Where the tooth lets go

An escapement is built from four numbers: how many teeth the wheel has, how far the pallet arbor sits above its centre, how many tooth pitches the two pallets span, and how deep the lock is. Everything else in the mechanism is a consequence, and the consequences include most of what horology argues about.

The half is the mechanism #

The pallets touch the 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 not a whole number of tooth pitches. It is a whole number and a half:

$$\Lambda = \left(k + \tfrac{1}{2}\right)\frac{2\pi}{N}$$

which on thirty teeth and $k = 4$ is 54°.

The half is why the mechanism works at all. With a whole number of pitches the two pallets meet the teeth in step with each other: a tooth arrives at both at once, both lock together and both release together, and the wheel is either arrested by two pallets or by none. With the extra half, the second pallet is exactly out of step with the first, so a tooth arrives at it precisely when a tooth leaves the other, and the wheel advances by that half pitch.

Build the wrong one and the arithmetic says so immediately. A pair of pallets spanning a whole four pitches gives a beat that advances the wheel by $-0.000000000°$ — the impulse takes it forward 2.267° and the drop takes it back 2.235°, and the mechanism ends the beat where it started. That is not a badly proportioned escapement. It is a mechanism that cannot escape.

So the wheel advances exactly $\pi/N$ per beat, six degrees on thirty teeth, and it does so by counting rather than by design: no dimension enters, and the same is true of an escapement of any proportions whatever.

The four things that happen to one tooth #

Follow a single tooth through one beat.

Lock. The tooth arrives on a pallet’s locking face and is arrested. The pendulum has not finished its swing, so the pallet continues to move for a while afterwards; that continuation is the supplementary arc, and what the wheel does during it is the subject of the next essay.

Unlocking. The pendulum comes back. The pallet swings the other way, the tooth slides along the locking face towards the corner where that face meets the impulse face, and reaches it. On a face cut as an arc about the arbor the wheel does not move at all during this; on any other face it does, and the amount it moves is the lock-in run, which appears in the budget below as a third term.

Impulse. Past the corner the tooth is on an inclined flank, and the wheel’s own torque now drives the pallet. This is the only part of the whole cycle in which the train does anything to the pendulum. The wheel turns through the impulse arc.

Drop. The tooth reaches the end of the flank and leaves. The wheel is now touching nothing, and it turns freely until the next tooth lands on the other pallet’s locking face. That free flight is the drop.

The budget closes #

Those three wheel motions — lock-in run, impulse, drop — are the whole of the wheel’s travel for the beat, so they must add to half a pitch. They do, and it is worth saying how the check is made, because the three are computed from three different contacts on two different pallets.

The impulse arc comes from the left pallet’s impulse face at two pallet angles. The drop comes from the right pallet’s locking face, with the tooth that is half a pitch behind. The lock-in run comes from the right pallet again, between the angle the tooth lands at and the angle its corner sits at. Three independent solves, and their sum:

$$2.0806° + 3.9194° + 0.0000° = 6.000000000°$$

against a half pitch of exactly 6°. Across face shapes, draw angles, tooth counts and spans the identity closes to twelve figures every time.

The deadbeat’s zero in the third term is the previous essay cashed out. Its locking face is an arc about the pallet arbor, so the pallet’s rotation carries the face into itself and the wheel does not move while the tooth slides along it. That is why a deadbeat’s budget has two terms and everything else’s has three.

Drop is the remainder #

Now the consequence, which is the sentence this essay exists for.

If the wheel’s travel per beat is fixed at half a pitch by the counting, and the impulse is chosen by the designer, then the drop is whatever is left. It is not a clearance to be set, not a gap to be adjusted, and not something a careful workshop can reduce. Lengthen the impulse and the drop shortens by exactly as much:

Every row sums to six degrees. The check that this is an identity rather than a coincidence is to take the differences between successive rows: each degree added to the impulse is a degree taken from the drop, and the sum of the two changes moves by less than $10^{-12}$ of a degree.

