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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trispectral distribution of the HP series also covers trispectral space where no transmission in the transfer spectrum occurs. 4.2. Comparison of CN and HP response trispectra Applying linear system theory (Sec. 3.2), structural responses 4 ( ) ( 1 , 2 , 3 ) and 4 ( ) ( 1 , 2 , 3 ) for the HP and CN load series are calculated using a set of real and synthetic transfer functions, resp. more precisely their corresponding transfer spectra 4 ( ) ( 1 , 2 , 3 ) . The reader is reminded that since both loads cannot be differentiated by their PSD (Figure 2), both response PSDs 2 ( ) ( ) are also the same. In contrast, Figure 6 shows the response cumulant trispectra on the basis of the transfer spectrum (ideal bandpass) shown in Figure 5. This simplification is kept as it conveys the essential results but remains comprehensible within the scope of this paper. Because of the fact that the relevant trispectral entries of the HP load are distributed within the entire frequency space, only a small percentage of these correlate with the transfer spectrum and are transfered through the bandpass. As a result, the response cumulant spectrum (Figure 6) contains only few and small entries above the common threshold. In contrast, due to the common arrangement of the excitational CN trispectrum and the transfer spectrum along the planes , the response spectrum contains a larger amount of relevant ‘response non - Gaussianty’ . To convey this visual impression in terms of kurtosis, the transmissibility of the kurtosis is evaluated (Sec. 3.3). In order to convey the essential differences between the two load series and effects on structural response of dynamic systems in a meaningful way, the bandwidth of the ideal bandpass is varied (Figure 7a). It is increased incrementally until it covers the full frequency space. The results for the transmissibility of the excess kurtosis Ξ are shown in Figure 7 b. It shows that the CN’s transmissibility is independent of the bandwidth – the non-Gaussianity transfers into the response. The small differences from Ξ = are due to the statistical variability caused by error noise. The HP load transmissibility instead is dependent on the bandwidth, which for mechanical systems is governed by the damping. For an exemplary bandpass of Δ = 20 Hz it has a value of Ξ = 0,3 , which increases until reaching Ξ = (full transmission). The latter can be interpreted as an ideal stiff system with no dynamic behavior ( ( ) = ). (a) (b)

H xy [-] H xy [-] H xy [-]

f [Hz]

f Ny

f [Hz]

f Ny

f [Hz]

f Ny

Figure 7: (a) Ideal bandpass with increasing bandwidth and (b) corresponding transmissibility of excess kurtosis

5. Conclusion This paper addresses the use of higher-order spectra (HOS) to investigate non-Gaussian loading and its effects on structural responses of linear structures. A review of visualization and estimation techniques is meant to aid in applying HOS analysis. Further this paper covers how linear systems theory can be employed to the fourth-order spectrum – the trispectrum, which is directly related to the kurtosis. This involves the irrevocable path to understand what mechanisms govern the non-Gaussianity of random loading and whether this non-Gaussianity transfers through arbitrary linear structures. Two different kurtosis control mechanisms, generating non-Gaussian load series of same PSD and kurtosis, are applied to exemplify this path. It shows that non-Gaussian effects governing structural response and thus fatigue damage, are not adequately represented by the scalar-valued kurtosis of the excitation. As such, non Gaussianity due to non-stationarity fully transfers through linear structures, while nonlinear-transformed non