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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in the resulting lifetime predictions. Further, the phase of the individual harmonics is varied in ϕ = [0 , π ] to verify that it does not a ff ect the error. Lastly, but most importantly, this analysis investigates whether a description of the sampling error for frequency holds the promise to be directly correlated with the PSD of the time series. Reminding that the PSD represents the distribution of the mean squared amplitudes for frequency (Eq. 3), the fundamental idea is that an error function for frequency may serve to be directly multiplied with the PSD. A subsequent integration of the PSD weighted by this error may provide a good error estimate for processes with arbitrary PSDs. Fig. 2(a) schematically visualizes this analysis for a single configuration of { ϕ, f f Ny } and an exemplary five periods. It includes the samples which determine the peaks (red) identified by the counting algorithm (RFC). To relate the sampling errors to vibration fatigue, the conformity of pseudo damage (CPD) is introduced as the ratio of pseudo damage calculated from the sampled and the referencing load spectrum. For the analysis of individual harmonics the latter is given by the real, unit-level spectrum (black, Fig. 2c,d). When performing this investigation with a relevant number of periods (here 1000 periods), the phase proves to be irrelevant for the analysis. Fig. 2(b) shows the conformity of pseudo damage for increasing ratios of f f Ny and di ff erent Miner exponents. With increasing frequency, the Miner exponent k has relevant e ff ects on this ratio. Interestingly, a large Miner exponent is favorable for the CPD in the chosen setting of pseudo damage via Palmgren-Miner elementary rule (Eq. 11). Further the functions show an extraordinary behavior at certain ratios f f Ny = { 1 1 . 5 , 1 2 . 5 , 1 3 . 5 , ... } . These ratios can be interpreted as having a CPD that is independent of the Miner exponent. A corresponding load spectrum ( f / f Ny = 1 / 2 . 5 = 40%) is shown in Fig. 2(c). With its step, it clearly deviates from the exemplary spectrum ( f / f Ny = 50%) plotted in Fig. 2(d). This e ff ect can also be attributed to the sampling. For the above ratios, the sampled series shows a unique interference — a beat. Counted by RFC leads to the phenomena at those ratios. The functions of Fig. 2(b) were subsequently used to analyze their applicability for arbitrary narrow- and broadband random processes using large sets of PSDs. As the subsequent sections will also demonstrate, they could not be used to adequately estimate the error for general processes. 3.3. Analysis of synthetic PSDs Since an individual evaluation of harmonics was not successful for a general estimation of the error due to insu ffi cient sampling, a data-driven approach is pursued. Therefore, three di ff erently generated synthetic CPD ( k ) = s eq,sampled ( k ) s eq,real ( k ) (12)

(a) unimodal PSD

(b) bimodal PSD approaching

2 f c,start

2 f c,start

G xx [(-)²/Hz]

G xx [(-)²/Hz]

f c

f c,start

f c,start

f c

f Ny

f Ny

f [Hz]

f [Hz]

(c) bimodal PSD distancing

3 2

f c,start

G xx [(-)²/Hz] f c 3

f c

f c,start

f Ny

f [Hz]

Fig. 3: Synthetic PSDs