I think what is happening in a "shimmy" can be explained without too much engineer-speak. A shimmy is like a resonant frequency - some sort of perturbation starts a vibration at just the right frequency and it continues to grow unless something is changed.

In the simplest terms, the natural frequency of a system (w) is defined as

w=sqrt(k/m)

where k is the stiffness of the system and m is the mass.

So to eliminate the shimmy you have two options:
1)change speed to alter or eliminate the input signal (vibration) that is causing the system to resonate at its natural frequency
2)change the natural frequency of the system by either altering its stiffness or mass

Examples of changing the stiffness and mass have already been given in this thread:

When you clamp the top tube with your knees, you are effectively stiffening the system. (If you want to play semantics, technically you are not "damping" any vibrations; you are stiffening the system and changing its natural frequency so that it no longer resonates in response to whatever input it is experiencing)

The member who posted about seeing a motorcycle shimmy so hard it bucked its rider off, then continued straight (without shimmy) for quite a distance gave us an example of changing the mass: when the rider was bucked off, the mass of the system changed and thus so did the natural frequency, and the shimmy disappeared.

I didn't staty at a Holiday Inn Express last night, but I did take a few mechanical engineering grad courses in dynamics and vibrations...

Mac