A joint that works one way

A ratchet is two moving parts and a spring, and it does something no linkage on this site can do: it transmits motion in one direction and refuses it in the other. Every mechanism in the first eleven fields is reversible. Drive a four-bar’s rocker and the crank turns; drive a gear train backwards and it runs backwards; drive an arm’s tool and the joints move. A ratchet cannot be driven backwards at all, and that asymmetry is not a property of any equation the mechanism satisfies.

It is a property of what a contact can do, which is push and not pull.

The question, stated so that it has an answer #

The wheel tries to turn the blocked way. The tooth face is a surface; the pawl’s tip rests against it; the pawl pushes the tooth along the face’s normal, in whichever of the two directions opposes the motion the pawl exists to stop. That fixes the direction of the push without any appeal to how hard anything is pushing — a normal is a property of a surface and the sense that opposes a motion is a property of the motion.

By the third law the tooth pushes the pawl along the other one. Now ask what that push does to the pawl, which turns about a pivot. It has a moment about that pivot, and the moment turns the pawl one way or the other; the pawl’s tip, being in contact, slides along the face as it turns. Two possibilities and no third:

• the tip slides down the face towards the root, which drives the pawl deeper into the tooth, and the ratchet holds;
• the tip slides up the face towards the tip of the tooth, which lifts the pawl off, and the wheel goes.

Nothing about magnitude appears anywhere in that. What is being asked for is the sense of a moment, which is the sign of a cross product, and a cross product of two directions with a length in it is a length with a sign. This site has kept a boundary against statics since its foundation, and this is the same instrument that gets used on the other side of it: a transmission angle is the direction of the force a coupler can transmit, computed without ever evaluating a force, and a toggle is where that direction becomes degenerate.

Two cross products, and therefore two lines #

Write the two conditions out. Let $\mathbf{q}$ be the vector from the contact to the pawl’s pivot, $\hat{\mathbf{f}}$ the direction of the face from root to tip, and $\hat{\mathbf{n}}$ the direction the tooth pushes the pawl. The moment’s sense is the sign of $\mathbf{q} \times \hat{\mathbf{n}}$, and the sense that engages is the sign of $\mathbf{q} \times \hat{\mathbf{f}}$, and the pawl holds when they disagree.

Each of those vanishes on a line through the contact: the first where the pivot lies along the normal, the second where it lies along the face. The face and the normal are perpendicular, so the two lines are perpendicular, and the plane around the contact is cut into four quadrants of which two — opposite ones — hold.

The rule that gets quoted in a workshop is the first of those two lines: the tooth face, extended, must pass on the correct side of the pawl’s pivot. It is a good rule and it is the one that matters in any sensible design, because the other boundary is far away from anywhere a pawl is actually put. It is also not the whole rule, and over the square above it is wrong about 53.1% of the positions — a number that mostly measures how much of that square nobody would use, but which is exactly zero for the two-line test.

A check that could not have failed #

The one-line version was in this library first, with an assertion behind it, and the assertion was worthless in a way worth recording because the shape of the mistake is general.

It swept the map, and for every cell whose verdict disagreed with the one-line prediction it recorded cell − |distance to the line|, then required the largest such value to be no more than one cell. The intent was: a cell may only disagree if it is within a cell of the boundary. What it actually says is nothing at all — cell − |d| is at most cell for every non-negative $|d|$, so the maximum over any set of disagreements is at most cell however many there are and however far away. Half of that map was predicted wrongly and the gate was green.

The repair is 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. The difference between the two checks is that the second one can lose.

This is the fourth time this fleet has found a check of exactly this shape — an assertion whose quantity is bounded above by its own tolerance — and the tell is always the same: the check passed the first time it was run, on the first thing it was pointed at.

The rule is about directions, so it forgets the size #

Sweep the tooth face angle with the pivot held and the pawl’s verdict flips twice, once at each boundary: at −16.19°, where the normal’s line sweeps through the pivot, and at 69.94°, where the face’s own line does.

Now do it on a wheel four hundred times larger. The critical angles are −16.187588825° and 69.937°, to every digit that came out the first time, on wheels of 0.13, 1, 7 and 55 units of radius. Do it on wheels of eight, twelve, twenty and twenty-eight teeth, and they are the same again.

That is not a coincidence and it is not a scaling law that happens to hold: the test is a comparison of directions, and the pivot’s position here is specified in wheel radii, so the direction from contact to pivot is the same on every one of them. Nothing in the criterion knows how large the mechanism is. A ratchet the size of a watch escapement and a ratchet on a ship’s windlass hold or slip by the same arithmetic, and the design rule is the same drawing.

Where along the face the pawl bears #

One boundary moves with the contact and the other does not, and both halves are worth having.

The face’s line is the same line wherever along the face the pawl’s tip happens to sit, because sliding a point along a line does not change the line. Measured across five positions from the root to the tip, the pivot’s distance from it is 1.080355576650 every time, to twelve figures. That invariance is what makes the workshop rule quotable at all: it can be drawn once on a design and read off, and a pawl worn a little shorter does not fall outside it.

The normal’s line is not the same line, because it passes through the contact and the contact has moved. Over the same five positions the pivot’s distance from it runs from 0.573 to 0.418 — a quarter of it gone as the pawl bears from the root to the tip. On this pawl that is margin being spent and nothing more, since the verdict is unchanged at all five. On a pawl set closer to that boundary it would not be, and the failure would present as a ratchet that works when new and slips when worn, which is a description a great many people have of a great many ratchets.

Getting into the notch is a third condition #

Holding is one question and riding out is another, and there is a third that only appears once the pawl is asked to find the notch rather than being placed in it.

