Key Inertia

[Sarah Fox]

Hi Mark,

>3) Removing mass from the key and wippen will make the action more
>efficient.

<<Hmm.  Starts to get fuzzy here.  If you keep the same speed, so less
energy is wasted, then I agree. But if you remove mass and play the key with
the same force, you will have more speed, not the same speed.>>

You are defining the situation with a constraint of constant force.  This is
very useful:  No matter the mass of the key, wippen, hammer, or whatever -- 
no matter what the circumstances may be -- energy is force times distance.
Constant force * constant key dip = constant energy input.

First apply force to a heavy keystick.  It goes into slow motion.  The
hammer hits the string in slow motion.  Everything is slow.  Not much energy
transferred to the string.

Now remove keystick mass.  Less mass means the key/action moves faster with
the *same force*.  That means the hammer moves to the string faster and
strikes the string harder.  More energy transferred to the string for the
same amount of input energy (again, equal to force times keydip).  That
makes it more efficient.


>4) Given our ideal action with no bending and friction losses changing
>the hammer mass does not effect the efficiency of the action.

<<Same problem as 3.  If you don't change the mass of hammer and wippen, and
keep the same key speed, you will waste the same amount of energy.  But
again, if you add mass to the hammer and keep the same force on the key, you
will have less acceleration, less speed, less key/wippen energy wasted.>>

But if the hammer is more massive, it doesn't need ot hit the string with as
much velocity to transfer the same amount of energy.  Energy imparted to the
hammer shank is jack force times jack travel.  This is converted to kinetic
energy of the hammer, which is proportional to mass.  Twice the hammer mass,
while preserving the same jack force and stroke, would mean 2^.5 the
velocity, true, but when the hammer hits the string, the same amount of
kinetic energy is transferred to the string.

Same jack travel, same jack force, same kinetic energy delivered to the
string.  Same efficiency, provided we're talking about the hammer hitting a
noncompliant object.

Now to argue against my point, the string does have a mass, and it moves and
vibrates.  There are two possible sources I can see for inefficiency of
*energy transfer* to the string.  (Mind you, I stand by my point that the
hammer has the same kinetic energy for the same input force either way!)

First, higher partials might be muffled if the hammer is too slow to get out
of the way
of the vibrating string fast enough.  A heavier hammer undoubtedly sounds a
bit darker than a lighter one.

Second, and less obviously, there is an impedance matching issue here.
Impedance is pretty difficult to explain in lay terms, but it has a lot to
do with the inertia (NOT TO BE CONFUSED WITH *MOMENTUM* OR *KINETIC ENERGY*)
of one object or medium with respect to that of another object or medium.
To illustrate how this works, dive into a pool.  Have someone talk to you
from above the water's surface.  It's very hard to hear, right?  That's
because the air vibrations don't apply enough force against the water to get
it vibrating very well.  It's also because the water vibrations have too low
of an amplitude to vibrate the ear structures very well, even though they
are quite forceful.  Repeat the experiment the other way, now:  Put a water
resistant speaker underwater, hook it up to a stereo amp, and blast away!
It's not very loud, is it?  That's because the forceful but low amplitude
water-born vibrations transfer to not very forceful and low amplitude
airborn vibrations.  This sort of impedance mismatch problem ALWAYS results
in inefficiency of energy transfer.

Ideally mechanical elements have the same impedance.  If the impedance is
exactly matched, the efficiency of energy transfer will be maximized.  If it
were not for the fact the string is vibrating (i.e. if it would just move
forward and keep moving), the optimally efficient hammer would stop dead in
its tracks when it collides with the strings, and the strings would bound
forth at exactly the same velocity as the hammer.  However, there's the
rebound, in which the strings would whack the stationary hammer, sending it
back into motion at an equal and opposite velocity.  In the process, the
strings would become stationary again.  This is the equivalent of two
strikes exactly 180deg out of phase.  Not a good scenario.

The hammer must actually be of a lower mechanical impedance than the
strings, so that it will bounce off of the strings with enough velocity to
clear the string upon the return vibration.  I think that would be fun to
model!  A light hammer will rebound too much without imparting much of its
kinetic energy to the strings.  The wasted energy would be expended against
the backcheck.  As hammer mass is increased, efficiency is increased, to the
point that the hammer gets too slow to clear the string on the return
vibration.  Then efficiency would fall off dramatically.  Oddlly, this is
one energy interface that cannot be 100% efficient.

More on other posts later...  but first a movie....  Spirit of St. Louis...
one of my favorites!  :-)

Peace,
Sarah
Peace,
Sarah