A flat holds the wheel to second order

A Geneva drive has two halves, and the disc decides the pin count measured the half that is usually left out of the arithmetic. The pin and the slot turn the wheel. Between turns, the wheel is held still by a disc on the driver sitting in a concave arc cut into the wheel’s rim, and that disc has to be cut away wherever the wheel passes through it while the pin is driving. Swept in the driver’s own frame, the cut-away came out far wider than the pin’s index: 236° of the driver’s turn on a six-slot wheel, for an index of 120°. Two cut-aways and a lock between them do not fit in one turn, so a Geneva of four or more slots can carry one pin, whatever the slot arithmetic allows.

That essay ended with the other way to hold a wheel. Between each pair of slots, cut a flat chord across the rim instead of a concave arc, and let the driver’s disc press on the flat. A flat removes much less of the wheel, so the slot mouths survive a larger lock. The question left was whether the cut-away falls far enough to let a second pin back in, and whether the flat still holds.

It does the first, and it does the second only in a weaker sense.

How large a disc a flat allows #

The arc’s disc was bounded by the slot. At a locked position the arc is centred on the driver’s axis, and an arc deeper than the crank radius would reach the nearest slot and cut it open. So the arc’s disc can be as large as the crank.

A flat is bounded by the slot too, but at its ends. The flat is a chord across the rim, square to the line of centres at the locked position, and it must stop short of the two slot mouths on either side of it. Those mouths open at the rim, at an angle π/n either side of the flat’s middle, so the chord can come no nearer the wheel’s centre than $r_w \cos(\pi/n)$. The disc touches the flat at the foot of the perpendicular from the driver’s axis, so its radius is the distance from that axis to the flat:

$$r_d \le a - r_w\cos\frac{\pi}{n} = c \sin\frac{\pi}{n},$$

where $a$ is the centre distance, $r_w$ the wheel’s rim and $c$ the crank. On six slots that is half the crank, and on twelve about a quarter. The bound is checked the direct way: at the largest flat, the chord’s half-length equals the distance from its middle to the slot mouth to 10⁻⁹, so a disc one per cent larger would open the slots.

The wheel this gives looks different from the classical one. With a flat between every pair of slots, the six-slot wheel is a hexagon with its corners at the slot mouths, not a star with scalloped arcs. The driver’s disc is small and sits close to its own axis.

The cut-away, swept again #

The measurement is the one the earlier essay made, with one line changed. The wheel’s material is everything inside its rim, clear of its slots, and short of every flat. A point fixed in the driver is swept through the index into the wheel’s frame at the angle the Geneva’s own law gives, and the cut-away is the angular extent of the driver-frame points the wheel ever occupies.

The flat’s cut-away is below half a turn at every slot count: 74° at three slots, 107° at four, 127° at six, 136° at twelve. The arc’s runs from 119° to 299°. Two things make the difference, and the second is the one worth noticing.

The first is size. A disc a quarter or half the radius spends less of the index inside the wheel’s rim, because the rim’s closest approach to the driver’s axis sits well outside a small disc at most driver angles.

The second is that, from eight slots, the flat’s cut-away is narrower than the index sweep itself: 133° against 135° at eight slots and 136° against 150° at twelve. For the arc that was impossible, because the pin drives for the whole index and the disc must be open for all of it. For the flat it is possible, because a disc this small fits inside the slot’s own gap. At the start and end of the index the slot facing the driver is the part of the wheel nearest the disc, and a slot is empty. The disc has to be open only while the wheel’s solid parts pass it, and near the ends of the index nothing solid does.

The pins come back #

The pin count follows as before. With $p$ pins the driver indexes $p$ times a turn, and the $p$ cut-aways must fit in a turn with a locking arc of the disc kept between each pair. Twenty degrees of it, as in the earlier essay, so that the disc holds over a stretch of its own rim and not at a sliver.

With an arc the disc is the binding limit at every count, and the only multi-pin Geneva is the three-slot wheel with two pins. With a flat the disc allows three pins at three slots and two at every other count to twelve. From six slots up that is exactly what the slot arithmetic allows, $p < 2n/(n-2)$, so the disc stops binding. At four and five slots the slots would allow three and the flat’s disc allows two.

