Two pins and no dwell at all

The mechanism that waits fixed the Geneva’s own arithmetic and corrected a number that had been wrong. A six-slot wheel indexes by 60° at a time, and the fraction of the driver’s turn during which it is being driven is not one sixth but one third: the pin is in a slot while the driver sweeps $180° - 360°/n$, so the driven fraction is $(n-2)/2n$ and the two expressions agree at four slots and nowhere else.

That is what one pin costs. A driver can carry more than one, and the mechanism is then indexing several times a turn — which is what a machine wanting a fast index and a slow driver would ask for.

Everything about whether it works is arithmetic in two integers.

One line #

With $p$ pins equally spaced round the driver there are $p$ engagements in a turn and each costs $(n-2)/2n$ of it, so the wheel is being driven for

$$\frac{p\,(n-2)}{2n}$$

of the turn and resting for the rest. The pins do not interact and the geometry of each engagement is exactly the single-pin geometry: the centre distance is still crank ÷ sin(180°/n), the pin still enters along the slot, the wheel still starts and stops from rest.

Each line is straight through the origin, because the pins are independent, and the only thing that differs between slot counts is the slope. Above one there is nothing left of the turn to rest in.

The independence is worth a sentence because it is what makes the arithmetic legitimate rather than a first approximation. Two pins on one crank are two rigid points on one body, so while one is in a slot the other is somewhere on its own circle and is doing nothing; there is no force path between them and no interaction to model. The wheel’s motion during an engagement is the function the Geneva’s own sweep integrates, and it is the same function whichever pin is doing the driving. So the only thing the pin count does is count.

The two conditions that are one #

A multi-pin Geneva looks as though it needs two things. There must be a dwell, which is the line above. And two pins must never be in slots at the same time, because two pins in two slots is two constraints on a wheel with one freedom and the machine seizes.

The second condition is that the engagement arc $180° - 360°/n$ be shorter than the $360°/p$ between pins. Multiply that out:

$$p\left(180° - \frac{360°}{n}\right) < 360° \quad\Longleftrightarrow\quad \frac{p\,(n-2)}{2n} < 1 .$$

It is the first condition. Checked over fifty-six pairs, from three to thirty-seven slots and one to nine pins, the two tests agree in every row — which they must, and which is worth measuring because a designer who believes there are two conditions will go looking for the second after satisfying the first and will not find it.

So a multi-pin Geneva has exactly one thing that can go wrong.

There is a reading of that identity worth keeping, because it explains why the two tests looked different. One is a statement about time — how much of the cycle is left over — and the other is a statement about space, about where two pins are relative to each other. They coincide because a Geneva’s driver turns at a constant rate, so an arc of the driver and a share of its cycle are the same quantity in different units. A mechanism whose driver did not turn steadily would have two genuinely different conditions, and the identity here is a consequence of the one thing about a Geneva that is never in question.

Three machines that never stop #

The boundary is the equality, and the equality is a Diophantine condition:

$$p\,(n - 2) = 2n \quad\Longleftrightarrow\quad n\,(p - 2) = 2p \quad\Longleftrightarrow\quad n = \frac{2p}{p-2}.$$

There are three solutions with at least three slots: six slots with three pins, four slots with four pins, and three slots with six pins. And there is no fourth, because $2p/(p-2)$ falls below three as soon as $p$ exceeds six, and $p = 1$ and $p = 2$ give no solution at all.

Each of those three drives its wheel for the whole of the driver’s turn: one pin leaves a slot exactly as the next enters, and the wheel never stops. They are not fast indexers. They are continuous drives with a varying ratio — mechanisms of a different kind, sitting on the boundary of this one, and a designer who arrives at six slots and three pins by adding pins one at a time has walked out of the family without being told.

The three of them are a small, closed list, which is the shape this field’s results keep taking. A clock is a factorisation reduces a gear train to an arithmetic of integers; the resolution is the pitch reduces an indexer’s accuracy to a count. Here a mechanism’s whole viability reduces to whether two integers satisfy an inequality, and the boundary between the viable and the impossible turns out to be inhabited by exactly three machines.

What a wheel can take #

Putting the condition to every reasonable pair gives a table rather than a formula, and the table is short.

