Six things an indexer is not

The mechanism that waits opened a field of machines that are still for most of a turn and moving for the rest: ratchets, escapements, Geneva wheels, cam indexers. They advance film, count time, index machine tables and hold hoists, and they are old enough to have gathered a body of things people say about them. Six of those things are collected here. Each is what a careful designer might conclude from one part of the mechanism, and each has been measured in the timing essays and found false, incomplete or misapplied.

Six things a cam drive is not found that every claim about cams failed in one way, by reading the drive at first order when it is decided at second. The claims below also fail in one way, and it is a different one.

One: more pins make a Geneva smoother #

Put a second pin on a Geneva’s crank, opposite the first, and the wheel indexes twice a turn. Asked to index at the same rate as before, the driver can turn at half the speed, and since every acceleration in a mechanism scales as the square of its input speed, the wheel’s peak falls to a quarter. Three pins give a ninth. A designer who wants a gentler wheel adds pins.

Two pins and no dwell at all measured the exchange and it is real: on three slots, 1.000 at one pin, 0.250 at two, 0.111 at three, 0.063 at four. But two things travel with it. The first is the dwell. Each pin’s index takes the same share of the driver’s turn, so the share of the cycle spent at rest falls in a straight line, from 83.3% to 16.7%. More pins are not a smoother version of the same machine. They are a machine that rests less. The second is that nothing has happened to the index itself. Each pin enters its slot exactly as the single pin did, with the step of $\tan(\pi/n)$ times the driver’s speed squared, and the step falls by the same square as the peak. Measured against the motion’s own peak, the jump at every start is exactly as large as before.

And the machine may not exist. The disc decides the pin count found that the arithmetic of the slots, which allows two pins on six slots and up to twelve, is not the binding condition. The wheel must be held between indexes by a disc on the driver sitting in a concave arc cut into its rim, and that disc has to be cut away wherever the wheel swings through it. The cut-away is wider than the index at every slot count. Counted with a locking arc of twenty degrees each, the disc allows two pins on three slots and one pin from four slots upward. A wheel locked the usual way cannot carry the second pin at all. A flat holds the wheel to second order got it back by changing the lock, which is the third claim’s subject.

The flat chord’s disc can be no larger than the crank times $\sin(\pi/n)$, and its cut-away falls to between 107° and 136° of the driver’s turn. That lets two pins in at every count from four slots to twelve, and three at three slots, so from six slots up the slot arithmetic and the disc agree. The second pin is bought with the lock’s order of hold, which falls from first to second. Every claim in this essay is tied to the others in the same way. A designer who adds a pin for smoothness has also chosen how still the wheel stays between indexes, without having asked about it.

So the verdict is misapplied. More pins lower the peak for a given rate, at the cost of the dwell, with the step unchanged in proportion, and only after the lock has been redesigned to let them in.

Two: a finer ratchet is a more precise one #

A ratchet’s output can stop only where its pawl drops into a tooth. Cut more teeth and there are more places to stop, so the ratchet seems to hold the load closer to where it was released. The fix for a ratchet that slips back too far is a finer tooth.

The resolution is the pitch found that a ratchet’s step and its error are the same number. On 24 teeth the pitch is 15°, and an input moved by up to 15° can leave the output where it was. The number is the same on a perfectly made ratchet, which is what separates it from the play a measurement of a worn one would find. Two pawls offset by half a pitch halve it and three divide it by three, since whichever pawl is over a root drops first. So far the claim is merely incomplete: finer teeth do help, in proportion.

The trouble is how far they can go. A tooth needs a depth to hold a pawl, and with a working tooth depth of 0.16 of the wheel’s radius the tooth profile stops closing at twenty-nine teeth: there is no room on the rim for a thirtieth tooth of that depth. The wall is geometric, not a matter of workmanship, and it moves with the depth. Halve the depth and the profile closes up to sixty-four teeth; deepen it to a quarter of the radius and it stops at sixteen. A shallower tooth is finer and gives its pawl a smaller face to hold on, and how much hold the pawl keeps is a question about its lines of action, which the depth alone does not settle. At 0.16 of the radius the finest ratchet’s pitch is 12.4°, and three pawls on 24 teeth already do better, at 5°. Past the wall, precision comes from more pawls, which divide the pitch by a whole number, or from no teeth at all. A ratchet with no teeth measured a roller clutch whose lost motion is a clearance divided by the wedge’s slope, 1.4° against the 24-tooth ratchet’s 15°. That is a length, not a fraction of a pitch, and adding rollers does not divide it.

