page_title: Epicyclic gearing
section_title: Gear ratio of standard epicyclic gearing
hierarchical_section_title: Epicyclic gearing / Gear ratio of standard epicyclic gearing
caption_attribution_description: English: A Sturmey-Archer AM bicycle hub gear mechanism with the gear ring removed to show the compound (stepped) planet gears.
context_page_description: An epicyclic gear train consists of two gears mounted so that the center of one gear revolves around the center of the other. A carrier connects the centers of the two gears and rotates to carry one gear, called the planet gear or planet pinion, around the other, called the sun gear or sun wheel. The planet and sun gears mesh so that their pitch circles roll without slip. A point on the pitch circle of the planet gear traces an epicycloid curve. In this simplified case, the sun gear is fixed and the planetary gear roll around the sun gear.
  An epicyclic gear train can be assembled so the planet gear rolls on the inside of the pitch circle of a fixed, outer gear ring, or ring gear, sometimes called an annular gear. In this case, the curve traced by a point on the pitch circle of the planet is a hypocycloid.
  The combination of epicycle gear trains with a planet engaging both a sun gear and a ring gear is called a planetary gear train. In this case, the ring gear is usually fixed and the sun gear is driven.
  Epicyclic gears get their name from their earliest application, which was the modelling of the movements of the planets in the heavens.
context_section_description: 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 three basic components of the epicyclic gear are:
  Sun: The central gear
  Carrier: Holds one or more peripheral Planet gears, all of the same size, meshed with the sun gear
  Ring or Annulus: An outer ring with inward-facing teeth that mesh with the planet gear or gears
  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:
  where
   is the angular velocity of the Ring, Sun gear, Planet gears and planet Carrier respectively, and
   is the Number of teeth of the Ring, the Sun gear and each Planet gear respectively.
  from which we can deduce that:
  or
  Considering .
  Alternatively, if the number of teeth on each gear meets the relationship , this equation can be re-written as the following:
  where
  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 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 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 ring gear (not depicted in diagram), in a corresponding ratio. If the ring gear has Nᵣ teeth, then the ring will rotate by Nₚ/Nᵣ turns for each turn of the planet gears. For instance, if the ring gear 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 ring gear. 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 ring gear
  So, with the planetary carrier locked, one turn of the sun gear results in  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+Nᵣ/Nₛ.
  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/(1 + Nᵣ/Nₛ) which can also be written as 1:(1 + Nᵣ/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+Nᵣ)/Nᵣ. The number of teeth in the planet gear is irrelevant.
caption: Compound planets of a Sturmey-Archer AM bicycle hub (ring gear removed)
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