The resolution is the pitch

Push a ratchet’s lever by a small amount and the wheel does not move. Push it further and the wheel does not move. Push it past a certain point and the wheel jumps by a whole tooth. There is no amount of care in manufacture that changes this, and no amount of it that makes the jump smaller.

That is a strange kind of error to have in a mechanism, and it is worth separating from the kind this site spent a whole phase measuring. A four-bar’s output is uncertain because its links have tolerances and its pins have holes; shrink the tolerances and the uncertainty shrinks with them, in proportion, to nothing. A ratchet’s is not like that at all. It is a property of the drawing.

The step and the error are one number #

A ratchet’s output can only be at a tooth. Between two teeth there is no configuration for it to occupy, because there is nowhere for the pawl to be except in a root; so the output’s position takes $n$ values and no others, and an input that has moved by less than one pitch has moved the output by nothing.

The worst case is therefore one pitch — $360°/n$ — and the average, over an input that arrives anywhere with equal likelihood, is half of that. On a twenty-four-tooth wheel: 15.0° worst, 7.5° mean. On twelve teeth, 30° and 15°. On eight, 45° and 22.5°.

Set that beside the other four rows of this field’s ledger and the ratchet is alone.

The reason for the difference is that the other four are driven to their positions and the ratchet is driven to somewhere near one. A Geneva’s pin carries the wheel through exactly one index and puts it down; there is no question of stopping short. A pawl is pushed until it drops in, and where it started from decides how much pushing was wasted.

More pawls divide it, and the division is exact #

The standard answer is more than one pawl, offset by a fraction of a pitch, so that whichever pawl is nearest a root drops first.

With $p$ pawls at offsets $0, 1/p, 2/p, \ldots$ of a pitch, the worst case is $360°/np$ and the mean is half of it. The division is exact rather than statistical: it is not that a second pawl usually catches sooner, it is that the input can never travel more than $1/p$ of a pitch without one of the pawls being over a root. Three pawls on a twenty-four-tooth wheel give 5.0° worst case, which is the resolution of a seventy-two-tooth ratchet with one.

That is worth a moment, because it is a mechanism buying resolution for nothing. The pawls do not have to be accurate relative to each other in any demanding sense — an offset that is out by a tenth of a pitch degrades the worst case by a tenth of a pitch and no more. And there is no equivalent trick anywhere else in this field: a Geneva cannot be given a second pin to make its index finer, because its index is fixed by the slot count and a second pin indexes it again rather than more finely.

Why not simply cut more teeth #

The obvious alternative is a finer wheel, and it runs into a wall.

A ratchet tooth has a depth — the face has to run from the root out to the tip circle, and the pawl’s tip has to sit inside that. The length of the face is set by the depth and by the angle it is cut at; the room available for it is one pitch of arc, and that is set by the tooth count. Ask for both and they collide.

At the depth this field draws its ratchets to — 0.16 of the wheel’s radius — the faces stop fitting between their neighbours at twenty-nine teeth, and the library refuses to build the thirtieth rather than drawing teeth that overlap. Halve the depth and it gets to sixty-four; take the depth to a quarter of the radius and it stops at sixteen.

The wall is geometric, which is what makes it interesting. It is not that a shallower tooth is weaker — that is a statement about forces and is outside this site — but that beyond a certain fineness there is no room for the tooth to be. A ratchet’s resolution and its tooth depth are a single trade, and the trade is drawn rather than calculated.

There is a third variable, and it is the one designers actually move: the wheel’s diameter. Nothing in the argument above knows about size, so doubling the wheel and keeping the depth in absolute terms halves the depth as a fraction and doubles the available tooth count. A ratchet with a fine index is a large ratchet, and that is why the fine ones sit where there is room for them.

Two lost motions on one mechanism #

Everything so far assumes a mechanism made exactly. Give it the treatment the practice field gives everything else and a second term appears, and the two behave completely differently.

