Getting the tire into its window
So far we have treated the friction coefficient as a property of the tire. In reality the same tire can be fast or hopeless depending on how it is operated: the angle it meets the road at, how hard it is inflated, and how hot it is. Race engineers talk about getting the tires “in the window”, and a good part of every test day goes into finding that window and keeping the tires in it. This lesson covers the three biggest levers, then shows how engineers turn tire test data into the kind of curves you have been using.
Camber and camber thrust
Camber \(\gamma\) is the tilt of the wheel from vertical, seen from the front of the car. By convention, negative camber means the top of the wheel leans in toward the car's centerline.
A tilted, rolling tire produces a lateral force even at zero slip angle: camber thrust. For small angles it adds to the slip-angle force,
where the camber stiffness \(C_\gamma\) is much smaller than the cornering stiffness \(C_\alpha\) for a car tire. For race cars, camber matters even more through what it does to the contact patch. A tire makes its best grip when the patch is loaded evenly, which for most race tires means the heavily loaded outside tire runs at a small negative camber at the limit.
The catch is body roll. When the car rolls in a corner, the outside wheel tends to tilt with the body, toward positive camber, onto the outer edge of the tread. Engineers therefore set negative static camber so that, after roll, the outside tire ends up close to its best angle:
Too much static camber has costs too. The tire rides on its inner edge in a straight line, which hurts braking and traction, overheats the inside shoulder and wears it quickly. Suspension geometry (Module 8) controls how much camber the wheel gains or loses as the car rolls.
Inflation pressure
A crude but useful model treats the tire as a pressurized membrane: the air pressure carries the load, so the contact patch area is about
(\(p\) is gauge pressure). Real carcasses carry some load themselves, but the trend is right. Lower pressure gives a bigger, softer patch, which grips better up to a point. Too low and the sidewalls flex too much, the response goes vague and the tire overheats. Too high and the patch shrinks and the tire skips over bumps.
The air inside heats up with the tire, so pressure rises during a run. Treating the air as an ideal gas at fixed volume, absolute pressure is proportional to absolute temperature. Teams set cold pressures so that the hot pressures land in the window.
Temperature and compound
Tread rubber is viscoelastic: partly spring, partly damper, and its properties change strongly with temperature. Grip comes from the rubber gripping the road surface (adhesion) and from the rubber deforming around the road's texture and losing energy as it does (hysteresis). Both peak in a temperature range that the compound is designed for. A cold tire is hard and slippery; an overheated tire turns greasy and wears fast.
Tire makers trade this off with the compound. A softer compound typically grips more and works at lower temperatures but wears faster; a harder compound needs more heat but lasts longer. Getting heat into the tires on the out-lap, and keeping it from building too far, is part of the driver's job.
- With zero static camber, what camber does the outside tire actually run at the limit? Find the static camber that puts it at its optimum.
- Raise the camber change in roll to 3°, as a softly sprung car might. How much more static camber do you need? What would that cost in a straight line?
- Bring the tread temperature into the window. Which variable was costing the most grip at the start?
- Switch to the hard compound without changing the temperature. Why might a team still choose it for a long race?
From test data to tire models
Tire forces are measured on specialist test machines that run a tire against a moving road surface (often a flat steel belt) while controlling its slip angle, slip ratio, load, camber and pressure. The result is a large cloud of data points. To use it in calculations and simulations, engineers fit it with a model. The Magic Formula from Lesson 2.1 is the most widely used:
It is an empirical curve fit, not a physical derivation, but its main parameters still have physical meaning: \(D\) is the peak force, the slope \(BCD\) is the cornering stiffness, \(C\) controls the overall shape and \(E\) the curvature around the peak. Full versions add many more coefficients to capture load, camber and combined-slip effects.
A powerful trick for comparing data at different loads is normalization: divide the force by the load, \(F_y / F_z\). If a tire were not load-sensitive, curves at every load would fall on top of each other. How far they don't collapse shows the load sensitivity directly.
- Fit the 3000 N data. Set the peak \(\mu\) and the initial slope first, then refine C and E. Can you get below 2% error before revealing the best fit?
- Fit the 1500 N and 4500 N sets. How do the best-fit \(\mu\) values compare? Which lesson predicted that?
- Switch to Normalized Fy/Fz. Do the three data sets collapse onto one curve? Where do they differ most: at the slope or at the peak?
