Re: Force Versus Distance of a Pneumatic
By Robbie Rhodes
John Grant, thanks for your reply. I think it was 30 years ago (!) I glued a nipple into a scrap DuoArt action pneumatic and performed some simple experiments. I mounted the pneumatic so the moving board would push upward, just as in a piano, and I hung a string from it upon which I had a stack of steel washers and nuts. There definitely was more force available to lift the weights when the pneumatic was open.
Let us define that 100% opening is when the pneumatic is completely open -- the cloth is stretched taut. Assume that in the piano action stack the pneumatic motion occurs between the limits of 20% open (the note is held "on") and 80% open (note at rest). When the note is played the hammer velocity results from the integral of the pneumatic force times the effective hammer mass, taken over the span of the motion of the pneumatic.
Now consider how lifting the hammer rest rail alters things. The Soft Pedal moves the rest rail so that the hammer travel distance is reduced to one-half. We assume that the travel of the key and action pneumatic is also halved, and therefore the pneumatic is working between 50% open (note at rest) and 20%. The integrated force in this operating range is considerably less than half of normal, and the result is that the hammer velocity is much less than it would be if the force had remained constant.
You questioned, John, if halving the hammer blow distance predicts a 3-decibel reduction in sound intensity. You bet it does, and the equations are simple, so let's look at it.
Kinetic energy is proportional to velocity-squared, written V^2 on a typewriter. (The circumflex ^ says that the exponent follows.) One of the definitions of decibels is that decibels (db) is ten times the logarithm of the energy, thus
db = 10*log[V^2]
If you will hang a microphone near the soundboard of the piano and observe the sound intensity with an old-fashioned VU meter you can read the decibels sound energy. (Use the meters on an old hi-fi reel-to-reel tape recorder, John. I bet you still have one _somewhere_ in your garage!)
Newton's Laws of Motion say that the kinetic energy of a mass (the hammer) is proportional to velocity-squared, and V^2=2as, where "a" is the acceleration and "s" is the distance traveled. If the acceleration is constant and the distance is halved then V^2 is halved, and
db = 10*log[0.5] = -3 db = 3 decibels reduction.
But the above assumes that the pneumatic force is constant, yielding constant acceleration. In reality, because the pneumatic is operating in the "mostly closed" region, the acceleration term "a" is also reduced, and the result is that the Soft Pedal produces much more than 3 db intensity reduction.
I'd like very much to read the articles in AMICA Bulletin that you mention. Can someone help me obtain them?
-- Robbie Rhodes |
(Message sent Sat 23 Mar 1996, 00:24:21 GMT, from time zone GMT-0800.) |
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