# Table of Geartrains

Certain projects I have considered doing require the use of unusual gear ratios. While some simple gear ratios, such as 0.600 (3/5) or 0.667 (2/3), are easy to construct with just two gears, most require 4 or more gears and many ratios that might be interesting (like the golden ratio) appear to be pretty much impossible to achieve. This results mainly from the rather small selection of available gear sizes.

The original Technic gears had 8, 24 and 40 teeth (introduced in 1977 in sets such as 956, Auto Chassis) and in 1979 the 16-tooth gear was added (e.g. set 857). These gears were typically joined in combinations that add up to a multiple of 16, such as 8 to 24, 16 to 16, or 8 to 40. This allows the gears to be mounted with the axles spaced an exact multiple of a stud spacing apart.

In 1993 piece 6541 was added making it a bit easier to create gear combinations that add to a multiple of 8 but are not a multiple of 16, such as 8->16, 16->24, 16->40, etc.

In 1994 the old 14-tooth bevel gear was replaced with the current 12-tooth bevel gear. Anyone who has the old bevel gears (until 2005 the old ones were still available from PITSCO) can join two of the 14-tooth bevel gears face-to-face on an axle, with a gap so their total width is same as a normal gear. This arrangement can then be meshed with any of the new double-bevel gears (12-, 20- and 36- teeth) or two of the 12-tooth or 20-tooth bevel gears, to produce reliable gear pairs with ratios of 6:7, 10:7 and 18:7.

In 1999 LEGO started producing 12-tooth and 20-tooth "double sided bevel" gears, which are capable of being used either as bevel gears or as normal straight gears. These gears first appeared in set 9748, Droid Developer Kit and various Bionicle kits, then appeared in various other new Technic sets. Typically they are used in combinations that add to a multiple of 16, such as 12 to 36 in set 3806, Spybotics Gigamesh.

In 2002 LEGO added the 36-tooth double sided bevel gear. It is in set 8270, Rough Terrain Crane, set 7659, Imperial Landing Craft and set 10076, TECHNIC Gear Wheels.

Used as straight gears, the 12- 20- and 36-tooth gears can be meshed with any of the standard Technic straight gears (available in 8, 16, 24 and 40 teeth). This allows a greater variety of gear reduction ratios to be produced with gear trains, and in a smaller space or with fewer gears. For example, the 20-tooth gear means that ratios involving a factor of 5 no longer need to use the large 40-tooth gear.

Used as bevel gears, the new gears can also be meshed with a pair of the 14-tooth bevel gears to produce ratios of 12 to 14, 20 to 14 or 36 to 14.

These new gears are a little more difficult to use as straight gears because the required axle spacing is unusual and doesn't directly fit the standard Technic beams. However, there are a number of ways to get around this.

Probably the simplest way to do axle spacing is to use two cross blocks attached to an axle and spaced an appropriate distance apart. You can slide the cross blocks to arbitrary positions and achieve any spacing you want.

You can also use technic beams and plates to construct gearboxes with precisely spaced axle-holes. This is the standard method and is what was used in most of the official building instructions for Technic models in the 1980's and 1990's.

For gear-pairs with a total number of teeth that is a multiple of 8, the gears can be positioned an integer or half-integer number of LSS apart, using normal beams (e.g. parts 3701, 3895, etc.) and the two special parts with the hole on stud center (6541 and 32000).

For other gear-pairs you need to go vertical or diagonal, sometimes using a combination of Technic bricks and plates. I have found that the spacing can be about 0.3 LGDU too small, or about 0.8 LGDU too big, and it will still work okay.

The tables pointed to by this page show all the gear trains that can be constructed using the 7 available gear sizes and the 14-tooth bevel gear trick.

## How to Use the Table

Here is a small bit of the table as an example:

Ratio Length S Geartrain 0.5486 6.1 +16->20+16->20+12->14+ 0.5510 5.0 +12->16+12->14+12->14+ 0.5556 4.0 s +8->12+20->24+ 0.5556 4.0 ** +8->24+20->12+ 0.5556 4.8 s +8->12+8->12+20->16+Ratio: The first column gives the overall gear reduction ratio. This is a fraction between 0 and 1. It tells how far the final axle turns for each revolution of the first axle. For example, a ratio of 0.3333 means that the final axle turns 1/3 of a revolution for each revolution of the first axle.

Notice that with many ratios you have two or more different gear trains to choose from. In most cases the different choices use different size gears and have a different total overall length.

If you want a very small ratio, consider using a worm gear in the first stage.

If you want a ratio greater than 1, use one of the geartrains in this table and just drive it from the other end. To find the ratio you want, look up the reciprocal of the desired ratio. For example, if you want a ratio of 1.500, use the geartrains listed under 0.6667 (which is 1 divided by 1.5). Very large ratios probably won't work too well because of friction.

