A steered pin keeps only the window

A curved slot moves the step bent a Geneva’s slots every way it could and found the same two numbers at the ends of every index. The wheel starts with an acceleration of $\tan(\pi/n)(1-\hat\kappa)$ and stops with $\tan(\pi/n)(1+\hat\kappa)$, so the two always add to $2\tan(\pi/n)$, and the slot’s bend decides only how the total is divided. It traced the total to one fact. The pin is on a circle, so at entry and exit it carries the crank’s centripetal acceleration square to the slot, and a slot is a single channel that the pin leaves by the mouth it came in by.

That essay closed by naming the obvious experiment. Keep the slots straight and radial, and take the pin off its circle. Carry it in a radial guide on the driver and let a stationary cam set its radius as the driver turns, so the pin follows any path the cam is cut for. This essay builds that drive, asks what the steering buys, and finds the one thing it cannot buy.

Two lines decide where the pin is #

While the pin is in a straight radial slot, the wheel’s angle is the direction of the slot. The pin is somewhere on the slot’s line, and it is also somewhere on the driver’s ray at the driver’s current angle, because the guide that carries it is radial. Two lines in a plane meet at one point. So for a wheel angle chosen at every driver angle there is exactly one place the pin can be, and the radius of that place, read off the driver’s ray, is the radius the steering cam must hold it at.

With the driver at the origin, the wheel’s centre a unit away, the driver at angle $\varphi$ and the slot pointing from the wheel’s centre at angle $\psi$, the meeting point is

$$\rho = \frac{\sin\psi}{\sin(\psi-\varphi)}, \qquad t = \frac{\sin\varphi}{\sin(\psi-\varphi)},$$

where $\rho$ is the pin’s radius on the driver and $t$ its distance from the wheel’s centre along the slot. Any law the wheel is asked to follow, written as its angle against the driver’s, becomes a radius against the driver’s angle. That radius is a cam profile.

The construction was checked in both directions. Fed the crank pin’s own law — the angle a Geneva’s wheel actually turns through as a crank of radius $\sin(\pi/n)$ carries its pin — it returns a pin radius of $\sin(\pi/n)$ at every sample, to $5\times10^{-14}$. Fed a chosen law, it returns a path, and the wheel’s angle recomputed from that path about the wheel’s centre reproduces the law to $3\times10^{-15}$ radians at every slot count tried. Twice-differenced, that recomputed angle follows the law’s own acceleration to within one per cent of its peak, ends included.

At the middle of the index the two lines coincide. The slot points straight back along the line of centres and so does the driver’s ray, and the formula reads nought over nought. The limit is short to take and worth stating, because it says something physical. The pin’s radius there is $\chi'/(\chi'+1)$, with $\chi'$ the wheel’s speed as a multiple of the driver’s. A law that moves the wheel faster through the middle carries the pin further out on the driver and so deeper into the slot.

Steered to a law with no step #

A cam indexer’s designer chooses a law that starts and ends with no acceleration, so that the output joins its dwells without a jump. Three such laws are standard, and each was given to the steered pin over exactly the window the crank pin uses, $\pi - 2\pi/n$ of driver angle for the wheel’s $2\pi/n$. The modified trapezoid holds its acceleration flat for most of each half and has a peak coefficient of 4.888. The modified sine trades some of that peak for a gentler change of acceleration and has 5.528. The cycloidal law is a single sine of acceleration, with 6.283.

On a six-slot wheel the crank pin jumps to 0.577 of the driver’s speed squared at entry, which is $\tan 30°$, climbs to a peak of 1.350, and drops from 0.577 to nought at exit. Each steered law starts at nought and ends at nought, so there is no step at either end. The modified trapezoid peaks at 1.167, fourteen per cent below the crank pin. The modified sine peaks at 1.320, two per cent below, and the cycloidal at 1.500, eleven per cent above.

The step has not been divided between the two ends, as a bent slot divides it. It has gone from the wheel altogether. The two-line construction shows where. At the mouth of a resting slot the pin is at the crank pin’s entry point, and the slot’s line there is square to the driver’s radius. A straight line square to a radius, written in polar form about the driver, has a radius whose second derivative at the foot equals the foot’s distance from the driver. So a pin running along the slot’s line has its radius accelerating outward at $a\omega^2$, where $a$ is the crank radius and $\omega$ the driver’s speed. That is precisely the centripetal acceleration the crank pin hands the wheel. The steered pin hands it to the steering cam’s follower instead, which in the measurement is 0.495 at the mouth of a six-slot wheel against a crank of 0.5. Outside the wheel the pin’s path is free, so the cam can bring that radial acceleration up gradually while the pin is still on its way to the mouth. The crank pin cannot, because its radius is a fixed length.

How little the pin has to move #

The surprising number is not the peak but the size of the steering.

For the modified trapezoid on six slots the pin’s radius stays within $-0.53\%$ and $+0.72\%$ of the crank’s. The pin is never more than 0.004 of the centre distance off the crank pin’s circle, and the steering cam’s pressure angle, the angle between the path’s normal and the guide, never passes 2.6°. The cycloidal law keeps the pin outside the circle by up to 2.1% with a pressure angle of 4.3°. The modified sine is the odd one: its acceleration changes slowly at the ends and so it moves the wheel more slowly through the middle, and the pin has to sit 6.4% inside the circle there, with a pressure angle of 7.3°.

