Traction, steering geometry, and stability

Requirement

Push a loaded shopping cart at a jog and try to corner it: it plows straight on, wheels turned and ignored. Push an empty one the same way and it darts wherever the handle points. Same cart, same wheels, same floor; only the mass and where it rides have changed. The analogy breaks at the source of the force: a cart is steered by hands that shove it from outside, while a vehicle acts on the road only through its tire contact patches. That is the design requirement in one line: corner and brake without losing grip or rolling over, using only the friction the tires can muster.

Variables

The vehicle has mass m, moving at speed v around a turn of radius r. The steering angle delta is the angle of the front wheels from straight ahead, in radians. The wheelbase L is the distance between front and rear axles; the track width t is the side-to-side distance between contact patches; h is the height of the center of mass above the road. Each axle presses on the road with a normal load N, its share of the vehicle's weight, and mu is the coefficient of static friction between tire and road; g is the gravitational acceleration, about 9.8 meters per second squared. The grip limit of an axle, the horizontal force it can transmit before its tires slide, is mu times N.

Design choices

Static friction caps the horizontal force a loaded tire transmits at roughly mu N, and the cap has no preferred direction: braking, drive thrust, and cornering all draw on one reserve. If an axle brakes with force F_b while cornering with side force F_c, it is the vector sum, the square root of F_b squared plus F_c squared, that must stay under mu N. Plotted in the plane of horizontal forces, the limit is a circle of radius mu N, and engineers name it the friction circle: one budget of grip per axle, spent on turning and braking together.

Next, where the vehicle goes when steered. At walking pace the tires roll where they point, so the two axles' headings fix a shared turning center, and geometry gives the turn radius as approximately r = L / delta for small steering angles. Because the inner and outer front wheels ride circles of different radii, they want slightly different angles, an arrangement named Ackermann geometry after the linkage that supplies it. At speed the picture shifts: tires generate side force by running at a small slip angle to their own heading, so the true radius drifts away from L over delta, which remains the low-speed limit.

Cornering itself is a demand on the budget. A body holding a circle of radius r at speed v needs an inward force of m v squared over r, the result the circular-motion entry derives; the tires must supply it or the circle is not held.

Now the load shifts. During braking at deceleration a, the road pushes rearward on the contact patches at ground level while the vehicle's inertia acts at the center of mass, a height h above. Take torques about the line of the front tire contacts: the braking force m a acting at height h is a torque m a h that the normal loads, separated by the wheelbase, must balance. So a load delta-N = m a h / L leaves the rear axle and lands on the front. The same balance turned sideways gives the cornering version: a lateral acceleration of v squared over r moves a load of m (v squared over r) h / t from the inner wheels to the outer ones.

Here is the subtlety the naive account misses. In steady cornering with no braking and no thrust, a moment balance about the center of mass makes each axle's share of the cornering demand equal its share of the static weight, and its grip limit is mu times that same weight. Demand and supply scale together, so the static weight split alone predicts nothing about which end saturates first. What breaks the tie is how the budget is spent. Braking mid-corner transfers m a h / L off the rear axle, shrinking the rear's grip circle, while the rear's geometric share of the cornering demand is set by where the mass sits along the wheelbase and does not change; on top of that, the rear brakes spend part of what remains. Drive thrust works the same way on the driven axle: front-drive under power spends part of the front circle, rear-drive part of the rear circle. So rear-drive power-on and mid-corner braking both push the rear axle toward saturation, and front-drive thrust pushes the front. When the front axle saturates first, its tires run at their limit and the vehicle runs wider than steered; this is named understeer, the push of the loaded cart. When the rear saturates first, the tail slides outward and the vehicle rotates more than steered; this is oversteer, and unchecked it becomes a spin. A deeper mechanism, tire load sensitivity, by which grip grows a little less than proportionally with load so that transferring weight costs total grip, is real and shifts this balance too, but it lies beyond this entry's uniform-friction model and is deferred.