The verdict is incomplete. A finer ratchet is more precise until the tooth runs out of room, and the precision a ratchet can reach is decided by its pawls and its tooth depth, not by how finely anyone is willing to cut.

Three: a locked wheel is a still wheel #

Between indexes a Geneva’s wheel is locked: a disc on the driver sits against a face on the wheel and prevents it from turning. The wheel is then in its indexed position until the next pin arrives, which is the whole point of an indexer.

The disc cannot touch the face exactly, because something has to run. There is a running clearance between them, and through it the wheel can turn until its material meets the disc. How far depends on the shape of the face, and the two shapes give different orders.

The classical concave arc lets the wheel rock in proportion to the clearance, by $c/(a\sin\psi)$ with $\psi$ the arc’s half-span. The flat chord, which shrinks the disc enough to let a second pin in, lets it rock as the square root, by $\operatorname{acos}(1 - c/a)$, close to $\sqrt{2c/a}$. At a clearance of 0.02 on a crank of 30, the arc holds the wheel to 0.022° and the flat to 1.48°, sixty-seven times as much. Both were measured by turning the wheel’s actual material against the disc and finding the first contact, not by the formulas, and the measured slopes on log axes are 1.000 and 0.500. A V holds the wheel to first order found the lock that keeps both the small disc and the first-order hold, and it still rocks: 0.386° at a clearance of 0.1.

Measured against what the wheel is for, the numbers are not small. A six-slot wheel indexes in steps of 60°, and a flat-locked wheel at a clearance of 0.02 sits anywhere within 1.48° of its station, two and a half per cent of a step. A machine that drills a hole in each station of an indexing table, or a projector that registers a film frame, feels that as positional error. A clearance ten times larger, which is an ordinary running fit on a worn machine, makes it 4.7°.

So the claim is false. A locked wheel is a wheel held within an angle, and the angle depends on the shape of the lock’s face to first or second order in the clearance. A designer choosing a lock is choosing that order, usually without being told so.

Four: a curved slot gives a smooth start #

A Geneva’s wheel starts with a jump in acceleration of $\tan(\pi/n)$ times the driver’s speed squared, because at tangential entry the pin’s centripetal acceleration is square to the slot and all of it goes to the wheel. A slot whose mouth bends with the pin’s circle takes some of that acceleration up in its own curvature, and the wheel starts more gently. Curved-slot Genevas are drawn in mechanism catalogues for exactly this reason.

A curved slot moves the step confirmed the start and found the finish. The pin leaves by the mouth it entered, and there the crank’s circle bends the other way relative to the slot, so the curvature that cancelled part of the entry step adds the same amount to the exit step. Solved at four slot counts and seven curvatures, the steps are $\tan(\pi/n)(1 \mp \hat\kappa)$ to $3\times10^{-6}$ of $\tan(\pi/n)$, and they always sum to $2\tan(\pi/n)$. A slot bent exactly to the pin’s circle at its mouth enters with no step and leaves with twice the straight slot’s. And every bend raises the peak: a six-slot slot bent to half the circle’s curvature peaks 22% higher, and of 246 slots straight at the mouth and bent inside, none peaked below the straight slot.

The verdict is misapplied. A curved slot does give a smooth start, and it pays for it at the finish, with a higher peak in between. Removing the step from both ends needs the pin taken off its circle, which a steered pin keeps only the window did.

Five: a cam indexer is always gentler than a Geneva #

A Geneva’s motion law is forced by its slot count. A cam indexer’s is chosen, and indexers are cut to laws designed for low acceleration and no jumps. A law designed for the job should beat a law that is an accident of geometry, so on peak acceleration the cam should win.

When the index law becomes a choice gave the cam the Geneva’s own index angle and step, which is the fair comparison, and found that the cam wins on peak only below a slot count that depends on its law: 5.19 for cycloidal and 6.23 for modified sine. The modified trapezoid, with a flatter acceleration, holds out to 8.30. Above 8.30 slots the Geneva’s peak is lower than all three. Measured in the unit that puts every law on one scale, the step over the index angle squared, the Geneva’s peak coefficient falls steadily as its slots multiply, from 5.653 at six slots downward, and it passes below each cam law’s fixed coefficient in turn. Simple harmonic motion, with $\pi^2/2 = 4.935$, is passed at 8.06 slots. Only the constant-acceleration law, with 4, stays below the Geneva at every count, and it does so by jumping from nought to its full acceleration at both ends, which is the very thing a cam indexer was chosen to avoid.

