The
gear ratio of 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 four basic components of the epicyclic gear are: •
Sun gear: The central gear •
Carrier frame: Holds one or more
planetary gear(s) symmetrically and separated, all meshed with the sun gear •
Planet gear(s): Usually two to four peripheral gears, all of the same size, that mesh between the
sun gear and the
ring gear •
Ring gear,
Moon gear,
Annulus gear, or
Annular gear: An outer ring with inward-facing teeth that mesh with the planetary gear(s) The overall gear ratio of a simple planetary gearset can be calculated using the following two equations, --> representing the sun-planet and planet-ring interactions respectively: : \begin{align} N_\text{s}\,\omega_\text{s} + N_\text{p}\,\omega_\text{p} - \left( N_\text{s} + N_\text{p}\,\right)\,\omega_\text{c} &= 0 \\ N_\text{r}\,\omega_\text{r} - N_\text{p}\,\omega_\text{p} - \left( N_\text{r} - N_\text{p} \right)\,\omega_\text{c} &= 0 \end{align} where : \omega_\text{r}, \omega_\text{s}, \omega_\text{p}, \omega_\text{c} are the
angular velocities of the
ring gear,
sun gear,
planetary gears, and
carrier frame respectively, and N_\text{r}, N_\text{s}, N_\text{p} are the number of teeth of the
ring gear, the
sun gear, and each
planet gear respectively. from which we can derive the following: : N_\text{s}\,\omega_\text{s} + N_\text{r}\,\omega_\text{r} = ( N_\text{s} + N_\text{r})\,\omega_\text{c} : \omega_\text{s} = \frac{\, N_\text{s} + N_\text{r} \,}{ N_\text{s} } \,\omega_\text{c} - \frac{\, N_\text{r} \,}{ N_\text{s} }\omega_\text{r} : \omega_\text{r} = \frac{\, N_\text{s} + N_\text{r} \,}{ N_\text{r}}\,\omega_\text{c} - \frac{\, N_\text{s} \,}{ N_\text{r} } \,\omega_\text{s} : \omega_\text{c} = \frac{ N_\text{s} }{\, N_\text{s} + N_\text{r} \,}\,\omega_\text{s} + \frac{ N_\text{r} }{\, N_\text{s} + N_\text{r} \,} \, \omega_\text{r} and : -\frac{\, N_\text{r} \,}{ N_\text{s} } = \frac{\, \omega_\text{s} - \omega_\text{c} \,}{ \omega_\text{r} - \omega_\text{c} } only if \omega_\text{r} \neq \omega_\text{c} ~. In many epicyclic gearing systems, one of these three basic components is held stationary (hence set \omega_\text{...} = 0 for whichever gear is 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 of the gears, and upon which component is held stationary. Alternatively, in the special case where the number of teeth on each gear meets the relationship \, N_\text{r} = N_\text{s} + 2 \, N_\text{p} \;, the equation can be re-written as the following: : n\,\omega_\text{s} + (2 + n)\,\omega_\text{r} - 2(1 + n)\,\omega_\text{c} = 0 where : n = \tfrac{\, N_\text{s} \,}{ N_\text{p} } \; is the sun-to-planet gear ratio. These relationships can be used to analyze any epicyclic system, including those, such as hybrid vehicle transmissions, where 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 in the diagram above) is held stationary, and the sun gear (yellow) is used as input. In that 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 \, N_\text{s} \, teeth, and each planet gear has \, N_\text{p} \, teeth, then the ratio is equal to -\tfrac{\, N_\text{s} \,}{ N_\text{p} }\;. For instance, if the sun gear has 24 teeth, and each planet has 16 teeth, then the ratio is , or ; 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. Rotation of the planet gears can in turn drive the ring gear (not depicted in diagram), at a speed corresponding to the gear ratios: If the ring gear has \, N_\text{r} \, teeth, then the ring will rotate by \, \tfrac{\, N_\text{p} \,}{N_\text{r} } \, turns for each turn of the planetary gears. For instance, if the ring gear has 64 teeth, and the planets 16 teeth, one clockwise turn of a planet gear results in , or clockwise turns of the ring gear. Extending this case from the one above: • One turn of the sun gear results in \, -\tfrac{\, N_\text{s} \,}{ N_\text{p} } \, turns of the planets • One turn of a planet gear results in \, \tfrac{\, N_\text{p} \,}{ N_\text{r} }\, turns of the ring gear So, with the planetary carrier locked, one turn of the sun gear results in \; -\tfrac{\, N_\text{s} \,}{ N_\text{r} } \; turns of the ring gear. The ring gear 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 + \tfrac{\, N_\text{r} \,}{ N_\text{s} } = \tfrac{\, N_\text{s} + N_\text{r} \,}{ N_\text{s} } ~. If the ring gear 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/\left( 1 + \tfrac{\, N_\text{r} \,}{ N_\text{s} }\right) = \tfrac{ N_\text{s} }{\, N_\text{s} + N_\text{r} \,} \; , which may also be written as \; N_\text{s} : N_\text{s} +N_\text{r} ~. 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 \tfrac{\, N_\text{s} + N_\text{r} \,}{ N_\text{r} } ~. The number of teeth in the planet gear is irrelevant. AM bicycle hub (ring gear removed) == Accelerations of standard epicyclic gearing ==