A body that bounces leaves a second swelling in the rows, and one pass reads both

One pass reads the road’s slope as well as its spring took a two-slit scroll camera on a sprung vehicle along a rough road and read the suspension from the spectrum of the rows of a line of posts: the scroll’s own answer to the road, a hump fixed by the posts’ depth and the wheelbase, plus a swelling where the spring rings. One 800-metre pass read the spring’s frequency, its damping and the road’s spectral slope together, and a second speed added only what a second pass adds.

Every reading treated the body as one oscillator in pitch. A car on four springs has at least two motions worth the name. It pitches about its middle, nose down and tail up, and it bounces up and down as a whole, at a somewhat different frequency, and a road’s unevenness excites both. The scroll’s rows read the camera’s pitch directly. A bounce changes the eye’s height rather than its pitch, and a scroll of a climbing road measures its grade found the rows answering a change of height only through the height the eye climbs between the slits’ two moments — a weaker answer, but not none. The essay ended by asking whether the rows’ spectrum then carries two swellings or one, where a fit of one oscillator lands — between them, it guessed, or on the pitch alone — and whether a fit of two reads both frequencies from a single pass.

Two modes, and the camera’s lever arm #

The vehicle is the earlier essays’: wheels 2.7 metres apart, a two-slit scroll camera looking sideways at a line of posts twelve metres out every quarter metre, rows read to a tenth of a pixel. Its body now has two modes. The pitch follows the chord between the wheels through a spring of its own, as before. The bounce follows the mean of the road under the two wheels through another. And the camera stands some distance ahead of the body’s centre, as a dashboard camera does, so that when the body pitches the camera rises and falls with the nose as well as turning.

Every part of that enters the rows linearly. Against the rigid vehicle every earlier essay used — eye at the road’s height plus a constant, turned to the chord — the rows gain the pitch’s departure from the chord through the camera’s turning, and the eye’s departure from its rigid height through the camera’s lifting: the bounce’s own, plus the lever arm times the pitch. The rows’ gain for a lift is a measured quantity like the others, a sinusoid in the eye’s height over a level road carried through both slits, and it turns out to be of the same order as the pitch’s: a metre of lift moves the rows by between a third and the whole of what a radian of pitch does, depending on the wavelength, and the bounce moves the body by about a metre per metre of road near its resonance while the pitch turns it by less than a radian. A bounce should therefore be visible.

It is. The body here pitches at 1.4 hertz and bounces at 1.0, both damped 0.3 — a car’s usual order, bounce below pitch — and at ten metres a second they ring at 0.14 and 0.10 cycles a metre of road, three or four of the spectrum’s points apart. With the camera 1.5 metres ahead of the centre the rows’ spectrum carries a broad swelling with a shoulder on its long-wave side; the two-mode fit draws through every point of it and reads the pitch at 1.402 hertz damped 0.295 and the bounce at 1.022 damped 0.280. The one-oscillator fit draws a single swelling at 1.420 hertz damped 0.270 that misses the shoulder. With the camera over the centre the two swellings are closer in strength, and the one oscillator falls between them.

One oscillator lands where the camera puts it #

The earlier essay guessed the one-oscillator fit would land between the modes. Whether it does depends on where the camera is.

Over twenty roads, with the camera over the body’s centre, one oscillator reads 1.22 ± 0.01 hertz and a damping of 0.19 — between the bounce at 1.0 and the pitch at 1.4, as guessed, and far too lightly damped for the single broad swelling it is trying to cover. Half a metre ahead, 1.30; a metre, 1.36; a metre and a half, 1.41; two and a half, 1.47. The lever arm adds the pitch’s own lift to the pitch’s turn, so the pitch’s swelling grows with the arm until the single oscillator sits on it — and then past it, because the lever arm itself biases a one-oscillator fit, as a later section shows. The damping stays low throughout, 0.23 to 0.26, because the oscillator is folding the bounce’s power into a width it can only explain as a light damping.

