Page - 136 - in Maximum Tire-Road Friction Coefficient Estimation

B Tire modelTMsimple and tire dynamics dynamic forceFDy is given with FDy = cT,y ·ye+dy · y˙e. (B.6) The partial derivative of the lateral force Fy with respect to the lateral velocity vy in the contact point can be calculated with ∂Fy ∂vy = ∂Fy ∂sα ∂sα ∂vy = ∂Fy ∂sα −1 re · |ωr|, (B.7) including an alternative definition of the lateral slip sα which is related to the side slip angle by sα = tanα, [Ril06]. According to Rojas Rojas, the partial derivative of the lateral tire force with respect to the slip angle α can be approximated by their global derivatives for small values ofα, [Roj12, p.29], and hence reads ∂Fy ∂sα ≈ Fy sα . (B.8) Inserting Equations B.6, B.7 and B.8 in Equation B.5 gives 1 cT,y · ( Fy sα 1 re · |ωr| ) · y˙e+ye= Fy cT,y , (B.9) including the term before y˙e, which gives a formulation of τ that can be used in the simulation model. Thus, the time function ultimately reads τ= 1 cT,y · ( Fy sα 1 re · |ωr| ) . (B.10) 136