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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mostly bears load data for which the probability density function (PDF) does not show a Gaussian distribution (Sweitzer, 2006; Wolfsteiner and Breuer, 2013). Fundamental for examining non-Gaussian loading is the kurtosis, which is the normalized fourth-order central moment. Various kurtosis control algorithms (Halfpenny and Kihm, 2010; Zheng et al., 2021) have been proposed to artificially create and modify excitational load series with a given kurtosis. Hereby it is crucial to understand the kurtosis as a metric with a spectral decomposition (Trapp and Wolfsteiner, 2019). This spectral decomposition – the trispectrum – is provided by the theory of higher-order spectral analysis (HOS) (Nikias and Petropulu, 1993). It offers essential insights into the specific nature of the non-Gaussianity and can be processed by linear systems theory to estimate the non-Gaussianity of structural responses (Ikelle and Amundsen, 2018; Trapp and Wolfsteiner, 2021b). However, HOS are rarely used in random vibration fatigue, which seems to be related to their complexity. The trispectrum is a function of three frequency arguments decomposing the kurtosis into cross-frequency correlation. It compromises redundancies (symmetries) and considerable computational effort. Therefore, and to provide simple access, this paper summarizes challenges of HOS and discusses options for their visualization (Section 2). From this consideration, in the following Sec. Error! Reference source not found. we propose a representation of the structural transfer behavior for an input- trispectrum (the spectral decomposition of the loading’s kurtosis). This provides the basis for calculating response output-trispectra and thus the degree of non-Gaussianity transferring through linear structures. A formal descriptor is introduced as the ‘ transmissibility of the kurtosis ’ . Popular kurtosis control algorithms are employed to discuss what has become known as the Papoulis rule – the tendency of non-Gaussian loading to cause Gaussian responses. Results (Sec. 4) are demonstrated showing that for common mechanical structures with pronounced resonances the nature of the non-Gaussianity is crucial for the transmission of the kurtosis and the resulting fatigue damage. These support experimental data (Palmieri et al., 2017; Zeng et al., 2021) that have been published in recent years. They show that a loading ’ s kurtosis cannot directly be related to the fatigue life of mechanical structures. Foremost, the herein presented consideration of linear systems theory for higher-order spectra makes the expensive experimental testing of the transfer behavior of the kurtosis obsolete. Nomenclature kurtosis CLT Central limit theorem ( ) ( 1 , … , −1 ) ℎ -order transfer excess kurtosis ƒ””‹‡” ‘‹•‡ spectrum mean HP Hermite polynomials ( ) transfer function ℎ -order central HOS higher-order spectra ℎ -order moment moment PDF probability density Planes (trispectrum, / 2 standard deviation/ function Gaussian manifold) variance SDOF single-degree-of- freedom ( ) probability density function Ξ transmissibility of the ℎ -order cumulant ( ) ( 1 , … , −1 ) ℎ -order auto-correlation kurtosis ( ) ℎ -order cumulant function Ξ transmissibility of the spectrum ( ) ( 1 , … , −1 ) ℎ -order moment excess kurtosis [⋅] expected value spectrum Nyquist frequency ( ) stochastic process Δ frequency resolution ( ) Fourier transform 2. Fundamentals In this section several relevant fundamentals are outlined, including the statistical characterization of random loading (Sec. 2.1), higher-order spectral analysis (Sec. 2.2) and its estimation (Sec. 2.3), and ultimately two kurtosis control algorithms in Section 2.4.