---- David Love <davidlovepianos at comcast.net> wrote: > But the radii are not the same, are they? And therefore they travel different distances. Balance rail/key interface to tip of capstan is not the same distance as wippen heal/contact interface to wippen center? Years ago, when I had only 2-D CAD software to work with, I reasoned that if I could manipulate the radius of the wippen heel such that the length of the cord along which the capstan would move across the heel would be equal to the cord along the capstan which the heel would follow, then there would be no sliding motion, only rolling. In other words, if the distance from the beginning to the end of the capstan's motion with respect to its contact point with the heel is equal to the corresponding distance along the wippen heel, there would be only rolling and no sliding. To achieve this, the radii would not be equal, nor would they be at the same ratio as the lever arm lengths. I am sure there is a more elegant mathematical solution, but I simple altered the radius of the wippen heel, by trial and error, until I ended up with the length of the capstan's excursion across the heel being pretty nearly equal to the length of the heel's excursion across the capstan. This discussion caused me to go back and reconsider my thinking, and I found it to be flawed. To others on the list with a better education in such things, and are more familiar with terms like "involutes," this may have already been obvious. Quite often I find myself reinventing the wheel, just to prove something to my own satisfaction. Today, with the benefit of 3-D modeling software, I created a simplified 3-D model of the contact between the capstan and wippen heel. By the way, I don't much care for the term "magic line," but that is how it is now known. I arranged the magic line at the middle of the range of motion for the model. I defined a motion cycle that I could observe and measure close-up. There was sliding going on through the entire motion cycle, but in a much more complex way than I had imagined. During about the first third of the motion, the capstan slides rather quickly toward the wippen center (axel), while the capstan's point of contact remains almost unchanged. Through the mid-range, the sliding was barely detectable. Through the final range, the capstan slides forward rather quickly, while the heel's point of contact with the capstan hardly moves. The "curve" scribed by the movement of the point of contact was essentially a straight line. The length of the cord along the surface of the wippen heel that the capstan followed toward the wippen center in the first half of its travel was 5.58 mm. The distance that it returned in the opposite direction in the last half of its travel was only 1.25 mm. The excursion of the wippen heel along the capstan in the first half was merely 0.04 mm. In the second half, the heel move 3.52 mm along the curve of the capstan. If the lever lengths and radii had been equal, the capstan might have returned the same distance as in the first half. With the lengths and radii more typical of a grand action configuration, the return excursion was only about 22% of the initial backward excursion with respect to the capstan's travel along the surface of the heel. Sorry if this description is difficult to follow. It's one of those things that you just have to see it to believe it. Frank Emerson
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