PSI - Issue 78

Ana Avramova et al. / Procedia Structural Integrity 78 (2026) 1633–1640

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then a different structural condition concerning the training period can be detected. As an alternative to the control chart based on PCA and Mahalanobis distance, the cointegration procedure (Cross et al., 2011) can be adopted. Given M non-stationary time-series (vectors) Y j  N (e.g., N observations of M natural frequencies), those vectors are said to be cointegrated if a stationary linear combination  k of Y j exists: =∑ =1 (2) The unknown coefficients  j ( j = 1, … , M ) in Eq. (2) can be found using the Johansen procedure (Johansen, 1988) during a training period in which the structure is supposed to be in normal condition under typical EOV. Once the unknown coefficients are determined, the resulting linear combination, denoted as εₖ, forms a new time series referred to as the cointegration residual. This residual is essentially free from the common trends affecting the natural frequencies, which are primarily associated with EOV, and an anomaly is considered to occur when the cointegration residual departs from the stationary behaviour established during the training period.

f B1 = 1.406 Hz

Mode T 1

f T1 = 1.664 Hz

Mode B 1

Mode B 2

f B2 = 1.854 Hz

Mode T 2

f T2 = 2.096 Hz

Mode B 3

f B3 = 2.393 Hz

Mode T 3

f T3 = 2.620 Hz

Mode B 4

f B4 = 2.726 Hz

Mode T 4

f T4 = 2.927 Hz

Mode B 5

f B5 = 4.794 Hz

Mode T 5

f T5 = 5.129 Hz

Mode B 6

f B6 = 5.504 Hz

Mode T 6

f T6 = 5.757 Hz

Figure 3. Selected reference modes identified (SSI-Cov) at the beginning of the continuous monitoring (08/06/2024, h 16:00).