All three meet the circle at the two mouths, because the mouth is where the slot’s line and the crank’s circle touch. Between the mouths they wander by a few per cent. A six-slot Geneva is already almost a modified trapezoid indexer. The crank pin’s law and the trapezoid’s agree in their window, their step and their speed through the middle, and the circle misses the trapezoid only by the shape of the acceleration, which the cam corrects with a radius change of about a hundredth.

That changes with the slot count. Fewer slots give the crank pin a short, violent index — a three-slot wheel peaks at 31 times the driver’s speed squared — and the steered pin has to bend the path a long way to tame it. The modified trapezoid’s cam then moves the pin 7.6% of the crank with a pressure angle of 15.5°. More slots give the crank pin a gentle index already, close to simple harmonic, and a law with a flat stretch of acceleration has to push the pin away from the circle to get it. At sixteen slots the trapezoid’s cam travel is 13.9% of the crank. Six slots sit at the bottom of that valley, where the circle is nearly the law a designer would have chosen.

The window cannot be stretched #

A cam indexer has one more freedom that the comparison in when the index law becomes a choice priced carefully: it can index over more or less of its input’s turn. Lengthening the window is the cheapest way to lower a peak there, since the peak falls as the square of the window. The steered pin looks as if it should have the same freedom, since its path is free.

It does not, and the reason is geometric. A resting slot runs from its mouth on the rim to near the wheel’s centre. Seen from the driver, that segment spans the rays from the crank pin’s entry direction, which is tangent to it at the mouth, round to the line of centres. A ray at any driver angle before the crank pin’s entry passes outside the mouth, so a pin on that ray cannot be in the slot. On a six-slot wheel a ray one degree early meets the slot’s line 0.875 from the wheel’s centre, outside the rim at 0.866; two degrees early, 0.883; ten degrees early, 0.954. At three, four, five, eight and twelve slots the same holds. So a steered pin cannot engage the wheel before the crank pin would, and by symmetry it must leave no later. Its index window is at most $\pi - 2\pi/n$, the circle’s.

Inside that window it can do anything, and outside it nothing. Of all the crank pin’s properties, the steering takes away the law, the steps, the shape of the path and the depth the pin reaches. It leaves the window, which is a property of where the slot’s mouth sits on the rim and not of the circle at all.

Where the circle is still the better law #

With the window fixed, the comparison between a steered pin and a crank pin is the comparison the earlier essay made between a cam indexer and a Geneva at the Geneva’s own index angle.

On three slots every steered law is far gentler: the trapezoid peaks at 0.297 of the crank pin, the modified sine at 0.336 and the cycloidal at 0.382. As the slot count rises the crank pin’s own law improves faster than any of them, and each crosses one where its peak coefficient equals the circle’s. The cycloidal crosses at 5.19 slots and the modified sine at 6.23, the crossovers already found for cam indexers. The modified trapezoid, not in that comparison, holds out to 8.30. Above 8.30 slots the circle’s peak is lower than any of the three.

What the steered pin keeps at every slot count is the absence of a step. On sixteen slots the crank pin’s step is 0.199 against a peak of 0.232, eighty-six per cent of the largest acceleration arriving in one jump. A steered trapezoid there peaks nine per cent higher and arrives with nothing. Which matters more is a question about the load. A film gate or an indexing table set ringing by a jump at every start cares about the step, and a heavy table at a high indexing rate cares about the peak. The circle wins one of the two at high slot counts and never wins the other.

The three crossovers fall in the order of the laws’ peak coefficients, the lowest coefficient crossing at the most slots. They are measured by bisection on a real-valued slot count, with the crank pin’s law sampled at three thousand points. The ordering is part of the check rather than a reading of the figure.

The slot can hold the wheel, at a price #

A V holds the wheel to first order ended by proposing that a pin in its slot could hold the wheel between indexes, doing without the locking disc and its cut-away. A slot’s walls have normals that miss the wheel’s centre by the pin’s own distance from it, so a pin at rest in a slot would hold to first order. The curved slot showed that a crank pin can never do it, because a pin on a circle is moving along the slot only at the instants of entry and exit, and at every other instant it is turning the wheel.

A steered pin can do it. Let the driver turn a few degrees past the crank pin’s entry while the wheel stays still. The pin is then on the resting slot’s line, and the two-line construction gives its radius on each ray as it runs inward along the slot. The wheel is held by the slot’s walls the whole time, with a hold whose lever is the pin’s distance from the wheel’s centre. Then the index begins, later than the circle’s, and at its end the pin runs out along the exit slot the same way.

