The disc decides the pin count

When the index law becomes a choice compared a Geneva against a cam indexer given the same step and the same index angle, and found the crossovers: a cycloidal cam has the lower peak acceleration below 5.19 slots, a modified sine below 6.23, simple harmonic motion below 8.06. Above those the Geneva wins, and what it can never escape is the step at entry — an acceleration of exactly $\tan(\pi/n)$ at the instant the pin arrives, never nought.

All of that is about the pin and the slot. A Geneva has another half, and none of the comparison mentions it. Between one index and the next the wheel has to be held, and it is held by a disc on the driver sitting inside a concave arc cut into the wheel’s rim. The disc and the wheel occupy the same region near the line of centres, so the disc has to be cut away wherever the wheel is ever there while the pin is driving — and what is left has to be enough to lock with.

That is a second inequality on the pin count and the field did not have it. With $p$ pins the driver has $2\pi/p$ of its own turn per index; the cut-aways take what they take and the locking arcs need the rest. Two pins and no dwell at all is the limit of the condition the field already had, and it is about the slots. Whether it bites before the slot arithmetic does is the question, and it is not a question a formula answers, because the cut-away is a region and not an angle until somebody measures it.

The wheel, drawn as what is left of it #

The measurement needs the wheel’s material rather than its outline, so it is worth being exact about what a Geneva wheel is.

In the wheel’s own frame, with its origin at the wheel’s centre and the driver’s axis at distance $a = c/\sin\beta$ with $\beta = \pi/n$: the rim is the circle of radius $r_w = a\cos\beta$, which is where the slot mouths open and is fixed by the requirement that the pin enter along the slot, since an entry at any other angle is a step in velocity rather than in acceleration. There are $n$ radial slots, and there are $n$ concave locking arcs of radius $r_d$ centred at distance $a$ from the wheel’s centre, one between each pair of slots, placed so that one of them is centred on the driver’s axis at each locked position. Material is inside the rim, outside every arc, and clear of every slot.

The disc’s radius $r_d$ is the one free number, and the bound on it is not the one it looks like. Writing it down as $a - r_w$ — the rim’s nearest approach to the driver’s axis, on the reasoning that a bigger disc would foul the wheel — is wrong by a factor of four and makes the whole model inert: at six slots it gives 8.04 against a crank of 30, the disc never reaches the wheel at all, and the cut-away comes out as nought degrees at every slot count. The arc is cut into the rim, so the rim does not bound the disc. The slot does. At a locked position the arc is centred on the driver’s axis, and the nearest slot runs radially at $\beta$ from the line of centres; the perpendicular distance from the axis to that slot is $a\sin\beta$, which is the crank radius. An arc deeper than the crank radius would cut the slot open and the wheel would lose its own drive.

So the disc may be anything up to the crank radius, which is a large fraction of the machine, and that is why a real Geneva’s locking disc looks the size it does. It is also why the disc is a body worth computing rather than drawing, which is the habit a ratchet with no teeth needed for its wedge and for the same reason.

The cut-away is a sweep, not an angle #

With the wheel’s material defined, the cut-away is a question about two bodies and is answered the way such questions are answered here.

A point fixed in the driver at polar coordinates $(\rho, \psi)$ sits, at driver angle $\alpha$, at fixed-frame angle $\psi + \alpha$. It has to be absent from the disc if the wheel is ever there while the pin is driving. So the cut-away is the swept image of the wheel in the driver’s own frame, taken over the whole index sweep — and its angular extent is what a pin costs the disc.

Nothing in that derives an angle. It samples the driver’s disc on a polar grid, runs the index, maps each grid point into the wheel’s frame at the wheel angle the Geneva’s own law gives, and asks whether the wheel is there. What comes back is a set of directions, and the width of the largest run of them is the answer.

