Machine Kinematics
solved, then drawn
Concept

Identifiability — where it appears

Whether the quantities a measurement is meant to recover are fixed by what it actually records. Three wheels or three loads identify a road's power law because three counts fix three unknowns; two would leave one of them free to take any value.

Named by 2 essays across one field — each of them below, with the objects they name alongside it.

Three wheels on one road, each bridging a different share of its texture. Two metres of a class C road built from 110 sines of random phase from 1 cm to 90 m wavelength, heights in millimetres and drawn far taller than they are, and the lowest points of three wheels of 0.3, 0.6, 1.2 m radius rolled along it together. Where a wheel's line lies on the road the wheel touches it; where it lifts off, the wheel bridges a trough. On this stretch they touch 4%, 2%, 1% of the way. Each counts its own centre's path, so the three counts differ by how much texture each bridges, and the differences carry the road's spectrum.

Three wheels read the road they measure

A wheel's overcount on a real road falls as a power of its radius that the road's spectrum sets, so two wheels cannot extrapolate it away without being told the power. Three can fit it. Rolled together over 150 m of a class C road textured to 1 cm, wheels of 0.3, 0.6 and 1.2 m read a power of 0.64 against the law's 2/3, read the road's class from its level, and extrapolate the distance to 2 × 10⁻⁵ where the smallest wheel alone is out by 10⁻³. The standard band's floor, which no practical wheel bridges, turns out to lie on the same law and is extrapolated with it. The method is worth five times on a smooth road and five hundred on a rough one, and near the end of the road's texture it reads the end instead.

rolling · Rolling One tyre pressed harder: its patch grows and averages the road it used to ride. 0.6 m of a class C road built from 110 sines from 1 cm to 90 m, heights in millimetres and drawn far taller than they are, under a 0.3 m tyre. The tyre is a circular tread on a bed of vertical springs; at each position its centre height is the one at which the springs carry the load. The grey line is a rigid wheel's lowest point, riding the peaks. The coloured line is the level the loaded tread's patch rests at, 9 mm deflection, a patch 146 mm long on a flat floor: it passes over the short texture as an average of it instead of following its tops.

A load reads the road a radius cannot

Press a tyre harder and its contact patch grows, and a patch averages the road where a rigid wheel rides its peaks. Averaging is linear, so the overcount it leaves falls as the patch's first power and scales with the road's roughness in exact proportion, not as the two-thirds power three wheel sizes read. One 0.3 m tyre at deflections of 3, 9 and 27 mm fits that law and extrapolates the distance to 3 × 10⁻⁷ on a smooth road and 6 × 10⁻⁶ on an average one, against 2 × 10⁻⁵ for three rigid wheels. On the roughest roads its lightest patch lifts off in places, and it does worse than they do.

rolling · Rolling

Named alongside it

The objects these essays reach for when they reach for this one.

ApproximationCalibrationRolling constraintRolling without slippingScale invariance

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