Check your understanding
A car's outside front tire has −3.0° of static camber. At the limit the body rolls 2.0°, and the suspension geometry recovers 0.5° of camber per degree of roll (so the wheel tilts only half as much as the body). What camber does the tire run at the limit?
Roll tilts the outside wheel toward positive camber. The positive change is the body roll times (1 − recovery).
\(\Delta\gamma = 2.0 \times (1 - 0.5) = +1.0^\circ\), so \(\gamma = -3.0 + 1.0 = -2.0^\circ\).
A tire is set to 120 kPa gauge at 20 °C. On track the air inside reaches 80 °C. Assuming the volume stays constant and atmospheric pressure is 101.3 kPa, what is the hot gauge pressure?
Use absolute pressure and absolute temperature: \(p_2 = p_1\,T_2/T_1\). Convert back to gauge at the end.
\(p_1 = 221.3\) kPa abs, \(T_1 = 293.15\) K, \(T_2 = 353.15\) K, so \(p_2 = 221.3 \times 353.15/293.15 = 266.6\) kPa abs \(= 165.3\) kPa gauge. A 45 kPa (≈6.5 psi) rise: that is why teams set cold pressures low.
Using the membrane model \(A \approx F_z/p\), estimate the contact patch area of a tire carrying 3000 N at 150 kPa gauge.
1 kPa = 1000 N/m², and 1 m² = 10 000 cm².
\(A = 3000 / 150\,000 = 0.02\ \text{m}^2 = 200\ \text{cm}^2\), roughly a 14 cm × 14 cm square.
Why do race cars usually run negative static camber?
- To make the car more stable in a straight line.Negative camber generally hurts straight-line braking and traction slightly. It is a cornering compromise.
- Body roll tilts the outside tire toward positive camber, so negative static camber leaves it near its best angle at the limit.The heavily loaded outside tire does most of the cornering work, so its camber at the limit matters most.
- Because camber thrust is larger than cornering force.Camber stiffness is much smaller than cornering stiffness. The main benefit is contact-patch loading.
A driver leaves the pits on new tires and spins at the first corner, at a speed they took easily on the previous lap. The most likely tire explanation is:
- The tires were over-inflated.Possible, but the most common out-lap problem is temperature.
- The tires were below their temperature window, so peak grip was much lower than the driver expected.Cold rubber grips poorly. Drivers must build temperature before pushing on new or cooled tires.
- New tires have too much load sensitivity.Load sensitivity does not change much between new and used tires. The dramatic change is temperature.
After normalizing lateral-force data from three loads (plotting \(F_y/F_z\)), the highest-load curve sits clearly below the others near the peak. What does this show?
- The test machine was miscalibrated at high load.The gap is the expected behavior of a real tire, not an error.
- The tire is load-sensitive: its friction coefficient falls as load rises.If \(\mu\) were constant, normalized curves would collapse onto one line. The gap measures load sensitivity.
- The tire has more cornering stiffness at high load.Absolute cornering stiffness may rise with load, but a lower normalized peak is about \(\mu\), not stiffness.
Going further with Milliken
Read: RCVD Chapter 2, Tire Behavior, on camber, inflation pressure and temperature effects. Then read Chapter 14, Tire Data Treatment, on how raw test data is processed and normalized.
As you read, look for answers to these questions:
- How large is camber thrust compared with slip-angle force in the examples given?
- What exactly is normalized in the book's approach (force, slip angle, or both), and what does each normalization reveal?
- What are the risks of using a tire model outside the range of the data it was fitted to?
Summary
| Idea | Key relation |
|---|---|
| Camber thrust (linear) | \(F_y \approx C_\alpha \alpha + C_\gamma \gamma\), with \(C_\gamma \ll C_\alpha\) |
| Camber at the limit | \(\gamma_{\text{static}} + \Delta\gamma_{\text{roll}}\); aim near the tire's optimum |
| Contact patch (membrane model) | \(A \approx F_z / p\) |
| Hot pressure | \(p_{2,\text{abs}} = p_{1,\text{abs}}\,T_2/T_1\) (absolute units) |
| Normalization | Plot \(F_y/F_z\); curves that don't collapse show load sensitivity |
That completes the tire. Module 3 moves up to the whole car: how load moves between the four tires, and why that changes the car's balance.