## 5 Energy Characteristics
### Damping Ratio  
The damping ratio reflects the hysteresis of the rock during cyclic loading. It is caused by the internal friction effect during the deformation of the rock as well as the viscous consumption of energy by the fluid. The damping ratio \(\lambda\) is calculated by the following formula [31]:  
\[\lambda=\frac{1}{\pi}\frac{A}{4A_{s}} \tag{11}\]  
where \(A\) is the area of the hysteresis loop, and \(A_{s}\) is the area enclosed by the peak stress corresponding to the strain and the center of the ellipse.  
Some scholars have modified the above equation from the energy point of view, based on the definition of damping ratio [32]:  
\[\lambda_{i}=\frac{1}{\pi}\frac{U_{di}}{4U_{ci}} \tag{12}\]  
where \(\lambda_{i}\) is the damping ratio of the ith cycle, \(U_{di}\) is the dissipated energy of the \(i\)th cycle, and \(U_{di}\) is the elastic strain energy of the \(i\)th cycle.  
From Figure 13, it can be noticed that there is a sudden increase in the damping ratio of the first cycle of each level under each cyclic period. The damping ratios of the other cycles fluctuated between 0 and 0.5, showing a trend of decreasing and then rebounding with the increase in the stress level, and the degree of fluctuation of the damping ratio of the  
Figure 12: Energy comparison at different cyclic periods.  
final level was greater than that of the first two levels; for example, the average damping ratios of the specimen with a cyclic period of 120 s were 0.02, 0.01, and 0.03 at each stress level, respectively. Analyzing the same stress level, the viscous interactions of the particle structures diminish upon compaction of the specimen, leading to a decrease in the energy dissipated via damping. At stress level 2, the effect of viscous forces between the mineral particles is not significant, and the damping ratio maintains a relatively stable decreasing trend. At stress level 3, the fatigue damage condition of the specimen is aggravated, and the energy consumption caused by the viscous damping effect when the crack expands is more obvious, which makes the damping ratio increase at the later stage.  
The damping ratios at different cyclic periods were analyzed, and it was found that the acceleration of the cyclic period increased the damping ratio of the specimens under the same stress conditions. The variation of the average damping ratio with a cyclic period in each stress level is shown in Figure 13. The relationship between the average damping ratio and the cyclic period satisfies a linear variation when in stress levels 1 and 2 and then changes to an exponential relationship in stress level 3. The average damping ratios of each stress level except the first cycle were analyzed, and the average damping ratios at stress level 1 were 0.021, 0.016, and 0.008 for the cyclic periods of 120 s, 300 s, and 480 s, respectively, with changes of 23.81% and 50.00%. At stress level 2, the average damping ratios were 0.014, 0.009, and 0.004, respectively, with changes of 35.71% and 55.56%; and at stress level 3, the average damping ratios were 0.033, 0.013, and 0.007, respectively, with changes of 60.61% and 46.15%. This may be due to the rapid loading and unloading of bias stresses that weaken the microstructural stability and increase the misalignment friction between the mineral particles, while the crack extension zone is subjected to a greater impact force and the presence of a relatively large number of coal fragments and particles between interfaces, leading to a relative increase in the energy used to overcome the damping force.  
Figure 13: Change curve of damping ratio.