Calculations

Take a small vehicle: m = 10 kilograms, L = 0.5 meters, t = 0.4 meters, h = 0.15 meters, mu = 0.8, weight split evenly so each axle carries N = 49 newtons.

Steering geometry sets the circle: at delta = 0.1 radians, r = L / delta = 5 meters.

Steady cornering: the demand is m v squared over r and the total supply is mu m g, about 78 newtons. They meet when v squared over r = mu g, so v = the square root of (0.8 times 9.8 times 5), about 6.3 meters per second. Each axle's demand share equals its load share in this model, so both axles reach their limits together at that speed.

Add braking at a = 2 meters per second squared. The transfer is 10 times 2 times 0.15 / 0.5 = 6 newtons off the rear: rear load 43 newtons, rear budget 0.8 times 43, about 34 newtons. At v = 5.5 the total cornering demand is 10 times 30.25 / 5, about 60 newtons; the rear's geometric share is half, about 30 newtons, and the rear also supplies half the 20-newton braking force, so 10 newtons. The vector sum, the square root of (30 squared plus 10 squared), is about 32 newtons, just inside the budget. By 5.8 meters per second it passes. So a corner that holds to 6.3 taken clean saturates the rear near 5.8 once the brakes come on: the oversteer risk is now a number.

Tipping is a separate threshold. The inner wheels unload completely when the lateral transfer m a_lat h / t, where a_lat is the lateral acceleration, equals the m g / 2 they started with, giving a_lat = g t / (2h). Here that is 9.8 times 0.4 / 0.3, about 13 meters per second squared, or 1.33 g, while sliding begins at mu g = 0.8 g: this vehicle drifts wide before it can roll. Stack the same mass high and narrow, h = 0.3 meters on a 0.3 meter track, and the tip threshold falls to 0.5 g, below the 0.8 g grip limit: that vehicle rolls before it slides.

Failure modes

Tip-over from a high narrow stance: whenever t / (2h) is less than mu, the rollover threshold sits below the sliding threshold, and the vehicle lifts its inner wheels while the tires still hold.

Snap oversteer on lift-off or braking mid-corner: forward weight transfer cuts the rear axle's grip circle while its geometric share of the cornering demand stays fixed, and the rear lets go abruptly at a speed that felt safe moments before.

Grip lost to combined demands: braking and cornering that would each fit the budget alone can exceed it together, because the friction circle caps the vector sum rather than each force separately.

Limits and boundary conditions

The analysis treats the vehicle as a rigid body in steady cornering on uniform friction. It ignores suspension compliance, which spreads load transfer out in time; tire load sensitivity, named above, which erodes total grip as transfer grows; and aerodynamic load, which adds grip at speed without adding mass. The geometry r = L / delta holds in the low-speed rolling regime; at higher speeds slip angles govern and the geometric radius is only the starting point. The thresholds are estimates, since real tires give up progressively around mu N rather than switching cleanly at it.

Build with it

Take a small vehicle you can measure and test, a kart, an RC car, or a robot chassis. Establish its numbers: mass and its front-rear split by weighing each axle, wheelbase L, track t, and center-of-mass height h from the axle-load change when one end is raised a known amount, and mu from a simple drag test. Then predict before driving: compare t / (2h) with mu to call the failure mode, slide or tip; compute the threshold speed for a marked circle of radius r from v = the square root of (mu g r) or the tipping analog, whichever bound is lower; and from the per-axle budget arithmetic under the test's braking or thrust, call understeer or oversteer. Then run the marked circle at gradually rising speed. Success has two parts: the measured threshold, the speed at which the vehicle slides out or lifts an inner wheel, falls within a stated tolerance of the prediction, fifteen percent is reasonable; and the understeer-or-oversteer call was written down in advance and justified by the grip-budget numbers for each axle, with the observed behavior matching the call, argued from arithmetic rather than from feel.