The wheel that goes backwards

A pendulum does not stop when the escape wheel does. The tooth lands on a pallet, the wheel is arrested, and the pendulum carries on for another few degrees before turning round — the supplementary arc, which on a well-set clock is most of the swing and on a badly set one is nearly all of it.

The pallet is attached to the pendulum, so during the supplementary arc the pallet is moving with a tooth resting on it. Something has to happen to the wheel.

The answer that is not a small number #

Cut the locking face as an arc of the pallet arm’s own radius, centred on the pallet arbor, and the wheel does nothing at all.

The reason is the arc’s own property: a rotation about the arbor carries that arc into itself, so the tooth’s resting place is a point of the arc before and after, and the point of the wheel’s tip circle that it can rest at does not move. Both curves are unchanged by the pallet’s motion, so their intersection is unchanged, and the contact solve — which is an intersection of two circles and has a closed form — returns the same number.

The measurement is worth stating precisely because it is stronger than the usual kind. Sixty-one samples across the supplementary arc, and every returned wheel angle is the same double. Not agreeing to some tolerance; identical. This site normally reports an agreement with the resolution it was measured at, and here there is no resolution to report, because nothing was measured against anything: the same arithmetic was performed sixty-one times on inputs that differed only in a quantity the arithmetic does not contain.

The face that is flat, and where it is dead #

The face a workshop actually produces is a flat, because a flat is what a straight tool cuts and because the arc and its tangent agree to first order. The question is how much the second order costs, and the answer is a good deal more than it looks.

Put the tooth exactly at the corner — the point where the locking face meets the impulse face, and the point the tangent is tangent at — and the flat behaves perfectly: its draw is $1.3 \times 10^{-16}$ per unit of wheel torque, which is zero, and the recoil starting from there is second order.

Each doubling of the arc multiplies the recoil by 4.02, 4.04, 4.09 and 4.20 — a quadratic, exactly as a tangent’s error should be.

But a tooth does not rest at the corner. It rests at the lock, a degree or so below it, because a lock of zero is a lock that any disturbance releases; and a degree below the corner the flat is no longer tangent to anything. Its inclination relative to the arc there is, to a very good approximation, the lock angle itself:

With a lock of 1.2° the flat face’s recoil acquires a linear term on top of its quadratic one, and the numbers stop being tidy: 0.0130°, 0.0306°, 0.0800°, 0.2381°, 0.8120° across the same five arcs, with successive ratios climbing from 2.36 towards 4. That is a mechanism with two error terms of comparable size, which is the least convenient thing a mechanism can have, because neither can be reasoned about without the other.

The face that is tilted on purpose #

Every escapement in the field’s ledger has a locking face tilted by a degree or two from the concentric, and the reason is the subject of the next essay: without a tilt, the lock has no tendency to hold itself and the smallest disturbance opens it.

The tilt costs recoil, and the cost is linear:

Successive ratios 2.000, 1.999, 1.997, 1.992. That is a straight line through the origin, and it is a completely different curve from the flat face’s — same order of magnitude at four degrees, and a factor of three apart at eight.

The recoil is linear in the tilt as well: 0.0351°, 0.0706°, 0.1064°, 0.1427°, 0.2163°, 0.4471° for tilts of 0.5°, 1°, 1.5°, 2°, 3° and 6°. So an escapement with an arc-and-tilt locking face has recoil proportional to the product of two small angles, and both of them are known to the drawing.

That is the practically important form of the result and it is worth stating as a rule. A deadbeat escapement is not a mechanism with no recoil. It is a mechanism whose recoil is a controlled first-order quantity rather than an uncontrolled second-order one, and the reason to cut the face as an arc is not to make the recoil zero — the tilt makes sure it is not — but to remove the term that nobody chose.

How this claim could have been wrong #

A claim of exactness is the easiest kind to make badly, and there are three ways this one could have been an artefact. All three are things this site has done before, in earlier phases, which is why they are checked for here.

The sweep could have been too coarse. A recoil measured at four sample points across the supplementary arc, on a face whose recoil happens to be small, would report zero for a mechanism that recoils. The measurement here is sixty-one points and the result is not “small at all of them” — it is the identical double at all of them, which no coarseness produces from a varying quantity.

