A wheel held by its slots needs three pins

A steered pin keeps only the window took a Geneva’s pin off its circle. Carried in a radial guide on the driver, with a stationary cam setting its radius, the pin is wherever the driver’s ray meets the slot’s line. The wheel can then be given any index law at all, with no step at entry or exit. What the steering cannot change is the window. A resting slot, seen from the driver, spans only the rays from the crank pin’s entry direction round to the line of centres, so no pin can reach the wheel earlier than the crank pin does or leave it later.

That essay found one more thing the steered pin can do. It can run along a resting slot, holding the wheel by the slot’s walls with a lever as long as the pin’s distance from the wheel’s centre. That is tighter than the locking disc, the flat or the V. Every degree of run, though, came out of the index, and a single pin could afford only a few per cent of the dwell before its index was worse than the crank pin’s. The essay’s closing question was whether a second pin could take the rest. One pin might hold the wheel while the other indexes it, the two handing the wheel from slot to slot through the whole dwell, with no disc at all.

Two pins cannot, at any slot count, and the reason is the window again. Three can.

A pin is in a slot for exactly the window #

Follow one steered pin through a turn of the driver. It reaches the wheel at the entry mouth of a resting slot, on the ray the crank pin would arrive on. From there it is in a slot until its ray reaches the exit mouth of the slot it has moved, on the ray the crank pin would leave on. In between it can do three things in any proportion. It can run inward along the resting slot while the wheel stays still, index the wheel, and run outward along the moved slot while the wheel stays still again.

What it cannot do is leave early. A pin is on its driver’s ray, and while it is in a slot it is also on the slot’s line, so its position is fixed by the driver’s angle and the wheel’s. It can only leave by a mouth, and the ray reaches the exit mouth at one angle only. Nor can it arrive late or leave early by any other route. The window bound says no ray meets a resting slot before the entry ray, and by symmetry none after the exit ray. Every pin is therefore in some slot for exactly $W = 180° - 360°/n$ of every turn: 120° on six slots, 90° on four. It spends no more and no less, whatever its steering cam does.

That was checked directly rather than taken from the formula. Every driver angle’s ray, at 7,200 angles round the turn, was tested against both resting slots of the wheel, and a ray counts only if it meets a slot’s line between the mouth and the wheel’s centre. The rays that reach a slot form a single arc of $W$ at every slot count from three to twelve: two pins half a turn apart leave exactly $360°/n$ of each half turn uncovered, to the size of the test’s step.

Two windows cannot cover a turn #

For slots alone to hold the wheel, some pin must be in some slot at every moment of the driver’s turn, indexing or holding. With $p$ pins each in a slot for $W$, the windows can cover the turn only if together they are at least a turn long:

$$pW \geq 360°, \qquad p \geq \frac{2n}{n - 2}.$$

The window approaches 180° as the slot count grows and never reaches it, so two pins never cover a turn. Two pins half a turn apart leave a gap of $360°/n$ every half turn, 90° on four slots, 60° on six, 30° on twelve. Spacing them unevenly cannot help, since two arcs shorter than a half can never cover a whole. The ray test finds the same gaps to the size of its step.

Three pins suffice from six slots up, with nothing to spare at six. Four are needed on four slots and on five, and six on three. On six, four and three slots the count is exact. The pins’ windows meet end to end, and the hold passes from one pin to the next at a single instant, when one pin is at the exit mouth of the slot it has just moved and the next is at the entry mouth of the next slot. One more pin than the least buys an overlap of $W - 360°/p$, during which two slots hold the wheel together: 30° of every 90° with four pins on six slots, 18° of every 72° with five on four.

Two pins and no dwell at all found that more crank pins on one driver are bounded by an equation in integers, because their circles’ windows must not overlap. The steered pins are bounded the other way: their windows must overlap, or at least touch, for the slots to take the disc’s place. The same window sets both bounds.

