## 1 Introduction
These challenges highlight the need for a more comprehensive approach to modeling \(R_{\text{n}}\), which this study aims to address by investigating the interactions between multiple controlling factors and developing a more accurate prediction model. Numerous studies, including theoretical analyses [8, 9, 10, 11, 12, 13, 14], laboratory experiments [15, 16, 17, 18, 19, 20, 21, 22], and field tests [23, 24, 25, 26, 27, 28], have been conducted to estimate \(R_{\text{n}}\) under various conditions. In theoretical analyses, several scholars have developed models for \(R_{\text{n}}\) based on classical theories. For instance, [PERSON] [8] and [PERSON] [9] respectively derived the formulas forestimating \(R_{\rm n}\) under ideal elastic-plastic conditions based on Hertz contact theory. Similarly, [PERSON] et al. [10] derived the normal impact velocity in the initial yield stage considering the mechanical properties of slope soil and proposed an estimation method for \(R_{\rm n}\). Moreover, [PERSON] and [PERSON] [11] approximated the falling rock as an ellipsoid, derived velocity expressions of various falling rock kinematics and the corresponding \(R_{\rm n}\). In addition, [PERSON] et al. [12] employed the Logistic equation to fit the normal falling rock velocity curve, deriving \(R_{\rm n}\) based on quasi-static contact mechanics theory. It is important to note that while this theory provides useful insights into rockfall collisions, our study does not directly apply this approach. Instead, our research focuses on experimental modeling of rockfall dynamics, primarily investigating the interactions between key parameters (such as slope surface material, falling rock shape, and initial velocity) to estimate \(R_{\rm n}\). [PERSON] et al. [13; 14] derived the \(R_{\rm n}\) formula for spheres using viscoelastic contact theory and developed a viscoelasticoplastic contact model. While these theoretical models are valuable for understanding rockfall behavior, we have not adopted these models in our study. Instead, we focus on direct laboratory measurements of \(R_{\rm n}\) and the interactions among main controlling factors, aiming to refine prediction methods and improve the accuracy of rockfall hazard assessment. In laboratory tests, many researchers have investigated the influence of various factors on \(R_{\rm n}\) focusing on the characteristics of falling rock, kinematics, and slope. For example, [PERSON] [15] developed an empirical formula, incorporating Schmidt hardness and slope angle, to estimate \(R_{\rm n}\) based on laboratory spherical falling rock free-fall tests. In the tests, rocks are released freely from a specified height to impact a flat surface or cushioning material, and the resulting rebound velocities are measured to calculate \(R_{\rm n}\). Meanwhile, [PERSON] and [PERSON] [16] conducted half-scale falling rock free-fall tests on sandy slopes, which investigated the effects of slope angle, falling rock mass, shape, and falling height on \(R_{\rm n}\). The tests provided controlled conditions to study how different parameters influence \(R_{\rm n}\) during the collision. Additionally, [PERSON] et al. [17] examined the effects of falling rock size on \(R_{\rm n}\) through laboratory spherical falling rock free-fall tests. The tests were designed to capture the behavior of the rockfall's collision with the surface and its energy dissipation characteristics. Moreover, [PERSON] and [PERSON] [18] analyzed the effects of impact velocity, falling rock mass, and material hardness on \(R_{\rm n}\), and they also proposed a new semi-empirical formula based on the correlative factors. Similarly, [PERSON] et al. [19] examined the interactions among seven factors and their nonlinear correlations with \(R_{\rm n}\). [PERSON] et al. [20] performed laboratory falling rock free-fall tests to investigate the effects of slope angle, kinetic energy, and falling rock shape on \(R_{\rm n}\).