PSI - Issue 54

Arvid Trapp et al. / Procedia Structural Integrity 54 (2024) 521–535 Arvid Trapp / Structural Integrity Procedia 00 (2023) 000–000

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3. Data-driven damage estimator

This section describes the ANN model employed for defining the damage estimator based on kurtosis, respectively its spectral representation in the form of the NSM. This presentation covers the rationale for the input-output selection, the data-generation process, the architecture of the ANN, and the training and testing procedures. When approaching the design of the ANN and the relevant features for damage estimation, an intuitive understanding emerges: In the context of fatigue damage, not just the amplitudes of a stress series matter, but also the stress series frequency charac teristic, defining the occurrences of stress amplitudes. Simply put, the greater the frequency content the more fatigue damage, due to the increase in cycles. Consequently, spectral moments (Eq. 3) play a decisive role in current damage estimators, which o ff er scalar-valued descriptors of the PSDs distribution along frequency. This understanding leads to the question if and how the distribution of the NSM behaves in a similar way — if it a ff ects fatigue damage, when the non-stationary characteristic tends to higher frequencies. And further, as a two-dimensional representation, if fatigue damage may also be a ff ected by the NSM’s orientation for the diagonal. This characteristic relates to the correlation of modes. Specifically diagonal entries without relevant o ff -diagonal contributions mean that the non-stationary exci tation of structural modes occurs rather independent of another and not synchronous. Fig. 2 provides a brief glimpse into these e ff ects, summarizing one of our studies. The figure contrasts two switching processes: Both have the same average PSD and identical kurtosis (or fourth-order moment), but they di ff er in the spectral distribution of the latter. The first process (Fig. 2 (b) ) is composed of two components where both frequency bands are active simultaneously. In contrast, the second process (Fig. 2 (f) ) involves four separate processes, ensuring the frequency bands are never concurrently active. As a result, the latter lacks o ff -diagonal elements that indicate synchronicity. Figure 2 (d) also includes their load spectra, revealing subtle di ff erences in their shapes and the pseudo-damages (Eq. 1). This dis crepancy becomes more pronounced when the second band becomes parameterized by its mid-frequency f m (Fig. 3). While pseudo-damage increases with frequency across all three settings (including the stationary Gaussian), the dis parities between the two NSM configurations widen. Even though this parametric analysis leaves the fourth-order and consequently kurtosis unchanged, the NSM as its spectral decomposition, o ff ers the ability to capture these e ff ects. However, for a concise characterization and to design generalizeble ANN we drew inspiration from the PSD-based spectral moments as a role model, and introduce NSM-based spectral moments Λ Θ , n = f 1 0 f 2 0 (2 π f 1 ) n + (2 π f 2 ) n M xx ( f 1 , f 2 )d f 1 d f 2 Λ I , n = f 1 0 f 2 0 (2 π f 1 ) n (2 π f 2 ) n M xx ( f 1 , f 2 )d f 1 d f 2 (5) The chosen frequency arguments are inspired from well-established concepts, such as the product and moment of inertia. Fig. 4 depicts the spectral weighting functions of the first two orders, which are clearly capable of di ff er entiating between the two NSM-configurations introduced in Fig. 2. In particular, Λ Θ , n accentuates the o ff -diagonal entries, while Λ I , n emphasizes entries in proximity to the main diagonal. This methodology promises an e ff ective and e ffi cient assessment of the NSM avoiding the need for more complex and computationally intensive ANN models. Figure 5 shows the methodology we propose for assessing structural fatigue damage under non-stationary vibration loading with the help of an ANN. The loads PSD and NSM are estimated and transferred through the structure for each transfer function of interest using linear systems theory. Subsequently, the fourth-order spectral moments (Eq. 5) are derived from each response NSM, while the equivalent stress amplitude s ( DK ) eq is obtained by applying the Dirlik method on the response PSD. The ANN then predicts the ratio between the Gaussian and the desired damage that would have been derived by rainflow counting (RFC). Relating this to s ( DK ) eq gives the anticipated rainflow damage s ( NN ) eq ≈ s ( RFC ) eq .

3.1. Data-generation — ’Train-on-Dirlik’

In the process of developing a data-driven damage estimator, we considered di ff erent data bases, which include real recorded data and synthetic models for generating non-stationary data. Here, we want to highlight another ap proach, which is extremely e ffi cient in computation and has been used for the results presented herein. The idea is to