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Harmonic Oscillators:
 
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This oscillator is identidal to the simple Mass Spring save one
addition: it is damped using a damping factor zeta.
Here is the second-order differential equation for the damped mass
spring:
The damping factor zeta is in fact the ratio of actual
damping to that of critical damping. This equation represents the
motion of a mass constrained by a linear elastic spring in parallel
wiht a dashpot full fo oil, which is assumed to provide a force opposing
the instantaneous velocity. When zeta is >1 we get critical damping and
the system becomes a point-attractor. That is, the system doens't oscillate
at all. When zeta is <0 the system becomes a repeller and the oscillator
diverges.
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.
- Now run the oscillator with default settings for a few timesteps.
- Click the
Layer Button to run a second simulation
- This time try increasing
damping to a value greater than 1
and then click Start to run the new simulation for a few timesteps.
This system should damp so well that it doesn't oscillate at all. Rather the system
moves directly to a single point at (0,0); it is a point attractor
- Now click the
Layer Button again to run a third simulation
- This time try setting
damping to a small negative number, say
-2 and then click Start to run the new simulation.
This time the system should repel due to the negative damping.
- Don't be confused by the constant resetting of the display windows. This is normal.
Why? As the oscillator repels further and further away from (0,0).
Vibe automatically zooms out.
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