Differential on Split Grip

What is modelled

A differential lets the two driven wheels turn at different speeds, which a car needs in every corner because the outer wheel travels further. The price: an open differential always sends the same torque to both wheels, so when one wheel is on ice the torque that wheel can carry limits the torque of the other, and the car barely moves. Locking devices move torque from the faster wheel to the slower one and give the grip back.

Equations

• Speed rule ωL + ωR = 2 ωc; torque of an open differential TL = TR = Tc/2.
• Locking transfers ΔT from the faster to the slower wheel: TL = Tc/2 − ΔT, TR = Tc/2 + ΔT, with ΔT ≤ Tpre + f Tc/2.
• Tyre force F = μ N tanh(s/0.08), slip s = (ω r − v)/max(ω r, v), wheel load N = 0.55 m g / 2; engine torque falls to zero at 300 rad/s.
• Open differential on split grip: steady thrust 2 μweak N; locked: (μL + μR) N; cornering vout/vin = (R + t/2)/(R − t/2).

Checks run in the page

• Energy balance of a launch with engine work, kinetic energy, tyre slip heat, clutch heat and drag.
• Wheel path lengths in a corner by integrating positions.
• Steady thrust of the open and locked differential against the closed forms.
• Thrust ordering open ≤ clutch, bias ≤ locked and the side torques adding up to the engine torque.

Honest simplifications

Wheel load is fixed (no load transfer), the tyre is a single friction curve without lateral force, the clutch and bias devices are reduced to a torque-transfer limit, the engine is a linear torque droop, and the locked differential is a rigid link. Real bias differentials depend on the torque ratio of the gears, which is only represented by the fixed factor f.

• Press Ice under one wheel, open: the weak wheel races and the car creeps.
• Press Same ice, limited-slip and Same ice, locked to compare the thrust.
• Press Dry road, open differential: with equal grip the open differential loses nothing.