PSI - Issue 78

Livia Fabbretti et al. / Procedia Structural Integrity 78 (2026) 823–830

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from the actual value (identified: 0.0949, actual: 0.03), with a relative error exceeding 200% (Table 4). This behavior is consistent with what was observed in the sensitivity analysis, which highlighted the marginal role of rp on the global dynamics of the system: the objective function can be minimized with appropriate combinations of μ fast and a K even in the presence of substantial deviations in rp .

Table 4. Parametric identification results with comparison between real and estimated values. Parameter “Real” value Identified value Relative error (%) μ fast 0.06 0.0600 0 rp 0.03 0.0949 +216.3 a K 51 51.0100 +0.0196

The cost function at the end of the optimization assumes a very low value (1.9165 × 10⁻⁴), and the convergence curve in Figure 4a shows a rapid decrease during the first generations followed by stabilization around the final value. Direct validation of the time series provides the most rigorous evidence of the proposed methodology’s accuracy: the nearly perfect overlap between the simulated signals and the reference ones, both for the relative isolator displacements and the accelerations, confirms the model’s ability to faithfully reconstruct the dynamic behavior of the isolators, even in the presence of noise. An example of the comparison between the simulated and “real” isolator deformations, as well as between the accelerations at one of the nodes, is presented in Figure 4b. The quantitative indicators calculated for each signal (Table 5) demonstrate that the model ensures excellent reconstruction of the overall response, as evidenced by the Normalized Root-Mean-Square Error (NRMSE), which is below 1% for each signal, and the Pearson correlation coefficients ( R ), always greater than 0.993. Peak estimation is less accurate in the y direction due to numerical approximations creating punctual discrepancies in instantaneous values. These differences become percentually significant at low acceleration values and are amplified by noise.

Fig. 4. (a) Convergence curve generated by the genetic algorithm; (b) Comparison of simulated and “real” isolator deformation (above) and acceleration at node 3 (below), both in x direction and with Gaussian noise.

Table 5. Quantitative comparison of results: NRMSE, Peak error, and Pearson R for each monitored signal. Signal NRMSE (%) Peak error (%) Pearson R Joint 2_Acc. x 0.50 7.37 0.9956 Joint 2_Acc. y 0.54 15.93 0.9962

Joint 3_Acc. x Joint 3_Acc. y Link 1_Def. x Link 1_Def. y

0.48 0.65 0.88 0.67

18.11 77.59 3.79 19.75

0.9937 0.9936 0.9940 0.9956