Machine Kinematics
solved, then drawn
Concept

Lock — where it appears

How far below the corner of a pallet a tooth comes to rest, measured as an angle of the pallet and chosen to be as small as will hold. It is a safety margin: too little and a jolt unlocks the wheel, too much and the impulse is wasted unlocking it again, so it is chosen as the smallest angle that holds.

Named by 4 essays across one field — each of them below, with the objects they name alongside it.

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 How far back the wheel is pushed. Recoil against the part of the pendulum's swing that happens after the tooth has landed. The arc cut concentric with the pallet arbor is the flat line on zero, and it is zero as a matter of arithmetic rather than of smallness: rotation about the arbor carries that arc into itself, so the tooth's resting place does not move and every sampled value is the same double. Every other face rises. The flat face with no draw at all is the interesting one — it starts at zero and curves, because it agrees with the arc to first order and not to second, which is precisely the amplitude sensitivity a deadbeat exists to remove.

The wheel that goes backwards

While a pendulum finishes its swing the escape wheel is doing something, and what it does is decided entirely by the shape of the face the tooth is resting on. An arc about the pallet arbor sends it nowhere — not nearly nowhere, the same double at every sample. A flat cut tangent to that arc is dead at exactly one point of itself, and it is the one point the tooth never rests on.

timing · Intermittent One angle, read twice. Draw against the angle the locking face is tilted by. It is not a separate design quantity from recoil: the pallet's moment per unit of wheel torque is minus the rate at which the wheel is driven backwards, which is virtual work and holds to nine figures. So an escapement cannot be made to hold its own lock without also being made to push its train back — and at δ = 0 the arc has neither, which is a lock that any disturbance opens. The flat face starts above zero because the tooth rests below the corner, where a straight face is no longer tangent to anything.

The angle that holds the lock

A locking face cut exactly concentric with the pallet arbor has no tendency to hold itself: the tooth's push aims straight at the pivot and its moment is zero, so the smallest disturbance opens the lock. Tilt the face and the moment becomes ρ sin δ — and the same tilt, by virtual work, is exactly the rate at which the wheel is driven backwards. Draw and recoil are one angle read twice.

timing · Intermittent A period, in the space the mechanism lives in. One whole period of an escapement, plotted as wheel angle against pallet angle. Give a four-bar its crank angle and its coupler is somewhere definite; give this its pallet angle and the wheel may be in any of 3 places, because what settles it is which face of which pallet a tooth is against. The path crosses itself and no amount of solving removes the crossing. The vertical jumps are the drops, drawn at the pallet angle of the release because during a drop nothing is touching anything and where the pallet is by then is a question about torque. Over the period the wheel advances 12.000000°, which is one tooth exactly.

Where the input stops deciding

Give a four-bar its crank angle and its coupler is somewhere definite. Give an escapement its pallet angle and the wheel may be in any of three places, because what settles it is which face of which pallet a tooth is against. The path crosses itself and no amount of solving removes the crossing — the state of these mechanisms has a discrete part, and that is what makes them a different kind of object.

timing · Intermittent

Named alongside it

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

EscapementLocking facePalletDeadbeatDrawDropEscape wheelImpulseIntermittent motionRecoilAssembly branchBeat

All concepts