PSI - Issue 62

Israel Alejandro Hernández-González et al. / Procedia Structural Integrity 62 (2024) 879–886 Hernández-González/ Structural Integrity Procedia 00 (2019) 000 – 000

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(80%), validation (10%), and test sub-sets (10%). The loss function in Eq. (10) is iteratively computed using forward propagation, followed by the updating of the weights using the Adam back-propagation algorithm. The training convergence results are depicted in Fig. 3 (b), where it is noted that the model convergence is optimally reached at epoch 100. It is important to remark that while the initial SOBI takes 9 minutes and 53 seconds, the source identification by MTL-DNN in the prediction step only takes 0.39 seconds, representing a reduction of roughly 99.93%. The resulting modal identification results and their comparison with the CoV-SSI results are reported in Table 1. For the CoV-SSI, the time lag used for the estimation of the Toeplitz matrix of the cross-correlations between the measurement channels has been set to 1 second, and the system matrices and the corresponding modal features have been estimated considering model orders varying from 40 to 60 with steps of 2. In this case, the identification of stable poles is conducted through a stabilization diagram. To this aim, the certainly spurious/mathematical poles are identified considering a set of hard criteria, including maximum damping value max , minimum Mode Phase Collinearity MPC min , and maximum Mode Phase Deviation (MPD max ) with values of 10%, 50%, and 70%, respectively. The results in Table 1 reveal close agreements between the estimates by CoV-SSI and the proposed MTL-DNN. Notably, the relative differences are exceptionally small in terms of frequencies, with a maximum value of 0.55%. To gain a deeper understanding of the model's performance, Fig. 4 (a) furnishes the first six mode shapes estimated as the weights between the last two layers of the MTL-DNN. The modes have been labelled using “F and “ T ” to denote the vertical bending and torsional modes, respectively. The accuracy of these mode shapes is compared against the CoV-SSI mode shapes using the Modal Assurance Criterion (MAC) as shown the Fig. 4 (b). Remarkably, the MAC values are all very close to 1.00, confirming an almost perfect correlation between the CoV-SSI mode shapes and those estimated by the MTL-DNN. Note that the identified mode shapes are indeed complex-valued. This is particularly evident in the complexity plots of Modes 4, 5, and 6, for which MPC values of 97.0%, 77.3%, and 54.3% are obtained, respectively.

Fig. 3. (a) MTL-DNN architecture for case of study: Méndez-Núñez Bridge. (b) Convergence of the loss function.

Table 1. Comparison between CoV-SSI and estimated modal parameters using MTL-DNN.

Frequency [Hz]

Damping [%]

Mode

CoV-SSI

MTL-DNN

Rel. Diff. [%]

CoV-SSI MTL-DNN

Rel. Diff. [%]

1 2 3 4 5 6 7 8 9

2.88 3.81 4.53 5.08 6.10 6.51 6.86 7.27 8.22 9.50

2.87 3.79 4.54 5.06 6.12 6.49 6.82 7.26 8.25 9.45

5.72 4.42 4.41 4.67 4.38 4.99 5.45 2.58 2.78 2.64 2.76 2.64 2.76 3.03

5.38 4.69 4.23 4.45 4.02 5.03 5.12 2.41 2.52 2.71 2.56 2.54 2.54 2.88

-6.3% 5.8% -4.3% -4.9% -9.0% 0.8% -6.4% -7.1% 2.6% -7.8% -3.9% -8.7% -5.2% -10.3%

-0.32% -0.44% 0.15% -0.39% 0.33% -0.31% -0.55% -0.11% 0.36% -0.53% -0.19% -0.18% 0.50% 0.19%

10 11 12 13 14

10.16 10.82 11.92 13.14

10.14 10.80 11.98 13.16