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Vibe:Harmonic Oscillators:Simple Pendulum

This is a simple pendulum. It differes from the Simple Linear Oscillator in that it accounts for the nonlinear performance inherent in high-magnitude swings of a stiff pendulum.

The Simple Linear Oscillator is


This pendulum adds a sine function which changes the response of the system at its maxima

Note that the 4Pi2 is simply added to keep the time equal to unity, meaning that the oscillator will orbit 1.0 times in 1.0 timesteps.

A Conundrum

The simple pendulum does not preserve periodicity in the same way as the Simple Linear Oscillator. That is, if you start the system with a position of, say, 3 (representing a fairly high-magnitude swing) it will have a markedly slower period than if you start it with a period of, say, 1. The conundrum is this: if the period is varible depending upon how high the pendulum is swinging, how can such a pendulum be used to keep time? After all, if the period of a clock pendulum changes as the mainspring unwinds (lessening amplitude) then shouldn't the clock slow down?

Well, yes and no. First, the clock makers try as hard as they can to keep the amplitude of the swing constant by using a mechanism that releases and then catches the pendulum at the same height. The spring powers the pendulum so that it can make it back up to that height again, but there is no gradual degredation in amplitude. But more importantly (at least for our topic) the swing of the pendulum on a clock is of relatively low magnitude. Why is this important? Look at the model. The nonlinearity which makes the Simple Pendulum not a Simple Linear Oscillator is sin(position). This term is vanishingly small as position approaches zero. So with a low-magnitude swing (yielding low-magnitude position values) the nonlinearity is small and the Simple Pendulum acts just like the Simple Linear Oscillator.

Vibe Hints and Experiments
  • First, click on the Launch Simulator! button in the upper left. If your browser supports Java, you will get the Vibe simulator in a separate window. If you don't see a Launch Simulator button, you are either browsing Vibe with Java turned off or, alternately, Java is not working on your browser.
  • To see the large-amplitude swings, try running the system with Start X=2, Start Y=0
  • Click the Layer Button to run a second simulation
  • This time try Start X=3, Start Y=0
  • Notice that the period of the oscillator changes as X changes.

Vibe Oscillator Simulator © 2000-2003 Douglas S. Eck.
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Thu Jul 24 09:12:34 EDT 2003