Moment of Inertia of grand action parts.

[Robin Hufford]

Hello John,
     It is interesting to me to see that your experimental results are as
close as they are to the calculated result.  A #40 keystick will probably
have a minimal level of flaring compared to others further away from the
central part of the keyboard in both directions and hence may be more
closely approximated by the formula for I(g) for a symmetric rod than may be
the case with many other keys.
     Still, I think it would be even more of a job using the torsional
pendulum, which I think you used in your experimental determination of
this,   than calculating these values, so I am looking for a program to do
this.   I know you are aware of this but to those technicians that are still
trying to understand the importance of  I  in the question of action
behavior I am going to offer my own repetition of definitions learned
elsewhere on this subject.
     In the rather heated discussion of inertia a week or two ago on this
subject the point was made that inertia is not exactly an engineering
quantity and lacks units per se which is precisely right.  As was said by
several, the term and concept of inertia is merely a way to indicate the
tendancy of a body to resist acceleration.
     As far as I understand the nature of motion determines how this
inertial tendancy will be quantified and expressed.  In translatory motion
where the particles comprising a body move in parallel paths the effects of
inertia working to resist a change in direction or velocity, that is
acceleration, can be quantified and referred to as the mass of the body.
This mass is a measure of the amount of matter in the body itself.  It
correlates in a gravity field with the weight of a body but exists
independantly of gravity.   Most people are aware of this and of the F = MA
equation which relates force, mass and acceleration.
     Where rotation about a fixed axis is concerned such as approximately
occurs in a piano action,  the collective effects of the mass of the parts
and its distribution  about the axis of rotation must be given due account.
As the particles may be closer or further to the axis of rotation their
inertial effects differ.  Collectively, the measure of these effects  is
termed the moment of inertia.  One must arbitrarily impose an axis of
rotation.  The concept has no meaning without one.  Also,  this axis must be
fixed and perpendicular to the plane of rotation.
     This is precisely the analogue of the term mass used  in the case of
translatory motion  where you had f = ma.  Now, however, a new but similar
set of terms is used to decribe these events in rotary motion about a fixed
axis.  Where force was equal to mass times acceleration,  the rotational
analogue of force, torque,  is equal to the moment of inertia times angular
acceleration and f=ma becomes (here the keyboard I am using lacks  the
correct characters) alpha (torque) equals I(moment of inertia) times angular
acceleration(omega).
     Despite the similarity of the two expressions, there are, however, some
important differences. When conceiving of the moment of inertia of an object
one must actively impose an axis of rotation which will then imply an I.
Taking a different axis will result in a different value for the same
object.  These axes may be represent by a subscript following the symbol I.
One frequently encounters I(g) which is an axis through the center of
gravity, also, I(x), I(y) or I(z).   Mass, on the other hand, is constant at
ordinary speeds.  So, we have the possiblity of one and the same rigid body
having differing values emanating from inertial effects which depend upon
the nature of the motion and the nature of the axes chosen.
Regards, Robin Hufford

> Inertia Heads,
>
> I have a little more to report on measuring the MOI of the action parts.
> A friend brought this method to my attention and I am very grateful for
> his help. A method based on the principles of a physical pendulum can be
> use to measure the MOI of odd shaped parts.
>
> I have been trying this out on some grand keys. I think it may be more
> accurate than the estimating method I proposed before. I compared using
> the estimate with the pendulum method and found a discrepancy of 8.5%.
> So our estimate is not too bad after all.
>
> Note C40 M&H BB with two leads in the key.
>
> Estimated MOI = 25077gmcm^2
> Measure MOI = 27389gmcm^2
>
> More on this later.
>
> John Hartman RPT
>
> John Hartman Pianos
[link redacted at request of site owner - Jul 25, 2015]
> Rebuilding Steinway and Mason & Hamlin
> Grand Pianos Since 1979
>
> Piano Technicians Journal
> Journal Illustrator/Contributing Editor
[link redacted at request of site owner - Jul 25, 2015]
>
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