PSI - Issue 37

Arvid Trapp et al. / Procedia Structural Integrity 37 (2022) 622–631 Author name / Structural Integrity Procedia 00 (2019) 000 – 000

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3.3. Transmissibility of the (excess) kurtosis Lastly, it becomes of interest to relate the detailed transfer spectrum to the extent of which the kurtosis transfers through structures into the response. Therefore, we introduce a new metric defined as the ratio of the structural response and the excitational kurtosis. It is named ‘ transmissibility of the kurtosis ’ and assigned by the greek letter Ξ (Eq. (12). Accordingly, for the cumulant trispectrum the ‘t ransmissibility of the excess kurtosis ’ is defined. = ; = (12) Due to the non-zero value of the ‘Gaussian kurtosis’ = 3 , a meaningful evaluation of the transmissibility preferably uses the excess kurtosis which centers around the Gaussian excess-kurtosis = 0 . In the following some significant cases are explained: • Ξ > : increase of non-Gaussianity in structural response • Ξ = : full transmission of the kurtosis through the structure with unaltered kurtosis value • Ξ < : increase of Gaussianity in structural response towards a Gaussian process (‘Papoulis rule’) • Ξ = 0 : Gaussian structural response, elimination of the non-Gaussianity within the structure • Ξ < 0 : non-Gaussian structural response with a change of the kurtosis characteristic (leptokurtic to platykurtic or vice versa)

(a)

(b)

4 ( ) [ 2 4 ]

4 ( ) [ 2 4 ]

Figure 6: Cumulant response spectra (a) HP response and (b) CN response

4. Comparison of stationary non-Gaussian and non-stationary non-Gaussian load models 4.1. Trispectra of HP and CN load series (excitation)

In this section, the cumulant trispectra of both excitational loads are visualized, analyzed, and compared. For the latter the color bars in Figure 4 share the same scale. The HP cumulant trispectrum (Figure 4a) shows trispectral values in dark blue – i.e. they are rather small – and evenly spread across the trispectral frequency space. In contrast, for the CN spectrum (Figure 4b) those trispectral values located off the planes P are mostly below the threshold. However, the remaining values are much higher than the HP load and show a clear pattern in the trispectral space – they are distributed along the three planes . The highest values are depicted in yellow and are found along the intersections of the planes (lines). Although both load series have the same overall moments 4 resp. kurtosis, its spectral decomposition is distributed differently. While the HP series distributes (here = 0 ) evenly for the full trispectral set, the CN series accumulates = 0 solely on the planes . This becomes crucial when considering how the non Gaussianity transfers through linear structures, i.e. the structural transfer spectrum. Fundamental is that the CN load series and the transfer spectrum of the ideal bandpass coincide in their spectral distribution on the planes, while the