I'd like to thank David Stang for his original question/statement, with out it I wouldn't have gotten to read Greg Grahams great reply below. Great thread, thanks Dave. To paraphrase and encapsulate, more weight on a shorter arm is better. Weight x Arm = Moment. Correct? Fenton ----- Original Message ----- From: "Greg Graham" <grahampianos at yahoo.com> To: <pianotech at ptg.org> Sent: Monday, September 29, 2008 10:31 PM Subject: What's all this I hear about Inertia ? > > David Stang wrote: "But I will reiterate my original point. Mass is the > same as inertia. And mass is proportional to weight. So on earth, two > things with the same weight have the same inertia. That's the point I had > been kind of hung up with." > > David, as I will try to demonstrate below, Inertia is NOT the same as > mass. > > David also wrote: "Rotational movement is another kettle-o-fish > altogether, and has nothing to do with the point I (admittedly ineptly) > was trying to make." > > David, when you are talking about pianos, rotational movement has > everything to do with the point you are trying to make. Piano parts > rotate. Keys rotate around the balance pin hole. Hammers rotate around > the shank center pin. > > Here are a couple important things to consider: > > Force equals mass multiplied by acceleration (F = M x A). A key will > require more force when you play loudly because you are trying to move the > key faster (greater acceleration). But... to complicate things, the > distance from the key lead to the balance pin has a big influence on the > force. > > So, for example... would it be better to have a lead with mass of 1 gram > 10 centimeters from the balance pin, or 2 grams 5 cm from the balance pin? > >>From a static (non-moving) balance point of view, they both do the same >>thing. The force of the lead is equal to its mass times the acceleration >>of gravity (g = 9.8 m/s^2). This force, commonly known as "weight", acts >>straight down. The static torque about the balance point is the weight >>(force) times the distance between the lead and the balance pin. 2 x 5 x >>g = 10g, or 1 x 10 x g = 10g. You can cancel the g. The acceleration of >>gravity is the same no matter where you put the lead. I think this is >>what you were suggesting. > > Here is the fine point you may have missed: The acceleration due to key > movement is NOT the same everywhere on the keystick. > > When you start to move the key, you are adding new acceleration to the > key. The velocity of the lead 5 cm from the balance is half what it is at > 10 cm, because it is only moving half the distance in the same amount of > time. Velocity = distance divided by time (V=d/t). Acceleration is the > change in velocity over time (A=V/t) which is the same as distance divided > by time squared (A=d/t^2). > > At the front of the key, you could have a dip of 10 mm. A very loud blow > would be fast and cover this distance in a short time. Let's guess a > tenth of a second, or 0.1 sec. The acceleration of a lead halfway between > the keyfront and balance pin would be 5 mm / 0.1 sec / 0.1 sec, which is > 500 mm per second squared, or 0.5 meters per second squared (0.5m/s^2). A > lead three quarters of the way from the keyfront to the balance pin would > accelerate 2.5mm/0.1sec/0.1sec = 0.25m/s^2. > > The 1 gram lead at 10 cm requires a force from the key to cause it to > accelerate 0.5m/s^2 which we can calculate using F=ma, or 1 gram x > 0.5m/s^2 = 0.5 gram-meters per second squared. The 2 gram lead at 5 cm > likewise is calculated, 2 gram x 0.25m/s^2 = 0.5. Same force, right? > Yes, but acting at different lever lengths. The torque equals force x > distance. The lead at 10 cm produces torque of 0.5 x 10 = 5, but the lead > at 5 cm produces only 0.5 x 5 = 2.5. > > This is the importance of inertia. The key with heavier lead closer to > the balance pin feels lighter at high volumes than the key with lighter > lead farther from the balance pin. > > Static (weight) balance involves mass times distance. > > Dynamic (Inertial) balance involves mass times distance SQUARED. > > The faster you try to move, the less important gravity becomes, and the > more important inertia becomes. > > But this is all theory. For practical application, please see > http://www.stanwoodpiano.com/articles.htm > and read how all this can be used in real piano work. > > But please don't say mass is the same as inertia. > > One last thing: You said "There would be less wasteful bending and > more dynamic range if there were less weight in front. Correct me if > I'm wrong here,...." > > Well, think of this: If all the lead was at the front of the key, your > finger would be acting directly on the lead, so the key would not have to > bend at all to get it moving. Bending is not the issue. Acceleration is > the issue. The lead at the front of the key would have to move the entire > distance of the keystroke, rather than a fraction of that if placed closer > to the balance pin. > > Regarding dynamic range, there is only one way to increase it: Make it > possible to play louder. The bottom end is zero, and never changes. > Silent = zero. Maximum volume comes with maximum hammer momentum, which > is mass times velocity. Heavier hammers moving faster. High inertia in > the keys may limit the player's ability to accelerate a heavy hammer to > maximum velocity, but heavy hammers require heavy counterweights in the > keys to keep static touch weight reasonable. Maybe higher static > touchweight would be OK if the trade off was lower dynamic touchweight > (lower inertia)? Maybe key bending and hammer shank bending can be made > to work with the performer to whip the hammer to higher velocity than a > rigid lever would? > > This is a deep topic, and truly IS worthy of study, Journal articles, test > equipment, and all of that. > > As in most of life, there is a simple answer for everything, and it is > usually wrong. > > Greg Graham, RPT (and Mechanical Engineer 20 years ago) > > > >
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