The drives, cycle by cycle #

A drive with $p$ pins indexes its wheel $p$ times a turn, once for each pin, so its cycle is $360°/p$ of driver. For a fair comparison with an ordinary Geneva, each drive was given the Geneva’s own share of index to dwell. A six-slot Geneva indexes for a third of its cycle and dwells for two thirds; a four-slot one indexes for a quarter. So three pins on six slots index for 40° of each 120° cycle, and each pin runs 40° in along its slot before indexing and 40° out after. Four pins on six slots index for 30° of each 90°, with runs of 45°. Four pins on four slots index for 22.5° of 90°, with runs of 33.8°, and five pins for 18° of 72°, with runs of 36°.

The timelines were sampled at 2,400 points a cycle. At every instant the wheel is either being indexed by exactly one pin or held by at least one slot. No instant has two pins indexing, a pin indexing while another holds a resting slot, or nothing holding the wheel. In the drives with a spare pin, two slots hold together for exactly the overlap the window arithmetic predicts.

The index’s peak acceleration is no worse than the crank pin’s. Each index is a steered modified sine over a window a $p$-th as long as the circle’s, so in the driver’s own speed squared its peak is $p^2$ times the single steered pin’s: 11.9 against 1.32 on six slots with three pins. But the driver turns $p$ times more slowly for the same rate of indexing, and in the cycle’s own time the factor cancels exactly. The six-slot drive peaks at 1.320 against a crank pin’s 1.350, and the four-slot drive at 3.52 against 5.41. Both start and stop with no step, as every steered law does.

The disc’s work goes to the steering cam #

So the handover costs nothing at the wheel. It costs a great deal at the steering cam, and one fact of geometry makes the cost easy to state.

At the entry mouth a resting slot’s line meets the driver’s ray square. That is why the crank pin enters tangentially, and why a curved slot moves the step found the crank pin’s centripetal acceleration landing square to the slot there. A point that runs along a line from the place where a ray meets it square is, by the time the ray has turned through an angle $r$, at a place where the line crosses the ray at $90° - r$. So the pin’s radius on its guide grows at $\tan r$ of itself per radian of driver, which is the definition of a pressure angle of $r$. On a run, the steering cam’s pressure angle is the run’s own angle from the mouth, exactly. The measurement agrees to within a tenth of a degree on every drive and share, and on the slot’s line the two are equal to the last digit. The index adds a fraction of a degree on top.

At the Geneva’s own share the runs are long, so the cam is steep: 40.3° for three pins on six slots, 45.2° for four. On four slots it is 33.8° with four pins and 36.0° with five. A cam designer usually holds a pressure angle to about 30°, and a lift is a size and a law is a shape found that the angle is what a cam’s size buys down. A stationary steering cam has no such freedom here: its angle on a run is set by the run, whatever its size. Holding the runs to 30° means making the index longer and the dwell shorter. The run is half of what the window leaves after the index, so a 30° run on six slots needs the index to take at least 60° of the 120° cycle. That leaves half the cycle as dwell, where the Geneva itself dwells for two thirds, and the index’s own contribution then lifts the cam to about 30.5°. Four slots with four pins can keep about two thirds of the cycle as dwell, where the Geneva keeps three quarters. A spare pin makes this worse, because the window stays the same while the cycle shortens, and the runs lengthen to fill it: four pins on six slots need the index to take 70% of the cycle before the cam comes under 30°.

So the overlap that makes a handover robust is bought with pressure angle. The exact drives hand over at an instant, with nothing to absorb an error in where a pin is mounted. The drives with a spare pin have a margin of 30° or 18°, and pay for it with a steeper cam or a shorter dwell.

The slot is cut deeper #

The second cost is in the wheel. A short index moves the wheel fast compared with the driver, and the earlier essay found that the pin’s radius at the middle of an index is $\chi'/(\chi'+1)$, with $\chi'$ the wheel’s speed over the driver’s. A fast wheel puts the pin far out on the driver and so deep in the slot. The six-slot drive with three pins takes the pin to 0.275 of the centre distance from the wheel’s centre, where a crank pin goes no deeper than 0.500. With four pins it goes to 0.221. On four slots it reaches 0.124 against the crank pin’s 0.293.

The disc decides the pin count found that wheel material is what limits a Geneva: the hub must carry the shaft, the slots must leave room for it, and the disc’s cut-away takes room from the driver. The disc-free drive gives back the driver’s room, since there is no disc to cut away, and takes more of the wheel’s. On six slots its slots must reach 0.591 of the centre distance in from the rim, where a crank pin’s reach 0.366, 1.6 times as far into the wheel. Lengthening the index helps here too. At an index share of a half the three-pin, six-slot drive’s pin reaches only 0.362.

