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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are shown in Fig. 8(a). The mean deviations of the fit are below one percent for the considered range of Miner exponent resp. in the interval of k = [1 . 5 , 8] for the single term equation (Eq. 14). To interpret these deviations of the fit in the context of the corrected error ε due to insu ffi cient sampling, Fig. 8(b) provides a representation of the formulas e ff ectiveness. Therefore, the deviations of the fit (root MSE) are related to the error ε corrected. Fig. 8(b) shows the mean value per Miner exponent, whereby e.g. 90% can be interpreted as correcting the error of insu ffi cient sampling with 10% deviations about the ’true’ referencing value.

(a) root mean squared error (MSE) of fit

(b) mean error of fit vs. error ε corrected

fit MSE

| fit|

Fig. 8: Evaluation of approximation formulas for error due to insu ffi cient sampling

4. Conclusion

This paper covers an investigation into the e ff ects of insu ffi cient sampling on counting algorithms in vibration fatigue. Therefore, it includes a consideration of adequate descriptors for systematic errors due to insu ffi cient sampling and applies them to analyze a variety of synthetic and realistic processes. Due to unique sampling e ff ects that occur for individual harmonics, meaningful analysis are carried out for actual random vibration. These are employed to derive an approximation function for the error due to insu ffi cient sampling. For the included processes the proposed approximation functions show a high e ff ectiveness for a large set of Miner exponents (Fig. 8b). Since this investigation is based on the elementary Palmgren Miner rule, further analysis may consider di ff erent damage accumulation rules but also non-Gaussian stress series. A first set of investigations into non-Gaussian series has shown that the CPD show a stronger spread but their mean values tend to be the same. However, such databases should be supplemented by synthetic non Gaussian models [19]. Another applicability of the presented investigation may be the down-sampling of load series for faster evaluation via cycle-counting algorithms. [1] C. Lalanne. Specification delevopment: Mechanical vibration and shock analysis vol. 5 . ISTE, London, 2009. [2] P. Wolfsteiner and A. Trapp. Fatigue life due to non-Gaussian excitation – An analysis of the fatigue damage spectrum using higher order spectra. International Journal of Fatigue , (127):203–216, 2019. [3] T. Dirlik. Application of computers in fatigue analysis . PhD Thesis, University of Warwick, 1985. [4] M. Palmieri, M. Cˇ esnik, J. Slavicˇ, F. Cianetti, and M. Boltezˇar. Non-Gaussianity and non-stationarity in vibration fatigue. International Journal of Fatigue , (97):9–19, 2017. [5] A. Trapp, M. J. Makua, and P. Wolfsteiner. Fatigue assessment of amplitude-modulated non-stationary random vibration loading. Procedia Structural Integrity , (17):379–386, 2019. [6] A. Trapp and P. Wolfsteiner. Fatigue assessment of non-stationary random loading in the frequency domain by a quasi-stationary Gaussian approximation. International Journal of Fatigue , (148):106214, 2021. [7] V. Rouillard and M. Lamb. On the e ff ects of sampling parameters when surveying distribution vibrations. Packaging Technology and Science , (21):467–477, 2008. [8] JME. Marques, D. Benasciutti, and R. Tovo. Variance of fatigue damage in stationary random loadings: comparison between time- and frequency-domain results. Procedia Structural Integrity , (24):398 – 407, 2019. [9] L. Lutes and S. Sarkani. Random vibrations: Analysis of structural and mechanical systems . Butterworth Heinemann, 2004. References