The vibration that turns part of the way

The compass that finds the axis the Earth turns about built a north-finder from a spinning wheel, and closed by asking how small a gyroscope could be made before its own drifts swamped the Earth’s rotation. Every gyroscope in that essay had something spinning in it. The answer the instrument makers found to the question of smallness was to stop spinning anything. A ring or a shell that merely rings, like a wine glass struck with a fingernail, feels its rotation through the same Coriolis force that pushes a spinning wheel’s axle sideways, and the way it shows that rotation is one of the odder facts in mechanics. Its pattern of vibration turns with it, but only part of the way.

A glass that beats when it turns #

In 1890 George Bryan, a Cambridge mathematician, published an analysis of a puzzle anyone can reproduce with a wine glass. Strike the rim so that it rings, then turn the glass slowly on the table. The note, which was steady, begins to beat — to swell and fade at a slow, regular rate. Nothing about the glass has changed except that it is turning.

The beating means that the pattern of the vibration is moving past the listener. A struck glass vibrates in its lowest mode by squashing its rim alternately into two ellipses at right angles, so that four points round the rim — the nodes — stay still while four points between them swing in and out. An ear beside the glass hears the rim at its location: loud near a swinging point, silent near a node. If the pattern turned exactly with the glass, the ear would always face the same part of the rim and hear a steady note. If the pattern stayed fixed in the room, the glass’s turning would not matter at all. The beats say the pattern moves relative to the room — and Bryan found that it does so at a definite fraction of the rotation.

For a thin ring the pattern of the lowest mode turns through three-fifths of the ring’s angle. Measured from the ring, it slips backward by two-fifths: a quarter turn of the ring leaves the pattern 36° behind where the material it started on has gone. The fraction does not depend on how fast the ring turns, on how hard it was struck, on the material, or on the size. It depends only on which vibration it is.

The beats can be counted. Turn a glass once round every ten seconds and its pattern turns through three-fifths of that, twenty-two degrees a second, past the listener’s ear. The sound reaching the ear from the nearest part of the rim falls silent each time a node passes, and the nodes of the lowest mode are forty-five degrees apart, so the note dies away about once every two seconds — slower than the turning itself, and a direct measurement of the fraction for anyone with a watch. Bryan noticed the same beats in rotating cylinders and in bells; he was explaining why a vibrating object turned in the hand does not keep its note.

Two vibrations of the same pitch #

The reason begins with a symmetry. On a perfectly round ring, a vibration pattern can sit at any angle: rotate the elliptical pattern by any amount and it is an equally good vibration with exactly the same frequency. Any one of them can be written as a mixture of two — one with its swinging points at 0° and 90°, the other at 45° and 135° — so the ring has a pair of modes with the same frequency, a degenerate pair, and the angle of the pattern says in what proportion they are mixed.

The two pendulums that swap found that two oscillators of equal frequency, coupled by anything at all, no longer keep their separate motions: the coupling picks out two combinations that keep their shape, at slightly different frequencies, and any other motion beats between them. Rotation supplies such a coupling. Each bit of the rim is moving — outward and inward, and also along the rim, since an inextensible ring that bulges in one place must shuffle material sideways to do it — and in a turning frame the forces that are not there include a Coriolis force on every moving bit, at right angles to its motion. The Coriolis force on one mode’s motion has exactly the shape of the other mode, so it feeds each into the other.

The combinations that keep their shape under that coupling are not standing patterns. They are waves running round the ring, one forwards and one backwards, and the Coriolis force shifts their frequencies in opposite directions. A standing pattern is the sum of the two, and when one runs slightly faster than the other the place where they add constructively — the pattern’s antinode — drifts round the ring. That drift is the pattern’s slip. Working it through for the nth mode of a thin ring, Bryan found that the pattern slips backward relative to the ring by

$$K = \frac{2}{n^2 + 1}$$

of the ring’s rotation, so that in space it turns through $(n^2 - 1)/(n^2 + 1)$ of it.

Where the two-fifths comes from #

The factor can be seen without solving anything. In the nth mode of a thin ring, a point on the rim moves outward and inward by an amount that varies round the ring as $\cos n\theta$, and because the rim does not stretch, the same point also moves along the rim by an amount $1/n$ as large, varying as $\sin n\theta$. Most of the motion is radial; a fraction is tangential, and the fraction shrinks as $n$ grows. Of the kinetic energy, $n^2/(n^2+1)$ is in the radial motion and $1/(n^2+1)$ in the tangential.

