Phillip, Your example is perfect, but neglects one important thing. The piano key moves approximately 2 degrees while the piano hammer moves approximately 20 degrees. That's a factor of 10 difference in angular acceleration. In a coupled system such as this where the parts move at different angular speeds, the inertia (i.e. the inertia of the hammer as felt through the key) must be adjusted by the square of the speed ratio. That's a factor of 100. I agree that the inertia of the hammer hasn't changed, but it feels larger due to the action leverage, and that gearing is part of what causes the energy to go into the hammer rather than the key. So now your inertias become Inertia of hammer about its center = 10 x 13 x 13 x 100 = 169000 g cm^2 Inertia of key lead about its center = 50 x 23 x 23 = 26450 g cm^2 Think about a flywheel turned by a gear with a crank attached. If you change the gear ratio, the resistance to the crank (which is all due to inertia) changes even though the inertia of the flywheel has not changed. That's whats going on here. Now I'm not really trying to say keyleads are insignificant (well I guess I did), just trying to make the point that they're not AS significant as people assume. -Mark ====================================== Phillip Ford fordpiano@earthlink.net Mon, 15 Dec 2003 00:29:56 -0700 Previous message: Machine shop charges Next message: Letting tomatoes fly (was re: adjusting wippen...) Messages sorted by: [ date ] [ thread ] [ subject ] [ author ] You make an interesting point about energy. I can't agree with you about inertia though. You seem to be implying that key inertia is relatively insignificant in the total inertia picture. This is counter to my experience. I know that I can easily lead up a key so that the action is unplayable. I also know that if I take an action that has a ton of lead in the keys, and make some changes to it, without reducing SW, so that the amount of lead in the keys has been reduced, that it will make a marked difference in the way the action plays. On the other hand I can increase SW rather significantly, and while noticeable, it won't render the action unplayable. To do some different math let's assume: SW = 10 g Hammer CG is 13 cm from its center WW = 18 g Wippen CG is 7 cm from its center Assume SWR of 5 - For a massless key this would imply 50 g of lead (or whatever) at the measuring point, which we'll assume is 23 cm from the key balance point. Inertia of hammer about its center = 10 x 13 x 13 = 1690 g cm^2 Inertia of wippen about its center = 18 x 7 x 7 = 882 g cm^2 Inertia of key lead about its center = 50 x 23 x 23 = 26450 g cm^2 The key doesn't look so insignificant to me. Admittedly this is a severe example - the key itself has no distributed mass and the lead is all concentrated out at the end. But still, if the key inertia is insignificant then these should be quibbles. The hammer assumes high velocity from the key input because of all the leverage. I don't see that all the leverage is affecting the hammer inertia though. To accelerate the hammer, a torque has to be applied to the hammer shank. This torque is being provided by jack force at the knuckle. This force is being reacted back through the mechanism to the capstan. If you want more acceleration then more force has to be applied by the capstan. The magnitude of the force is going to be dependent on the action geometry. But changing this geometry (changing the amount of leverage) isn't changing the inertia of the hammer and shank, but changing the force at the capstan that's a manifestation of its inertia. Its inertia is still small compared to that of the key. Phil Ford -- Phillip Ford Piano Service and Restoration 1777 Yosemite Ave - 130 San Francisco, CA 94124 Previous message: Machine shop charges Next message: Letting tomatoes fly (was re: adjusting wippen...) Messages sorted by: [ date ] [ thread ] [ subject ] [ author ]
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