# Assessment of Tire Features for Modeling Vehicle Stability in Case of Vertical Road Excitation
## 2 Investigation the Tire-Road Interaction  
### Investigation of the Tire Properties  
When dealing with the vehicle stability tasks, the issue of tire-road interaction is one of the most complex tasks. Analysis of the vehicle wheel-to-road contact requires taking into account the deformation properties of the tire. The complexity of the task stems from the uncertainty of the operating conditions of the tire and the complex structure of the tire. The complexity of the problem is defined by the mere fact that no consensus has been reached on the shape of the tire. The performed studies of the tire-road interaction [36] confirmed that it would be necessary to consider the deformation properties of the band in the investigation of the compensating function of the band. For this purpose, an additional element is added to the quarter-car model (Figure 2). Additional studies are needed in order to identify the deformation properties of the element. Figure 1: Influence of external factors on vehicle directional stability. The methodology may be used where deformation of an additional element, i.e., the tread, is needed in the quarter-car model (Figure 2). Whereas both tread layers (with and without the pattern) are not made of composite material, the equations applicable to rubber parts are used here as they consider that \(\
u=0.5\)[37]. The tread model is comprised of two consistently joined layers: the one with a pattern and the one without a pattern. The deformation of the part with a pattern was assessed using three models. This paper analyzes radial tires of three types that differ considerably in terms of structure: passenger car (175/70R13; 195/50R15), light-duty truck (185/75R14C), and heavy-duty truck (12.00R20) tries. In the assessment of the vehicle tire-road interaction, it is first necessary to describe the deformation properties of the tire under the vertical load. The static load consists of the vehicle and cargo mass. The loaded tire is subject to deformation when forming the contact area with the road pavement. The tire-road contact is depicted in Figure 3, and tire element characteristics are presented in Table 1. The following assumptions were used when determining the area of the contact with the road:  
* the contact with the road has an elliptical shape and is limited by the tire width and length of the contact with the road;
* the layers under the tread are sufficiently stiff, and their length does not change during deformation of the tire (\(l_{d}\approx l_{0}\)) (Figure 3a);
* there is the linear dependency between the radial load of the tire and radial deformation \(\Delta l\);
* the pressure is distributed evenly across the contact surface. The second and third assumptions enable determining the length of the tire-road contact:  
\[L_{k}/2=r\cdot\arccos((1-C_{R}R_{z})/r) \tag{1}\]  
where \(C_{R}\)--radial stiffness of the tire and \(r\)--free radius of the tire. More accurate information on the tire element loads is provided by the model 'elastic ring with the outer elastic layer' (Figure 4a,b) [38]. The ring is considered to be the elastic band; the tensile force \(N_{0}\) caused within it by the inner pressure does not change during deformation; and the ring points are displaced only in the radial direction by the measure \(w\)  
Figure 2: Quarter-car model: \(L_{k}\)—tire-road contact length; \(m_{1}\), \(c_{1}\) and \(k_{1}\)—mass, stiffness, and damping ratio of the tire tread part; \(c_{2}\), \(k_{2}\)—tire stiffness and damping ratio; \(c_{3}\), \(k_{3}\)—suspension stiffness and damping ratio; \(m_{2}\)—unsprung mass; \(m_{3}\)—sprung mass.