## 6 Conclusions  
1. With the increase in stress level, the shape of the axial hysteresis loop of the specimen changes from the \"shuttle\" type to the \"streamline\" type, and the shape of the annular hysteresis loop mainly shows an elastic effect, which is of the \"streamline\" type. The shape of the loop direction hysteresis loop mainly shows the elastic effect, and both of them are of the \"streamline\" type. With the increase in the cyclic period, due to the slowing down of the loading rate, the structural stability of the specimen at the cyclic unloading place is enhanced, the angle of the stress-strain curve at the partial stress unloading place is gradually increased, and the elastic response sensitivity is changed from an \"obtuse angle\" to an \"acute angle\". The sensitivity of the elastic response is enhanced. 2. The elastic modulus decreases step by step with the number of cycles in a \"increase-stabilization-decline\" pattern. The Poisson's ratio rises incrementally with increasing stress levels, and the gradual elevation of Poisson's ratio over several cycles prior to damage might be considered a forerunner to the destabilization of coal rock. This is due to the increase in shear stress that intensifies the specimen's internal crack development and expansion behavior, along with the stiffness of the coal specimen and its resistance to deformation. As the cyclic period accelerates, the elastic modulus decreases and the Poisson's ratio increases, which is due to the faster loading rate resulting in increased frictional slip between particles within the coal sample and the increased radial expansion of the specimen. 3. The cumulative irreversible deformation increases, step by step, with the number of cycles in a slope-like manner. Faster cycle periods have a greater impact on the specimen, intensify joint and fracture development, and increase the irreversible deformation of coals. According to the damage variable, \(D_{s}\), defined by the deformation parameter, the test process can be divided into three stages of \"deceleration-stabilization-acceleration\" according to the rate of change, and in the accelerated accumulation stage, the rate of damage accumulation accelerates with the cyclic period. 4. The cumulative dissipated energy increases stepwise with the number of cycles, and in the last few cycles, the cumulative dissipated energy increases gradually due to the continuous expansion of the plastic damage zone, and the cumulative dissipated energy increases significantly when the stress ratio is greater than 3. The majority of the input energy in each cycle is conserved as elastic energy, with the dissipated energy ratio resembling a \"L\" shape at stress levels 1 and 2. At stress level 3, due to the rapid propagation of cracks, the dissipated energy ratio takes on a \"U\" shape and evidently increases. As the cyclic period increases, the kinetic damage and friction-slip effects on the specimens diminish, resulting in a drop in the average dissipated energy ratio at each stress level. 5. The relationship between the damping ratio and the number of cycles resembles the proportion of dissipative energy. The rapid loading and unloading during the cyclic period exacerbates misalignment friction, resulting in a greater impact force on the crack extension area. Additionally, the presence of increased coal debris between the interfaces enhances the specimen's damping characteristics. Consequently, the relationship between the average damping ratio and the cyclic period evolves into an exponential form as the stress level rises. The damage variable, \(D_{d}\), defined based on the cumulative dissipated energy, can also characterize the three damage stages of the specimen under cyclic loading conditions. **Author Contributions:** Conceptualization, Y.J. ; methodology, X.Y. and D.Z. ; writing--original draft preparation, Y.J. and X.M. ; writing--review and editing, Y.J. ; visualization, X.Y. ; supervision, Y.J. All authors have read and agreed to the published version of the manuscript.