A rough road rings the suspension well enough to read, over four hundred metres

A scroll camera rings with its vehicle’s suspension put a two-slit scroll camera — two slits leaning ten degrees either way of upright, drawing the roadside as the vehicle drives — on a body sprung at 1.3 hertz and damped 0.3, and drove it over a step from level to six per cent. The body met the step with its front wheels, lagged its wheelbase chord, overshot it and rang down, and the rows of a line of roadside posts parted with every swing. Read to a tenth of a pixel, forty metres of posts gave the suspension back: 1.302 ± 0.011 hertz and a damping of 0.303 ± 0.008. A sharp change of grade, crossed briskly, was the best calibration a scroll camera’s suspension reading could have.

The essay ended on the input every road supplies without being asked. A road surface is uneven everywhere by a few millimetres over a few metres, and a body on springs responds to all of it continuously. The question was whether the body’s frequency can be found from the rows’ own spectrum — a peak at the speed over the frequency along the road — and how long a stretch of posts is needed before the peak stands above the reading error. If it could, a scroll camera would read its vehicle’s suspension on any road at all.

It can, from a few hundred metres of a good road. But not from the peak.

A road that is uneven everywhere #

The road is level on average, and its height carries random unevenness with the spectrum road surveys use: the power in each band of wavenumber falls as the square of the wavenumber, so long gentle swells are much larger than short ripples. Three levels are drawn, the ones a survey would call very good, good and average, with displacement spectra of 16, 64 and 256 millionths of a cubic metre at a tenth of a cycle a metre; a good road’s height wanders by about eight millimetres overall. The vehicle is the earlier essay’s, wheels 2.7 metres apart, at ten metres a second, and posts stand twelve metres out every quarter metre, drawn by the two-slit scroll a scroll through two slits ranges in a straight line built.

The rows such a road leaves are random too, and the first thing to settle is whether anything simpler than simulating them can describe them. It can. For unevenness this small the scroll is a linear system: a sinusoid in the road’s height comes back as a sinusoid in the rows, of the same wavenumber, with a gain and a phase that depend on the wavenumber alone. The gain has two parts. One is a rigid vehicle’s — the rows a body pitched exactly to its wheelbase chord would leave, which is the earlier essay’s wheelbase reading answering a sinusoid. The other is the body’s departure from its chord, which the spring sets, times the rows’ answer to a sinusoidal pitch. Both parts are computed from the scroll’s exact projection, once each; summed over the road’s sinusoids they reproduce the simulated rows of a random road to three ten-thousandths of a pixel, against a signal of a seventh of one.

The dots are the rows’ spectrum along the road, from 400 metres of posts read to a tenth of a pixel, averaged over windows of fifty metres. The dashed line is what a rigid vehicle would leave at the same level of unevenness, and the solid line what a sprung body leaves at the frequency and damping the fit chose: 1.283 hertz and 0.253, for this one road. The spring’s contribution is the difference between them, a broad swelling near 0.13 cycles a metre, which is the speed over the frequency — one ringing every 7.7 metres of road.

The slider changes the road. A very good road reads 1.275 hertz and 0.232; an average one, 1.288 and 0.258. Each is one road’s reading, and the figures below say how such readings scatter from road to road; all three land within two per cent of the true frequency and within a quarter of the damping.

The peak is a mixture #

The proposal was to find the frequency from the spectrum’s peak. The rows’ gain says why that would not work.

The scroll answers a rough road by itself, without any spring. A rigid vehicle’s rows, for posts twelve metres out, respond most to unevenness about six metres long. The rows read the difference between the wheelbase chord and the eye’s chord between the slits’ two moments, as the wheelbase essay found, and that difference answers some lengths of unevenness more than others; for this vehicle and these posts the strongest answer falls near six metres. The response peaks at 0.165 cycles a metre, close to where the spring rings. The spring’s own swelling is added on top, and the sum peaks somewhere between: at 0.120 cycles a metre for a damping of 0.3, at 0.105 for 0.5, and only for a damping as light as 0.15 does the peak sit at the spring’s own 0.13.