So a second pin comes back at every slot count the earlier essay had taken it from. Two pins and no dwell at all found that a second pin quarters the wheel’s acceleration for the same output rate. When the index law becomes a choice found that below about eight slots a cam indexer beats a one-pin Geneva on peak acceleration. With flats, those slot counts are the ones where a second pin can now be fitted. On the arithmetic, the flat reopens the comparison the disc had closed. Whether it should depends on the hold.

An arc holds at first order #

A lock holds a wheel still against a torque. The question is how far the locked wheel can turn before the disc stops it, when the disc runs in its lock with a clearance, as every running fit does. The arc and the flat give answers of different kinds.

Take the arc first. At the locked position the concave arc is centred on the driver’s axis and the disc sits inside it, smaller by the clearance $c$. Turn the wheel through a small angle δ about its own centre and the arc’s centre moves off the driver’s axis by $a\delta$, square to the line of centres. The gap between the disc and the arc closes fastest where the arc’s own radius points most nearly along that movement, at the arc’s end, where it meets the rim at a half-span ψ from the line of centres. It closes at the rate $a\delta\sin\psi$, so the wheel can turn by

$$\delta_{\text{arc}} = \frac{c}{a \sin\psi}.$$

That is a first-order hold: the rock is proportional to the clearance. It is the property the arc that is concentric with the pivot is about, seen from the other side. An arc about the driver’s axis lets the driver turn and holds the wheel, because turning the wheel moves the arc’s centre and turning the driver does not.

A flat holds only at second order #

Now the flat. At the locked position it is a chord square to the line of centres, at distance $a - r_d$ from the driver’s axis, with the disc touching it at the foot of the perpendicular. Turn the wheel through δ and the flat turns about the wheel’s centre. Its distance from the driver’s axis becomes $a\cos\delta - h$, with $h$ the flat’s distance from the wheel’s centre. That distance does not change at first order, because the cosine’s derivative vanishes at nought. The flat moves towards the disc only as $a(1 - \cos\delta)$, so a clearance $c$ lets the wheel turn by

$$\delta_{\text{flat}} = \arccos\!\left(1 - \frac{c}{a}\right) \approx \sqrt{\frac{2c}{a}}.$$

That is a second-order hold. The flat is tangent to the circle it would have to move along, so a small turn of the wheel slides it across the disc instead of pressing it in. This is the same geometry as a mechanism that moves to first order and not at all, turned round. There, two bars in line allowed a first-order motion that stretched them at second order. Here, the lock admits a first-order turn that closes the clearance only at second order.

Both closed forms are checked against a measurement that uses neither. The wheel’s material, with rim, slots and lock together, is turned about its centre in small steps, and every point of the disc’s boundary facing it is tested for contact. The first turn at which any point of the disc is inside the wheel is found by bisection, either way round. Across three decades of clearance on a six-slot wheel, the arc’s measured rock agrees with $c/(a\sin\psi)$ to 1.3 × 10⁻⁴, with ψ = 59.8°. The flat’s agrees with $\arccos(1 - c/a)$ to 1.2 × 10⁻⁴. The slopes on log axes are 1.000 and 0.500.

The first version of the arc’s closed form was the textbook one, $c/a$ in effect, which assumes the disc can meet the arc a quarter of the way round from the line of centres. The measurement came back 1.157 times larger at every clearance. That factor is $1/\sin 59.8°$: the arc does not reach a quarter of the way round, because the rim cuts it off first, and its end is where it meets the disc.

How much each lock lets the wheel move #

The difference is large at any clearance a machine would use. On a crank of 30, a running clearance of 0.02 is a close fit. It lets a six-slot wheel locked on arcs turn by 0.022°, and one locked on flats by 1.48°, sixty-seven times as much. At the wheel’s rim, 52 from its centre, the arc allows 0.02 of movement and the flat 1.34, about as much as the clearance multiplied by seventy.

With the clearance exaggerated to 1.2 so that the geometry can be seen, the arc lets the wheel turn 1.33° and the flat 11.48°. In the close-up the arc’s end reaches the disc almost at once. The flat slides across the disc’s face, its distance from the axis barely changing, until it has turned far enough for its cosine to matter.

