Moment of Inertia of grand action parts.

[Mark Davidson]

Let's look at it a little more.
Forces on key are:
 
FF: finger force
F: friction
G: gravity
S: spring
 
FF is self-explanatory
G: is basically the same magnitude as BW, but pushes UP.  
 It is the net force measured at the key front, due to all 
 the gravitational forces and leverage of the action.
F: friction is always opposite the direction of motion.  If
 we define down as positive direction, then F is negative when
 key is going down and positive when key is going up.
S: any spring/magnet forces, as measured at the key front.
 These normally help push the key down, so are positive.

also:
AA: angular acceleration
KFA: key front acceleration
L: radius of key front
 
Total force is FF + S - G - F.  Note that 
DW + S - G - F = 0, or
DW = -(S - G - F), so 
total force is FF-DW.
 
Torque T is force * L (key front length)
T = (FF-DW) * L.

key front acceleration KFA:
 
KFA = AA * key radius = (T / I) * key radius 
 
gives
 
KFA = (T / I) * L, which in turn gives
 
*** KFA = (FF-DW)*L^2/I ***
 
This gives us the vertical key front acceleration,
as a function of the force on the key, total reflected inertia
and key front radius.

What it says is that if I is small, then
KFA will be big, and vice versa.  (L^2/I) 
is basically the quantity we've been looking 
for, that lets you compare how easy or difficult 
it is to accelerate a key for a given amount 
of force.  Comparing I alone between two pianos 
doesn't tell you much.  We need to compare
L^2/I to make comparisons of key acceleration, or 
I/(L^2) to make meaningful comparisons of
inertia.
 
The analogy is you have two rocks, with mass M1 and M2.
And two levers, L1 and L2.  And I ask you which lever
is easier to push down, by only looking at M1 and M2.
Well, you have to look at the lever lengths also.  
Same thing in piano action. We have I, but 
without L, it doesn't tell us much.
  
-Mark