A load reads the road a radius cannot

What a wheel counts on a rough road established that a rigid wheel’s odometer measures the length of the path its centre follows. That path lies above the road wherever the wheel bridges a trough. A real road sets its own power found that on a road whose spectrum falls in the standard way, the excess of that length over the true distance falls as the wheel’s radius to the power two thirds. Three wheels read the road they measure then rolled wheels of 0.3, 0.6 and 1.2 m over the same road. Their three counts fixed the power, the road’s roughness and the true distance, and extrapolated the distance to about $2 \times 10^{-5}$ on an average road.

That essay ended on an engineering inconvenience. Three wheels of very different sizes make an awkward instrument, and a tyre can change what it bridges without changing its size, because pressed harder it flattens and its contact patch grows. Run one wheel at three loads, and the patch might do what three radii did.

It does better than that on most roads, and the reason is that the patch is a different kind of instrument from a larger wheel. A larger rigid wheel rides higher peaks. A loaded patch averages. The difference changes the law the counts follow, and it is also what makes the method fail on the roughest roads.

A tread on a bed of springs #

The tyre here is the simplest model with a patch in it. A circular tread of radius 0.3 m sits on a bed of independent vertical springs, all of one stiffness, sampled every 0.4 mm. Wherever the undeformed tread would sit below the road it is pushed up by the overlap, and the springs together carry the load. The load is named by the deflection it would cause on a flat floor: 3, 9 and 27 mm, which make flat patches 85, 146 and 249 mm long. At each position along the road the centre height is the one at which the springs carry exactly that load. The sum of the overlaps is convex and piecewise linear in the height, so Newton’s method lands on it in about two steps, and the model checks that every position balances its load to $10^{-9}$.

With no load the model is the rigid wheel, and that is its first check. The earlier essays computed a rigid wheel’s path by a different route, as the road dilated by a disc, in separate code. On all ten roads, classes A to E with two random realizations each, the two agree on the excess to $10^{-6}$.

The roads are the earlier essay’s roads exactly: 150 m each, built from 110 sine waves from 1 cm to 90 m long with random phases and the ISO 8608 spectrum, each class four times the last in spectral level. Any difference between the tyre’s readings and the three wheels’ comes from the instrument, not from the road it was run on.

One assumption has to be stated rather than modelled. A rigid wheel’s count is exactly its centre path’s length, because its contact point is its instantaneous centre of rotation. A tyre’s rolling radius depends on its load, and the counts here assume each load calibrated on a flat floor first. The count then measures the centre path’s length at that load, with the tread’s own strain and slip on texture left out.

The overcount falls as the patch’s first power #

On the flat the tyre’s centre is lower by its deflection and its count is calibrated to the distance. On a rough road the centre path is longer than the distance, as a rigid wheel’s is, but less so.

On class A to C roads the excess falls as the patch half-length to a power between 0.988 and 0.999, measured over the three heaviest loads with the true distance known. That is a power of one, where rigid wheels gave two thirds. The model checks it within 0.015 of one on all six of those roads.

The reason is the averaging. A rigid wheel rests on the highest points under it. Which troughs it bridges depends on how deep they are as well as how wide, since a deep narrow trough and a shallow wide one can both be bridged. That dependence on amplitude is what puts two thirds into the rigid law. A tyre whose whole patch is pressed onto the road rests at the patch’s average height, weighted by how hard each spring pushes. Averaging over a window of length about $a$ passes the waves longer than the window and suppresses the shorter ones by an amount that depends only on their length. It is the same moving average a helical gear’s face makes of its spur pair’s contact count in the face carries the least line, and a twisted rotor makes of its profile in a twist steadies what it cannot tighten, applied here to a road.

On the standard spectrum every octave of waves adds excess in proportion to its frequency, so the excess the tyre still sees is set by the shortest waves it passes, which go as $1/a$. The exponent is one. The road’s own roughness only scales the result.

The patch reads the level in proportion #

Averaging is linear, and that has a second consequence that is easy to check. If the tyre’s centre path is a linear filter of the road, its slopes are in proportion to the road’s slopes, and its excess, which goes as the square of slope, is in proportion to the road’s spectral level.

