The kick that does not change the beat

A pendulum on its own is a perfect clock for about a minute. The period that depends on the swing found its period fixed by its length and by gravity, with a small correction for the size of the swing; but every swing loses a little energy, to air and to the flexing of its suspension, and the swing dies away. A clock has to push it. The mechanism that does the pushing — and that also counts the swings, by releasing a toothed wheel by one tooth each time — is the escapement.

The push is the clock’s central contradiction. The pendulum is there because its period does not depend on anything but its own length, and the escapement must reach in, once a swing, and disturb it. The curve that does not ask where it started ended by naming the escapement as the problem that replaced Huygens’s cycloidal cheeks: a control problem rather than a geometry one. The first person to state that problem precisely, and to solve its simplest form, was George Biddell Airy, then a young professor at Cambridge and later Astronomer Royal. His paper of 1826 asked a single question: if a pendulum is given a small impulse, how much does it change the time of its next beat?

A clock’s state on a circle #

The cleanest way to see the answer is to draw the pendulum’s state as a point. A small-amplitude pendulum’s angle and angular speed, each scaled by the size of its swing, trace a circle once a period: at the bottom of the swing the angle is zero and the speed is greatest, at the ends the speed is zero and the angle greatest. The angle round the circle is the pendulum’s phase, and the time of its next beat is set by how far round the circle the point has to travel.

An impulse from the escapement increases the pendulum’s speed in the direction it is already moving, and on this diagram that is a straight jump upward. Where the jump happens decides what it does. At the top of the circle — the bottom of the swing, where the pendulum is at the centre and moving fastest — the jump points straight away from the circle’s centre. The point lands on a bigger circle at the same angle round it. The swing is larger and the phase is unchanged: the next beat comes exactly on time.

Anywhere else the jump has a component round the circle. Before the bottom of the swing, the upward jump carries the point forward in phase, and the pendulum arrives at its next beat early: the clock gains. After the bottom, the same jump carries it backward, and the clock loses. At the very ends of the swing, where the pendulum is momentarily still, a kick changes the timing and adds hardly any energy at all.

Airy’s two curves #

The geometry can be made exact for small impulses. If the pendulum’s motion is $\theta = A\sin\varphi$, so that $\varphi = 0$ is the bottom of the swing moving forward, a kick that raises its angular speed by $\Delta v$ shifts its phase by

$$\delta\varphi = -\sin\varphi\,\frac{\Delta v}{A\omega},$$

and adds energy in proportion to the speed at that moment, $A\omega\cos\varphi$.

The two curves are Airy’s theorem in one picture. Energy is best supplied at the bottom, where the pendulum is moving fastest, because the work an impulse does is force times distance and the distance the pendulum covers during a brief push is greatest there. Timing is least disturbed at the bottom, because the push is then purely radial on the circle. The best place for both purposes is the same place. That is not a coincidence: both follow from the fact that at the bottom the pendulum’s displacement is zero and only its speed is non-zero, and a push changes only speed.

The figure’s curves were computed for the idealised circle, and the essay’s first claim was checked against the real thing. A full, non-linear pendulum integrated step by step through a kick at three different points of its swing shifts its next beat by the predicted amount within a few per cent; kicked at the bottom, it shifts by less than one per cent of the shift a kick half-way out produces. The small-angle circle is a good map of a real pendulum’s timing.

Why a clock settles at one swing #

A pendulum that is pushed every swing and loses energy every swing does not need its pushes to be adjusted to keep a steady amplitude. It finds one. The energy a push adds is the impulse times the speed at which the pendulum meets it, and the speed at the bottom grows in proportion to the amplitude; the energy lost by air drag and by the flexing suspension, for a pendulum of fixed $Q$, grows as the square of the amplitude. At small amplitudes the pushes win and the swing grows; at large ones the losses win and it shrinks. Where the two lines cross is a stable amplitude, and a clock disturbed from it — by a door slammed, by a jolt to the case — returns to it within a few hundred swings.

That self-adjusting amplitude is what makes a clock an oscillator that keeps itself going rather than a pendulum that is merely kept going, and it is the reason every minimum is a parabola could describe an escapement as a device for holding the amplitude constant. It is also why Airy’s theorem matters so much. The amplitude a clock settles at depends on how hard it is driven, and the drive is never perfectly constant; if the push is off-centre, each change in the drive’s force becomes a change in the beat directly, and if it is centred, only through the amplitude.

