# Adaptive Probabilistic Optimization Approach for Vibration-Based Structural Health Monitoring
## 4 Results and Discussion  
Three damage cases of a simply supported beam with different characteristics and different levels of noise have been considered to evaluate the effectiveness of the probabilistic optimization algorithm. The details of the three cases are provided in Table 1.  
User inputs for the adaptive algorithm are listed in Table 2. The user input values are subject to change and could vary with applications of interest, quality of measurements, and the precision required in the detection of damage characteristics. The segment size limits in Equation (3) are set to 0.1 and 0.5 for \(\delta_{ub}\) and \(\delta_{lb}\), respectively. Thus, the segment size increases when the absolute difference between measured and calculated FRFs, \(H(\omega_{k})\), is less than 0.1, and decreases when \(H(\omega_{k})\) is greater than 0.5.  
The probability constraints in Equations (4) and (5) are set to 15%. This implies that the probabilities of crack size ratio and crack location ratio are optimized such that the predicted values are within 15% of the beam dimensions. A 60% weighting factor is assigned to \(\eta_{d}\) in Equation (6) for the probability of damage existence. This implies that the probability of crack location contributes 60% to the probability of damage existence and the remaining 40% is assigned to the probability of non-zero crack size. The adaptive algorithm is set to terminate when the difference between the current stage probability of damage existence \(\left(\mathbf{P}_{d}\right)_{q}\) and the previous stage probability \(\left(\mathbf{P}_{d}\right)_{q-1}\) is less than 5%. For the sake of simplicity, algorithm results were limited to two stages only for each case considered in the study.  
The beam used for analysis made of aluminum has a rectangular cross-section with a length of 1000 mm and cross-sectional dimensions of 5 mm \(\times\) 10 mm. A Young's modulus of 70 GPa and a density of 2800 kg/m\({}^{3}\) are used for analysis. The forcing frequency range for each case was kept unchanged throughout the analysis. The excitation force has a frequency ranging from 1 Hz to 32 Hz. Similarly, the applied force is located at 300 mm from the left end of the beam. The random noise level that is added to the simulated frequency response is increased from Case 1 to Case 3, using 1%, 9%, and 15%, respectively.  
\begin{table}
\begin{tabular}{c c c c c} \hline \multirow{2}{*}{**Noise Level**} & **Sensor Location** & **Crack Size** & **Crack Location** \\  & **(mm)** & **(mm)** & **(mm)** \\ \hline Case 1 & 1\% & 350 & 3.0 & 400 \\ Case 2 & 9\% & 500 & 3.0 & 725 \\ Case 3 & 15\% & 350 & 2.8 & 500 \\ \hline \end{tabular}
\end{table}
Table 1: Damage cases of a simply supported beam.  
\begin{table}
\begin{tabular}{c c c c c c} \hline \multirow{2}{*}{\(\delta_{lb}\)} & \multirow{2}{*}{\(\delta_{ub}\)} & \multirow{2}{*}{\(\eta_{a}\)} & \multirow{2}{*}{\(\eta_{\beta}\)} & \multirow{2}{*}{\(\eta_{d}\)} & \multirow{2}{*}{\(\eta_{\beta}\)} \\ \hline
0.1 & 0.5 & 15\% & 15\% & 60\% & 5\% \\ \hline \end{tabular}
\end{table}
Table 2: User inputs for the adaptive algorithm.  
Figure 3 shows the frequency response signal collected from a node located at 350 mm from the origin.