Detached, and safe while detached

An oscillator keeps time well when it is left alone. Every escapement is an interference with the thing it is counting, and the whole history of the subject is a search for ways to make the interference briefer.

An anchor escapement’s pallets are on the pendulum’s own arbor. They are in contact with the escape wheel for the entire beat: a tooth is on a pallet at every instant, and the train is leaning on the pendulum from one end of its swing to the other. A deadbeat is better only in that the leaning is steadier.

A detached escapement is a different arrangement altogether, and the number that says so is a fraction.

The pair, which has been on this site since its foundation #

The fork is a slot and the balance carries a pin, and the pin runs in the slot. That is the whole kinematic connection between the two, and it is not a new object.

A pin entering a radial slot is the pair a Geneva drive is made of: a driving crank carries a pin, a driven wheel carries a slot through its own pivot, and while the pin is in the slot, the slot must point at the pin. There is nothing to iterate — the follower’s angle is the direction of the line from its pivot to the pin — and this field solves both mechanisms with the same eight lines.

Which is a claim rather than a resemblance, so it is checked. The same function is handed a Geneva drive and required to reproduce the independently written solution the cams field has used since its own phase: sixty positions across a full engagement, agreeing to $10^{-14}$ radians.

The difference is what each mechanism wants from the pair. A Geneva wants the pin to enter along the slot, so the driven wheel starts from rest; that requirement fixes its centre distance and everything downstream of it. A lever wants the pin to enter, deliver an impulse and get out again as fast as possible, and does not care about starting from rest at all, because the thing being started is not at rest — it is a balance going past at full speed.

The sine rule decides how long they touch #

Three lengths make the connection: the roller radius $r$ on which the pin sits, the fork radius $f$ at which the slot’s mouth is, and the distance $c$ between the two pivots. The pin is in the fork while its distance from the lever’s pivot is less than $f$, and the two instants at which that distance equals $f$ are the entry and the exit.

At those two instants the lever, the balance’s centre and the pin form a triangle with sides $c$, $r$ and $f$. The lever’s half swing is the angle at the lever’s pivot; the balance’s half swing is the angle at the balance’s. Two angles of one triangle, so the sine rule ties them:

$$\frac{\sin A}{\sin B} = \frac{r}{f}$$

and it holds to twelve figures across every proportion tested.

That is a design law rather than a rule of thumb, and it says something quite strong. The pallets fix $A$ — the lever must swing exactly far enough to carry the pallets through their lock and their lift, which for the escapement here is $2(1.75° + 1.5°) = 6.5°$ and nothing else. So the only remaining freedom is the ratio $r/f$, and it alone decides $B$.

The lever swing column is the check: every row delivers exactly the 6.5° the pallets asked for, which is the condition the centre distance was solved from.

Forty-four degrees #

The middle row is worth stopping on.

A mechanical watch’s lift angle — the angle the balance turns through while the escapement is acting on it — is a specification number. It is stamped in service manuals, it is what a timing machine is set to before it can read an amplitude, and for a Swiss lever it is 44° or thereabouts. It is the sort of number that looks like a convention.

It is not. Put a roller of 0.15 of the fork’s radius on a lever whose pallets need 6.5° of swing, solve the triangle, and the balance’s lift comes out at 44.41°. Nothing in that calculation was told what the answer should be: the inputs were a tooth count, an arbor position, a lock, a lift and one ratio of two lengths, and the output is the number the industry quotes.

What the ratio 0.15 is, is where two pressures meet, and they pull in opposite directions.

Upwards, because a larger roller detaches the balance for longer — that is the sine rule, and the table shows it: 87.2% at 0.10, 91.8% at 0.15, 96.6% at 0.35.

Downwards, because a larger roller barely enters the fork. The pin’s deepest point below the fork’s mouth, as a fraction of the roller’s own radius, runs 0.19 at $r/f = 0.10$, 0.085 at 0.15, 0.041 at 0.22 and 0.018 at 0.35. At the top of that range the pin grazes the mouth rather than sitting in the slot, and an engagement that shallow is one a shock can throw out of. And there is a hard floor at the bottom: below $r/f = 0.057$ the roller cannot swing the fork as far as the pallets need at all, and the library refuses to build the mechanism rather than returning a smaller swing.