The tolerance could have been the answer. An assertion written as recoil is below $10^{-9}$ passes for a face that recoils by $10^{-10}$ and for a face that does not recoil at all, and cannot tell them apart. The assertion here is that every sampled value is bit-for-bit equal to the first, which is a test with something to lose: run it on the flat face and it fails at the second sample.

The solve could have agreed with itself. A contact solved by seeding an iteration with a closed form and then reporting that the iteration did not move is a check that has run nothing — this site did exactly that to Bennett’s linkage in its expansion phase, and the assertion passed with the solver never having iterated. Here the closed form and the Newton solve are separate code, the Newton solve is started a milliradian off, and they agree to $8.9 \times 10^{-16}$ radians after three iterations.

There is a fourth possibility that is not a mistake but is worth naming: the exactness is a property of the model, and the model has rigid bodies and no clearance. A real pallet has a pivot with a hole in it, and a real escape wheel’s teeth are not all at the same radius. What the arc removes is a term that is present in the ideal mechanism and is nobody’s fault; what it does not remove is everything the practice field measures, and on a rough clock those may be larger. The claim is that one specific term has been taken to zero, not that the mechanism is exact.

The zero is a coincidence of two centres #

There is one more thing the exactness depends on, and naming it says what a deadbeat needs that an anchor does not.

The arc’s property is that a rotation about the pallet arbor carries the arc into itself. That is true of a rotation about the centre the arc was struck from, and about no other point. The face is cut on one centre and the pallet turns about another — the pivot in its hole — and the whole of the deadbeat’s exact zero is the assertion that those two points coincide.

They coincide as well as the workshop makes them coincide. A pivot running in a worn hole does not turn about a fixed point at all; it turns about wherever the load has pushed it, which moves with the direction of the force and therefore differs between the two halves of a beat. Displace the effective centre and the locking face is an arc about a point the pallet is not turning about, so a rotation no longer carries it into itself, and recoil appears at first order in the displacement.

That is the structural difference between the two faces, and it runs opposite to their nominal ranking. The flat face’s recoil is quadratic in the supplementary arc and has no exact zero to lose; moving the centre perturbs an error that was already there. The arc’s recoil is exactly nothing and is exactly nothing only while the centres agree — so the deadbeat’s advantage is not a property of its shape alone but of its shape together with its pivots, and it is the one of the two that degrades as a clock wears.

The size of that degradation is not measured here and would need a model with a displaced arbor in it, which is a fifth face rather than a variation on the four. What the geometry does establish without any measurement is the order: linear in the displacement for the arc, and no new term at all for the flat. That is enough to say which escapement is asking more of its bushings, and it is the kinematic half of a fact usually given as workshop lore.

Recoil as something a designer might want #

It would be tidy if recoil were simply a defect, and it is not, which is why anchor escapements went on being made for two centuries after Graham.

The kinematic fact is the one already measured: an escapement with an inclined locking face has a wheel position that depends on how far the pendulum has swung. Push the pendulum harder and the wheel is driven further back; let the pendulum die down and the recoil goes with it. The mechanism’s geometry is a function of its amplitude, and that is unusual — a Geneva’s geometry is not a function of how fast the driver is turning, and a four-bar’s is not a function of anything but its crank.

What that buys is a matter of forces and is not settled here. What can be said is where the coupling comes from and how large it is: the recoil is proportional to the supplementary arc, with a constant of proportionality that is the draw, and both quantities are on the drawing. An anchor escapement at 4° of supplementary arc drives its wheel back 1.14°, and at 2° it drives it back 0.57°.

The other thing recoil buys is more prosaic and entirely kinematic: an escapement whose wheel is pushed backwards cannot overbank. A deadbeat’s tooth sits on a face that goes nowhere, so a pendulum given a large enough push can swing past the end of the locking face and let the tooth escape at the wrong moment; a recoil escapement’s tooth is being pushed harder into its face the further the pendulum goes, and the geometry closes rather than opening. Whether the clock survives being knocked is a question about which of those two things happens, and it is a question about the length of a face and the size of an arc.

What recoil costs in travel #

There is one consequence of recoil that stays entirely on this side of the site’s boundary, and it is the one that shows up as wear rather than as rate.