A wheel of six slots or more can take two pins and no more. Three, four and five slots can take three. Only a three-slot wheel can take five.

That is a strong restriction and it is worth seeing why. The driven fraction per pin, $(n-2)/2n$, rises towards a half as the slot count grows — a twelve-slot wheel is being driven for 42% of the turn by a single pin, and a twenty-slot wheel for 45%. So the dwell a single-pin Geneva leaves is already barely more than half the turn at large slot counts, and a second pin consumes almost all of what is left. The finer the index, the less room there is for pins, which is the opposite of the intuition that a fine index leaves the wheel more idle.

So the multi-pin Geneva is not a family with a parameter to tune. It is a short list, and most of the list is two pins.

That restriction has a consequence for how the mechanism is usually met. Textbook Genevas have four or six slots, and a six-slot wheel is at the edge of what a second pin allows: it drives for two thirds of the turn and rests for a third. Eight slots with two pins drives for three quarters and rests for a quarter; twelve slots with two pins rests for a sixth. Somewhere along that sequence the dwell stops being long enough to do anything in, and where that is depends on the machine the wheel is carrying rather than on the wheel. So the table’s “most pins it can take” is an upper bound on a quantity a designer will usually want smaller.

What the second pin buys #

Two pins is the case that matters, and it is worth pricing properly.

The index itself is unchanged — the wheel still advances one slot pitch, still starts and stops from rest, and the shape of its motion through the index is a property of the slot count alone. What changes is how fast the driver has to turn. Asking for a stated number of indexes a second with $p$ pins needs a driver turning $p$ times slower, and the wheel’s angular acceleration is the shape’s own second derivative times the square of the driver’s speed.

So the acceleration falls as $1/p^2$: to 0.250 at two pins, 0.111 at three, 0.063 at four. The dwell falls linearly: 83.3% to 16.7% over the same range on a three-slot wheel.

The second pin costs a sixth of the cycle and saves three quarters of the acceleration. That is a good exchange, and it is why the arrangement exists — the Geneva’s standing complaint, the one the mechanism that waits ends on, is that its acceleration is finite but large and is what tears film sprocket holes. Halving the driver’s speed is the only way to attack it that does not change the mechanism, and a second pin is how a driver is halved without halving the output.

It is also the answer to a question the gear with its teeth cut away raises from the other side. A mutilated gear indexes by having teeth over part of its circumference and none over the rest, and its driven fraction is whatever the designer cuts — there is no arithmetic limiting how often it may index, because the driving and the dwelling are the same body doing two things. A Geneva’s engagement is a fixed arc it cannot shorten, which is why its pin count is bounded and a mutilated gear’s tooth count is not. The Geneva buys its exact start and stop with a constraint on how often it may be asked for one.

What the pins are not #

They are not resolution. The resolution is the pitch: a Geneva’s index is $360°/n$ and it is exact, and adding pins changes how often that index happens rather than how large it is. A machine wanting a finer step still needs more slots, and more slots is exactly what leaves no room for pins.

They are not a second mechanism. Each engagement is the same engagement. Nothing about the entry, the exit, the starting acceleration or the locking arc depends on how many pins the driver carries, which is what makes the whole question arithmetic rather than geometric.

They are not a finer index in disguise. A wheel indexing twice a turn is not a wheel with twice as many slots. Two pins on a six-slot wheel index by 60° twice a driver turn; one pin on a twelve-slot wheel indexes by 30° once. The first advances the wheel 120° per driver turn and the second 30°, and the two motions have nothing in common but the number of events.

They do not change the ratio. The wheel still turns once for every $n$ indexes. What changes is that $n$ indexes now take $n/p$ driver turns rather than $n$, so the overall ratio between driver and wheel is $n/p$ to one — and there is the second reason to want them, which is a reduction without a gear.

The same question asked of a ratchet #

A ratchet is the other mechanism in this field that indexes, and the pin count has an analogue there that behaves in exactly the opposite way.

The resolution is the pitch measured what several pawls buy a ratchet: with $m$ pawls offset by a fraction of a tooth, the lost motion falls to a pitch divided by $m$, and there is no upper limit on $m$ but the room to mount them. A ratchet’s pawls do not have to take turns — they are all riding the same teeth all the time, and only one of them happens to be the one that catches.

