## 1 Introduction  
Rockfill dams, a type of embankment dam constructed using rockfill materials, are increasingly preferred in Southwest China due to their strong adaptability to geological environments, the feasibility of using local materials, lower construction costs, and superior seismic performance [1, 2]. The complex geological conditions and frequent seismic activities in this region make seismic performance a critical factor for the safety of these dams. A failure in seismic resistance could lead to catastrophic societal consequences. Thus, optimizing the seismic design of rockfill dams and enhancing their seismic performance are of paramount importance. In this context, the dynamic elastic modulus and damping ratio, as key parameters for soil dynamics, are the focal points in this analysis. Coarse-grained soil, often employed as a filling material in rockfill dam projects, is chosen for its excellent engineering characteristics like high permeability, strength, and minimal compression deformation. Accurately determining its dynamic elastic modulus and damping ratio is crucial for the seismic design of rockfill dams. Consequently, in-depth research into these parameters holds significant theoretical and engineering value [3]. There has been substantial research by international scholars on the dynamic elastic modulus, the dynamic shear modulus, and damping ratio of coarse-grained soils. Studies from [PERSON] [4], [PERSON] [5], [PERSON] [6], [PERSON] [7], and others have indicated that the dynamic elastic modulus and damping ratio all trend to increase when pressure and frequency increase. The maximum dynamic elastic modulus of coarse-grained soil was empirically formulated by [PERSON] [8], considering the confining pressure and frequency, based on extensive data from large-scale dynamic triaxial tests. [PERSON] [9] conducted tests on limestone aggregate using a static and dynamic simple shear testing device, developing an estimation formula for the maximum dynamic shear modulus that considers the effects of confining pressure, consolidation ratio, gradation, and porosity characteristics. [PERSON] [10] investigated the influence of factors like the consolidation ratio on the dynamic modulus and damping ratio of materials used in dam construction, including dam shell, heart wall, and dam foundation fault materials, through dynamic triaxial tests. Research by [PERSON] [11] and [PERSON] [12], and subsequent work by [PERSON] [13], indicated that higher confining pressure and vibration frequencies increase soil stiffness and stabilize soil dynamics, leading to the development of a dynamic strain backbone curve model that considers vibration cycles and confining pressure. Overall, the investigations into the dynamic modulus and damping ratio of coarse-grained soil have primarily focused on factors like confining pressure, consolidation ratio, vibration, and porosity characteristics. However, studies specifically addressing the impact of relative density on these parameters in coarse-grained soil are less frequent. Limited research, such as that by [PERSON] [14] and [PERSON] [15], has provided only qualitative insights, leaving the quantitative relationship between relative density and the dynamic modulus and damping ratio of coarse-grained soil largely unexplored. This gap highlights the current inadequacy in understanding the influence of relative density on these parameters, emphasizing the need for more quantitative research. [PERSON] [16; 17] underscored the significance of relative density as a mechanical parameter reflecting the physical state of soil, heavily influencing the mechanical characteristics of coarse-grained soil. Therefore, quantitative experimental research into how relative density affects the dynamic modulus and damping ratio of coarse-grained soil is imperative. Therefore, this study utilizes a GDS large-scale dynamic-static triaxial apparatus for dynamic triaxial testing on compacted samples of a specific coarse-grained soil.