## 3 Theoretical Analysis
### Model Derivation
The actual rolling distance and the extent of damage can then be estimated based on the specific shape of the falling stone and slope characteristics, as shown in Figure 22.  
In the calculation of the rolling speed of blocks with different length-to-width ratios (\(\lambda\)), it is necessary to consider the situation where the rolling axis changes during the rolling process. For falling stones with 0.33 \(<\lambda<\) 1, it is necessary to calculate the moments of inertia for both the minimum and maximum rolling axes, along with the corresponding rolling friction coefficients and minimum circumscribed circle radii of the two-dimensional cross-sectional shape. These values are used to determine the upper and lower bounds of the rock's rolling speed. For falling stones with \(\lambda<\) 0.33, only the parameters corresponding  
Figure 21: Illustration of three categories of rock scales [45]. to the rolling axis with the minimum rolling resistance need to be determined as the theoretical values for calculating the rock's rolling speed. The formula for calculating the angular velocity of a rolling stone is as follows:  
\[\omega^{\prime}=\frac{\sqrt{v_{\mathrm{x}}^{2}+v_{\mathrm{y}}^{2}}}{R^{\prime}}= \frac{v}{R^{\prime}} \tag{8}\]  
\(v_{\mathrm{x}}\), \(v_{\mathrm{y}}\)--Velocities of the rolling stone in the x and y directions (m/s). \(v\)--Rolling speed of the rolling stone (m/s). \(R^{\prime}\)--Minimum circumscribed circle radius of the two-dimensional cross-section perpendicular to the slope (m). For non-circular cross-sectional shapes of rolling stones, the following modification can be made:  
\[\omega=\frac{\omega^{\prime}}{d_{k}} \tag{9}\]  
\(d_{k}\)--Circularity of the cross-section, \(d_{k}=2\pi R^{\prime}/c\). \(c\)--Circumference of the minimum circumscribed circle of the cross-section (\(m\)). By combining Formulas (6) and (A3) to solve Formula (A1), the range of the rolling speed of the rolling stone (\(v_{1}<v<v_{2}\)) can be obtained, where \(v_{1}\) and \(v_{2}\) represent the upper and lower bounds of the rolling stone's rolling speed, respectively:  
Figure 20:0  
Figure 20:1  
\(d_{k1}\), \(d_{k2}\)--Maximum and minimum circularities of the two-dimensional cross-sectional shape perpendicular to the rolling axis. \(\psi_{1}\), \(\psi_{2}\)--Maximum and minimum rolling friction coefficients corresponding to the shape shape. \(I_{1}\), \(I_{2}\)--Maximum and minimum moments of inertia corresponding to the shape (kg/m\({}^{2}\)). \(R_{1}\), \(R_{2}\)'--Minimum circumscribed circle radii corresponding to the maximum and minimum circularities of the shape (m). Similarly, the maximum and minimum rolling energies of the rolling stone (\(E_{1}\) and \(E_{2}\)) can be obtained by Formulas (A4) and (A5). Specific calculation parameters for the test blocks are provided in Figure 20:2  
Figure 20:3 Schematic illustration of determining rolling distance and damage impact range. 1. In the process of formula derivation, sliding friction losses during the rolling process were not considered. However, it was observed in the experiments that as the slope angle increases, some blocks with relatively high rolling friction coefficients (e.g., the error for Block B at a release height of 0.45 m and a slope of 45\({}^{\circ}\) is 3.39%. When the slope increases to 60\({}^{\circ}\), the error becomes 7.87%.) tend to experience a small amount of sliding friction when rolling on the slope. These blocks have higher overturning moments, and when the slope angle approaches the sliding friction angle, the forces acting on the slope surface after rolling can result in a small amount of sliding friction. This phenomenon gradually decreases with increasing rotational kinetic energy. 2. For some blocks with relatively low rolling friction coefficients, during the process of shifting direction until stabilization, the two-dimensional cross-section used for calculation is not exactly perpendicular to the actual rolling axis, leading to a slightly larger actual rolling resistance than the calculated value.