In a turning frame the Coriolis force on a moving bit of rim is perpendicular to its velocity, so it turns radial motion into a tangential push and tangential motion into a radial one. Its effect on the pattern comes from the product of the two — the radial swing of one mode pushed sideways into the shape of the other, and the other’s tangential swing pushed back — and adding the two cross terms round the ring gives a coupling of $2n/(n^2+1)$ times the rotation rate, between amplitudes whose pattern on the ring turns at $1/n$ of their own phase angle. The slip of the pattern is the coupling divided by $n$: $2/(n^2+1)$. For the ellipse, four-tenths.

The ingredients are the ring’s own geometry and nothing about its material. A steel ring and a glass one, a ring a metre across and one a millimetre across, have the same factor, because inextensibility fixes the ratio of tangential to radial motion and the Coriolis force does the rest. That is why the factor is a reliable constant of an instrument rather than something to be calibrated: it is fixed when the shape is.

From a pendulum to a high note #

The formula has a revealing first entry. The $n = 1$ “mode” of a ring is not a bending at all: it is the whole ring swinging bodily back and forth, which is a pendulum, or a mass on springs free to swing in any direction. Its pattern — the direction of its swing — follows none of the rotation. It stays fixed in space while the support turns beneath it, which is precisely Foucault’s pendulum at the North Pole: the floor turns once a day and the plane of the swing does not. The deflection that closes on itself found the same rule for any body moving freely in a rotating frame. Bryan’s factor is the generalisation of Foucault’s pendulum to an object that is not free, and the higher modes follow the material more because more of their motion is bending of the rim in place, which the Coriolis force does not see, and less is the sweeping of material round the ring, which it does.

The figure’s dots are not drawn from the formula. They come from integrating the two equations that describe the ring’s pair of modes in a turning frame, starting a pattern at one angle and reading its angle back after six hundred cycles; the integration gives 0.000, 0.600, 0.800 and 0.946 for $n = 1, 2, 3$ and 6, the formula’s values to three figures. The arithmetic of the coupled equations is all the effect is.

Real objects are not thin rings. A wine glass is a shell with a stem; its lowest mode still squashes the rim into ellipses, but some of the motion is in the bowl’s sides, and the factor is different. For a hemispherical shell fixed at its pole — the shape the most precise vibrating gyroscopes use — the corresponding slip factor for the lowest mode is about 0.28 rather than 0.4. What does not change is that the factor belongs to the shape of the vibration and nothing else.

An instrument that reads an angle #

That last property is what makes a ringing shell an instrument of a special kind.

A gyroscope of the familiar kind, or the tiny vibrating ones in a telephone, measures a rate of rotation: a signal proportional to how fast it is turning, which electronics then add up over time to give the angle turned. Every error in the rate signal is added up with it, and the angle wanders. A freely ringing symmetric shell does something different. Its pattern angle, read from pickups fixed to its case, is minus the slip factor times the angle the case has turned, at every instant, however the turning was done — fast or slow, forwards and back. The physics does the adding up. If the instrument is turned by 90° and back by 135°, the pattern ends 18° from where it began, without any rate having been measured.

Such rate-integrating gyroscopes — hemispherical resonators of fused quartz a few centimetres across, ringing at a few thousand hertz in a vacuum, their patterns sustained by gentle electrostatic pushes and read by electrodes round the rim — are among the most stable rotation sensors made. With nothing to wear, no bearings and no spinning mass, they run for decades, and they have flown on many spacecraft, among them the James Webb Space Telescope, holding patterns with drifts of thousandths of a degree an hour. Silicon rings a few millimetres across, made by the same etching processes as computer chips and running on the same principle, steer cars and stabilise cameras.

On a turning planet #

An instrument fixed to the ground turns with the Earth, once a day about the polar axis, and a ringing resonator with its axis vertical feels the component of that rotation along its axis, the Earth’s rate times the sine of the latitude. Foucault’s pendulum turns its plane at that rate relative to the floor, about eleven degrees an hour in Paris. A ringing ring in the same place slips its pattern back at four-tenths of that, four or five degrees an hour, and a hemispherical shell at about 0.28 of it. That slip, for an instrument on a stationary vehicle, is how it finds which way is north: tilt its axis in different directions and the rate is largest along the Earth’s axis, as the compass that finds the axis the Earth turns about found for a spinning wheel.

