# Driving Safety Improved with Control of Magnetorheological Dampers in Vehicle Suspension
## Vehicle Model and Its Coupling with Road and Environment
### Burckhardt Model of Tire to Road Friction  
Tire-to-road friction is a complex phenomena, which, on the microscopic scale, is mainly based on rubber hysteresis caused by its continuous compression and expansion, and a rubber adhesion to the road surface, as discussed in Section 1. Thus, a macroscopic Burckhardt tire friction model was selected in order to avoid a further significant increase of the tire model complexity. The Burckhardt tire friction model that is presented in Figure 4 was described, e.g., in [33; 46] and it is defined for tire slip ratio \(sr\) by the following function:  
\[\mu_{r}(sr)=\text{sign}(sr)\cdot\left[f_{b0}\cdot(1-e^{-f_{b1}\cdot|sr|})-f_{ b2}\cdot|sr|\right], \tag{9}\]  
where parameters of the model were selected for a wet concrete surface, as: \(f_{b0}=0.4004\), \(f_{b1}=33.7080\), \(f_{b2}=0.1204\). It can be seen that \(\mu_{r}(sr)\) increases with the increase of the slip ratio up to the maximum possible tire friction. A further increase of the slip ratio deteriorates friction coefficient and, consequently, tire adhesion to the road surface. However, the macroscopic tire model needs to be linked with the microscopic approach dedicated to normal tire forces presented in the previous section in order to apply Burckhardt model. Thus, the inclination \(\gamma\) of the averaged road profile is defined based on direction of resultant normal forces as \(\gamma=\text{atan2}(-F_{nx},F_{nz})\). Moreover, the absolute value of the resultant normal force is obtained as \(F_{n}=\sqrt{F_{nx}^{2}+F_{nz}^{2}}\). Assuming that both the translational and rotational movements of the wheel are projected on an averaged road profile defined by angle \(\gamma\), the slip ratio \(sr\) is evaluated, as follows:  
\[sr= \begin{cases}\frac{v_{u,\text{friction}}-r_{e}\cdot\varphi_{u}}{|v_{u, \text{friction}}|}&\text{for}\ \ |v_{u,\text{friction}}|\geq|r_{e}\cdot\varphi_{u}|\\ \frac{v_{u,\text{friction}}-r_{e}\cdot\varphi_{u}}{|r_{e}\cdot\varphi_{u}|}& \text{for}\ \ |v_{u,\text{friction}}|<|r_{e}\cdot\varphi_{u}|\end{cases}  \tag{10}\]  
where \(v_{u,\text{friction}}=\cos(\gamma)\cdot\dot{x}_{u}+\sin(\gamma)\cdot\dot{x}_{u}\) denotes wheel velocity that is projected on the average road surface and \(sr\) needs to be additionally limited to the range from \(-1\) to \(1\). As a result, a tire to road friction force \(F_{r}\), perpendicular to force \(F_{n}\), can be evaluated, as follows:  
\[F_{r}=-\mu_{r}\cdot F_{n}, \tag{11}\]  
where its \(x\) and \(z\) components are obtained based on averaged inclination angle \(\gamma\): \(F_{rx}=\cos(\gamma)\cdot F_{r}\), \(F_{rz}=\sin(\gamma)\cdot F_{r}\). The effective wheel radius \(r_{e}\) that is used in Equation (10) is a conventional quantity that is commonly evaluated in an ideal case of flat road. In the case of significant unevenness of the road profile, a single effective radius cannot be indicated, since wheel dynamics are partially influenced by numerous fragments of road for which slip occurs in varying degrees. Different methods of calculation of the effective radius for a Burckhardt friction model were tested and analysed. Consequently, it was decided that the effective radius is calculated as an average of projections of all radii that were obtained for individual points of the road surface on a direction defined by a resultant normal wheel force (perpendicular to the averaged road profile defined by \(\gamma\)). This relationship can be concluded by the following formula:  
\[r_{e}=\frac{1}{N}\sum_{x_{i}}^{(-r_{u},r_{u})}r_{i}\cdot\cos(\alpha_{i}+\gamma), \tag{12}\]  
where \(N\) is a number of points on the road surface that are in contact with a tire. As a validation, it should be noted that the presented approach for the evaluation of the effective radius gives reasonable  
Figure 4: Characteristics of tire to road friction force on slip ratio for the Burckhardt model and wet concrete surface. results for the standard case of a flat road surface.