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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2.2. Signal reconstruction The resolution ∆ t of a time series is provided by the sampling frequency f s

1 f s

T N

(8)

=

∆ t =

It relates the observation time T to the the number of discrete samples N . Signal reconstruction basi cally supplements the existing number N by additional samples M so that ∆ t becomes smaller. A popular technique to increase the number of samples is zero-padding [11]. Its fundamental mechanism is to add additional frequencies above the Nyquist frequency f Ny with an amplitude of zero. A time series complying with the Nyquist-Shannon theorem is fully composed by frequencies up to the Nyquist-frequency f Ny . Fol lowing the assumption that the Nyquist-Shannon theorem is upheld, zero-padding sets all frequencies above to zero. Due to the additional samples, the resolution ∆ t improves, which better resolves the peaks of a time series. The up-sampled time series, composed of N + M samples, is denoted x zp ,ζ ( t ), whereby ζ defines the up-sampling factor.

M N

(9)

ζ =

The implementation of zero-padding is very basic. In comparison to other reconstruction algorithms it fur ther allows for choosing non-integer ζ [12]. An alternative reconstruction algorithm is the sinc-reconstruction (Whittaker-Shannon interpolation formula) which uses the analytical sinc-function [13]

sin( t ) t

sinc( t ) =

(10)

The sinc-function belongs to the signal reconstruction filters, such as the box-, Mitchell-Netravalli-, cone-, Gaussian- and the Lanczos filter [14, 15]. However, even though the sinc( t ) decreases in ± t , in theory it is unbound. This may motivate for signal reconstruction using splines. These are defined by the n -th order polynomials that compose the spline for up-sampling a time series [16].

3. Analysis of insu ffi cient sampling via signal reconstruction

Signal reconstruction, such as zero-padding, enables the recovery of insu ffi cient sampling, provided that the Nyquist-Shannon sampling theorem is upheld. Therefore, this section makes use of signal reconstruction to analyze the error in the fatigue damage that results from insu ffi cient sampling. The pseudo damage is introduced to define these errors in the context of fatigue damage predictions (Sec. 3.1). Employing the pseudo damage, it is begun with analyzing errors due to insu ffi cient sampling by taking discrete, individual harmonics into account (Sec. 3.2). Subsequently, random vibration is investigated by considering time series of synthetic (Sec. 3.3) and realistic PSDs (Sec. 3.4). Even though signal reconstruction provides a path to resolve insu ffi cient sampling, appending samples compromises additional computational e ff ort for counting algorithms. When analyzing entire FE-models or when searching for critical planes, counting algorithms such as rainflow counting require substantial computational times in random vibration fatigue. Therefore, this section further investigates whether the di ff erence of the reconstructed, zero-padded data in comparison to the insu ffi ciently sampled (original series) may provide the chance of deriving a meaningful error function (Sec. 3.5). This circumvents the possible necessity of up-sampling by simply including the error-correction formula. It aims to enable more accurate results without increasing computational times. 3.1. Pseudo damage The outcome of lifetime predictions for random vibration is considerably a ff ected, if up-sampling sub stantially improves the resolution of peaks and turning points of a stress series. Hereby, structural responses are processed by counting-algorithms — foremost rainflow counting — to extract load spectra. Recon structed time series lead to load spectra which increase in fullness — they cover a larger area than the referencing load spectrum due to larger peaks and — possible — a larger amount of cycles (e.g. Fig. 2c,d).