The two-mode fit, told where the camera stands, reads the pitch at 1.40 and the bounce at 1.00 or 1.01 at every place, the pitch to ±0.02 hertz and the bounce to between ±0.015, over the centre, and ±0.04, two and a half metres ahead. The bounce is read best with the camera over the centre, where nothing but the bounce lifts the eye; ahead of the centre the pitch’s lift mixes into it.

So the one-oscillator readings of the earlier essays were readings of the pitch only for a camera well ahead of the centre, and even then the damping was off. For a camera near the centre, the frequency they reported was neither mode.

How close the two modes can be #

The two modes here are four tenths of a hertz apart. Real cars sit anywhere from a few tenths to nearly none.

With the bounce at 0.8 or 1.0 hertz the fit reads it to ±0.03. At 1.2 the spread is ±0.10, at 1.3 ±0.15, at 1.35 ±0.17: a tenth of a hertz apart, two swellings each as wide as a damping of 0.3 makes them overlap into one, and their two halves trade against each other. The pitch is read to ±0.02 throughout, because with the camera ahead of the centre its swelling is the stronger, and the bounce is read from what the pitch does not explain. So the two-mode reading has a resolution, set by the damping: modes closer than the width a damping gives each are seen as one broad swelling and a guess at how to split it.

The lever arm biases even a body that does not bounce #

The lever arm turned out to matter on its own, and it is worth a figure because it reaches back to every earlier reading.

A body that pitches at 1.4 hertz and has no bounce at all, its camera over the centre, is read at 1.40 by the earlier essays’ fit, as it should be. Put the camera half a metre ahead and the same fit reads 1.47; a metre and a half ahead, 1.50; two and a half, 1.52. The camera ahead of the centre rises and falls with the nose, the rows answer the rise through the height climbed between the two moments, and a fit that does not know about the rise reads its extra answer as a stiffer spring. Told the lever arm, the fit reads 1.39 or 1.40 at every place.

The earlier essays placed the camera at the body’s centre, so none of their numbers carries this bias; a real installation, behind a windscreen or on a roof rack, would. The lever arm is a length a tape gives, and once known it costs the reading nothing — which is the same lesson a vehicle’s pitch lags the road by its wheelbase drew about the wheelbase: a length of the vehicle enters the rows, and a fit that assumes the wrong one reads the error as the spring.

One pass says whether the body bounces #

A fit with more numbers always fits at least as well, and a reader should ask whether the second mode is there or merely fitted.

The pass answers decisively. Each pass is read twice, by the pitch alone with the camera’s lever arm known and by pitch and bounce together, and the difference in their log-likelihoods is counted in independent windows of road. With a bounce at 1.0 hertz and the camera 1.5 metres ahead, the bounce mode adds between 92 and 125. Over the centre, between 51 and 69. With the bounce at 1.3 hertz, close to the pitch and read only to ±0.15, it still adds between 93 and 123: the fit cannot say exactly where the bounce is, but it can say there is one. On a body with no bounce, the two-mode fit’s best is the stiff bounce its model contains as a limit, and it finds it on every road: the bounce mode adds nothing.

That is the property a reading needs before anybody should trust its second number. A camera count needs a tolerance made the general point for counting cameras behind a picture: two of anything fit better than one, and the count exists only against what the extra freedom buys by chance. Here the extra freedom buys nothing on a body without the mode and a great deal on a body with it, so the count of modes is read as well as their frequencies.

Why the rows can see a lift at all #

It is worth saying why a sideways-looking scroll sees the camera’s height at all, because the earlier essays treated the pitch as the only motion that mattered.

The two slits of the scroll draw each post at two moments, a little apart along the road, and the rows record the difference between the post’s two drawings. A camera that turns between the two moments moves every post’s drawing by the turn, and the rows answer that strongly: that is the pitch. A camera that rises between the two moments sees every post from a little higher at the second, and a post’s foot and top both drop a little in the picture, by an amount set by how far away the post is. A scroll of a climbing road found exactly that response to a steady climb; a bounce is a climb that goes up and down, and the rows answer it at each wavelength of road by the height gained over the distance between the two moments.

