# Driving Safety Improved with Control of Magnetorheological Dampers in Vehicle Suspension
## Vehicle Model and Its Coupling with Road and Environment
### Wheel and Tire Model  
The experimental ATV is equipped with two types of wheels of different widths that were equal to \(w_{uf}\) = 0.2 m (8 inches) for the front wheel and \(w_{ur}\) = 0.254 meter (10 inches) for the rear wheel. Other dimensions of both wheels remain the same, i.e., radius of the unloaded wheel \(r_{uf0}=r_{ur0}\) = 0.317 m and radius of the rim \(r_{wf0}=r_{ur0}\) = 0.152 m (6 inches). A tire aspect ratio being a ratio of tire sidewall height to tire width is a key parameter defining a type of tire. For standard tires, this ratio varies commonly from 0.5 to 0.7. In the experimental vehicle, this parameter takes a value of 0.65 for the rear tire and 0.81 for the front. Thus, the front tire appears to be slimmer in comparison to standard tires and the rear wheel is assumed as a reference in order to estimate moment of inertia of both wheels, denoted as \(I_{uf}\) and \(I_{ur}\), respectively. The diagram of the moments of inertia of the wheels for standard vehicles was presented with respect to a dynamic wheel radius (i.e., for a moving vehicle) in [43]. These data were compiled for the purpose of this study with a comparison between the dynamic radii and radii of unloaded wheels that were obtained experimentally and discussed in [44]. Consequently, the moment of inertia of the standard rear wheel was evaluated as \(I_{ur}\) = 0.9 kg m\({}^{2}\) with respect to the aspect ratio. The moment of inertia of the front wheel was scaled down with respect to the ratio between wheel weights and evaluated as \(I_{uf}\) = 0.6 kg m\({}^{2}\), since the moment of inertia depends on a weight of the object with the same shape and size of the cross-section. The goal of the tire model is to evaluate the resultant force \(F_{n}\) and torque \(T_{n}\) acting on the wheel axle at a given time instant. These forces are related to ground force generated as a reaction to tire stiffness \(F_{kii}\) and damping forces \(F_{cui}\) in a given point of road profile, as noted in Figure 3. The tire stiffness comes mainly from the compressed air of pressure \(p_{u}\) inside the tire, which acts on a road surface. Tire damping is defined here as a force which counteracts deflection of the tire and is directly proportional to the velocity of deflection denoted as \(\dot{n}_{ci}\). Ground forces are transferred through the stiff sidewalls of the tire to the wheel axle. The rim was assumed as a rigid component, while the outer surface of the tire is assumed to be deformable, ideally adapting to uneven terrain. The evaluation of wheel forces is performed in several stages for a given moment in time. Firstly, for a given location of wheel and known road profile an area of tire cross-section, denoted as \(\Delta A_{u}\), is calculated, which shows how much the road deflects the tire. Next, instantaneous tire air pressure multiplied by \(w_{uf}\) is calculated according to the general gas Clapeyron's equation assuming a constant temperature:  
\[(p_{u0}\cdot w_{u})\cdot A_{u0}=(p_{u}\cdot w_{u})\cdot A_{u}, \tag{3}\]  
where \(A_{u0}=\pi\cdot(r_{u}^{2}-r_{w}^{2})\) denotes an initial area of tire air and it is equal to the difference of wheel and rim areas. Symbol \(p_{u0}\) denotes initial tire air pressure, while the method of its estimation multiplied by \(w_{u}\) is later described in the subsection. The instantaneous area of tire air is denoted as \(A_{u}=A_{u0}-\Delta A_{u}\).