Machine Kinematics
solved, then drawn
Concept

Tooth pitch — where it appears

The angle between neighbouring teeth on a wheel, which is a turn divided by the tooth count. It is the unit everything about a mesh is measured in: the contact ratio counts how many pitches of contact path there are, and the base pitch is the same angle measured along the line of action.

Named by 3 essays across 2 fields — each of them below, with the objects they name alongside it.

Lost motion on a 24-tooth ratchet. A pawl can only drop into a tooth, so an input that has moved by less than one tooth pitch has moved the output by nothing at all. On 24 teeth the pitch is 15.0°, and that is the worst case with one pawl. Two pawls offset by half a pitch halve it and three thirds it, because whichever pawl is over a root drops first. None of this is a manufacturing question: the numbers are the same on a perfectly made ratchet, which is what separates them from the lost motion an as-built measurement would.

The resolution is the pitch

A ratchet's step and a ratchet's error are the same number. Nothing about how well it is made improves that, more pawls divide it by a whole number, and the obvious remedy — cut more teeth — runs into a wall that is geometric rather than practical: at a tooth depth of 0.16 radii the construction stops at twenty-nine.

timing · Intermittent 30 teeth and two pallets. An escape wheel of 30 teeth and a pair of pallets spanning 4 tooth pitches. The heavier line at each pallet is the locking face, here an arc about the arbor; the lighter one is the impulse face the tooth slides along once it is let go. The wheel is drawn where the contact puts it, not where it looks well: at this pallet angle the tooth in play sits on the lock face and the wheel is -0.0323° from it. Dragging the pallet through its whole engagement moves the wheel by 0.107° of recoil. Of the 6.0° the wheel turns each beat, -37.3% is drop and does nothing.

The escapement that could not alternate

Four drawings of intermittent mechanisms that appear everywhere and would not work: pallets spanning a whole number of teeth, a pawl whose pivot is on the wrong side of the tooth face, a Geneva at the wrong centre distance, and an intermittent gear with no locking arc. Each one is put through the library that draws the working version, and each returns a number.

wrong · Misconception 30 teeth and two pallets. An escape wheel of 30 teeth and a pair of pallets spanning 4 and a half tooth pitches. The heavier line at each pallet is the locking face, here an arc about the arbor; the lighter one is the impulse face the tooth slides along once it is let go. The wheel is drawn where the contact puts it, not where it looks well: at this pallet angle the tooth in play sits on the lock face and the wheel is -0.0320° from it. Dragging the pallet through its whole engagement moves the wheel by 0.106° of recoil. Of the 6.0° the wheel turns each beat, 65.9% is drop and does nothing.

Where the tooth lets go

A pair of pallets spanning a whole number of tooth pitches and a half advances the escape wheel exactly half a pitch every beat, and that half pitch divides into the impulse and the drop with nothing left over. Drop is not chosen. It is whatever the impulse leaves, and on a thirty-tooth wheel it is two thirds.

timing · Intermittent

Named alongside it

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

Intermittent motionDropDwellEscapementthe Geneva mechanismLocking facePalletPawlRatchetBacklashBeatClearance

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