# Adaptive Probabilistic Optimization Approach for Vibration-Based Structural Health Monitoring
## 4 Results and Discussion
#### Case 1  
In Case 1, a 3 mm open crack (\(\alpha=0.3\)) was inflicted in the simply supported beam at 400 mm from the left boundary, as shown in Figure 2. The frequency response signal was collected from a node located at 350 mm. The proposed algorithm provided salient information about the damage characteristics in terms of damage location and severity. Table 3 shows the adaptive algorithm outputs including the probability of damage existence, probability of crack size, and probability of crack location for each stage. In stage 1, the algorithm predicted the existence of damage with a probability of 66.1%. The depth of the crack was predicted to be between 6.33 mm and 6.93 mm with an approximately 50% probability. The predicted crack size was more than twice the actual crack size. However, the crack location was predicted within 20 mm of the damage location with a 55% probability. The frequency diagrams of crack size and crack location for two stages are plotted in Figure 4. The \(y\)-axis represents the number of occurrences, n. The frequency diagram of crack location suggests that the crack location is within \(\pm\)40 mm range from 450 mm. In stage 1, however, the frequency diagram of crack size in Figure 4b indicates that there were three potential crack depths with average values of 0.5 mm, 3 mm, and 6.5 mm. The optimal range of crack size reported by the algorithm represents the occurrences falling within that range, making 6.5 mm a strong candidate in stage 1. Figure 3: Simulated frequency response of a simply supported beam. \begin{table}
\begin{tabular}{c c c c c c c c} \hline \hline \multirow{2}{*}{**Stage**} & \multirow{2}{*}{\(\epsilon\)} & \multirow{2}{*}{\(\boldsymbol{P_{d}}\)} & \multicolumn{3}{c}{**Crack Size (mm)**} & \multicolumn{3}{c}{**Crack Location (mm)**} \\ \cline{3-8}  & & & **Actual** & **Predicted** & \(\boldsymbol{P_{\alpha}}\) & **Actual** & **Predicted** & \(\boldsymbol{P_{\beta}}\) \\ \hline
1 & \multirow{2}{*}{1\%} & 66.1\% & \multirow{2}{*}{3.0} & 6.23–6.93 & 50.4\% & \multirow{2}{*}{400} & 36–387 & 55.0\% \\
2 & & 89.0\% & & 2.73–3.42 & 73.6\% & & 37–408 & 90.7\% \\ \hline \hline \end{tabular}
\end{table} 
Table 3: Results summary for Case 1. In stage 2, the algorithm yielded a significant improvement in the probabilities and the predicted ranges for crack size and crack location. As can be seen from Table 3, the probability of damage existence increased to about 89%. The actual crack location was detected with a high probability. The damage was predicted between 377 mm and 408 mm with a probability of 90.7%. Crack location range increased slightly from 2.2% in stage 1 to 3.1% of the beam length in stage 2. However, the range remained within the design constraint (\(\eta_{\beta}\)), which was set at 15% of the beam length. The probability improved significantly, and the actual crack location was included within the predicted range in stage 2. The probability of crack size also improved by more than 20%, and the predicted crack size was much closer to the actual value. It is worth noting that the frequency diagrams for both stages in Figure 4b show no occurrences of crack size greater than 7.5 mm, as well as other smaller ranges around 1.5 mm and 4.5 mm. Likewise, all occurrences of damage location in both stages fell between 300 mm and 450 mm. This demonstrates that the adaptive algorithm can also be used to eliminate regions that might be less likely to be damaged and consequently assist in directing inspection resources to more likely regions of expected areas. Limiting potential solutions to a few scenarios is an appreciable advantage of the probabilistic approach over the deterministic approach. This case was repeated with the same damage characteristics and zero noise. The results which were not included here yielded the exact crack location and crack size as the predefined values in the simulation model with about 100% probability. Furthermore, several trials of this case have been repeated with a noise level of 30%.