What the cam always removes is the step. On sixteen slots the Geneva’s entry step is 0.199 of the driver’s speed squared against a peak of 0.232, so most of its largest acceleration arrives at once. The cam’s arrives gradually and peaks nine per cent higher. The verdict is misapplied: a cam indexer is gentler at the instant of entry at every slot count and gentler at its peak only at low ones. A designer who compares peaks and forgets the step will choose a Geneva at high counts and be surprised by the noise.

Six: if every state is in one piece, every state can be reached #

Where the input stops deciding and the configuration-space essays judged whether a mechanism can go from one position to another by asking whether the two are in the same connected piece of its free space. A four-bar that cannot be reassembled without being taken apart has two pieces, and nothing moves it from one to the other. A mechanism with a single piece seems to be one in which anything can get anywhere.

One piece, and still not reachable found the case that breaks it. A cable tie’s free space is the whole interval of how far it is pulled in, with no barrier anywhere: one piece. But it can only be pulled tighter. From any state the reachable set is everything forward and, backward, only as far as the tooth the pawl has dropped into. Of all ordered pairs of states 52.1% can be joined by an allowed motion, and 4.2% both ways. Connectivity is symmetric, because a path run backwards is a path. Reachability is an order, because a one-way mechanism’s paths cannot be run backwards.

The same fact appears in the escapement, from the other side. Where the input stops deciding found that one pallet angle occurs at three different wheel angles in a single period, and the three are not three roots of one equation but three different contacts, each with its own equation. A ratchet’s state needs to know which tooth the pawl is in, and an escapement’s which face of which pallet the tooth is against. Neither can be read off a position.

The claim is false. A component count answers whether two states are joined by any path. A one-way mechanism asks which way the path is allowed to go, and the question cannot be answered from the free space’s shape at all.

The way they fail #

Each claim judges the mechanism by its motion. More pins are judged by the peak, the ratchet by its teeth per turn, the locked wheel by its locked position, the curved slot by its start, the cam law by its peak, the cable tie by its free space. In each the motion is right, and in each the answer was somewhere the motion does not show.

That somewhere is the switch: the instants at which the mechanism changes which contact is driving it. A pin enters a slot and the wheel changes from held to driven, and the step is the jump across that change. A pawl drops into a tooth and the ratchet changes from free to holding, and the resolution is how far apart those drops are. A disc takes over from a pin, and the rock is the clearance the takeover leaves. Where the input stops deciding found that these mechanisms’ state has a discrete part, which contact is active, as well as a continuous one, and every claim above reasons about the continuous part only.

The switch is also why the claims survive. A plot of a Geneva’s motion through its index, of a ratchet’s output against its input, or of a locked wheel’s angle against time draws the switch as one point among thousands, an instant of no duration. A peak, an average or a resolution measured in teeth per turn is computed from the thousands and is blind to the one. The things people say about indexers are said from those plots, and the plots are right about everything except the instant that matters.

Six things a cam drive is not found cam claims failing at second order, in accelerations and curvatures. Indexer claims fail at the switch, which is a zeroth-order matter: not how fast something changes but that it changes discontinuously at all. The repair is the same as for the cams. Ask the question at the place the mechanism decides, which for an indexer is entry, exit, drop and lock, and measure there rather than in the motion between.

Still open: whether the switches can be moved off the load #

All six answers locate the trouble at a switch, and the switches are always at the instants the load changes hands. A steered pin moved the Geneva’s step off the wheel and onto its own small follower, and the roller clutch moved a ratchet’s resolution from its pitch into a clearance.

The distinct argument would be a census of where each indexer’s switches put their jumps: on the output, on the input, or on a small part of the mechanism whose mass does not matter. Each would be measured as the jump in acceleration or force carried by each body at the instant of switching. The measurement would cover the Geneva, the steered Geneva, the cam indexer, the escapement and the ratchet. It would ask whether there is a rule — every indexer must put the jump somewhere, and the only choice is where — and whether any mechanism in the field puts it nowhere.