PSI - Issue 38

Arvid Trapp et al. / Procedia Structural Integrity 38 (2022) 260–270 A. Trapp / Structural Integrity Procedia 00 (2021) 1–11

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them requires counting algorithms to identify damaging cycles, which are collectively represented as a load spectrum. Among the counting algorithms, rainflow counting is certainly the most popular. However, the processing of sampled stress states bears pitfalls that can have considerable e ff ects on structural lifetime predictions. The assessment of rapidly varying stress states, e.g. due to vibration or shocks, places certain requirements on the sampling, which have not been su ffi ciently specified yet. Since counting algorithms count peak-to-peak stresses, these peaks must be su ffi ciently resolved by the sampling. Insu ffi cient sampling (Fig. 1a) results in lower stress amplitudes (peak is in between samples) and lower cycle counts (turning points in between samples). Throughout this paper, insu ffi cient sampling is understood in the sense that the sampling complies with the Nyquist-Shannon theorem but yet is insu ffi cient to prevent systematic errors when deriving load spectra via cycle-counting. While the discrete Fourier transform requires at least two samples per period to obtain the correct amplitudes in the frequency domain (Nyquist-Shannon theorem), it requires substantially more samples to resolve the correct peaks in the time domain.

(a) insu ffi cient sampled time series

(b) Fatigue Damage Spectrum of uniform colored noise

stress amplitude [MPa]

time [s]

Fig. 1: Insu ffi cient sampling in random vibration fatigue

Fig. 1(b) illustrates possible consequences for random vibration fatigue by employing the Fatigue Dam age Spectrum (FDS) [1, 2]. It shows discrepancies in the fatigue damage of stationary Gaussian response states resolved for a set of eigenfrequencies f D . This is visualized by the FDS in the form of the pseudo damage per eigenfrequency (Fig. 1b), which is derived by rainflow counting (RFC; sampling-based) resp. es timated via the Dirlik formula (DK; statistical) [3]. Commonly the discrepancy between RFC and DK is investigated in the research field of non-Gaussian random vibration fatigue [4, 5, 6]. Hereby, a statistical evaluation is generally bound to a stationary Gaussian assumption and thus estimates less fatigue damage for non-Gaussian loading. However, in this example the discrepancies lead to opposite results due to in su ffi cient sampling in the time domain. These discrepancies (in Fig. 1(b) of up to 25%) vanish when the time series is up-sampled. The up-sampling can be accomplished e.g. by zero-padding (Fig. 1(b), x zp , 2 ( t ) and x zp , 10 ( t ) with twice resp. ten times the number of samples). Signal reconstruction techniques allow to fully recover insu ffi cient sampling provided that the Nyquist-Shannon sampling theorem is upheld. This motivates for a detailed consideration of errors in fatigue calculations due to insu ffi cient sampling. In this context Fig. 1(b) provides an intuition for a basic property of insu ffi cient sampling for random vibration fatigue. For increasing eigenfrequencies, which means that the response states are composed of a higher frequency content, the discrepancies between RFC and DK increase. This indicates a clear path which con sists in relating the frequency distribution of the response states to their sampling property — the Nyquist frequency f Ny . A general discussion on sampling parameters of random loading is included in [7]. Further, [8] covers an analysis of the variability of stationary Gaussian realizations. As motivated by the introduction of this passage, insu ffi cient sampling may e ff ect the comparability of sampling- and statistical-based fatigue approaches and can influence such analyses. The aim of this paper is to investigate and to quantify the e ff ects of insu ffi cient sampling on predicted fatigue damage. In Sec. 2 we give a short review on the characterization of random time series in the