An epicyclic gear train (also known as a planetary gearset) is a gear reduction assembly consisting of two gears mounted so that the center of one gear (the "planet") revolves around the center of the other (the "sun"). A carrier connects the centers of the two gears and rotates to carry the planet gear(s) 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 that 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.J. J. Uicker, G. R. Pennock and J. E. Shigley, 2003, Theory of Machines and Mechanisms, Oxford University Press, New York.B. Paul, 1979, Kinematics and Dynamics of Planar Machinery, Prentice Hall. By choosing to hold one component or anotherthe planetary carrier, the ring gear, or the sun gearstationary, three different gear ratios can be realized.
One important property of planetary gear systems is their ability to combine multiple rotational inputs into a single output. This behavior can be understood through an equivalent linear (rack-and-pinion) analogy, which simplifies the kinematic relationships.
Object A → upper rack corresponds to one gear element (e.g., sun gear) Object C → lower rack corresponds to another gear element (e.g., ring gear) Object B → represents the planet carrier If Object A is fixed and Object B moves one inch, Object C moves two inches.
C moves twice as far as B, because rotation and translation combine.
Kinematic equations
A simple planetary gear train can be analyzed using an equivalent linear rack-and-pinion model. In this analogy, rack A corresponds to one gear element, such as the sun gear; rack C corresponds to another gear element, such as the ring gear; and body B represents the planet carrier.
Let
be the velocity of rack A, be the velocity of carrier B, be the velocity of rack C, be the angular velocity of the planet gear, and be the pitch radius of the planet gear.
If rack A is fixed, the no-slip condition at the upper contact point is:
Therefore,
At the lower contact point, the velocity of rack C is the sum of the carrier translation and the rotational contribution of the planet gear:
Substituting gives
Thus, if carrier B moves one unit while rack A is fixed, rack C moves two units.
More generally, the no-slip conditions at the two contact points are
and
Adding these equations gives
or
For a planetary gear train, the corresponding rotational quantities are as follows: corresponds to for the sun gear, corresponds to for the ring gear, and corresponds to for the carrier. Here, is the number of teeth on the sun gear, is the number of teeth on the ring gear, and , , and are the angular velocities of the sun gear, ring gear, and carrier, respectively.
Substituting these quantities gives the standard kinematic relationship
This equation describes how the angular velocities of two gear elements determine the angular velocity of the third. If only one element is driven and the others are unconstrained, the motion is not uniquely determined. A fixed member, load condition, or directional constraint, such as a ratchet or one-way clutch, is needed to define a unique output motion. Without a ratchet mechanism, the higher torque input may influence the lower-torque input, resulting in unintended reverse motion or energy transfer.
For example, let
Solving for the ring gear speed,
so
and
The ring gear therefore rotates with an angular velocity of , indicating rotation in the negative direction under the chosen sign convention.
A related relationship applies to a symmetric differential, where the carrier speed is the average of the two side-gear speeds:
Solving for the right-side gear gives
For example, if
and
then
Modified planetary gear systems have also been proposed for wave-energy applications, where they may help convert irregular, low-speed, pulsating motion into steadier one-directional rotation.
Overview
Epicyclic gearing or planetary gearing is a gear system consisting of one or more outer 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 (or epicyclic 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 (there are at least two more planets in mesh with each other in each planet train), stepped-planet (there exists a shaft connection between two planets in each planet train), and multi-stage structures (the system contains two or more planet sets). 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 (see below). Further, the sun, planet carrier and ring axes are usually coaxial.
Another configuration of epicyclic gearing consists of a sun gear, a carrier, and two planets that mesh with each other. One planet meshes with the sun gear, while the second planet meshes with the ring gear. In this configuration, when the carrier is fixed, the ring gear rotates in the same direction as the sun gear, providing a directional reversal compared to standard epicyclic gearing.
History Around 500 BC, the Greeks invented the concept of epicycles, which are circles travelling on 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 that 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. (The Greeks interpreted the motion they saw, not as elliptical, but rather as epicyclic motion.)
In the 2nd century AD treatise The Mathematical Syntaxis (a.k.a. Almagest), 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.
The 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,
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 design the gearing arrangement 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 (10° equiangular), 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(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:
where
are the angular velocities of the ring gear, sun gear, planetary gears, and carrier frame respectively, and
are the number of teeth of the ring gear, the sun gear, and each planet gear respectively.
from which we can derive the following:
and
only if . In many epicyclic gearing systems, one of these three basic components is held stationary (hence set 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 , the equation can be re-written as the following:
where
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 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(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 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 have 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