The design term is the pitch: it is the same for every specimen off the line, it does not depend on the direction of travel, and it goes to zero only if the tooth count goes to infinity. The built term is clearance: it varies from specimen to specimen, it reverses with the direction of travel, and it goes to zero as the tolerances do. On the same wheel they simply add — the input must first take up the clearance and then travel to the next root — so a ratchet’s total lost motion is a fixed number plus a variable one, and a measurement of one specimen cannot tell them apart.

Which is why a ratchet that is measured and found to have twelve degrees of lost motion is not necessarily a badly made ratchet. On a thirty-tooth wheel it is a perfect one.

The sizes are worth putting side by side, because they are not close. A four-bar with a hundredth of a unit of clearance in every joint — the mechanism the practice field measures — loses about 2.7° of crank at a typical position, rising without bound near a limit position where the linkage’s own leverage does the rest. A twenty-four-tooth ratchet with no clearance at all loses up to 15°. So on any ratchet coarse enough to be worth calling one, the design term is several times the built term, and a workshop that responds to sloppy indexing by tightening the pins is chasing the smaller of the two.

The exception is the ratchet near its own limit, and it is the same exception the four-bar has: as the pitch is made fine, the design term falls towards the built one and eventually below it, and a very fine ratchet is a mechanism whose lost motion is a tolerance question after all. Where the crossover sits is a division: at a hundredth of a unit of clearance and this geometry, somewhere around a hundred and thirty teeth — which the tooth-depth wall above says cannot be cut on a wheel of this proportion at all.

What buys the division, and what limits it #

Multiple pawls divide the pitch exactly, and it is worth asking what is actually being paid for that, because the answer is not obvious and it is the reason the trick is used so widely.

Cutting more teeth asks for angular accuracy on the wheel: every tooth must be at its nominal place, and a finer wheel demands the same absolute accuracy over a smaller pitch, which is the dividing engine’s problem and the wall this essay is about. Adding pawls asks for something entirely different — positional accuracy on the pawl pivots, which are a few holes in a plate at stated angular offsets. Those are two different manufacturing operations with two different tolerances, and the second is very much the easier of them.

That is the whole economics of the trick. Three pawls at thirds of a pitch turn a fifteen-degree step into a five-degree one using a wheel nobody had to cut more finely, and the accuracy demanded of the plate is a fraction of a pitch rather than a fraction of a tooth. A ratchet resolution problem is thereby moved from the part that is hard to make to the part that is easy.

It does not move indefinitely, and the limit is worth stating because it is the same limit arriving from the other side. With $p$ pawls the sub-pitch is $360°/np$, and each pivot must sit within a fraction of that to keep the pawls from arriving together — so the accuracy demanded of the plate tightens in proportion to $p$. Past some point the plate is being asked for the same precision the wheel was, and nothing has been gained except that the difficulty has been relocated.

There is a second and cruder limit that usually bites first. Each pawl needs a pivot, a spring and room to swing, and they all have to fit around one wheel without fouling one another. Three or four is comfortable, six is crowded, and a dozen is a mechanism whose pawls are the size of the teeth they are engaging. So the practical range of the trick is a factor of three or four on the resolution, which is real and is not the order of magnitude a finer wheel would have given if the geometry had allowed one.

Between the two, the design reading is that pawls and teeth are alternative purchases of the same quantity, priced in different currencies, and that a designer should exhaust the cheap one before starting on the expensive one. That is the reverse of the order the obvious answer suggests.

What the limit case is #

Take the tooth count up and the lost motion down, and at the end of that line is a mechanism with no teeth at all: a wheel and a shoe pressed against it, holding one way and slipping the other. Its lost motion is zero, its resolution is continuous, and it will hold any position at all.

It is also not a ratchet, and the reason is exactly the boundary this site keeps. What holds a toothed ratchet is a geometry — the two lines of the previous essay, and a verdict that does not care how hard anything is pushed. What holds a friction clutch is a coefficient. One of those can be drawn and the other cannot, and the whole difference between a mechanism that indexes and a mechanism that grips is which of the two is doing the work.

So the ratchet’s coarse resolution is not a defect that a better ratchet would fix. It is the price of the property that the mechanism was chosen for: that whether it holds is decided by a drawing.

