# Driving Safety Improved with Control of Magnetorheological Dampers in Vehicle Suspension
## Vehicle Model and Its Coupling with Road and Environment
### Wheel and Tire Model
Stiffness \(F_{hui}\) and damping forces \(F_{cui}\) that are generated by each fragment of the road profile contribute to the resultant force \(F_{nx}\) and \(F_{nz}\) in \(x\) and \(z\) directions, respectively, as well as torque \(T_{n}\), as follows:  
\[F_{nx} =\sum_{x_{ri}}^{(-r_{u},r_{u})}\sin(\beta_{i})\cdot\max(0,F_{hui} +F_{cui}),\] \[F_{nz} =\sum_{x_{ri}}^{(-r_{u},r_{u})}\cos(\beta_{i})\cdot\max(0,F_{ Kui}+F_{cui}), \tag{4}\] \[T_{n} =\sum_{x_{ri}}^{(-r_{u},r_{u})}\cos(\alpha_{i}+\beta_{i})\cdot r _{i}\cdot\max(0,F_{hui}+F_{cui}),\]  
where angle \(\alpha_{i}=\text{atan}2(x_{u}-x_{ri},z_{u}-z_{ri})\) describes orientation of radius \(r_{i}\) with respect to the vertical axis. Angle \(\beta_{i}=\text{atan}(\frac{dz_{ri}}{dx_{ri}})\) denotes an inclination of the road profile in the given point and it can be calculated based on a derivative of the road profile height \(z_{ri}\) with respect to its coordinate \(x_{ri}\). The forces that are presented in Equation (4) are calculated for the road surface coordinates \(x_{ri}\) varying from \(-r_{u}\) to \(r_{u}\), which is the radius of unloaded wheel. Using the max function means that Equation (4) is only considered for \(F_{kui}+F_{cui}\) not lower than zero, which indicates detachment from the road surface. The wheel radius of a given point denoted as \(r_{i}=\sqrt{(x_{u}-x_{ri})^{2}+(z_{u}-z_{ri})^{2}}\) is calculated as a distance between the road profile point and a center of the wheel. Forces \(F_{kui}\) and \(F_{cui}\) are defined for the given point of road profile, as follows:  
\[F_{kui} =\Delta l_{i}\cdot(p_{u}\cdot w_{u}), \tag{5}\] \[F_{cui} =\Delta l_{i}\cdot(c_{A}\cdot w_{u})\cdot\dot{n}_{ci}.\]  
Figure 3: Tire model for uneven road: (**a**) tire subjected to road irregularities and ground normal forces, (**b**) tire stiffness and damping forces. where the length of the unit fragment of road profile \(\Delta l_{i}=\Delta x\cdot\tan(\beta_{i})\) and \(\Delta x=\frac{1}{f_{ps}}\) depends on the spatial frequency \(f_{ps}\). The velocity of a given point of tire \(\dot{n}_{ci}\) parallel to the normal direction of the road is evaluated based on wheel translational and rotational velocities, as follows:  
\[\dot{n}_{ci}=\sin(\beta_{i})\cdot\dot{x}_{u}-\cos(\beta_{i})\cdot\dot{z}_{u}- \sin(\alpha_{i}+\beta_{i})\cdot r_{i}\cdot\dot{\varphi}_{u}. \tag{6}\]  
The initial tire air pressure \(p_{u0}\cdot w_{u}\) and tire damping parameter \(c_{A}\cdot w_{u}\) were calculated based on vertical tire stiffness \(k_{u}\) and damping parameter \(c_{u}\), measured for static tire load. Measurements that were taken from the experimental vehicle indicate that static deflections of the front and rear tires \(z_{tf,static}\) and \(z_{tr,static}\) are approximately equal to 14 and 13 millimeters for wheel normal forces \(F_{kuf,static}\) and \(F_{kur,static}\) that are equal to 986 and 941 Newtons, respectively. The reference tire damping parameter \(c_{u}\) was evaluated assuming that the damping ratio is close to 0.05 for the majority of tires, as stated in [45]. As a result, parameters \(c_{uf}\) and \(c_{ur}\) take values of 84 and 102 Nsm\({}^{-1}\) for the front and rear tire, respectively. The tire model and evaluation of tire stiffness and damping forces was linked with the reference parameters \(k_{u}\) and \(c_{u}\) on the basis of the assumption that these parameters were obtained for a flat road surface, similarly as during laboratory measurements. In the case of reference tire stiffness, the following relationship is valid based on the integral evaluated for a continuous road profile:  
\[F_{ku,static}=k_{u}\cdot z_{t,static}=\int_{-x_{r,static}}^{x_{r,static}}(p_{ u0}\cdot w_{u})\cdot dx, \tag{7}\]  
which gives \(p_{u0}\cdot w_{u}\) equal to 5234.1 and 5113.6 Nm\({}^{-1}\) for the front and rear tire, respectively.