###### Abstract  
Rockfalls pose significant threats to infrastructure, transportation routes, and human safety in mountainous regions, making them a critical concern in natural hazard and risk management. Accurate prediction of rockfall behavior is essential for designing effective mitigation strategies. The normal coefficient of restitution (\(R_{\text{n}}\)) is a key kinematic parameter for modeling falling rock dynamics, specifically quantifying the energy retained after collision between a rock and a slope surface. While this parameter is not directly used in prevention design, it is crucial for predicting the movement and trajectory of falling rocks and can indirectly support the development of more effective hazard mitigation strategies. However, \(R_{\text{n}}\) is influenced by multiple factors, including slope angle, surface material, falling rock shape, and initial velocity. The interactions among these factors make a precise prediction of \(R_{\text{n}}\) particularly challenging. Existing theoretical and empirical formulas typically consider individual factors in isolation, often neglecting their interactions, which leads to significant discrepancies in the results. To address this gap, we conducted a series of laboratory physical model tests to investigate the interactions among highly sensitive controlling factors and improve the accuracy of \(R_{\text{n}}\) prediction. A self-designed release apparatus, coupled with a high-speed recording and analysis system, was used to capture full kinematic data during rockfall collisions on slopes. This study not only examined how the main controlling factors and their interactions affect \(R_{\text{n}}\) but also developed a multi-factor interaction regression model, which was verified using on-site test data. The results show that the effect of the main controlling factors decreases in the following order: falling rock shape, slope surface material, initial velocity, and slope angle. Considering that falling rock shape and slope surface material cannot be quantitatively evaluated, the shape factor (\(\eta\)) and material factor (\(A_{\text{slope}}\)) are proposed to represent two controlling factors, respectively. Specifically, increases in \(\eta\), \(A_{\text{slope}}\), initial velocity, and slope angle are negatively correlated with \(R_{\text{n}}\). Highly significant interactions were observed among falling rock shape-slope surface material, falling rock shape-initial velocity, falling rock shape-slope angle, slope surface material-initial velocity, and falling rock shape-slope surface material-initial velocity. These interactions mitigate the \(R_{\text{n}}\) reduction, resulting in a weaker effect than the stacking effect of the individual factors. The phenomenon is primarily attributed to the fact that high-level \(\eta\), \(A_{\text{slope}}\), initial velocity, and slope angle diminish the effect of intersecting factors. Finally, a comparison of the multi-factor interaction model with on-site tests and empirical formulas revealed the accuracy of the proposed model.  
rockfall collisions; normal coefficient of restitution; multi-factor interaction; full-range analysis; regression model +
Footnote †: journal: Applied Sciences  
Article