## 3 Test Results and Analysis
### Stress Wave Propagation Characteristics of Bolt Tension Process
#### 3.2.2 Time and Frequency Domain Analysis of Stress Wave Signals at Different Stages of Bolt Tension  
After filtering the original signals of the stress wave obtained during the bolt tensioning process by using the complementary ensemble empirical mode decomposition method, the reconstructed signals obtained were characterized in the time and frequency domains as follows. 1 Time domain feature analysis  
Figure 9 shows the time domain diagram of the stress wave signal at different stages of bolt tension. \begin{table}
\begin{tabular}{c c c c c c c c} \hline
**IMF Serial Number** & **IMF1** & **IMF2** & **IMF3** & **IMF4** & **IMF5** & **IMF6** & **IMF7** & **R** \\ \hline cross-correlation coefficient & 0.1107 & 0.7858 & 0.728 & 0.1393 & 0.0026 & \(-\)0.0034 & 0.0017 & \(-\)0.0013 \\ energy ratio & 0.73\% & 55.00\% & 41.55\% & 2.23\% & 0.21\% & 0.15\% & 0.08\% & 0.05\% \\ \hline \end{tabular}
\end{table} 
Table 2: The cross-correlation coefficient between each IMF and the original signal and their respective energy ratios. Figure 8: CEEMD decomposition results of mg1-0 raw data. Figure 9a-i show the time domain plots of the stress wave signals of the bolt stretching to the elastic stage, yielding stage, strengthening stage, and necking fracture stage, respectively. By observing the time domain graph of the stress wave when different axial displacements are applied to the bolt, the amplitude of the stress wave decay continuously with time. The amplitude indicates the change of energy when the stress wave propagates in the bolt because the deformation of the bolt makes the stress wave propagate in the bolt with different degrees of reflection and transmission, resulting in the attenuation of energy. In order to show more intuitively the energy changes of stress wave propagation during bolt stretching, the incident energy, reflected energy, and attenuating energy of stress waves are calculated for different axial displacements of bolt stretching. The incident and reflected energy magnitude of the stress waves could not be directly obtained because the bottom-end reflection could not be directly identified after the bolt stretched into plastic deformation. Therefore, the stress wave velocity in the free bolt was calculated first, and the bolt wave velocity was determined to be 5170 m/s. Then, the incident and reflected wave time intervals were determined by the bolt stretching length, and the time domain signals in the time range were integrated to obtain the incident and reflected energy magnitudes of the stress wave. The stress wave energy change curve during bolt tension is illustrated in Figure 10.  
Figure 9: Time domain diagrams of the stress wave signal at different stages of bolt tension. (**a**) Unstretched; (**b**) Elastic stage; (**c**) Yield stage; (**d–f**) Strengthening stage; (**g–i**) Necking fracture stage. Figure 10 shows that the incident energy, reflected energy, and attenuating energy curves of the stress wave are consistent with the continuous application of axial displacement during the tensioning of the same bolt the incident energy increases rapidly and linearly in the tensile-elastic phase of the bolt and remains constant after reaching the plastic deformation phase with a slow increase in the incident energy. The reflected energy change curve of the stress wave is complicated, mainly due to the deformation occurring during the bolt stretching process, resulting in different degrees of attenuation of the stress wave energy. It can also be seen that the attenuating energy is increasing with the increase in tensile axial displacements. In the elastic phase, the bolt stretching makes the bolt thinner, the wavefront surface diffusion energy of the incident wave decreases, the sensor receives an increased signal, the incident energy increases, and the reflected energy attenuation increases accordingly. However, the incident energy, reflected energy, and attenuating energy increase linearly due to the reversible elastic deformation of the bolt, and no internal damage occurs.