The pawl’s tip is a point at a fixed distance from its pivot, so it runs on a circle; the wheel’s surface is what it rests on; and where it rests is the first place that circle meets the surface, coming down from clear. Follow that from frame to frame as the wheel turns — each position continued from the one before, which is how this site solves every sweep — and the pawl is lifted up a tooth’s back, carried over the crest, and dropped into the next notch. One click.

That is what the figures here are drawn from, and it took two wrong models to get to. Dropping the pawl from fully lifted at every frame answers a different question and answers it differently: on an undercut tooth the outside of the profile is disconnected along the pawl’s circle, so a pawl let fall from clear lands on the crest beside the notch and never reaches the notch at all. A pawl in a ratchet is never fully lifted; it rides on the surface continuously, and continuity is the whole of the model.

The third condition is what that turns up. Rake the tooth face forward instead of undercutting it and the pawl never drops in. At a face of −10° the holding test still says the pawl holds — the pivot is comfortably on the correct side of both lines — and the ride never falls into a notch at all: the tip skims the crests, the click never happens, and a mechanism that passes the first two tests is not a ratchet.

So the tooth’s face angle is doing two separate jobs. It sets which side of the boundary the pivot has to be for the pawl to hold, which is the subject of this essay. And it decides whether the pawl can reach the place where that verdict applies, which is a different question with a different answer.

The other condition, which is not idle #

A pawl has a second job. It must hold when the wheel is driven the blocked way, and it must ride out of the way when the wheel is driven the free way, climbing the back of each tooth and dropping into the next root.

That is the same test again, with the back’s surface and the back’s normal instead of the face’s, so the admissible region for the pivot is the intersection of two conditions rather than one. It could easily have been idle — the backs of ratchet teeth are shallow ramps, and a shallow ramp lifts almost anything. It is not: on the default ratchet here it removes 4.5% of the region the first condition allows, and the removed part is the interesting kind of failure. A pivot in it gives a pawl that holds perfectly against the load and then jams solid the moment the wheel is turned the way it is supposed to go, which is a mechanism that has to be taken apart to be understood and reads as a manufacturing fault rather than as a design one.

The pawl that is drawn #

There is a way of drawing a ratchet that appears everywhere, and it is worth putting through the test rather than complaining about.

The pawl is drawn as a hooked lever reaching over the wheel from a pivot on the far side of the contact — the side the blocked tooth is moving away from rather than towards. It looks right. The hook is in the tooth, the geometry is tidy, and the direction of free rotation is obvious from the shape of the teeth.

The mechanism does not hold. 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 such a drawing shows is a ratchet that would work perfectly if the wheel were turned the other way — which is to say, a ratchet whose teeth are the wrong way round for its pawl.

The failure is not marginal, and that is the useful part. Measured against the face’s line, the working pawl stands 1.080 wheel radii on one side and this one 1.197 on the other; there is nothing close about it, and nothing that a deeper tooth or a stronger spring or a more accurate machine would change. A ratchet drawn this way is not a ratchet that needs adjusting.

What the pawl’s margin is, though, is 0.175 rather than 1.197 — because the margin is the distance to the nearer of the two boundaries, and the nearer one here is the normal’s line. The two lines are perpendicular and the pivot is a long way past one of them and close to the other, which is a fair description of most bad designs: comfortably wrong about the thing that was checked, and marginal on the thing that was not.

A pivot with nothing in hand #

Between the two is the case that a check has to be able to catch: a pivot close to a boundary.

A pawl here is a mechanism whose verdict is decided by quantities nobody controls to that accuracy: the exact length of the pawl, where its tip has worn to, how much the pivot hole has opened out. The site has a name for this shape of problem — fragility has a direction — and the direction here is legible, because the boundaries are lines and the margin is a distance to them. Moving the pivot along a boundary buys nothing at all; moving it across one buys everything.

The same test, later #

This is the first of three places in this field where the same cross product decides a mechanism, and it is worth flagging the other two now, because the resemblance is the argument rather than a curiosity.

And the third is the transmission angle, which is nine essays and six phases older than either. All three ask which side of a pivot a contact normal passes, all three are answered by a cross product of two directions, and none of the three evaluates a force. Whether that is a deep fact or a shallow one is a fair question; what it is not is a coincidence, since the direction a contact can push in is the only thing a contact has.

What the mobility count cannot see #

It is worth putting the ratchet through the count the first field of this site is built on, because the count comes back entirely reasonable and entirely silent about the property the mechanism exists for.

Three links — frame, wheel, pawl — with a revolute at the wheel’s shaft, a revolute at the pawl’s pivot, and one higher pair where the tip bears on the face. Grübler gives

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

one degree of freedom, which is right: the wheel turns and the pawl rides. The count says the same for the ratchet that holds and the ratchet whose pawl is on the wrong side of both lines and is levered out on the first tooth. It says the same for a ratchet driven the free way and the blocked way. It has no way not to, because a mobility count knows how many freedoms a joint removes and nothing whatever about the direction a contact can push in.

That is the general shape of this field’s relation to the site’s first one. Counting freedoms was always a statement about how many independent motions exist. Everything here is about which of them can actually be taken, and the two questions have different answers whenever a joint is a contact rather than a pin.

What a pawl costs #

Nothing in this essay has been about a spring, and a real pawl has one. Its job is to keep the tip against the wheel while the mechanism is unloaded, so that the pawl is in a tooth when the load arrives rather than resting somewhere above it — and the moment the load does arrive the spring is irrelevant, because the holding is decided by the two lines and not by anything pressing the pawl down.

Which is the honest reason a spring can be left out of a kinematic account of a ratchet, and the honest reason it cannot be left out of a real one. The geometry decides whether a loaded pawl holds. What the spring decides is whether the pawl was in the tooth when the load came, and that is a question about a mechanism moving in time under forces — the boundary this site keeps, met here in one turn of a screwdriver.