Two thirds of the escape wheel’s motion, on the escapement this field is arranged around, is free flight in which the mechanism does nothing at all. That is a large number to have arrived at by counting.

What buys it back, and what it costs #

There are two ways to reduce the fraction and both have a price attached.

Take a longer impulse. The table above: 5° of lift instead of 3.4° brings the drop from 65% to 52%. What it costs is pallet swing — the pendulum has to travel further while it is engaged, so more of its arc is spent being interfered with, and the mechanism becomes less like a free oscillator and more like a driven one.

Take more teeth. A finer wheel has a smaller half pitch, and the impulse arc does not shrink with it, because the impulse arc is set by the pallet’s geometry and the pallet’s geometry is set by the arbor distance rather than by the tooth count.

The impulse arc barely moves across that whole range — 1.93° to 2.08°, a variation of eight per cent while the tooth count triples — and the drop follows the half pitch down. What more teeth cost is the drop’s absolute size, which is the mechanism’s tolerance to being made badly, and that is an essay of its own.

Two routes to every contact #

Every position in this essay is the intersection of two circles, or of a circle and a line, and both have closed forms. That is exactly the situation in which a site whose habit is to check things is most tempted not to, so the check is kept.

Each contact is solved twice: once in closed form, and once by Newton on the residual — the signed distance from a tooth tip at wheel angle $\theta$ to the pallet’s face — started deliberately a milliradian off the answer. The two agree to $8.9 \times 10^{-16}$ radians over the whole engagement, on both faces, and Newton takes three iterations to get there.

The value of the second route is not the eight and a half digits. It is that a closed form and an iteration fail differently: a sign error in the closed form is invisible to itself and is caught immediately by a residual that will not go to zero, and a badly seeded iteration converges to the wrong branch and is caught immediately by a closed form that disagrees. This site has been caught by exactly the first and by exactly the second, in two different phases.

How far out the arbor may go #

The arbor distance is the one dimension of the four that is genuinely free, and it is bounded — not by anything practical, but by the budget refusing to balance.

Push the arbor further from the wheel’s centre and the pallet arms get longer, the pallets meet the tip circle at a shallower angle, and the impulse arc grows:

Follow the last column down and it reaches zero, at an arbor distance of 2.2633 wheel radii. There the impulse arc is exactly the half pitch and the drop is $6 \times 10^{-15}$ degrees, which is to say nothing: the tooth is still on the impulse face at the instant the next tooth is supposed to land on the other pallet.

Past that the arithmetic returns a negative drop, and a negative drop is not a small drop — it is a mechanism in which two teeth are in contact at once and the wheel is locked solid. So the escapement’s proportions are bounded above by a condition that has nothing to do with strength or wear or space: the impulse cannot be allowed to consume the whole of the wheel’s travel, because something has to be left for the next tooth to arrive in.

That is the same shape of constraint as the ratchet’s tooth depth two essays ago, and the same shape as the interference limits the gear field is built on. A mechanism’s proportions are bounded by geometry before anything else gets a say.

Drop is a tolerance, not a waste #

Two thirds of the wheel’s motion spent in free flight reads as an indictment, and the arithmetic above is what makes it something else entirely.

Every quantity in the beat budget is exact. The wheel advances $\pi/N$ per beat by counting, the impulse arc is fixed by the pallet geometry, and the drop is the remainder — which means the budget has no slack anywhere except in the drop. That is not an aesthetic observation. It is the whole reason the mechanism can be manufactured.

Consider what has to be absorbed. The wheel’s teeth are cut to a pitch that is right to some tolerance and not exactly; the pallet span $k + \tfrac{1}{2}$ is a nominal that a real pair of pallets meets to a few minutes of arc; the arbor sits at its distance to within the accuracy of a plate and a bushing; and the whole assembly runs with pivots in holes, so the arbor distance is not even constant through a beat. Each of those perturbs where a tooth arrives on a locking face. The counting still says half a pitch — the counting cannot be perturbed, being a statement about integers — and the impulse arc is set by the pallet’s own faces. So every error in the mechanism lands in the one term that is free to move, and the drop is where it goes.

