User:Prof McCarthy/epicylicgearing

The oldest known example of a differential was once thought to be in the Antikythera mechanism. It was supposed to have used such a train to produce the difference between two inputs, one input related to the position of the sun on the zodiac, and the other input related to the position of the moon on the zodiac; the output of the differential gave a quantity related to the moon's phase. It has now been proven that the assumption of the existence of a differential gearing arrangement was incorrect.

Fixed Carrier Train Ratio
A convenient approach to determining the various speed ratios available in a planetary gear train begins by considering the speed ratio of the gear train when the carrier is held fixed. This is known as the fixed carrier train ratio.

In the case of a simple planetary gear train formed by a carrier supporting a planet gear engaged with a sun and annular gear, the fixed carrier train ratio is computed as the speed ratio of simple gear train formed by the sun, planet and idler on the fixed carrier. This is given by,
 * $$R= \frac{\omega_a}{\omega_s}=\frac{N_s}{N_a}.$$

In this calculation the planet gear is an idler gear.

The fundamental formula of the planetary gear train with a rotating carrier is obtained by recognizing that this formula remains true if the angular velocities of the sun, planet and annular gears are computed relative to the carrier angular velocity. This becomes,
 * $$R= \frac{\omega_a-\omega_c}{\omega_s-\omega_c}=\frac{N_s}{N_a}.$$

This formula provides a simple way to determine the speed ratios for the simple planetary gear train under different conditions: 1. The carrier is held fixed, ωc=0,
 * $$ \frac{\omega_a}{\omega_s}=R.$$

2. The sun gear is held fixed, ωs=0,
 * $$ \frac{\omega_a-\omega_c}{-\omega_c}=R, \quad \mbox{or}\quad \frac{\omega_a}{\omega_c}=1-R.$$

3. The annular gear is held fixed, ωa=0,
 * $$ \frac{-\omega_c}{\omega_s-\omega_c}=R,\quad \mbox{or} \quad \frac{\omega_s}{\omega_c}=\frac{R-1}{R}.$$

Each of the speed ratios available to a simple planetary gear train can be obtained by using band brakes to hold and release the carrier, sun or annular gears as needed.