Open Differential on a Split Surface

What is modelled

An open differential splits the input torque equally between the two driven wheels and lets them turn at different speeds; the carrier speed is the average of the two. Each tyre can transmit at most a force mu N (its adhesion limit). If one wheel is on ice, that wheel can only push on the road with a small force, so the same small torque is all the other wheel receives, and the ice wheel spins up while the car barely moves.

Equations

• Kinematics: ωcarrier = (ωL + ωR) / 2, spider gears turn at (ωL - ωR) · 24/28.
• Torque: TL = TR = Tshaft / 2 at every instant.
• Wheel dynamics I dω/dt = Tshaft/2 - r F - b ω, tyre force F = μ N tanh(slip speed / vc).
• Vehicle m a = FL + FR - Froll - Fair.

Checks run in the page

• Split-mu run: both wheels receive the same shaft torque and the ice wheel sits at its adhesion limit.
• Rolling acceleration against the formula with the wheels' rotational inertia.
• Energy balance: input work equals kinetic energy plus slip, road, air and bearing losses.

Honest simplifications

A single smooth friction law, no weight transfer or suspension, no steering, tyre temperature or lateral forces, and a simplified engine torque roll-off, so the numbers are illustrative orders of magnitude. The page integrates the original solver with a 0.33 ms step because the stiff tyre law chatters at the 3.3 ms step of the 3D page when grip is high.

• Choose Split adhesion: the left wheel spins at thousands of rpm while the right wheel barely turns.
• Now raise the left grip slider step by step: the two wheel speeds converge and the car accelerates.
• Choose Wet road: both wheels are limited equally and the car moves slowly but without wheel-speed difference.