A scroll camera rings with its vehicle's suspension

A vehicle’s pitch lags the road by its wheelbase put a scroll camera — two slits leaning ten degrees either way of upright, drawing the roadside as the vehicle drives, the instrument a scroll through two slits ranges in a straight line built — on a vehicle that pitches with the chord between its front and rear wheels rather than with the road under its camera. On a straight grade the chord is the grade and the rows of the two slits’ drawings agree, as they did for an eye riding the road. Over a step in the grade the chord turns over the wheelbase’s length, and the rows part: by 2.18 pixels at most for posts twelve metres out over a step to six per cent, and back to zero once the vehicle is on the new grade. One line of posts read the wheelbase back.

That vehicle was rigid. The essay ended on the one a road actually carries: a body on springs, which meets the step with its front wheels, compresses, pitches past the chord and back, and rings down over a second or so. The camera’s pitch then departs from the chord by an amount that rings after every change of grade. The question was whether the rows ring with it, above the reading error, and whether the ringing’s period and decay can be read back — which would make a scroll camera a record of how the vehicle’s suspension was set.

A body that follows its chord on springs #

The model is the simplest that rings. The body’s pitch is driven towards the wheelbase chord’s pitch by a spring and slowed by a damper: an oscillator with a natural frequency and a damping ratio, the chord as its input and time as distance over speed. A passenger car’s body pitches at around one to one and a half hertz with a damping ratio of a few tenths; the vehicle here is sprung at 1.3 hertz, wheels 2.7 metres apart, and drives at ten metres a second across a step from level to six per cent. The scroll camera’s frame is the body’s, at every moment of the drive: a scroll is a camera that moves, and here it moves with the body’s pitch as well as along the road.

The sprung vehicle’s rows are not the rigid one’s, and they differ in two ways. Near the step they part further, to 2.64 pixels at a damping of 0.3 against the rigid vehicle’s 2.18, because the body lags the chord as the wheels cross the step: the camera is still pointing along the old grade when the eye’s two moments straddle the new one. Past the step they do not come back to zero. Eight metres on, where the rigid vehicle’s rows have returned exactly, the sprung one’s are still parted by a third of a pixel, rising and falling with a length along the road of the speed over the frequency — 7.7 metres.

The slider changes the damping, and the damping trades one of these departures against the other. Lightly damped, at 0.15, the body swings back up past the chord almost as soon as it has fallen behind it: the rows near the step part by only 1.78 pixels, less than the rigid vehicle’s, and eight metres on they are still parted by 0.59, ringing for several crests. Heavily damped, at 0.8, the body creeps onto the new grade without overshooting: the rows near the step part by 4.33 pixels, twice the rigid vehicle’s, and eight metres on by only 0.06. Every damping leaves a signature, at the step or after it.

The lag and the overshoot are one response #

The two departures are not separate effects. A sprung body’s pitch is the chord’s pitch passed through the spring, and a spring both delays what passes through it and lets it overshoot; the damping decides the share of each. In the rows the delay shows where the step is — the body still pointing along the old grade while the eye’s two moments straddle the new one — and the overshoot shows after it, as the ringing. A rigid vehicle has neither, and its rows are confined to the stretch over which its own wheels and the eye’s two moments straddle the step.

That makes the stretch past the step the cleanest place to read the spring. There the rigid model’s rows are exactly zero, so whatever the rows show is the body’s own motion: its overshoot, its period along the road and its decay. Near the step the spring’s signature is mixed with the step’s own excursion, which the step’s size and place have to account for, and a fit has to separate the two. The read-back below uses both, because the lag carries the damping as clearly as the decay does.

What no rigid vehicle can take #

A sprung vehicle’s rows look like a rigid vehicle’s that has been moved and stretched, and the first thing to ask is whether a rigid vehicle with some other step can imitate them. If it could, the ringing would be invisible in practice: a reader would fit the earlier essay’s model and find a slightly different step.