The push is not the only way to feed a swing. The swing that is pumped, not pushed found a child on a playground swing adding energy by standing up at the bottom and crouching at the ends — changing the pendulum’s length, not pushing it — and that parametric pumping, too, works best at the bottom, where shortening the pendulum does the most work against the tension. The child, like the escapement, has to act at the right moment of the swing, and a child who stands at the wrong moment takes energy out instead.

There is also the question of where on the pendulum the push should land. An impulse applied to a rigid body that is free to pivot jolts its pivot unless it lands at the centre of percussion, which the point that takes the blow without a jolt located for a swinging rod. An escapement’s pallets act near the top of a pendulum rod, far from that point, so every push sends a small reaction into the suspension. A spring suspension takes the reaction by flexing, and the energy that flexing dissipates is one of the losses the escapement then has to replace. Precision clocks with their impulse applied low on the rod, or through the suspension itself, as Sigmund Riefler’s escapement of 1889 did, were attacking that loss.

What a push costs, and what hides it #

An escapement pushes once each swing, and the size of each push is not free: it must replace exactly the energy the pendulum loses in that swing, or the amplitude would grow or decay. The three ways of coming to rest measured that loss with the quality factor $Q$: a pendulum loses $2\pi/Q$ of its energy each period. Replacing it with two pushes a period at phase $\varphi$ needs $\Delta v/A\omega = \pi/2Q\cos\varphi$, and the phase shift that comes with those pushes makes the clock’s fractional rate error

$$\frac{\delta T}{T} = \frac{\tan\varphi}{2Q}.$$

The $Q$ in the denominator changes how the escapement problem should be thought about. A pendulum that loses little energy needs only a small push, and a small push can only shift the beat by a little, wherever it lands. An ordinary longcase clock pendulum has a $Q$ of a few thousand; a precision regulator in a sealed case, ten thousand or more; a pendulum swinging in a partial vacuum, with a flexing spring suspension instead of a knife edge, a hundred thousand. The same imperfect escapement on each would cost seconds a day, tenths of a second and hundredths. Much of the history of precision clocks is the history of raising $Q$, so that the escapement’s inevitable faults matter less.

The same reasoning applies to the push’s variations. No escapement delivers exactly the same impulse every swing: the force reaching it through the clock’s gear train changes as the driving weight descends, as oil thickens, as the teeth mesh slightly differently. If the push lands off-centre, every change in its size is a change in the timing. A push at the bottom, by Airy’s theorem, turns variations in force into variations in amplitude only.

The circular error that amplitude brings #

Variations in amplitude are not harmless either, because a pendulum is not perfectly isochronous.

The period of a pendulum swinging through an amplitude $A$ is longer than that of an infinitely small swing by about $A^2/16$, the circular error that the period that depends on the swing computed exactly and that Huygens’s cycloidal cheeks were designed to remove. A constant amplitude gives a constant error, which the clockmaker removes by adjusting the bob; it is changes in amplitude that matter, and at a swing of two degrees a change of a tenth of a degree costs two-thirds of a second a day. A push at the bottom, with a drive whose force wanders, turns every wander into a change of amplitude, and every change of amplitude into a change of rate through the circular error.

That is why precision pendulums swing through small arcs — a degree or two rather than the ten of a domestic clock — where the circular error and its slope are small. And it opens a more subtle possibility.

Two errors set against each other #

An escapement whose push lands at a fixed point of the swing, slightly after the bottom, has an error that changes with amplitude too. The push lands at a fixed angle $\theta_i$, and as the swing grows, that angle is a smaller fraction of it, $\sin\varphi = \theta_i/A$, so the delay each push causes shrinks. More drive means a wider swing, a larger circular error making the clock slower, and a smaller escapement error making it less slow. The two can cancel.

With the push at the bottom, a 20 per cent change in drive moves the rate by over five seconds a day, entirely through the circular error. With the push placed where the slopes of the two errors cancel, about three-tenths of the swing after the bottom for this pendulum, the same change of drive moves the rate by just over a second, and what is left is curvature that neither error can remove. With the push the same distance before the bottom, the escapement’s error adds to the circular one and the clock becomes twice as sensitive as the centred one. The cancellation needs a low enough $Q$ that the escapement’s error is large enough to matter; for a pendulum of very high $Q$ the push after the bottom that would be needed lies beyond the end of the swing, and the only option is to hold the amplitude constant.

