## 3 Results
### The Interactive Effects of Main Controlling Factors on \(R_{\rm n}\)
#### 3.4.5 The Interaction of Falling Rock Shape, Slope Surface Material, and Initial Velocity  
The study of the interaction mechanisms among the main controlling factors reveals significant interaction among falling rock shape, slope surface material, and initial velocity. As shown in Figure 16, the contour decreases with \(v\) increases under the interaction between falling rock shape and slope surface material. The interaction is consistent with the results of the two-factor interaction analysis, with the effect of falling rock shape being stronger than that of slope surface material. High-level \(v\) increases collision kinetic energy and reduces the kinematic behaviors, such as rolling, sliding and collision. When \(v\) is 6 m\(\cdot\)s\({}^{-1}\), the collisions between high-level \(\eta\) and \(A_{\text{slope}}\) result in \(R_{\text{n}}\) remaining constant. The local contact deformation areas increase, while the impact force decreases. At this point, the energy dissipation is mainly dominated by the falling rock shape and the initial velocity, while the effect of slope surface material on \(R_{\text{n}}\) is weakened, as shown in Figure 17.  
By analyzing the effect of multi-factor interactions on \(R_{\text{n}}\), it is found that falling rock shape influences the contact pattern and energy dissipation. The falling rock with low-level \(\eta\), which is round or smooth, has a smaller contact area when it hits the slope. This means it slides more easily with less friction, allowing more of its kinetic energy to be preserved, resulting in a higher \(R_{\text{n}}\). In addition, the falling rock with high-level \(\eta\), with sharp edges or irregularities, creates a larger contact area with the slope, leading to more friction and greater energy dissipation during the collision. As a result, the rock retains less kinetic energy, resulting in a lower \(R_{\text{n}}\). In natural geological contexts, this means that smoother rocks (like those that have undergone natural weathering) will rebound more efficiently when they hit the ground, while rougher, angular rocks will experience more energy loss and bounce less. This behavior is important when designing rockfall mitigation strategies, as the shape of the falling rocks and the type of slope surface they collide with play a key role in how far they travel after collision. The strength and hardness discrepancies in slope surface materials affect \(R_{\text{n}}\). However, \(R_{\text{n}}\) is less sensitive to \(A_{\text{slope}}\) under high-level \(\eta\). As \(v\) increases, the increase in impact energy dissipation reduces \(R_{\text{n}}\), which reaches its minimum for high-level \(\eta\). The normal velocity component increases with the angle, while \(R_{\text{n}}\) decreases for low-level \(\eta\). In contrast, for high-level \(\eta\), the movement distance is shortened, reducing plastic deformation energy dissipation and increasing \(R_{\text{n}}\). The three-factor interaction analysis reveals the synergistic mechanism among the three main controlling factors and provides an accurate basis for predicting and controlling falling rock kinematic behavior. However, the model's applicability and reliability need to be verified. Therefore, to assess the predictive ability of the estimation model in engineering, this study validates the proposed multi-factor interaction estimation model combined with on-site experiments, considering multi-factor conditions during falling rock movement. Figure 16: Contour plot for the interaction effects of falling rock shape and slope surface material at different initial velocity: (**a**) \(v=0\) m\(\cdot\)s\({}^{-1}\); (**b**) \(v=2\) m\(\cdot\)s\({}^{-1}\); (**c**) \(v=4\) m\(\cdot\)s\({}^{-1}\); and (**d**) \(v=6\) m\(\cdot\)s\({}^{-1}\). The plot illustrates the same data in two dimensions, providing a clearer view of the interaction.