So 44° is not a compromise between two arbitrary preferences. It is where a curve that wants the roller large meets a curve that wants it small, and both curves are trigonometry on three lengths.

The idealisation, stated #

The fork here is a radial slot, which is exactly what makes it the Geneva’s pair, and a real fork’s slot has straight parallel sides. The pin bears first on one side and then on the other, and the difference between the two models is a fraction of the pin’s own diameter.

It does not change either angle quoted above, because both are decided by where the pin enters and leaves, and the mouth of a straight-sided notch is at the same radius as the mouth of a radial one. What it changes is the detail of the impulse in between, which is the part of a lever escapement most carefully drawn on a real design and least relevant to what this essay measures.

Worth saying because the alternative — quietly modelling a radial slot and calling it a fork — is how a resemblance becomes a claim without anybody deciding.

Safe while detached #

For 91.8% of every beat, the escape wheel is held by a pallet on a lever that nothing is holding. Something has to keep the lever where it is, or the wheel escapes at the wrong moment and the watch gains a beat.

Three pieces of geometry do it, and all three are lengths and angles.

The first is draw, and this is why a lever escapement carries ten to fifteen degrees of it where a clock’s deadbeat carries one and a half. The tilt of the locking face means the wheel’s own torque pulls the lever hard against its banking pin and holds it there. The previous essay showed that draw costs recoil — and here it costs nothing, because the lever is not moving. Recoil is expensive when the thing recoiling is attached to the oscillator. In a detached escapement it is a single settling movement as the tooth lands, and then stillness.

The second is the banking: two pins that stop the lever’s travel just beyond the position the fork needs to reach. Draw pushes the lever onto them and there it stays.

The third is the guard pin and safety roller, and it is the one that has no analogue anywhere else in this field. A pin on the lever runs close to a disc on the balance staff; if the lever tries to leave its banking while the balance is away, the guard pin meets the disc and cannot pass. The disc has a small crescent cut in it, positioned so that the guard pin is opposite the crescent exactly when the impulse pin is entering the fork — which is the only moment the lever is supposed to move.

That is a higher pair used as an interlock, and its whole specification is two clearances: how close the guard pin runs to the roller, and how wide the crescent is. Both are lengths, both are drawn, and the mechanism’s most safety-critical behaviour is decided by them.

What the lever is, counted #

It is worth putting the mechanism through the site’s first instrument, because a lever escapement has more parts than anything else in this field and the count comes out unremarkable.

Four moving links — the escape wheel, the lever, the balance, and the frame they all turn in. Three revolutes, one at each arbor. And, at any given instant, either one higher pair (a tooth on a pallet, with the fork empty) or two (a tooth on a pallet and the pin in the fork). Grübler’s planar count gives

$$M = 3(4-1) - 2(3) - j_2 = 9 - 6 - j_2$$

which is two freedoms while only the pallet is in contact and one while the pin is in the fork as well.

Two freedoms is the correct and slightly surprising answer for the detached phase: the balance and the lever are genuinely independent then, and the mechanism is not one machine but two that happen to be near each other. It is also why the safety action has to exist. A mechanism with two degrees of freedom and only one of them being driven has a second one that will do whatever it likes, and the guard pin is the constraint that decides what that is.

Three states, and the transitions between them #

The lever escapement is the mechanism in this field with the most contact states, and listing them is the shortest description of what it does.

Locked and detached. A tooth on a locking face, the lever hard against a banking pin, the fork empty, the balance swinging freely somewhere out in its arc. This is 91.8% of the beat.

Unlocking. The impulse pin has entered the fork and is driving the lever away from its banking; the tooth is sliding along its locking face towards the corner, and the wheel is recoiling by the amount the draw dictates. This is the balance paying for the safety of the previous state.

Impulse. The tooth is on the impulse face and the wheel is driving the lever, which is driving the fork, which is driving the pin, which is driving the balance. The train’s whole contribution to timekeeping happens here.

Then the tooth drops, the lever hits the other banking pin, the pin leaves the fork, and the mechanism is in the first state again with everything mirrored.