It cannot. The best rigid vehicle — its step sized and placed to fit the sprung rows as closely as possible — leaves 0.25 pixels root mean square over forty metres of posts at a damping of 0.15, 0.38 at 0.3 and 0.63 at 0.5. Every one of those is several times a tenth of a pixel, the reading the earlier essays took as ordinary. The lightly damped leftover is the ringing itself, a wave that runs on past the step. The heavily damped one is a different shape, one long swell where the body lags the chord by more than any shift of the step can imitate, and it is the larger of the two. A spring with any damping in this range leaves a signature in the rows that the rigid model cannot absorb.

Reading the suspension back #

The rows are a function of the step, the wheelbase, the speed, the frequency and the damping. With the wheelbase and the speed known — the vehicle’s own odometry gives the second, its specification the first — the step, the frequency and the damping can be fitted to the rows of one line of posts.

Read to a tenth of a pixel, one line of posts gives the body’s frequency as 1.302 ± 0.011 hertz and its damping ratio as 0.303 ± 0.008, against a truth of 1.3 and 0.3. Read to a twentieth, 1.300 ± 0.004 and 0.300 ± 0.004. Even read to four tenths of a pixel the frequency comes back to within five per cent and the damping to within an eighth. At a tenth of a pixel the frequency is known to under one per cent and the damping to under three, which is finer, in proportion, than the earlier essay read the wheelbase itself from the same posts: to about eight centimetres in 2.7 metres, three per cent. The spring is a quantity the rows carry more of than the geometry that produced them, because it shapes forty metres of rows where the wheelbase shapes a few.

The two are read from different parts of the rows. The frequency is set by where the ringing’s crests fall along the road, and forty metres of posts every quarter metre place them closely. The damping is set by how fast the crests shrink, which depends on the few crests that stand above the reading error, and is fixed less well in proportion. That is the same split the earlier essay found for the wheelbase and the step: the shape of a feature says one thing, its height another, and the rows carry both.

Faster is better #

The speed changes how hard the step kicks the body. The wheels cross the step in the wheelbase over the speed: 0.54 seconds at five metres a second, 0.08 at thirty-two, against a ringing period of 0.77 seconds. Slow, the chord turns over most of a period and the body follows it almost as it goes, ringing hardly at all; the frequency is read to ±0.016 hertz. Fast, the chord turns in a tenth of a period, the body is kicked as if by a blow, and it rings hard; at thirty-two metres a second the frequency is read to ±0.003 hertz, five times better, even though one ringing then stretches twenty-five metres along the road and the forty metres of posts hold fewer than two of them.

That is the opposite of what the posts’ spacing alone would suggest, and it says what the scroll is measuring. It is not counting crests along the road; it is reading the body’s response to a sudden input, and a sharper input excites more of the response. A road whose grade changes abruptly — a ramp onto a bridge, a kerb, a dip — driven at speed, is the best calibration a scroll camera’s suspension reading could have.

Near and far posts #

The step’s own excursion grows with the posts’ distance from the track, from 2.11 pixels for posts six metres out to 2.96 for posts twenty-four metres out: the rows read the camera’s pitch against the eye’s chord between the two slits’ moments, and those moments are further apart for a further post, so a pitch error is read over more road. What the spring adds does not follow it. It stays between 0.38 and 0.71 pixels root mean square at every distance, least for posts about twelve metres out and larger both nearer and further. The dependence on how far apart the slits’ two moments fall, against the ringing’s length along the road, was not worked out here; what matters for a reading is that the spring’s signature stays several tenths of a pixel at every distance a roadside offers.

What a scroll records #

Put together, the rows of a scroll camera on a sprung vehicle record the vehicle as well as the road. A scroll of a climbing road measures its grade found the rows parting by the grade itself for a camera fixed to the track; an eye that pitches with the road keeps its rows found them closing again for a camera that pitches with the road beneath it; the wheelbase essay found them parting by the difference between the eye’s chord and the wheels’. A body on springs adds one more term — the difference between the wheels’ chord and where the body actually points — and that term rings with the body’s frequency and dies with its damping.