The angle that holds the lock met the same family of question on an escapement. A locking face cut exactly concentric with the pallet arbor has no tendency to hold itself, because the tooth’s push aims at the pivot. The flat has the matching property against the disc: at the locked position the contact normal passes through both the driver’s axis and the wheel’s centre. A torque on the wheel then has no first-order component pressing the flat into the disc, and the hold comes only from the curvature of the flat’s path.

The trade, stated as one #

A Geneva locked on flats carries the pins its slots allow, and holds its wheel to second order in its clearance. A Geneva locked on arcs holds to first order and carries one pin. Neither geometry has both, and the reason is the same fact seen twice. A lock that holds to first order needs its face to move off the disc as soon as the wheel turns, which means curving round the driver’s axis. A face that curves round the driver’s axis reaches deep into the wheel, which forces a large disc and a large cut-away.

For an indexer that has to place its output precisely, such as a film transport, a machine-tool turret or a filling line, a second-order lock is not a lock in the sense the application needs. The flat’s allowance of 1.34 at the rim, against 0.02 for the arc, is lost motion in exactly the place the Geneva is supposed to eliminate it. Backlash is an allowance described clearance as something a designer chooses to spend. Here the lock’s geometry decides how much of the output it costs, and the flat multiplies it by a large factor that grows as the clearance shrinks.

The application that made the Geneva famous shows why. Stopping thirty times a second described the projector’s intermittent: the wheel carries a sprocket, the sprocket carries the film, and the frame’s position at the projector’s aperture is the wheel’s angle times the sprocket’s radius, read at the instant the shutter opens. Nothing after the lock corrects it. A lock that lets the wheel rock by a degree and a half moves every frame by that angle at the sprocket, visibly and in a different direction each time, depending on which way the film’s tension happens to push. The same wheel on arcs moves it by a sixty-seventh as much. The second pin’s quartered acceleration would buy a quieter, faster pull-down at the cost of the one thing the mechanism is there to guarantee.

The gear with its teeth cut away met the same kind of trade from the gear side. The locking segment that holds a mutilated gear still is the material that must be absent for the driving teeth to pass, and here too the part that holds the output and the part that drives it compete for the same space. In both, what the output can do is limited by the part of the mechanism that does not move it.

This fits the earlier essay’s observation that the multi-pin Genevas that exist have three or four slots. Three is the one count where even the arc’s disc leaves room for a second pin, and four is the lowest count where a flat does. Where a wheel is held by a detent, by friction or by its own load rather than by the disc’s geometry, the flat’s weak hold matters less, and that case is outside what the geometry here can say.

What is not modelled #

Friction. A flat pressed by a disc holds by friction to first order, and nothing here computes friction. A lightly loaded flat lock can be adequate for that reason. The claim is geometric: without friction, and at a stated clearance, the flat lets the wheel rock by $\sqrt{2c/a}$.

The disc’s own shape. The driver’s locking element is taken to be a disc about its axis, cut away where the wheel passes. A flat could instead be pressed by a pad of another shape, a roller on a spring, or a second flat, and each would change both the cut-away and the hold. Only the disc is measured here.

Wear. A flat and a disc touch at a point that moves along the flat as the wheel rocks, and an arc and a disc touch along a line that does not move. How each wears, and whether a worn flat loses more hold than a worn arc, is not addressed.

Still open: a lock that is both small and first order #

The trade above is between two shapes, and the two properties it separates are not obviously bound together. First order needs a face that moves off the disc when the wheel turns. The arc does it by curving round the driver’s axis. A shallow V or a pair of short flats angled towards each other does it too, because turning the wheel then drives one face into the disc at first order, and such a V need not reach deep into the wheel.

The distinct argument there would be the V lock: two short faces meeting at a shallow angle between each pair of slots, pressed by a disc on the driver. Two measurements would decide it. The rock against clearance would show whether the slope returns to one, and at what constant, which the V’s angle should set. The cut-away and the pin count would show whether a V shallow enough to keep the disc small still gives a first-order hold. If both come out favourably, the flat’s trade was a choice between two particular shapes and not a law of the mechanism.