# Prediction of Damage to the Vehicle Underbody due to Stone Chipping
## 3 Results
### Numerical Simulation for Impact Prediction with the Kinetic Energy
(**d–f**): The effect of tire pressure on stone chipping zone at the vehicle speed of 110 kph for different stone masses; (**d**) 1 g; (**e**) 5 g; and (**f**) 10 g.  
As observed in Figure 7, the stones have the same radial distribution for all driving speeds ranging from 60 to 110 kph. Certain critical zones exhibit differences in the density; however, their shapes and sizes are identical. This finding suggests that for vehicles with a certain driving speed range, the critical zone is located in the same area. Thus, this method can be used for determining the design criteria for the critical zone, and the weight for robust vehicle component design. The empty area of the stone chipping zone in Figure 7 is located between 1500 to about 1800 mm vehicle length. As the tangential angle in Equation (3) varies with the tire pressure, the chipping area locates along the vehicle length from the front-axle to the rear one. This means that stones with high tangential angles (low tire pressure) impact below 1500 mm vehicle length. Considering in combination with the code angles of the tire belt from \(-5^{\circ}\) to \(20^{\circ}\), they spread out widely over 1800 mm vehicle length. Considering the discrete intervals of tire pressure (70, 75, 80, 85, 90 kPa) in the numerical simulation, this empty area is expected to have low possibility for stone chipping, even at the smaller intervals of tire pressure. In general, the position at a given time can be calculated using Equation (11) for each stone, but for random stone chipping, the position range must be predicted by statistical probability. In this study, density estimation was performed to determine the probabilistic distribution characteristics of the stone chipping zone in Figure 8 by applying the extensively used kernel density estimation method [14]; i.e., after obtaining the predicted histogram by non-parametric density estimation, the histogram was normalized, and the probability density function was used. By setting the bandwidth of the kernel density function, the stone chipping probability density was obtained, as depicted in Figure 8.  
Figure 6: Stone chipping zones for different ground clearance (ride height) (**a**) 0.25\(R\); (**b**) 0.5\(R\); (**c**) 0.75\(R\); and (**d**) \(R\); respectively. Figure 8 demonstrates that the probability density of stone chipping is high behind the front-axle and tends to be low between the front and rear-axle, as suggested by Figure 7. However, the probability density is relatively high in the rear-axle zone. As observed in Figure 8, random stone chipping can cause concentrated damage to the rear-axle zone, as well as the rear parts and suspension parts. Hence, the chipping zone is large, when the road clearance is 75-100%. Furthermore, the stones that are chipped and fly affect the kinetic energy and impact zone, depending on the driving speed of the vehicle and the gravitational acceleration. Using Equation (20), the kinetic energy of each stone was calculated to be approximately 50 J. Figure 9 demonstrates that the vehicle speed and gravitational acceleration do not substantially affect the impact zone, but serious damage may result, when the front-axle is impacted. However, it is predicted that the rear-axle, with relatively large portions in the critical zone, will be continuously impacted. When a metal surface with a large area is continuously impacted, shape changes and extreme fractures occur due to fatigue. Figure 7: Predicted stone chipping zones of all the cases (5 tire pressures, 4 ground clearances, 10 stone masses) for different vehicle speeds of (**a**) 60 kph; (**b**) 80 kph; and (**c**) 110 kph; (**d**) combination of (a–c). As shown in Figures 7-9, the numbers of stones that can reach the critical zone and impact the vehicle, as well as the potential energy, were calculated, as shown in Table 3, dividing the critical zone into 500 mm interval sections from the front-axle.