What an exact index costs instead #

Set the ratchet beside the Geneva and the trade becomes legible, because the Geneva pays for its exact index with something a ratchet never spends.

A Geneva’s index is $360°/n$ and it is exact. Not exact to a tolerance — exact in the sense that the pin enters a slot, carries the wheel through one index and leaves, and the wheel is then held by a locking arc in the position the geometry put it in. There is no accumulation and no drift: a Geneva that has indexed a million times is one index from where it started, not a million errors from it.

What it spends is freedom of proportion. A ratchet designer chooses the tooth count, the depth, the face angle, the number of pawls and where each pivot goes, and every one of those is a real choice. A Geneva designer chooses $n$. The centre distance is not free, the wheel radius is not free, the dwell fraction is $(n-2)/2n$ and cannot be moved, and the acceleration is whatever those produce.

So the two mechanisms sit at opposite ends of a single trade: the ratchet has every proportion free and a resolution it cannot improve, and the Geneva has an exact index and no proportions at all.

Resolution in a mechanism that is counting #

The escapement’s row in the ledger reads half a tooth pitch a beat, and it belongs to a different conversation than the other five, because an escapement is not positioning anything.

An indexer’s job is to put something in a place. An escapement’s job is to let a train advance a fixed amount and stop it again, over and over, so that a wheel further down the train has turned by an amount proportional to the number of beats. The quantity it is accurate about is not a position but a count, and a count has no resolution error at all: the wheel has either advanced $k$ half-pitches or it has not.

That is why the escapement’s index of 6° is not comparable with the ratchet’s 15° even though both appear in the same column. Six degrees is the size of the step; the accuracy of the mechanism is a different quantity, and it is the accuracy of the thing being counted — which is a question about a train of gears and about arithmetic, and not about the escapement at all.

The route that buys any resolution at all #

There is one mechanism in the field with no resolution limit, and it is the one that has to be cut rather than assembled.

A cam indexer’s step is whatever the profile says, its dwell is however much of the turn is a circular arc, and the two can be chosen independently — which none of the other five allows. The motion law can be chosen too, so the accelerations are a design variable instead of an inheritance.

The price is that the accuracy of the output is the accuracy of the surface. A ratchet’s output is right because a pawl is in a root, and a root is a wide, forgiving thing; a cam’s output is right to whatever the profile was cut to. That is a completely different kind of error — continuous, one-sided in no particular direction, and improved by money.

Which is the shape of the whole ledger, really. The mechanisms with discrete stops have coarse resolution and are right regardless of how well they are made; the mechanism with continuous positioning has any resolution at all and is exactly as right as its manufacture. There is no row with both.

What is exact here and what is not #

Two of this essay’s numbers are exact and one is not, and it is worth saying which.

The lost motion of a perfect ratchet is exact: $360°/np$ is arithmetic on a tooth count, with no measurement in it. The division by extra pawls is exact for the same reason.

The tooth-count limit is not, and the table above should be read with that in mind. It comes from requiring the face to occupy less than nine tenths of the arc between neighbouring roots, and nine tenths is a drawing convention rather than a theorem — a wheel cut with a form tool leaves a fillet at the root that eats into the same space, and a pawl with a rounded tip needs a root wide enough to take it. What is not a convention is the shape of the constraint: face length is fixed by the depth, room is fixed by the count, and the two are in direct competition however the margin is drawn.

There is a third purchase available and it is worth naming for completeness, because it is the one that escapes the trade entirely. Nothing requires the pawls to be equally offset: with pawls at arbitrary fractions of a pitch the worst case is set by the largest gap between consecutive offsets, so equal spacing is simply the arrangement that minimises the worst case for a given number of pawls. A designer who cares about the average rather than the worst case, or who has room for pawls only on one side of the wheel, has a genuine choice to make there — and it is the same choice, with the same arithmetic, that the planetary field’s unequally spaced planets present. Both are a question about how to place a few things on a circle when the number that divides evenly is not the number available.