A Geneva’s pins do have to take turns, because a pin in a slot is the constraint, and two constraints on one freedom is a seizure rather than a redundancy. So the two mechanisms give opposite answers to the same design move. Adding a second holding element to a ratchet improves it with no arithmetic to satisfy; adding a second driving element to a Geneva improves it only while an inequality in two integers holds, and past that boundary it destroys it.

That is the same trade the resolution is the pitch drew between the two mechanisms, arriving one level down: the ratchet has every proportion free and a resolution it cannot improve, and the Geneva has an exact index and almost nothing free.

Where a designer lands #

Reading the three results together gives a short procedure, which is worth writing out because the arithmetic above is easy to admire and easy to misapply.

Start from the index the machine needs, which fixes the slot count: $n = 360°/\text{index}$, and it must be an integer. That decides everything else. The driven fraction per pin is $(n-2)/2n$ and is not negotiable. The most pins the wheel can take is the largest $p$ with $p(n-2) < 2n$, which is two for any wheel of six slots or more, three for four or five slots, and five for three.

Then the choice is between one pin and the maximum. One pin gives the longest dwell and the fastest driver; the maximum gives the shortest dwell and the gentlest acceleration. There is nothing in between to tune, because the pin count is an integer and the list is two or three items long.

And if the dwell the maximum leaves is not long enough for whatever the machine does during it, the pins cannot help and the mechanism has to change. A cam indexer’s dwell is a programme and can be made any length; a Geneva’s is $1 - p(n-2)/2n$ and is arithmetic. That is the price of an index that is exact without being measured.

What this does not settle #

The locking disc. A Geneva’s wheel is held during the dwell by a disc on the driver with a cut-away that admits the wheel’s own locking arcs. With $p$ pins there must be $p$ cut-aways, and whether the disc has enough material left between them is a condition on the widths of those cut-aways that nothing here computes. It is a plausible place for a second condition genuinely to exist, and it would be a condition on the locking geometry rather than on the counts.

Nothing here is a force. The acceleration quoted is angular acceleration of the wheel, not a torque, a contact stress or a pin bending load. A driver with two pins carries the same peak torque at half the speed, so the pin loads fall too — but by how much depends on what is being indexed and its inertia, which is a dynamics question.

The pins are all the same radius. Two pins at different radii would enter different slots at different driver angles and would need different centre distances for a tangential entry, which the one centre distance cannot provide — so the arrangement is impossible rather than merely unmeasured, and saying so is worth more than sweeping it.

The pins are equally spaced. Unequal spacing is available and changes the problem entirely: the engagements no longer come at regular intervals, the dwells between them differ, and the single condition above becomes one condition per gap. Whether unequal spacing can fit more pins in is not asked.

Reversal is not asked about. Where the input stops deciding shows that an escapement’s output depends on which contact it is on rather than on the input angle alone. A Geneva’s does not: give the driver its angle and the wheel is somewhere definite, and adding pins does not change that, because only one pin is ever engaged. What does change is that there are now $p$ driver angles giving each wheel position, which is a statement about the map’s many-to-oneness and not about its determinacy.

Internal Genevas are absent. The mechanism that waits draws the external Geneva, whose driven fraction is under a half. An internal Geneva’s is over a half, so it has less room for pins, and the boundary equation would have a different form.

Still open: the locking disc as the real limit #

The condition measured here is about the pins and the slots, and it is complete as far as it goes. The mechanism has a third part that has not been counted: the locking disc, which holds the wheel still through the dwell and must be cut away wherever a pin is engaging.

Its distinct argument would be that geometry as a second inequality — the angular width each cut-away must have for the wheel’s locking arc to pass, against the $360°/p$ available between pins — and whether it ever bites before the pin condition does. Two things would come out of it. A number: the largest pin count a real Geneva can carry, which may be smaller than the arithmetic above allows and would then be the answer a designer actually needs. And a statement about the three continuous-drive machines: at the boundary the dwell is nought and the locking disc has nothing to do, so it should vanish entirely — and whether it vanishes smoothly, or whether there is a range of pin counts just short of the boundary where the disc is impossible while the pins are fine, is what would decide whether the family’s real edge is where the arithmetic puts it.