Read that way, the two thirds is a clearance budget, and the negative drop at an arbor distance of 2.2633 radii is what happens when it is spent. A mechanism at zero drop is not efficient; it is one in which any error at all puts two teeth in contact and stops the clock. The margin between the design point and that limit is the margin against everything a workshop cannot control.

It also settles what the two ways of reducing drop are really buying and paying. Lengthening the impulse from 3.4° to 5° takes the drop from 65% to 52%, and the thing being consumed is not waste — it is the tolerance the escapement is built with. A finer wheel does the same more sharply: tripling the tooth count shrinks the half pitch threefold while the impulse arc moves by eight per cent, so almost the whole reduction comes out of the clearance. That is a straightforward reason why very fine escape wheels are the province of good workshops rather than a free improvement available to anybody, and it is a kinematic reason rather than a metallurgical one.

The same reading explains why drop is quoted as a fraction of the half pitch rather than as an angle, which otherwise looks like an odd convention. An angle at the wheel means nothing on its own — six degrees is generous on a thirty-tooth wheel and impossible on a ninety-tooth one — whereas the fraction is directly comparable across escapements of any size, and it is the fraction that says how much of the mechanism’s one elastic term is still unspent.

There is a limit to how far this reading goes, and it is worth marking. Drop is the kinematic clearance and says nothing about whether a tooth arriving after it has fallen through four degrees of free flight lands gently. It arrives with whatever speed the wheel’s torque gave it over that arc, and the impact is a matter of momentum and of what the pallet is made of — the subject that begins exactly where this field’s boundary is. So the argument here establishes that drop must exist and roughly how much, and not that more of it is better: the clearance is a floor set by tolerance, and above that floor the reasons to keep it small are ones this essay cannot see.

The other budget, which does not close #

The wheel’s ledger closes. The pallet’s does not, and the reason it does not is exactly where this site’s boundary runs.

Over one beat the pallet swings from one extreme to the other, and its travel divides into: the supplementary arc it has left over after the tooth lands, the lock run from the landing to the corner, the lift during the impulse, and then whatever it travels while the wheel is in free flight. That last term is not a geometric quantity. It is how far the pallet has got by the time the wheel has turned through its drop, which depends on how fast the wheel is turning, which depends on the torque on it and the inertia of the train.

So the term is reported as absent rather than estimated. beatBudget returns the pallet’s three geometric pieces and null for the fourth, and says that the ledger does not close.

That is an honest limit and it is a productive one. Everything an escapement is specified by — lock, lift, drop, draw, total swing — is on the geometric side of it, which is why an escapement can be designed on a drawing board at all. What is on the other side is how the mechanism behaves at a particular amplitude with a particular weight on it, which is what escapement makers spent two centuries measuring rather than calculating.

The dimensions that are not free #

One last consequence of the four numbers, and it is the one a beginner is most likely to want to argue with.

The pallet arm’s length is not a design variable. Given the tooth count, the span and the arbor distance, the two contact points are fixed on the wheel’s tip circle, and the arm is the distance from the arbor to them: 0.859 radii for the mechanism in the first figure, 0.905 at fifteen teeth, 0.876 at forty-eight. It comes out where it comes out.

Nor is the angle between the pallets, nor the position of the corner, which sits at $(\text{lift} - \text{lock})/2$ from the pallet’s centre position — 1.1° here — precisely so that one pallet’s release and the other’s arrival are the same instant. Get that offset wrong and the escapement still runs, badly: one beat is longer than the other, and a clock whose beats are unequal is a clock that sounds wrong before it keeps time wrong.

An escapement is therefore a mechanism with four inputs and about a dozen dimensions, and the eight or so that are not inputs are solved for rather than drawn. Which is the site’s usual arrangement, arriving in a field where it is least expected: a mechanism that looks like it was designed by eye, and was not.