## 4 Numerical Simulations  
The simulation example features a fixed-fixed beam modeled with ten nodes and eight identical lumped masses, as shown in Figure 1. The flexural rigidity, EI for the beam is taken as \(5.1752\times 10^{5}\) kN.m\({}^{2}\) so that the fundamental frequency is 1 Hz in the undamaged state. Sensors measuring acceleration in the vertical direction are assumed to be present at each of the lumped mass locations. Viscous dissipation is included in the form of proportional damping with a magnitude of 5% of critical. Data for the baseline state, necessary for initial training and subsequent testing, is generated using the finite element model of the beam by assigning an initial velocity at each sensor location, one at a time. Acceleration measurements are simulated at these positions with a sampling time of 0.01 s, over a 30-s period. This data is then contaminated with varying levels of noise, randomly distributed within the range of 3% to 5%. The resulting signal pool includes 120 sets of acceleration measurements utilized to compute the transmissibility functions and extract the DSFs as defined in Equation (7). From this pool, the first 20 sets are used to train the classification model, while the remaining 100 sets are reserved for validation. Damage states for testing are simulated by introducing a 20% reduction in flexural stiffness at each segment between the sensor locations, one at a time. Given the sensor arrangement, this results in seven different damage cases, each with a different damaged element. Acceleration measurements are then simulated and contaminated in the same manner with again 100 simulations conducted for each damage scenario. This process leads to a total of 700 sets of simulated damaged data for the beam. The natural frequencies of the system associated with these damage cases, as well as the baseline state, are displayed in Table 2. Table 3, which lists the changes in natural frequencies across the inflicted damage scenarios, indicates that the most significant frequency change is less than 2%. This makes the proposed approach more attractive than alternative methods that employ identified modal parameters as DSFs. \begin{table}
\begin{tabular}{l l l l l l l l l} \hline \hline \multirow{2}{*}{**State**} & \multicolumn{2}{l}{**Damage btw**} & \multicolumn{6}{l}{**Natural Frequencies (Hz)**} \\ \cline{2-9}  & \multicolumn{1}{c}{**Sensors**} & \multicolumn{1}{c}{**Mode 1**} & \multicolumn{1}{c}{**Mode 2**} & \multicolumn{1}{c}{**Mode 3**} & \multicolumn{1}{c}{**Mode 4**} & \multicolumn{1}{c}{**Mode 5**} & \multicolumn{1}{c}{**Mode 6**} & \multicolumn{1}{c}{**Mode 7**} & \multicolumn{1}{c}{**Mode 8**} \\ \hline Baseline & - & 1.000 & 2.755 & 5.389 & 8.847 & 12.983 & 17.425 & 21.446 & 24.161 \\ D. Case1 & (1–2) & 0.996 & 2.743 & 5.316 & 8.710 & 12.833 & 17.184 & 21.059 & 23.962 \\ D. Case2 & (2–3) & 0.997 & 2.701 & 5.320 & 8.787 & 12.813 & 17.256 & 21.073 & 23.847 \\ D. Case3 & (3–4) & 0.987 & 2.717 & 5.356 & 8.682 & 12.837 & 17.222 & 21.251 & 23.744 \\ D. Case4 & (4–5) & 0.981 & 2.750 & 5.270 & 8.785 & 12.799 & 17.154 & 21.354 & 23.700 \\ D. Case5 & (5–6) & 0.987 & 2.717 & 5.356 & 8.682 & 12.837 & 17.222 & 21.251 & 23.744 \\ D. Case6 & (6–7) & 0.997 & 2.701 & 5.320 & 8.787 & 12.813 & 17.256 & 21.073 & 23.847 \\ D. Case7 & (7–8) & 0.996 & 2.743 & 5.316 & 8.710 & 12.833 & 17.184 & 21.059 & 23.962 \\ \hline \hline \end{tabular}
\end{table} 
Table 2: Natural frequencies of the system for the baseline and the damaged states. Figure 1: Simulated structure with sensor locations. Next, the proposed approach was tested with the noise-polluted data. The classification algorithms used for anomaly detection, including OCSVM, iForest and LOF require certain amount of historical data to train the model for the baseline (healthy system). DSFs extracted from the first 20 simulations out of the total of 120 generated for the baseline state were utilized to train these models, while the remaining 100 simulations were reserved for validation.