VM-YMD

What each point is

Each intersection of the lines of the diagram is a point, which is what the tyres do if the car is held at that body slip angle β and steer angle δ, at a constant longitudinal speed u. The car only remains in steady state on the x-axis, M_z = 0. Anywhere else the yaw moment induces an oversteer (+) or understeer (−) yaw acceleration on the chassis. Even so, every point is calculated as if the car remained in steady state, that is, as if an external M_z, coming from somewhere magic, always cancelled the yaw moment of the tyres.

Assumptions

• Steady state. No dampers and no relaxation length, since they change how fast the car gets to a point and leave the point where it is.
• Rear wheel drive. The front wheels roll free, without rolling resistance.
• Rigid chassis. Springs and bars are wheel rates and react the roll moment about the roll axis. Heave and pitch change the loads but not the geometry, so there is no camber or ride height change from them.
• Parallel steering, no Ackermann, no compliance steer and no roll steer.
• Aero with constant coefficients, no ride height sensitivity. Drag acts at the height of the centre of gravity.
• Tyre with load sensitivity and combined slip. Camber enters as an offset of slip angle and does not change the peak grip. No pressure or temperature effects.
• The differential as a saturated coupling, with the demand worked out with a linear longitudinal tyre.

Wheel kinematics

Wheel k is at (x_k, y_k), with the turn to the right: e is the outer wheel on the left, i the inner one on the right. Toe in steers each wheel towards the centre line.

Tyre

The tyre gives less grip per kilogram as the load increases.

Lateral and longitudinal slip share the same grip, so both are divided by the slip at the peak and combined as one slip, s = |(α/α*, κ/κ*)|. Camber moves the slip angle by r_γ γ, where the camber at the ground changes with roll as

Vertical loads

Each wheel is loaded by its static load, its part of the downforce, the longitudinal transfer and the lateral transfer of its axle j. The lateral transfer travels by three paths, through springs and bars (elastic), through the roll centre (geometric) and through the unsprung mass.

Throttle and differential

With “hold speed” the rear axle pushes exactly what keeps u constant. With “engine torque” the rear axle gets T_d = T_e i and the car speeds up or slows down, which adds longitudinal transfer.

The differential is a saturator. The demand is the force difference the tyres ask for when both rear wheels turn at the same speed. The capacity is what the diff can hold. If the demand is below the capacity the diff is locked, otherwise it slips at its capacity.

ε_d takes its drive value when T_d > 0 and its coast value when T_d < 0. M_d > 0 is oversteer.

Yaw moment

The superscript b means the force turned into the body axes by the wheel steer angle. Positive M_z turns the car further into a right hand corner.

The numbers on the right

• Held: the highest a_y on the trim line, M_z = 0. It is the maximum the car will hold in steady state, white dot.
• Peak: the highest a_y of the whole diagram, the tip. The tyres reach it, but a yaw moment M_z is left, so the car would start to rotate. Positive M_z is oversteer, and negative is understeer, gold dot.
• Lost: peak minus held, the grip the balance leaves unused.
• Balance: M_z at the peak, in kN·m.
• Entry is the trim point at 0 g, apex the trim point 0.05 g below held. At each one, ∂M_z/∂δ is the control the driver has and ∂M_z/∂β the stability, positive is stable. Both by central differences of ±0.1°.

The model and its equations

The full version shows every assumption and every equation behind the diagram: tyre, load transfer by its three paths, throttle and differential, and how each number on the right is read.