## 7 Numerical Results of NSDI
### 31-Bar Planar Truss  
The standard finite element (FE) approach is used to model the 31-bar planar truss in Figure 4 without internal nodes, resulting in 25 degrees of freedom [45]. The first five vibrating modes are used for NSDI in this case. The elasticity modulus and density of the material are 70 GPa and 2770 kg/m, respectively. Two distinct damage scenarios are implemented in the structure, as shown in Table 2, and the suggested approach is evaluated in each case. The impact of noise on the accuracy of NSDI based on modal data is studied in this section. The uncertainties of the measured mode shapes of structures are typically larger than the uncertainties of the frequencies in modal testing [1]. In this study, uncertain frequencies and mode shapes are considered as the normal distributed random variables defined in Equations (6) and (7). The mean values of the relative random noises \(X_{\lambda i}\) are zeros, and standard deviations \(\xi_{\lambda}\) indicate the noise level. Tables 3 and 4 show PODEs and DMIs of the damage scenarios 1 and 2, respectively. The noise level is set as 2% and 5% in the frequencies and mode shapes, respectively. In scenario 1, higher PoDE values are obtained at element numbers 1 and 2 compared to the undamaged elements. Moreover, the PoDE values of elements 1 and 2 are higher than those of the undamaged elements. The DMI value of element 1 is similarly higher than that of element 2; both of these elements are correct damage locations with varying severity conditions. The same condition can be seen for scenario 2, where higher PoDE values observed at segments 11 and 25 and the DMI values are also higher at the elements with higher severity. \begin{table}
\begin{tabular}{c c c c} \hline \hline \multicolumn{1}{c}{**Case 2**} & \multicolumn{2}{c}{**Case 1**} \\ \hline
**Element Number** & **SRF** & **Element Number** & **SRF** \\ \hline
11 & 0.25 & 1 & 0.30 \\
25 & 0.15 & 2 & 0.20 \\ \hline \hline \end{tabular}
\end{table} 
Table 2: Damage scenarios of the planar truss. Figure 4: The 31-bar planar truss. To further prove the capability of the proposed method, Tables 5 and 6 compare the PoDE values obtained by the proposed method with the PDE values for the same damage cases calculated using a probability-based method developed by [PERSON] et al. [33,46]. [PERSON] et al. [33] uses the probability density function of the existence of undamaged and damaged states to determine the probability of damage existence (PDE). The main algorithm consists of three interconnected loops based on coupling metamodeling techniques with a metaheuristic optimization algorithm for probability-based damage detection (PBDD) of structures. These results show that the proposed technique outperforms the PBDD method by providing a more meaningful damage severity indicator than the statistical surrogate model method, which simply displays damage severity in terms of the probabilities. [21,26]. Furthermore, compared to the aforementioned surrogate methods [33,47], the proposed non-probabilistic method provides an advantage since it does not require surrogate models to predict the upper and lower boundaries of the ESP values. Surrogate models impose some uncertainties on the process, which increase the prediction error. The standard deviation of the probability density function (PDF) used to calculate the probability of \begin{table}
\begin{tabular}{c c c} \hline \hline
**Element Number** & **ESMA (PoDE) \%** & **ESMA (PDE) \%** \\ \hline
10 & 0.00 & 7.00 \\
11 & 100.00 & 96.00 \\
12 & 8.00 & 20.00 \\
13 & 0.00 & 0.00 \\
14 & 2.00 & 0.00 \\
24 & 0.00 & 5.00 \\
25 & 95.00 & 92.30 \\
26 & 0.00 & 12.00 \\ Total time (s) & 175 & 301 \\ RMSE & \(1.2\times 10^{-3}\) & \(1.0\times 10^{-2}\) \\ \hline \hline \end{tabular}
\end{table} 
Table 6: PoDEs and PDEs of case 2 for the 31-bar planar truss.