Hello Stephane, The best way to make sense of this problem is to separate it into two simpler ones. Michael Spalding correctly pointed out that we tend to become careless in the distinction between mass and weight. In a grand action both play a role, but under different playing conditions. I just glanced at Stephen's paper, and it's very good reading! In fact, I may be repeating some of what he covered in the paper. Take the action out of the piano and tilt it so the action frame is vertical, with the narrow side resting on the floor. Measuring "balance weight" is impossible because the key no longer springs back. By doing this, you have removed the "weight" component out of the picture. Note that we haven't taken any mass out of the action. With your finger push the hammer against the rest rail. The key front will be ready for you to "play" a note. If you push the key very slowly, it will take almost no effort. You will be overcoming the small amount of friction in the mechanism, which is just a fraction of the force you would need when the action is in its normal horizontal position. Repeat the experiment, but this time try to press the key as fast and hard as you can. There will be considerable force at the finger tip. You are observing the effect of the inertia without any contribution from the weight. The amount of inertia is determined by the mass of the various parts and its physical distribution within. This component of the required force will not change when you put the action back into its normal horizontal position. It's not as easy to create an experiment to demonstrate the effect of weight in the grand action without that of inertia. One could get the point across by moving the experiment to another planet that is much more massive than Earth so that its gravitational force is huge even for a small mass. In other words, we would have to move the experiment to a planet where our 12g lead weighs much more, let's say 12kg, and a hammer weighs 10kg instead of 10g! In such a place, with the action in its normal horizontal position, it would be difficult to push the key down at any speed. The inertial effect in the action would be unchanged by the move to the massive planet, but it would be overshadowed by the gravitational effect. Now, to answer some of your questions: The question of work or energy has to be answered separately for the gravitational and inertial effects: In the gravitational (weight) portion of the problem, pushing the key down _very slowly_ puts some potential energy into the hammer by lifting it. The amount of this energy (or work) is roughly equal to: A=d*BW where A = work done d = key dip BW = balance weight The energy expended in overcoming the inertial effects is determined from the product of the torque on the key and the arc that the key travels. The torque in turn is determined from the moment of inertia and angular acceleration of the key. The result is that the calculation for the gravitational (weight) effect is not really related to the calculation for inertial effects, except that both use the mass of the components in the action in their respective equations. This is the one common point between the two subjects, but that is pretty much where the commonality ends. For example, static balancing with the lead follows a formula that only requires the product of weight and distance from the balance rail hole to be constant. As long as you satisfy that requirement, you can place the weight anywhere along the key and the balance weight will come out the same. When you look at inertia however, the formula contains the "r-squared", so the two are not the same. The answer to your first question then is: if you move a key _very_ slowly, the work done moving the key with the 24g weight through 5mm is the same as the work done moving the 12g weight 10mm. There is no significant inertial effect when you move the key slowly. As soon as you start pressing the key faster, you will have to confront the inertia of the mechanism and the amount of work required will in general depend on the acceleration profile. Now, to your question about "why r-squared" in I=m*r^2. This is derived from the energy conservation principle, but I think I can give you an intuitive explanation: Let's say, the key dip is 10mm, and the distance from the front of the key to the balance rail hole is 250mm. If you place a lead at r=50mm, it will travel only one fifth of the key dip, that is 2mm. Note the time it takes to complete the key stroke on a hard blow and save the number for later. Move the lead to r=100mm. In order to apply the same force at the lead as you did in the first part of the experiment, you will have to double the force at the front of the key where your finger is. This is similar to what you know from static balancing, and that is the origin of one of the "r"s in the r-squared. There is a however also a second effect at work. The lead at r=100mm has to travel further along its arc (4mm instead of 2mm) in the same amount of time as we measured with r=50mm. That means, we have to accelerate it more, so it can complete the trajectory on time. This further increases the force required and is the origin of the second "r". What is ideal with regard to inertia cannot be answered simply. I think, it will vary from pianist to pianist and will also be a function of the music being played. I have some open ended comments here, maybe the discussion will reveal something. - I wonder if it is necessary that all 88 keys be adjusted to the same balance weight. Perhaps a better action can be made if the bass starts at say 55g balance weight and the keys progress along a smooth curve to something like 35g at the top. It seems to me that balance weight receives a lot of attention because it is easy to measure. Inertia is harder to measure, so we don't do it. - Does it matter where in the action the inertia originates? Imagine playing a vibraphone with heavy mallet sticks and really light heads, vs. heavy heads and light sticks. Maybe the inertia contributed by key lead is less desirable than inertia contributed by the hammer. We are ultimately trying to control the manner in which the hammer moves, and the capstan is not tied to the whippen heel - they do separate at times. - We can only accelerate the hammer by means of the key but we can't slow it down. If it starts going too fast, the most we can do is let go of the key and hope that the hammer slows down. Inertia works against that. Gravity does help slow the hammer down, and so does a measured amount of friction in the center. There is a delicate balance of parameters at work in a great action. Vladan ======================================== Stéphane Collin wrote: Hi Vladan. Most interesting comments from you, as always. Reading this, I have some intuitive reactions that I would like to share, for the case it can rise a better understanding of the matter. How does moment of inertia relate to work ? I would have thought that 24 gram moved (was it in a circular way) by 5 mm (or an arc of circle whose ray would be 5 mm) needs the same work as 12 gram moved by 10 mm. Is this right ? Does this apply to our key weight matter ? Isn't it the work to accomplish that causes fatigue to the pianist ? I thought, ok, if there is much lead in the key, there is much work to accomplish, thus the heavy feel. But I'm still to understand why the same [weight/distance from balance] figure would give a different feel. It is the square of ray parameter that I fail to figure out. Now, from a pianist point of view, I have thought uptill now that as long as the action has a decent repetition capability and an acceptable heavyness feel, the more inertia would be the better (up to a certain point) as this inertia helps alot smoothing out the minute discrepancies between the pianist's fingers, by giving all along the 10 mm dip some feedback information to the pianist's finger about how much force he is really applying to the key, and letting him (intuitively) adapt this, in order to produce the exact desired sound, favouring for example perfect legato, and avoiding the "ow, I didn't want this" effect. But then, at what point does the extra inerty start to impede repetition (by making the change of direction of move of the key more difficult) ? I want the key to stick to my finger for the fastest that I can repeat the note with that finger. But I heard comments from other pianists that this is already too much for good feel. Any comments about all this ? Best regards. Stéphane Collin. __________________________________________________ Do You Yahoo!? Tired of spam? Yahoo! Mail has the best spam protection around http://mail.yahoo.com
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