The escapement that could not alternate

This site’s sixth field exists because mechanisms get drawn confidently and wrongly, and because the difference between a drawing that would work and one that would not is usually invisible to the eye and never invisible to a computation. Five of them from the new field, each measured against the library that draws the working version.

None of the five is a strawman. All of them are arrangements that appear in published diagrams, in textbook illustrations and in the enormous body of clip-art that any search for “ratchet mechanism” returns.

One: the pallets that span a whole number of teeth #

An escapement’s two pallets touch the escape wheel’s tip circle at two points, and the angle those points subtend at the wheel’s centre is the mechanism’s central decision. It is a whole number of tooth pitches and a half.

Drop the half and the mechanism stops working, completely rather than badly.

The arithmetic says so immediately and says it in the mechanism’s own units. The beat budget — the wheel’s travel from one pallet’s corner to the next — comes out at

$$-0.000000000°$$

against 6° for the correct span. The impulse carries the wheel forward 2.267° and the drop carries it back 2.235°; the beat ends where it began, and the escapement never escapes.

That is a mechanism whose failure is a zero rather than a small number, which is the most useful kind for a check to catch. A gate that asked whether the budget was close to half a pitch would have to choose a tolerance; a gate that asks whether it is a quarter of the half pitch or less catches this and nothing legitimate.

The half tooth is also the single hardest thing to see in a drawing. Fifty-four degrees against forty-eight, on a wheel with thirty teeth: an illustrator drawing an escapement by eye has no way of noticing, and the two pictures are the same picture to within the thickness of a line.

Two: the pawl on the wrong side #

The commonest drawing of a ratchet has the pawl as a hooked lever reaching over the wheel, pivoted on the far side of the contact from the direction the load is trying to move it.

The holding test is the sign of one cross product, and this pivot has crossed the tooth face’s line: it stands 1.197 wheel radii on the far side of it, where the working pawl in the same figure stands 1.080 on the near side. The tooth’s push has a moment about that pivot which slides the tip up the face, and a tip sliding up a face reaches the top of the tooth and leaves.

What the drawing shows is a ratchet whose teeth are the wrong way round for its pawl — a mechanism that would work perfectly if the wheel turned the other way, which is exactly the kind of error that survives being looked at, because it is internally consistent.

The failure is not marginal either. Neither pawl is anywhere near the boundary, so the drawing cannot be rescued by a stronger spring, a deeper tooth or a more accurate machine.

Three: the Geneva at the wrong centre distance #

A Geneva drive has one design requirement — the pin must enter the slot along the slot — and it fixes the centre distance at $c = r / \sin\beta$ exactly. Nothing about that is negotiable, and it is the requirement most often lost when a mechanism is redrawn from a photograph.

Move the centres and the pin arrives across the slot instead of along it. The driven wheel’s angular velocity is then not zero at entry: the wheel is struck rather than accelerated, which is the defect the whole mechanism exists to avoid.

The middle row is exactly zero — the numerator of the derivative collapses to $c^2\cos^2\beta - c^2\cos^2\beta$ — and everything else is not. A five per cent error in the centres, which no drawing would betray, gives an entry velocity a sixtieth of the peak, which is a velocity step where the mechanism promises none.

Worth noticing that this failure is a degradation rather than a refusal, which makes it more dangerous than the first two. The wrong Geneva still indexes, still dwells, still locks. It merely does the thing it exists to do slightly badly, forever, and nothing in the mechanism reports it.

Three kinds of failure, and only one needs a threshold #

The four drawings fail in ways that are worth sorting, because the sorting says which of them a check can catch cheaply and which needs a judgement.

A failure that returns a zero. The whole-tooth escapement advances the wheel by 0° per beat, against 6° for the correct span. Nothing has to be decided about how small is too small: the mechanism does not index at all, and any test comparing the advance against the half pitch reports a total failure with no tolerance in it. This is the easiest kind to catch and the hardest kind to see in a drawing, which is exactly the combination that makes a check worth having.

