Epicyclic gearing


An epicyclic gear train is a gear reduction assembly consisting 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 the planet gear around the sun gear. The planet and sun gears mesh so that their pitch circles roll without slip. If the sun gear is held fixed, then a point on the pitch circle of the planet gear traces an epicycloid curve.
An epicyclic gear train can be assembled so the planet gear rolls on the inside of the pitch circle of an outer gear ring, or ring gear, sometimes called an annulus gear. Such an assembly of a planet engaging both a sun gear and a ring gear is called a planetary gear train. By choosing to hold one component or anotherthe planetary carrier, the ring gear, or the sun gearstationary, three different gear ratios can be realized.

Overview

Epicyclic gearing or planetary gearing is a gear system consisting of one or more outer, or planet, gears or pinions, revolving about a central sun gear or sun wheel. Typically, the planet gears are mounted on a movable arm or carrier, which itself may rotate relative to the sun gear. Epicyclic gearing systems also incorporate the use of an outer ring gear, which meshes with the planet gears. Planetary gears are typically classified as simple or compound planetary gears. Simple planetary gears have one sun, one ring, one carrier, and one planet set. Compound planetary gears involve one or more of the following three types of structures: meshed-planet, stepped-planet, and multi-stage structures. Compared to simple planetary gears, compound planetary gears have the advantages of larger reduction ratio, higher torque-to-weight ratio, and more flexible configurations.
The axes of all gears are usually parallel, but for special cases like pencil sharpeners and differentials, they can be placed at an angle, introducing elements of bevel gear. Further, the sun, planet carrier and ring axes are usually coaxial.
Epicyclic gearing is also available which consists of a sun, a carrier, and two planets which mesh with each other. One planet meshes with the sun gear, while the second planet meshes with the ring gear. For this case, when the carrier is fixed, the ring gear rotates in the same direction as the sun gear, thus providing a reversal in direction compared to standard epicyclic gearing.

History

Around 500 BC, the Greeks invented the idea of epicycles, of circles travelling on the circular orbits. With this theory Claudius Ptolemy in the Almagest in 148 AD was able to approximate planetary paths observed crossing the sky. The Antikythera Mechanism, circa 80 BC, had gearing which was able to closely match the Moon's elliptical path through the heavens, and even to correct for the nine-year precession of that path.
In the 2nd century AD treatise The Mathematical Syntaxis, Claudius Ptolemy used rotating deferent and epicycles that form epicyclic gear trains to predict the motions of the planets. Accurate predictions of the movement of the Sun, Moon, and the five planets, Mercury, Venus, Mars, Jupiter, and Saturn, across the sky assumed that each followed a trajectory traced by a point on the planet gear of an epicyclic gear train. This curve is called an epitrochoid.
Epicyclic gearing was used in the Antikythera Mechanism, circa 80 BC, to adjust the displayed position of the Moon for the ellipticity of its orbit, and even for its orbital apsidal precession. Two facing gears were rotated around slightly different centers; one drove the other, not with meshed teeth but with a pin inserted into a slot on the second. As the slot drove the second gear, the radius of driving would change, thus invoking a speeding up and slowing down of the driven gear in each revolution.
Richard of Wallingford, an English abbot of St. Albans monastery, later described epicyclic gearing for an astronomical clock in the 14th century. In 1588, Italian military engineer Agostino Ramelli invented the bookwheel, a vertically revolving bookstand containing epicyclic gearing with two levels of planetary gears to maintain proper orientation of the books.
French mathematician and engineer Desargues designed and constructed the first mill with epicycloidal teeth.

Requirements for non-interference

In order that the planet gear teeth mesh properly with both the sun and ring gears, assuming equally spaced planet gears, the following equation must be satisfied:
where
are the number of teeth of the sun gear and the ring gear, respectively and
is the number of planet gears in the assembly and
is a whole number
If one is to create an asymmetric carrier frame with non-equiangular planet gears, say to create some kind of mechanical vibration in the system, one must make the teething such that the above equation complies with the "imaginary gears". For example, in the case where a carrier frame is intended to contain planet gears spaced 0°, 50°, 120°, and 230°, one is to calculate as if there are actually 36 planetary gears, rather than the four real ones.

Gear speed ratios of conventional epicyclic gearing

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 symmetrically and separated, all meshed with the sun gear
  • Planet gear: 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
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
from which we can derive the following:
and
only if
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 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 the 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 one arrangement, the planetary carrier is held stationary, and the sun gear is used as input. In that case, the planetary gears simply rotate about their own axes at a rate determined by the number of teeth in each gear. If the sun gear has teeth, and each planet gear has teeth, then the ratio is equal to 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 about its axis.
Rotation of the planet gears can in turn drive the ring gear, at a speed corresponding to the gear ratios: If the ring gear has teeth, then the ring will rotate by 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 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
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 which may also be written as 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 The number of teeth in the planet gear is irrelevant.
Image:1948amhub.jpg|thumb|Compound planets of a Sturmey-Archer AM bicycle hub

Accelerations of standard epicyclic gearing

From the above formulae, we can also derive the accelerations of the sun, ring and carrier, which are:

Torque ratios of standard epicyclic gearing

In epicyclic gears, two speeds must be known in order to determine the third speed. However, in a steady state condition, only one torque must be known in order to determine the other two torques. The equations which determine torque are:
where: — Torque of ring, — Torque of sun, — Torque of carrier. For all three, these are the torques applied to the mechanism. Output torques have the reverse sign of input torques. These torque ratios can be derived using the law of conservation of energy. Applied to a single stage this equation is expressed as:
In the cases where gears are accelerating, or to account for friction, these equations must be modified.