Tire Behavior

This page explains how a tire generates force and moment, and what that means for a vehicle model. For the wider vehicle model, see Vehicle Dynamics.

Force as a map

A tire is a nonlinear force law, not a scalar friction coefficient. $\mu$ is not constant. Think of it as a map:

$$\mu =\mu (F_z,\alpha ,\kappa ,\gamma ,T,p,\text{history},\text{wear})$$

$F_z$ is normal load, $\alpha$ is slip angle, $\kappa$ is slip ratio, $\gamma$ is camber, $T$ is temperature, and $p$ is pressure.

Slip and force buildup

Slip angle and slip ratio are not literal rubber-deformation measurements. They are practical coordinates that correlate with deformation and force buildup. Simplified forms:

$$\alpha \approx {\tan }^{-1}\left(\frac{-V_y}{|V_x|}\right)\kappa \approx \frac{R\omega -V_x}{max(|V_x|,ϵ)}$$

In the small-slip region, force buildup is roughly linear:

$$F_y\approx C_{\alpha }\alpha$$

This is spring-like: the tire deforms, and force grows with a deformation-like input. As slip increases, force builds more slowly. The tire then plateaus or falls to a lower force level. The three regions are:

Region	Behavior
Linear	mostly static-friction-like
Transition	mixed static and sliding
Saturated	sliding-dominated

Real tires add adhesion, hysteresis, tread deformation, carcass behavior, pressure, temperature, compound, road surface, and wear. The simple model still explains why a tire feels linear, then nonlinear, then saturated.

Combined slip

Combined slip matters because a tire cannot spend all of its longitudinal and lateral capacity at once. Slip ratio and slip angle together change the deformation pattern and the total force available. The tire is one deforming structure with a coupled force and moment response, not two independent force generators.

Pneumatic trail and scrub

Tires also shift their effective point of force application. Pneumatic trail and scrub are moment arms from tire deformation, not extra forces.

Pneumatic trail is usually the fore-aft offset of the lateral force resultant. It is a primary source of aligning moment:

$$M_z\approx -F_y{t}_p+M_{z,res}$$

$t_p$ is pneumatic trail. $M_{z,res}$ is the residual aligning moment that the offset picture does not capture. The sign convention depends on the tire coordinate system. Mechanical trail, pneumatic trail, scrub, caster, KPI, and compliance together decide how that moment becomes steering torque and upright load.

Pneumatic scrub is the lateral version: the effective force application point moves sideways as the patch deforms. Under braking, drive, and combined slip, that shift changes how longitudinal and lateral forces feed moments into the wheel, upright, steering system, and suspension.

Many empirical tire models output forces and moments about a common tire origin. There, trail and scrub are fitted internal quantities, and the result is still a force-and-moment system. A force-only tire match can reproduce lateral acceleration and still miss steering torque, compliance loading, and the loss of feel near saturation. Near the limit, pneumatic trail can collapse while lateral force stays high. The driver and suspension feel that change.

Relaxation, pressure, temperature, and wear

Tire force does not appear instantly. Relaxation length $\sigma$ is roughly the distance a tire must roll to build 63.2% of its steady-state force after a slip input. At vehicle speed $V$, it gives an approximate time constant:

$$\tau \approx \frac{\sigma }{V}$$

First-order relaxation model

In distance:

$$\frac{d{F}_y}{ds}=\frac{F_{y,ss}(\alpha )-F_y}{\sigma }$$

$s$ is distance traveled. Relaxation length also relates to structural stiffness:

$$\sigma \sim \frac{C_{\alpha }}{k_y}$$

$C_{\alpha }$ is cornering stiffness and $k_y$ is lateral shear stiffness.

Higher pressure tends to stretch and stiffen the tire structure. That reduces deformation and often speeds up force buildup. Check this against data rather than assume it.

Temperature can change peak force, stiffness, relaxation behavior, and wear at the same time. Tire data is the only reliable way to know. For FSAE teams, the practical sources are TTC data and published Magic Formula fits from experienced fitting groups, such as Stackpole Engineering Services.

Wear can help or hurt capability, depending on the tire, compound, surface, and operating window. Test at several life points. Track lifetime power output to correlate drive-day usage with controlled force-and-moment testing over time.

Empirical tire models

Magic Formula and Pacejka-style models are empirical force laws, not physics. That is acceptable: tires are complex, and the goal is to reproduce measured behavior across the operating region you care about.

Know the valid range: normal load, camber, pressure, temperature, slip angle, slip ratio, surface condition, and tire age and wear state. Outside that range, a good fit can give confident wrong answers.