A failure that flips a sign. The pawl on the wrong side of the tooth face is caught by the sign of one cross product, and a sign is a discrete quantity. There is no marginal case: either the pivot is on the side that lets the pawl bite or it is on the side that lets the tooth push it clear, and the test that separates them was already written for the working mechanism. Again no threshold, and again a defect that a drawing renders perfectly plausibly.

A failure that returns a wrong count. The intermittent gear with no locking arc has two degrees of freedom and one input. That is an integer, it comes from the mobility arithmetic the site has run since its first field, and the verdict is that the drawing has an uncontrolled member in it. No threshold once more, and the instrument is one that existed already.

And a failure that merely degrades. The Geneva at the wrong centre distance is the one that is different in kind. The pin enters the slot at an angle rather than along it, the entry acceleration rises from exactly zero to something finite, and the mechanism works — less well, more noisily, with a shock at every index. There is no zero, no sign and no wrong integer; there is a continuous quantity that ought to be nought and is not.

That one therefore needs a threshold, and it is the only one of the four that does. How far from the correct centre distance is too far is a question with an engineering answer rather than a kinematic one, and any gate written around it has to choose a number and defend it.

Which is why it is the most dangerous of the four rather than the least. The other three are drawings of mechanisms that do not function, and a prototype settles them in an afternoon. The Geneva at the wrong centres is a drawing of a mechanism that functions, passes a prototype, ships, and wears out its pin and its slot faster than it should — and the reason is a dimension that is nearly right and a derivative that is not zero. A defect that refuses is found; a defect that degrades is lived with, and the difference between them is not how wrong the drawing is but whether the wrongness has anywhere to hide.

Four: the intermittent gear with no locking arc #

A mutilated gear is drawn as a gear blank with teeth over part of its circumference and a plain hub over the rest. The drawing is complete as far as it goes, and what it leaves out is a component.

Without a locking arc — an arc about the driver’s shaft bearing on a concave face on the driven wheel — the driven wheel is not held during the dwell. It is free: it keeps whatever motion it had, and the next engagement catches it wherever it happens to be. That is not an intermittent drive, it is a drive with a gap in it.

The mobility count says exactly this and says it in one line. With the locking arc, the mechanism during its dwell is frame, driver and driven wheel with a revolute at each shaft and one higher pair:

$$M = 3(3-1) - 2(2) - 1 = 1$$

one degree of freedom, driven by the input. Take the contact away and the count is

$$M = 3(3-1) - 2(2) - 0 = 2$$

two degrees of freedom with one input, which is a mechanism with something in it that nothing controls.

Reading a drawing without the arc as an intermittent drive is reading in a component that is not there, and it is easy to do because the arc is drawn as a plain circular edge and looks like the blank.

Five: the flat face called deadbeat #

A fifth, and it is the one that is wrong in a book rather than in a picture.

A deadbeat escapement is routinely described as one whose pallets have “dead” locking faces set at the correct angle, and drawn with those faces as flats. A flat is what a straight tool cuts and it is tangent to the right curve, so the description sounds like a description of the mechanism.

It is not. The face that makes a deadbeat dead is an arc struck from the pallet arbor, and a flat tangent to it agrees only to first order. Measured: the arc’s recoil over a four-degree supplementary arc is zero — the identical double at all sixty-one samples — and the flat’s is 0.238°, which is more than twice the recoil of an arc deliberately drawn at a degree and a half.

And there is a sharper version of the same point. A flat face has no draw at the corner and gains draw in proportion to how deep the tooth rests — 0.0053 per unit of wheel torque at a lock of 0.3°, 0.0214 at 1.2°. So a flat-faced escapement is deadbeat at exactly one point of itself, and it is the one point the tooth never rests on. A drawing that shows the flat and calls the mechanism dead is showing a mechanism that is dead nowhere it operates.

What the correct drawings carry #

The four working mechanisms each have one number that a drawing has to get right, and listing them is the shortest form of this essay.

