Machine Kinematics
solved, then drawn
Concept

Assembly condition — where it appears

The requirement that a planetary's sun and ring counts sum to a multiple of how many orbiting gears are wanted. It is an integer condition rather than a tolerance: if it fails, the last orbiting gear has nowhere to go and no accuracy of manufacture helps.

Named by 4 essays across 4 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 One member of the family is a pair of wheels. The same demand — one plus 0.4 sin φ — scaled by a constant, and for each the angle the driven wheel is out by after one turn of the input. It has to be zero, or the teeth do not line up with themselves and there is no wheel. Only the unscaled member closes; five per cent either way leaves the output 18° out, which is a third of a tooth on a thirty-tooth wheel and a mechanism that seizes on its second turn. Non-circular gearing is a search rather than a drawing for exactly this reason: the closure condition is one equation on a whole function, and almost no function satisfies it.

The demand that cannot be met

Ask for an output rate and the two pitch curves follow with no design step in between — so the interesting question is not how to draw them but which demands admit any pair of wheels at all. The answer is one equation on a whole function, it is about the demand's mean and nothing else, and five per cent of error leaves the output eighteen degrees out after a turn.

meshing · Meshing Neither one comes out, and the two of them do. Two congruent Z-shaped parts in a tray that is open at the top. Each has a step that lies over the other's, so part A's four contacts with part B have normals at all four points of the compass and leave it no free direction at all — and the same is true of B, for the same reason and by symmetry. The blocking is mutual and there is no order in which the two can be taken out one at a time. Together they have 6 contacts, all of them with the tray, and exactly one direction out: straight up. So the removal cone of a set of parts is not built from the removal cones of its members, and which part comes out first is a question with no answer here. positioned by solving, not by drawing.

Neither part comes out first

Two congruent Z-shaped pieces in a tray open at the top. Each has four contacts with the other, with normals at all four points of the compass, so each alone is blocked in every direction there is — and the pair lifts straight out. The removal cone of a set of parts is not built from the removal cones of its members, and *which part comes out first* is a question with no answer.

holding · Restraint 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.

EnumerationUndercuttingDesign ruleEpicyclicPlanet spacingVelocity ratioBlockingCentre distanceClosure conditionConjugate-actionContact normalDisassembly

All concepts