So the highest point of the rows’ spectrum is not the spring’s frequency, and a reading that took it for one would read 1.2 hertz for a body sprung at 1.3, or 1.05 for a well-damped one. The spring is in the shape of the whole curve against the scroll’s own, and reading it means fitting that shape. The fit here knows the scroll’s gains, the vehicle’s wheelbase and speed, and the road spectrum’s slope — the square of the wavenumber, which surveys find for almost every road — and it does not know the road’s level or its particular unevenness. It chooses the frequency, the damping and the level that make the predicted spectrum most likely to have produced the measured one, with the reading error’s floor known from the reading.

How much road it takes #

A hundred metres of posts is two windows’ worth of road, and it is not enough: on a good road the frequency scatters by ±0.3 hertz and the mean sits twelve per cent high, because a spectrum averaged over so little is too noisy for its swelling to be told from the scroll’s own. Two hundred metres gives ±0.053, four hundred ±0.018, eight hundred ±0.014. The fall is steep while each added length adds windows to average and slow once there are a dozen, where the remaining spread is set by how well the swelling’s shape is drawn at all. From 400 metres on, the mean is within one per cent of the truth on every road. An average road reads ±0.009 hertz from 800 metres; a very good one, ±0.027.

The step of the earlier essay gave ±0.011 hertz from forty metres. A good road’s own unevenness comes within twice that from 400 metres and near it from 800 — ten to twenty times the road for the same reading. The difference in size is plain in the rows themselves. Over the step, the earlier essay’s rows parted by up to 2.6 pixels within a few metres; along a good road, with no step, the rows wander by about a sixth of a pixel everywhere, a sixteenth of the step’s excursion. A reading’s information grows as the square of its signal and in proportion to how much of it there is, so a signal sixteen times smaller has to be read over a few hundred times more road to say as much; that it needs only ten or twenty times is the spectrum’s doing, which gathers every ringing the road provokes rather than the one a step provokes. That is the price of an input that is everywhere and small rather than once and large: a step kicks the body with all of its energy at once, and the rows record a clean ringing; a rough road kicks it gently all the time, and the ringing has to be found statistically, in the spectrum, against the scroll’s own response to the same road.

A window short enough to average is too short to resolve #

The rows’ spectrum has to be estimated by cutting the line of posts into windows and averaging their spectra. Short windows give many to average and a smoother spectrum; long windows give fewer and sharper ones. And a window has a resolution of its own: features of the spectrum narrower than about one over its length cannot be seen in it.

The body’s swelling at ten metres a second is about twice the damping times 0.13 cycles a metre wide, eight hundredths of a cycle a metre. A twenty-five-metre window resolves four hundredths, so it broadens the swelling by about half again. A fit that compares that broadened spectrum with a sharp model reads the broadening as damping: thirty-eight per cent too much, with the frequency pushed seven and a half per cent high. A fifty-metre window biases the damping by seven per cent, a hundred-metre window by three.

The bias is not a property of the road or the rows; it is a property of how the spectrum was estimated, and it can be removed where it was made. A fit that spreads its own model by the window’s power response before comparing it with the measured spectrum — blurs the prediction exactly as the window blurs the measurement — reads the damping within two per cent at every window, 0.307, 0.297 and 0.302 against 0.3, and the frequency within half a per cent. Every reading in this essay outside this figure uses that fit.

What the window still decides is the spread. Twenty-five-metre windows give more to average and a wider spread in the frequency, ±0.026 hertz against ±0.016 at fifty; a hundred metres gives ±0.017, as few long windows fix the swelling’s shape no better than many middling ones. Fifty to a hundred metres of window, several times one ringing’s length along the road, is the length to use.

The damping is read eight times worse than the frequency #

The frequency and the damping are not read equally well. From 400 metres of a good road read to a tenth of a pixel, the frequency’s spread is a little over one per cent of itself and the damping’s about ten per cent. The frequency sets where along the spectrum the body’s swelling sits, and its position is fixed by the whole swelling at once; the damping sets the swelling’s width, which competes with the window’s own blurring and with the scroll’s own response, whose shape near the swelling is not flat. Width is the harder thing to measure in a noisy curve, and the earlier essay found the same split for a step: the shape of a feature says one thing well, its decay another less well.

