# Prediction of Damage to the Vehicle Underbody due to Stone Chipping
## 2 Definition and Prediction of Stone Chipping
### Matrix Definitions for Numerical Simulation  
To apply a statistical estimation model for random stone chipping, the variables shown in Equations (12)-(16) were defined, where \(\alpha_{i}\) is the tangential angle of the stone, \(M_{pj}\) is the mass of the stone, \(V_{k}\) is the vehicle speed, and \(\beta\) is the code angle of the tire belt.  
\[\begin{array}{c}A_{i,j,k,l}=\left(\alpha_{i},M_{pj},V_{k},\beta_{l}\right) \\ \alpha_{i}=\left[\alpha_{i}\right],\quad i=1,2,\ldots,p\\ M_{pj}=\left[M_{pj}\right],\quad j=1,2,\ \ldots,q\\ V_{k}=\left[V_{k}\right],\quad k=1,2,\ldots,r\\ \beta_{l}=\left[\beta_{l}\right],\quad l=1,2,\ \ldots,s.\end{array} \tag{12}\]  
\[0.0097\leq\alpha_{i}\leq 0.0293. \tag{13}\]  
\[1\leq M_{pj}\leq 10. \tag{14}\]  
\[60\leq V_{k}\leq 110. \tag{15}\]  
\[-5\leq\beta_{l}\leq 20. \tag{16}\]  
Here the letters \(p\), \(q\), \(r\) and \(s\) correspond to the number of levels of each variable. In this study, we set them as 5, 10, 3 and 51 with discrete intervals, respectively. In this simulation, the tangential angle \(\alpha_{i}\) of a stone is assumed to be the nominal exit angle of the stone having the magnitude equal to the vehicle speed \(V_{k}\). Equation (12) was reconfigured and expressed as a projection matrix (3D space matrix). First, the tangential angle of the stone from Table 2 was included as the \(x\)-axis of the matrix, and the mass of the stone from Equation (14) was included in the \(y\)-axis. Then, the vehicle speed component along the \(z\)-axis was set to 60, 80 and 110 kph to configure a \(5\times 10\times 3\) projection matrix. Furthermore, Equation (16) substituted in the projection matrix, and as \(\beta\) was distributed across the overall width  
Figure 4: Local reference frame for the absolute speed of the stone.  
of the underbody corresponding to belt code angles of \(-\)5-20\({}^{\circ}\), Equation (12) was expanded into Equation (17) as a projection matrix for each belt code angle.  
\[A_{i,j,1,1}=\left[\begin{array}{ccc}A_{1,1,1,1}&\cdots&A_{p,1,1,1}\\ \vdots&\ddots&\vdots\\ A_{1,q,1,1}&\cdots&A_{p,q,1,1}\end{array}\right]\quad\cdots\quad A_{i,j,k,1}= \left[\begin{array}{ccc}A_{1,1,k,1}&\cdots&A_{p,1,r,1}\\ \vdots&\ddots&\vdots\\ A_{1,q,k,1}&\cdots&A_{p,q,k,1}\end{array}\right] \tag{17}\]  
\[A_{i,j,1,l}=\left[\begin{array}{ccc}A_{1,1,1,s}&\cdots&A_{p,1,1,s}\\ \vdots&\ddots&\vdots\\ A_{1,q,1,s}&\cdots&A_{p,q,1,s}\end{array}\right]\quad\cdots\quad A_{i,j,k,1}= \left[\begin{array}{ccc}A_{1,1,k,s}&\cdots&A_{p,1,r,1}\\ \vdots&\ddots&\vdots\\ A_{1,q,k,1}&\cdots&A_{p,q,k,1}\end{array}\right].\]  
To calculate the impact zone over time and thereby identify the critical stone chipping zone, Equation (11) was applied, and the time was calculated using the following matrix equation:  
\[\begin{bmatrix}\begin{bmatrix}\begin{smallmatrix}1\\ 2\\ \end{smallmatrix}&\begin{matrix}V_{abs}k&(H-R(1-\cos\alpha))\\ \vdots&\cdot&\cdot\end{matrix}\end{bmatrix}\begin{bmatrix}\begin{smallmatrix}t \\ t\\ t\\ \end{smallmatrix}\end{bmatrix}=\begin{bmatrix}0\\ 0\end{bmatrix}. \tag{18}\]  
The energy, immediately after chipping from tire, can be calculated using Equation (19), which can be formulated from Equations (8)-(10). The kinetic energy, immediately before collision, can be calculated using Equation (20) with \(V_{abs}k\) and \(t\), which can be calculated using Equations (10) and (11).  
\[E_{p0}(i,j,k,l)=\frac{1}{2}M_{pj}(\left[V_{abs}i(i,j,k,l)\right]^{2}+[V_{abs} j(i,j,k,l)]^{2}+[V_{abs}k(i,j,k,l)]^{2}), \tag{19}\]  
\[E_{p1}(i,j,k,l)=\frac{1}{2}M_{pj}(\left[V_{abs}i(i,j,k,l)\right]^{2}+[V_{abs} j(i,j,k,l)^{2}\right]+[(V_{abs}k(i,j,k,l)-gt(i,j,k,l)]^{2}), \tag{20}\]  
where \(E_{p0}\) and \(E_{p1}\) are the kinetic energies of the stone after chipping and immediately before collision, respectively. The matrix for the kinetic energy of the stone can be defined as Equation (20) by substituting the 3D space matrix for random stone chipping, which can be determined using Equation (17) and the time, which can be determined using Equation (18).