# Pipeline Damage Identification Based on Additional Virtual Masses
## 3 Numerical Simulation
### Sensitivity Analysis with Virtual Masses  
By adding mass \(m\) to the middle of the 10 substructures, the new structure is defined as virtual structure \(G_{i}(\mu,m)\), (\(i=1,2,3\)\(\dots\) 10). Figure 3 depicts the results of the first four frequencies with the relative sensitivity of substructure \(i\). Relative sensitivity is defined as  
\[\eta_{ki,I}=\frac{R_{ki,I}(\mu,m)}{\widetilde{\omega}_{ki}(\mu,m)}, \tag{11}\]  
and is the result of sensitivity to the normalization of frequency, which is beneficial to the comparison and analysis.  
Figure 3 shows that relative sensitivity varies for every pair of mass and sub-structure due to the symmetry of the pipeline constraints. Figure 3a shows the relative sensitivity cure for substructure 1/2/9/10. In the figure, we see that the relative sensitivity of the first and third frequencies increases with the mass. However, the relative sensitivity of the third frequency rises sharply from the initial stage to the peak, whereas the first rises and falls relatively gently with a peak of 7 kg. The relative sensitivity of the second frequency increases slowly with the additional mass. The fourth remained constant from the beginning to the lowest.  
\begin{table}
\begin{tabular}{c c c c c} \hline \hline
**Order** & **1** & **2** & **3** & **4** \\ \hline
**Frequency** & 182.24 & 476.79 & 795.07 & 814.57 \\ \hline \hline \end{tabular}
\end{table}
Table 3: The first four natural frequencies after reduction/Hz.  
Figure 2: The first four mode shapes. (**a**) The 1 st mode shape; (**b**) The 2 nd mode shape; (**c**) The 3 rd mode shape; (**d**) The 4 th mode shape.  
In combination with the above analysis and considering the ideal mode of the pipeline that can be obtained by the subsequent test, two virtual structures can be selected for the substructure near the bearing: one is the first frequency corresponding to \(G_{1}(\mu,m_{11})\), and the other is the second frequency corresponding to \(G_{1}(\mu,m_{21})\). In selecting additional mass value, the mass should be as small as possible to avoid impact on the structure itself and to ensure the improvement of relative sensitivity. Therefore, according to the sensitivity curve in Figure 3a, combined with the above analysis, the first frequency is selected as the modal information for the subsequent damage identification for substructure 1/2/9/10. The selected additional virtual mass (\(m_{11}\)) is 7 kg.  
Based on the analysis of relative sensitivity in Figure 3, 10 groups of virtual structures can be obtained on the order of G1-G10. Table 4 shows the additional mass values and selected frequencies of the virtual structures.  
\begin{table}
\begin{tabular}{c c c c c c c c c c c} \hline
**Virtual Structure** & **G1** & **G2** & **G3** & **G4** & **G5** & **G6** & **G7** & **G8** & **G9** & **G10** \\ \hline
**Substructure** & 1 & 2 & 3 & 4 & 5 & 6 & 7 & 8 & 9 & 10 \\
**Order** & 1 & 1 & 1 & 1 & 1 & 1 & 1 & 1 & 1 & 1 \\
**Mass (kg)** & 7 & 7 & 5 & 6 & 8 & 8 & 6 & 5 & 7 & 7 \\ \hline \end{tabular}
\end{table}
Table 4: Ten groups of virtual structures.  
Figure 3: Relationship between relative sensitivity and mass. (**a**) Substructure: 1/2/9/10; (**b**) Substructure: 3/8; (**c**) Substructure: 4/7; (**d**) Substructure: 5/6.