Underdamped Systems

Have a look at this:

Article: Second Order Systems

And this:

p://www.che.ttu.edu/pcoc/software/ppt/Chap05.ppt

what system? You specified "dampening" but didnt explain what "system"

There is no such thing, for example, as an oscillator that is not under-damped.

It wont oscillate.

1. I could not get your first document Lyn, it seems it is no longer in the list on that webpage.
2. The advantage of an under-damped system response is that the target set-point value will be reached more quickly, but with some overshoot. This is usually OK for a speed governor, where near 10% overspeed will not cause any risk of damage. However, for a furnace control, any overshoot might damage the product or even the furnace.
3. A damping ratio of 0.7 in the theory of "second order systems" is that for which the target value of output is just not exceeded. Lower damping gives overshoot, higher damping takes longer to reach the final value
4. If the graph of response for second order systems, for different damping values, in Lyn's second document is examined, points 2, 3 above can be seen.