This discussion was first introduced by Warren Officer on a Facebook
page and then it showed up here.  I responded initially on Facebook.
Now, after reading many theories, I will respond again.

MMDers have blamed two conditions for the shimmy.  Those conditions
are (1) out-of-balance pump or motor, and (2) loose contact with the
floor supporting the piano.  The loose contact with the floor can be
due to loose glue joints, loose leg screws, lack of caster contact, and
caster movement.

I would like to announce that all responses are correct.  (There is one
exception: this isn't a motor problem as the motor is somewhere between
900 and 1200 rpm and the piano is doing the "Swing and Sway with Sammy
Kaye" at a slow rate that can't come from a motor at 1150 rpm.)

The key to this whole discussion is that there are no in-balance pumps.
There is no counter-balance system in the Ampico pump.  The internal
eccentric, called the "spider" by some, is an off-center rotating device
without counter-balance.  Coupled with that, the four bellows oscillate
in two planes at 90-degree angles, two bellows in each direction.

Because the pump rotates/oscillates slowly, this isn't a problem if
the pump is securely fastened, which it is.  But it is fastened to the
piano, thus the piano has to be securely fastened.  That means solid
contact with the floor.

Solid floor contact is most often lost due to loose leg joints.  Glue
joints where the legs join the capital become loose.  If the piano
shimmies while it is playing, place your finger on a front leg where
it joins the capital.  If the joint is loose, your finger can feel the
movement between those two pieces.

Until permanent repairs are made, wedging something under the pedal
lyre, making solid floor contact, can help.  Also, reposition the
casters to so that they do not align with the movement.

Bob Taylor
Missouri