Moment of Inertia of grand action parts.

[Mark Davidson]

Some notes on action ratio vs. gear ratio.

First, let's define these so we're on the same page.  By gear ratio (R) I
mean the ratio of angular speeds.  If moving the key by 2 degrees around the
balance rail results in the hammer moving 20 degrees around its center, then
the ratio of angles is 2:20 or 1:10.  Angle, angular speed and angular
acceleration are all proportional.  That is, if the gear ratio is 1:10 and
the key accelerates at 1 degree per second per second, then the hammer
accelerates at 10 degrees per second per second.

By action ratio (A) I mean the familiar concept of how far the hammer moves
vs how far the key moves.  So if the key moves 1 mm and the hammer moves 5
mm, then the action ratio is 1:5.  In a perfect world, with horizontal lever
arms and perfect action efficiency, the action ratio is the same as the
weight ratio.  In reality, these will normally differ by a few percent.

While gear ratio and action ratio may not at first appear to be closely
related, consider the following.  The distance that the key moves is equal
to the key radius times the change in key angle (in radians).  Call this
Kr*Ka.  The distance the hammer moves is the hammer radius times the change
in hammer angle - Hr*Ha.  but Ha is just Ka times the gear ratio (call this
R), so the hammer distance is Hr*Ka*R.  This gives a distance ratio  of A =
Hr*Ka*R / Kr*Ka =  R*Hr/Kr.

Likewise, we can get the gear ratio from the action ratio: R = A*Kr/Hr.

This is a very useful result because it means that if we know the action
ratio and Hr and Kr we can calculate the gear ratio from these, without
having to know the length of all the internal lever arms (balance rail to
capstan, wippen heel to wippen center, etc.).  It also suggests that the
reflected inertia of the hammer is is related to SWR^2.

-Mark