The price is the window. Every degree of run at each end is a degree the index loses, so the peak rises as the square of the window that is left. On six slots, four degrees of run at each end is 3.3% of the dwell held by the slot, and it lifts the modified sine’s peak from 1.320 to 1.515, already past the crank pin’s 1.350. Twelve degrees, a tenth of the dwell, gives 2.062. Thirty degrees, a quarter of the dwell, gives 5.279, four times the crank pin. The slot can hold the wheel, but only for a few per cent of the dwell before its index is worse than the one it replaced.

What the slot gives back is a tight hold. The wheel rocks in its slot by the running clearance divided by the pin’s distance from the wheel’s centre. On a crank of 30 that is 0.0192 radians per unit of clearance at the mouth, rising to 0.0288 at the deepest point of a thirty-degree run. At a running clearance of 0.1 the slot holds the wheel to between 0.11° and 0.17°. The two-pin V lock of 14.1° that the V lock’s own measurement fitted to a six-slot wheel rocks 0.386° at the same clearance, and the flat chord in a flat holds the wheel to second order 3.31°. For the short stretches where it can be afforded the slot is the tightest lock in the field. It is not a replacement for the disc, which in any index a designer would accept must still hold the wheel for all but a few per cent of every dwell.

The handover is where a slot hold earns its place. At the instant the disc releases the wheel and the pin takes over, both are briefly in play, and every lock in the disc decides the pin count was shaped around that overlap. A degree or two of slot hold at each end gives the disc’s cut-away somewhere to begin and end that is not also the start of the index. That is a design detail rather than a measurement here, and it is named as one.

What the steering cam costs #

The steered pin turns a Geneva into something between a Geneva and a cam indexer, and the honest comparison is with a cam indexer outright. A barrel or plate cam indexer drives its output directly through followers on the cam, and it too can be cut to any law in any window. It is limited by the follower’s pressure angle and by the cam’s size, not by a slot’s geometry.

Against that, the steered Geneva keeps three things. Its slot carries the wheel’s load through a pin in a slot, at a lever of a fifth to four fifths of the centre distance. Its cam carries only the share of that load that lies along the driver’s radius, which is nought at the middle of the index where the slot and the ray coincide, and it accelerates the pin and its guide rather than the wheel. How large that share is through the rest of the index is not computed here. And the cam’s demands are small where Genevas are usually built. On six slots it moves the pin 1.2% of the crank at 2.6°, on five 3.2% at 5.0°, on eight 4.5% at 3.7°.

What it adds is a second higher pair, a stationary cam whose follower rides on the turning driver. That follower sees the pin’s radial acceleration plus the driver’s centripetal and Coriolis terms, and it is lubricated and worn like any other cam contact. Neither the follower’s own dynamics nor its wear is computed here. The ledger also shows the trade turning: the steering is smallest exactly where the circle is almost the right law, and grows at the slot counts where the circle no longer needs help on its peak.

Two further readings are in the last column. A steered pin at three slots comes no nearer the wheel’s centre than 0.200 of the centre distance, where the crank pin reaches 0.134. At eight slots and above it goes slightly deeper than the crank pin, because the trapezoid moves the wheel faster through the middle, and the radius formula carries a faster wheel deeper. Slot depth is wheel material, and the disc’s essay found material to be what limits a Geneva. The steered pin spends a little of it at high counts and saves some at low.

What the steering does not settle #

The cam’s own dynamics. The steering cam’s follower is the pin and its guide, a small mass on a turning driver. Its radial acceleration in the driver is computed, and at the mouth it equals the crank radius times the driver’s speed squared. The follower’s spring or groove, its contact force under the driver’s centripetal load and its own wear are not computed.

The path outside the wheel. Between the exit of one index and the entry of the next the pin’s radius is free, and the essay assumes it is brought from the exit’s radial acceleration to the next entry’s smoothly. Any smooth path that stays outside the rim will do, but no particular one is drawn or priced.

Laws chosen for a window. The three laws are the standard ones, used as an indexer’s designer would use them. The least peak with no step is not a law but a limit: the constant-acceleration law has coefficient 4 and steps at both ends, and any law without steps lies above 4. The trapezoid’s 4.888 is one practical point on that approach, not the end of it.

Loads and play. Everything is kinematic, in units of the driver’s speed squared. The pin’s clearance in its slot enters only in the hold, and the running clearance in the guide and at the cam contact is not modelled.

Still open: what a second pin does with a free path #

Every finding here is for one pin. Two pins and no dwell at all found that more pins on a crank are bounded by an equation in integers, since their circles’ windows must not overlap, and the disc decides the pin count that the locking disc binds before the windows do. A steered driver has a pin guide for each pin, each with its own path. The windows are still the circle’s, so their arithmetic is unchanged, but nothing requires two pins on one driver to follow the same path.

The distinct argument would be a two-pin driver whose pins are steered to different laws, or one pin steered to hold the wheel while the other indexes it. The measurement would take a four- and a six-slot wheel, give each pin a window and a run, and ask whether a pair of steered pins can hold the wheel through the whole dwell by handing it from slot to slot, with no disc at all. The rock would be the slot’s clearance over the pin’s distance throughout, and the price the two indexes’ shortened windows. Whether a wheel held only by its slots can index at a peak a real machine would accept is the question, and the run measurement here says it will be close.