Inertia, was "Grand Touch"

[V T]

Jude,

I need to correct something you wrote;  mass x
velocity is momentum rather than inertia.

There are a few important things to know about lead
placement.  We are interested in something called
"moment of inertia" which is a measure of how hard it
is to set the key, or for that matter the entire
mechanism including the hammer into motion.

For simplicity, let's look only at the key stick with
one lead in it.  Let's also make the extreme
simplification that the wood in the key doesn't weigh
anything at all, and so we are looking only at the
effect of the one piece of lead.  The moment of
inertia caused by that lead is equal to:

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

=================================
Jude Reveley wrote:

David hits the nail on the head. As inertia is defined
as mass x velocity, there is an equal trade off in the
weight verse the position of leads. So we should be
talking about the limit of acceptable FW. In my
experience if the lowest key starts with front weight
in the mid-forties (grams) where the FW matches the
strike weight in its curve, there will be that inertia
dampening effect (pardon the Star Trek jargon).


Jude Reveley, RPT

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