Machine Kinematics
solved, then drawn
Concept

Superposition — where it appears

Adding two motions a linear mechanism permits to obtain a third, which is what makes the tabular method for a planetary correct. It works because the relation between the members' rates is linear, so two admissible motions add to a third — which is exactly what the tabular method assumes.

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

The lever of a simple planetary. Each member sits at a position on the lever fixed by the tooth counts alone, and its speed is the height of one straight line over that position. The line here is drawn through sun and ring; every other member is plotted where the train's null space puts it, and lands on the line exactly — the residual is zero in rationals, and 2.2e-16 once the coordinates have been rounded to doubles for the drawing. Where the line crosses the axis is the member that is standing still, and that is what a brake does: it pins the line to the axis at one position and leaves it free to pivot there. The planets are on the lever too, off the end of it, which is where they belong — they are members of the train and are not shafts anybody can reach.

The lever that is the gearset

The lever diagram of an epicyclic is usually offered as a mnemonic. It is exact, and the reason is a fact about the null space: a gearset whose frame carries no teeth can turn as a block, and that one motion supplies the coordinate every member is plotted at. Where the line crosses the axis is the member standing still, and the ordering of the members on the lever settles which gears are reductions and which run backwards, without a formula anywhere.

transmission · Transmission Sun 24, ring 72, planet 24. An epicyclic train has three shafts and one equation relating them, so fixing any one gives a different ratio from the same gears. With the ring held, a sun input turns the carrier at 1 + 72/24 = 4.000 to one. Willis's equation and the tabular superposition method are independent derivations and both are computed here; the site requires them to agree, because planetary ratios are the most commonly mis-stated numbers in mechanism work and the sign errors are notorious.

Epicyclic ratios, two ways

An epicyclic train has three shafts and one equation relating them, so fixing any one gives a different ratio from the same gears. The sign errors are notorious, so every ratio here is computed by Willis's equation and by the tabular method, and the two are required to agree.

gears · Tooth

Named alongside it

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

EpicyclicMobilityCarrierConstraintExact arithmeticInvoluteLever analogyMeshNull spacePlanetaryRatioTransmission relation

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