Page - 134 - in Maximum Tire-Road Friction Coefficient Estimation

B Tire modelTMsimple and tire dynamics proposedbyHirschberg et al. asshowninEquationB.3isused, [HRW07]. This isdoneby adjustingthemaximumforceYmax andthesaturationforceY∞bytheproportional factor µmax µ0 , whereµ0 is the nominal value of the friction potential for which the respective tire parameters have been measured. The initial stiffness dY0 remains unaltered. Ymax(Fz) = ( a1 Fz Fz,nom +a2 ( Fz Fz,nom )2) ·µ max µ0 dY0(Fz) = b1 Fz Fz,nom +b2 ( Fz Fz,nom )2 (B.2) Y∞(Fz) = ( c1 Fz Fz,nom +c2 ( Fz Fz,nom )2) ·µ max µ0 The coefficientsa1 to c2 in Equation B.3 are needed to consider the decreasing influence of the tire loadFz on the horizontal tire forces. These coefficients depend on measured values for the nominal tire loadFz,nom and the doubled nominal tire load 2·Fz,nom. The values for a1 and a2 are given, for example, by a1 = 2 ·Y1− 1 2 Y2 and a2 = −Y1 + 1 2 Y2, (B.3) with Y1 = Ymax(Fz,nom) and Y2 = Ymax(2 ·Fz,nom) at 2 ·Fz,nom. Accordingly, the coefficients b1 and b2 are calculated using the respective initial stiffness values, and c1 and c2 are calculated using the respective saturation values. B.2. Modelling time function τy When a rolling tire is being steered, the time until a lateral tire contact force is built up can be described by a first order system, as shown in Section 3.3.2. According to Rill, the time function τ in Equation 3.29, which describes the dynamic processes that occur in the contact patch, depends on two factors, [Ril06]. The first factor is the transport velocity re·|ωr|withwhich theparticles in the tire treadaremoving throughthecontact patch. The second factor is the relaxation length rα that describes the distance that a particle in the tire tread travels from when a change ofα occurs until the full force is built up. The time function τ is given by τ= rα re · |ωr|. (B.4) 134