The camera’s lift and the camera’s turn are thus two separate channels into the rows, with different shapes across the spectrum. The road’s unevenness drives both, the pitch through the chord between the wheels and the bounce through their mean, and those two have different shapes again: the chord is strongest for unevenness about two wheelbases long and blind to unevenness exactly one wheelbase long, while the mean is strongest for long swells and blind at two wheelbases. So the two modes are not merely two swellings at two frequencies; each comes through the scroll with its own signature across the spectrum, and the fit is reading signatures, not peaks. That is the reason the earlier essay’s one pass could read the road’s slope too: a rough road rings the suspension for the rows in several shapes at once, and the whole spectrum tells them apart.

What a fleet’s camera should fit #

For anybody using a scroll camera’s rows to watch a vehicle’s suspension — a fleet checking its dampers from the dashcams it already has — the practical reading is short, and it reverses one piece of the earlier advice. Fit two modes, not one: the one-oscillator reading is a reading of neither mode unless the camera is well ahead of the centre, and even then its damping is a quarter low, which is the number a damper check most needs. Measure the camera’s lever arm once, with a tape, and give it to the fit; without it, a perfectly ordinary pitch spring reads seven per cent stiff. And read the evidence for the bounce before trusting the bounce’s number: when the two modes sit within a tenth of a hertz of each other, the pass will say confidently that there are two and only roughly where the second one is.

The reversal is in what the earlier essay’s one pass was trusted to do. It read the road’s slope because a straight line on logarithmic axes resembles neither the scroll’s hump nor the spring’s swelling. It could not have read a second mode it was not given, and nothing in its residual would have said so: the proportion is the assumption is the collection’s standing case of an answer that is a function of a model nobody had asked the picture about. Here the model was the number of springs.

What the fit takes from outside the rows #

Where the camera stands. The two-mode fit is told the lever arm, and the one-oscillator fit is shown to be biased without it. The arm is a length on the vehicle, measured once; a fit could also be asked for it, as the earlier essay asked for the road’s slope, and whether one pass reads it was not measured.

The road’s slope, and its being one straight line. Taken here as the survey standard, as the earlier essays took it before freeing it; a road whose spectrum bends near the swellings would give the fit a shape it has no model for.

Two modes and no more. A real body also rolls, and its wheels hop on their tyres at ten hertz or so. The roll moves the camera sideways and turns it about the line of sight, which this measurement does not model; the wheel hop is far above the band read here. A body with a third mode in band would leave a third signature, and the two-mode fit would fold it into whichever of its two modes it resembled.

The modes are independent. The body’s pitch and bounce are taken to answer the road separately, through the chord and the mean. A real car whose springs differ front and rear couples them, so that a bump excites both at once in a fixed proportion; the rows would then carry the coupled modes, and the fit’s two frequencies would be theirs rather than a pure pitch and a pure bounce.

What the second mode was worth #

A scroll camera rings with its suspension began this line of reading with one step in the road and one oscillator; a scroll is a camera that moves is the reason any of it is possible, since the rows record the camera’s motion before they record the scene. The motion of a real vehicle has more than one part, and the rows record every part that turns or lifts the camera. The one-oscillator fit was not wrong about what it saw; it was wrong about what it was seeing, folding two modes into one at a frequency that belonged to neither unless the camera happened to be well ahead of the centre. With the second mode in the model, one pass reads both frequencies and both dampings, says whether the bounce is there, and exposes a lever-arm bias the earlier readings would have carried on any real installation.

Still open: whether the rows read the camera’s lever arm as well as the modes #

The two-mode fit here was told how far ahead of the body’s centre the camera stands, and a fit not told it reads a 1.4 hertz pitch as 1.5. The arm enters the rows in a particular way — it adds the pitch’s own swelling to the lift channel, in proportion — and that is a shape, like the road’s slope, which the whole spectrum might draw plainly enough to be read.

The measurement that settles what that is worth makes the lever arm a fifth unknown of the two-mode fit, reads twenty roads at arms of nought to two and a half metres, and asks how well one pass reads the arm, whether reading it costs the two frequencies anything, and whether a body with no bounce, where the lift channel carries only the pitch through the arm, reads its arm better or worse than one that bounces — since there the arm is the only thing the lift channel has to say.