Key Leads and Inertia

[Phillip Ford]

>I lost track of the various posts from Richard and Phil, but I think 
>I can clear up a few points by referring to the latest of those from 
>Phil....
>
>>My understanding was that M was the combined mass of the keystick 
>>and action components located at their combined center of mass.  I 
>>think this might have been clearer if Stephen had called out a mass 
>>for the action components, showed where he located it (I assume at 
>>the capstan), called out a mass for the key stick, showed where he 
>>located it, and then done the arithmetic to arrive at the combined 
>>mass M and the center of mass C.
>
>Yes and no. From the point of view of the pianist's finger what is 
>happening at the backside of the key is irrelevant. So there is no 
>need to separate the action & key bits. Everything that has to be 
>rotated at the fulcrum by the action of the finger force can be 
>lumped as a single mass effect....

I agree that from the point of view of the pianist's finger that what 
is happening at the backside of the key is irrelevant.  But from the 
point of view of Phil Ford (or anyone else reading this paper and 
trying to understand what you did) it is relevant.  To determine the 
mass and center of mass of the keystick itself seems relatively 
straightforward.  Not so with the action mass as I see it.  The mass 
of the action components is not all supported by the key.  Some of it 
is supported by the action rails.  So, how much of the mass of the 
action components is included in your reference key?  And where did 
you put it?  The action components' weight is acting at the capstan. 
I doubt that the center of mass of the action components is right 
over the capstan.  So where did you locate this 'mass'?  What I have 
said before is that what I would do for a first cut is take a scale 
reading of the action components installed on the action rails with 
the whippen sitting on the scale approximately at the capstan contact 
point.  Then treat this as a pseudo mass that would be applied to the 
keystick at the capstan location.  Is this what you did?

>
>>For action components lumped at the capstan I suppose this is a 
>>good assumption (we probably can't easily assume anything else for 
>>a first cut).  For a large or long item like a key this is a poor 
>>assumption.  To idealize the key by lumping it's entire mass at one 
>>point and saying its inertia is mass x radius^2 is an 
>>oversimplification....
>
>The inertial effects of the action components and key stick are not 
>lumped at a point....only the effect on troque coming from the mass. 
>This is exactly what I did...the leadless key+action [reference] is 
>treated as a distributed mass rigid body system in the most general 
>case. That mass contributes to the equations of motion in two ways:
>1) through its moment of intertia with respect to the fulcrum,
>and
>2) through it weight causing a moment (torque).
>In these kinds of analyses you calculate  acceleration from inertia 
>and torque - the latter consisting of the torque which comes from 
>the applied force (finger) + the torque that comes from the action 
>of the weight (2 above). It can be shown that the effect of the 
>weight force is as if all the mass of the body is concentrated at 
>the centre of mass, which is where the rk comes from. This is not a 
>simplification though - it just works to do it this way. The 
>key+action "body" is perfectly general and can describe any piano 
>action we might come across.

Perhaps it's a matter of definition of terms.  Locating all of the 
mass at a point to do the statics is a simplification compared to 
doing the summation (integration) of all infinitesimal points of mass 
throughout the body with their associated moment arms.  Whatever - I 
agree it works for statics, but not for dynamics.  Locating all the 
mass at a point and deriving the inertia from mr^2 will give the 
wrong answer.  But you've said that you didn't do this.  It wasn't 
clear to Richard and me.  The choice of the identical subscript k for 
Ik and rk suggested to us that perhaps there was a connection between 
the two and that you had derived I in this way.  Perhaps rk should be 
rc or rm or some such.

You say that the leadless key + action is treated as a distributed 
mass rigid body system.  Would you explain how you arrive at the 
inertia about the key balance point for the key and action components 
combined?  For the key it seems relatively straightforward, since it 
is rotating about the balance point.  The action components are 
rotating about their own centers.  So how do you arrive at an inertia 
for the action components about the key balance point?  I have two 
reasons for asking:

1.  I want to understand what you did.
2.  If we eventually want to try to use this information in the real 
world we need to have a method of arriving at inertias that it simple 
enough for the tech in the field to use.  This either needs to be a 
simple calculation method or a measurement method.  I am not aware of 
equipment for measuring inertias that falls within the budget of a 
lowly piano technician.  Perhaps you know of some inexpensive 
equipment.  Could you enlighten me?

>
>On the question of significance the jury has to remain firmly out on 
>that one ...You know the breakpoint is somewhere on the horizontal 
>line at acceleration of (r/rb)g, a function of lead location and 
>finger force location. [incidentally i"m convinced this is why 
>playing at the backs of the keys feels so different].

Could you expand on that a little?  It feels so different because you 
have a different breakpoint?

>  A quick calc. puts the breakpoint somewhere in the mid-dynamic 
>range for a mid lever lead position (~20m/s^2).

Would you mind sharing a sample calc with us?  Telling us what values 
you're assuming?

>This proves nothing about whether the breakpoint is actually 
>important from the pianist's point of view, or about the control 
>aspect, both of which need to be investigated experimentally before 
>conclusions can even be hinted at.
>
>Stephen

And when do the games begin?

Phil F