Moment of Inertia of grand action parts.

[Ron Nossaman]

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>Now we just need the velocity of the hammer.

Except it doesn't work that way in a real action. The peak velocity of a 
hammer in a real action is limited by the compliance of the parts, not the 
input impulse. That is primarily, but not entirely, key flex. So beyond a 
certain level of input impulse, hammer velocity won't increase no matter 
how hard the key bottoms out on the punching. Action saturation, and hence 
maximum hammer velocity, depends on cumulative and interactive effects of 
the mass, stiffness, moment arms, compressibility of felt and leather, of 
every affected part for each hammer. Factors influencing saturation point 
of any given  position in the scale include the hammer mass, shank length, 
mass and flexibility, knuckle placement and compressibility, hammer rail 
stiffness, wippen mass, beam stiffness, capstan pad compressibility, angle 
and positioning, wippen rail stiffness, key mass, stiffness,  and moment 
arms, balance rail punching compressibility, and balance rail stiffness. 
Key bed flex is also present, but it's usually way down on the list. All 
mass measurements include considering MOI, naturally, and I probably missed 
some things, but the point is that this isn't something you are going to 
casually calculate to any degree of accuracy beyond a very rough estimate.

This is the way I see it, for whatever that might be worth, with more than 
a few decimal points lopped off. Once the basic geometry is established and 
the friction is under control, the hammer weights are the priority. High 
hammer weights need more key lead to get static weights in the ball park of 
usability. The excess key leads are perceived as being the cause of the 
inertia problem, when they are only there because the hammers are already 
too heavy. Moving the leads back toward the center of the key and adding 
more does change the inertial effects of the key leading. It also increases 
the flexibility of the key making it somewhat easier to bottom the key 
before the hammer moves significantly, giving the impression of playing 
easier as the more flexible key lowers the saturation point and limits 
hammer velocity. It isn't the Moonlight Sonata crowd that notices the 
difference. It's the aggressive pianist.

Adding a wippen assist spring does a number of things. It provides static 
balance compensation for excess hammer weight, which allows lead removal 
from the keys, which lowers the overall inertia of the action. The spring 
obviously has no direct affect on the inertia of anything in the action. 
When we set repetition springs on the bench, we typically use hammer rise 
as an indication of spring strength. In play, the hammer doesn't rise for 
the jack to reset. The key does, lifted by that little rep spring. With no 
wippen assist spring, there is somewhat of a correlation between hammer 
weight and key weight as seen by the rep spring, but the addition of a 
wippen assist spring changes that relationship considerably. So less key 
mass means faster jack reset and higher repetition rates even if the 
hammers ARE still too heavy. Less key mass will translate to slightly 
higher key down velocities for a given input even though the hammers still 
contribute most of the overall inertia. Fewer leads mean fewer holes, 
therefor stiffer keys, making terminal hammer velocity more a function of 
input, and less of action compliance. It raises the saturation point. In 
some instances, this will mean that the pianist needn't pound the keys as 
hard to get the high end, and should get better control over a wider range.

I don't know if the rep spring will lift a center weighted key easier and 
quicker than an end weighted key, but I suspect so, even if it is only that 
the key flex means that weights toward the ends of the keys travel 
proportionately farther than weights placed toward the center, and so have 
to be moved proportionally farther to reset the jack. Again, this is a real 
concern only with high amplitude, quick repetition playing, but that's 
where this stuff is noticed, so I think it's a factor.

As long as I'm already here and have some on me, I'd also like to comment 
on the idea that key lead inertia doesn't happen until the downward 
acceleration exceeds gravitational acceleration. I disagree. The system is 
counterweighted. The key weights are not in free fall, so for mass being 
pushed down, mass is being levered up. Gravity only counts in static 
balance measurements.

No math, no minutia, just my attempt to step back for an overview of my 
own. There are a whole lot of minute details being discussed here, all of 
which are worth defining and clarifying to the degree that it's possible, 
but the original questions of how these things fit together in an action 
are of more general interest and use, and that still isn't being addressed. 
The problems we deal with, and the confusion that remains is still how all 
this stuff mechanically interacts in a piano action.

Ron N

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