## 3 Results
### The Interactive Effects of Main Controlling Factors on \(R_{\rm n}\)
#### 3.4.2 The Interaction of Falling Rock Shape and Initial Velocity  
Figure 13 shows \(R_{\rm n}\) decreases with \(\eta\) and \(v\) increase. This decrease is 69.07%, which is lower than the combined independent effect of the two factors (70.15%). The interaction between falling rock shape and initial velocity is slightly weaker than the independent superimposed effects. The effect of falling rock shape on \(R_{\rm n}\) varies significantly with \(v\). At low-level \(v\), weak collisions occur, causing a 58.33% decrease in \(R_{\rm n}\) (from blue to orange) as \(\eta\) decreases. In contrast, at high-level \(v\), heavy collisions occur, resulting in a smaller 35.02% reduction in \(R_{\rm n}\) (from yellow to red). In this case, the effect of falling rock shape on \(R_{\rm n}\) is less pronounced than low-level \(v\). At low-level \(v\), the collision kinetic energy is small, and changes in \(\eta\) substantially affect the contact area and friction force, influencing the energy dissipation. Under low-level \(\eta\), rockfall has a smaller contact area, lower friction force, less energy dissipation, and higher \(R_{\rm n}\). However, under high-level \(\eta\), rockfall has a larger contact area, increased friction force, and significantly increased deformation energy dissipation, resulting in a decrease in \(R_{\rm n}\). At high-level \(v\), the collision kinetic energy increases significantly, leading to greater impact energy dissipation and diminishing the effect of falling rock shape on \(R_{\rm n}\) as \(\eta\) rises. This occurs because high-level \(v\) causes more collision kinetic energy to be converted into deformation and impact energy, attenuating the effect of falling rock shape on \(R_{\rm n}\). Figure 12: The interactive effects of falling rock shape and slope surface material on \(R_{\rm n}\): (**a**) Three-dimensional surface plot showing the variation in \(R_{\rm n}\) with different \(\eta\) and \(A_{\rm slope}\). 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 harder slope surface materials; (**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 \(A_{\rm slope}\) that correspond to notable changes in \(R_{\rm n}\). In addition, the correlation between initial velocity and \(R_{\rm n}\) varies significantly depending on \(\eta\). For low-level \(\eta\), fewer contact areas occur, leading to a 52.40% reduction in \(R_{\rm n}\) (from blue to yellow) as \(v\) increases. However, for high-level \(\eta\), more contact areas occur, resulting in a smaller 25.77% reduction in \(R_{\rm n}\) (from orange to red). In this case, the effect of falling rock shape on \(R_{\rm n}\) is approximately half of that observed for low-level \(\eta\). Under low-level \(\eta\), there is less contact area, smaller local deformation, and smaller collision kinetic energy dissipation. Low-level \(v\) results in higher \(R_{\rm n}\). However, as \(v\) rises, the collision kinetic energy increases significantly, and the ratio of plastic deformation energy dissipation in the collision process increases significantly, leading to a decrease in \(R_{\rm n}\), and the high-level \(\eta\) has more contact area, resulting in larger local deformation energy dissipation. As \(v\) increases, the rising collision kinetic energy causes local deformation and friction energy dissipation, leading to a decrease in \(R_{\rm n}\). However, the effect of initial velocity on \(R_{\rm n}\) is relatively small because the high-level \(\eta\) already causes large energy dissipation. In conclusion, high-level \(\eta\) significantly reduces the effect of initial velocity on \(R_{\rm n}\).