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Harmonic Oscillators:

 

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Vibe:Harmonic Oscillators:Simple Linear Oscillator

This is a simple linear oscillator. In it's simplest form, the equation for this oscillator is

So that time is equal to unity (meaning that the oscillator will make exacly 1.0 orbits in 1.0 timesteps) we multiply by (2*Pi)2 = 4*pi2, yielding

The solution for this differential equation is sin(t). Hence, it is considered a harmonic oscillator, though there are more complex models of a harmonic oscillator.

This oscillator will model a pendulum to some degree. However, there are two important problems with this: first, there is no damping in the model, so the pendulum never stops. Second, there is no nonlinearity in the model, which in the case of the pendulum means that very high magnitude swings look just like small magnitude swings. In fact this is not the case with a real pendulum. See the pendulum model for more on this.

Note that, unlike a real pendulum on a grandfather clock, this oscillator has a period which is independant of amplitude. No matter how hard you "hit" the pendulum, it swings at the same rate. (If it seems curious to you that grandfather clocks don't work this way, then read more about it in the pendulum mode. You can explore this and more by running Vibe with different starting X=position, Y=velocity values and noting that the model oscillates at the same rate for all values of X and Y.

Still interested? Consider Thompson and Stewart's excellent dynamics book, Nonlinear Dynamics and Chaos. Their discussion of this topic greatly influenced this and many models in Vibe.


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