Field

Motion that stops

Every other mechanism here moves whenever its input does. A ratchet, an indexer and an escapement are still for most of a turn and moving for the rest, and what decides which is not an equation running out of answers but a tooth arriving at a face. Whether one holds is the sign of a lever arm; what it wastes is two thirds of its own travel.

$Every way of stopping, on the same four questions. Six mechanisms that all turn a continuous input into an output that moves and then waits. Index is how far the output steps. Moving is the fraction of the input's turn the output is actually going for; the rest is dwell. From rest says whether the output starts and stops at zero velocity, and acceleration whether its acceleration is a number at all. Every entry is computed from the mechanism's own library, which matters for two of them: a Geneva's moving fraction is (n − 2)/2n and not 1/n, and its entry rate is zero in closed form rather than to the accuracy of a sampled sweep. The three rows whose acceleration is not a number are not badly made — they are mechanisms whose output velocity has a step, and no tolerance improves that.$

The mechanism that waits

Every mechanism in this collection so far moves whenever its input moves. A ratchet, an indexer and an escapement do not: they are still for most of a turn and moving for the rest, and what decides which is not an equation running out of answers but a tooth arriving at a face.

A 12-tooth ratchet, holding. A ratchet wheel with a 8° tooth face and its pawl. The long line is the face extended; the short one through the contact is the normal the tooth pushes along, and both belong to the wheel and turn with it. The pawl holds, and the reason is which side of those two lines its pivot is on: 1.080 wheel radii from the face's line and 0.495 from the normal's, and on opposite sides of them. Nothing about how hard anything pushes enters the question: a normal is a direction and a moment arm is a length with a sign. Dragging turns the wheel the way it may go, and the pawl's tip is solved onto the wheel's surface at every frame: it is lifted up a tooth's back and dropped into the next notch, once per click.

A joint that works one way

A pawl either holds a ratchet or is levered out of it, and which one happens is decided by two lines through the contact. One of them is the rule a workshop quotes. The other is the boundary that rule leaves out, and a check written to confirm the quoted rule turned out to be incapable of failing.

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.

$The curves that do not move. A rotation about a point carries a curve into itself if and only if the curve is an arc of a circle centred on that point, and that one sentence is every exact dwell on this site. Each bar is how far a curve moves when it is turned two degrees about the axis its mechanism turns about, as a fraction of its own radius. The two arcs about their own centres — a cam's dwell and a deadbeat's locking face — sit at the sampling floor, which is the sagitta of the polyline they are measured as and not a property of the geometry; the number is quoted with the floor beside it because an agreement quoted without its resolution is a mistake this site has already made once. Everything else is orders of magnitude above it, including the near-circular stretch of a coupler curve that a six-bar builds its approximate dwell out of.$

The arc that is concentric with the pivot

A rotation carries a curve into itself exactly when the curve is an arc of a circle about the centre of rotation. Every exact dwell on this site is that one sentence applied — a cam's dwell, a Geneva's locking disc, a deadbeat escapement's locking face — and the six-bar dwell that is merely very good is what happens when the curve is nearly one.

A quantity that is not there, refusing to converge. A second central difference of a function with a bounded second derivative settles as the step is halved; across a step in the first derivative it doubles, every time. The rising line is a mutilated gear at the instant its teeth engage, where the driven wheel goes from stopped to full pitch-line speed: its successive estimates grow by a factor of 2.000000, which is the signature and not an accident of the step. The flat line is a six-slot Geneva at the same point in its cycle, whose pin enters along the slot and whose acceleration is a number. This is the only way to report an acceleration that does not exist: not by quoting a large one, but by showing the measurement refuse.

The gear with its teeth cut away

Leave teeth on part of a gear's circumference and take the rest off, and the output turns for part of the input's revolution and stops for the rest. It is the cheapest intermittent drive there is and it engages at full speed, so its output's velocity has a step and its acceleration is not a large number — it is not a number, and the way to report that is to watch a difference quotient refuse to converge.

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.

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.

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.

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.

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.

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.

One piece, and still not reachable

The site decides whether two configurations can be joined by asking whether they are in the same connected component, and that relation is symmetric because a path run backwards is a path. A one-way mechanism breaks the symmetry and leaves the connectivity alone: its free space is a single interval with no barrier anywhere in it, and about half of the ordered pairs of states cannot be joined by any admissible motion.

One pin and two, on the same wheel. A 6-slot Geneva wheel at one instant of its index, driven by a crank carrying one pin and by the same crank carrying two. The wheel, the slots and the centre distance are identical; only the pin count differs. Each pin drives the wheel through one slot pitch while the driver sweeps 120°, so 1 pin gives 1 index a driver turn and leaves 67% of it at rest, 2 pins give 2 indexes a driver turn and leave 33% of it at rest. Nothing else about the mechanism changes, which is why the whole question is arithmetic.

Two pins and no dwell at all

Put a second pin on a Geneva's crank and the wheel indexes twice a turn instead of once, at a quarter of the acceleration for the same output rate. Put a third on a six-slot wheel and it never rests; put a third on an eight-slot wheel and two pins meet in two slots and it jams. Both of those look like separate conditions and are one, and the boundary between them is an equation in integers with exactly three solutions.

A ratchet with no teeth

A roller clutch holds one way and runs free the other with nothing on it a tooth could be called. Whether it grips is a question about friction, which no drawing settles. How far it moves before it does is geometry: a clearance divided by the gap's own slope, so its lost motion is a length rather than a fraction of a pitch — 1.4° against a 24-tooth ratchet's 15° — and unlike a ratchet's it is not divided by adding more holding elements.

A Geneva's acceleration against three cam laws, at the same index angle. The output's acceleration through one index, divided by the step over the square of the input angle it takes — the unit in which a cam law's acceleration coefficient is quoted — so every law is one fixed curve whatever the step. The dark curve is a Geneva of 6 slots: its step is 360°/6 and its input turns 120° while indexing, and a cam indexer is given exactly those two numbers. The Geneva's coefficient is 5.653 against cycloidal 6.283, modified sine 5.528 and simple harmonic 4.935, so at 6 slots modified sine and simple harmonic have the lower peak. The Geneva's curve starts and ends away from nought, at tan(π/6) = 0.577 in absolute terms: its acceleration steps the instant the pin enters. Dragging the slot count moves only the Geneva.

When the index law becomes a choice

A Geneva's motion law is forced by its slot count and a cam indexer's is chosen, so the fair comparison gives the cam the Geneva's own index angle and step. On peak acceleration the cam wins only below a slot count that depends on the law — 5.19 for cycloidal, 6.23 for modified sine, 8.06 for simple harmonic — and above it the Geneva does. What no slot count removes is the step: the pin arrives with an acceleration of exactly tan(π/n).

The disc decides the pin count

A Geneva's pin count is usually bounded by the slots: p indexes must not overlap, so fewer than 2n/(n−2) pins fit. The other half of the mechanism has its own inequality and nobody had measured it. The locking disc must be cut away wherever the wheel passes through it, that cut-away is wider than the index sweep at every slot count, and it is the binding condition everywhere — one pin only, from four slots upward.

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