The answer is larger than the index sweep at every slot count, and by a great deal. A three-slot wheel indexes over 60° of driver and needs 119° cut away. A six-slot wheel indexes over 120° and needs 236°. A twelve-slot wheel indexes over 150° and needs 299°, which is five sixths of a whole turn for one pin.

The reason is not subtle once it is measured. The pin leaves the slot when the wheel has finished turning, but the wheel’s rim is still where it ended up, and where it ended up is inside the circle the disc wants to occupy until the wheel has carried its next locking arc round to the axis. The disc cannot be there while the rim is. The index sweep is how long the pin drives; the cut-away is how long the wheel is in the way, and the second is the longer.

What the cut-away is made of #

The cut-away is wider than the index and it is worth seeing where the extra comes from, because the two contributions behave differently as the slot count rises.

The first part is the index itself. While the pin is in the slot the wheel is turning through $2\beta$, and every position it passes through has to be clear of the disc. That contributes the index sweep $\pi - 2\beta$ of driver angle, and it is the part the slot arithmetic already accounts for.

The second part is the approach and the departure, and it has nothing to do with the pin. Before the pin arrives, the wheel is sitting at its locked position with a locking arc centred on the driver’s axis — and the next arc along, the one that will take over after the index, is still well away from the axis. As the driver turns towards entry, the rim between those two arcs swings across the region the disc wants. It is already in the way before the pin reaches the slot mouth, and the mirror image is true after the pin leaves.

That second part is what makes the measured widths so large, and it grows with the slot count for a reason that is visible in the geometry. A wheel of many slots has a rim close to the driver’s axis — $a - r_w = a(1 - \cos\beta)$ falls as $n$ rises — so more of its rim spends more of the turn inside the disc’s circle. At three slots the rim’s nearest approach is 17.3 on a centre distance of 34.6; at twelve slots it is 3.9 on 115.9. The wheel of many slots is the one whose rim is almost touching the driver all the time, and a disc has almost nowhere to be.

Two inequalities, and the one that bites #

Now the two conditions can be put side by side, which is the whole point of measuring the second one.

The slot condition is the field’s own. With $p$ pins the wheel indexes $p$ times per turn of the driver, each index consumes $(n-2)/2n$ of that turn, and the indexes must not overlap: $p(n-2)/2n < 1$, so $p < 2n/(n-2)$. It allows five pins at three slots, three at four and five, and two from six slots upward.

The disc condition is that $p$ cut-aways must fit in a turn with something left between them. A disc reduced to slivers is not a disc that holds anything, so the bare $p \times \text{width} \le 2\pi$ is too generous; requiring twenty degrees of locking arc at each index as well gives the honest number. It allows two pins at three slots and one pin at every count from four upward.

So the disc is the binding condition at every slot count tried, and strictly so at every one. The largest pin count a buildable Geneva can carry is the disc’s number, and from four slots it is one.

That is a design conclusion and it matches what gets built. Multi-pin Genevas are rare, and the ones that exist have three or four slots. The usual explanation is that more pins shorten the dwell until there is none left, which is the slot arithmetic, and the slot arithmetic says a six-slot wheel takes two pins perfectly well. It does not, and the reason is on the other side of the machine. A mechanism whose two halves answer to different inequalities is one where the binding condition can sit where nobody is looking — the same shape as a dwell made from a curve, where what decides the dwell is not the quantity the dwell is specified by.

A better lock is a bigger cut #

The one free number is the disc’s radius, and sweeping it shows why the inequality cannot simply be designed around.

At 30% of the crank radius the cut-away is 166°; at 90% it is 236°. The disc’s whole job is to hold the wheel, and a bigger disc holds it better — over a longer arc of contact and at a shallower angle, so a given load on the wheel presses the arc against the disc rather than trying to lever it off. The same size is exactly what makes it collide with the wheel for longer.

There is no setting that escapes both. A disc small enough to need little cut-away is a disc that barely reaches the wheel; a disc large enough to lock properly needs two thirds of the turn removed. And the axis stops at one, because past the crank radius the arc eats the slot.

