# Adaptive Probabilistic Optimization Approach for Vibration-Based Structural Health Monitoring
## 4 Results and Discussion
#### Case 1
\begin{table}
\begin{tabular}{c c c c c c c c} \hline \multirow{2}{*}{**Stage**} & \multirow{2}{*}{\(\epsilon\)} & \multirow{2}{*}{\(\mathbf{P_{d}}\)} & \multicolumn{3}{c}{**Crack Size (mm)**} & \multicolumn{3}{c}{**Crack Location (mm)**} \\ \cline{3-8}  & & & **Actual** & **Predicted** & \(\mathbf{P_{a}}\) & **Actual** & **Predicted** & \(\mathbf{P_{\beta}}\) \\ \hline
1 & \multirow{2}{*}{9\%} & 67.3\% & \multirow{2}{*}{3.0} & 2.5-3.0 & 50.4\% & \multirow{2}{*}{725} & 683–779 & 72.1\% \\
2 & & 65.8\% & & 2.5-2.9 & 53.5\% & & 713–831 & 67.4\% \\ \hline \end{tabular}
\end{table} 
Table 4: Results summary for Case 2. **Case 3**  
This case further examines the ability of the adaptive optimization approach developed herein to provide probabilistic details for damage characteristics in a simply supported beam. A noise of 15% was introduced into the simulated frequency response of a damaged beam with a smaller crack of 2.8 mm inflicted at the middle of the beam. To avoid overlap of damage and sensor locations, the frequency response data were collected from a node at 350 mm, identical to the sensor location of Case 1. A direct comparison can be made between the two cases due to the sensor location. The results of both stages for Case 3 are presented in Table 5.  
As can be seen from the results in Table 5, the probability of damage existence is more than 70% in both stages. This percentage is significant enough to declare the status of the damage. The algorithm successfully predicted the range of the crack size between 2.2 mm and 2.9 mm with the actual crack size being within the predicted range, but the range was slightly shifted and increased in stage 2. However, the algorithm improved the probability of crack size noticeably in the second stage to 55% and the probability of crack location increased to more than 78%. The ranges of crack location were also predicted within the design constraints of 15%. Figure 5: Frequency diagrams of (**a**) crack location and (**b**) crack size for Case 2. \begin{table}
\begin{tabular}{c c c c c c c c} \hline \hline \multirow{2}{*}{**Stage**} & \multirow{2}{*}{\(\epsilon\)} & \multirow{2}{*}{\(P_{d}\)} & \multicolumn{3}{c}{**Crack Size (mm)**} & \multicolumn{3}{c}{**Crack Location (mm)**} \\ \cline{3-8}  & & & **Actual** & **Predicted** & \(P_{a}\) & **Actual** & **Predicted** & \(P_{\beta}\) \\ \hline
1 & \multirow{2}{*}{15\%} & 71.6\% & \multirow{2}{*}{2.8} & 2.5–3.0 & 51.9\% & \multirow{2}{*}{500} & 486–579 & 70.5\% \\
2 & & 72.7\% & & 2.2–2.9 & 55.0\% & & 432–564 & 78.3\% \\ \hline \hline \end{tabular}
\end{table} 
Table 5: Results summary for Case 3. The probability distributions shown in Figure 6 provide meaningful details about damage characteristics even in the presence of significant noise while the crack size is relatively small. Despite the modest improvement observed in Figure 6a, several occurrences for damage location beyond 800 mm were eliminated in stage 2, with most occurrences predicted within \(\pm\)100 mm range from the actual damage location. In Figure 6b, there are three potential crack sizes substantially apart in value at about: 0.25 mm, 3 mm, and 8.5 mm. However, the candidate solution calculated by the algorithm was in the vicinity of 3 mm. The algorithm has made a major shift of the 3 mm peak with a minor increase in the number of occurrences. In order to test the algorithm's ability to predict undamaged status, this case was repeated with no damage. In this case, the adaptive algorithm yielded a crack size of nearly zero with about 90% probability. It should be noted that the sensor location must be carefully identified in order to avoid vibrational nodes, especially when high frequencies are being considered. Sensor location plays a significant role in accuracy and precision during structural health monitoring. Sensor placement is often informed by several factors, including system dynamics, types of expected defects, number and type of sensors, etc.