Ultimate Goal: Calculate the mass moment of inertia that a finger experiences as it depresses a piano key.
Background on the question is at the end if you need, but I'll keep this high-level so that you don't need to be a Registered Piano Technician to answer this.
Imagine you have two interconnected levers (such that the output of one is mechanically linked to the input of the other). Each lever, which carries arbitrary weights, has its own moment of inertia about its pivot point. These individual moments of inertia can easily be calculated using thin rod approximations, parallel axis theorem, etc.
The tricky part to me comes in when you try to calculate the total moment of inertia of the system about the pivot of the first lever. Recall that, since the levers are connected, the first lever's I will be subject to the I of the second lever in some way. What I'm looking for is, as the title suggests, how this moment of inertia transfers.
At first I thought I could just use the the basic definition of moment of inertia:
with V as the solid lever system (though difficult by hand, this is easily done with a SolidWorks model), but then I read this article about reflection of moment of inertia through a system with a motor, two gears, and a load at the other end. It suggested that the total I at the motor is:
With I1 being the moment of inertia of the motor about its CM, I2 being the moment of inertia of the load about its CM, and N being the gear ratio. Since levers are analogous to gears, I predicted that the transfer of moment of inertia through our imagined 2-lever system would be the same as the above equation, only instead of a gear ratio, we'd have:
where B1 is the length of the rear segment of lever 1, A1 is the length of the front segment of lever 1, and so on.
However, when I compare this to calculations done by some guy on this website (85% of the way down the page) for the same purpose, he got (adapted to fit this example):
which makes his N
So it seems like, to calculate his gear ratio (or rather, mechanical advantage) he only involved the input segment of the second lever, and the output segment of the first lever, whereas I involved all segments.
So, finally, the questions:
- Is it indeed incorrect to use the simple integration form of moment of inertia (shown above) for a lever system?
- Is it incorrect to extend this gear-train moment of inertia formula to levers?
- If none of the above formulae are correct, what is the correct way to calculate the moment of inertia about the first lever's pivot in a system of levers?
I think I might be misunderstanding the conversion from gear ratio to the analogous gear ratio for levers, but I'm surprised I haven't been able to find any good sources about this online.
The grand piano action consists of a system of 3 interconnected levers that looks like this:
Where L1 is input segment of lever 1, L2 is output segment of lever 1, L3 is input segment of lever 2, and so on. Levers 1, 2, and 3 are blue, red, and green, respectively. Finger presses down on L1, this transfers force through levers that eventually throws the hammer (end of L6) towards the string.