The measurement agrees. From class to class the tyre’s excess at 9 mm grows by 4.00, 4.00, 3.97 and 3.84, against the factor of four the classes differ by. The rigid wheel’s grows by 2.61, 2.56, 2.54 and 2.55, close to $4^{2/3} = 2.52$, the same two-thirds power in another guise. The model checks on both realizations that the tyre’s factor is nearer four than the rigid wheel’s at every step. The departure on the roughest step, 3.84, is the first sign of the failure below.

A profiler gains something from this beyond the distance. The earlier essay read a road’s class from the level of its three-wheel fit through a calibration, because the rigid wheels’ level enters as a power of the roughness. A loaded tyre’s level is the roughness itself, up to a constant, so one road’s calibration carries to every other road without an exponent.

Three loads and the distance #

With the law’s power near one, three loads fit it the way three radii did. The counts give $C_i = D'(1 + c'\,a_i^{-p})$ at three patch half-lengths. The ratio of their differences fixes $p$, and $D'$ is the distance the law extrapolates to as the patch grows without limit.

On the smooth roads the result is a large improvement. On class A the three loads are out by $2.9$ and $3.3 \times 10^{-7}$ on the two realizations, against $2.7$ and $2.8 \times 10^{-5}$ for three rigid wheels, nearly a hundred times better. On class B they are out by $8 \times 10^{-7}$ and $1 \times 10^{-6}$, 25 to 50 times better. On the average road, class C, they are out by 6 and $7 \times 10^{-6}$ against 2 and $6 \times 10^{-5}$, three to eight times better. The model requires at least twenty times on A and B and three times on C.

The reason is that the fitted law is nearly exact. On a smooth road the three loads sit on a straight line of slope one, so the fit’s constant term, the extrapolated distance, is pinned by three points on the line it actually lies on. The rigid wheels’ law is two thirds only on average, bending slightly as each wheel bridges a different part of the spectrum, and that bend is what limited their extrapolation.

The limit the fit extrapolates to deserves a sentence of its own, because it is the same limit the rigid wheels reached by a different road. A rigid wheel of unlimited radius follows no wave at all and runs level, so its centre path is the horizontal distance. A patch of unlimited length averages every wave away, the 90 m ones included, and its centre also runs level. The two instruments disagree at every finite size and agree at infinity, which is why both can be extrapolated to the distance, and why the one whose law is cleaner gets there more accurately.

On class D the three loads are out by $1.3 \times 10^{-4}$ against $1.6 \times 10^{-4}$, level. On class E they are out by $2.6 \times 10^{-3}$, more than a hundred times worse than the three rigid wheels.

Why rough roads defeat it #

The averaging argument assumed the whole patch is pressed onto the road. On a rough road it is not.

At 3 mm of deflection on class A to C roads the patch is in contact over its whole length, to within a per cent. On class D 95% of it is, and on class E only 81%. Where the road’s texture is taller than the tyre’s deflection, the tread bridges between peaks inside its own patch. That part of the tyre rides the road as a rigid wheel does, and the two-thirds law leaks back in. At 9 mm the patch is still 96% in contact on class E, and at 27 mm over 99%, so the heavy loads still average while the light one does not.

The three counts then sit on no single law. Measured with the distance known, the power drops from 0.99 on class C to 0.94 on D and 0.83 on E. The three-count fit is more sensitive still, because it must also place the constant term: it reads 0.95 on C, 0.79 on D and 0.41 on E, and the distance it extrapolates to carries the error. The model checks that the lightest patch loses contact on the rough classes, that the power falls away with it, and that on class E the three loads do more than ten times worse than three radii.

The rule this gives a profiler is simple. The loads must be heavy enough that every patch stays down, which means a deflection well above the height of the road’s texture. A passenger-car tyre deflects of the order of 20 mm under its working load, so the method suits a measuring wheel run as heavily loaded as a road wheel, not a light trailing wheel.

Heavier windows read rougher roads #

The rule can be tested directly by moving the window of loads.

