Planetary Gearbox Speed Ratio Formula

The speed ratio of a planetary gearbox (also called an epicyclic gearbox) is determined by the relationship between the number of teeth (or diameters) of the gears involved. The formula for calculating the speed ratio depends on which gear (sun gear, planet gear, or ring gear) is fixed and which is used as the input or output.

Formula:

Speed Ratio = [1 + (N_ring / N_sun)] / Fixed Gear Component Factor

Where:

• N_ring = Number of teeth on the ring gear.

• N_sun = Number of teeth on the sun gear.

• The Fixed Gear Component Factor depends on the component that is fixed:

• If the sun gear is fixed: The factor is 1.

• If the ring gear is fixed: The factor is -1.

• If the planet carrier is fixed: The factor is 0, meaning the system does not provide any ratio.

Example Scenarios:

1. If the ring gear is fixed and the sun gear is the input:

The speed ratio can be calculated as:

Speed Ratio = 1 + (N_ring / N_sun)

This results in a positive ratio, meaning the planet carrier will rotate slower than the input (sun gear).

2. If the sun gear is fixed and the ring gear is the input:

The speed ratio becomes:

Speed Ratio = -[(N_ring / N_sun) + 1]

This negative result indicates that the output rotates in the opposite direction to the input.