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Harmonic Oscillators:

 

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Vibe:Harmonic Oscillators

This section is devoted to mass springs and pendulums (or penduli if you prefer). At their simplest, these are harmonic oscillators. We will see, however, that certain nonlinearities enter the picture when we try to model some properties of these physical systems.

Linear Systems

The Simple Linear Oscillator is in fact a pure sine wave generator. It is interesting, and perhaps not entirely surprising, that sine waves can be generated by simply pitting acceleration against position. The equation acceleration+position=0 generates sinusoidal limit cycles. As acceleration gets large, position must get small. And as position changes to account for acceleration in one direction, it in turn generates acceleration in the opposite direction.

Pendulums

Though one might suspect that this simple linear oscillator is a good model of a pendulum, it fails in a crucial way: it cannot account for large amplitude swings. Think of when you swing on a swingset; when you swing as high as you can it feels like there is a momentary slow down at the highest points. In fact, with a sinusoid as a baseline, there is a slow down. The Simple Pendulum adds to the equation of acceleration+position=0 to account for it. However, this model does not account for other aspects of a simple stiff pendulum (and certainly doesn't model swingsets since it doesn't account for the rider). One conspicous omission is that of damping from air drag. Our simple pendulum never, ever stops! A second Damped Pendulum is offered to show the effects of air on a pendulum.

Mass Springs

Mass spring dynamics are, at the simplest level, just like pendulum dynamics. That is, mass springs and pendulums both oscillate. But a mass spring model is different in that it must account for the stiffness of the spring. The Simple Mass Spring model allows you to adjust mass and stiffness and see how it affects the oscillation. Like the Simple Pendulum, the simple mass spring would run forever (or until the Java Virtual Machine crashed...) The Damped Mass Spring adds damping that would come from a dashpot of oil.

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