# Robust Structural Damage Detection Using Analysis of the CMSE Residual's Sensitivity to Damage
## 5 Experimental Validation
### Results and Discussions  
In this experiment, the first 15 modes of the undamaged beam calculated from the updated baseline FE model as well as the first 3 modes of the damaged beam identified from the measured data are used for damage identification. That is to say, a total of 45 CMSE equations can be formed. Therefore, for the classical CMSE method, 45 equations are always used to identify the damage, whereas for the RCMSE method, an analysis of the residual function's sensitivity to damage is performed first to discriminate redundant equations. For damage case I (1/4-thickness notch), Figures 20 and 21 give the 42 equations that are most appropriate to solve the actual location and extent of the damage. Then, these equations are selected to construct the CMSE system. \begin{table}
\begin{tabular}{c c c c c c} \hline \hline \multirow{2}{*}{**Case**} & \multirow{2}{*}{**Location**} & \multirow{2}{*}{**Extent**} & \multicolumn{3}{c}{**Natural Frequencies (Hz)**} \\ \cline{3-6}  & & & **1 st** & **2 nd** & **3 rd** \\ \hline
**Undamaged** & N/A & N/A & 5.524 & 34.711 & 97.200 \\
**I** & 5 & 1/4 thickness & 5.473 & 34.768 & 96.874 \\
**II** & 5 & 1/2 thickness & 5.299 & 34.755 & 95.089 \\
**III** & 5 and 14 & 1/2 and 1/2 thickness & 5.289 & 33.837 & 91.485 \\ \hline \hline \end{tabular}
\end{table} 
Table 4: Simulated damage cases and measured natural frequencies in the experiment. \begin{table}
\begin{tabular}{c c c c c c c} \hline \hline \multirow{2}{*}{**Data Type**} & \multicolumn{3}{c}{**Natural Frequencies (Hz)**} & \multicolumn{3}{c}{**MACs**} \\ \cline{2-7}  & **1 st** & **2 nd** & **3 rd** & **1 st** & **2 nd** & **3 rd** \\ \hline
**Measured** & 5.524 & 34.711 & 97.200 & & & \\
**Analytical** & 5.523 & 34.644 & 97.079 & & & \\ \hline \hline \end{tabular}
\end{table} 
Table 5: Measured and analytical natural frequencies and MACs of the undamaged beam. Figure 22 shows a comparison of the resulting location and extent of damage by using the classical CMSE and the RCMSE methods. It is evident that the damage of element No. 5 can be located by the RCMSE method but cannot be located by the CMSE method, indicating an enhancement of damage identification by eliminating three redundant equations. It is worth mentioning that the performance of the CMSE and RCMSE methods with regard to damage severity estimation is not discussed here. This is because the actual stiffness reduction of the beam is unknown. For damage case II (1/2-thickness notch), 43 equations are used to identify damage for the RCMSE method. One can observe from Figure 23 that damage at element No. 5 also can be located clearly by the RCMSE method. However, for the classical CMSE method, although the actually damaged element can be identified, a false-alarm of damage at element No. 15 is also produced. The difference between the results, again, illustrates the effectiveness of the elimination of redundant equations. Figure 21: Fitness function of damage case I, for the ITR method. Figure 22: Damage identification results of damage case I, with an iterative Tikhonov solution: (a) CMSE method; (b) RCMSE method. Furthermore, by comparing Figures 22b and 23b, it can be seen that RCMSE also produces a false-positive damage at element No. 1 for damage case I, hinting that smaller damage always results in a smaller change in the structural modal parameter (see Table 4) and leads to lower damage detectability. For damage case III (1/2-thickness notches), 23 equations are used to identify damage for the RCMSE method. Figure 24 shows that both methods cannot yield a sparse solution. In other words, both methods produce several false-positives of damage except for the correct indications of damage at elements Nos. 5 and 14. This illustrates that double-damage is difficult to identify for both methods. However, it is also shown that the estimated extents of damage at the actually damaged locations by the RCMSE method are greater than others. In other words, the actually damaged elements are not masked by false-positives.