A fleet that wants to watch a damper wear is asking for the hard half. A damper that has lost a fifth of its damping moves the swelling’s width by a fifth and its position hardly at all, and at ten per cent a pass, a single drive over 400 metres says little; a week of daily passes, averaged, says it clearly. A scroll can be asked its own radius found the scroll answering a question about its own track from a single pass; this one answers about its springs, and asks for repetition.

The reading error’s floor #

The swelling has to stand above the floor that the reading error lays across the whole spectrum, and the floor rises as the square of the error. Read to a twentieth of a pixel, 400 metres of a good road give the frequency to ±0.012 hertz and the damping to ±0.02; to a tenth, ±0.016 and ±0.03; to a fifth, ±0.030 and ±0.07. At four tenths of a pixel the reading comes apart: ±0.27 hertz, the damping 0.33 ± 0.21, the swelling hardly distinguishable from the floor. An average road, with four times the unevenness, holds on longer: ±0.12 hertz at four tenths. A good road wants the rows read about as well as the earlier essay’s step did, to a tenth of a pixel or better.

What the road’s own roughness can tell #

Put together, a scroll camera on a sprung vehicle can read its suspension on a road with no ramp at all. The road’s ordinary unevenness rings the body, the body pitches the camera, and the rows’ spectrum carries the spring’s swelling on top of the scroll’s own answer to the road. Fitted against both, knowing only the road spectrum’s slope, 400 metres of posts along a good road read to a tenth of a pixel give the frequency to two hundredths of a hertz and the damping to about a tenth of itself; 800 metres to a hundredth and a half.

The price against a step is length: ten to twenty times the road for a comparable reading. The gain is that the road need not have anything on it. A scroll of a climbing road measures its grade read a road’s geometry from the rows; the wheelbase essay read the vehicle’s; the earlier essay read its springs from a single event. This reads them from the road’s texture, which every road has — and so a fleet whose cameras drive the same few hundred metres of ordinary road every day could watch a damper wear without ever looking for a ramp.

An eye that pitches with the road keeps its rows found the rows insensitive to a camera that pitches exactly with the ground under it; a sprung body pitches with neither the ground nor its chord, and the rows read the difference. The general point is the one a scroll is a camera that moves began from: a scroll camera’s picture is a record of its own motion as much as of the scene. Here the motion is a response to the scene — the road — and the reading has to separate what the road did to the picture directly from what it did through the spring. The separation is possible because the two have different shapes in the spectrum, and impossible from the spectrum’s highest point, where they are added.

What the fit takes from outside the pictures #

The road’s spectrum falls as the square of the wavenumber. The fit knows the slope and not the level. Roads depart from that slope, especially at the long wavelengths a vehicle’s springs respond to, and a slope wrong by a little tilts the predicted spectrum against the measured one; the fit would then put part of the tilt into the frequency. How large a slope error the reading survives was not measured here.

The body pitches as one oscillator. As in the earlier essay, a real vehicle bounces and pitches at two frequencies, and the rows would carry two swellings; a fit of one would land between them.

The vehicle’s speed is steady. A scroll’s paper advances with distance, as a map along, and a picture across found, so the rows are a function of where along the road each post stands; the spring’s swelling sits at the speed over the frequency along the road, so a speed that changes over the 400 metres smears it along the spectrum; a speed known from the odometer frame by frame can be used to resample the rows to time rather than distance, where the swelling stays put.

The scroll’s gains are computed for one depth of post. Posts at varied distances from the track have gains of their own, and a roadside of mixed furniture needs each feature’s gain at its own depth; the rows of a fence twelve metres out and a hedge twenty metres out do not share a spectrum.

Still open: whether two passes at two speeds separate the spring from the road #

A road’s unevenness is fixed in the road; the spring’s swelling sits at the speed over the frequency, so it moves along the road’s spectrum when the speed changes. The scroll’s own response to the road does not move. Two passes over the same few hundred metres, at two speeds, would see the scroll’s response in the same place both times and the spring’s swelling in two places.

The measurement that settles what that is worth drives the same stretch of rough road at two speeds, say eight and fifteen metres a second, fits the two spectra together with one frequency and one damping, and asks whether the second pass narrows the reading more than doubling the length of a single pass would — and whether it frees the fit from needing the road spectrum’s slope, since a road’s unevenness, seen twice at two speeds, can be read from the part of the rows that does not move.