## 1 Introduction  
Earthquakes are among the most destructive geological hazards, with the resulting ground motion characteristics being closely related to the dynamic parameters of the rock mass [1], including shear modulus [2] (modulus of elasticity) and damping ratio [3], which are essential for exploring the intrinsic connections between earthquakes and rock behavior. Shear modulus measures a rock mass's resistance to deformation under seismic forces, and this relationship is proportional; that is, a larger shear modulus indicates higher material stiffness [4; 5; 6]. Theoretical analyses indicate that seismic waves propagate with reduced energy attenuation in materials of higher stiffness, consequently increasing the amplitude of vertical ground fluctuations [7; 8; 9]. The damping ratio quantifies a material's energy dissipation during vibration; a higher damping ratio indicates effective energy dissipation, which suppresses the amplitude of vibrations caused by earthquakes and reduces their destructive impact on rock structures [10; 11; 12]. Studying the dynamic properties of rocks under various cyclic modes holds both theoretical value and practical significance for enhancing the safety of underground engineering constructions. In mechanical tests, the shear modulus and damping ratio of rocks are influenced by factors such as material composition, stress amplitude, and loading frequency [13; 14; 15]. Scholars have conducted extensive theoretical and experimental research on the fatigue deformation [16], strength, and damage of intact rocks. In studies on the fatigue of soft rocks, such as sandstone, shale, and salt rock, water content and stress amplitude significantly influence the dynamic shear modulus and damping characteristics. Under conditions of low strain amplitude, the effect of water content on the dynamic shear modulus is subtle, but it significantly affects the damping ratio and damping coefficient [17]. However, as stress amplitude increases, the impact of water content on the dynamic shear modulus becomes more pronounced. This indicates a complex interaction between water content and stress amplitude that collectively determines the dynamic response characteristics of soft rocks [18; 19]. Under graded cyclic loading, the damping parameter of soft rocks increases with rising stress amplitude, and this trend varies under different stress paths. Specifically, the damping parameter values measured under stress paths with incrementally increasing stress levels are higher than those under paths with incrementally increasing stress amplitudes. This may be attributed to the nonlinear response of the internal structure of the rock and the associated damage accumulation mechanism [20; 21]. Under constant amplitude cyclic loading, the damping ratio and damping coefficient of soft rock change as the number of cycles increases. These changes adhere to a specific evolutionary law, particularly when the stress level surpasses the fatigue damage threshold of the rock. The evolution of the damping parameter exhibits a three-stage semi-U curve, encapsulating the entire process from the rock's initial elastic response through the accumulation of fatigue damage to its ultimate destruction. This phenomenon encapsulates the entire process from the rock's initial elastic response to the accumulation of fatigue damage, culminating in its destruction [22; 23; 24]. For hard rocks, such as marble, granite, and schist, their mechanical fatigue properties are influenced by cyclic loading conditions. Hard rocks typically exhibit higher shear moduli and lower damping ratios, attributed to their internal structural compactness and mineral composition. Under cyclic loading, the fatigue damage and damping properties of hard rocks demonstrate an evolutionary pattern distinct from that of soft rocks. The fatigue damage process in hard rocks is often more abrupt, with stress-strain curves displaying a more pronounced linear elastic phase before the onset of damage.