Key Leads and Inertia

[Stephen Birkett]

Yo - action dudes:

Well it's about time I earned those oatmeal chocolate chip cookies 
which Bill very kindly sent  mmmm. Thanks much Bill for the treat.

I have revised the slides (on my website), converting into a very 
brief article about inertia/leads, adding some discussion about the 
general case of a real piano [as opposed to ye massless beam etc.], 
and clarified the concept of breakpoint. I also included a page with 
the simple derivation of the relevant formulas on which the graphs 
are based, in case anyone wants to go through those - nothing more 
than simple high school physics here. 
http://real.uwaterloo.ca/~sbirkett/key_balance.pdf

Someone commented:
>At one point it is said:  "moving the location of the lead moves the 
>breakpoint (along the
>red line)"  thoough there is no relationship (formula) given. I am 
>wondering just
>what this might be.
>I imagine that it wouldn't be too difficult to come up with a 
>formula to relate break point to lead location, but it probably 
>would be for a specific key configuration (a specific location of 
>finger force and a specific location of action mass).

The added stuff gives the relevant formulas for the general case of 
adding lead(s) to a real piano key (with all action junk). The 
breakpoint acceleration can be identified according to the location 
of the lead in relation to the point of application of the finger 
force - nothing else is required. Only when you want to find the 
corresponding breakpoint force do you need to examine the 
distribution of mass in the key and action components.

In general the force/acceleration graph for a given key/lead 
configuration has 2 parameters: the intercept (static balance force), 
and the slope (dynamic sensitivity). These are both affected 
simultaneously by lead location and mass, so you can't say mass 
controls one and location the other. The graphs I described are 
actually based on the formulas which are easily derived, but I hadn't 
included those in the article to keep it simple.

>As the lead is moved closer to the balance point the slope of the 
>balanced line becomes closer to the unbalanced line, so that the 
>response of the key must become more like an unbalanced key.

Yes.

>  >Its interesting that the use (or not) of leads changes absolutely
>  >nothing relative to the division between hard/soft play.
>I don't understand what you mean here.  The charts show that the 
>location of the leads affects the break point, or the level of force 
>required to go from soft play to hard play.  At a given lead 
>location, the break point is independent of the amount of lead - 
>perhaps that's what you mean.

Yes. That's what I mean.

>One thing that I question about the charts is the meaning of the 
>break point between soft and hard play.  Stephen's conjecture is 
>that in the soft play area the action is 'harder to control'. 
>You'll notice that the break point for the unbalanced key moves with 
>the lead location under discussion, and yet the unbalanced 
>configuration hasn't physically changed at all.  Why should the 
>location of the breakpoint, or transition from the hard to control 
>zone to the easier to control zone be changing for the unbalanced 
>key?

Because it's related to the force for which any lead added at a given 
location gives the same acceleration, i.e. a comparative concept. 
It's not an attribute of the unleaded key per se. It's only defined 
when you pick a location to put a lead in that key.

>greater for the non balanced line. This seems at odds with the idea 
>that the key would be more difficult to control in the soft zones 
>for the balanced key then the non balanced key.
>It would seem (intuitively) to me that one would have better control 
>when the slope of the acceleration is slight. And this regardless of 
>which zone we are in. ....

another someone commented on this:
>This is an interesting point.  I have to agree with you that 
>intuitively it would seem to me that the action setup with the most 
>shallow slope would be easiest to control.  Less change in 
>acceleration per change in force.  The action would be less 
>'touchy', so to speak.

Yes. This is one certainly possible interpretation - I added some 
comments on the subject of control in the article version. I did say 
in the original slides that the question of "more or less difficult" 
control was "arguable" ["some might believe" etc.] It really depends 
on whether it's easier to control smaller absolute forces over a 
shallower slope vs larger forces over a steeper slope [this in the 
soft zone]. The answer is not unequivocal and needs investigation.

Stephen
-- 
Dr Stephen Birkett
Associate Professor
Department of Systems Design Engineering
University of Waterloo
Waterloo, Ontario
Canada N2L 3G1

Davis Building Room 2617
tel: 519-888-4567 Ext. 3792
PianoTech Lab Ext. 7115
mailto: sbirkett[at]real.uwaterloo.ca
http://real.uwaterloo.ca/~sbirkett