On class A the light window does as well as the heavy one, out by $1.9$ and $2.9 \times 10^{-7}$, because even 1 mm of deflection keeps the patch down on so smooth a road. From class C up the light window fails first. On class C it reads a power of 0.74 and is out by $7 \times 10^{-5}$, no better than three rigid wheels, where the heavy window reads 0.95 and is out by $6 \times 10^{-6}$. On class D the light window is out by $2 \times 10^{-3}$, and on class E its three counts fit no power at all. The model checks that from class C up the lighter window always does worse.

So the window of loads plays the part the window of radii did in three wheels read the road they measure, where wheels too close to the texture’s short end read the end instead of the law. Here a load too light for the road’s texture reads the rigid wheel instead of the patch.

What a load window costs #

Loads are a cheaper knob than radii in one respect and a dearer one in another. A patch’s half-length grows only as the square root of the deflection, $a \approx \sqrt{2R\delta}$, so tripling the deflection lengthens the patch by 1.73 times, and the window of 3 to 27 mm spans patch half-lengths of 42 to 124 mm, a factor of 2.9. The three rigid wheels spanned a factor of four in radius. To match that, the deflections would have to span a factor of sixteen, and the heaviest load would flatten a 0.3 m tyre by a sixth of its radius.

The narrower span is affordable because the law it has to fit is cleaner. A fit through three points on a straight line places its constant term well whatever the spacing, and on smooth roads the three loads are on a straight line of slope one. The rigid wheels needed their wider spacing to average over the bend in their law. What the narrower span cannot afford is a lightest load that is too light. On a rough road that load’s point falls off the line, and with the points close together nothing else in the fit can compensate.

There is also a reason the rigid wheels win on the roughest road. On class E the two-thirds law holds well: the three rigid wheels read a power of 0.67 and 0.68, close to the law’s two thirds, because the texture is tall enough that every wheel rides its peaks in the regime the law describes. Roughness that defeats the patch is exactly the roughness that suits the rigid wheel.

A rolling constraint with a spring in it #

The earlier essays in this field treated rolling as a constraint between a rigid wheel and a rigid surface. A constraint that takes nothing away was about what such a constraint permits, and a wheel that cannot report its radius about the length a rolling constraint carries. A tyre adds compliance to the contact, and that compliance changes the measurement from a nonlinear operation, the dilation, into a nearly linear one, the average.

That is why the result runs against intuition. A tyre is softer and less precise than a steel wheel, and the expected effect of softness is noise. Here the softness removes the amplitude dependence that made the rigid wheels’ law uncertain. A soft enough instrument reads the road’s spectrum more faithfully, as long as it stays in full contact.

It also explains why a second wheel cannot correct the first on a rigid frame. Two rigid wheels at different places on the same frame both dilate the road at the same radius, and their errors are the same error. Two loads on one tyre are two different averaging windows, and their errors differ in a known way.

What this does not settle #

A real tyre’s patch. The bed of independent springs ignores the tread’s bending stiffness, which spreads a load sideways along the tread and keeps it from following texture inside the patch as closely as these springs do. A real tyre’s patch is somewhere between this model and a flat plank. The law’s power is set by the patch averaging at all, so it should survive, but the level and the contact share at which it fails will move.

The count itself. The counts here are centre-path lengths, assuming each load was calibrated on a flat floor. A loaded tyre’s effective rolling radius also depends on tread strain and on slip in the patch, both of which vary with texture. If that variation is comparable to the excess, $10^{-5}$ on an average road, it would swamp the method on smooth roads, where the gains are largest.

Loads that change during a run. Each load here is held constant. A profiler that changes its load mid-run has to know the patch at each moment, and a vehicle’s own pitching changes the load on every wheel continuously.

Still open: a load chosen by the road #

The method fails when the lightest load is too light for the road’s texture, and the share of the patch in contact measures that directly. A profiler that measured its own contact share could choose its loads to keep the patch down.

The distinct argument would make the window of loads adaptive. The instrument would read the contact share at a trial load, raise the load until the patch stays down over a set share, and then fit the law over loads above that threshold. The measurement would ask whether an adaptive window holds the power near one and the distance error near the smooth roads’ $10^{-6}$ across all five classes. It would also find the deflection, as a multiple of the texture’s height, at which a patch first stays down, which is the single number a profiler’s design would need.