Three states in a cycle of six, each with its own contact set and its own equations, and the transitions between them fixed by geometry: the pin’s entry by the fork’s mouth radius, the unlocking’s end by the corner of the pallet, the impulse’s end by the length of the impulse face. Every boundary is a length.

The fraction carries an amplitude with it #

The 91.8% is quoted as a property of the escapement and it is not one on its own: it is a ratio between an angle the escapement fixes and an angle the balance chooses, and only the first is in the mechanism.

The escapement acts over the lift angle, 44.4° here, and that number is settled by the sine rule from three lengths — it is the escapement’s own. The balance’s beat carries it from one extreme of its swing to the other, which is twice the amplitude. So the detached fraction is one minus the lift angle over twice the amplitude, and the 91.8% is what that expression returns at an amplitude of about 270°, which is the figure a healthy watch runs at and is nowhere in the escapement’s geometry.

Follow it down and the consequence is direct. At 200° of amplitude the balance’s beat is 400° and the detached fraction falls to 88.9%. At 180° it is 87.7%. A watch losing amplitude — a mainspring running down, a thickening oil, a knock that has cost it some swing — is a watch becoming progressively less detached, and it becomes less detached in the region where its timekeeping is already suffering for other reasons.

That is worth having explicitly, because the usual account of why low amplitude spoils rate reaches for the balance spring or for isochronism and does not mention this. The escapement’s interference is a fixed angle, and the fraction of the beat it occupies grows as the amplitude shrinks. Both quantities are geometric and both are computable here; only their ratio is the number anybody quotes.

It also gives the lift angle a second reading as a design number. A small lift angle is worth having not merely because it detaches the balance for longer at full amplitude but because it degrades more slowly as amplitude falls: the fraction lost is the lift angle divided by the swing, so halving the lift angle halves the loss at every amplitude at once. That is an argument for the small roller ratio that sits alongside the two the sine rule already supplies, and it points the same way as one of them and against the other.

And it explains why the lift angle is the number a timing machine has to be told. A machine measuring a watch’s rate infers its amplitude from the escapement’s own signals, and the inference needs the lift angle as an input — the same 44.4° computed here, entered by hand for each calibre. The number is on a specification sheet because it is the one piece of the escapement’s geometry that an instrument outside the watch cannot see and cannot do without, which is an unusually direct use for a quantity derived from three lengths and a sine rule.

What the fraction does not say #

The 91.8% is a fraction of angle, and the honest caveat is that a beat is not swept at a uniform rate: the balance is slowest at its extremes and fastest through the middle, and the fork engagement is centred on the middle, where it is fastest. So the fraction of time the balance is detached is higher than 91.8%, and computing it needs a velocity, which needs a torque, which is not here.

That is a rare case on this site of the kinematic number being the conservative one. The angular fraction understates how detached the balance is, so a design decision made on it errs in the safe direction, and the dynamical correction only ever improves the answer.

The other thing the fraction does not say is anything about how much the escapement disturbs the balance while it is engaged, which is the question the whole arrangement exists to answer and is entirely a question about forces. What geometry settles is when the disturbance can happen and how long it lasts. What it costs while it lasts is somebody else’s subject.

That division is worth one more sentence, because it is unusually clean here. Everything a watchmaker adjusts on a lever escapement is a length or an angle: the depth of lock, the draw angle, the banking pin positions, the guard pin clearance, the crescent width, the roller radius. Everything a watchmaker measures is a rate or an amplitude, and neither is a length. The mechanism sits precisely on the boundary this site keeps, with its whole specification on one side of it and its whole performance on the other — which is why it can be designed on a drawing board and cannot be finished on one.

A small note on how the fraction should be quoted, following from that. An escapement’s own number is the lift angle, and it is the one that belongs on a data sheet; the detached fraction is a derived figure that needs an amplitude beside it or it is not a statement about anything. Quoting 91.8% without the 270° is the same species of omission this site’s wrong field collects — a value of a function, offered as though the function were a constant — and it is worth noticing that it happens here in a mechanism whose other numbers are unusually well behaved.