PSI - Issue 84

F. Foria et al. / Procedia Structural Integrity 84 (2026) 645–652

648

7. further execution of modal identification processes using new pairs of Monte Carlo generated parameters values and other sets of records and comparison of the result with those of the database through the Mahalanobis distance and Cosine Similarity metrics; 8. filtering of all pairs of model order - time lag that exceed a certain acceptance threshold for distance according to Cosine Similarity and Mahalanobis metric; 9. interpolation of admissible pairs through the GPR 10. final selection of a fixed number of optimal model order - time lag pairs belonging to the GPR interpolation curve and analysis of new sets of recordings for validation; 11. final identification of modal quantities, such as those with the highest multiplicity of occurrence In Step 4, the selection of true modes excluding spurious ones introduced by noise data or not optimal parameter selection is performed using stability diagrams identifying stable poles of the sequence of covariance matrices, identifying stable poles as those that satisfy stability tests on frequency, damping and modal shapes between successive iterations of the covariance matrix [8]: [1 − ( ( ); ( + 1))] < 0.02 Frequency criterion (multiplicity criterion) involves the number of frequencies repeated within the same analysis. If this number exceeds a certain threshold, the frequency is accepted; otherwise, it is excluded. Damping criterion rejects modes with excessive dispersion of values or unrealistic values compared to expected values for the analyzed structure. Modal shape criterion involves the application of the Modal Assurance Criterion (MAC), indicating whether two modal shapes are equal or not. In this case, frequencies with two identical modal shapes are unified. It is underlined that all described operations are performed automatically, without intervention of the operator’s judgement. Steps 1 to 8, which take the longest time to complete, are only be performed once at the beginning of the monitoring process and should not be repeated. Once the optimal pairs of model order - time lag parameters has been selected, only these are used for successive identifications on future recordings. The selection of optimal pairs include an efficiency criterion, that exclude parameters requiring long time for the identification procedure: therefore the successive analysis are quite fast and can be performed almost in real time. The results are returned in tables, which show the average modal frequency, its occurrence multiplicity during the analysis, and the modal damping ratio. The analysis output also returns the modal displacement components at the points where the accelerometers are physically located. This output, omitted here for brevity, allows the identification of modal shapes. The procedure is illustrated with reference to a simulated model consisting in a tower with the following characteristics: • Cross section 4.90 x 4.00 m 2 , thickness 1.00 m • Height:15.20 m • Young modulus 2800x10^6 N/m² • Specific weight: 18000 N/m³ The model has been simulated with a FEM code, and the modal analysis provides the natural frequencies shown in Table 1 and the modal displacements shown in Table 2 at four nodes located at 2.3 m, 6.3 m, 10.3 m and 15.2 m along the height. 3.1 A simple illustrative example ( ( ) − ( () + 1) < 0.01 ( ( ) − ( + 1) ( ) )<0.05