PSI - Issue 37

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

624

3

2.1. Statistical characterization of random loading In theory any stochastic process ( ) can be fully specified by its set of probability density functions (PDF) in time ( , ) . Practical applications, however, are simplified by assuming a time-independent PDF ( ) to realize reasonable and meaningful analyses. Nevertheless, the challenge of handling high amounts of data remains due to a large set of required realizations. If the generating process of a load series further conforms with ergodicity, only the presence of a single realization is obligated to derive the sought statistical characterization. The PDF of a stationary Gaussian process is defined by the parametric Gaussian PDF: ( ) = √2 1 2 (− ( 2 − 2 ) 2 ) (1) The Gaussian PDF has two parameters – variance 2 and mean – which belong to the statistical moments. These divide into statistical and central moments , whereby the former measures spread about zero and the latter about the mean (equal to first statistical moment 1 = ) = [ ( )] = ∫ − ∞ ∞ ( ) = [( ( ) − [ ( )]) ] = ∫ ( − ) − ∞ ∞ ( ) (2) The variance 2 belongs to central moments and corresponds to the second-order 2 . According to Parseval´s theorem the PSD 2 ( ) ( ) (frequency-domain) is linked to the time-domain via the second-order central moment 2 = 2 = ∫ ( ( ) − ) 2 ∞ − ∞ = ∫ 2 ( ) ( ) ∞ − ∞ (3) Apart from being the fundamental indicator of a process ’ spread, the variance 2 also standardizes higher-order ( > 2 ) moments. The consecutive (standardized) moments are the skewness (third-order, only relevant for asymmetric spread), and the kurtosis (fourth-order) = [( − ) 4 ] = 4 4 = 4 22 (4) To quantify deviations from the Gaussian PDF, statistical moments are elemental descriptors. Cumulants are closely related to statistical moments and embody an interesting alternative – namely, cumulants hold advantageous mathematical properties. For instance, for the Gaussian PDF all higher-order cumulants hold the value zero = 0; > 2 . Consequently, cumulants provide transparent information for the deviation from a Gaussian process and are statistically independent from cumulants of other order. Whereas the kurtosis of a stationary Gaussian process holds the value = 3 , the standardized fourth-order cumulant is zero for a Gaussian process = 0 . It commonly denotes the ‘ excess kurtosis ’ = 4 22 = 4 − 3 22 22 = − = − 3 (5) Eq. (3) states that the PSD’s integral corresponds to the second-order central moment 2 resp. the variance 2 . The PSD provides their spectral decomposition, whereby it resolves the contribution to the load series ’ variance 2 frequency . In the following this core idea is elaborated for defining spectral decompositions of higher-order moments , introducing higher-order spectral analysis ( ) ( 1 , … , −1 ) . This extends the PSDs application to general, non-Gaussian processes. 2.2. Higher-order spectra (HOS) The central limit theorem (CLT) in probability theory states that the sum of a large amount of independent random variables approaches the Gaussian PDF (Eq. (1). In reality, this independency of random variables is hardly ever the case. Instead, several process mechanisms are interacting in a barely observable manner. The auto-correlation functions of th -order ( ) ( 1 , … , −1 ) describe how random variables interact in time. The frequency-domain analogue (Wiener-Khinchin theorem) is the spectral analysis ( ) ( 1 , … , −1 ) , which provides these interactions as cross-frequency correlation. Spectral analysis processes the Fourier transform ( ( ) – finite Fourier transform). ( ) ( 1 , … , −1 ) = →∞ 1 [ ( 1 ) … ( −1 ) ∗ ( 1 + … + −1 )] (6)