# Prediction of Damage to the Vehicle Underbody due to Stone Chipping
## 2 Definition and Prediction of Stone Chipping
### Numerical Definition of Stone Chipping Based on Vehicle Geometry  
The vehicle driving environments in this study were assumed to be irregularly cleaned roads with speed limits of 60 kph (downtown), 80 kph (urban expressway), and 110 kph (expressway), respectively, and the masses of the dropped objects or stones that are highly likely to exist on these roads were assumed to be approximately 1-10 g. The road driving environment and dropped objects that are highly likely to be encountered by vehicles in the development stage were considered. The escape direction of a dropped object that has passed under the contact surface of a rotating tire is tangential to the tire from the escape point, which depends on the air pressure and slip angle of the tire. Thus, the equation for the tire contact area (Equation (1)) can be used for calculating the tire contact length, \(D_{h}\), depending on the travel direction. \(D_{h}\) was calculated as a function of the tire pressure by selecting the height, based on the tire radius, \(R\), as shown in Table 2. In Equation (3), \(\alpha\) is the tangential angle of the stone at the escape point, and \(D_{h}\) is the vertical length of the tire contact area. \[\alpha=\sin^{-1}\ \frac{D_{h}/2.}{R} \tag{3}\]  
Under the given tire conditions, the escape point of the dropped object or stone is determined by \(D_{h}\), as given in Equation (3). Thus, the tangential angle of escape and the escape point of the dropped object can be identified. As seen in Figure 3, the range of the stone chipping zone as a function of the tire pressure can be defined by \(\theta_{1}\) and \(\theta_{2}\), which are the tangential angles of the dropped object, and \(L\) is the overall length \begin{table}
\begin{tabular}{c c c c c c} \hline \hline
**Code Angle of the Tire Belt (\({}^{\circ}\))** & \(\mathbf{-20}\) & \(\mathbf{-10}\) & \(\mathbf{0}\) & \(\mathbf{10}\) & \(\mathbf{20}\) \\ \hline Weight Distribution (kPa) & 6 & 4 & 8 & 4 & 6 \\ Weight Distribution Ratio & 0.75 & 0.5 & 1 & 0.5 & 0.75 \\ \hline \hline \end{tabular}
\end{table} 
Table 1: Weight distribution of the tire by spline interpolation [13]. \begin{table}
\begin{tabular}{c c c c c c} \hline \hline
**Tire Pressure (\%)** & \(\mathbf{70}\) & \(\mathbf{75}\) & \(\mathbf{80}\) & \(\mathbf{85}\) & \(\mathbf{90}\) \\ \hline Height (\(D_{h}\), cm) & 22.31 & 18.59 & 14.88 & 11.16 & 7.44 \\ Tangential Angle (\(\alpha\), rad) & 0.2972 & 0.2465 & 0.1965 & 0.1469 & 0.0978 \\ \hline \hline \end{tabular}
\end{table} 
Table 2: Tire contact length and tangential angle for various tire pressures [13]. Figure 2: (a) Underbody geometry and effective stone chipping area due to the code angle of tire belt; (b) the weight distribution of the tire. of the vehicle. In this study, the ratio of the potential energy of the tire slip transferred to the stone was defined as in Equation (4), where \(\eta\) is the kinetic energy ratio (%) delivered by the slip described above. \[\eta=1-\text{Slip}(\%), \tag{4}\]  
\[V_{\Delta}=\sqrt{\eta\frac{M_{\varphi}gR\alpha^{2}}{4M_{p}}}. \tag{5}\]  
Furthermore, Equation (5) defines the linear velocity of the stone [4], where \(M_{\varphi}\) is the empty weight of the vehicle, \(M_{p}\) is the mass of the stone, and \(V_{\Delta}\) is the amount by which the linear velocity of the stone increases. Applying the weight distribution ratio (_WD_ratio_) from Table 1, \(V_{\Delta ratio}\) depending on the escape angle of the dropped object can be calculated as follows:  
\[V_{\Delta ratio}=WD\_ratio\cdot V_{\Delta}. \tag{6}\]  
The absolute speed, \(V_{abs}\), of the stone immediately after it passes below the tire can be determined using the local reference frame (LRF), as illustrated in Figure 4 [4]. Furthermore, \(V_{abs}\) can be expressed in terms of its \(i,j\) and \(k\) vector components, based on ISO Standard 4130; the functions for the absolute speed are given by Equations (7)-(10), where \(V_{a}\) is the driving speed of the vehicle.