Logs / Route 005
Six tyre models later
The long version of how the tyres work, with the real numbers. A brush in the contact patch, a carcass that winds up, rain, heat and mud, and the parked cars that wouldn't stay parked.
The tyre in Sandbox Drive today is the sixth one. The first, back in January, was a simple brush-friction curve. The current one is a brush model too, which means seven months and four other models went into walking in a circle. It’s a much better circle, though. This is how it works, and what it cost.
(The timeline has all six, for anyone who wants to watch the project change its mind.)
A brush in the contact patch
Picture the tread as a brush, made of countless short rubber bristles. As the wheel rolls, each bristle touches down at the front of the contact patch, gets dragged along as the tyre slips, and lets go at the back once it can’t hold on any more. At small slip angles almost the whole patch is stuck to the road. Push harder and the sliding zone creeps forward from the back until everything is sliding and the driver is a passenger.
The model doesn’t fit a curve to measured tyre data, which is the common approach and was the third model here. The brush is solved directly, with a pressure under the patch that peaks in the middle. Nobody put the grip peak there. It exists because rubber grips better stuck than sliding. The sports street tyre, for example, has a static friction of 1.15 and a sliding friction of 0.95, and it peaks at 9.5% slip under braking and about 9° of slip angle in a corner.
Braking and cornering are solved together, as one slip vector and one force. Brake hard mid-corner and the two fight over the same grip, like they do on a real car. That sounds obvious. It took until the fifth model to get it right.
Why grip takes a moment
Behind the tread is the carcass: sidewalls and belts. It’s modelled as a spring with two damped arms, so it stores energy and loses a bit each cycle, like rubber does. On turn-in, the carcass has to wind up before the full cornering force arrives, and the distance the tyre rolls while that happens is called the relaxation length.
That isn’t tuned either. It falls out as the brush stiffness divided by the carcass stiffness, so it grows with load the way a real tyre’s does: about 0.23 m on a lightly loaded wheel, up to 0.73 m on a heavily loaded one. It used to be a fixed number. More on that below.
Hot, cold, soft, hard
A tyre has two temperatures that matter, and they move at very different speeds.
The surface heats up in seconds from friction, every time the tyre slips. That’s what changes grip. The sports street tyre is happiest at 70 °C. At 10 °C it’s lost 7%, and at 115 °C it’s lost 14%.
The carcass and the air inside heat up over minutes, from the tyre flexing as it rolls. That raises the pressure, about 11 kPa for every 10 °C, so a tyre set to 220 kPa cold reads about 230 kPa warm. Pressure has its own sweet spot: within 30 kPa either side there’s full grip, under-inflated costs up to 15%, over-inflated up to 10%. Airflow cools everything down again, so a hard run followed by a gentle drive home really does bring the tyres back.
Rain
Rain works on two levels and they stack. A wet film costs grip at any speed, down to 78% of dry when the road is properly wet. Standing water is the dangerous one. It builds up wherever rain beats the drainage: heavy rain, 50 mm an hour, settles at about 4 mm deep, never more than 8. When the water is deeper than the tread can clear (3 mm for a street tyre, 7 for a mud tyre, zero for a slick) and the car is going fast enough, it aquaplanes, and grip drops to as little as a tenth. Worn tyres keep a fifth of their clearance. Higher pressure buys a little more speed before it happens.
Mud
On soft ground the brush is replaced by the soil itself. How much a tyre can pull is down to the soil’s strength, its cohesion and friction, times what the tread can bite into. A mud-terrain tyre gets 1.4 times the grip on loose ground and twice as much on soil. A racing slick gets half or less, which is about what anyone would expect.
The wheel also sinks, and here tyre pressure is a proper lever. A tyre can’t press on the ground much harder than its own inflation pressure, so letting air out makes the patch longer and the wheel floats. Sinking is permanent: 30% happens straight away, the rest the longer the car sits there, and it never springs back. Spinning the wheels digs them in a few millimetres deeper for every second of wheelspin. There are nine soils, from dry sand to bog mud.
Through the steering wheel
The same forces drive the steering wheel. The front tyres try to straighten themselves (the aligning torque), and that torque fades as they get close to their limit. So the wheel goes light a moment before the front washes wide. With the steering geometry added, it tops out at 8 N·m at the rim.
Fast enough
Physics runs at 60 steps a second, but the tyres sub-step four times inside each one, so they effectively run at 240 Hz. A tyre changes its mind much faster than a car body does. It’s all native C++ now. The script version was deleted in August.
What it cost
The switch to the brush came on 13 August, with a note in the project saying to rip the old model out “if it makes the game currently broken, so be it”. It did, in three known ways. The funniest: parked cars wouldn’t stay parked. One crept 1.71 m across the test pad while its tyre force flipped direction 82 times, because the solver’s answer was being thrown away and rebuilt from a simpler slip value. Two days later a single “standstill bristle” per wheel went in, and parked cars went back to being parked.
Then there was the wobble. At 15 km/h, cars developed a gentle 2.9 Hz shimmy, the body bouncing on its tyres and feeding itself. Adding the carcass’s second damped arm nearly doubled the damping at exactly that frequency. It also fixed a problem that had been open for months: cars braking hard with ABS at low speed had veered almost 6° off line. Afterwards, under 1°.
Rain had its own disaster. For a while, any rain at all capped cars at 80 to 100 km/h, because at 120 a commuter tyre kept 7.5% of its grip. That’s worse than ice. Two wet models had been stacked on top of each other, and one of them fed on wheel spin, so it made itself worse. Pulling them apart into the film and depth described above fixed it.
And the relaxation length: until mid-September it sat at 0.456 m whatever the load, because the carcass stiffness had absorbed all of the load dependence without anyone noticing. Now it’s derived from the load, like everything else in there.