Inertia, was "Grand Touch"

[Michael Spalding]

William,

Vladan is correct.  The mass vs  force concepts are, unfortunately, not
intuitive.  It's not made any easier by the fact that we use the same units
for both mass and force.  (real physicists use different units)  The
readings you refer to, taken at the front of the key, are force (or weight,
if you will).  A given value of key front weight could be the result of an
infinite number of combinations of mass and distance from the fulcrum. 
Might be a large mass close to the fulcrum, might be a smaller mass further
from the fulcrum.  So measuring front weight does not tell us the mass.  To
determine the rotational moment of inertia, it is necessary to know where
the mass is, and how big it is.  

Computing the rotational moment of inertia of a complex solid involves
breaking the solid up into tiny bits, determining the mass and radius of
each bit, multiplying the mass of each bit times the square of the radius,
and adding these products all together.  Taken to the extreme of an
infinite number of infinitessimal bits, this is called integral calculus.

In our discussion of key leading, we simplify.  We consider each lead as a
mass at a single point, and multiply its mass by the square of the radius
to the center of the lead.  We consider the moment of inertia of the key to
be insignificantly small, or at least fixed and unalterable therefore
irrelevant.   Not a problem, since we are interested in the practical
effects of changing the location and or quantity of lead in an existing key.

Follow the link to Dr. Birkett's paper.  
http://real.uwaterloo.ca/~sbirkett/key%20balance.pdf
This was a revelation to me.  I had assumed that minimizing inertia was the
goal, but as he points out, player preference based on control, power,
fatigue, etc. may point to some other configuration.

Mike


> [Original Message]
> From: William R. Monroe <pianotech at a440piano.net>
> To: Pianotech List <pianotech at ptg.org>
> Date: 7/12/2006 7:01:23 AM
> Subject: Re: Inertia, was "Grand Touch"
>
> Vladan,
>
> I don't have my book handy, is this the formula for tangential MOI, or
> linear?
>
> I'm assuming tangential.
>
> Also, I would question your results (in a friendly way).  I would think
you
> would be taking the mass reading for the formula at the front of the key,
as
> you would take FW.  Thus, Mass would be mass of the keystick, not mass of
> the individual lead.  In this case, regardless how much mass you use
(more,
> near the balance hole, or less near the front), the Mass as measured at
the
> front of the key is going to be the same, thus, equivalent inertia at the
> front of the key.
>
> Comments?
>
> William R. Monroe
>
> ----- Original Message ----- 
> From: "V T" <pianovt at yahoo.com>
>
> > I=m*r^2
> >
> > I=moment of inertia
> > m=mass of the lead
> > r=distance from the balance rail hole to the lead
> >
> > Note that the formula takes the square of "r".
> >
> > Now, let's look at our options for lead placement.
> > Suppose that we can use a 12 gram weight placed 10 cm
> > away from the balance rail hole to get the desired
> > balance weight.  As an alternative, we could also use
> > a 24 gram weight placed 5 cm away from the balance
> > rail hole and still have the same balance weight.  Is
> > there a difference between the two?
> >
> > Yes, there is.  Looking at the formula above, the
> > inertia increases with "r" squared.  In the first
> > case, the inertia will be 12*10*10=1200 gcm.  In the
> > second case, the inertia will be 24*5*5=600 gcm.
> > Placing the 24g weight 5cm away from the balance rail
> > hole reduced the moment of inertia by 50%.  That's a
> > lot.  You can minimize the inertia by placing the
> > weights closer to the balance rail hole and increasing
> > the amount of lead accordingly so that the balance
> > weight meets the target.
> >
> > There is a fundamental trade-off between balance
> > weight and key stick inertia.  If we add a lot of lead
> > to make the balance weight low, the action will feel
> > very light, but only when you move the key slowly.  If
> > you try to play a loud note, all that lead will have
> > to be accelerated and the key will feel heavy.
> >
> > If you put very little (or no) lead into the key and
> > accept a high balance weight, the action will be heavy
> > when you measure it with your weights, but it will
> > feel light when you play fast/hard/loud.
> >
> > The question is: What is better?  My own preference is
> > for a heavier balance weight with less inertia.
> >
> > Vladan
>