Epicyclic Gearing - Gear Ratio

Gear Ratio

The gear ratio in an epicyclic gearing system is somewhat non-intuitive, particularly because there are several ways in which an input rotation can be converted into an output rotation. The three basic components of the epicyclic gear are:

• Sun: The central gear
• Planet carrier: Holds one or more peripheral planet gears, all of the same size, meshed with the sun gear
• Annulus: An outer ring with inward-facing teeth that mesh with the planet gear or gears

In many epicyclic gearing systems, one of these three basic components is held stationary; one of the two remaining components is an input, providing power to the system, while the last component is an output, receiving power from the system. The ratio of input rotation to output rotation is dependent upon the number of teeth in each gear, and upon which component is held stationary.

In other systems, such as hybrid vehicle transmissions, two of the components are used as inputs with the third providing output relative to the two inputs.

In one arrangement, the planetary carrier (green) is held stationary, and the sun gear (yellow) is used as input. In this case, the planetary gears simply rotate about their own axes (i.e., spin) at a rate determined by the number of teeth in each gear. If the sun gear has Ns teeth, and each planet gear has Np teeth, then the ratio is equal to -Ns/Np. For instance, if the sun gear has 24 teeth, and each planet has 16 teeth, then the ratio is -24/16, or -3/2; this means that one clockwise turn of the sun gear produces 1.5 counterclockwise turns of each of the planet gear(s) about its axis.

This rotation of the planet gears can in turn drive the annulus (not depicted in diagram), in a corresponding ratio. If the annulus has Na teeth, then the annulus will rotate by Np/Na turns for each turn of the planet gears. For instance, if the annulus has 64 teeth, and the planets 16, one clockwise turn of a planet gear results in 16/64, or 1/4 clockwise turns of the annulus. Extending this case from the one above:

• One turn of the sun gear results in turns of the planets
• One turn of a planet gear results in turns of the annulus

So, with the planetary carrier locked, one turn of the sun gear results in turns of the annulus.

The annulus may also be held fixed, with input provided to the planetary gear carrier; output rotation is then produced from the sun gear. This configuration will produce an increase in gear ratio, equal to 1+Na/Ns.

These are all described by the equation:

where n is the form factor of the planetary gear, defined by:

If the annulus is held stationary and the sun gear is used as the input, the planet carrier will be the output. The gear ratio in this case will be 1/(1+Na/Ns). This is the lowest gear ratio attainable with an epicyclic gear train. This type of gearing is sometimes used in tractors and construction equipment to provide high torque to the drive wheels.

In bicycle hub gears, the sun is usually stationary, being keyed to the axle or even machined directly onto it. The planetary gear carrier is used as input. In this case the gear ratio is simply given by (Ns+Na)/Na. The number of teeth in the planet gear is irrelevant.