<!doctype html public "-//w3c//dtd html 4.0 transitional//en"> <html> Ok Thanks Stephen I was kind of looking for a way of starting to plug in values easily measured and looking at some things right away. But what you say below would seem to make things a bit more difficult. I suppose you could take an average of the mass of all keys, and an average for all whippens and shanks, and then figure in for each of the 88 keys the variance in hammer weight to find M for each key. And I'd rather thought you could come up with a very good guestimation of rk as since it is the location of the center of mass.... which would mean the key would balance at that point.   We are correct in assuming that rk is the distance the center of mass is from the fulcrum  arent we ???  not just simply << the center of mass >> But since you say you cant take  Mrk2 for Ik then theres not much point in doing either of the above, and certainly the example numbers I solved with the other day must be seen as meaningless as they were bassed on the asummptions I ask about below. Thanks for taking the time to clear this up for me Stephen. RicB   Stephen Birkett wrote: <blockquote TYPE=CITE>Ric asked: >Well.. if you could just answer two rather straightforward >questions. Can M  be taken to be << rougly >> the weight of the >action components on the capstan, and if so can Ik then be << >roughly >> found by taking the product of this M and rk2 .....as in >Mrk2  ? No........ M = the mass of the key stick, capstan, whippen, hammer et al ...i.e. *everything* that the poor pianist's finger has to move, with the exception of any key keads that I add separately.  The position of the capstan in this analysis is irrelevant [except insofar as the small mass of the brass capstan itself will be involved in the moment of inertia of the whole works, and a capstan further back will contribute more to the moment of inertia than one closer to the front]. You can find M simply by putting all those key and action parts on a weighing scale, but you have to "remove" the mass of any lead in the key stick. and No..... Ik has to be found by considering the distribution of M with respect to the fulcrum point. You can't simply concentrate the mass at the centre of mass. The relationship between the whippen heel and capstan, key ratio, etc etc. has nothing to do with Ik. All that matters is physically where the mass of these parts is located with respect to the fulcrum of the key. [The simplification as noted is that the real action parts do not move in an arc with respect to the fulcrum, but rather move in more complicated trajectories, so the moment of inertia is actually changing through the key stroke.] So where does rk the centre of mass fit into this? You can determine it simply enough experimentally. And you *can* consider all the mass M acting at rk when calculating the torque it contributes due to its weight. Remember M contributes to the equations in two ways: (1) through its moment of inertia and (2) through its weight.  And these are separate effects. One factor that does have a serious impact on all of this is the mass of the dampers, all the more so because it is relatively far away from the fulcrum. These do seem to be treated in rather cavalier fashion in discussions of key balancing [it never seems to be made explicitly clear whether the dampers are on or off when doing measurements, like a balck sheep relative in jail who never gets mentioned in polite company]. The *inertial* effects of dampers being lifted vs pedal up are certainly quite significant. Again you can feel this in playing. Why do you suppose most modern pianists habitually keep the pedal down? Hmmmm. Stephen -- Stephen Birkett Fortepianos Authentic Reproductions of 18th and 19th Century Pianos 464 Winchester Drive Waterloo, Ontario Canada N2T 1K5 tel: 519-885-2228 mailto: sbirkett[at]real.uwaterloo.ca <a href="http://real.uwaterloo.ca/~sbirkett">http://real.uwaterloo.ca/~sbirkett</a> _______________________________________________ pianotech list info: <a href="https://www.moypiano.com/resources/#archives">https://www.moypiano.com/resources/#archives</a></blockquote> -- Richard Brekne RPT, N.P.T.F. UiB, Bergen, Norway <A HREF="mailto:rbrekne@broadpark.no">mailto:rbrekne@broadpark.no</A> <A HREF="http://home.broadpark.no/~rbrekne/ricmain.html">http://home.broadpark.no/~rbrekne/ricmain.html</A> <A HREF="http://www.hf.uib.no/grieg/personer/cv_RB.html">http://www.hf.uib.no/grieg/personer/cv_RB.html</A>  </html>