## 3 Results
### The Interactive Effects of Main Controlling Factors on \(R_{\rm n}\)
#### 3.4.3 The Interaction of Falling Rock Shape and Slope Angle  
Figure 14 shows that \(R_{\rm n}\) decreases with increasing \(\eta\) and \(\theta\), but the reduction (51.58%) is less than the combined independent effect of the two factors (59.05%). This indicates that the interaction between falling rock shape and slope angle is significantly weakened.  
The effect of falling rock shape on \(R_{\rm n}\) significantly relies on \(\theta\). With low-level \(\theta\), weak collisions occur, resulting in a 64.25% decrease in \(R_{\rm n}\) (from blue to red) as \(\eta\) decreases. However, with high-level \(\theta\), heavy collisions occur, leading to a 32.11% reduction in \(R_{\rm n}\) (from green to orange). In this case, the effect of slope angle on \(R_{\rm n}\) is less than that observed with low-level \(\theta\). At low-level \(\theta\), the normal velocity component is small, and the falling rock movement distance is long. Low-level \(\eta\) has low energy dissipation and high \(R_{\rm n}\) due  
Figure 13: The interactive effects of falling rock shape and initial velocity on \(R_{\rm n}\): (**a**) Three-dimensional surface plot showing the variation in \(R_{\rm n}\) with different \(\eta\) and \(v\). The color scale indicates the corresponding \(R_{\rm n}\), with blue representing high \(R_{\rm n}\) and red representing low \(R_{\rm n}\). The plot highlights the positive interaction effect on \(R_{\rm n}\), with higher \(R_{\rm n}\) observed for more spherical falling rocks and higher initial velocities; (**b**) Contour plot illustrating the same data in two dimensions, providing a clearer view of the interaction. The contour lines represent different \(R_{\rm n}\) values, and the feature points on the plot indicate combinations of \(\eta\) and \(v\) that correspond to notable changes in \(R_{\rm n}\).  
to the small contact area. In contrast, high-level \(\eta\) disperses the impact force, leading to increased plastic deformation energy dissipation and a decrease in \(R_{\rm n}\). At high-level \(\theta\), the normal velocity component increases, and the energy dissipation becomes complicated as \(\eta\) increases. However, the energy dissipation decreases due to the shortened falling rock movement distance. Therefore, the effect of falling rock shape on \(R_{\rm n}\) is smaller than that of the low-level \(\theta\).  
In addition, the correlation between slope angle and \(R_{\rm n}\) varies significantly with \(\eta\). With low-level \(\eta\), few contact areas occur, resulting in a 28.68% decrease in \(R_{\rm n}\) (from blue to green) as \(\theta\) increases. In contrast, with high-level \(\eta\), more contact areas occur, leading to a 35.42% decrease in \(R_{\rm n}\) (from red to orange). In this case, the effect of falling rock shape on \(R_{\rm n}\) is more than that observed with low-level \(\eta\). Low-level \(\eta\) has fewer contact areas and concentrated impact force. The collision energy mainly exists in the elastic recovery form. Therefore, with \(\theta\) increases, the normal velocity component gradually rises and \(R_{\rm n}\) decreases due to increased normal impact energy dissipation. In contrast, high-level \(\eta\) has larger contact areas and dispersed impact force. The energy dissipation mainly occurs through local deformation and contact friction. When \(\theta\) increases, the rising normal velocity component increases the ratio of plastic deformation energy dissipation. However, the shortened movement distance reduces energy dissipation and increases \(R_{\rm n}\).  
To sum up, high-level \(\eta\) significantly enhances the effect of slope angle on \(R_{\rm n}\). Conversely, high-level \(\theta\) diminishes the effect of falling rock shape on \(R_{\rm n}\). The interaction suggests that the effect of falling rock shape on \(R_{\rm n}\) (\(-\)28.68%-35.42%) is more pronounced than that of slope angle (\(-\)62.25%--32.11%). When interacting with high-level \(\eta\), the influence of slope angle on \(R_{\rm n}\) has changed, exhibiting the opposite effect compared to low-level \(\eta\).