Machine Kinematics
solved, then drawn
Concept

Planet spacing — where it appears

The angular stations orbiting gears occupy round a sun, admissible only when the sun and ring counts divide by how many there are. Equal spacing balances the gearset and is only available when the arithmetic permits it, which is why some tooth-count choices force unequal spacing or fewer planets.

Named by 2 essays across 2 fields — each of them below, with the objects they name alongside it.

Six gearsets, three conditions. Every one of these can be drawn, and five of the six are drawn in some textbook or other. The columns are the three conditions a planetary has to satisfy: that a whole planet fits between the sun and the ring, that the sun and ring teeth add to a multiple of the planet count, and that the planets clear each other. The last column is how far out of mesh the worst planet station is, in teeth — a quantity that is zero or is not, and that no drawing shows, because a drawing of a planetary at this scale draws circles.

The gearset that could not be assembled

A planetary drawing shows a sun, a ring and three or four planets between them, and if the circles are the right sizes at the right stations it looks right. The condition that decides whether the second planet can actually be dropped in is arithmetic — the sun and ring teeth must add to a multiple of the planet count — and it appears in no drawing, at no scale, in any style.

wrong · Misconception Everything one planetary can do, and the gap in the middle. A single epicyclic has three shafts, so there are six ways of choosing which is held, which is driven and which comes out. Each gives a band of reductions as the tooth counts run over every design that can be cut, assembled with three planets and kept clear of undercutting. The bands above 1 are drawn; between them is a gap running from 1.6304 to 2.5862 that no single planetary reaches in any configuration — and a reduction of exactly 2, which is the most ordinary thing anybody asks a gearbox for, is inside it. The gap's width as a factor is exactly the smallest achievable ring-over-sun ratio, 1.5862, which is a statement about how small a planet may be and how large a sun may be.

The reductions a planetary cannot give

One epicyclic offers six ratios, and the formula for each of them suggests the whole positive line is available. Sweep every design that can actually be cut and assembled and the reachable set has a hole in it running from 1.630 to 2.586 — the width of which has a closed form — and a reduction of exactly 2, the most ordinary thing anybody asks a gearbox for, sits in the middle of it.

transmission · Transmission

Named alongside it

The objects these essays reach for when they reach for this one.

Assembly conditionDesign ruleEnumerationEpicyclicUndercuttingCentre distanceMeshMisconceptionReachable setReductionToothTransmission relation

All concepts