Each stage of that sequence has been a statement that the scroll measures differences of pitch, and each has added a new reason the pitch could differ. The suspension is the first that is about the vehicle rather than the geometry — a scroll can be asked its own radius asked the scroll about its track, and this asks it about its springs — and it is also the first that a fleet operator might want measured: a damper that has worn shows as a damping ratio that has fallen, and a scroll camera driving over the same ramp every day would see it fall.

A reading plan #

The measurements say how a suspension reading should be taken, and none of the conditions is demanding. The road should change its grade sharply — a ramp onto a bridge, a driveway’s edge, a speed table — because a sharp change kicks the body and a gentle one does not. The vehicle should cross it briskly, because the kick is the change of chord over the time the wheels take to cross it, and at thirty metres a second that time is a tenth of the body’s period. The roadside should carry a line of posts or any vertical features dense enough to sample a ringing a few metres long, read to a tenth of a pixel or so, for some tens of metres past the change. And the wheelbase and speed should be known, which they are.

With those, one pass gives the frequency to about one per cent and the damping to about three. A fleet that drives the same ramp every day would see a worn damper as a damping ratio that falls by more than that from week to week, from the same camera that is already recording the road.

What a reading has to know #

The fit above was given the wheelbase and the speed. The speed is the scroll’s own: a scroll camera’s paper advances with distance, and its rate in time is the vehicle’s speed, which the camera’s line clock and the odometer share. The wheelbase is the vehicle’s, a fixed number from its specification.

What the fit was not given is the step: its size, its place, and that it was a step at all. A real road’s change of grade is a vertical curve of some length, and the wheelbase essay found the rows over a long vertical curve much smaller than over a step, falling as one over its length. A gentle curve is a gentle input, and like a slow crossing it excites little ringing. The suspension is read best where the road changes abruptly, and a reading should be taken there.

What was assumed #

The body pitches as one oscillator. A real vehicle’s front and rear suspensions are separate springs, and its body bounces as well as pitching, so two modes ring together, at two frequencies. The rows would then carry a beat of two ringings, and a fit of one would find a frequency between them; a fit of two would need a longer line of posts.

The step’s place is known. The read-back fitted the step’s size, the frequency and the damping, and was told where along the road the grade changes. A real reading would fit the place as well, from the same rows. The rigid fit above did fit it, but the sprung read-back’s spreads were measured with the place given, and with it free they would be somewhat wider by an amount not measured here.

The tyres are rigid. A tyre is a stiff spring under the suspension’s soft one, and it rings at ten hertz or more — about a metre along the road at ten metres a second, which posts a quarter of a metre apart would sample. It was left out; its deflections are small beside the suspension’s, but the rows would carry them as a fine ripple on the body’s ringing.

The camera is fixed to the body. A camera on a bracket flexes, and a bracket’s own ringing — tens of hertz, lightly damped — has a length along the road near the posts’ own spacing, where posts this far apart would alias it into a slower ripple that a fit could mistake for the body’s.

The posts’ rows are read with independent errors. A scroll camera’s row error comes partly from its own line-by-line jitter, which is correlated along the scroll; a correlated error over the length of a ringing would look like the ringing, and the read-back’s spread here is for independent errors only.

Still open: whether a real road’s own unevenness is enough to read the suspension #

The readings here needed a step in the grade — a single sharp input. A road surface has small unevenness everywhere, a few millimetres over a few metres, and a body on springs responds to all of it continuously.

The measurement that settles whether that is enough drives the sprung vehicle along a level road whose height carries a random roughness of stated size and length, records the rows of a long line of posts, and asks 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 is, a scroll camera would read its vehicle’s suspension on any road at all, without waiting for a ramp.