The wheel’s net travel per beat is half a tooth pitch, whatever the faces do — that is the counting, and it closes to twelve figures. Its path length is not. A wheel that is driven back by $r$ and comes forward again travels $\pi/N + 2r$, and every bit of that extra travel is sliding at a contact under load.

On the deadbeat with 1.5° of draw at a 4° supplementary arc: 6.213° travelled for 6° gained, an excess of 3.5%. On the anchor escapement: 8.27° travelled for 6° gained, an excess of 38%. And the extra travel is concentrated at exactly the two places on the mechanism that are hardest to lubricate — the tips of the wheel teeth and the working faces of the pallets, which is where an old anchor clock shows its wear and why a deadbeat’s pallets are usually jewelled and an anchor’s usually are not.

That number is a kinematic one throughout. No force appears in it, and it does not need one: sliding distance at a contact is a length, and lengths are what this site computes.

What the excess travel adds up to #

The 3.5% is a fraction of a beat, and a clock is a mechanism for turning a fraction of a beat into a large number.

A seconds pendulum beats twice a second, so the escape wheel makes 172,800 advances a day, each of six degrees on a thirty-tooth wheel. That is 2,880 revolutions a day of net travel — and with 1.5° of draw at a 4° supplementary arc, an extra 0.213° per beat that is travelled and then given back. Over a day the extra is 36,800°, a further 102 revolutions; over a decade, some 373,000 revolutions of motion that advance nothing.

None of that motion is wasted in the sense of costing time, because the counting is exact and the net travel is half a pitch whatever the faces do. What it costs is contact. Every degree of it is a tooth tip sliding on a locking face under the full force of the train, in both directions, at the two places on the mechanism where the pressure is highest and the lubrication worst. A deadbeat’s locking faces see the lock-in run and nothing else; a recoiling escapement’s see that plus a hundred extra revolutions a day of reciprocating rub.

That is the sharp form of the difference between the two escapements, and it is not the one usually given. The textbook contrast is about rate — a recoiling escapement’s period depends on the driving force, a deadbeat’s much less so — and it is a statement about forces, which is outside this field. The contrast that stays inside it is this one: the two mechanisms count identically, advance identically, and differ by a factor of a hundred revolutions a day in how much their faces are rubbed to do it.

It also gives the exactness of the deadbeat’s zero a practical meaning it would otherwise lack. A recoil of $10^{-16}$ degrees and a recoil of $10^{-4}$ degrees are both, on any single beat, nothing at all; the difference between them is invisible in one swing and in a thousand. It becomes visible only when multiplied by a number the size of a decade of beats, which is exactly what a clock is. An exact zero is worth having because the thing it is a property of runs a hundred million times, and a quantity that is merely small is a quantity that accumulates.

The wear itself is not a kinematic quantity and nothing here predicts a rate of it. What the geometry supplies is the sliding, in degrees, and that is the multiplicand — the other factor is a matter of load, materials and oil, and belongs on the far side of this field’s boundary. But the multiplicand is the one that separates the two escapements, and it is computable from four numbers and a face profile.

What it looks like in the state space #

That picture is the cleanest statement of the difference. A deadbeat’s path in the state space has flat treads: intervals of the pendulum’s motion over which the wheel’s coordinate is constant. A recoil escapement’s has sloped ones, and the slope is the same number as the draw, which is the next essay.

It is also where the amplitude enters. The recoil at the end of a supplementary arc is proportional to that arc, so a clock swinging further recoils further; the deadbeat’s flat treads are the same flat treads at any swing at all. What that does to the rate of the clock is a dynamical question and is not answered here. What can be said, and is a fact about the drawing, is that one mechanism’s geometry depends on the amplitude and the other’s does not.

Where 1715 comes in #

George Graham’s deadbeat escapement is usually described as an improvement on the anchor, and the description that gets given is that the pallets were made “dead” so the wheel would not recoil. That is true and it leaves out the part that makes it a piece of geometry rather than a piece of craft.

The change was to cut the locking faces as arcs struck from the pallet arbor. Not flats at a better angle, not shallower flats, not the same flats made more carefully: a different curve, chosen because rotation about a point carries circles about that point into themselves and carries nothing else. Everything in the table above follows from that one substitution, and none of it is available to a mechanism that keeps the flats.