That trade between where to push and how strongly the pendulum resists being pushed is the whole design space of escapements. The recoil escapement of the 1670s pushes the gear train backwards at each tick and pushes over much of the swing, with errors that partly compensate the circular error of its wide arc. George Graham’s deadbeat escapement of 1715 locks the train without recoil and pushes nearer the centre, and became the standard for observatory regulators. Gravity escapements, used in the clock of the Palace of Westminster from 1859, give the pendulum a push from a small weight lifted by the train, so the push’s size no longer depends on the train’s force at all.

The free pendulum #

The logical end of Airy’s analysis is a pendulum that is pushed as rarely and as cleanly as possible. William Hamilton Shortt’s free-pendulum clock of 1921 kept its master pendulum in a partial vacuum, touched it only once every thirty seconds, with a small gravity arm that dropped onto it near the bottom of its swing, and did all the counting and the work of driving the hands with a second, slave pendulum synchronised to the first by an electrical signal. Its error was about a second a year, and it was the most accurate clock in the world until quartz oscillators overtook it in the 1940s.

Every element of that design is one of the figures. The vacuum raises $Q$, so the pushes can be small. The push is given near the bottom, so it changes amplitude rather than timing. It is given only every thirty seconds, so the pendulum is free for nearly all its swings. And the pendulum does no work for the clock except to be timed, so nothing it drives can disturb it. The two pendulums that swap found that clocks hung on the same wall pull each other into step; Shortt turned that coupling into a deliberate correction, with the slave driven to follow the master.

The same theorem applies to the balance wheel of a watch, which is a pendulum whose restoring force comes from a spiral spring instead of gravity. A watch’s lever escapement is designed to give its impulse as near the centre of the balance’s swing as possible, and the residual errors are adjusted out by the watchmaker’s art of “isochronism”, which is circular error by another name. The precision a clock pays for in heat showed that any clock, however good, must dissipate energy to keep time; the escapement is where that energy goes in.

Small swings, single kicks and a clean pendulum #

The figures assume a pendulum whose swing is small enough to be drawn as a circle, given instantaneous impulses at a single point of its swing, with its losses described by a single quality factor. Real escapements push over a few degrees of the swing, not at a point, and the effect of a push spread out is the average of the curves drawn here over the push; real escapements also exert friction on the pendulum as they lock and unlock, which acts as an impulse opposite to the motion and has its own timing error. The rate figures take one push each swing, symmetric on the two sides, and the balance figure varies only the escapement’s force, not the losses. The circular error is computed exactly for a simple pendulum; a real compound pendulum, with a suspension spring that stiffens as it bends, has a slightly different amplitude dependence that clockmakers measure rather than compute. The domain is pendulums swinging through a few degrees with quality factors from a thousand to a hundred thousand, and pushes delivered well inside the swing.

Still open: how far a mechanical clock can be pushed #

Mechanical clocks were overtaken by quartz and then by atoms, but the question of how good a pendulum can be was never quite closed. In 2015 a clock built to a design by John Harrison, whose eighteenth-century claims of a second in a hundred days had been dismissed for two centuries, ran for a hundred days at the Royal Observatory and kept within about a second. It does not use a free pendulum or a vacuum; it uses a light escapement acting over a wide swing and pushes that partly cancel the circular error, much as the last figure here suggests. Whether further gains are possible with modern materials, or whether the slow drifts of gravity, temperature and suspension set a floor near a second a year, has not been settled, and it is argued by a small community of horologists who build the clocks to find out.

The theory under all of it fits in one statement. An impulse $\Delta v$ given to a pendulum at phase $\varphi$ from the bottom shifts its beat by $-\sin\varphi\,\Delta v/A\omega$ and adds energy in proportion to $\cos\varphi$, so a push at the bottom changes the amplitude and not the timing; an escapement that must replace losses of $2\pi/Q$ each period then costs $\tan\varphi/2Q$ of rate — 8.8 seconds a day at a fifth of the swing with $Q = 1{,}000$ — and a push placed after the bottom can cancel the circular error’s dependence on the drive, cutting a 5.3-second spread to 1.3. A clock can push its pendulum without disturbing it, if it pushes in the right place.