11.50 By t 7 s, the
pendulum is still swinging, i.e., little damping.
?
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Normalized Damping as a function of Micrometer
Setting
?
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Normalized Torque as a function of Applied Voltage
?
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?
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