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.... >I read the above to say that M is the mass of the key and action >components (whippen, shank, and hammer) without any balancing leads. .... >'m assuming that the action components are being idealized as a mass >that would be located at the capstan point. The key stick itself >has a mass and a center of mass. Combine the two and you get total >mass for the 'reference key' M with its center of mass located at rk. >... >So Mg is the weight of the unleaded key stick and action components >together. rk is their center of mass. Both of these are constant >for a given note of an action and are independent of the balance >leading. I believe that he is treating the balance leading >separately rather than changing the mass and center of mass of the >reference key every time the leading is changed. Yes. Correct. I've taken everything together without leads (key, action bits, etc) and called it the "reference key" - this is the part that is fixed and presented for balancing. Hence reference. To this are added balancing leads which I have treated separately. Terminology may be a little misleading...maybe "reference key assembly " or something is bettter. The point is that it's a reference which remains the same, while leads are added to it. >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.... >The formula for moment of inertia = mass x radius^2 only applies to >point masses. For key leads this is a pretty good assumption. Yes. And this is what I 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, perhaps too much so even for simple cases like >the ones we're looking at. For example, if the key was a perfectly >symmetrical rectangle in profile that was centered on the balance >point, its center of mass would be at the balance point. Looking at >mass x radius^2 would indicate that it had no inertia. This is >certainly not the case. (For example, for a rectangle of length l >and height h, it's moment of inertia would be lh/3(l^2 + h^2) if I >remember correctly). A better assumption might be to say that Ik is >t! >he moment of inertia of the >keystick, whatever it is by calculation, plus action mass x capstan >location^2. 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. >The breakpoint is located on the acceleration line of the unbalanced >key at an acceleration of (r/rb)g. The slope of that acceleration >line is determined by Ik. If you change the inertia of the >unbalanced key then you change the slope of the acceleration line, >which in turn changes the force corresponding to the point (r/rb)g. >For a given unbalanced key acceleration line, then the point where >the acceleration lines of the keys with balancing lead intersect it >is independent of the slopes of the >balanced key lines. >My point was that at a given acceleration you either measure the >force applied to get that acceleration, or having measured some >other force and acceleration points and established a line, you just >read the force off the chart at the given acceleration. You either >use the slope of the line to determine the force or you use the >measured forces and accelerations to determine the slope of the line. Yes. All the above is a good summary of what I meant Phil. On the question of significance the jury has to remain firmly out on that one - well actually the trial hasn't even begun yet. We can certainly do some quick ballpark estimates as Ric started a few messages back, to get a feel for what dynamic levels these breakpoints come at. It's actually simpler than proposed, because you only need to look at the acceleration --- dynamic level is controlled by velocity and that's determined by acceleration. The force required to get that acceleration is irrelevant. 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]. A quick calc. puts the breakpoint somewhere in the mid-dynamic range for a mid lever lead position (~20m/s^2). Given the significant effect of moving the lead location it isn't too difficult to move the breakpoint well into the louder dynamic accelerations, or well into the pp dynamic range, according to where you put the lead. 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. Ric - are there any points still unclear? 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
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