## 3 Numerical Simulation and Experimental Analysis
### Crack Depth
#### 3.3.2 Experimental Analysis  
Experiments are performed in the rig according to the conditions listed in Table 2. The propagation characteristics of the elastic waves in the real shaft can be deduced by comparing the spectrum of the signals. The results are shown in Figure 11.  
\begin{table}
\begin{tabular}{c c} \hline \hline
**Depth (\(a/R_{c}\))** & **Stopbands (Hz)** \\ \hline
0.1 & [102; 134], [480; 534], [1060; 1261] \\
0.2 & [80; 126; 412; 518], [918; 1261] \\
0.3 & [58; 124], [372; 516], [860; 1261] \\
0.4 & [42; 124], [354; 516], [838; 1261] \\
0.5 & [34; 124], [346; 516], [828; 1261] \\ \hline \hline \end{tabular}
\end{table}
Table 5: Numerical Stopbands derived at different crack depths.  
Figure 10: Numerical propagation constants derived from different crack depths.  
Figure 11 shows the experimental stop bands derived at different crack depth (\(a/R_{c}=0.1\), 0.2, 0.3, 0.4). As mentioned before, the uniform shaft having the same mass with the non-uniform shaft is also studied here for deducing the propagation characteristics by comparison. Similarly, the time histories of the acceleration signals are truncated to 0.01 s. The real lines stand for the uniform shaft while the dotted ones stand for the non-uniform shaft. As for the crack depth \(a/R_{c}=0.5\), the experimental results have been shown in Figure 7. All the experimental results are listed in Table 6.  
As can be seen in Table 6, there are only two stopbands for the cases 0.1 and 0.2, while the number of the stopbands becomes three for the cases 0.3, 0.4 and 0.5. All the three stopbands become wider with the increase of the depth, and the central frequency will be lower. The results of the experiments match well with the numerical ones. For the case 0.1 and 0.2, the lower stopbands are missing, the main reason is due to the constriction of experimental conditions. Thus, it can be verified that the  
\begin{table}
\begin{tabular}{c c} \hline
**Depth (\(a/R_{c}\))** & **Stopbands (Hz)** \\ \hline
0.1 & [455, 540], [1000, 1270] \\
0.2 & [421, 522], [920, 1270] \\
0.3 & [42, 143, 337, 528, 880, 1270] \\
0.4 & [35, 130, 335, 525, 850, 1275] \\
0.5 & [30, 120], [335, 520], [810, 1275] \\ \hline \end{tabular}
\end{table}
Table 6: Experimental stopbands derived at different crack depths.  
Figure 11: Experimental stopbands derived from different crack depth, where (**a**) is for the case \(a/R_{c}=0.1\), (**b**) is for the case \(a/R_{c}=0.2\), (**c**) is for the case \(a/R_{c}=0.3\) and (**d**) is for the case \(a/R_{c}=0.4\).  
bandwidths of the three stopbands become wider with the increase of the depth of the transverse crack, while the central frequencies of the three stopbands become lower and lower at same time.  
The change in the central frequencies and the bandwidths of the three stopbands can help to detect a transverse crack online in a real practice. The central frequencies and the bandwidths of the stopbands for a normal shaft can be obtained firstly, the central frequencies and the bandwidths of the three stopbands will change if a transverse crack occurs. As the central frequencies and the bandwidths vary with the depth of the crack, they can also provide a reference to study the depth growth too.