Length: This is the distance between the first and last axle, assuming the entire gear train is laid out with all axles in the same plane. The length is given in terms of the standard "horizontal" LEGO® unit length, which is about 7.985 millimeters. In many cases there are many available lengths for a given ratio. Longer geartrains are generally better because they can handle more torque and have less friction. Shorter geartrains are better because they take up less space, and the smaller gears are more common in LEGO® sets.

S : This column contains two asterisks ** if the geartrain can be built using the standard spacings of LEGO® units and half-units. This is true if every pair of meshed gears adds up to some multiple of 8 teeth. The S column contains "s" for "standard" if the geartrain doesn't use the 14-tooth bevel gear trick.

Geartrain: The gear train itself is given in the form of a string of numbers linked with + signs and -> signs. Each + sign represents an axle (because + looks like the cross section of an axle) and each -> represents two gears that mesh with each other (you can think of the - as being a tooth of one gear and the > as being two teeth of the other gear; or think of it as an arrow showing the direction the energy flows while the gears are turning). Here is an example from the sample above:

+8->12+20->24+ example A

This is a geartrain with 4 gears and three axles. The first axle has an 8-tooth gear. The second axle has a 12-tooth gear (driven by the 8-tooth gear) and a 20-tooth gear. The third axle has a 24-tooth gear (driven by the 20-tooth gear).

## Alternate Geartrains

Notice that most geartrains can be rearranged and still produce the same ratio. Example A can be changed to:

+20->24+8->12+ example B

which uses the same 4 gears in a different order to produce the same ratio of 0.5556. This rearrangement is considered equivalent so it isn't listed in the table. In your designs you might find it useful to try both rearrangements and pick the one that fits best. In most cases, the table lists the rearrangement that has the greatest reduction in the first two gears, which is usually best because it has the least friction. Examble A appears in the table and example B does not because 8/12 is less than 20/24.

Less obvious rearrangements are listed in the table explicitly because you might not think of them. For example, the following rearrangement is listed in the table:

+8->24+20->12+ example C

which is another way to use the same 4 gears to produce the same ratio. This time, the 24 and 12 gears were interchanged and the 8 and 20 gears stayed in the same position. This gear train is substantially different from example B because the spacing between the axles is different. In example C both axle spacings are exactly 2.0 units, allowing it to be built in the "standard" way with axles going through the holes in a Technic beam (which is indicated by the ** marker in the S column). Examples A and B have strange axle spacings (1.25 and 2.75) and are therefore more difficult to build.

There are also ways to add extra gears without changing the ratio and without the extra gears directly cancelling each other out. For example, a little further down in the table you'll find:

0.5556 6.0 ** +16->24+20->12+8->16+which is another rearrangement of examples A, B and C with an extra 16-tooth gear added at each end. This would usually be a waste of two 16-tooth gears, but it is useful when you need a certain ratio and you also need a certain size gear at the end of the geartrain. For example, that 16-tooth gear at the end might be the special 16-tooth gear with a round center, or the 16-tooth part of a differential housing. It also may be useful because the direction is reversed.

## Directionality, Idlers and Bevels

The direction of rotation will reverse with each "->" symbol. If you want the other direction, insert an idler.

An idler is a gear that meshes with exactly two other gears, and is not tied to any other gears through its axle; and is driven by exactly one of the gears it touches. An idler can be inserted at any point in any geartrain without changing the effective ratio. Symbolically, this is done by replacing any "->" symbol with "->N->" or "->N+N->", where N is any number.

To change axle orientation (from vertical to horizontal, etc.) use a pair of bevels. In the table, a pair that mesh can be beveled if either one of them is 24 (using the 24-tooth crown gear) or if both of them are in the set {12, 14, 20, 24, 36}.

## Limits and Statistics

Using 2, 4 or 6 gears, with 8 different gear sizes, it would
theoretically be possible to construct 8^{2}+8^{4}+8^{6}=266304
different gear trains. However, most of these are of little interest
because of one or more of the following:

- They have two gears in a row with the same number of teeth, OR
- They are identical to another geartrain except for a trivial reordering of the gears, OR
- They have two or more gears that effectively cancel each other out, OR
- They have bevel gears meshing with non-bevel gears (which doesn't work physically)

My tables list only the 8124 gear trains that remain after all of these are eliminated.

If you limit it to geartrains of 2 or 4 gears, the number drops to 565.

Of the 8124 geartrains in the table, there are 5490 that do not use any 14-tooth gears. Of these, 469 can be built with the standard axle spacings.

The following table shows how these statistics have increased over time:

when | total | 2 or 4 gears | no-14 | std-spacing | notes |

1977 | 146 | 39 | 146 | 146 | original set of 4 gear sizes: z8, z16, z24 and z40 |

1995 | 631 | 99 | 440 | 146 | after introduction of 12-tooth bevel gear |

1999 | 3628 | 297 | 2502 | 249 | after addition of 12- and 20-tooth double bevel gears |

2002 | 8124 | 565 | 5490 | 469 | after addition of 36-tooth double bevel gear |

The graph paper in my newer photos is ruled at a specing of 1 LSS, which is about 7.99 mm.

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