Vibe is a Java-based simulator, but you now have Java turned off. You may
turn on Java at any time.
     
Harmonic Oscillators:
 
Other Categories:
|
  |
   
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.
|