# Prediction of Damage to the Vehicle Underbody due to Stone Chipping
## 2 Definition and Prediction of Stone Chipping  
### Tire Geometry and Numerical Model  
In a typical road environment, an automobile may drive over dropped objects or stones above a certain size or weight. To predict the impact area and energy that may be transmitted to the underbody of the vehicle resulting from tire slip, mathematical models for the initial slip speed and progression direction in stone chipping were analyzed in this study. A statistical model, based on the established mathematical model, was applied to predict the areas where impact could occur, as well as the kinetic and impact energies of the stones. The vehicle specifications, layout, and tires were defined for a compact hybrid electric vehicle (HEV) of the H Company, Korea, which has a diverse lineup including internal combustion engine vehicles, HEVs, plug-in hybrid electric vehicles, and electric vehicles (EVs). The empty mass of the target vehicle was approximately 1400 kg, the tire size was 195 mm/65/15 inch, and the tire pressure was 241.3 kPa. The tire contact area, geometry, and variables for the weight distribution on the contact surface, which were used for calculating the escape speed of a stone due to tire slip, are shown in Figure 1.  
The lateral force characteristics of a rotating tire are determined by the constant load and slip angle conditions in steady state. For an initial slip angle, \(S_{\alpha}\), range of 5-10\({}^{\circ}\), the lateral force is directed straight forward, whereas for an \(S_{\alpha}\) range of 15-20\({}^{\circ}\), the contact surface increases but the lateral force tends to decrease. Therefore, the range of \(S_{\alpha}\) was set to \(\pm\)20\({}^{\circ}\)[12]. In Figure 1, the tire contact area \(S\), is defined as in Equation (1), and can be calculated by substituting the abovementioned vehicle specifications in this equation:  
\[S=\frac{F}{P}, \tag{1}\]  
where \(F\) is the empty weight of the vehicle (kN) and \(P\) is the tire pressure (kPa). The acceleration and braking force of the tire are caused by the difference between the rotational and driving speeds. Consequently, minor slippage occurs due to the difference between the frictional and propulsive forces of the contact surface, resulting in energy loss. The slip is defined as a dimensionless quantity, which is a percentage of the forward speed, as in Equation (2). Hence, the potential energy ratio delivered to the stone can be calculated using the slip ratio:  
\[\text{Slip }(\%)=(1-\frac{r\omega}{V})\times 100, \tag{2}\]  
where \(r\) is the effective rotational radius of the tires, \(\omega\) is the angular speed of the wheels, and \(V\) is the driving speed of the vehicle. In this study, the linear speed, immediately after the escape of the dropped object, was assumed to be equal to the vehicle speed multiplied by the slip ratio. The tire contact area was defined as per the definition in ISO 4130. Furthermore, when the vehicle turned, the belt code angle was assumed to range approximately between \(-\)5\({}^{\circ}\) and 20\({}^{\circ}\) as shown in Figure 2a. The effective stone chipping area was defined by the range, because the stones below the angle \(-\)5\({}^{\circ}\) does not affect the underbody damage, considering the slip angle of \(\pm\)20\({}^{\circ}\). The escape direction and potential energy of a dropped object generally vary, depending upon the tire contact weight distribution. In general, the contact load is the highest at the center of the tire contact area, increasing again at both its ends. As the vehicle moves, the code angle of the tire belt maintains under 20\({}^{\circ}\) in order to stabilize the contact force against lateral deformation of the tire. The contact force of tire, which is caused by shear stress on the contact surface, does not occur at one point but on the whole contact surface [13]. In this study, the cross-section of the tire contact weight distribution was calculated using spline interpolation, considering the continuity.