## 3 Results  
The analyses showed a large variety of trends and dependencies on the accuracy of the estimated frequencies and measured rotations. The proposed parametric analysis only relates to the testing dataset of each neural network training process. Referring to the identification of the presence of damage, the quality of the classification between damaged and undamaged was measured through the misclassification error. Figure 3 plots the confusion matrix for \(a_{freq}\) = 0.01 Hz and \(a_{rot}\) = 0.1\({}^{\circ}\). The testing dataset was constituted by 1455 undamaged and 1545 damaged beams, for a total of 3000 structures. In the green cells, the number of true positives (the NN predicts damage on a structure that is really damaged) and true negatives (the NN predicts no damage on a structure that is really undamaged) are reported. The red cells highlight the false positives (the NN predicts damage on a structure that, on the contrary, is undamaged) and false negatives (the NN predicts no damage on a structure that, on the contrary, is damaged). The misclassification error, i.e., the percentage of false values (199) with the respect to the number of tested cases (3000), was 6.63% and is reported in red in the bottom-right cell. The misclassification error was adopted as an indicator of the performance of the capacity of the system to identify whether the beam is damaged. Figure 2: Scheme of the datasets adopted for the training of the neural networks for each step of the structural monitoring. The misclassification error was evaluated for various accuracies of the vibration frequencies and rotations. Figure 4 depicts a contour plot of the error for the considered ranges of variables. Contour lines, which identify an equal amount of error, parallel to one axis denote that the variable represented in the axis does not influence the performance of the classification. That is, for precise rotation measurements or for definite vibration frequencies estimation, the influence of the accuracy of the other data is negligible, as the contour lines are parallel to the X and Y axes, respectively. Referring to Step 2 and Step 3 simulations, the root-mean-squared error (RMSE) is the parameter that describes the precision in the location of the damage and its magnitude. The RMSE is defined as  
\[\text{RMSE}\ =\ \frac{1}{N}\Sigma_{i=1}^{N}(O_{i}-T_{i})^{2}, \tag{3}\]  
where \(N\) is the number of testing entries (3k for the present case), \(T\) is the target value, and \(O\) is the output value from the trained neural network. Figure 5 illustrates the regression plot related to \(a_{freq}\) = 0.001 Hz  
Figure 4: Misclassification error (in percentage) as a function of the accuracy in the estimation of the vibration frequency and in the measure of the rotation on log-log axes. Figure 3: Example of a confusion matrix for \(a_{freq}\) = 0.01 Hz and \(a_{rot}\) = 0.1\({}^{\circ}\). Dam and Undam refer to the damaged and undamaged structure, respectively. The percentages in the grey cells can be interpreted as “precision”, “specificity”, and so forth (check [28] for each single term). and \(a_{rot}=0.001^{\circ}\). Each circle corresponds to a couple (_T;O_)\({}_{i}\). The dashed red line relates to the perfect fitting, i.e., the output equals the target. The squared error tends to reduce for damage located at the midspan (target). The same analysis was repeated, varying the accuracy variables in the aforementioned range. As can be seen from Equation (2), the root-mean-squared error for damage location is a quantity with a physical dimension. The parameter related to damage location is in meters, while the parameter related to damage magnitude has the same physical dimension as \(d\), i.e., it is dimensionless. Figure 6 reports the values of the errors for damage location and damage magnitude. It can be noted that damage location largely depends on the accuracy in the measure of the rotation of the beam, while it is roughly unaffected by the accuracy in the frequency estimation.