Foucault’s own pendulum is the $n = 1$ member of the family, and the fact that its plane turns at the sine of the latitude rather than at the full daily rate is a statement about geometry: the plane of swing is carried round a cone as the Earth turns, and it comes back after a day rotated by an angle set by the area that cone encloses, the effect the phase that is only a shape describes for light. The ringing ring’s slip has the same geometric origin with a factor attached. Optical gyroscopes measure the same rotation a third way, by the ring where the two beams disagree: light sent both ways round a loop takes different times, and the difference is a rate. The loop reports a rate, which must be added up; the pendulum and the ringing shell report an angle, and the shell does it in a few centimetres.

The flaw that holds the pattern #

Everything above assumed a perfect ring, whose pattern can sit at any angle at the same frequency. A real ring is slightly heavier on one side, or slightly stiffer in one direction, and then the two orientations of the pattern have slightly different frequencies.

The split acts like a pair of preferred angles that the pattern prefers to sit at, and the rotation tries to drag it past them. For slow rotations the flaw wins: the pattern rocks back and forth about the stiff axis and makes no progress at all, and the instrument is blind. The threshold is where the rotation’s drag, $K\Omega$ on the ring, equals the split divided by twice the mode number. Above it the pattern turns again — at first faster than a perfect ring’s would, since the flaw’s beating adds to the drift once it no longer holds it, the excess fading as the rotation grows.

That dead band is the gyroscope designer’s main enemy. A rotation of the Earth’s rate, fifteen degrees an hour, is a few parts in a hundred million of a few-kilohertz vibration frequency, so for the instrument to see it at all, the frequencies of its two modes must agree to better than that. Shells are made as symmetric as possible and then trimmed — by removing tiny masses from the rim, by laser or by etching, while the frequencies are measured — until their two modes match to parts in a million or better, and electronics apply small electrostatic corrections for the rest. The same coupled-oscillator mathematics that the two pendulums that swap worked through in a laboratory exercise decides whether a navigation instrument can find north.

What the thin ring leaves out #

The model behind every figure is a thin, perfectly elastic ring vibrating in its own plane, with its rim inextensible, its two modes undamped except where a split is introduced, and a rotation small compared with the vibration frequency. Shells, bells and glasses vibrate in three dimensions, their walls stretch a little, and the factor $2/(n^2+1)$ becomes a number that must be calculated for each shape — about 0.28 for a hemisphere, different again for a goblet with a thick stem. Damping matters too: a ringing glass dies away in a second or two, a quartz resonator in a vacuum rings for minutes, and in an instrument the vibration is kept going by drive electrodes, which themselves can push the pattern if they are not perfectly symmetric. The drift of a real hemispherical gyroscope is set by these residual asymmetries in damping and drive, not by the Coriolis physics, which is exact.

The figures also show only the slow motion of the pattern. The pattern’s slip is a small effect riding on a fast vibration — a few thousand cycles a second against a rotation that, for navigation, is a fraction of a cycle an hour — and the averaging that turns the coupled equations into a smooth drift assumes the rotation changes slowly compared with the vibration. For violent manoeuvres that is not true, and the full equations, not the slip factor, govern what the instrument reports.

The domain of the argument is a symmetric resonator whose two orientations of one mode share a frequency to within the rotation’s coupling, turned slowly compared with its vibration. Within it, the pattern follows a fixed fraction of the rotation, and the fraction belongs to the mode.

Still open: how small a resonator can keep its symmetry #

The best vibrating gyroscopes are centimetre-sized shells of fused quartz, made and trimmed with great care, and they are expensive. The question for the next generation is whether chip-scale resonators — millimetre rings and shells of silicon or glass, made by the million — can be made symmetric enough, and lossless enough, to integrate angle as well. Small resonators have more surface for their size, and surface losses and asymmetries grow in relative importance as the size shrinks; three-dimensional micro-shells blown from glass and trimmed by laser have reached frequency matches and ringing times that would have been thought out of reach a decade ago. Whether they can approach the stability of their larger cousins, and so put a north-finder of the kind the gyrocompass essay asked about into something the size of a fingernail, is being decided now.

The physics they all exploit is Bryan’s. The two equal-frequency orientations of a ring’s vibration are coupled by the Coriolis force, so when the ring turns, the pattern slips back by $2/(n^2+1)$ of the rotation and turns in space through $(n^2-1)/(n^2+1)$ of it — none for a swinging pendulum, 0.60 for the ellipse a struck glass makes, 0.95 for the sixth mode — and a resonator left ringing reads the angle it has turned through as an angle, provided its two orientations share their pitch. A struck wine glass turned on a table is, in principle, a gyroscope; the beats are its readout.