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
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