What's all this I hear about Inertia ?

[Fenton Murray]

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)
>
>
>
>