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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(a) Behavior of ν ∗ 0 , ν ∗ p , and α 2 for di

ff erent center frequencies f c

(b) CPD for bimodal PSD (approaching) and di ff erent k

Fig. 4: Central parameters for the description of the error due to insu ffi cient sampling for synthetic PSDs

sets of PSDs (Fig. 3) are considered. Each set of PSDs is parameterized by its center frequency f c and relates to the Nyquist frequency by f c f Ny . The starting value further specifies the width of each mode ∆ f = 2 f c , start . The unimodal PSD in Fig. 3(a) is swept for [ f c , start , f Ny − f c , start ]. Fig. 3(b,c) show the bimodal PSDs, whereby the modes approach resp. distance from another while being swept. These configurations of PSDs are uniquely defined and well distinguishable and reproducible, so uni- and bimodal processes are represented. Using the set of synthetic PSDs allows to investigate how the error behaves for a variation of center frequencies f c and across the di ff erent synthetic models. Therefore, the CPD is calculated for the original series with a referencing up-sampled series using zero-padding. For the analysis presented herein a considerable ratio of ζ = 20 was implemented — as the prior Fig. 2(b) shows, the error vanishes quickly and asymptotically for low ratios of f f Ny . To describe the individual PSDs in terms of their decomposition for frequency and in relation to the Nyquist frequency new parameters are considered, which provide the basis for the following investigations and the definition of an error function. Spectral moments describe the decomposition of the PSD for fre quency by weighting the frequency with di ff erent exponents (Eq. 4). The upward zero-crossing rate ν 0 and peak rate ν p relate those spectral moments of di ff erent order (Eq. 6). Both are of the unit ’per time instance’, e.g. per second. To relate them to the sampling via the Nyquist frequency they are slightly modified by a standardization using the Nyquist frequency The asterisk indicates the standardized rates ν ∗ 0 and ν ∗ p which are dimensionless and range between [0 , 1]. Using the introduced synthetic PSDs allows to generate a large set of combinations for ν ∗ 0 and ν ∗ p (Fig. 4). The behavior of the parameters ν ∗ 0 and ν ∗ p , but also their ratio α 2 = ν ∗ 0 /ν ∗ p (Eq. 7), are shown for two models in Fig. 4(a). It is included here to provide an intuition for the introduced descriptors — foremost for the fact that the rate of peak occurrences ν ∗ p depends heavily on the contribution of high-frequency components. Fig. 4(b) shows the CPD for ν ∗ p , exemplary for the bimodal set of PSDs (approaching). In its essence it compares well to the prior representation of the error for single harmonics (Fig. 2b). The error increases smoothly with ν ∗ p and weights less for larger Miner exponents. However, it di ff erentiates in that this more realistic analysis is not subjected to the prior sampling e ff ect (’beat’) occurring for individual harmonics. In order to determine whether ν ∗ 0 and ν ∗ p are general descriptors for the error due to insu ffi cient sampling, the di ff erent definitions of synthetic PSDs are compared. Therefore, Fig. 5 shows the CPD for ν ∗ 0 , ν ∗ p and two Miner exponents k = { 1 , 5 } . Interestingly, for the fullness of the actual load spectrum ( k = 1), ν ∗ p appears to be a promising descriptor, since all parametric models align. However, considering actual damages e.g. via a Miner exponent k = 5, clearly shows, that the parametric PSD models deviate. Also ν ∗ 0 shows deviations between the di ff erent processes — already for k = 1. ν ∗ 0 = ν 0 f Ny ; ν ∗ p = ν p f Ny (13)