Physics 101- Introduction to Mechanics
April 29- SHO & Pendulums

Chaos in Pendulums

Chaotic systems exhibit irregular, unpredictable behavior.  A classic example of this is what's known as the butterfly effect- due to nonlinearities in weather processes, a butterfly flapping its wings in Tahiti can, in theory, produce a tornado in Kansas.

Chaos in a Double Pendulum

Here we don't have any external forces other than gravity, yet we see chaotic motion.

Now we pull the bob to one side and let go.

Here are graphs of the two angles versus time.

Chaos in a Driven Pendulum

If we look at the actual equation for the pendulum, no small oscillation approximations, we see that it is very non-linear.

Okay, now what if there is a torque that pushes on the pendulum in a periodic manner?

Now what if there is a dampening force that is proportional to the angular velocity?  (air drag)

Wow. This is a mess, but that's the beauty of it.  This is a much more correct model than our simple harmonic oscillations, consequentially it also means that our motion prediction is closer to the actual motion.

From the equations one sees that we get chaos.  The motion is no longer periodic.

Let's look at a few different systems, each with different driving and dampening torques.

 Simple harmonic motion- no driving torque, no drag torque.  Since the length is chosen to be 9.8m, the angular frequency is simply 1/ sec.

 "Slightly more complicated"- there is an oscillating drive torque (whose frequency is 0.689/ sec versus the pendulum's natural frequency of 1/ sec), and there is a drag torque.

 "Even more complicated"- same drive torque amplitude, but the angular frequency is slightly different (now 0.694 / sec), the drag torque is the same as in "double"

 Finally we reach a chaotic system- the drive torque is now greater amplitude and lower frequency (now 0.54 / sec), the drag is just as in the previous two.

Notice that by making just slight changes to our ideal system, we produced complicated, even chaotic, motion.

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 Jeff Phillips phillips@lmu.edu Loyola Marymount University Spring 2002