Tire Dynamics

At present, two groups of models can be identified, handling models and structural or high frequency models. Structural tire models like RMOD-K or F Tire are very complex. These models are computer time consuming and they need a lot a data. Handling models like the MF-formula or TMeasy rely also on measured and observed force-slip characteristics.

Steady State Tire Forces and Torques

For the calculation of the contact patch geometry the tire is considered as a rigid body. Then, the tire deformation, the orientation of the local contact area, the location of the contact point, and the contact point velocities can be calculated from the momentary state of the wheel rim and the description of the road surface. The rolling resistance torque T_y which is less important for vehicle dynamics may be approximated by a rolling resistance lever. Within handling models the steady state tire forces in longitudinal and lateral direction are approximated by appropriate functions

(3.1)

(3.2)

which mainly depend on the longitudinal and lateral slip s_x and s_y. The steady state torque T^S_z around an axis perpendicular to the local road plane consists of the self aligning torque  and the bore torque T^S_B

(3.3)

where c_o = c_o (s_y) names the tire caster and s_B denotes a bore slip.

Simple Dynamic Extension

Measurements show that the dynamic reaction of the tire forces and torques to disturbances can be approximated quite well by first order systems. Then, the dynamic tire torque T^D_z are given by first order differential equations

(3.4)

(3.5)

(3.6)

which are driven by the steady values F^S_x, F^S_y and T^S_z. The tread particles of a rolling tire move with the transport velocity r_D|Ω| through the contact patch, where r_D and Ω denote the dynamic rolling radius and the angular velocity of the wheel. Now, time constants τ_i, can be derived from so called relaxation lengths r_i

(3.7)

But it turned out that these relaxation lengths are functions of the longitudinal and lateral slip s_x, s_y and the wheel load F_z, Figure 2. Therefore, constant relaxation lengths will approximate the real tire behavior in zero order approximation only. An appropriate model for the dynamic tire performance would be of great advantage because then, the cumbersome task of deriving the relaxation lengths from measurements can be avoided.