Held tighter than a V lock #

What the slots give in return is the hold. A wheel held by a pin in a slot rocks by the running clearance divided by the pin’s distance from the wheel’s centre, so the hold is tightest when the pin is at the mouth and loosest at the deepest point of its run. With three pins on six slots, a crank of 30 and a clearance of 0.1, the rock runs from 0.110° at the mouths to 0.214° where each run turns into an index. A V holds the wheel to first order fitted a two-pin V lock to the same wheel and found 0.386° at the same clearance. A flat holds the wheel to second order found 3.31° for the flat chord. The slots are tighter than either at every instant of the dwell. With four pins the worst is 0.260°, still under the V. On four slots the rock is larger, 0.574° at worst with four pins, because the runs go deeper.

A slot’s hold is also a different kind of hold from a disc’s. A disc holds the wheel by the rim, through a concave arc on the wheel bearing on the disc, and it must be cut away wherever a pin passes. A slot holds it by the same walls that index it, so there is nothing to cut away and nothing to hand over between a lock and a drive, only between one pin and the next.

The ledger #

Read across, three pins on six slots is the drive the earlier essay hoped for, with one large qualification. It indexes on a law with no step, 2% under the crank pin’s peak, and holds the wheel through the whole dwell tighter than a V lock, with no disc and no cut-away. Its steering cam works at 40°, its slots go 1.6 times as far into the wheel, and its handovers happen at an instant with no margin. Each of those can be eased by giving up dwell. A designer who accepts a dwell of half the cycle instead of two thirds gets a 30° cam, a slot 0.362 deep and a peak, now spread over the longer index, of 0.587 in the cycle’s own time.

When the index law becomes a choice priced a cam indexer against a Geneva and found that a cam indexer can index over any share of its input’s turn. The disc-free drive sits between the two. Like a cam indexer it trades dwell for gentleness. Like a Geneva it drives the wheel through pins in slots, and its slots lock it. It has a stationary steering cam for every pin rather than a cam on the wheel’s own shaft.

What the slots do not settle #

The steering cam’s forces. Pressure angles are kinematic. A 40° cam driving a pin and its guide against the driver’s centripetal and Coriolis terms carries side loads that a 30° one would not, and neither the forces nor the guide’s friction is computed. Three pins need three guides and three followers on one stationary cam track, and the track’s shape between windows, where the pins run clear of the wheel, is free and not drawn.

The handover instant. In the exact drives the hold passes from one pin to the next at the instant both sit at their mouths. A pin mounted a fraction of a degree late leaves the wheel unheld for that fraction, and a pin mounted early enters a slot that the other pin still holds, which is harmless. The tolerance a real drive could accept, and whether a little mouth chamfer would give it, is not modelled.

Impact at entry. A steered pin enters its slot with no acceleration step on the wheel. It still meets the slot’s wall with whatever relative velocity its free path brings it in at, and that path is not designed here.

Wear. A slot that holds the wheel for two thirds of every cycle sees its pins sliding along its walls under the wheel’s holding load for all of that time. A locking disc’s arc sees the same duty and is replaced; a wheel’s slots are not.

Still open: two pins and a disc that holds only the gap #

Two pins leave a gap of $360°/n$ in every half turn, 60° on six slots, and three pins pay for closing it with a steep cam. A locking disc need not hold the whole dwell. A disc that held the wheel only through the gap, a short arc where two pins cannot reach, would have a much shorter cut-away.

The distinct argument would be that hybrid: two steered pins running as far along their slots as a 30° cam allows, and a partial locking disc covering only what they leave. The measurement would find the arc the disc must span on four- and six-slot wheels, and ask whether a disc that short still binds the pin count, as the disc decides the pin count found the whole disc does. It would also ask how the disc’s handovers to the pins compare with the slots’ handovers to each other. If a short arc is enough, two steered pins and a sliver of disc may be the practical form of a disc-free drive, with the disc doing only what the window will not let the slots do.