Keyweights and moment of inertia

[Robert Scott]

Rob Kiddell wrote:

>    Recently, I published  an article on grand piano action 
> diagnosis, and one of the conclusions I came to was re-balancing 
> keys with many 1/2" leads per key. Basically I remove larger leads 
> near the balance rail and insert smaller leads near the end of the 
> key. This seems to make the actions feel inertially lighter, and 
> there is less 'thumping' from heavy keys on balance rails. However, 
> it was brought to my attention that (I quote here)

>   When the key is moving the "moment" of inertia becomes much 
>   more important. Unfortunately it is affected by the distance
>   from the balance rail by a factor of R squared.

Rob,

It is true that the moment of inertia for a constant weight varies 
as the square of the distance to the pivot point.  Therefore if 
you move a weight away from the balance rail and reduce its weight
proportionally (to maintain static weight-off), then the effect on 
the moment of inertia due to that weight will be increased only 
proportionally to the distance to the balance rail (not the square).
So mathematically, the action should have gotten a little harder to 
play when you moved the weights away from the balance rail.

But let's keep things in perspective.  The total moment of inertia
that resists playing the notes is dominated by the inertia of the
hammer as reflected back through the action.  Consider that the
hammer blow distance is approximately five times as much as the
key dip.  And because of letoff and some slight lost motion at
the beginning, not all of the offical key dip is actually used to
move the hammer.  This means that the effective motion ratio
between hammer blow and key press is probably even greater than
5 : 1.  This is equivalent (as far as moment of inertia is
concerned) to having the hammer mounted to the key at a distance 
greater than five times the distance between the front rail 
and the balance rail.  If the hammer weighs 10 grams, then 
its contribution to the moment of inertia is 25 times what a
10-gram lead weight would be if that weight were placed at the
front rail.  In other words, it would contribute as much inertia 
as a 250-gram lead weight at the front rail.  So even though you
increased the moment of inertia when you moved the weights away
from the balance rail, you did not increase it very much compared
to the inertia of the hammer.

As to why you observed the piano to be easier to play after moving
the weights, I have no idea.  Considering that the feel of the
keyboard is affected by many factors, including psycological ones
(i.e. how hard do I have to play to get the sound I want), it is
possible that it might actually feel easier to play with a
slightly higher moment of inertia.

By the way, can someone fill me in on the state of the art in
measurement devices for down-weight, up-weight, and moment of
inertia?  I know that down-weight and up-weight can be measured
by trial and error using calibrated weights placed on the key,
but is there any more automated way to do it?  Also, does anyone
make a device that measures total moment of inertia?

Bob Scott
Ann Arbor, Michigan