Machine Kinematics
solved, then drawn
Concept

Draw — where it appears

The moment a resting tooth's push exerts about a pallet's arbor, which carries the pallet deeper into its lock and is zero for a face concentric with the arbor. It is what keeps a pallet against its stop between beats, and it is computed as the sign of the moment the tooth's push makes about the pallet arbor.

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

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 The fork and the roller, which is a Geneva pair. A lever escapement's fork with the balance's impulse pin inside it. The pin is on a roller of 0.150 against a fork of 1.00, and the slot points at the pin — which is the entire kinematic relation and is the same one a Geneva drive's pin and slot obey. The same eight lines answer both, and the check is that they reproduce the cams field's independently written Geneva solution to fourteen figures. The pin is in the fork for 44.4° of the balance's swing out of 540°, so the balance is left alone for 91.8% of the beat. Drag the balance.

Detached, and safe while detached

A lever escapement touches its balance for a twelfth of each beat and leaves it alone for the rest. What connects the two is a pin entering a radial slot — the same pair a film projector's Geneva drive is made of, solved by the same eight lines — and the sine rule then fixes the balance's lift angle at 44.4°, which is the number a Swiss lever is specified at.

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.

One test, three mechanisms

Whether a pawl holds, whether an escapement's lock draws itself deeper, and how much a four-bar's coupler can do for its rocker are the same question asked three times: on which side of a pivot does a contact normal pass? All three are one cross product, none evaluates a force, and the three answers are used for completely different things.

timing · Intermittent

Named alongside it

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

EscapementContact normalDeadbeatLever armLockLocking facePalletRecoilTransmission angleVirtual-workBalanceBanking

All concepts