The trade has the shape this field keeps meeting: a quantity that is good for one half of a mechanism and is the same quantity that is bad for the other, with no third parameter to separate them. The gear with its teeth cut away meets it at the same joint — the locking segment that holds a mutilated gear still is the material that has to be absent for the driving teeth to pass.

The question that was left open, closed #

The comparison ended by asking whether a multi-pin Geneva could beat a cam at a slot count where a one-pin Geneva cannot, and noted that the answer depended on how far the locking disc let the pin count rise.

It lets it rise to two, at three slots, and nowhere else.

That collapses the question rather than answering it in the expected direction. The slot counts where a Geneva loses to a cam are three to eight — below 5.19 for a cycloidal cam, below 6.23 for a modified sine, below 8.06 for simple harmonic motion. Those are exactly the counts where a second pin might have been asked to help. The disc refuses one at four, five, six and eight, so the family of cases to check is not a family. It is a single machine: a three-slot wheel with two pins.

And the reason is worth separating from the arithmetic that was suspected of it. The slot condition would happily allow three pins at four slots and two at six; what forbids them is a region swept by a rim, which no count of indexes per turn can see. An escapement that could not alternate failed for the same kind of reason: a count that was right and a geometry that was not. The pin count is limited by the half of the mechanism that does not move the wheel.

What is not modelled #

The lock is geometric. Whether a given arc of contact actually holds the wheel against a given load is a statics question with friction and stiffness in it, and none of those appears here. The twenty degrees of locking arc required at each index is a stated convention rather than a computed requirement, and the pin limits move with it: at ten degrees the six-slot wheel still takes one pin, and at forty the three-slot wheel loses its second.

There is no clearance anywhere. The disc, the arcs and the slots are at their nominal sizes, so the cut-away measured is the smallest that could possibly work. A real mechanism cuts further back for a running fit, which makes the disc’s inequality bite harder and never softer.

The wheel is drawn from three rules. Rim, arcs and slots, with the slots given the pin’s own half-width and no entry radius. A real wheel has a relieved slot mouth and a chamfer — the same allowance a tooth that lives on a sphere needs at its own ends — both of which remove material near the rim — which is where the cut-away is decided — and both of which would reduce it a little.

One crank radius carries the sweep. The disc’s radius is measured as a share of the crank radius, and the cut-away’s angular width is scale-free in the sense that doubling the whole machine changes nothing; what is not checked is a Geneva whose pin is a different fraction of its centre distance from the tangential-entry value, which is not a Geneva in this field’s sense but is built.

The cut-away is one connected run. Its width is taken as the largest run of blocked directions, which is right for a single-pin driver and is an assumption for a multi-pin one, where two pins’ cut-aways could in principle merge into one. The pin limits quoted are therefore upper bounds on what the disc allows.

The grid is a grid. Directions are sampled 480 to a turn and radii sixteen deep, so a cut-away is resolved to three quarters of a degree. The pin limits are integers and none of them is within a degree of changing.

Still open: the wheel that locks on a flat #

Everything above gives the wheel a concave arc of the disc’s own radius, which is the classical construction and the reason the disc can be as large as the crank radius. It is not the only way to hold a wheel still.

Its distinct argument would be the same measurement for a wheel locked on a flat — a chord across the rim between each pair of slots, held by a driver carrying a pad rather than a disc. A flat removes far less material from the wheel, so the slot mouths survive a larger lock; but a pad has to approach the flat rather than rotate into an arc, so what it sweeps through the driver’s frame is a different region and possibly a smaller one. Two things would come out of it. Whether the cut-away falls enough to let a second pin into a six-slot wheel, which would put the multi-pin question back into a family; and whether the flat’s own hold survives the comparison, since a pad on a chord resists a torque by friction where an arc in a disc resists it by shape, and what is computed here is shape and not friction.