## 3 Theoretical Analysis  
### Model Derivation  
The current study has clarified that the shape of falling rock will affect its rolling characteristics and has pointed out that falling rocks with different shapes will experience a different rolling resistance during rolling. However, no clear definition of rolling resistance has been provided. To accurately describe the influence of rockfall shape on its rolling characteristics, two concepts related to rolling--rolling resistance and rolling resistance distance--need to be established [44]. For the same test block placed stationary on the slope, its rolling resistance arm \(\delta\) is determined by the size of the contact surface between the test block and the slope, as shown in Figure 20.  
Through the analysis of data obtained from model experiments, the mechanisms through which the rock shape influences its rolling speed have been identified. Consequently, this paper defines a shape parameter that reflects the overall shape and local convex-concave characteristics of a rolling stone. This parameter is based on the difference between the two-dimensional cross-sectional area perpendicular to the shortest and longest dimensions of the falling stone and the area of its minimum circumscribed circle, as seen in  
Figure 19: Mean level of each influencing factor. \begin{table}
\begin{tabular}{c c c c c c c} \hline \hline
**Item** & \(\psi\) & **Release Height** & **Mass** & **Slope** & **Roundness** & \(R^{\prime}\) \\ \hline \(K1\) & 78.33 & 58.944 & 39.84 & 146.999 & 115.99 & 39.84 \\ \(K2\) & 83.05 & 71.608 & 129.043 & 130.714 & 113.42 & 27.64 \\ \(K3\) & 37.66 & 82.419 & 37.66 & 129.353 & 34.51 & 37.66 \\ \(\mathbf{R}\) & 0.765 & 1.183 & 0.791 & 0.294 & 0.687 & 0.827 \\ convert coefficient \(d\) & 0.32 & 0.4 & 0.34 & 0.52 & 0.37 & 0.31 \\ \(R^{\prime}\) & 1.095 & 2.84 & 1.275 & 1.185 & 1.393 & 1.088 \\ Factor primary and secondary order & & & & & & \\ \hline \hline \end{tabular}
\end{table} 
Table 3: Rolling speed range analysis results. Figure 20: Schematic representation of block rolling. Figure 21. The rolling resistance coefficient \(\psi\), stemming from the shape of the rolling stone, is defined as follows. \[\psi=\frac{u_{1}-u_{2}}{u_{1}} \tag{5}\]  
\(u_{1}\)--The area of the smallest circumscribed circle of the 2D cross-section of the rolling stone perpendicular to the slope. \(u_{2}\)--The area of the 2D cross-section of the rolling stone perpendicular to the slope. Additionally, the rolling resistance of a rock also arises from factors such as slope angle and roughness. Taking into account the rolling friction coefficient related to the roughness of the slope [46], the rolling resistance \(f\) for rock rolling is defined as:  
\[f=(\mu+\psi)N \tag{6}\]  
\(\mu\)--Rolling friction coefficient related to the roughness of the slope. The assumptions made during this derivation are as follows:  
(1) The rolling stone experiences rolling but not sliding on the slope. The slope angle \(\theta\) must satisfy the following condition:  
\[arc\tan\frac{H}{\delta}<\theta<arc\tan\mu \tag{7}\]  
where: \(\delta\)--Rolling resistance arm of the rolling stone. \(H\)--Distance from the center of gravity of the rolling stone to the slope. \(\mu\)--Rolling friction coefficient related to the roughness of the slope. (2) The rolling stone and the slope do not deform during the rolling process. (3) The rolling stone remains intact while rolling on the slope. According to the principle of conservation of energy, the energy calculation formula for the rolling stone on a rigid slope can be derived as ((A1), refer to Appendix A):  
In experiments, the actual rolling distance of the test block can be determined by analyzing the entire process of the block's rolling, captured by a high-speed camera during model tests. In practical applications, combined with Formulas (A2) and (A3), the shortest and longest rolling distances \(S_{1}\) and \(S_{2}\) of the rolling stone on the slope can be estimated based on the stone's height and the corresponding maximum offset \(\eta\)[47].