The escapement: the pallet span, which is $(k + \frac{1}{2})$ tooth pitches at the wheel’s centre. Nothing else in the mechanism can compensate for it.

The ratchet: which side of the tooth face’s line the pawl’s pivot is on. A drawing that extends the face across the picture answers the question by inspection, and a drawing that does not cannot be checked at all.

The Geneva: the centre distance, $r / \sin\beta$, which follows from the slot count and from nothing else. Draw the slot radial at the moment of entry and the centres are right by construction.

The intermittent gear: the locking arc, drawn as an arc about the driver’s shaft rather than as the blank’s edge.

Each of the four is a single scalar that decides whether the mechanism functions at all, each is invisible to inspection, and each is the kind of thing a computation cannot avoid getting right — because a mechanism drawn from a solve has to be handed the number before it can be drawn.

What they have in common #

Three of the first four are invisible to the eye, the fourth is invisible because it looks like nothing, and the fifth is invisible because it is in the words rather than in the picture.

The half tooth is six degrees on a fifty-four degree span. The pawl’s pivot is on the wrong side of a line nobody draws. The Geneva’s centres are five per cent out. The locking arc is a circular edge that a blank would have anyway. Not one of the four could be caught by looking harder at the picture, and all four are caught immediately by a computation that has to place a part rather than a pen.

That is the argument this field has been making since the foundation, in the sharpest form it has yet taken. A mechanism drawn from a solve cannot contain an error of this kind, because there is no way to express one: the pallet span is a number the geometry uses, the pawl’s verdict is returned before the drawing starts, the Geneva’s centre distance is computed from the slot count, and a locking arc that is not modelled produces a mobility of two and a figure that refuses to build.

Six, and it is this site’s own #

Fairness requires one from the home team, because this field’s own library shipped a mistake of exactly the kind it is complaining about.

The first version of the ratchet’s holding test predicted each cell of the pivot map from the tooth face’s line alone — the rule a workshop quotes — and had an assertion behind it. The assertion recorded, for every cell whose verdict disagreed with the prediction, the quantity cell − |distance to the line|, and required the largest such value to be no more than one cell.

That quantity is at most one cell for every non-negative distance, so the maximum over any set of disagreements is at most one cell however many there are and however far away they lie. The check could not fail. Half of that map was predicted wrongly and the gate was green, because the tooth face is only one of two boundaries and the contact normal is the other.

The repair was to count rather than to bound: take the cells that disagree and lie more than one cell from both boundaries, and require there to be none. There are none, and the check can now lose.

That is worth recording next to the four drawings, because the shape is identical. A drawing that is internally consistent and wrong, and a check whose arithmetic is internally consistent and vacuous, fail in the same way: both look like they are saying something and neither is constrained by anything. The tell, in both cases, is that the thing passed the first time it was tried on the first object it was pointed at.

What a checking gate can and cannot do for this #

It is worth being honest about which of the four a gate catches automatically and which needs somebody to think.

The whole-tooth span is caught by machinery: the beat budget’s sum is a computed quantity and a mechanism that returns nought where half a pitch is expected fails a check that exists. The Geneva’s centres are caught by construction, since this library computes the centre distance from the slot count and offers no way to set it wrongly except through a function written to do so.

The pawl’s pivot is caught only if somebody asks. The library will happily build a ratchet whose pawl does not hold and draw it perfectly; what it will not do is claim that it holds, since the verdict is computed and appears in the caption. The figure of the wrongly drawn pawl above is a correct drawing of a mechanism that does not work, which is what this field is for.

The missing locking arc is not caught at all, because a component that is absent cannot be checked. The mobility count returns two rather than one and nothing is watching the mobility count. That is the honest gap, and it is the general one: a gate can test what a model contains and cannot test what a model was never given.

Which is the same lesson as a search that agreed with itself and as every other absence this site has found. Every check here asks whether a computed thing is right. None of them asks whether a thing that should have been computed was.